) V 1 5 3 1 _ J R E T U R N I N G M A T E R I A L S : P l a c e i n b o o k d r o p t o L J B R A R J E S r e m o v e t h i s c h e c k o u t f r o m 1 - - z y - . - L . y o u r r e c o r d . F I N E S w i l I b e c h a r g e d i f b o o k i s r e t u r n e d a f t e r t h e d a t e s t a m p e d b e l o w . A S I M U L A T I O N M O D E L O F F I E L D D R Y I N G A L F A L F A W I T H I N T H E S O I L - S W A T H - A T M O S P H E R E C O N T I N U U M B Y Y i C h e n A D I S S E R T A T I O N S u b m i t t e d t o M i c h i g a n S t a t e U n i v e r s i t y i n p a r t i a l f u l f i l l m e n t o f t h e r e q u i r e m e n t s f o r t h e d e g r e e o f D O C T O R O F P H I L O S O P H Y i n A g r i c u l t u r a l E n g i n e e r i n g D e p a r t m e n t o f A g r i c u l t u r a l E n g i n e e r i n g 1 9 8 7 A B S T R A C T A S I M U L A T I O N M O D E L O F F I E L D D R Y I N G A L F A L F A W I T H I N T H E S O I L - S H A T H - A T H O S P H E R E C O N T I N U U M B Y Y i C h e n F i e l d d r y i n g i s a p r i m a r y s t e p i n a l f a l f a h a y p r o d u c t i o n . A m o d e l w a s d e v e l o p e d t o c l o s e l y s i m u l a t e t h e d r y i n g p r o c e d u r e i n o r d e r t o b e t t e r u n d e r s t a n d t h e p r o c e s s . T h e m o d e l w a s b a s e d u p o n t h e p h y s i c a l a n d p h y s i o l o g i c a l p r i n c i p l e s g o v e r n i n g h e a t a n d m o i s t u r e t r a n s f e r i n t h e s o i l - s w a t h - a t m o s p h e r e . c o n t i n u u m . S h o r t - w a v e a n d l o n g - w a v e r a d i a t i o n , w i n d s p e e d , s o i l t e m p e r a t u r e , e x c h a n g e s b e t w e e n l e a f s u r f a c e s a n d s u r r o u n d i n g a i r , b e t w e e n a i r s p a c e s , a n d b e t w e e n t h e s o i l s u r f a c e , t h e s w a t h a n d t h e a m b i e n t a i r w e r e m o d e l e d a n d i n t e g r a t e d v i a e n e r g y a n d m o i s t u r e c o n s e r v a t i o n . T h e M o n t e i t h - P e n m a n e q u a t i o n w a s a d a p t e d t o c h a r a c t e r i z e e x c h a n g e s a t l e a f s u r f a c e s w i t h c o n s i d e r a t i o n o f t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y o f l e a v e s u n d e r s e v e r e w a t e r s t r e s s . L e a f i n t e r n a l r e s i s t a n c e t o m o i s t u r e r e m o v a l ( r 1 ) w a s ' m o d e l e d t h r o u g h a n e x p o n e n t i a l f u n c t i o n o f t h e l e a f m o i s t u r e c o n t e n t ( M ) : 1 , 1 : 1 0 3 - b M / M 0 P a r a m e t e r s a a n d b w e r e s t a t i s t i c a l l y d e t e r m i n e d w i t h t h e u s e o f f i e l d - c o l l e c t e d d a t a f o r n i n e c u t t i n g d a t e s , t w o l o c a t i o n s a n d t h e e n t i r e d a t a s e t . F i e l d e x p e r i m e n t s w e r e c o n d u c t e d o v e r a w i d e r a n g e o f s w a t h A P P R O V E D B Y M l é fi 7 C , d e n s i t i e s ( 0 . 2 6 2 t o 2 . 5 “ 8 k g d r y m a t t e r / m 2 ) , i n i t i a l m o i s t u r e c o n t e n t s ( 2 . 6 “ 5 t o u . u 9 5 k g w a t e r / k g d r y m a t t e r ) , m a t u r i t y l e v e l s ( 0 t o 1 0 0 % b l o o m ) a n d m e t e o r o l o g i c a l c o n d i t i o n s f o r E a s t L a n s i n g , M i c h i g a n ( 1 9 8 R ) a n d F r e s n o , C a l i f o r n i a ( 1 9 8 5 ) . T h e o v e r a l l a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n o f t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t w a s 0 . 9 3 6 ( b y n i n e c u t t i n g d a t e s ) , 0 . 9 1 “ ( b y t w o l o c a t i o n s ) a n d 0 . 9 0 6 ( b y t h e e n t i r e e x p e r i m e n t ) . F o r a p a r t i c u l a r l o c a t i o n , o n e s e t o f t h e p a r a m e t e r s a a n d b u n d e r n o r m a l p r o d u c t i o n c o n d i t i o n s w o u l d b e s u f f i c i e n t t o m a k e r e a s o n a b l e p r e d i c t i o n s . T h e m o d e l w a s m o s t s e n s i t i v e t o c h a n g e s i n p a r a m e t e r a . O t h e r i m p o r t a n t p a r a m e t e r s i n c l u d e d p a r a m e t e r b , l e a f a r e a i n d e x , s w a t h d e n s i t y , l e a f m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n a n d a c o n d i t i o n i n g f a c t o r . F u r t h e r i n v e s t i g a t i o n o f l e a f r e s i s t a n c e t o m o i s t u r e l o s s i s e s s e n t i a l f o r i m p r o v i n g t h e m o d e l . T h e m o d e l w a s w r i t t e n i n s t a n d a r d F O R T R A N - 7 7 a n d u s e d o n C D C - 7 5 0 , P R I M E - 7 5 0 , H P - 9 0 0 0 , V A X - 8 6 0 0 / 7 5 0 , S U N - 3 / 1 8 0 a n d I B M - P C c o m p a t i b l e c o m p u t e r s . W e a t h e r d a t a , i . e . a i r t e m p e r a t u r e a n d h u m i d i t y , w i n d s p e e d , p r e c i p i t a t i o n a n d s k y c l o u d f r a c t i o n , w e r e r e q u i r e d t o d r i v e t h e m o d e l . T h e s e c a n b e m e a s u r e d a t a l o c a t i o n o r o b t a i n e d f r o m a l o c a l w e a t h e r s e r v i c e . A P P R O V E D B Y C 1 4 / 7 4 . 4 % M a j o r P r o f é S s o ¥ D e p a r t m e n t C h a i r m a n T o m y f a m i l y i v A C K N O W L E D G M E N T S T h e a u t h o r w i s h e s t o e x p r e s s d e e p e s t a p p r e c i a t i o n t o D r . C . A l a n R o t z ( U n i t e d S t a t e s D a i r y F o r a g e R e s e a r c h C e n t e r , A g r i c u l t u r a l E n g i n e e r a n d A d j u n c t A s s o c i a t e P r o f e s s o r , D e p a r t m e n t o f A g r i c u l t u r a l E n g i n e e r i n g ) f o r h i s e n c o u r a g e m e n t , g u i d a n c e a n d s u p p o r t a s t h e m a j o r p r o f e s s o r a n d c h a i r m a n o f t h e d o c t o r a l p r o g r a m c o m m i t t e e . S p e c i a l g r a t i t u d e i s t o w a r d D r . M e r l e L . E s m a y ( P r o f e s s o r E m e r i t u s , D e p a r t m e n t o f A g r i c u l t u r a l E n g i n e e r i n g ) f o r h i s c o n t i n u a l a d v i c e a n d h e l p d u r i n g t h e a u t h o r ' s g r a d u a t e s t u d y a t M i c h i g a n S t a t e U n i v e r s i t y . T h e a u t h o r i s v e r y g r a t e f u l f o r h a v i n g D r . J a m e s V . B e c k ( P r o f e s s o r , D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g ) , D r . J . W i l l i a m T h o m a s ( P r o f e s s o r E m e r i t u s , D e p a r t m e n t o f A n i m a l S c i e n c e ) a n d D r . J . R o y B l a c k ( P r o f e s s o r , D e p a r t m e n t o f A g r i c u l t u r a l E c o n o m i c s ) o n t h e d o c t o r a l a d v i s o r y c o m m i t t e e , a n d D r . F r e d W . B a k k e r - A r k e m a ( P r o f e s s o r o f A g r i c u l t u r a l E n g i n e e r i n g ) a s t h e o u t s i d e e x a m i n e r . M a n y t h a n k s a r e d u e t o M r . R a n d a l l D a v i s , M r . D e n n i s B u c k m a s t e r a n d o t h e r p e r s o n s w o r k i n g i n t h e f o r a g e r e s e a r c h p r o j e c t f o r t h e i r a s s i s t a n c e i n f i e l d e x p e r i m e n t s . T h e a u t h o r i s i n d e b t e d t o t h e f a c u l t y , s t a f f a n d s t u d e n t s i n t h e D e p a r t m e n t o f A g r i c u l t u r a l E n g i n e e r i n g , o t h e r D e p a r t m e n t s , M . S . U . C o m p u t e r L a b o r a t o r y a n d A l b e r t H . C a s e C o m p u t e r C e n t e r o f t h e C o l l e g e o f E n g i n e e r i n g f o r t h e i r h e l p d u r i n g t h e a u t h o r ' s s t u d y o n t h e S p a r t a n c a m p u s . T A B L E O F C O N T E N T S C h a p t e r P a g e L I S T O F T A B L E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x i L I S T O F F I G U R E S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x i i 1 I N T R O D U C T I O N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . 1 k c k 8 r o u n d O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 1 1 . 2 O b j e c t i v e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 2 L I T E R A T U R E R E V I E W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 S Y S T E M D E F I N I T I O N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 3 . 1 S y s t e m B o u n d a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 3 . 2 S u b s y s t e m s a n d C o m p o n e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 3 . 3 S y s t e m L i n k a g e a n d S t a t e V a r i a b l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 A M O D E L D E V E L O P M E N T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 " . 1 S h o r t - N a v e R a d i a t i o n P r o f i l e s i n s i d e t h e S w a t h . . . . . . . . . . . . . . . 2 2 4 . 1 . 1 S o l a r I r r a d i a n c e a t t h e S w a t h T o p . . . . . . . . . . . . . . . . . . . . . . . . 2 3 N . 1 . 1 . 1 E s t i m a t i o n o f t h e S o l a r I r r a d i a n c e . . . . . . . . . . . . . . . . . . . 2 5 u . 1 . 1 . 2 S o l a r E l e v a t i o n A n g l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 fl . 1 . 2 O p t i c a l P r o p e r t i e s o f t h e S w a t h . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8 N . 1 . 2 . 1 S c a t t e r i n g C o e f f i c i e n t o f L e a v e s . . . . . . . . . . . . . . . . . . . . . 2 8 N . 1 . 2 . 2 S c a t t e r i n g C o e f f i c i e n t o f a S w a t h L a y e r . . . . . . . . . . . . . . 3 0 fl . 1 . 3 R a d i a t i o n F i e l d d u e t o B e a m R a d i a t i o n . . . . . . . . . . . . . . . . . . . . 3 2 fl . 1 . 3 . 1 A t t e n u a t e d B e a m R a d i a t i o n i n s i d e t h e S w a t h . . . . . . . . . . . 3 3 v i 4 . 1 . 3 . 2 E s t i m a t i o n o f t h e L e a f A r e a I n d e x . . . . . . . . . . . . . . . . . . . . 3 3 4 . 1 . 3 . 3 C o m p l e m e n t a r y R a d i a t i o n d u e t o B e a m R a d i a t i o n . . . . . . . . 3 5 4 . 1 . 4 R a d i a t i o n F i e l d d u e t o D i f f u s e S k y R a d i a t i o n . . . . . . . . . . . . . 3 7 4 . 1 . 4 . 1 A t t e n u a t e d D i f f u s e S k y R a d i a t i o n . . . . . . . . . . . . . . . . . . . . . 3 7 4 . 1 . 4 . 2 C o m p l e m e n t a r y R a d i a t i o n d u e t o D i f f u s e S k y R a d i a t i o n . 3 8 4 . 1 . 5 T o t a l S h o r t - W a v e R a d i a t i o n i n t h e S w a t h . . . . . . . . . . . . . . . . . . 3 9 4 . 2 L o n g - W a v e R a d i a t i o n P r o f i l e i n s i d e t h e S w a t h . . . . . . . . . . . . . . . . . 4 1 4 . 2 . 1 E m i s s i v i t i e s o f t h e A t m o s p h e r e , A l f a l f a a n d S o i l . . . . . . . . . 4 4 4 . 2 . 2 L o n g - N a v e R a d i a t i o n F i e l d i n s i d e t h e S w a t h . . . . . . . . . . . . . . . 4 6 4 . 2 . 3 T o t a l L o n g - R a v e R a d i a t i o n i n s i d e t h e S w a t h . . . . . . . . . . . . . . . 4 8 4 . 3 E x c h a n g e s a b o v e t h e S w a t h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0 4 . 3 . 1 V i r t u a l T e m p e r a t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 4 . 3 . 2 Z e r o P l a n e D i s p l a c e m e n t a n d R o u g h n e s s P a r a m e t e r . . . . . . . . . . 5 3 4 . 3 . 3 R a t e E q u a t i o n s o f M o m e n t u m , H e a t a n d M b i s t u r e T r a n s f e r . . . 5 4 4 . 3 . 4 M i x i n g L e n g t h a n d F r i c t i o n S p e e d . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 4 . 3 . 5 A t m o s p h e r e S t a b i l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 4 . 3 . 6 N e u t r a l C o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0 4 . 3 . 7 S t a b l e C o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 4 . 3 . 8 U n s t a b l e C o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 4 . 3 . 9 R e s i s t a n c e t o M o i s t u r e T r a n s f e r . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 4 . 4 E x c h a n g e s i n s i d e t h e S w a t h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7 4 . 4 . 1 N e u t r a l C o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 8 4 . 4 . 2 U n n e u t r a l C o n d i t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 0 4 . 4 . 3 R e s i s t a n c e a t t h e S o i l S u r f a c e . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 4 . 5 E x c h a n g e s o n L e a f S u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 4 . 5 . 1 S e n s i b l e H e a t T r a n s f e r o n L e a f S u r f a c e s . . . . . . . . . . . . . . . . . . 7 5 4 . 5 . 2 M o i s t u r e T r a n s f e r o n L e a f S u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . 8 0 v i i 4 . 5 . 2 . 1 R e s i s t a n c e t o M o i s t u r e T r a n s f e r i n t h e B o u n d a r y L a y e r 8 1 4 . 5 . 2 . 2 S u r f a c e R e s i s t a n c e t o M o i s t u r e T r a n s f e r . . . . . . . . . . . . . . 8 3 4 . 5 . 3 R e s p i r a t i o n o n L e a f S u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6 4 . 5 . 4 E n e r g y B a l a n c e o n L e a f S u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7 4 . 5 . 4 . 1 T h e L a t e n t H e a t F l u x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 0 4 . 5 . 4 . 2 E q u i l i b r i u m R e l a t i v e H u m i d i t y . . . . . . . . . . . . . . . . . . . . . . . . 9 1 4 . 5 . 4 . 3 M o d i f i e d M o n t e i t h - P e n m a n E q u a t i o n . . . . . . . . . . . . . . . . . . . . 9 4 4 . 5 . 4 . 4 L e a f T e m p e r a t u r e 9 7 4 . 6 R e w e t t i n g a n d R e d r y i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 4 . 6 . 1 R a i n I n t e r c e p t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 0 4 . 6 . 2 D e w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 1 4 . 6 . 3 R e w e t t i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 2 4 . 6 . 4 I n t e r n a l M o i s t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 3 4 . 6 . 5 S u r f a c e M o i s t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 5 4 . 6 . 6 R u n a b l e M o i s t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 6 4 . 6 . 7 D e t e r m i n a t i o n o f M o i s t u r e C o n t e n t s . . . . . . . . . . . . . . . . . . . . . . . . 1 0 8 4 . 7 B a l a n c e o f M o i s t u r e a n d E n e r g y i n A i r S p a c e s . . . . . . . . . . . . . . . . . 1 0 9 4 . 7 . 1 M o i s t u r e B a l a n c e i n A i r S p a c e o f a S w a t h L a y e r . . . . . . . . . . . 1 0 9 4 . 7 . 2 S e n s i b l e H e a t B a l a n c e i n A i r S p a c e s . . . . . . . . . . . . . . . . . . . . . . 1 1 2 4 . 8 E x c h a n g e s i n S o i l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 4 . 8 . 1 E n e r g y B a l a n c e a t S o i l S u r f a c e . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 7 4 . 8 . 2 R e l a t i v e H u m i d i t y a t S o i l S u r f a c e . . . . . . . . . . . . . . . . . . . . . . . . 1 1 8 4 . 8 . 3 S o i l T e m p e r a t u r e P r o f i l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 0 4 . 9 M o d e l S t r u c t u r e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 4 . 9 . 1 I n i t i a l i z a t i o n S e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 4 . 9 . 2 S i m u l a t o r S e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 7 4 . 9 . 3 O u t p u t S e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 1 v i i i 5 6 7 F 5 5 5 P 6 6 6 M 6 6 7 I . . . . . A . . . O . 7 . 1 1 4 E 5 5 5 R 5 5 2 5 5 3 2 3 4 3 D 1 7 7 2 7 7 7 7 . L . . . . . . . A E . . . . . . 2 2 2 2 2 2 2 D 9 M M L M M T O 1 L A M R 1 1 1 S 2 2 . . e i o E h e r . . . . . . . e o . . . . e . 4 E a c 1 2 3 4 5 6 7 T t c e d V M s d 2 3 4 1 n 2 1 X s r a E h o u e i A s u o T M R P A S M n R C o d l l l P P L V V V V i E T s E r m e i e y n o e a o d i t I a a t a a h f y R E e c e m l r e C i a m o f r s D V l l l l i e f m T e e t m o g t i t i A S a N P a i i i i c I S m r t p a a n l p l l s i y A a i i i i v e h M T e o e e t n o u e i l o o T l d d d d i c r E n o l r i o m M m e g r d I L f I i a a a a t t o N M e t r o a v m e e o a x y O d t t t t y s E f f e l g u s o t t e e e o T i s o t o o a o o A o c . r g r e r u o t u O p - a N o a . t f H I f m t o e r t l y e r n r A i n n n n n f t S m i i s r g l S N o P m n D . . b b b b l T o c c e i w i P P f a q e e y y y y y w f . . . . . . . . . a d r r a c x r S u t s o L . . . . t t l o i S o a a a e E . t N T U i . I i . . . h h b t e c m S l r r N i w s s h S . . . n b . M . e y e e . n e e e S n o e e . . t i . r e D . . s . t s s d . a e . I E I . . a t . . n a a r P . o u P s r T . . e . E o L t i . r y s r . r r r r c . . . . . f t I E . C o n f i a . y . i u o . . . s e . o . o h s i V . u c t s l . . . . . . . n r b b m I . r E t t a i t . . . . . D . . . . . . g e e l a T . . m s P i t t r i . . i M s . m e a . t r m t Y . i i e c . . . . . . f o . e . m . . . . . . . . t i o a i n o a . f i . . . . . . . . . E e s . n . i m c t o A g n l . . . . . t s . . o i n a e N . . . . . . . . x r t . . . . . . . . . . . P u . . . . n o t d A D p G e . . . . . . . . . . . . . i u n . L a r e n . a r . . . . . . . . . . o r Y t o r t . r e . . . . . . . . . . . . . . . . . . . . n e o S e u i . . a . . . . . . . . . . . f I p m . . . . . W C . m . . . . . . . . . . . . . . . . S . G s e . o e . e . . . . . . . . . . . . . . . . . r n P . a t . n . . . . . . . . . . . . . . . . . . . a . o t t . e t . . . . . . . . . . . . . . . . . . . . u h . r r e . . . . . . . . . . . . a a s n . . p e . . . . . . . . . . . . . . . . . . . . . m s r . s t . . . . . . . . . . . . . . . . . . . e . . . . . . . . . . . . . . . . . . . . . . . . . . t . . . . . . a D . . . . . . . . . . . . . . . e . . . . . . . a . . . . . . . . . . . . . . . r . W t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s . . h a . . . . . . . . . . . . . . . . . . . . . . . o . . . . . . . . . . . . . . . . . . . l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 3 3 4 3 4 4 4 4 5 5 6 8 8 8 8 9 9 9 6 0 0 1 1 2 2 5 6 8 8 9 9 0 7 2 8 8 4 7 9 3 3 5 8 0 5 7 2 4 i x 7 . 2 . 3 T h e E f f e c t o f S w a t h A r e a D e n s i t y . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 4 7 . 2 . 4 T h e E f f e c t o f t h e R a t i o o f L e a f A r e a I n d e x t o Y i e l d . . . . . . 2 0 6 7 . 2 . 5 T h e E f f e c t o f A l f a l f a M a t u r i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 7 7 . 2 . 6 T h e E f f e c t o f t h e C o n d i t i o n i n g F a c t o r . . . . . . . . . . . . . . . . . . . . 2 0 7 7 . 2 . 7 T h e E f f e c t o f L e a f M o i s t u r e C o n t e n t a t F u l l H y d r a t i o n . . . . 2 0 8 7 . 2 . 8 T h e E f f e c t o f S o i l S u r f a c e R e s i s t a n c e t o M o i s t u r e T r a n s f e r 2 0 9 7 . 2 . 9 T h e E f f e c t o f t h e R a t i o o f S w a t h T h i c k n e s s t o D e n s i t y . . . . 2 0 9 8 C O N C L U S I O N S A N D R E C O M M E N D A T I O N S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 1 8 . 1 C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 1 8 . 2 R e c o m m e n d a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 3 A P P E N D I C E S A F O R T R A N C o d e o f S i m u l a t i o n M o d e l A L F D R Y . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 4 B F O R T R A N C o d e o f P s y c h r o m e t r i c L i b r a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 8 C D r y i n g D a t a , F i l e D R Y C A S E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 5 0 M e a s u r e d W e a t h e r D a t a , F i l e V T R M I C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 1 E L o c a l W e a t h e r D a t a , F i l e W T R D A Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 5 F F O R T R A N C o d e o f S E A R C H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 8 G F O R T R A N C o d e o f A R N L I N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 7 H A S a m p l e o f t h e P a r a m e t e r D a t a F i l e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8 7 I A R N L I N O u t p u t s f o r N i n e S e l e c t e d T r i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8 8 J A L F D R Y O u t p u t f o r C u t t i n g D a t e G r o u p 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 8 K D e t a i l e d R e p o r t f r o m A L F D R Y f o r T r i a l 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0 1 B I B L I O G R A P H Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 4 T a b l e 1 1 O 1 1 1 2 1 3 L I S T O F T A B L E S T h e f r a c t i o n o f t h e d i f f u s e s k y r a d i a t i o n i n t h e s o l a r i r r a d i a n c e a s a f u n c t i o n o f t h e s o l a r e l e v a t i o n a n g l e . . . . . . . . . . . . . . . . . . . . . . A v e r a g e y i e l d o f h a y a n d l e a f a r e a i n d e x ( L A I ) o f a l f a l f a m v e s t e d t h r e e t 1 . e s m p s e a s o n 0 0 0 . . . . . . - . . . . . . O C O O O O O O O O O O O O O O A e r o d y n a m i c p a r a m e t e r s i n t h e a t m o s p h e r i c s u r f a c e l a y e r w i t h Z r = 0 0 6 . , 2 c = 0 0 1 . ’ T p = 3 w ° K 3 1 1 d u r = 1 e o I / S o e o o e e m e o e e e e e e e e e e e e E m p i r i c a l c o n s t a n t s i n T h o m p s o n ' s e q u a t i o n o f e q u i l i b r i u m r e l a t i v e h u m i d i t y b a s e d u p o n B a k k e r ' s d a t a ( 1 9 6 1 ) . . . . . . . . . . . . . . . E s t i m a t e d e q u i l i b r i u m r e l a t i v e h u m i d i t y o f m a t u r e a l f a l f a i n d e s o r p t i o n a t v a r i o u s t e m p e r a t u r e a n d m o i s t u r e c o n t e n t s . . . . . . . . . G r o u p i n g d r y i n g t r i a l s b a s e d u p o n t h e d a t e o f c u t t i n g . . . . . . . . . . . E s t i m a t e d p a r a m e t e r ( a a n d b ) w i t h 9 6 1 c o n f i d e n c e i n t e r v a l s , t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) a n d t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) f o r n i n e s e l e c t e d t r i a l s . . . . . . . . . . . . . . . . . . E l e m e n t s o f t h e P m a t r i x ( P a a v P a b a n d P ) i t h e c o r r e l a t i o n c o e f f i c i e n t ( r a h ) , t h e s t a n d a r d d e v i a t i o n s ) a n d t h e n u m b e r o f o b s e r v a t i o n s ( n ) f o r t h e n i n e s e l e c t e d t r i a l s . . . . . . . . . . . . . . . . M i n i m u m v a l u e s o f t h e c o e f f i c i e n t o f d e t e r m i n a t i o n ( r ‘ ) f o r d e c l a r a t i o n o f r e g r e s s i o n m o d e l s w i t h t w o p a r a m e t e r s . . . . . . . . . . . . S u m m a r y o f t h e t o t a l s u m o f s q u a r e s ( S S T O ) , t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) a n d t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 1 ) f r o m t h r e e t r e a t m e n t s o f v a l i d a t i o n . . . . . . . . . . . . . . . . . . . . . . . . T h e s e t s o f s t a t i s t i c a l p a r a m e t e r s ( a a n d 6 ) u s e d d u r i n g m o d e l v a l i d a t i o n . . . . . . O O O O O O O O O O O O 0 0 . 0 . 0 . 0 . . . O . . . . . . O O O O O O O O O O O O O O O O O O V a l u e o f t h e i n t e r c e p t ( a ) a n d s l o p e ( B ) o f E q u a t i o n [ 2 2 1 ] o b t a i n e d f o r s c a t t e r d i a g r a m s o f t h r e e v a l i d a t i o n t r e a t m e n t s . . . . S u m m a r y o f t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) a n d t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) f o r t h e E a s t L a n s i n g g r o u p b y u s e o f l o c a l w e a t h e r s e r v i c e d a t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x i P a g e 2 4 3 4 6 6 9 2 9 3 1 7 1 1 7 2 1 7 2 1 8 5 1 8 7 1 8 9 1 9 5 1 9 6 L I S T O F F I G U R E S F i g u r e P a g e 1 S y s t e m b o u n d a r i e s f o r t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m . . . . . . . 1 7 2 A s i m u l a t e d p r o f i l e o f n e t d o w n s h o r t - w a v e r a d i a t i o n o f 1 0 1 1 1 2 1 3 1 4 1 5 1 6 a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . . . . 4 2 S i m u l a t e d a l b e d o a n d t h e e f f e c t i v e t r a n s m i t t a n c e f o r a s w a t h i n E a s t m u s i n g , M 1 C h i g a n o n A u g u s t 1 u - 1 6 ’ 1 9 8 ” . . . . . . . O O O O O O O O O . “ 3 A s i m u l a t e d p r o f i l e o f n e t d o w n w a r d l o n g - w a v e i r r a d i a n c e f o r a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . . . . 5 1 A s i m u l a t e d w i n d p r o f i l e w i t h i n a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . 0 . 0 . 0 . 0 . . . O O O O O O O O O O O O O O O O O O O O O O O 7 3 A s i m u l a t e d p r o f i l e o f t h e a l f a l f a m o i s t u r e c o n t e n t w i t h i n a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . . . . . . 1 1 0 A s i m u l a t e d p r o f i l e o f a i r t e m p e r a t u r e f o r a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 5 A s i m u l a t e d p r o f i l e o f w a t e r v a p o r d e n s i t y i n t h e a i r s p a c e s o f a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . 1 1 6 A s i m u l a t e d p r o f i l e o f s o i l t e m p e r a t u r e u n d e r n e a t h a s w a t h i n E S S A : L e a n i n g , M I C h i g a n o n A U S U S U 1 1 9 - 1 6 , 1 9 8 “ . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 1 2 “ B l o c k d i a g r a m o f t h e A L F D R Y s i m u l a t i o n m o d e l . . . . . . . . . . . . . . . . . . . . 1 2 5 C o m p a r i s o n b e t w e e n m e a s u r e d a n d p r e d i c t e d d r y i n g c u r v e s w i t h a s l o w s t e p o f 1 h r a n d a f a s t s t e p o f 6 , 6 0 o r 6 0 0 s . . . . . . . . . . . . . 1 3 0 A t r a y w i t h a n a l f a l f a s a m p l e d r y i n g i n t h e f i e l d . . . . . . . . . . . . . . . 1 3 4 A r r a n g e m e n t o f s a m p l e t r a y s i n t h e f i e l d . . . . . . . . . . . . . . . . . . . . . . . . 1 3 4 A m i c r o l o g g e r w i t h a s s o c i a t e d s e n s o r s a n d a c a s s e t t e r e c o r d e r . . . 1 3 7 A s o i l m o i s t u r e b l o c k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 7 T h e c a l i b r a t i o n c u r v e f o r s o i l m o i s t u r e b l o c k s . . . . . . . . . . . . . . . . . . 1 4 1 x i i A r e l a t i v e h u m i d i t y p r o b e i n t h e t o p l a y e r o f a s w a t h . . . . . . . . . . . 1 4 3 1 7 1 8 1 9 2 O 2 1 2 2 2 3 2 4 2 6 2 7 2 8 3 0 3 1 3 2 3 3 A m i c r o w e a t h e r u n i t i n t h e f i e l d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 3 S o l a r i r r a d i a n c e a n d w i n d s p e e d a t a r e f e r e n c e h e i g h t o f 0 . 6 m m e a s u r e d i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . . . 1 4 4 T e m p e r a t u r e s a b o v e a s w a t h , a t t h e t o p a n d b o t t o m o f t h e s w a t h , a n d b e l o w t h e s o i l s u r f a c e u n d e r n e a t h t h e s w a t h m e a s u r e d i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . . . . . . . . . . . . . . . 1 4 5 R e l a t i v e h u m i d i t i e s a b o v e a s w a t h , a t t h e t o p a n d b o t t o m o f t h e s w a t h , a n d b e l o w t h e s o i l s u r f a c e u n d e r n e a t h t h e s w a t h m e a s u r e d i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . . . . . . . . . . . . . . . . . P r e d i c t e d a n d o b s e r v e d d r y i n g c u r v e s f o r T r i a l 4 w i t h p a r a m e t e r a = 4 . 5 3 9 a n d b = 1 . 4 6 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 5 . 0 9 4 a n d P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 4 . 8 3 4 a n d P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 4 . 8 7 9 a n d P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 4 . 4 2 7 a n d P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 3 . 9 6 3 a n d P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 4 . 2 3 8 a n d P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 4 . 9 7 5 a n d P r e d i c t e d a n d o b s e r v e d p a r a m e t e r s a = 4 . 2 3 2 a n d d r y i n g c u r v e s f o r T r i a l 5 w i t h b : 2 0 2 3 7 e e e e e m e m e e e e e e e e e e e m o e e e e e e e e e e e e e d r y i n g c u r v e s f o r T r i a l 9 w i t h b = 1 . 8 3 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d r y i n g c u r v e s f o r T r i a l 1 3 w i t h b = 1 0 7 6 0 0 . . . . . . O O O O O O O O O O O O O O O O . 0 0 0 0 0 0 0 0 0 0 d r y i n g c u r v e s f o r T r i a l 2 5 w i t h b = 1 0 1 0 2 0 . . . . . . O . . . . . . . . . O O O O O O O O O O . . . . . . O d r y i n g c u r v e s f o r T r i a l 3 2 w i t h b = 0 0 3 2 0 0 0 . . . . O O O O O O O O O O O O O O . O O O O O O O O O O O O O d r y i n g c u r v e s f o r T r i a l 3 5 w i t h b = 0 0 8 u 3 0 0 0 0 0 0 0 0 0 0 0 0 O . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 d r y i n g c u r v e s f o r T r i a l 3 8 w i t h 0 : 2 . 5 6 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d r y i n g c u r v e s f o r T r i a l 4 4 w i t h b = 2 0 n 5 6 0 . . . . . . . . . O O O O O O O O O O O O O O O . . . . . . . . 0 M e a s u r e d v e r s u s s i m u l a t e d a v e r a g e s w a t h m o i s t u r e c o n t e n t s w i t h n i n e s e t s o f s t a t i s t i c a l p a r a m e t e r s o v e r a r a n g e o f f i e l d c o n d i t i o n s i n E a s t L a n s i n g , M i c h i g a n a n d F r e s n o , C a l i f o r n i a . . . . M e a s u r e d v e r s u s s i m u l a t e d a v e r a g e s w a t h m o i s t u r e c o n t e n t s w i t h t w o s e t s o f s t a t i s t i c a l p a r a m e t e r s o v e r a r a n g e o f f i e l d c o n d i t i o n s i n E a s t L a n s i n g , M i c h i g a n a n d F r e s n o , C a l i f o r n i a . . . . . M e a s u r e d v e r s u s s i m u l a t e d a v e r a g e s w a t h m o i s t u r e c o n t e n t s w i t h o n e s e t o f s t a t i s t i c a l p a r a m e t e r s o v e r a r a n g e o f f i e l d c o n d i t i o n s i n E a s t L a n s i n g , M i c h i g a n a n d F r e s n o , C a l i f o r n i a . . . . x i i i 1 4 6 1 7 5 1 7 6 1 7 7 1 7 8 1 7 9 1 8 0 1 8 1 1 8 2 1 8 3 1 9 2 1 9 3 1 9 4 3 4 N o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s o f t w o s t a t i s t i c a l p a r a m e t e r s B X ( 1 ) a n d B X ( 2 ) f o r T r i a l 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 9 3 5 N o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s o f i n i t i a l m o i s t u r e c o n t e n t ( S M C O ) , s w a t h d e n s i t y ( S D E N ) , t h e r a t i o o f l e a f a r e a i n d e x t o y i e l d ( S L / Y ) a n d c r o p m a t u r i t y l e v e l ( S B L M ) f o r T r i a l 5 . . . . . . . . . 2 0 0 3 6 N o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s o f t h e c o n d i t i o n i n g f a c t o r ( S C M E ) , l e a f m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n ( S F M C ) , S o i l s u r f a c e r e s i s t a n c e ( S R V S ) a n d t h e r a t i o o f s w a t h t h i c k n e s s t o d e n s i t y ( 8 2 / 0 ) f o r T r i a l 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 1 x i v C H A P T E R 1 I N T R O D U C T I O N 1 . 1 B a c k g r o u n d F o r a g e s p l a y a v e r y i m p o r t a n t r o l e i n t h e a g r i c u l t u r a l e c o n o m y o f t h e U n i t e d S t a t e s . M a i n t a i n i n g a p r o d u c t i v e a g r i c u l t u r e i n t h e U n i t e d S t a t e s d e p e n d s i n p a r t o n t h e u t i l i z a t i o n o f f o r a g e c r o p s i n o v e r a l l f a r m i n g o p e r a t i o n s ( M i l l e r , 1 9 8 4 ) . F o r a g e c r a p s p r o v i d e 8 0 % o f t h e f e e d c o n s u m e d b y r u m i n a n t a n i m a l s ( c a t t l e a n d s h e e p ) w h i c h p r o d u c e 1 / 3 o f t h e A m e r i c a n f o o d s u p p l y . T h e r e f o r e , i n a n A m e r i c a n d i e t , 1 8 % o f t h e e n e r g y a n d 3 6 % o f t h e p r o t e i n o r i g i n a t e i n d i r e c t l y f r o m f o r a g e s . R e c e n t l y , t h e i m p o r t a n c e o f p a s t u r e s a n d f o r a g e s h a s i n c r e a s e d d u e t o t h e l a r g e r n u m b e r o f h o r s e s i n t h e U n i t e d S t a t e s . A y e a r - r o u n d f o r a g e p r o g r a m i n c l u d i n g p a s t u r e , h a y , h a y l a g e , a n d s i l a g e c a n r e d u c e t h e f e e d c o s t f o r g e s t a t i n g s o w s b y a s m u c h a s 5 0 % ( M i l l e r , 1 9 8 4 ) . M a n y f a r m e r s h a v e r e c e n t l y b e e n i n t e r e s t e d i n g r o w i n g f o r a g e a s a c a s h c r o p . D u e t o t h e e n e r g y c r i s i s , t h e c o s t o f n i t r o g e n f e r t i l i z e r h a s i n c r e a s e d , s o t h e p r o d u c t i o n o f f o r a g e c r o p s f o r n i t r o g e n f i x a t i o n h a s b e c o m e m o r e i m p o r t a n t f o r A m e r i c a n f a r m e r s . A l f a l f a i s t h e o n l y f o r a g e k n o w n t o b e c u l t i v a t e d b e f o r e t h e e r a o f r e c o r d e d h i s t o r y , 7 0 0 0 - 4 0 0 0 8 . 0 . ( H a n s o n , 1 9 7 2 ) . A l f a l f a i s t h e m o s t p r o d u c t i v e p e r e n n i a l l e g u m e g r o w n i n t h e U n i t e d S t a t e s , a n d i t h a s t h e h i g h e s t f e e d i n g v a l u e a m o n g a l l f o r a g e c r o p s . I t c o n t a i n s s u p e r i o r l 2 p r o t e i n a n d e n e r g y ; i t i s h i g h i n v i t a m i n a n d m i n e r a l c o n t e n t s . O n e t h o u s a n d k i l o g r a m s o f a l f a l f a ( d r y m a t t e r ) c o n t a i n s 1 8 0 k g o f p r o t e i n , w h i c h i s e q u i v a l e n t t o 8 1 0 k g o f c o r n g r a i n o r 1 8 0 k g o f s o y b e a n s , a n d 7 1 0 k g o f T o t a l D i g e s t i b l e N u t r i e n t s ( T D N ) , w h i c h e q u a l s 3 5 0 k g o f c o r n g r a i n o r 5 2 0 k g o f c o r n s i l a g e ( M i l l e r , 1 9 8 4 ) . A l f a l f a i s a l s o a g o o d c r o p f o r n i t r o g e n f i x a t i o n . A l f a l f a - g r a s s c o m b i n a t i o n s a r e e x c e l l e n t i n r e d u c i n g s o i l e r o s i o n a n d w a t e r r u n - o f f f o r s o i l c o n s e r v a t i o n . M a y i s t h e m a j o r f o r m o f h a r v e s t e d a l f a l f a . I n 1 9 8 5 , t h e t o t a l p r o d u c t i o n o f a l f a l f a h a y i n t h e U n i t e d S t a t e s w a s 8 6 m i l l i o n t o n s o n 1 1 m i l l i o n h e c t a r e s ( P e t r i t z , 1 9 8 6 ) . A l t h o u g h t e c h n i q u e s o f a r t i f i c i a l l y d r y i n g h a y h a v e b e e n d e v e l o p e d s i n c e 1 9 2 0 , m o s t a l f a l f a h a y i s s t i l l n a t u r a l l y d r i e d i n f i e l d s r a t h e r t h a n i n b a r n d r y e r s o r d e h y d r a t o r s . T h e g r e a t e s t c h a l l e n g e i n h a y - m a k i n g i s t o d r y a l f a l f a a s q u i c k l y a s p o s s i b l e i n o r d e r t o m i n i m i z e l o s s e s a n d p r e s e r v e f o r a g e q u a l i t y . A s h i g h a s 4 % o f t h e d r y m a t t e r y i e l d c a n b e l o s t e a c h d a y w h e n c u t a l f a l f a r e m a i n s i n t h e f i e l d ( W i l k i n s o n , 1 9 8 1 ) . D i s c o l o r i n g a n d n u t r i e n t c h a n g e s a l s o o c c u r d u r i n g f i e l d c u r i n g d u e t o e x c e s s i v e s u n s h i n e , r a i n a n d m i c r o b i a l a c t i v i t y . D u r i n g f i e l d d r y i n g , o u t a l f a l f a r e m a i n s i n a s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m . I n t h e f i e l d e n v i r o n m e n t , d r y i n g i s a c o m p l i c a t e d p r o c e s s g o v e r n e d b y p h y s i c a l a n d p h y s i o l o g i c a l p r i n c i p l e s . T h e m a j o r m e c h a n i s m s a r e h e a t a n d m o i s t u r e t r a n s f e r . M a n y p l a n t a n d e n v i r o n m e n t f a c t o r s i n f l u e n c e f i e l d d r y i n g , s u c h a s s o l a r r a d i a t i o n , s k y c l o u d c o v e r , a t m o s p h e r i c t e m p e r a t u r e a n d h u m i d i t y , w i n d s p e e d , r a i n f a l l , d e w , a l f a l f a v a r i e t y , l e a f a r e a , l e a f o r i e n t a t i o n , p l a n t g r o w t h s t a g e , p l a n t r e s i s t a n c e t o m o i s t u r e r e m o v a l , s w a t h d e n s i t y , s o i l t e m p e r a t u r e a n d s o i l m o i s t u r e l e v e l . 3 M a t h e m a t i c a l m o d e l s b a s e d u p o n t h e p h y s i c a l a n d p h y s i o l o g i c a l p r i n c i p l e s o f t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m p r o v i d e u s e f u l t o o l s t o s t u d y f i e l d d r y i n g a n d t o c o m p r e h e n s i v e l y u n d e r s t a n d t h e n a t u r a l d r y i n g p r o c e s s . T h r o u g h m o d e l i n g , t h e r e l a t i v e i m p o r t a n c e a n d i n t e r r e l a t i o n s h i p a m o n g v a r i o u s f a c t o r s c a n b e q u a n t i f i e d . A l t e r i n g i m p o r t a n t a n d c o n t r o l l a b l e f a c t o r s c a n s p e e d f i e l d d r y i n g a n d , t h u s , i m p r o v e h a y q u a l i t y . F o r i n s t a n c e , m e c h a n i c a l c o n d i t i o n i n g a n d c h e m i c a l t r e a t m e n t s c a n r e d u c e t h e p l a n t ' s r e s i s t a n c e t o m o i s t u r e r e m o v a l . T e d d i n g c a n m a n i p u l a t e t h e s w a t h s t r u c t u r e a n d e x p o s e u n d r i e d a l f a l f a t o d i r e c t s u n l i g h t . D i f f e r i n g f r o m a r t i f i c i a l d r y i n g i n a c o n t r o l l e d e n v i r o n m e n t , f i e l d d r y i n g t a k e s p l a c e i n a n a t u r a l e n v i r o n m e n t . T h e n u m b e r o f c o n t r o l l a b l e f a c t o r s i s l i m i t e d . F o r e x a m p l e , w e a t h e r a n d s o i l c o n d i t i o n s a r e r a r e l y c o n t r o l l a b l e . H o w e v e r , w h e n s t a t i s t i c a l d a t a o f t h o s e u n c o n t r o l l a b l e f a c t o r s a r e a v a i l a b l e , m a t h e m a t i c a l m o d e l s c a n b e e m p l o y e d t o p r e d i c t t h e r a t e o f f i e l d d r y i n g u n d e r c e r t a i n e n v i r o n m e n t a l c o n d i t i o n s . T h i s m a y e n a b l e s e l e c t i o n o f s u i t a b l e d a t e s f o r f i e l d c u r i n g o p e r a t i o n s . C o m p u t e r t e c h n o l o g y p r o v i d e s a p o w e r f u l t o o l f o r r a p i d c a l c u l a t i o n a n d m o d e l i n g o f c o m p l i c a t e d p h e n o m e n a . T h i s e x t e n d s t h e h u m a n c a p a b i l i t y o f p r o b l e m - s o l v i n g . B y u s e o f c o m p u t e r s , t h e d y n a m i c b e h a v i o r o f a r e a l p r o c e s s , w h i c h m a y t a k e p l a c e f o r h o u r s , d a y s , m o n t h s , o r y e a r s , c a n b e s i m u l a t e d i n s e c o n d s , m i n u t e s , o r h o u r s . T h e r e f o r e , e x t e n s i v e s t u d i e s a n d p r e d i c t i o n s b e c o m e p o s s i b l e . C o m p u t e r s i m u l a t i o n i s a n u m e r i c a l t e c h n i q u e f o r c o n d u c t i n g h y p o t h e t i c a l e x p e r i m e n t s o n m a t h e m a t i c a l m o d e l s w h i c h d e s c r i b e t h e q u a n t i t a t i v e b e h a v i o r o f d y n a m i c s y s t e m s . T h i s m e t h o d o l o g y i s v e r y s u i t a b l e f o r s t u d y i n g f i e l d d r y i n g o f a l f a l f a . 1 . 2 O b j e c t i v e s T h e o v e r a l l g o a l o f t h i s s t u d y w a s t o d e v e l o p a s i m u l a t i o n m o d e l o f t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m f o r f i e l d d r y i n g a l f a l f a . E m p h a s i s w a s o n t h e d y n a m i c b e h a v i o r 0 f t h e h e a t a n d m o i s t u r e s t a t u s w i t h i n t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m , a n d t h e a p p l i c a t i o n o f e n g i n e e r i n g p r i n c i p l e s t o b i o l o g i c a l s y s t e m s . T h e m o d e l w a s t o b e g e n e r a l e n o u g h t o r e p r e s e n t f i e l d d r y i n g o f a l f a l f a u n d e r v a r i o u s c o n d i t i o n s . S p e c i f i c o b j e c t i v e s w e r e : ( 1 ) t o f o r m u l a t e a c o m p u t e r m o d e l t o s i m u l a t e t h e p r o c e s s e s o f h e a t a n d m o i s t u r e t r a n s f e r i n t h e s y s t e m o f t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m w h e n c u t a l f a l f a i s d r i e d i n t h e f i e l d ; ( 2 ) t o v a l i d a t e t h e m o d e l b y c o m p a r i n g s i m u l a t e d r e s u l t s w i t h m e a s u r e d d a t a o f f i e l d d r y i n g a l f a l f a ; a n d ( 3 ) t o q u a n t i f y t h e i m p o r t a n c e a m o n g p a r a m e t e r s a n d v a r i a b l e s o f t h e m o d e l b y a s e n s i t i v i t y a n a l y s i s . C H A P T E R 2 L I T E R A T U R E R E V I E W T h i s c h a p t e r p r e s e n t s a b r i e f r e v i e w o f t h e l i t e r a t u r e o n p r e v i o u s e x p e r i m e n t a l a n d m o d e l i n g w o r k d e a l i n g w i t h f i e l d d r y i n g o f a l f a l f a . S o m e l i t e r a t u r e w i l l b e d i s c u s s e d m o r e e x t e n s i v e l y i n l a t e r c h a p t e r s . A l f a l f a w a s t h e f i r s t d o m e s t i c a t e d f o r a g e c r o p . H i s t o r y i n d i c a t e s t h a t a l f a l f a w a s u s e d a s a f o r a g e i n T u r k e y o v e r 3 3 0 0 y e a r s a g o ( M i l l e r , 1 9 8 4 ) . A l f a l f a w a s i n t r o d u c e d f r o m S o u t h A m e r i c a i n t o t h e U n i t e d S t a t e s d u r i n g t h e C a l i f o r n i a g o l d r u s h . I n t h e U n i t e d S t a t e s , t h e r e w e r e a f e w h e c t a r e s o f a l f a l f a c u t f o r h a y i n 1 8 5 4 ; s i n c e t h e n a l f a l f a p r o d u c t i o n r e a c h e d a p e a k o f 1 2 . 1 m i l l i o n h e c t a r e s i n 1 9 5 8 ( H a n s o n , 1 9 7 2 ) . T h e e a r l i e s t s t u d y o n i m p r o v e m e n t o f f i e l d d r y i n g o f h a y w a s r e p o r t e d i n t h e 1 9 3 0 ' s . J o n e s a n d h i s a s s o c i a t e s ( 1 9 3 2 , 1 9 3 3 , 1 9 3 4 , 1 9 3 6 , 1 9 3 9 a n d 1 9 4 1 ) c o n d u c t e d a s e r i e s o f r e s e a r c h s t u d i e s o n t h e i n f l u e n c e o f p l a n t p h y s i o l o g i c a l r e a c t i o n s o n f i e l d c u r i n g o f h a y . I n t h e i r e a r l y w o r k , t h e g o a l w a s t o i n v e s t i g a t e t h e b e h a v i o r o f s t o m a t a , l e a v e s a n d s t e m s r e g a r d i n g t h e i r e f f e c t s o n f i e l d d r y i n g . S i n c e t h e n , m a n y i n v e s t i g a t o r s h a v e f o c u s e d o n r e l a t e d t o p i c s . H a r r i s a n d T u l l b e r g ( 1 9 8 0 ) p r e s e n t e d a n e x t e n s i v e r e v i e w o f t h e l i t e r a t u r e i n t h i s a r e a . T h e y r e p o r t e d t h a t l e a v e s w h e r e s t o m a t a w e r e a b u n d a n t p l a y e d a n i m p o r t a n t r o l e i n p l a n t m o i s t u r e l o s s r a t h e r t h a n s t e m s w h e r e f e w s t o m a t a w e r e p r e s e n t . L e a v e s d r e w m o i s t u r e f r o m s t e m s c o n t i n u o u s l y u n t i l l e a v e s w e r e d r i e d . T h e l o s s o f p l a n t 5 6 m o i s t u r e w a s a t t r i b u t e d t o t h e o p e n i n g o f s t o m a t a . A b o u t 2 h o u r s a f t e r c u t t i n g , s t o m a t a c o m p l e t e l y c l o s e d t o p r e v e n t f u r t h e r m o i s t u r e l o s s ; t h e r e a f t e r , t h e p l a n t i n t e r n a l r e s i s t a n c e t o m o i s t u r e r e m o v a l i n c r e a s e d s u b s t a n t i a l l y . T h e p l a n t m o i s t u r e c o n t e n t a t w h i c h s t o m a t a w e r e c l o s e d w a s a b o u t 7 0 - 8 0 % o f t h e i n i t i a l m o i s t u r e l e v e l a t c u t t i n g . H o w e v e r , p r e c i s e a n d d e t a i l e d d a t a o n t h e p l a n t r e s i s t a n c e t o m o i s t u r e l o s s a s a f f e c t e d b y l i g h t i n t e n s i t y , t e m p e r a t u r e a n d p l a n t m o i s t u r e c o n t e n t , e s p e c i a l l y u n d e r s e v e r e w a t e r s t r e s s , i s n o t a v a i l a b l e . D u e t o t h e c o m p l e x n a t u r e o f f i e l d d r y i n g , m o d e l i n g o f f i e l d c u r i n g h a y w a s n o t a t t e m p t e d u n t i l s o m e d r y i n g t h e o r i e s o f a g r i c u l t u r a l p r o d u c t s w e r e d e v e l o p e d a n d w i d e l y a p p l i e d t o a r t i f i c i a l d r y i n g o f g r a i n a n d h a y . R e s e a r c h e r s h a v e a t t e m p t e d t o m o d e l f i e l d d r y i n g o f h a y w i t h t h e t h e o r y o f t h i n - l a y e r d r y i n g . A c c o r d i n g t o t h e t h e o r y , d r y i n g b i o l o g i c a l m a t e r i a l s d u r i n g t h e f a l l i n g - r a t e p e r i o d c a n b e e x p r e s s e d a s a n e x p o n e n t i a l l y d e c a y i n g f u n c t i o n ( H e n d e r s o n a n d P e r r y , 1 9 6 6 ) : M R = ( M - M e ) / ( M 1 - M e ) : A e ' k t [ 1 ] w h e r e M R = m o i s t u r e r a t i o , d e c i m a l , t = t i m e , M = m o i s t u r e c o n t e n t a t t i m e t , k g w a t e r / k g d r y m a t t e r , M e = e q u i l i b r i u m m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , M 1 = i n i t i a l m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , k = d r y i n g c o n s t a n t , r e c i p r o c a l t i m e , a n d A = c o e f f i c i e n t o f t h e m a t e r i a l s h a p e , d e c i m a l . T h e d i f f e r e n t i a l f o r m o f E q u a t i o n [ 1 ] i s : a M / a t = - k ( M - M e ) [ 2 ] w h e r e B M / B t : d r y i n g r a t e , k g w a t e r / k g d r y m a t t e r p e r u n i t t i m e . 7 M o d e l s o f f i e l d d r y i n g h a y h a v e n o r m a l l y b e e n m o d i f i c a t i o n s o f t h e s e b a s i c e q u a t i o n s . T h e d r y i n g c o n s t a n t k h a s b e e n m o d e l e d a s a f u n c t i o n o f v a r i o u s p l a n t a n d e n v i r o n m e n t f a c t o r s . K e m p e t a l . ( 1 9 7 2 ) p r o p o s e d a r e g r e s s i o n m o d e l i n w h i c h t h e d r y i n g c o n s t a n t k w a s a n o n l i n e a r f u n c t i o n o f t i m e a n d ' l a t e n t e v a p o r a t i o n ' . L a t e n t e v a p o r a t i o n w a s a m e t e o r o l o g i c a l p a r a m e t e r r e p r e s e n t i n g t h e e v a p o r a t i o n r a t e o f w a t e r i n t o t h e a t m o s p h e r e m e a s u r e d w i t h a s p e c i a l i n s t r u m e n t c a l l e d a n a t m o m e t e r . A l i m i t e d n u m b e r o f l a b o r a t o r y t r i a l s w e r e u s e d t o t e s t t h e m o d e l . H o w e v e r , n o v a l i d a t i o n w i t h f i e l d d r y i n g d a t a w a s r e p o r t e d . H a y h o e a n d J a c k s o n ( 1 9 7 4 ) s u g g e s t e d a r e p l a c e m e n t o f t h e e x p o n e n t t e r m , k t , i n E q u a t i o n [ 2 ] w i t h a s u m o f d a i l y d r y i n g i n d i c e s . T h e d a i l y d r y i n g i n d e x w a s d e f i n e d a s a l i n e a r r e g r e s s i o n f u n c t i o n o f t h e d a i l y p o t e n t i a l e v a p o r a t i o n a n d t h e d a i l y p r e c i p i t a t i o n . F o r d r y i n g a n d h a r v e s t i n g o p e r a t i o n s , p r e d i c t i o n s m a d e o n h o u r l y b a s e s w o u l d b e m o r e d e s i r a b l e t h a n t h o s e o n d a i l y b a s e s . I n t h e m o d e l , t h e e q u i l i b r i u m m o i s t u r e c o n t e n t M e w a s n o t i n c l u d e d . H i l l e t a l . ( 1 9 7 7 ) p r e s e n t e d a m o d e l i n w h i c h t h e d r y i n g c o n s t a n t k w a s a n o n l i n e a r f u n c t i o n o f t i m e a n d v a p o r p r e s s u r e d e f i c i t . T h e i r m o d e l w a s d e r i v e d f r o m l a b o r a t o r y e x p e r i m e n t s . T h e y e x p e c t e d t h a t m a j o r e r r o r s m a y o c c u r i f t h e v a p o r p r e s s u r e d e f i c i t i s c a l c u l a t e d f r o m a m b i e n t a i r c o n d i t i o n s , s i n c e s u r f a c e t e m p e r a t u r e o f s w a t h s i s s u b s t a n t i a l l y e l e v a t e d u n d e r s u n s h i n e . T h e y s u g g e s t e d s t u d y o f t h e e f f e c t s o f s o l a r r a d i a t i o n i n t e n s i t y a n d s o i l m o i s t u r e o n f i e l d d r y i n g . S a v o i e e t a l . ( 1 9 8 2 ) p r o p o s e d a n e m p i r i c a l m o d e l o f t h e f o r m o f E q u a t i o n [ 2 ] . T h e d r y i n g c o n s t a n t k w a s a l i n e a r c o m b i n a t i o n o f s o l a r r a d i a t i o n , a i r t e m p e r a t u r e , w i n d s p e e d , w i n d r o w a r e a d e n s i t y a n d i n d i c e s 8 f o r r a k i n g , c o n d i t i o n i n g , r a i n - d e w a n d d a y . T h e c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) o f t h e m o d e l w a s o n l y 0 . 3 6 , w h i c h w a s n o t a d e q u a t e t o a p p l y t h e m o d e l t o v a r i o u s d r y i n g p r a c t i c e s . R o t z a n d C h e n ( 1 9 8 5 ) i m p r o v e d S a v o i e s ' m o d e l b y u s i n g a n o n l i n e a r r e g r e s s i o n t e c h n i q u e . I n t h e i r e m p i r i c a l m o d e l , t h e d r y i n g c o n s t a n t k w a s a n a l g e b r a i c f r a c t i o n o f s o l a r r a d i a t i o n , v a p o r p r e s s u r e d e f i c i t , s w a t h a r e a d e n s i t y , a p p l i c a t i o n r a t e o f a c h e m i c a l c o n d i t i o n e r , ‘ s o i l m o i s t u r e a n d a d a y i n d e x . T h e e q u i l i b r i u m m o i s t u r e c o n t e n t M e w a s r e m o v e d f r o m t h e e x p r e s s i o n o f m o i s t u r e r a t i o M R ( s e e E q u a t i o n [ 2 ] ) , b e c a u s e t h e y f o u n d t h a t M e w a s v e r y s m a l l u n d e r f i e l d d r y i n g c o n d i t i o n s . T h e c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) w a s i n c r e a s e d t o 0 . 6 4 . P i t t ( 1 9 8 4 ) r e l a t e d t h e t e r m , k t , i n E q u a t i o n [ 2 ] t o p a n e v a p o r a t i o n t h r o u g h a m a t h e m a t i c a l a n a l y s i s . P a n e v a p o r a t i o n i s a c l i m a t i c p a r a m e t e r w h i c h m e a s u r e s t h e i n t e g r a t e d e f f e c t s o f s u n l i g h t , a i r t e m p e r a t u r e a n d h u m i d i t y o v e r t i m e . P a r a m e t e r s o f t h e m o d e l , i n c l u d i n g k , w e r e a s s u m e d t o b e c o n s t a n t . H o w e v e r , r e s u l t s s h o w e d t h a t t h e v a l u e s o f t h o s e p a r a m e t e r s v a r i e d f r o m o n e e x p e r i m e n t t o a n o t h e r , w h i c h i m p l i e d t h a t a p p l i c a t i o n o f t h e m o d e l w a s l i m i t e d . H e a l s o f o u n d t h a t t h e m o d e l w a s n o t s e n s i t i v e t o t h e c h o i c e o f t h e e q u i l i b r i u m m o i s t u r e c o n t e n t M e , a n d a z e r o v a l u e o f M e w a s a s s u m e d . T h e s e a f o r e m e n t i o n e d m o d e l s h a v e u s e d o n e e q u a t i o n t o d e p i c t a c o m p l i c a t e d s y s t e m d u r i n g f i e l d d r y i n g o f a l f a l f a . T h e r e f o r e , t h e s e m o d e l s c a n n o t e x p o s e t h e s y s t e m s s t r u c t u r e , o r t h e b e h a v i o r o f i n d i v i d u a l c o m p o n e n t s a n d t h e i r i n t e r a c t i o n s . T h e p a r a m e t e r s o f t h e s e m o d e l s w e r e d e r i v e d b y u s e o f s t a t i s t i c a l t e c h n i q u e s , w h i c h c a n l i m i t a p p l i c a t i o n s o f t h e s e m o d e l s t o o t h e r s e t s o f d a t a n o t u s e d i n m o d e l d e v e l o p m e n t . R e w e t t i n g d u e t o d e w o r p r e c i p i t a t i o n w e r e e x c l u d e d f r o m 9 t h e s e m o d e l s , e x c e p t t h e o n e b y H a y h o e a n d J a c k s o n ( 1 9 7 4 ) . M o r e c o m p l i c a t e d m o d e l s i n w h i c h t h e f i e l d d r y i n g p r o c e s s w a s i n t e g r a t e d s t e p - b y - s t e p o v e r a d r y i n g p e r i o d h a v e b e e n d e v e l o p e d ( D y e r a n d B r o w n , 1 9 7 7 ; a n d D a l e e t a l . , 1 9 7 8 ) . F o r e a c h t i m e s t e p , t h e d r y i n g r a t e o f a l f a l f a w a s c a l c u l a t e d a s a f u n c t i o n o f v a r i o u s e n v i r o n m e n t a n d p l a n t f a c t o r s . D y e r a n d B r o w n ( 1 9 7 7 ) p r o p o s e d a m o d e l ( F H A Y D ) . I n t h e m o d e l , t h e d a i l y d r y i n g i n d e x b y H a y h o e a n d J a c k s o n ( 1 9 7 4 ) w a s a d o p t e d t o c a l c u l a t e a d a i l y d r y i n g r a t e , a n d e m p i r i c a l r e l a t i o n s w e r e u s e d t o d e t e r m i n e a d a i l y r e w e t t i n g r a t e d u e t o r a i n a n d / o r d e w . T h e l e n g t h o f t h e t i m e s t e p w a s o n e d a y , a n d t h u s , t h e m o d e l c o u l d n o t s i m u l a t e t h e p r o g r e s s o f d r y i n g w i t h i n a d a y p e r i o d . D a l e e t a l . ( 1 9 7 8 ) d e v e l o p e d a d r y i n g m o d e l i n w h i c h a n h o u r l y d r y i n g r a t e w a s t h e r e s u l t o f e v a p o r a t i o n , r a i n a n d / o r d e w . T h e e v a p o r a t i o n r a t e w a s o b t a i n e d f r o m a n e m p i r i c a l e x p r e s s i o n i n c l u d i n g e f f e c t s o f s o l a r e n e r g y , e q u i l i b r i u m m o i s t u r e c o n t e n t , w i n d s p e e d , v a p o r p r e s s u r e d e f i c i t , w i n d r o w s u r f a c e a r e a a n d h a y s p e c i e s . N o n e o f t h e s e m o d e l s a b o v e w a s d e v e l o p e d b a s e d o n t h e p r i n c i p a l m e c h a n i s m - h e a t a n d m o i s t u r e t r a n s f e r - i n t h e f i e l d d r y i n g p r o c e s s . T h e s e m o d e l s d i d n o t s i m u l a t e t h e a c t u a l t r a n s f e r p r o c e s s i n t h e s y s t e m o f f i e l d d r y i n g o f a l f a l f a . T h e r e l a t i o n s o f v a r i o u s e n v i r o n m e n t a n d p l a n t f a c t o r s t o f i e l d d r y i n g i n t h e s e a b o v e m o d e l s w e r e d e s c r i p t i v e r a t h e r t h a n f u n c t i o n a l . B r u c k a n d v a n E l d e r e n ( 1 9 6 9 ) r e p o r t e d a t w o - l a y e r m o d e l i n w h i c h t h e d r y i n g r a t e o f h a y w a s c a l c u l a t e d b a s e d u p o n t h e m o i s t u r e a n d e n e r g y b a l a n c e b e t w e e n t h e a t m o s p h e r e a n d t h e u p p e r p o r t i o n o f a h a y s w a t h . T h e t h i c k n e s s o f t h e u p p e r l a y e r i n c r e a s e d o r d e c r e a s e d a s d r y i n g o r l O r e w e t t i n g a d v a n c e d . T h e l o w e r p o r t i o n o f t h e s w a t h w a s a s s u m e d t o b e n o t d i r e c t l y a f f e c t e d b y e n v i r o n m e n t a l c o n d i t i o n s . D u r i n g f i e l d d r y i n g , i n t e r a c t i o n s e x i s t b e t w e e n p l a n t s , s o i l a n d t h e a i r ; h o w e v e r , t h e i r m o d e l n e g l e c t e d t r a n s f e r p r o c e s s e s w i t h i n t h e s w a t h a n d b e t w e e n t h e s w a t h a n d s o i l . V a l i d a t i o n o f t h e m o d e l w a s l i m i t e d t o o n l y o n e f i e l d d r y i n g c a s e . N e v e r t h e l e s s , t h e i r w o r k w a s a n e a r l y m o d e l i n g a t t e m p t o f r e l a t i n g f i e l d d r y i n g t o h e a t a n d m o i s t u r e t r a n s f e r p r o c e s s e s i n h a y s w a t h s . P l a n t s c i e n c e h a s m a d e g r e a t a d v a n c e s s i n c e t h e i n t r o d u c t i o n o f t h e c o n c e p t o f t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m . T h e s y s t e m s a p p r o a c h h a s b e e n a p p l i e d t o l i v i n g v e g e t a t i o n o n l y . M a n y r e s e a r c h e r s h a v e s t u d i e d t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m s y s t e m s o f g r o w i n g p l a n t s . M O n t e i t h ( 1 9 7 3 a n d 1 9 7 5 ) , M i l t h o p e a n d M o o r b y ( 1 9 7 4 ) , C a m p b e l l ( 1 9 7 7 ) , L a n g e e t a l . ( 1 9 8 2 ) a n d R o s e n b e r g e t a l . ( 1 9 8 3 ) d i s c u s s e d v a r i o u s t o p i c s r e g a r d i n g t h e e n e r g y a n d w a t e r r e l a t i o n s o f t h e m i c r o c l i m a t i c e n v i r o n m e n t - a b o v e , w i t h i n a n d b e l o w p l a n t c a n o p i e s - i n s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m s y s t e m s . A n e q u a t i o n k n o w n a s t h e M o n t e i t h - P e n m a n e q u a t i o n h a s b e e n w i d e l y a p p l i e d i n m o d e l i n g t h e e v a p o t r a n s p i r a t i o n l o s s o f p l a n t m o i s t u r e i n v a r i o u s e n v i r o n m e n t s ( s e e C h a p t e r 4 f o r d e t a i l ) . I n r e c e n t d e c a d e s , t h e t e n d e n c y o f r e s e a r c h i n v e g e t a t i o n s c i e n c e h a s s h i f t e d f r o m d e s c r i p t i v e t o a n a l y t i c a l a n d f u n c t i o n a l d e v e l o p m e n t . R o s s ( 1 9 8 1 ) p r e s e n t e d a t h o r o u g h l y t h e o r e t i c a l a n a l y s i s o f t h e r a d i a t i o n r e g i m e a n d t h e a r c h i t e c t u r e o f p l a n t s t a n d s . H i s w o r k w a s a m a j o r c o n t r i b u t i o n t o a p p l i e d p l a n t s c i e n c e s , s i n c e s o l a r e n e r g y i s t h e f u n d a m e n t a l s o u r c e d r i v i n g v a r i o u s p h y s i c a l a n d b i o l o g i c a l p r o c e s s e s o n t h e e a r t h . l l S i m u l a t i o n m o d e l s h a v e b e e n d e v e l o p e d t o s t u d y p l a n t p h y s i o l o g y . D e W i t a n d K e u l e n ( 1 9 7 2 ) , a n d H i l l e l ( 1 9 7 7 ) r e p o r t e d t h e i r s i m u l a t i o n m o d e l s o f h e a t t r a n s f e r a n d w a t e r m o v e m e n t i n s o i l . G o u d r i a a n ( 1 9 7 7 ) a n d d e V r i e s e t a l . ( 1 9 8 1 ) p r o p o s e d s i m u l a t i o n m o d e l s f o r p l a n t g r o w t h a n d c r o p p r o d u c t i o n . T h e s e m o d e l s p r o v i d e d u s e f u l b a c k g r o u n d i n f o r m a t i o n f o r t h i s s t u d y . H o w e v e r , t h e m o d e l s c o u l d n o t b e d i r e c t l y a p p l i e d t o f i e l d d r y i n g a l f a l f a . C u t a l f a l f a i n t h e f i e l d h a s d i f f e r e n t c h a r a c t e r i s t i c s f r o m l i v i n g p l a n t s t a n d s . S e v e r a l m a j o r c h a n g e s t a k e p l a c e . F i r s t o f a l l , a l f a l f a p l a n t s l a y o n t h e g r o u n d , a n d f o r m t h i n s w a t h s . T h e w i n d s p e e d a t t h e t o p o f a s w a t h i s m u c h l e s s t h a n a t t h e t o p o f s t a n d i n g a l f a l f a , s i n c e t h e w i n d s p e e d d e c r e a s e s a s t h e h e i g h t a b o v e t h e g r o u n d d e c r e a s e s . W i t h i n a s w a t h , t h e r e i s l i t t l e a i r m o v e m e n t , a n d t h u s , c o n v e c t i v e t r a n s f e r o f h e a t a n d m o i s t u r e i s r e d u c e d . F o r a l i v i n g a l f a l f a c a n o p y , m o s t l e a v e s a r e l o c a t e d i n t h e u p p e r p o r t i o n , s o t h e l e a f d i s t r i b u t i o n i s h e i g h t - d e p e n d e n t . H o w e v e r , i n a s w a t h , l e a v e s a r e d i s t r i b u t e d m o r e e v e n l y a l o n g t h e s w a t h h e i g h t . T h i s a f f e c t s b o t h l i g h t p e n e t r a t i o n a n d a i r m o v e m e n t i n s i d e t h e s w a t h . S e c o n d l y , p l a n t g r o w t h s t o p s a f t e r c u t t i n g a n d l i v i n g a c t i v i t i e s g r a d u a l l y c e a s e . T h i r d l y , t h e w a t e r s u p p l y f r o m t h e s o i l t h r o u g h r o o t s t o s t e m s a n d l e a v e s i s b r o k e n . W a t e r s t r e s s c a u s e s s t o m a t a c l o s u r e w h i c h i n c r e a s e s t h e i n t e r n a l r e s i s t a n c e t o m o i s t u r e a n d g a s t r a n s f e r . H e a t a n d m o i s t u r e a c t i v e l y t r a n s f e r b e t w e e n t h e s o i l s u r f a c e a n d t h e p l a n t s a t t h e s w a t h b o t t o m . T h e s e a f f e c t f i e l d d r y i n g s i g n i f i c a n t l y . S i m u l a t i o n m o d e l s o f t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m f o r f i e l d d r y i n g p l a n t s s h o u l d r e f l e c t t h e s e f a c t o r s . T h o m p s o n ( 1 9 8 1 ) p r o p o s e d a b u l k m o d e l a n d a m u l t i - l a y e r m o d e l f o r 1 2 f i e l d d r y i n g o f h a y . A v e r a g e s w a t h m o i s t u r e c o n t e n t s ( d r y b a s i s ) e s t i m a t e d b y t h e b u l k m o d e l w e r e o v e r e s t i m a t e d c o m p a r e d t o t h o s e o b t a i n e d b y t h e m u l t i - l a y e r m o d e l . T h i s m a y l e a d t o e r r o n e o u s d e c i s i o n s i f t h e b u l k m o d e l i s a p p l i e d f o r m a n a g e m e n t p u r p o s e s . T h o m p s o n ( 1 9 8 1 ) i n d i c a t e d t h a t a b u l k m o d e l c o u l d n o t p r o v i d e a c o m p l e t e l y a d e q u a t e t r e a t m e n t o f t h e d r y i n g p r o c e s s . T h e m u l t i - l a y e r m o d e l t r e a t e d h e a t a n d m o i s t u r e t r a n s f e r b a s e d u p o n t h e d i f f e r e n c e s o f t e m p e r a t u r e a n d s p e c i f i c h u m i d i t y b e t w e e n l e a v e s o f e a c h l a y e r a n d t h e a m b i e n t a i r a b o v e t h e t o p . T h i s w a s s i m i l a r t o t h a t o f t h e b u l k m o d e l . T r a n s f e r p r o p e r t i e s o f h e a t a n d m o i s t u r e w i t h i n t h e s w a t h s h o u l d b e d e t e r m i n e d b y l o c a l c o n d i t i o n s i m m e d i a t e l y a r o u n d t r a n s f e r s u r f a c e s , e . g . l e a f s u r f a c e s a n d i n t e r f a c e s b e t w e e n a d j a c e n t ' l a y e r s ' . A l t h o u g h t h e s w a t h w a s m o d e l e d i n e i g h t - l a y e r d e t a i l , t h e m o d e l i n g o f r a d i a t i o n , w i n d s p e e d w i t h i n t h e s w a t h a n d l e a f r e s i s t a n c e s e e m e d t o b e o v e r s i m p l i f i e d . V a l i d a t i o n o f t h e t w o m o d e l s w a s a l s o l i m i t e d . S m i t h ( 1 9 8 5 ) p r o p o s e d a n o t h e r b u l k - s w a t h d r y i n g m o d e l i n w h i c h t h e e f f e c t o f h o r i z o n t a l a d v e c t i o n w a s t a k e n i n t o a c c o u n t . H e p r e s e n t e d s o m e a n a l y t i c d i s c u s s i o n s w i t h n o e x p e r i m e n t a l v e r i f i c a t i o n . T h e d i s a d v a n t a g e o f t h e b u l k m o d e l w a s p o i n t e d o u t b y T h o m p s o n ( 1 9 8 1 ) . T h e r e f o r e , i t w a s n e c e s s a r y t o d e v e l o p a f u n c t i o n a l m u l t i - l a y e r m o d e l t o c l o s e l y s i m u l a t e t h e p h y s i c a l a n d p h y s i o l o g i c a l p r o c e s s e s , e s p e c i a l l y h e a t , m o m e n t u m a n d m o i s t u r e t r a n s f e r , d u r i n g f i e l d d r y i n g o f a l f a l f a w i t h i n t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m . C H A P T E R 3 S Y S T E M D E F I N I T I O N A s y s t e m i s a n i n t e g r a t i o n o f i n t e r a c t i n g p a r t s w h i c h c a n b e d i s t i n g u i s h e d f r o m i t s s u r r o u n d i n g e n v i r o n m e n t b y e i t h e r p h y s i c a l o r c o n c e p t u a l b o u n d a r i e s , a n d w h i c h f u n c t i o n s a s a w h o l e t o r e a c h c e r t a i n g o a l s . A s y s t e m i n t e r a c t s w i t h i t s e n v i r o n m e n t t h r o u g h c e r t a i n e x c h a n g e s , e . g . e n e r g y a n d m a t t e r f o r b i o l o g i c a l s y s t e m s . T h e b o u n d a r i e s o f a s y s t e m a r e a r b i t r a r y a n d d e p e n d e n t o n t h e s u b j e c t t o b e s t u d i e d . T o a v o i d c o m p l i c a t i n g a p r o b l e m , b o u n d a r i e s o f a s y s t e m a r e o f t e n c h o s e n s o t h a t t h e i n t e r a c t i o n s b e t w e e n t h e s y s t e m a n d t h e s u r r o u n d i n g e n v i r o n m e n t c a n b e n e g l e c t e d . T h i s t y p e o f s y s t e m i s c a l l e d a c l o s e d s y s t e m . T h e p a r t s o f a s y s t e m i n t e r a c t w i t h e a c h o t h e r , a n d t h u s , l i n k t o g e t h e r t o f o r m t h e s y s t e m . S o m e p a r t s o f a s y s t e m c a n b e d e f i n e d a s s u b s y s t e m s ( f i r s t - l e v e l , s e c o n d - l e v e l , e t c . ) . T h e l o w e s t - l e v e l s u b s y s t e m s c o m p r i s e i n d i v i d u a l c o m p o n e n t s . T h e s c o p e o f s u b s y s t e m s a n d c o m p o n e n t s i s a l s o a r b i t r a r y a n d d e p e n d e n t u p o n t h e d e s i r e d d e t a i l o f a n a n a l y s i s . 3 . 1 S y s t e m B o u n d a r i e s T h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m s y s t e m s t u d i e d c o n s i s t e d o f t h r e e p a r t s : t h e s o i l , a n a l f a l f a s w a t h a n d t h e a t m o s p h e r e . T h e s o i l a n d t h e a t m o s p h e r e a r e i n f i n i t e l y l a r g e , b u t t h e s w a t h i s n o t . T h e 1 3 l 4 d e t e r m i n a t i o n o f t h e s y s t e m b o u n d a r i e s s h o u l d t h e r e f o r e f o c u s o n t h e s w a t h . I t w a s p r e f e r r e d t o c h o s e t h e s p a t i a l b o u n d a r i e s o f t h e s y s t e m s u c h t h a t t h e s y s t e m h a d l i t t l e i n f l u e n c e o n i t s s u r r o u n d i n g e n v i r o n m e n t , i . e . t o b e a p p r o x i m a t e l y a c l o s e d s y s t e m . T h e s w a t h h a s a l o n g , w i d e a n d t h i n c o n f i g u r a t i o n . I t s l e n g t h m i g h t b e h u n d r e d s o f m e t e r s d e p e n d i n g o n t h e f i e l d s i z e , i t s w i d t h i s a b o u t 2 m d e t e r m i n e d b y t h e s e t t i n g o f t h e m o w e r , a n d i t s t h i c k n e s s i s a b o u t 0 . 1 - 0 . 2 m s u b j e c t t o t h e a l f a l f a y i e l d a n d t h e m a c h i n e w i d t h s e t t i n g . S i n c e t h e l e n g t h a n d w i d t h o f t h e s w a t h a r e m u c h l a r g e r t h a n t h e t h i c k n e s s , t h e s w a t h m i g h t b e a b s t r a c t e d a s a t h i n b u t i n f i n i t e l y l a r g e s u b s t a n c e l a y i n g o n t h e g r o u n d . I n t h e c e n t r a l p o r t i o n o f t h e s w a t h , t h e t e m p e r a t u r e a n d m o i s t u r e s t a t u s c o u l d b e c o n s i d e r e d t o b e u n i f o r m i n t h e h o r i z o n t a l d i r e c t i o n , s o h o r i z o n t a l t r a n s f e r o f h e a t a n d m o i s t u r e w a s n e g l i g i b l e c o m p a r e d w i t h t h a t i n t h e v e r t i c a l d i r e c t i o n . A t t h e s w a t h e d g e s , l a t e r a l h e a t a n d m o i s t u r e t r a n s f e r ( a d v e c t i o n ) m i g h t b e s u b s t a n t i a l . T h e a b s t r a c t i o n o f a n i n f i n i t e l y l a r g e ' s w a t h ' l e a d s t o a v o i d i n g e d g e e f f e c t s , a n d t h u s , s i m p l i f i e s t h e s w a t h t o b e h o r i z o n t a l l y h o m o g e n e o u s . B a s e d u p o n t h i s a s s u m p t i o n , t h e h o r i z o n t a l b o u n d a r i e s o f t h e s y s t e m s t u d i e d w e r e s e t t o b e t h e b o r d e r s o f a s e c t i o n ( 1 m b y 1 m ) t a k e n f r o m t h e c e n t e r o f t h e i n f i n i t e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m . T h e a s s u m p t i o n o f h o r i z o n t a l h o m o g e n e i t y c o u l d a l s o b e a p p l i e d t o t h e s o i l u n d e r n e a t h t h e s w a t h , a n d t h e a t m o s p h e r e a b o v e t h e s w a t h . T h e h o r i z o n t a l d i m e n s i o n ( 1 m ) o f t h e s y s t e m w a s s o s m a l l c o m p a r e d t o t h e i n f i n i t e d i m e n s i o n s o f t h e s o i l a n d t h e a t m o s p h e r e , d i f f e r e n c e s i n t e m p e r a t u r e a n d m o i s t u r e i n t h e h o r i z o n t a l d i r e c t i o n i n t h e s o i l a n d t h e a t m o s p h e r e a r e m e a n i n g l e s s , o r n o r m a l l y n o t 1 5 d e t e c t a b l e i n s u c h a s m a l l s c a l e . P a s t r e s e a r c h ( M o n t e i t h , 1 9 7 3 . M o n t e i t h e t a l . 1 9 7 5 ; C a m p b e l l , 1 9 7 7 ; G o u d r i a a n , 1 9 7 7 ; R o s e n b e r g e t a l . , 1 9 8 3 ) i n d i c a t e d t h a t t r a n s f e r p h e n o m e n a i n t h e v e r t i c a l d i r e c t i o n a r e t h e m a j o r c o n c e r n i n b i o l o g i c a l e n v i r o n m e n t a l s c i e n c e s . R o s e n b e r g e t a l . ( 1 9 8 3 ) s h o w e d t h a t a t a g i v e n d e p t h b e l o w t h e s o i l s u r f a c e t h e r e i s a h e a t w a v e w i t h a d a i l y p e r i o d . T h e a m p l i t u d e o f t h e d a i l y h e a t w a v e i n s o i l d e c r e a s e s a s t h e d e p t h i n c r e a s e s . A t a b o u t 0 . 5 m b e l o w t h e s u r f a c e , t h e a m p l i t u d e i s c l o s e t o z e r o , i . e . t h e s o i l t e m p e r a t u r e a t t h a t d e p t h i s n e a r l y c o n s t a n t . T h i s i m p l i e s t h a t h e a t t r a n s f e r t o a n d f r o m d e e p e r s o i l l a y e r s i s n e g l i g i b l e . T h e l o w e r b o u n d a r y o f t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m w a s , h e n c e , s e t a t a d e p t h o f 0 . 5 m b e l o w t h e s o i l s u r f a c e . T h e u p p e r b o u n d a r y o f t h e s y s t e m m i g h t b e c h o s e n a t t h e h i g h e s t e l e v a t i o n o f t h e a t m o s p h e r e w h e r e e x c h a n g e s b e t w e e n t h e a t m o s p h e r e a n d t h e e x t r a t e r r i t o r i a l s p a c e c o u l d b e i g n o r e d . H o w e v e r , t h i s w o u l d b e i m p r a c t i c a l , b e c a u s e m e a s u r e m e n t s o f w e a t h e r d a t a u p t o t h a t u p p e r b o u n d a r y a r e e x t r e m e l y d i f f i c u l t , e v e n i m p o s s i b l e , a n d t h a t b o u n d a r y i s t o o . f a r a w a y f r o m t h e r e g i o n o f i n t e r e s t i n t h i s s t u d y . R o s e n b e r g e t a l . ( 1 9 8 3 ) p o i n t e d o u t t h a t i n t h e r e g i o n o f t h e f i r s t f e w m e t e r s o v e r t h e g r o u n d , v e r y l a r g e q u a n t i t i e s o f e n e r g y , m o m e n t u m a n d m o i s t u r e a r e e x c h a n g e d . T h e r a t e s o f t h e s e e x c h a n g e s w i t h r e s p e c t t o t i m e a n d e l e v a t i o n a r e m u c h g r e a t e r t h a n t h o s e a t h i g h e r e l e v a t i o n s . T h e r e f o r e , t h e u p p e r b o u n d a r y o f t h e s y s t e m w a s s e t a t a n e l e v a t i o n n e a r t h e g r o u n d w h e r e t h e c h a n g e s i n t e m p e r a t u r e , w i n d s p e e d a n d m o i s t u r e c a n b e e a s i l y m o n i t o r e d . T h i s e l e v a t i o n i s c a l l e d t h e r e f e r e n c e h e i g h t i n t h e l i t e r a t u r e . T h e s t a n d a r d r e f e r e n c e h e i g h t f o r m e t e o r o l o g i c a l o b s e r v a t i o n s i s 2 m f o r w e a t h e r s t a t i o n s . F i e l d s w h e r e a l f a l f a d r y i n g 1 6 t a k e s p l a c e a r e u s u a l l y n o t n e a r a l o c a l w e a t h e r s t a t i o n ; a n d c l i m a t i c m e a s u r e m e n t s s h o u l d b e m a d e a t t h e a c t u a l f i e l d . W h e n t h e r e f e r e n c e h e i g h t i s c l o s e t o t h e t o p o f t h e s w a t h ( o r a p l a n t c a n o p y ) , t h e i n f l u e n c e f r o m a d j a c e n t s w a t h s ( o r c a n o p i e s ) i s s l i g h t , h e n c e , t h e i n f l u e n c e c a n b e i g n o r e d ( G o u d r i a a n , 1 9 7 7 ) . I n t h i s s t u d y , t h e r e f e r e n c e h e i g h t w a s 0 . 6 4 m , a l e v e l a t w h i c h a p o r t a b l e m i c r o w e a t h e r s t a t i o n w a s p l a c e d ( s e e C h a p t e r 5 f o r d e t a i l ) . A t t h e r e f e r e n c e h e i g h t , t r a n s f e r o f h e a t , m o m e n t u m a n d m a s s b e t w e e n t h e s y s t e m a n d i t s s u r r o u n d i n g e n v i r o n m e n t c a n n o t b e i g n o r e d . T h e s t r e n g t h s o f t h e s e t r a n s f e r p r o c e s s e s d e p e n d u p o n t h e s t a t e o f t e m p e r a t u r e , w i n d s p e e d a n d h u m i d i t y a t t h e i n t e r f a c e . T h e r e f o r e , t h e s e q u a n t i t i e s a t t h e r e f e r e n c e h e i g h t w e r e u s e d a s f o r c e f u n c t i o n s f o r t h e s y s t e m s t u d i e d . I n a d d i t i o n , s o l a r r a d i a t i o n a n d s k y d i f f u s e r a d i a t i o n t r a v e l t h r o u g h t h e a t m o s p h e r e a n d t h e n a r e i n t e r c e p t e d , s c a t t e r e d a n d a b s o r b e d b y a l f a l f a l e a v e s . T h e s e r a d i a t i o n e n e r g y q u a n t i t i e s a r e m a j o r f o r c e f u n c t i o n s i n f i e l d d r y i n g . T h e a t m o s p h e r e i s n o r m a l l y a s s u m e d t o b e t r a n s p a r e n t , a n d h e n c e , t h e i n t e r c e p t i o n , r e f l e c t i o n , t r a n s m i s s i o n a n d a b s o r p t i o n t a k e p l a c e f r o m t h e t o p o f t h e s w a t h d o w n t o t h e g r o u n d , i . e . i n t h e r e g i o n w h e r e l e a v e s e x i s t . T h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m o f f i e l d d r y i n g a l f a l f a w a s t h e r e f o r e d e f i n e d a s a s e m i - c l o s e d s y s t e m w i t h o n e - d i m e n s i o n a l t r a n s f e r i n t h e v e r t i c a l d i r e c t i o n . T h e s y s t e m s t u d i e d w i l l b e r e f e r r e d t o a s t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m f r o m n o w o n . F i g u r e 1 s h o w s t h e s y s t e m b o u n d a r i e s o f t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m . m S F i g u r e 1 . S y s t e m b o u n d a r i e s f o r t h e S o i l - S w a t h - A t m o s p h e r e C o n t i n u u m . l 7 H e a t R a d i a t i o n I | A i r . M o i s t u r e I E l e r 1 ' ' S w a t h o . S l p r a c e ? L J ‘ J ’ fi - r : W . A l f a l f a u l L ' 7 ' ¢ — W " S o i l 3 _ I . . . . 1 — 1 1 V g | _ _ _ _ _ 9 3 . . . . I 6 4 1 » ( 1 0 l a y e r s ) I v u . . l % . 1 4 | ' 1 i I ' | o — _ . _ _ _ — _ _ _ — — _ — _ _ _ J L . 1 I V 1 m j 1 8 3 . 2 S u b s y s t e m s a n d C o m p o n e n t s S i n c e t h e t r a n s f e r r a t e s o f h e a t , m o m e n t u m a n d m o i s t u r e w i t h r e s p e c t t o e l e v a t i o n w e r e g r e a t i n t h e s y s t e m , i t b e c a m e m a n d a t o r y t o f u r t h e r d e f i n e s u b s y s t e m s s o t h a t t h e m o d e l c o u l d r e p r e s e n t t h e r e a l . s y s t e m m o r e a c c u r a t e l y . E a c h p a r t ( t h e s o i l , t h e s w a t h o r t h e a t m o s p h e r e ) o f t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m w a s c o n s i d e r e d a s a s u b s y s t e m , w h i c h w a s c o n c e p t u a l l y f o r m e d b y d i v i d i n g i t i n t o s e v e r a l t h i n l a y e r s a l o n g t h e v e r t i c a l d i r e c t i o n . T h e s o i l s u b s y s t e m o f 0 . 5 m i n d e p t h w a s d i v i d e d i n t o 1 0 l a y e r s o f 0 . 0 5 m t h i c k n e s s . I n e a c h l a y e r , s o i l t e m p e r a t u r e w a s a s s u m e d t o b e u n i f o r m ; a n d h e a t t r a n s f e r o c c u r r e d b e t w e e n a d j a c e n t s o i l l a y e r s . G e n e r a l l y s p e a k i n g , t h e t h i n n e r t h e l a y e r , t h e m o r e a c c u r a t e t h e s i m u l a t e d p r o f i l e o f s o i l t e m p e r a t u r e , b u t m o r e c o m p u t e r t i m e i s r e q u i r e d f o r t h e s i m u l a t i o n . I n a d d i t i o n , t e m p e r a t u r e d a t a m e a s u r e d u n d e r f i e l d c o n d i t i o n s a r e n o t s o p r e c i s e a s t h a t i n l a b o r a t o r i e s . A c c o r d i n g t o G o u d r i a a n ' s e x p e r i e n c e ( 1 9 7 7 ) , c h o o s i n g 1 0 s o i l l a y e r s w a s r e a s o n a b l e f o r n u m e r i c a l s t a b i l i t y . T h e a t m o s p h e r e s u b s y s t e m s c o m p r i s e d 4 l a y e r s o f e v e n t h i c k n e s s . I n e a c h l a y e r , t e m p e r a t u r e , w i n d s p e e d a n d h u m i d i t y w e r e a s s u m e d t o b e u n i f o r m . T h i s c h o i c e o f t h e n u m b e r o f l a y e r s i n t h e a t m o s p h e r e w a s d u e t o t h e e m p l o y m e n t o f a p r o c e d u r e b y G o u d r i a a n ( 1 9 7 7 ) t o d e t e r m i n e t h e w i n d s p e e d p r o f i l e a b o v e t h e s w a t h w h e n t h e a t m o s p h e r e w a s u n s t a b l e ( s e e S e c t i o n 4 . 3 f o r d e t a i l ) . T h e s w a t h s u b s y s t e m c o n s i s t e d o f 3 l a y e r s o f u n i f o r m t h i c k n e s s . F i e l d o b s e r v a t i o n s s h o w e d t h a t t h e r e w e r e s i g n i f i c a n t d i f f e r e n c e s i n t e m p e r a t u r e , a i r h u m i d i t y a n d a l f a l f a m o i s t u r e c o n t e n t f r o m t h e t o p t o t h e b o t t o m o f s w a t h s . H e n c e , d i s t i n c t i o n o f s w a t h l a y e r s w a s p r e f e r r e d . 1 9 S i n c e t h e s w a t h t h i c k n e s s w a s r e l a t i v e l y s m a l l ( 0 . 1 - 0 . 2 m ) , d i v i d i n g i t i n t o t o o m a n y l a y e r s w a s u n n e c e s s a r y . A s w a t h i s a p o r o u s s t r u c t u r e w h i c h c o n t a i n s a l f a l f a p l a n t s a n d s u r r o u n d i n g a i r s p a c e s . H e a t , m o m e m t u m a n d m o i s t u r e t r a n s f e r b e t w e e n t h e a l f a l f a a n d t h e a i r s p a c e s . M o i s t u r e l o s s f r o m a l f a l f a a c t u a l l y t a k e p l a c e t h r o u g h t h e a i r s p a c e s . H e n c e , e a c h s w a t h l a y e r w a s f u r t h e r c l a s s i f i e d a s a s e c o n d - l e v e l s u b s y s t e m w h i c h w a s m a d e u p o f a l a y e r o f a l f a l f a p l a n t s , a n d a n a i r s p a c e s u r r o u n d i n g t h e p l a n t s . T h e a l f a l f a p l a n t s m a y a l s o b e d e f i n e d a s a l o w e r - l e v e l s u b s y s t e m c o n s i s t i n g o f l e a v e s a n d s t e m s . H o w e v e r , a s i n d i c a t e d i n t h e l i t e r a t u r e , a l f a l f a l e a v e s a r e t h e m o s t i m p o r t a n t i n t h e d r y i n g p r o c e s s ; p l a n t m o i s t u r e i s r e m o v e d m a i n l y t h r o u g h l e a v e s r a t h e r t h a n s t e m s . T h u s , i t w a s a s s u m e d t h a t l e a v e s r e p r e s e n t e d t h e a l f a l f a p l a n t i n f i e l d d r y i n g . T h i s s i m p l i f i c a t i o n w a s a l s o n e c e s s a r y a s t h e s t u d y o f w a t e r t r a n s p o r t f r o m s t e m s t o l e a v e s w a s b e y o n d t h e s c o p e o f t h i s s t u d y . 3 . 3 S y s t e m L i n k a g e a n d S t a t e V a r i a b l e s T h e t r a n s f e r o f e n e r g y a n d m o i s t u r e w e r e t h e m a j o r l i n k a g e w i t h i n t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m . T h e r e w e r e s e n s i b l e h e a t t r a n s f e r a n d m o i s t u r e t r a n s f e r b e t w e e n a d j a c e n t l a y e r s i n t h e a t m o s p h e r e , b e t w e e n a d j a c e n t l a y e r s i n t h e s o i l , b e t w e e n t h e l e a v e s a n d t h e s u r r o u n d i n g a i r s p a c e i n e a c h l a y e r o f t h e s w a t h , b e t w e e n t h e t o p l a y e r o f t h e s w a t h a n d t h e a t m o s p h e r e , a n d b e t w e e n t h e b o t t o m l a y e r o f t h e s w a t h a n d t h e s o i l . T h e r e a l s o e x i s t s m o m e n t u m t r a n s f e r b e t w e e n a d j a c e n t l a y e r s o f t h e a i r w i t h i n a n d a b o v e t h e s w a t h . T h e w i n d p r o f i l e s w i t h i n a n d a b o v e t h e s w a t h w e r e d e t e r m i n e d b y t h e c h a r a c t e r i s t i c s o f m o m e n t u m t r a n s f e r . 2 0 T h e a i r i s t h e m a j o r m e d i u m o f t h e s e t r a n s f e r p r o c e s s e s . L e a v e s a n d s t e m s a r e s u r r o u n d e d b y t h e a i r . H e a t a n d m o i s t u r e t r a n s f e r b e t w e e n p l a n t s i n a d j a c e n t s w a t h l a y e r s , b e t w e e n p l a n t s a n d t h e s o i l w a s a s s u m e d t o b e c a r r i e d t h r o u g h t h e a i r s p a c e s o n l y . A n y d i r e c t c o n t a c t b e t w e e n t h e l e a v e s i n a d j a c e n t l a y e r s o f t h e s w a t h , o r b e t w e e n t h e l e a v e s a n d t h e s o i l s u r f a c e w a s i g n o r e d i n t h i s s t u d y . T r a n s p o r t o f s o l a r r a d i a t i o n a n d d i f f u s e r a d i a t i o n i s a l s o t h r o u g h t h e a i r . T h i s s t u d y f o c u s e d o n t h e t r a n s f e r o f h e a t a n d m o i s t u r e i n t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m , s o t e m p e r a t u r e a n d m o i s t u r e l e v e l ( w a t e r v a p o r d e n s i t y i n t h e a i r a n d i n t h e s o i l , a n d m o i s t u r e c o n t e n t o f a l f a l f a ) a t v a r i o u s l o c a t i o n s w i t h i n t h e s y s t e m w e r e d e f i n e d a s s t a t e v a r i a b l e s o f t h e s y s t e m . S i n c e t h e s e t r a n s f e r p r o c e s s e s t h r o u g h t h e a i r a r e s u b j e c t t o t h e a i r m o v e m e n t , w i n d s p e e d w a s a l s o d e f i n e d a s a s t a t e v a r i a b l e i n t h e a i r s p a c e s a n d i n t h e a t m o s p h e r e . I n s u m m a r y , t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m s y s t e m s t u d i e d w a s d e f i n e d a s a s e m i - c l o s e d s y s t e m w i t h o n e - d i m e n s i o n a l t r a n s f e r i n t h e v e r t i c a l d i r e c t i o n . T h e s y s t e m h a d n e g l i g i b l e i n t e r a c t i o n s w i t h t h e s u r r o u n d i n g e n v i r o n m e n t a t t h e l a t e r a l a n d l o w e r b o u n d a r i e s , b u t s i g n i f i c a n t e x c h a n g e s o f e n e r g y , m o m e n t u m a n d m o i s t u r e a t t h e u p p e r b o u n d a r y . S o l a r r a d i a t i o n , s k y d i f f u s e r a d i a t i o n i n t e r c e p t e d a t t h e s w a t h t o p , a n d t e m p e r a t u r e , w i n d s p e e d a n d h u m i d i t y a t t h e u p p e r b o u n d a r y o f t h e s y s t e m w e r e d r i v i n g f o r c e f u n c t i o n s o f t h e s y s t e m . T r a n s f e r o f h e a t , m o m e n t u m a n d m o i s t u r e t h r o u g h t h e a i r w e r e t h e l i n k a g e t h r o u g h o u t t h e s y s t e m . T e m p e r a t u r e , w i n d s p e e d , w a t e r v a p o r d e n s i t y i n t h e a i r a n d i n t h e s o i l , a n d a l f a l f a m o i s t u r e c o n t e n t s w e r e s t a t e v a r i a b l e s o f t h e s y s t e m . C H A P T E R 4 M O D E L D E V E L O P M E N T T h i s c h a p t e r p r e s e n t s t h e c o n s t r u c t i o n o f t h e m a t h e m a t i c a l m o d e l w h i c h s i m u l a t e d t h e e s s e n t i a l f e a t u r e s a n d b e h a v i o r o f t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m . N u m e r i c a l m e t h o d s w e r e e m p l o y e d i n t h e m o d e l f o r c o m p u t e r a n a l y s i s . T h e m o d e l ( A L F D R Y ) w a s m a d e u p o f s e v e r a l s u b m o d e l s . T h e s e s u b m o d e l s a n d c o r r e s p o n d i n g s u b p r o g r a m n a m e s ( i n c a p i t a l l e t t e r s ) a r e l i s t e d b e l o w : ( 1 ) t h e s h o r t - w a v e r a d i a t i o n p r o f i l e i n s i d e t h e s w a t h ( S W R A D ) , ( 2 ) t h e l o n g - w a v e r a d i a t i o n p r o f i l e i n s i d e t h e s w a t h ( L W R A D ) , ( 3 ) t h e w i n d p r o f i l e a n d a e r o d y n a m i c r e s i s t a n c e s t o h e a t a n d m o i s t u r e t r a n s f e r a b o v e t h e s w a t h ( A B O V E ) , ( 4 ) t h e w i n d p r o f i l e a n d a e r o d y n a m i c r e s i s t a n c e t o h e a t a n d m o i s t u r e t r a n s f e r i n s i d e t h e s w a t h ( I N S I D E ) , ( 5 ) e n e r g y a n d m o i s t u r e b a l a n c e o n l e a f s u r f a c e s ( D B L E A F ) , ( 6 ) r e w e t t i n g a n d r e d r y i n g o f a l f a l f a ( D B M C L ) , ( 7 ) e n e r g y a n d m o i s t u r e b a l a n c e i n a i r s p a c e s o f t h e s w a t h ( M A T R I X ) , ( 8 ) e n e r g y b a l a n c e o n t h e s o i l s u r f a c e ( G R O U N D ) , a n d t h e t e m p e r a t u r e p r o f i l e i n t h e s o i l ( D B S O I L ) . T h e s e s u b m o d e l s a r e d i s c u s s e d i n c o r r e s p o n d i n g s e c t i o n s , a n d a t t h e e n d o f t h i s c h a p t e r , t h e o v e r a l l m o d e l s t r u c t u r e a n d c h o i c e s o f i n t e g r a t i o n t i m e s t e p s a r e p r e s e n t e d . M a n y a s s u m p t i o n s w e r e i m p l e m e n t e d t o s i m p l i f y t h e m o d e l b a s e d u p o n 2 1 2 2 t h e d i s c u s s i o n i n t h e p r e c e d i n g c h a p t e r . T h e a s s u m p t i o n s w e r e : ( 1 ) t h e a l f a l f a p l a n t w a s r e p r e s e n t e d b y t h e l e a v e s ; ( 2 ) p r o p e r t i e s o f b o t h s i d e s o f t h e l e a v e s w e r e i d e n t i c a l ; ( 3 ) t h e l e a f a r e a d e n s i t y w a s i n d e p e n d e n t o f t h e e l e v a t i o n i n s i d e t h e s w a t h ; ( 4 ) t h e l e a f o r i e n t a t i o n d i s t r i b u t i o n w a s u n i f o r m ; ( 5 ) r a d i a t i o n e n e r g y w a s i n t e r c e p t e d , r e f l e c t e d , t r a n s m i t t e d a n d a b s o r b e d b y t h e l e a v e s o n l y ; ( 6 ) p h o t o s y n t h e s i s i n l e a v e s w a s n e g l e c t e d d u r i n g f i e l d d r y i n g ; ( 7 ) t h e a i r s p a c e s i n s i d e t h e s w a t h w e r e t h e m e d i a f o r t r a n s f e r o f h e a t a n d m o i s t u r e , e x c e p t r a d i a t i o n e n e r g y ; a n d ( 8 ) p h y s i c a l p r o p e r t i e s o f t h e l e a v e s , t h e a i r s p a c e s a n d t h e s o i l w e r e u n i f o r m w i t h i n e a c h l a y e r o f t h e s w a t h a n d t h e s o i l , r e s p e c t i v e l y . 4 . 1 S h o r t - N a v e R a d i a t i o n P r o f i l e i n s i d e t h e S w a t h S h o r t - w a v e r a d i a t i o n o r i g i n a t i n g f r o m t h e s u n i s t h e m o s t i m p o r t a n t e n e r g y s o u r c e f o r d a y t i m e d r y i n g . S h o r t - w a v e r a d i a t i o n ( w a v e l e n g t h 2 8 0 - 4 0 0 0 n m ) c a n b e r o u g h l y d i v i d e d i n t o t h r e e w a v e b a n d s ( w a v e l e n g t h 4 0 0 - 7 6 0 n m ) , a n d n e a r - i n f r a r e d ( w a v e l e n g t h 7 6 0 - 4 0 0 0 n m ) ( R o s e n b e r g e t a l . , 1 9 8 3 ) . T h e u l t r a v i o l e t p o r t i o n c o n t r i b u t e s o n l y a b o u t 1 % o f t h e t o t a l s o l a r r a d i a t i o n e n e r g y ; a n d t h e r e s t i s n e a r l y e q u a l l y c o n t r i b u t e d b y v i s i b l e a n d n e a r - i n f r a r e d r a d i a t i o n . S i n c e t h i s s t u d y n e g l e c t e d p h o t o s y n t h e s i s , n o s p e c i a l p r o v i s i o n w a s m a d e t o s e p a r a t e v i s i b l e a n d n e a r - i n f r a r e d r a d i a t i o n . S h o r t - w a v e r a d i a t i o n e m i t t e d d i r e c t l y f r o m t h e s u n i s c a l l e d d i r e c t s o l a r r a d i a t i o n , o r b e a m r a d i a t i o n , w h i c h c o n c e n t r a t e s i n a f i x e d 2 3 d i r e c t i o n c h a r a c t e r i z e d b y t h e s o l a r e l e v a t i o n a n g l e . W h e n d i r e c t s o l a r r a d i a t i o n t r a v e l s t h r o u g h t h e a t m o s p h e r e , i t i s a t t e n u a t e d b y c l o u d s a n d d u s t d u e t o s c a t t e r i n g a n d a b s o r p t i o n . T h e s c a t t e r e d s o l a r r a d i a t i o n i s c a l l e d d i f f u s e s k y r a d i a t i o n , w h i c h g o e s i n a l l d i r e c t i o n s . I n c o m i n g s o l a r r a d i a t i o n f o r a s w a t h i s a c o m b i n a t i o n o f b o t h d i r e c t s o l a r r a d i a t i o n a n d d i f f u s e s k y r a d i a t i o n . S h o r t - w a v e r a d i a t i o n i n s i d e a s w a t h ' d e p e n d s u p o n i n c o m i n g s o l a r r a d i a t i o n a t t h e t 0 p o f t h e s w a t h , a n d o p t i c a l p r o p e r t i e s o f t h e s w a t h w h i c h a r e a f f e c t e d b y t h e l e a f a r e a , l e a f o r i e n t a t i o n , l e a f d e n s i t y d i s t r i b u t i o n , a n d l e a f o p t i c a l p r o p e r t i e s . 4 . 1 . 1 S o l a r I r r a d i a n c e a t t h e S w a t h T o p S o l a r i r r a d i a n c e ( r a d i a t i o n e n e r g y p e r u n i t a r e a ) o n a h o r i z o n t a l p l a n e i s a n i n t e g r a t i o n o f d i r e c t s o l a r r a d i a t i o n a n d d i f f u s e s k y r a d i a t i o n a t t h e e l e v a t i o n o f t h e p l a n e . T h e s o l a r i r r a d i a n c e i s d e f i n e d a n d o f t e n m e a s u r e d p e r p e n d i c u l a r l y t o t h e h o r i z o n t a l p l a n e . B o t h t h e s t r e n g t h a n d t h e c o m p o s i t i o n o f t h e s o l a r i r r a d i a n c e a t t h e t o p o f a s w a t h a r e a f u n c t i o n o f t h e s o l a r e l e v a t i o n a n g l e a n d t h e s k y c o v e r f r a c t i o n . W h e n t h e s k y i s o v e r c a s t , o n l y d i f f u s e s k y r a d i a t i o n r e a c h e s t h e s w a t h , s o t h e f r a c t i o n o f d i f f u s e s k y r a d i a t i o n i n t h e s o l a r i r r a d i a n c e i s u n i t y . W h e n t h e s k y i s c l e a r , t h e f r a c t i o n o f d i f f u s e s k y r a d i a t i o n i n t h e s o l a r i r r a d i a n c e i s a f u n c t i o n o f t h e s o l a r e l e v a t i o n a n g l e 8 ( T a b l e 1 ) . N e a r s u n r i s e o r s u n s e t , t h e s o l a r i r r a d i a n c e i s s o l e l y d i f f u s e s k y r a d i a t i o n , w h i l e a t n o o n t i m e t h e p r i m a r y r a d i a t i o n i s d i r e c t s o l a r r a d i a t i o n . G o u d r i a a n ( 1 9 7 7 ) m e n t i o n e d t h a t t h e s o l a r i r r a d i a n c e o f a n o v e r c a s t s k y i s o n l y 1 7 % o f a c l e a r s k y . 2 4 T a b l e 1 T h e f r a c t i o n o f t h e d i f f u s e s k y r a d i a t i o n i n t h e s o l a r i r r a d i a n c e a s a f u n c t i o n o f t h e s o l a r e l e v a t i o n a n g l e + S u n E l e v a t i o n A n g l e D i f f u s e / T o t a l 8 f , ( d e g r e e s ) ( r a d i a n s ) ( d e c i m a l ) 0 ' 0 . 0 1 . 0 0 5 0 . 0 8 7 3 1 . 0 0 1 5 0 . 2 6 1 8 0 . 3 2 2 5 0 . 4 3 6 3 0 . 2 2 3 5 0 . 6 1 0 9 0 . 1 8 4 5 0 . 7 8 5 4 0 . 1 6 9 0 1 . 5 7 0 8 0 . 1 3 I F r o m G o u d r i a a n — ( 1 9 7 7 ) ‘ — F o r a p a r t l y c l o u d y s k y , t h e s o l a r i r r a d i a n c e w a s a s s u m e d t o b e t h e s u m o f a c o m p o n e n t u n d e r a c l e a r s k y , a n d a n o t h e r c o m p o n e n t u n d e r a n o v e r c a s t s k y . S 1 = 5 1 c + S i o = ( 1 ‘ 6 v ) $ c s + 0 . 1 7 c V S c s : ( 1 - 0 . 8 3 C V ) S C S [ 3 ] w h e r e 8 1 = s o l a r i r r a d i a n c e a t t h e t o p o f a s w a t h , W / m z , 5 1 c = c o m p o n e n t o f 3 1 u n d e r a c l e a r s k y , W / m z , $ 1 0 : c o m p o n e n t o f 8 1 u n d e r a n o v e r c a s t s k y , W / m z , S c , = c l e a r s k y i r r a d i a n c e , W / m z , a n d C v = s k y c o v e r f r a c t i o n , d e c i m a l . T h e s o l a r i r r a d i a n c e $ 1 a t t h e t o p o f a s w a t h w a s a l s o e x p r e s s e d a s : 5 1 = S d i r + S d i f [ “ 1 w h e r e S d i p = d i r e c t s o l a r i r r a d i a n c e , W / m z , a n d S d i f = d i f f u s e s k y i r r a d i a n c e , W / m z . S d i r = ( 1 ‘ f r ) 5 1 c [ 5 ] S d i f = f r s l c + 5 1 0 [ 6 ] 2 5 w h e r e f r : f r a c t i o n o f d i f f u s e r a d i a t i o n i n 8 1 , d e c i m a l . D a t a o f C V w a s o b t a i n e d f r o m t h e l o c a l w e a t h e r s e r v i c e . W i t h t h e s e f e u r e q u a t i o n s , t h e d i r e c t s o l a r i r r a d i a n c e S d i r a n d t h e d i f f u s e s k y i r r a d i a n c e S d i f c o u l d b e d e t e r m i n e d f r o m m e a s u r e d 5 1 . I n s u b r o u t i n e S O L A R , v a r i a b l e s S L O , S L D I R O , S L S K Y O , F R a n d C V s t o o d f o r $ 1 , S d i r : S d i f , f r a n d C V , r e s p e c t i v e l y . S d i r a n d S d i f w e r e c a l c u l a t e d a s : ( 1 — r , ) ( 1 - c , ) s l / ( 1 - 0 . 8 3 c v ) [ 7 ] S d i r S d i f S l - S d i r [ 8 1 4 . 1 . 1 . 1 E s t i m a t i o n o f t h e S o l a r I r r a d i a n c e W h e n m e a s u r e d 3 1 w a s n o t a v a i l a b l e , i t w a s e s t i m a t e d f r o m t h e b e a m i r r a d i a n c e . U n d e r a c l e a r s k y ( C V = O ) , S b O : ( 1 ‘ f r ) S c s = ( 1 - f r ) 5 1 / ( 1 - O . 8 3 C v ) w h e r e S b O = b e a m i r r a d i a n c e , W / m z . I n s u b r o u t i n e S O L A R , 8 1 w a s c a l c u l a t e d a s : 5 1 = ( 1 - 0 . 8 3 c v ) s b o / ( 1 - r r ) [ 9 ] F u n c t i o n s u b p r o g r a m B E A M C L c a l c u l a t e d t h e b e a m i r r a d i a n c e S b O f o r a s t a n d a r d c l e a r s k y w i t h a p r o c e d u r e b y D u f f i e a n d B e c k m a n ( 1 9 8 0 ) . S b O = t S c s i n 8 ( 1 - 0 . 0 3 3 c o s ( 2 n n / 3 6 5 ) ) [ 1 0 ] w h e r e r = a t m o s p h e r e t r a n s m i t t a n c e , d e c i m a l , S 0 = s o l a r c o n s t a n t , 1 3 7 5 W / m z , B = s o l a r e l e v a t i o n a n g l e , a n d n = d a y o f t h e y e a r , 1 t o 3 6 5 o r 3 6 6 . S u b r o u t i n e Y R D A T E w a s u s e d t o c o n v e r t c a l e n d a r d a t e s t o d a y s o f t h e y e a r . T h e a t m o s p h e r e t r a n s m i t t a n c e T w a s e s t i m a t e d a s : r = a 0 + a 1 e ' k / S i n e [ 1 1 ] 2 6 w h e r e a 0 = r o ( 0 . 4 2 3 7 - 0 . 0 0 8 2 1 ( 6 . 0 - A 1 t ) z ) a 1 = r 1 ( 0 . 5 0 5 5 + 0 . 0 0 5 9 5 ( 6 . 5 - A 1 t ) 2 ) k = r k ( 0 . 2 7 1 1 + 0 . 0 1 8 5 8 ( 2 . 5 - A 1 t ) 2 ) W h e r e A l t i s t h e a l t i t u d e o f t h e l o c a t i o n i n k m . C o r r e c t i o n f a c t o r s r 0 , r 1 a n d r k e q u a l e d 0 . 9 7 , 0 . 9 9 a n d 1 . 0 2 , r e s p e c t i v e l y , f o r m i d - l a t i t u d e r e g i o n s i n s u m m e r s e a s o n s . 4 . 1 . 1 . 2 S o l a r E l e v a t i o n A n g l e T h e s o l a r e l e v a t i o n a n g l e 8 w a s t h e a n g l e b e t w e e n t h e r a y o f b e a m r a d i a t i o n a n d i t s p r o j e c t i o n o n a h o r i z o n t a l p l a n e a t a g i v e n t i m e b e t w e e n s u n r i s e a n d s u n s e t . F r o m s u n s e t t o n e x t s u n r i s e , B w a s z e r o . I n t h e m o d e l , B w a s c a l c u l a t e d i n s u b p r o g r a m B E T A a c c o r d i n g t o : s i n B = s i n ¢ s i n 6 + c o s o c o s s c o s w [ 1 2 ] w h e r e o = l o c a l l a t i t u d e , 6 : s o l a r d e c l i n a t i o n a n g l e , a n d w = h o u r a n g l e . T h e s o l a r d e c l i n a t i o n a n g l e 6 w a s t h e a n g u l a r p o s i t i o n o f t h e s u n a t s o l a r n o o n ( 1 2 : 0 0 s o l a r t i m e ) w i t h r e s p e c t t o t h e e q u a t o r p l a n e . T h e a n g l e 6 w a s c a l c u l a t e d i n d e g r e e s a s : 6 = 2 3 . 4 5 c o s ( 2 n ( n - 1 7 2 ) / 3 6 5 ) ) [ 1 3 ] w h e r e n = d a y o f t h e y e a r . T h e h o u r a n g l e m w a s t h e a n g u l a r d i s p l a c e m e n t o f t h e s u n e a s t o r w e s t o f t h e l o c a l m e r i d i a n d u e t o t h e r o t a t i o n o f t h e e a r t h a t a s p e e d o f 1 5 ° p e r h o u r . w = 1 5 ( t s - 1 2 ) [ 1 n ] 2 7 W h e r e t s w a s t h e s o l a r t i m e , i n h o u r s , a n d m w a s i n d e g r e e s . T h e s o l a r t i m e t s w a s d e t e r m i n e d b a s e d o n t h e a p p a r e n t a n g u l a r m o t i o n o f t h e s u n a c r o s s t h e s k y . A t s o l a r n o o n , t h e s u n c r o s s e s t h e m e r i d i a n o f t h e o b s e r v a t i o n l o c a t i o n . t s = t + A t [ 1 5 ] l o c a l t i m e , h r , w h e r e t a n d A t t i m e i n c r e m e n t , h r . T h e t i m e i n c r e m e n t A t w a s d u e t o t h e d i f f e r e n c e b e t w e e n t h e l o c a l l o n g i t u d e a n d t h e m e r i d i a n o f t h e l o c a l s t a n d a r d t i m e . I t w a s c a l c u l a t e d a s : A t = ( 4 ( L 3 t - L 1 0 c ) + E ) / 6 0 [ 1 6 ] w h e r e L s t = m e r i d i a n o f t h e l o c a l s t a n d a r d t i m e , ° W e s t , l o c a l l o n g i t u d e , ° W e s t , L l o c a n d E e q u a t i o n o f t i m e , m i n u t e s . T h e e q u a t i o n o f t i m e E t o o k i n t o a c c o u n t t h e p e r t u r b a t i o n s o f t h e r a t e o f t h e e a r t h ' s r o t a t i o n ( D u f f i e a n d B e c k m a n , 1 9 8 0 ) . I t w a s c o m p u t e d a s : E 9 . 8 7 s i n 2 0 - 7 . 5 3 c o s C — 1 . S s i n C [ 1 7 ] w h e r e C 2 n ( n - 8 1 ) / 3 6 5 [ 1 8 ] w h e r e n = d a y o f t h e y e a r . T h e t i m e a n g l e s a t s u n r i s e a n d s u n s e t w e r e e s t i m a t e d f r o m E q u a t i o n [ 1 2 ] b y s e t t i n g 3 : 0 . c o s ' 1 ( - t a n ¢ t a n 6 ) [ 1 9 ] . m s [ 2 0 1 “ s “ r w h e r e m s = t i m e a n g l e a t s u n s e t , t i m e a n g l e a t s u n r i s e . a n d m p 2 8 T h e n , t h e l o c a l t i m e o f s u n r i s e a n d s u n s e t w a s d e t e r m i n e d f r o m E q u a t i o n s [ 1 4 ] a n d [ 1 5 ] . I n s u b m o d e l S W R A D , v a r i a b l e s H S U N , D , H , H S E T , H R I S E , T S O L A R , L A T , L S T , L O N a n d A L T d e n o t e d B , 6 , w , m s , w r , t 5 , ¢ . L s t , L 1 0 c a n d A l t : r e s p e c t i v e l y . 1 1 . 1 . 2 O p t i c a l P r o p e r t i e s o f t h e S w a t h W h e n s h o r t - w a v e r a d i a t i o n i s i n t e r c e p t e d b y l e a v e s , i t i s r e f l e c t e d , t r a n s m i t t e d , a n d a b s o r b e d b y t h e l e a v e s . T h e a b s o r b e d p o r t i o n o f t h e r a d i a t i o n i s u s e d t o v a p o r i z e m o i s t u r e , t o r e m o v e w a t e r v a p o r , a n d t o r a i s e p l a n t t e m p e r a t u r e . T h e s c a t t e r e d ( r e f l e c t e d a n d t r a n s m i t t e d ) p o r t i o n o f s h o r t - w a v e r a d i a t i o n m a y b e i n t e r c e p t e d b y o t h e r l e a v e s a g a i n , a n d t h e n , r e f l e c t e d , t r a n s m i t t e d a n d a b s o r b e d . T h i s i s a m u l t i - s c a t t e r i n g p h e n o m e n o n . O p t i c a l p r o p e r t i e s o f l e a v e s p l a y a n i m p o r t a n t r o l e i n t h e s c a t t e r i n g p r o c e s s . F u r t h e r m o r e , t h e r e a r e m a n y l e a v e s i n t h e s w a t h , s o t h e o p t i c a l b e h a v i o r o f t h e s w a t h i s n o r m a l l y n o t t h e s a m e a s t h a t o f a s i n g l e l e a f . 4 . 1 . 2 . 1 S c a t t e r i n g C o e f f i c i e n t o f ’ L e a v e s R o s s ( 1 9 8 1 ) h a s s u m m a r i z e d p a s t r e s e a r c h o n t h e o p t i c a l b e h a v i o r o f l e a v e s a n d p o i n t e d o u t t h a t : ( 1 ) P a r t o f t h e r a d i a t i o n i s r e f l e c t e d o n t h e l e a f s u r f a c e d u e t o s u r f a c e r o u g h n e s s a n d t h e r e f r a c t i v e i n d e x o f t h e c u t i c u l a r w a x o f t h e u p p e r e p i d e r m i s ; ( 2 ) I n t e r n a l s c a t t e r i n g i s d u e t o t h e a r r a n g e m e n t o f t i s s u e s , c e l l s a n d 2 9 i n t e r c e l l u l a r a i r s p a c e s ; ( 3 ) L e a f a b s o r p t i o n i s d e t e r m i n e d b y t h e c o m p o s i t i o n , a m o u n t a n d d i s t r i b u t i o n o f p i g m e n t s , a n d b y t h e c o n c e n t r a t i o n a n d d i s t r i b u t i o n o f l e a f m o i s t u r e . T h e a b s o r p t i o n d e c r e a s e s w i t h y e l l o w i n g o f l e a v e s ; a n d ( 4 ) L e a f a b s o r p t a n c e , r e f l e c t a n c e a n d t r a n s m i t t a n c e a r e l a r g e l y d e t e r m i n e d b y t h e m e a n o p t i c a l p a t h l e n g t h o f l e a v e s , w h i c h i s a f f e c t e d b y t h e l e a f t h i c k n e s s . D e s p i t e t h e c o m p l e x s t r u c t u r e o f a s i n g l e l e a f , t h e a p p a r e n t o p t i c a l p r o p e r t i e s o f t h e l e a f , a s a w h o l e , o b e y : 0 1 + 7 1 + T 1 = a 1 + w 1 = 1 . 0 [ 2 1 ] w h e r e c l = l e a f a b s o r p t a n c e , 7 1 = l e a f r e f l e c t a n c e , 1 1 = l e a f t r a n s m i t t a n c e , a n d m l : l e a f s c a t t e r i n g c o e f f i c i e n t . R o s s ( 1 9 8 1 ) h a s s h o w e d t h a t l e a f r e f l e c t a n c e i s w a v e l e n g t h - d e p e n d e n t , b u t i t c a n b e a s s u m e d t o b e a p p r o x i m a t e l y a c o n s t a n t f o r t h e w h o l e s o l a r s p e c t r u m . I d s o e t a l . ( 1 9 6 9 ) f o u n d t h a t a l b e d o ( s h o r t - w a v e r e f l e c t a n c e o f c a n o p i e s ) v a r i e d w i t h r e s p e c t t o t i m e , b u t i t c o u l d b e a v e r a g e d f o r a w h o l e d a y . T h e e f f e c t o f t h e m o i s t u r e c o n t e n t o f l e a v e s o n t h e o p t i c a l p r o p e r t i e s o f p l a n t s h a s b e e n s t u d i e d b y s o m e r e s e a r c h e r s . T h o m a s e t a l . ( 1 9 6 6 ) , W o o l l e y ( 1 9 7 1 ) a n d C a r l s o n e t a l . ( 1 9 7 1 ) r e p o r t e d t h a t t h e l e a f r e f l e c t a n c e o f c o r n , c o t t o n , s o y b e a n a n d s o r g h u m i n c r e a s e d a s l e a f ' s m o i s t u r e c o n t e n t d e c r e a s e d . T h o m a s e t a l . ( 1 9 6 6 ) m e n t i o n e d t h a t w h e n c o t t o n l e a v e s b e c a m e n e a r f u l l y t u r g i d , t h e l e a f r e f l e c t a n c e w a s i n v a r i a n t w i t h r e s p e c t t o t h e m o i s t u r e c o n t e n t . T h o m a s e t a l . ( 1 9 6 6 ) p r o p o s e d l i n e a r f u n c t i o n s 3 0 t o c h a r a c t e r i z e t h e r e l a t i o n s h i p b e t w e e n 7 1 a n d t h e m o i s t u r e c o n t e n t o f c o t t o n l e a v e s . T h e r e w a s a l a c k o f i n f o r m a t i o n , h o w e v e r , a b o u t l e a f r e f l e c t a n c e a n d i t s r e l a t i o n s h i p t o t h e m o i s t u r e c o n t e n t o f a l f a l f a . I n f u n c t i o n s u b p r o g r a m R E F L , t h e f o l l o w i n g e q u a t i o n w a s a s s u m e d : 0 . 4 5 - 0 . 0 5 2 5 M ( 0 . 0 g M g 4 . 0 ) Y 1 = i [ 2 2 ] 0 . 2 4 ( M 3 4 . 0 ) w h e r e M = l e a f m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r . I d s o e t a l . ( 1 9 6 9 ) u s e d a v a l u e o f 0 . 2 4 a s t h e a l b e d o o f a l f a l f a s t a n d s . T h e t r a n s m i t t a n c e o f l e a v e s i s c l o s e l y r e l a t e d t o t h e r e f l e c t a n c e , d u e t o a s i m i l a r m e c h a n i s m o f s c a t t e r i n g . D e W i t ( 1 9 6 5 ) m e n t i o n e d f o r a w i d e v a r i e t y o f l e a f s p e c i e s , t h e t r a n s m i t t a n c e w a s a b o u t t h e s a m e a s t h e r e f l e c t a n c e . M o n t e i t h ( 1 9 7 3 ) , G o u d r i a a n ( 1 9 7 7 ) a n d R o s s ( 1 9 8 1 ) a p p l i e d e q u a l r e f l e c t a n c e a n d t r a n s m i t t a n c e i n t h e i r a n a l y s e s . I n t h i s s t u d y , 1 1 a n d 7 1 w e r e a s s u m e d t o b e t h e s a m e , s o t h e s c a t t e r i n g c o e f f i c i e n t m l w a s e q u a l t o 2 1 1 o r 2 7 1 . 4 . 1 . 2 . 2 S c a t t e r i n g C o e f f i c i e n t o f a S w a t h L a y e r A l a y e r o f t h e s w a t h c o n s i s t s o f m a n y l e a v e s d i s t r i b u t e d r a n d o m l y w i t h v a r i o u s o r i e n t a t i o n a n g l e s . W h e n a l a y e r o f l e a v e s , a s a w h o l e , i s s t u d i e d , i t s o p t i c a l b e h a v i o r s h o u l d b e r e l a t e d t o t h e o p t i c a l p r o p e r t i e s o f t h e l e a v e s , a n d t h e l e a f o r i e n t a t i o n d i s t r i b u t i o n . R o s s ( 1 9 8 1 ) u s e d a G - f u n c t i o n t o c h a r a c t e r i z e t h e l e a f o r i e n t a t i o n d i s t r i b u t i o n . T h e G - f u n c t i o n w a s d e f i n e d a s t h e p r o j e c t i o n o f a f o l i a g e a r e a i n a g i v e n d i r e c t i o n a t a g i v e n h e i g h t i n s i d e a c a n o p y . I f a l l l e a v e s i n a u n i t a r e a a r e h o r i z o n t a l , t h e v a l u e o f t h e G - f u n c t i o n e q u a l s t h e s i n e o f t h e a n g l e b e t w e e n t h e r a y o f t h e d i r e c t i o n a n d t h e 3 1 h o r i z o n . T h i s e x a m p l e s h o w s t h e p h y s i c a l m e a n i n g o f t h e 0 - f u n c t i o n . I n a r e a l s w a t h ( o r c a n o p y ) , l e a v e s a r e n o t o r i e n t e d i n t h e s a m e d i r e c t i o n , b u t r a n d o m l y i n a l l d i r e c t i o n s . T h e o r i e n t a t i o n o f l e a v e s i s a p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n o v e r t h e w h o l e r a n g e o f i n c l i n a t i o n f r o m h o r i z o n t a l t o v e r t i c a l . F o r i n s t a n c e , i f a l l l e a v e s i n a c a n o p y a r e h o r i z o n t a l , t h e n t h e p r o b a b i l i t y o f h o r i z o n t a l l e a v e s i s u n i t y , a n d o f o t h e r a n g l e s - z e r o . I n t h e l i t e r a t u r e , t h e l e a f o r i e n t a t i o n d i s t r i b u t i o n h a s o f t e n b e e n a s s u m e d t o b e u n i f o r m , i . e . t h e p r o b a b i l i t y f o r a n y i n c l i n a t i o n a n g l e o f l e a v e s w a s e q u a l . T h i s c a n b e v i s u a l i z e d a s i f l e a v e s w e r e e v e n l y l o c a t e d o n a h e m i s p h e r e s u r f a c e . H e n c e , u n i f o r m o r i e n t a t i o n d i s t r i b u t i o n i s a l s o r e f e r r e d t o a s s p h e r i c a l d i s t r i b u t i o n . R o s s ( 1 9 8 1 ) d e r i v e d t h e G - f u n c t i o n f o r s p h e r i c a l l y ( u n i f o r m l y ) o r i e n t e d l e a v e s : c = 1 / 2 [ 2 3 ] B y u s e o f t h e h e m i s p h e r e a p p r o a c h , t h e p r o j e c t i o n o f t h e h e m i s p h e r e t o a n y i n c l i n a t i o n i s e q u a l t o t h e a r e a o f a c i r c l e w h i c h p a s s e s t h r o u g h t h e c e n t e r o f t h e s p h e r e . T h e a r e a o f t h e c i r c l e i s e x a c t l y o n e h a l f o f t h e s u r f a c e a r e a o f t h e h e m i s p h e r e . R o s s ( 1 9 8 1 ) p r o p o s e d a n a p p r o a c h t h a t a p l a n t s t a n d c o u l d b e c o n s i d e r e d a s a p l a t e m e d i u m . H e a s s u m e d t h a t t h e r e w a s n o m u t u a l s h a d i n g o f l e a v e s i n a n y d i r e c t i o n w i t h i n a n e l e m e n t a r y t h i c k n e s s o f t h e p l a t e , b u t t h e r e w e r e a s u f f i c i e n t n u m b e r o f l e a v e s i n t h e e l e m e n t s o t h a t t h e o p t i c a l p r o p e r t i e s o f t h e p l a t e w e r e h o m o g e n e o u s . H e s h o w e d t h a t t h e o p t i c a l p r o p e r t i e s o f t h e a b s t r a c t e d p l a t e m e d i u m w e r e e q u a l t o t h o s e o f a s i n g l e l e a f , i f b o t h s i d e s o f l e a v e s h a d T w w 2 = = 1 l T m T , , + a n d y = 2 7 Y o = = Y a 1 1 ! , [ 2 4 ] 3 2 i d e n t i c a l p r o p e r t i e s , a n d i f t h e l e a f o r i e n t a t i o n w a s s p h e r i c a l l y d i s t r i b u t e d . T h i s a p p r o a c h w a s a p p l i e d t o e a c h l a y e r o f t h e s w a t h i n t h i s s t u d y . T h e s c a t t e r i n g c o e f f i c i e n t ( m ) , r e f l e c t a n c e ( y ) , t r a n s m i t t a n c e ( I ) , a n d a b s o r p t a n c e ( a ) o f e a c h l a y e r o f t h e s w a t h w e r e e q u a l t h e c o r r e s p o n d i n g p r o p e r t i e s o f l e a v e s ( m l , Y 1 , T 1 a n d a l ) i n t h e s a m e l a y e r . 4 . 1 . 3 R a d i a t i o n F i e l d d u e t o B e a m R a d i a t i o n W h e n d i r e c t s o l a r r a d i a t i o n a t a g i v e n s o l a r e l e v a t i o n a n g l e i n c i d e n t s o n t h e t o p o f a s w a t h , a p a r t o f t h e r a d i a t i o n i s i n t e r c e p t e d ( s c a t t e r e d a n d a b s o r b e d ) b y l e a v e s . U n i n t e r c e p t e d d i r e c t s o l a r r a d i a t i o n p a s s e s t h r o u g h g a p s b e t w e e n l e a v e s d o w n w a r d s i n t o t h e s w a t h . I t i s s t i l l b e a m r a d i a t i o n ; h o w e v e r , i t s s t r e n g t h i s a t t e n u a t e d c o m p a r e d w i t h t h a t a t t h e t o p o f t h e s w a t h . T h e s c a t t e r e d r a d i a t i o n i s d i f f u s e r a d i a t i o n d u e t o v a r i o u s o r i e n t a t i o n s o f t h e l e a v e s . T h i s p o r t i o n o f t h e r a d i a t i o n g o e s i n a l l d i r e c t i o n s ( u p w a r d s a n d d o w n w a r d s ) . I t i s a g a i n i n t e r c e p t e d , s c a t t e r e d a n d a b s o r b e d b y o t h e r l e a v e s . T h e s e c o m p l i c a t e d p r o c e s s e s o f i n t e r c e p t i o n a n d p a s s i n g t a k e p l a c e t h r o u g h o u t t h e s w a t h . T h e r e f o r e , t h e r a d i a t i o n f i e l d f r o m d i r e c t s o l a r r a d i a t i o n c o n s i s t s o f a p o r t i o n o f a t t e n u a t e d b e a m r a d i a t i o n a n d a n o t h e r p o r t i o n o f c o m p l e m e n t a r y d i f f u s e r a d i a t i o n . 3 3 4 . 1 . 3 . 1 A t t e n u a t e d B e a m R a d i a t i o n i n s i d e t h e S w a t h T h e a t t e n u a t e d b e a m r a d i a t i o n i n t h e s w a t h w a s w r i t t e n a s : S b , i : a s ’ i s b o [ 2 5 ] w h e r e S b , i = a t t e n u a t e d b e a m i r r a d i a n c e i n t h e i t h l a y e r , W / m z , S b O = i n c o m i n g b e a m i r r a d i a n c e a t t h e t o p o f t h e s w a t h , W / m z , a n d a s , 1 = p e n e t r a t i o n f u n c t i o n d o w n t o t h e i t h l a y e r , d e c i m a l . F o r s p h e r i c a l l y o r i e n t e d l e a v e s : 3 3 , 1 : e - L i / Z s i n B [ 2 6 ] w h e r e L 1 = d o w n w a r d c u m u l a t i v e l e a f a r e a i n d e x , I n 2 l e a f / m 2 , a n d 8 = s o l a r e l e v a t i o n a n g l e . I n S W R A D , S L D I R ( I ) , S M U 2 a n d S U M L d e n o t e d S b , 1 , Z s i n B a n d L 1 , r e s p e c t i v e l y . T h e d o w n w a r d c u m u l a t i v e l e a f a r e a i n d e x L 1 w a s : 1 L 1 = I a t . n ( i = 1 , 2 a n d 3 ) n : 1 W h e r e a t “ w a s t h e l e a f a r e a i n d e x ( L A I ) i n t h e n t h l a y e r . W h e n t h e l e a f a r e a d e n s i t y w a s i n d e p e n d e n t o f t h e h e i g h t i n t h e s w a t h . T h u s , 3 L 1 = 3 L 2 = 3 L 3 = L a i / 3 = L 3 / 3 [ 2 7 ] W h e r e L 3 1 w a s t h e l e a f a r e a i n d e x ( L A I ) o f t h e w h o l e s w a t h , i n m 2 l e a f / m 2 . N O t e t h a t 8 L 1 w a s t h e e l e m e n t L A I i n t h e i t h l a y e r , b u t L 1 w a s t h e d o w n w a r d c u m u l a t i v e L A I t o t h e i t h l a y e r . 4 . 1 . 3 . 2 E s t i m a t i o n o f t h e L e a f A r e a I n d e x T h e l e a f a r e a i n d e x ( L A I ) o f a l f a l f a c o u l d b e e s t i m a t e d f r o m a l f a l f a y i e l d s . F u e s s a n d T e s a r ( 1 9 6 8 ) r e p o r t e d f i e l d e x p e r i m e n t a l d a t a o n t h e d r y m a t t e r ( D M ) y i e l d a n d L A I o f a l f a l f a ( T a b l e 2 ) . 3 4 T a b l e 2 A v e r a g e y i e l d o f h a y a n d l e a f a r e a i n d e x ( L A I ) o f a l f a l f a h a r v e s t e d t h r e e t i m e s p e r s e a s o n C u t t i n g A v e r a g e Y i e l d + A v e r a g e L A I + L A I / Y i e l d ( k g D M / m z ) ( m 2 l e a f / m 2 ) ( m z l e a f / k g D M ) 1 0 4 9 2 3 . 6 6 7 . 4 0 2 0 . 3 8 3 2 . 0 3 5 . 3 0 3 0 . 1 8 1 1 . 2 0 6 . 6 2 + F r o m F u e s s a n d T e s a r ( 1 9 6 8 ) T h e r e f b r e , t h e l e a f a r e a i n d e x ( L A I ) o f t h e s w a t h w a s e s t i m a t e d f r o m a k n o w n s w a t h a r e a d e n s i t y ( d r y m a t t e r , D M ) : L 3 1 = 1 . d e w [ 2 8 ] w h e r e L y = p r o p o r t i o n a l c o n s t a n t f r o m t h e l a s t c o l u m n o f T a b l e 2 , a n d d 3 " : s w a t h a r e a d e n s i t y , k g d r y m a t t e r / m 2 . I n t h e m o d e l , v a r i a b l e s L A I a n d D E N S d e n o t e d L a i a n d d 3 " . A s d r y i n g a d v a n c e d , p l a n t l e a v e s l o s t t u r g o r a n d w i l t e d d u e t o s e v e r e l o s s o f m o i s t u r e . T h i s o f t e n c o i n c i d e d w i t h s t o m a t a l c l o s u r e a t a c e r t a i n l e v e l o f t h e l e a f m o i s t u r e c o n t e n t ( B r a d f o r d a n d H s i a o , 1 9 8 2 ) . D u r i n g w i l t i n g , l a m i n a e a n d p e t i o l e s o f a l f a l f a ( d i c o t ) d r o o p e d d o w n w a r d s . T h i s r e s u l t e d i n a d e c r e a s i n g p r o j e c t i o n a r e a o n a h o r i z o n t a l p l a n e , t h u s , a r e d u c t i o n i n t h e l e a f a r e a i n d e x , e v e n t h o u g h t h e t o t a l l e a f a r e a m a y n o t c h a n g e . A d r o o p y s w a t h ( c a n o p y ) l a y e r e n h a n c e d p e n e t r a t i o n o f s u n l i g h t a n d w i n d , s o a l f a l f a i n d e e p l a y e r s c o u l d d r y f a s t e r . H o w e v e r , t h e d r o o p y l a y e r i n t e r c e p t e d l e s s r a d i a t i o n w h i c h l e d t o l e s s e v a p o r e s p i r a t i o n . T h e r e w a s l a c k o f i n f o r m a t i o n f o r m o d e l i n g t h e l e a f a r e a i n d e x a s a f u n c t i o n o f l e a f m o i s t u r e s t a t u s . I n t h i s s t u d y , a n a s s u m p t i o n w a s m a d e t h a t t h e l e a f a r e a i n d e x w a s a n e x p o n e n t i a l l y d e c a y i n g f u n c t i o n w i t h 3 5 r e s p e c t t o t h e l e a f m o i s t u r e c o n t e n t , w h e n t h e l e a f m o i s t u r e c o n t e n t d r o p p e d b e l o w 8 0 % o f i t s i n i t i a l l e v e l . I t w a s a l s o a s s u m e d t h a t a d e c r e a s e d l e a f a r e a i n d e x c o u l d n o t b e r e c o v e r e d w h e n r e w e t t i n g o c c u r r e d , i . e . t h e w i l t i n g p h e n o m e n o n w a s i r r e v e r s i b l e f o r o u t a l f a l f a . 3 L 0 ( M > 0 . 8 M o ) 8 L : { a L o e ‘ c ( 1 ’ M / M ° ) ( o t h e r w i s e ) [ 2 9 ] w h e r e 3 L = L A I o f a s w a t h l a y e r , m Z l e a f / m 2 , a t , = i n i t i a l L A I o f t h e l a y e r , m 2 l e a f / m 2 , M = m o i s t u r e c o n t e n t i n t h e l a y e r , k g w a t e r / k g d r y m a t t e r , M o : i n i t i a l m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , a n d c = c o e f f i c i e n t . A v a l u e o f 0 . 1 2 w a s a s s i g n e d t o c o e f f i c i e n t 0 w h i c h i m p l i e d t h a t t h e f i n a l l e a f a r e a i n d e x w o u l d b e 8 9 % o f i t s i n i t i a l v a l u e w h e n a l f a l f a w a s c o m p l e t e l y d r i e d . T h e c v a l u e w a s o b t a i n e d t h r o u g h a p r e l i m i n a r y p a r a m e t e r e s t i m a t i o n a f t e r t h e m o d e l w a s f o r m e d . F u r t h e r i n v e s t i g a t i o n o n t h i s t o p i c i s n e e d e d . 4 . 1 . 3 . 3 C o m p l e m e n t a r y R a d i a t i o n d u e t o B e a m R a d i a t i o n R o s s ( 1 9 8 1 ) d e r i v e d a p a i r o f d i f f e r e n t i a l e q u a t i o n s t o d e s c r i b e t h e g r a d i e n t s o f t h e a v e r a g e c o m p l e m e n t a r y i r r a d i a n c e s ( u p w a r d a n d d o w n w a r d ) o n a u n i t h o r i z o n t a l a r e a . I n t h i s s t u d y , t h r e e p a i r s o f d i f f e r e n t i a l e q u a t i o n s w e r e w r i t t e n f o r t h r e e s w a t h l a y e r s : 3 D 3 ’ 1 / 8 L 1 = ‘ ( 1 ' T i ) D s , i + 7 1 0 3 , ; + O . S T i S b ’ 1 [ 3 0 ] ( 1 : 1 t o 3 ) 8 U 3 ’ 1 / 8 L 1 = ‘ Y i D s , 1 + ( 1 ' T i ) u s , i - O ° S Y i S b , l [ 3 1 ] w h e r e 0 3 , 1 = d o w n w a r d c o m p l e m e n t a r y i r r a d i a n c e , W / m z , 0 3 , 1 u p w a r d c o m p l e m e n t a r y i r r a d i a n c e , W / m z , 3 6 3 L 1 = e l e m e n t o f L A I , m 2 l e a f / m 2 , 3 0 3 , , = i n c r e m e n t o f 0 3 , 1 w i t h i n 3 L , W / m z , 3 0 3 , 1 = i n c r e m e n t o f 0 3 , 1 w i t h i n 8 L , W / m z , T 1 = t r a n s m i t t a n c e o f t h e c a n o p y , d e c i m a l , 7 1 = r e f l e c t a n c e o f t h e c a n o p y , d e c i m a l , a n d S b , 1 = a t t e n u a t e d b e a m i r r a d i a n c e d o w n t o t h e e l e m e n t , W / m z . T h e s e e q u a t i o n s w e r e v a l i d w h e n t h e c o m p l e m e n t a r y r a d i a t i o n w a s i s o t r o p i c , b o t h s i d e s o f l e a v e s w e r e i d e n t i c a l , a n d r w a s e q u a l t o y . T h e b o u n d a r y a n d c o n t i n u i t y c o n d i t i o n s f e r E q u a t i o n s [ 3 0 ] a n d [ 3 1 ] w e r e : 0 3 , 0 = 0 a t t h e t o p o f t h e s w a t h D s 1 = D s 2 ’ ’ } a t t h e i n t e r f a c e o f l a y e r s 1 a n d 2 U s , 1 = U s , 2 D 3 2 = D s ' ’ 3 } a t t h e i n t e r f a c e o f l a y e r s 2 a n d 3 " 3 , 2 = " 3 , 3 U 3 , 3 = Y g ( D s , 3 + S b , 3 ) a t t h e s o i l s u r f a c e W h e r e 7 8 w a s t h e r e f l e c t a n c e o f t h e s o i l s u r f a c e , a n d a s s u m e d t o b e a c o n s t a n t ( 0 . 1 5 ) i n t h e m o d e l . S u b s c r i p t 0 m e a n t a t t h e s w a t h t o p . T h e a n a l y t i c s o l u t i o n s o f t h e s e t o f d i f f e r e n t i a l e q u a t i o n s [ 3 0 ] a n d [ 3 1 ] w e r e e x p r e s s e d a s : D 3 , 1 : A 1 / H 1 + 3 1 8 1 - F 1 0 1 [ 3 2 ] ( 1 : 1 , 2 a n d 3 ) U 3 , 1 = R 1 A 1 / H 1 + B i H i l R i + F 1 V 1 [ 3 3 ] w h e r e H 1 = e c l a L l F 1 = 0 . 2 5 w 1 u 3 8 b 0 e ' L i / “ s R 1 = ( 1 - C 1 ) / ( 1 + C 1 ) - ( C 1 8 L 1 + O . 5 R 1 ) / ( 1 - R 1 ) u 3 C 1 : 1 C : p . H ( 1 + u s ) / ( 1 - u § C § ) o t h e r w i s e 3 7 ( C 1 R 1 3 L 1 + 0 . 5 ) / ( 1 - R i ) u s c i : 1 V 1 = ( 1 - u 3 ) / ( 1 - u § C i ) o t h e r w i s e w i t h m i : 1 1 + Y i u s = 2 s i n 8 C 1 = / T T : E I 9 S i x i n t e g r a t i o n c o n s t a n t s ( A 1 a n d B i , i = 1 , 2 a n d 3 ) w e r e o b t a i n e d b y u s e o f a m a t r i x m e t h o d ( s u b r o u t i n e G A U S S ) t o s o l v e a s y s t e m o f s i x l i n e a r e q u a t i o n s a c c o r d i n g t o t h e b o u n d a r y a n d c o n t i n u i t y c o n d i t i o n s a b o v e a p p l i e d t o E q u a t i o n s [ 3 2 ] a n d [ 3 3 ] . T h e n , t h e s o l u t i o n s o f E q u a t i o n s [ 3 0 ] a n d [ 3 1 ] w e r e c a l c u l a t e d i n s u b r o u t i n e D I R D F . A r r a y s D F D I R D a n d D F D I R U s t o r e d v a l u e s o f 0 3 , 1 a n d 0 3 , 1 , r e s p e c t i v e l y . 4 . 1 . 4 R a d i a t i o n F i e l d d u e t o D i f f u s e S k y R a d i a t i o n S i m i l a r t o t h e r a d i a t i o n f i e l d o f d i r e c t s o l a r r a d i a t i o n d i s c u s s e d i n S e c t i o n 4 . 1 . 3 a b o v e , t h e r a d i a t i o n f i e l d o f d i f f u s e s k y r a d i a t i o n c o m p r i s e d a p o r t i o n o f a t t e n u a t e d d i f f u s e s k y r a d i a t i o n p l u s a p o r t i o n o f c o m p l e m e n t a r y d i f f u s e r a d i a t i o n . 4 . 1 . 4 . 1 A t t e n u a t e d D i f f u s e S k y R a d i a t i o n T h e a t t e n u a t e d d i f f u s e s k y r a d i a t i o n a v e r a g e d o n a u n i t h o r i z o n t a l a r e a i n s i d e t h e s w a t h w a s d e s c r i b e d a s : 5 f , i = a d , 1 3 r 0 [ 3 4 ] a t t e n u a t e d d i f f u s e s k y i r r a d i a n c e , W / m z , w h e r e S f ’ i d i f f u s e s k y i r r a d i a n c e a t t h e t o p o f t h e s w a t h , W / m z , S f O 3 8 a n d a d , i = p e n e t r a t i o n f u n c t i o n f o r d i f f u s e s k y r a d i a t i o n , d e c i m a l . ‘ I n t h e m o d e l , v a l u e s o f S f ’ i w e r e s t o r e d i n a r r a y S L S K Y . R o s s ( 1 9 8 1 ) p r o p o s e d a m o d e l t h a t a d , 1 w a s a n i n t e g r a t i o n o f t h e d i r e c t i o n a l p e n e t r a t i o n f u n c t i o n a s , i ( E q u a t i o n [ 2 6 ] ) o v e r a w h o l e h e m i s p h e r e . H e s h o w e d t h a t f o r s p h e r i c a l l y o r i e n t e d l e a v e s : a d , 1 = 2 E 3 ( L 1 / 2 ) . [ 3 5 ] w h e r e L 1 = d o w n w a r d c u m u l a t i v e L A I , m 2 l e a f / m 2 , a n d E 3 ( x ) = t h i r d - o r d e r e x p o n e n t i a l i n t e g r a l o f E n ( x ) . T h e n t h - o r d e r e x p o n e n t i a l i n t e g r a l ( B e y e r , 1 9 8 4 ) i s : E n ( x ) 4 f : y ' " e ' x y d y n / 2 I o c o s O ( s i n 0 ) n ’ 2 e ' x / S i n e d 0 [ 3 6 ] T h i s i n t e g r a l c a n n o t b e a n a l y t i c a l l y s o l v e d . I n t h e m o d e l , i t w a s n u m e r i c a l l y s o l v e d i n f u n c t i o n s u b p r o g r a m E ( N , X ) b y u s e o f t h e G a u s s i a n Q u a d r a t u r e f o r m u l a ( B e y e r , 1 9 8 4 ) , s e e f u n c t i o n s u b p r o g r a m G A U I N T . 4 . 1 . 4 . 2 C o m p l e m e n t a r y R a d i a t i o n d u e t o D i f f u s e S k y R a d i a t i o n R o s s ( 1 9 8 1 ) d e r i v e d a p a i r o f d i f f e r e n t i a l e q u a t i o n s f o r t h e c o m p l e m e n t a r y d i f f u s e i r r a d i a n c e a v e r a g e d o n a h o r i z o n t a l p l a n e d u e t o d i f f u s e s k y r a d i a t i o n . I n t h e m o d e l , t h r e e p a i r s o f d i f f e r e n t i a l e q u a t i o n s w e r e f o r m u l a t e d f o r t h e s w a t h : 3 0 f , 1 / 3 L 1 = - ( 1 - 1 1 ) D f , 1 + y i u f , i + m i s f 0 8 2 ( L 1 / 2 ) [ 3 7 ] 3 U f ’ 1 / 3 L 1 = ‘ Y i n , i + ( 1 - T 1 ) U f ’ i - Y i S f o E 2 ( L 1 / 2 ) [ 3 8 ] ( i = 1 , 2 a n d 3 ) w h e r e 8 L 1 = e l e m e n t o f L A I , n 2 l e a f / m 2 , D f 1 d o w n w a r d c o m p l e m e n t a r y d i f f u s e i r r a d i a n c e , W / m z , P 3 9 U f ’ i u p w a r d c o m p l e m e n t a r y d i f f u s e i r r a d i a n c e , W / m z , L 1 d o w n w a r d c u m u l a t i v e L A I , m 2 l e a f / m 2 , a n d E 2 ( x ) = s e c o n d - o r d e r e x p o n e n t i a l i n t e g r a l , s e e E q u a t i o n [ 3 6 ] . T h i s s e t o f d i f f e r e n t i a l e q u a t i o n s c o u l d n o t b e s o l v e d a n a l y t i c a l l y , b e c a u s e t h e y i n v o l v e d t h e f o l l o w i n g i n d e f i n i t e i n t e g r a l : I ( y ) = [ 5 2 ( x ) e i d e x [ 3 9 ] A n a p p r o x i m a t i o n w a s m a d e i n o r d e r t o s o l v e E q u a t i o n s [ 3 7 ] a n d [ 3 8 ] . N u m e r i c a l s o l u t i o n s w e r e o b t a i n e d i n s u b r o u t i n e S K Y D F w i t h t h e f o l l o w i n g s t r u c t u r e : D f , 1 : A 1 / H 1 + B i H i + s t P 1 [ 4 0 ] ( 1 : 1 , 2 a n d 3 ) u f , 1 = R 1 A 1 / H 1 + B i H 1 / R 1 + S f O Q i [ 4 1 1 W h e r e H 1 a n d R 1 w e r e t h e s a m e a s t h o s e i n E q u a t i o n s [ 3 2 ] a n d [ 3 3 ] . I n t e g r a t i o n c o n s t a n t s A 1 a n d 8 1 w e r e c a l c u l a t e d w i t h t h e m a t r i x m e t h o d ( s u b r o u t i n e G A U S S ) b a s e d o n t h e b o u n d a r y a n d c o n t i n u i t y c o n d i t i o n s w h i c h w e r e t h e s a m e a s t h o s e o f E q u a t i o n s [ 3 0 ] a n d [ 3 1 ] . F u n c t i o n s P a n d 0 w e r e d e c l a r e d f r o m n u m e r i c a l s o l u t i o n s o f E q u a t i o n s [ 3 7 ] a n d [ 3 8 ] . S u b r o u t i n e P Q c a r r i e d o u t t h e n u m e r i c a l c a l c u l a t i o n f o r P a n d Q . T h e v a r i a b l e n a m e s f o r D f , 1 a n d U f ’ i i n S K Y D F w e r e D F S K Y D a n d D F S K Y U , r e s p e c t i v e l y . 4 . 1 . 5 T o t a l S h o r t - W a v e R a d i a t i o n i n t h e S w a t h T h e t o t a l s h o r t - w a v e r a d i a t i o n i n e a c h l a y e r o f t h e s w a t h w a s t h e s u m o f s i x t e r m s : ( 1 ) t h e a t t e n u a t e d b e a m r a d i a t i o n ( S b , 1 ) , ( 2 ) t h e a t t e n u a t e d d i f f u s e s k y r a d i a t i o n ( S f , 1 ) , ( 3 ) t h e c o m p l e m e n t a r y d i f f u s e r a d i a t i o n d u e t o d i r e c t s o l a r r a d i a t i o n ( U s , i a n d 4 3 , 1 ) a n d ( 4 ) t h e c o m p l e m e n t a r y d i f f u s e r a d i a t i o n d u e t o d i f f u s e s k y 4 O r a d i a t i o n ( D f , i a n d U f , i ) . I f a p o s i t i v e s i g n i s d e s i g n a t e d t o d o w n w a r d r a d i a t i o n , t h e n t h e t o t a l s h o r t - w a v e r a d i a t i o n m a y a l s o b e c a l l e d n e t d o w n w a r d s h o r t - w a v e r a d i a t i o n . I n s u b r o u t i n e S W R A D , t h e n e t d o w n w a r d s h o r t - w a v e i r r a d i a n c e a t v a r i o u s i n t e r f a c e s o f t h e s w a t h w a s c a l c u l a t e d a s : $ 1 = S b , i + S f , i + D s , i + D f , 1 - U s , i - U f ’ i [ 4 2 ] ( 1 : 0 , 1 , 2 a n d 3 ) W h e r e 0 m e a n t a t t h e t o p o f t h e s w a t h , a n d 3 - a t t h e i n t e r f a c e b e t w e e n t h e s o i l a n d t h e b o t t o m l a y e r o f t h e s w a t h . I n S W R A D , a r r a y S L s t o r e d v a l u e s o f S i . T h e n o t a b s o r b e d s h o r t - w a v e r a d i a t i o n i n a l a y e r o f t h e s w a t h w a s t h e d i f f e r e n c e i n t h e n e t d o w n w a r d i r r a d i a n c e a t t w o i n t e r f a c e s . R n 3 , 1 : 3 1 - 1 - S i ( i = 1 , 2 a n d 3 ) [ 4 3 ] T h i s q u a n t i t y w a s o n e s o u r c e f o r t h e e n e r g y b a l a n c e o f l e a f s u r f a c e s . T h e n e t d o w n w a r d s h o r t - w a v e i r r a d i a n c e a t t h e s o i l s u r f a c e w a s : R n s g : S 3 [ 4 4 ] R n s g w a s u s e d f o r t h e e n e r g y b a l a n c e a t t h e s o i l s u r f a c e . T h e a l b e d o , i . e . t h e e f f e c t i v e r e f l e c t a n c e o f t h e s w a t h , w a s t h e r a t i o o f t h e t o t a l u p w a r d s h o r t - w a v e i r r a d i a n c e t o t h e i n c o m i n g s o l a r i r r a d i a n c e a t t h e t o p o f t h e s w a t h . I t w a s c a l c u l a t e d i n S W R A D a s : y e : ( U s , 0 + u f , O ) / S l [ 4 5 ] w h e r e y e = a l b e d o . T h e e f f e c t i v e t r a n s m i t t a n c e f o r t h e s w a t h w a s t h e r a t i o o f t h e t o t a l d o w n w a r d s h o r t - w a v e i r r a d i a n c e a t t h e b o t t o m o f t h e s w a t h t o t h e i n c o m i n g s o l a r i r r a d i a n c e a t t h e t o p o f t h e s w a t h . T h e e f f e c t i v e t r a n s m i t t a n c e w a s o b t a i n e d a s : T e = ( S b ’ 3 + S f ’ 3 + D s ’ 3 + D f , 3 / S l [ 4 6 ] 4 1 w h e r e T e = e f f e c t i v e t r a n s m i t t a n c e . I n s u b r o u t i n e S W R A D , a r r a y R N S W , a n d v a r i a b l e s R N S W G , R E a n d T E s t o r e d v a l u e s o f R s n , i v R n s g : y e a n d r e , r e s p e c t i v e l y . F i g u r e s 2 a n d 3 s h o w t h e s i m u l a t e d n e t d o w n w a r d s h o r t - w a v e r a d i a t i o n p r o f i l e , t h e a l b e d o a n d t h e e f f e c t i v e t r a n s m i t t a n c e f o r a n a l f a l f a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 o f 1 9 8 4 . 4 . 2 L o n g - W a v e R a d i a t i o n P r o f i l e i n s i d e t h e S w a t h L o n g - w a v e r a d i a t i o n i s a n o t h e r i m p o r t a n t e n e r g y t e r m w i t h i n t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m s y s t e m . T h e l a w s o f p h y s i c s s t a t e s t h a t a n y s u r f a c e e m i t s r a d i a t i o n a s l o n g a s i t s t e m p e r a t u r e i s a b o v e a b s o l u t e z e r o ( 0 ° K o r - 2 7 3 . 1 6 ° C ) . R a d i a t i o n i s g o v e r n e d b y : E = e o T “ [ 4 7 ] w h e r e E = i r r a d i a n c e , W / m z , e : e m i s s i v i t y , d e c i m a l , o = S t e f a n - B o l t z m a n n c o n s t a n t , 5 . 5 7 > < 1 O " 8 W / m 2 ° K 5 , a n d T = a b s o l u t e t e m p e r a t u r e o f t h e e m i t t i n g s u r f a c e , ° K . W i e n ' s l a w s t a t e s t h a t : A m a x : 2 8 9 7 / T [ 4 8 ] w h e r e A m a x = w a v e l e n g t h o f t h e m a x i m u m i n t e n s i t y e m i t t e d , p m . T h e s u r f a c e t e m p e r a t u r e o f t h e s u n i s a b o u t 6 0 0 0 ° K , s o t h e w a v e l e n g t h o f s o l a r r a d i a t i o n i s a r o u n d 4 8 0 p m w h i c h i s w i t h i n t h e s h o r t - w a v e s p e c t r u m . W i t h i n t h e e a r t h ' s a t m o s p h e r e , t e m p e r a t u r e o f n a t u r a l o b j e c t s i s u s u a l l y c l o s e t o t h e a t m o s p h e r i c t e m p e r a t u r e , w h i c h i s s e l d o m a b o v e 5 0 ° C . S o , w a v e l e n g t h s o f r a d i a t i o n e m i t t e d b y n a t u r a l s u r f a c e s a r e w i t h i n t h e l o n g - w a v e s p e c t r u m ( 4 - 3 0 0 0 u m ) . L o n g - w a v e r a d i a t i o n i s a l s o : . 0 : 8 r 1 : " . m ” : 1 C : : h 0 : : 2 f o n . o 4 A / : H 1 i 8 W / / ' I , t 9 a 1 i d , a 6 0 r 1 : - 4 6 e v 1 a w t - s t u 6 r g 0 1 o u : h A 0 s e n t d o \ r \ \ 0 a a n D w a n g w i \ \ : d o h \ 8 1 n d c \ 3 a i t M 3 e 1 ? . r ; k / / ' r n , u g 0 o : H f n o i s 2 1 ( e n l a i L Z e f / m o t / i r s p a 0 T E : d 6 e n t i a l h u t m a . 5 i w 0 s s : 1 / 0 A a 8 _ \ \ ~ \ . : \ 0 8 1 . 2 e r u g i F 3 ’ m . . 4 0 : 1 O 2 / 1 8 ( g m / N 1 ) e o u o g p o u l M O M - n o u s p i o m u m o q i e N ' i ' - fl E i : I I ” ° ' : i i ’ / / ' / 4 \ M 4 2 h t a w s a r o f e . c 4 n 8 a 9 t 1 t i , m 6 s 1 n - a 4 r 1 t t e s v u i g t u c A e f n f o e n e a h g t i h d c n i a M o , d g e n b i l s a n a d L e t t a s l a u E m i n S i . 3 e r u g i F 4 2 J 1 . . 0 0 1 ' 1 . 0 e o u o i o e u e g e n g i o e i i a O O U D l l l e U D J l e n g i o e i i g 0 . 0 J 0 . 6 1 O . 4 ~ 0 . 3 - 1 0 . 2 . 1 0 . 0 1 2 : 0 8 / 1 4 1 8 : 0 0 : 0 6 : 0 1 2 : 0 1 8 : 0 0 : 0 6 : 0 1 2 : 0 1 8 : 0 8 1 5 1 6 / T i m e ( H o u r a n d D a t e 4 3 4 4 c a l l e d i n f r a r e d r a d i a t i o n o r t h e r m a l r a d i a t i o n ; a n d i t i s d i f f u s e r a d i a t i o n . I n a d d i t i o n t o e m i s s i o n o f t h e r m a l r a d i a t i o n , t h e a t m o s p h e r e , p a r t i c u l a r l y w a t e r v a p o r , c l o u d s a n d c a r b o n d i o x i d e , a b s o r b s a b o u t 9 0 % o f t h e i n f r a r e d r a d i a t i o n e m i t t e d b y t h e e a r t h , a n d r a d i a t e s b a c k t o t h e e a r t h ( R o s e n b e r g e t a l . , 1 9 8 3 ) . T h e r e f o r e , l o n g - w a v e r a d i a t i o n p l a y s a n i m p o r t a n t r o l e i n t h e e n e r g y b a l a n c e w i t h i n t h e m i c r o c l i m a t e r e g i o n . P l a n t s g a i n t h e r m a l r a d i a t i o n e n e r g y a s w e l l a s s e n s i b l e h e a t w h e n t h e i r t e m p e r a t u r e i s l o w e r t h a n t h a t o f t h e a t m o s p h e r e o r t h e s o i l . H o w e v e r , p l a n t s l o s e t h e r m a l r a d i a t i o n e n e r g y a s w e l l a s s e n s i b l e h e a t w h e n t h e p l a n t s a r e w a r m e r t h a n t h e a t m o s p h e r e o r t h e s o i l . S u b m o d e l L W R A D w i t h a s s o c i a t e d s u b p r o g r a m s s i m u l a t e d t h e l o n g - w a v e r a d i a t i o n p r o f i l e i n s i d e t h e s w a t h . 4 . 2 . 1 E m i s s i v i t i e s o f t h e A t m o s p h e r e , A l f a l f a a n d S o i l S i m i l a r t o t h e o p t i c a l p r o p e r t i e s o f s h o r t - w a v e r a d i a t i o n , r a d i a t i v e p r o p e r t i e s o f o b j e c t s a l s o o b e y t h e l a w s t a t e d i n E q u a t i o n [ 2 1 ] . F o r s h o r t - w a v e r a d i a t i o n , t h e s c a t t e r i n g c o e f f i c i e n t ( r e f l e c t a n c e p l u s t r a n s m i t t a n c e ) i s d o m i n a t e , h o w e v e r , f o r l o n g - w a v e r a d i a t i o n a b s o r p t a n c e i s p r i m a r y . T h i s i s b e c a u s e m o s t s o i l s a n d v e g e t a t i o n b e h a v e n e a r l y a s b l a c k b o d y ( 2 : 1 . 0 ) e m i t t e r s ( R o s e n b e r g e t a l . , 1 9 8 3 ) . K i r c h o f f ' s l a w s t a t e s t h a t : e : a = 1 . 0 - y [ 4 9 ] w h e r e c = e m i s s i v i t y , d e c i m a l , a = a b s o r p t a n c e , d e c i m a l , a n d y = s c a t t e r i n g c o e f f i c i e n t , d e c i m a l . 4 5 T h e r e f o r e , t h e e m i s s i v i t y , a b s o r p t a n c e o f s o i l s a n d p l a n t s f o r t h e r m a l r a d i a t i o n a r e c l o s e t o u n i t y w h i l e t h e s c a t t e r i n g c o e f f i c i e n t i s n e a r z e r o , a n d n e g l i g i b l e . M o n t e i t h ( 1 9 7 3 ) l i s t e d l o n g - w a v e e m i s s i v i t i e s o f 8 s p e c i e s o f p l a n t s w h i c h r a n g e d f r o m 0 . 9 3 8 t o 0 . 9 9 5 , a n d h e m e n t i o n e d m o s t s o i l s h a v e t h e r m a l e m i s s i v i t y f r o m 0 . 9 0 t o 0 . 9 5 . C a m p b e l l ( 1 9 7 7 ) m e n t i o n e d t h a t t h e r m a l e m i s s i v i t i e s o f n a t u r a l s u r f a c e s a r e b e t w e e n 0 . 9 0 a n d 0 . 9 7 , a n d h e u s e d b o t h 0 . 9 5 a n d 0 . 9 7 f o r p l a n t l e a v e s i n s a m p l e p r o b l e m s . R o s s ( 1 9 8 1 ) u s e d 2 : 0 . 9 5 f o r b o t h p l a n t l e a v e s a n d s o i l s i n h i s m o d e l . T h e l i t e r a t u r e h a s n o t y e t p r o v i d e d d e t a i l e d d a t a o n t h e r a d i a t i v e p r o p e r t i e s o f a l f a l f a a n d s o i l s . I n t h i s s t u d y , i t w a s a s s u m e d t h a t : 2 1 = 0 . 9 6 ( e a c h l a y e r o f a l f a l f a ) 0 . 9 5 ( s o i l s u r f a c e ) 5 8 T h e l i t e r a t u r e s h o w s t h a t t h e l o n g - w a v e e m i s s i v i t y o f t h e a t m o s p h e r e i s m a i n l y a f f e c t e d b y t h e w a t e r v a p o r d e n s i t y i n t h e a i r a n d t h e s k y c o v e r f r a c t i o n ( M o n t e i t h , 1 9 7 3 ; C a m p b e l l , 1 9 7 7 ; R o s e n b e r g e t a l . , 1 9 8 3 ) . M o n t e i t h ( 1 9 7 3 ) a n d C a m p b e l l ( 1 9 7 7 ) r e v i e w e d p a s t r e s e a r c h o n d e v e l o p m e n t o f e m p i r i c a l m o d e l s t o e s t i m a t e t h e a t m o s p h e r i c t h e r m a l e m i s s i v i t y . T h e y i n d i c a t e d t h a t i n a c l e a r s k y , t h e a t m o s p h e r i c e m i s s i v i t y i s s t r o n g l y i n f l u e n c e d b y t h e a i r t e m p e r a t u r e a n d h u m i d i t y i n t h e f i r s t f e w h u n d r e d m e t e r s a b o v e t h e g r o u n d . B o t h o f t h e m s u g g e s t e d u s e o f p a r a m e t e r s m e a s u r e d a t a r e f e r e n c e h e i g h t ( 1 t o 2 m ) t o e s t i m a t e t h e a t m o s p h e r i c e m i s s i v i t y . A i r t e m p e r a t u r e , v a p o r d e n s i t y a n d v a p o r p r e s s u r e h a v e b e e n u s e d i n v a r i o u s e m p i r i c a l e q u a t i o n s . I n t h i s s t u d y , a m o d e l b y B r u t s a e r t ( 1 9 7 5 ) w a s a p p l i e d b e c a u s e o f i t s r e a s o n a b l e t h e o r e t i c a l j u s t i f i c a t i o n . 5 3 , , = 1 . 5 1 3 ( p w e 0 - 4 4 2 1 1 / 7 [ s o ] 4 6 w h e r e € a , c = t h e r m a l e m i s s i v i t y o f c l e a r s k i e s , d e c i m a l , p " = v a p o r d e n s i t y a t t h e l o c a t i o n , k g w a t e r / m 3 a i r , a n d z = a l t i t u d e a b o v e t h e s e a l e v e l , k m . F o r a n o v e r c a s t s k y , t h e a t m O S p h e r i c e m i s s i v i t y w a s v e r y c l o s e t o u n i t y . C a m p b e l l ( 1 9 7 7 ) p r o p o s e d a n e q u a t i o n : e a , o = 1 - 4 A T / T [ 5 1 ] W h e r e 2 3 , 0 w a s t h e a t m o s p h e r i c e m i s s i v i t y o f a n o v e r c a s t s k y , a n d A T i s t h e d i f f e r e n c e b e t w e e n a i r t e m p e r a t u r e a n d c l o u d b a s e t e m p e r a t u r e , w i t h a t y p i c a l v a l u e o f 2 ° K ( M o n t e i t h , 1 9 7 3 ) . F o r a p a r t l y c l o u d y s k y , t h e a t m o s p h e r i c e m i s s i v i t y w a s c a l c u l a t e d a s : ‘ e a = ( 1 - c , , ) e , , , c + C v e a ’ o [ 5 2 ] w h e r e c a = e f f e c t i v e a t m o s p h e r i c e m i s s i v i t y , d e c i m a l a n d C V = s k y c o v e r f r a c t i o n , d e c i m a l . C a m p b e l l ( 1 9 7 7 ) i n d i c a t e d t h a t E q u a t i o n [ 5 2 ] w a s i n s e n s i t i v e t o e r r o r s i n C V a n d A T s i n c e t h e r a n g e o f c a w a s r e l a t i v e l y s m a l l . I n s u b r o u t i n e L W R A D , v a r i a b l e s E M L , E M G a n d E M A s t o o d f o r c l , 2 8 a n d c a , r e s p e c t i v e l y . 4 . 2 . 2 L o n g - W a v e R a d i a t i o n F i e l d i n s i d e t h e S w a t h R o s s ( 1 9 8 1 ) d e r i v e d a p a i r o f d i f f e r e n t i a l e q u a t i o n s o f p r o j e c t i o n o f d i r e c t i o n a l l o n g - w a v e i n t e n s i t y o n a h o r i z o n t a l p l a n e f o r i s o t r o p i c e m i s s i v i t i e s o f t h e a t m o s p h e r e , p l a n t s a n d s o i l s a n d f o r s p h e r i c a l l y o r i e n t e d l e a v e s : ( a d t / B L ) s i n 0 = - 0 . 5 d t + n 1 [ 5 3 ] ( B u t / 3 L ) s i n 0 = 0 . 5 u t - n 1 [ 5 4 ] w h e r e 3 L = e l e m e n t o f L A I , m 2 l e a f / m 2 , d t : d o w n w a r d d i r e c t i o n a l t h e r m a l i n t e n s i t y , W / m z , 4 7 u p w a r d d i r e c t i o n a l t h e r m a l i n t e n s i t y , W / m z , “ t B d t i n c r e m e n t o f 9 t i n 3 L , W / m z , 3 0 , = i n c r e m e n t o f u t i n 8 L , W / m z , 0 = i n c l i n a t i o n a n g l e i n t h e u p p e r h e m i s p h e r e , 0 = i n c l i n a t i o n a n g l e i n t h e l o w e r h e m i s p h e r e , a n d n 1 = e m i s s i o n i n t e n s i t y f r o m o n e s i d e o f l e a v e s , W / m z . F o r s p h e r i c a l l y o r i e n t e d a n d i s o t r o p i c a l l y e m i t t i n g l e a v e s , n 1 : 0 . 5 E 1 / fl [ 5 5 ] w h e r e E 1 : l o n g - w a v e i r r a d i a n c e f r o m o n e s i d e o f l e a v e s , W / m z . . I n E q u a t i o n s [ 5 3 ] a n d [ 5 4 ] , n o t e r m d e a l i n g w i t h e i t h e r r e f l e c t i o n o r t r a n s m i s s i o n o f t h e r m a l r a d i a t i o n w a s i n c l u d e d b e c a u s e i t w a s n e g l i g i b l e . N o t e t h a t l e a v e s e m i t t h e r m a l r a d i a t i o n f r o m b o t h s i d e s , s o l o n g - w a v e r a d i a t i o n f r o m o n e s i d e o f l e a v e s a p p e a r e d i n b o t h E q u a t i o n s [ 5 3 ] a n d [ 5 4 ] . R o s s ( 1 9 8 1 ) s o l v e d t h e s e t r a n s f e r e q u a t i o n s , i n t e g r a t e d t h e d i r e c t i o n a l i n t e n s i t i e s o v e r t h e u p p e r a n d l o w e r h e m i s p h e r e s , r e s p e c t i v e l y , a n d o b t a i n e d t h e d o w n w a r d a n d u p w a r d t h e r m a l i r r a d i a n c e a v e r a g e d o v e r a u n i t h o r i z o n t a l a r e a . T h e s o l u t i o n e q u a t i o n s w e r e u s e d i n t h i s m o d e l : D t , i = 1 “ 1 1 : 1 , 1 4 1 + E 1 , 1 [ 5 6 ] ( 1 : 1 , 2 a n d 3 ) U t , i = 1 " M 1 4 1 3 1 + E 1 , 1 [ 5 7 ] w h e r e D t , i = d o w n w a r d t h e r m a l i r r a d i a n c e , W / m z , U t , i = u p w a r d t h e r m a l i r r a d i a n c e , W / m z , a d , 1 = d o w n w a r d p e n e t r a t i o n f u n c t i o n , d e c i m a l , 3 6 , 1 : u p w a r d p e n e t r a t i o n f u n c t i o n , d e c i m a l , A 1 a n d B i = i n t e g r a t i o n c o n s t a n t s , W / m z . T h e u p w a r d p e n e t r a t i o n f u n c t i o n w a s e x p r e s s e d a s : 4 8 a a , i = 2 E 3 ( L i / 2 ) [ 5 8 ] w h e r e L i = L 8 1 - L 1 [ 5 9 ] w h e r e L f : u p w a r d c u m u l a t i v e L A I f r o m t h e g r o u n d , m 2 l e a f / m 2 , L a i = t o t a l d o w n w a r d c u m u l a t i v e L A I , m 2 l e a f / m 2 . a n d L 1 : d o w n w a r d c u m u l a t i v e L A I f r o m t h e t o p , 1 1 1 2 l e a f / m 2 . I n t e g r a t i o n c o n s t a n t s A 1 a n d B i w e r e c a l c u l a t e d b a s e d u p o n b o u n d a r y a n d c o n t i n u i t y c o n d i t i o n s . D t , 0 = E a a t t h e t o p o f t h e s w a t h D t , 1 = D t , 2 } a t t h e i n t e r f a c e o f l a y e r s 1 a n d 2 u t , 1 = U t , 2 D t , 2 = D t , 3 } a t t h e i n t e r f a c e o f l a y e r s 2 a n d 3 U t , 2 = u t , 3 U t , 3 = E 3 + ( 1 - c g ) D t ’ 3 a t t h e s o i l s u r f a c e w h e r e E a : a t m o s p h e r i c t h e r m a l i r r a d i a n c e , W / m z , s o i l t h e r m a l i r r a d i a n c e , W / m z , 3 3 a n d a s t h e r m a l e m i s s i v i t y o f t h e s o i l . I n s u b r o u t i n e L W R A D , v a r i a b l e s E M G , E A O a n d E B O , a r r a y s E L , E D a n d E U s t o r e d v a l u e s o f £ 8 , E a , 8 8 ’ 3 1 , 1 1 D t , 1 a n d U t , 1 , r e s p e c t i v e l y . D t , 1 a n d U t , 1 w e r e c a l c u l a t e d i n s u b r o u t i n e T R M . 4 . 2 . 3 T o t a l L o n g - W a v e R a d i a t i o n i n s i d e t h e S w a t h I n L W R A D , t h e n e t d o w n w a r d l o n g - w a v e i r r a d i a n c e a t v a r i o u s i n t e r f a c e s w a s c a l c u l a t e d a s : E t , i = D t , i - U t , i ( i = 0 , 1 , 2 a n d 3 ) [ 6 0 ] w h e r e E t , i = n e t d o w n w a r d t h e r m a l i r r a d i a n c e , W / m z , 4 9 d o w n w a r d t h e r m a l i r r a d i a n c e , W / m z , D t , i a n d U t , 1 u p w a r d t h e r m a l i r r a d i a n c e , W / m z . F o r t h e t o p o f t h e s w a t h , 1 : 0 ; a n d f o r t h e i n t e r f a c e b e t w e e n t h e s o i l a n d t h e b o t t o m o f t h e s w a t h , 1 : 3 . T h e n e t a b s o r b e d l o n g - w a v e i r r a d i a n c e i n a l a y e r o f t h e s w a t h w a s : R n t ’ i = E t , 1 _ 1 - E t , 1 ( i = 1 , 2 a n d 3 ) [ 6 1 ] w h e r e R n t , i = n e t a b s o r b e d t h e r m a l i r r a d i a n c e , W / m z . T h i s q u a n t i t y w a s u s e d f o r t h e e n e r g y b a l a n c e o n l e a f s u r f a c e s . T h e n e t d o w n w a r d l o n g - w a v e i r r a d i a n c e a t t h e s o i l s u r f a c e w a s : R n t g = R n t , 3 [ 5 2 ] R n t g w a s u s e d a s a s o u r c e f o r e n e r g y b a l a n c e a t t h e s o i l s u r f a c e . T h e n e t a b s o r b e d i r r a d i a n c e i n a l a y e r o f t h e s w a t h w a s t h e s u m o f a b s o r b e d l o n g - w a v e a n d s h o r t - w a v e i r r a d i a n c e s i n t h e l a y e r . L i k e w i s e , t h e n e t d o w n w a r d i r r a d i a n c e a t t h e s o i l s u r f a c e w a s t h e s u m o f n e t d o w n w a r d l o n g - w a v e a n d s h o r t - w a v e i r r a d i a n c e s a t t h e i n t e r f a c e . I n L W R A D , a r r a y s E N E T a n d R N L W , a n d v a r i a b l e s R N L W G w e r e u s e d t o s t o r e d a t a o f E t , 1 , R n t , i a n d R n t g : r e s p e c t i v e l y . I t s h o u l d b e p o i n t e d o u t t h a t t h e l o n g - w a v e r a d i a t i o n p r o f i l e i n s i d e t h e s w a t h d e p e n d e d u p o n t h e t e m p e r a t u r e p r o f i l e a t v a r i o u s l o c a t i o n s o f t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m , b u t t h i s t e m p e r a t u r e p r o f i l e i t s e l f w a s d e p e n d e n t o n t h e t h e r m a l r a d i a t i o n p r o f i l e . T h e r e f o r e , a n i t e r a t i o n p r o c e d u r e w a s p r e f e r r e d ; h o w e v e r , i t r e q u i r e d r e p e t i t i o n o f a w h o l e m o d e l s i m u l a t i o n w h i c h w a s v e r y t i m e - c o n s u m i n g a n d c o s t l y . D e t a i l e d d i s c u s s i o n a b o u t t h i s i s p r e s e n t e d i n t h e l a s t s e c t i o n o f t h i s c h a p t e r . F i g u r e 4 s h o w s t h e s i m u l a t e d p r o f i l e o f n e t d o w n w a r d l o n g - w a v e r a d i a t i o n w i t h i n a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t o f 1 9 8 4 . 5 0 I n t h e f i g u r e , n e g a t i v e v a l u e s i n d i c a t e u p w a r d t h e r m a l r a d i a t i o n . 4 . 3 E x c h a n g e s a b o v e t h e S w a t h I n t h e s o i l - p l a n t - a t m o s p h e r e c o n t i n u u m , t r a n s f e r o f s e n s i b l e h e a t a n d m o i s t u r e b e t w e e n t h e a t m o s p h e r e a n d t h e c a n o p y ( s w a t h ) i s n o r m a l l y c o n v e c t i v e d u e t o a i r m o v e m e n t . H o r i z o n t a l w i n d s p e e d a b o v e t h e s w a t h g o v e r n s t h e c h a r a c t e r i s t i c s o f c o n v e c t i v e t r a n s f e r . V e r t i c a l a i r m o v e m e n t a l s o e x i s t s d u e t o e d d i e s a n d r i s i n g a i r p a r c e l s c a u s e d b y a t e m p e r a t u r e g r a d i e n t i n t h e v e r t i c a l d i r e c t i o n . F r o m f l u i d m e c h a n i c s , c h a n g e s i n f l o w s p e e d o v e r a s u r f a c e i s d u e t o m o m e n t u m e x c h a n g e b e t w e e n d i f f e r e n t l a y e r s i n a f l o w . T h e r e f o r e , e x c h a n g e a b o v e t h e s w a t h i s s i m u l t a n e o u s t r a n s f e r o f m o m e n t u m , h e a t a n d m o i s t u r e . K n o w l e d g e o f t h e w i n d p r o f i l e w i t h h e i g h t a b o v e t h e s w a t h e n a b l e s d e t e r m i n a t i o n o f w i n d s p e e d a t a n y h e i g h t l e v e l w i t h m e a s u r e d w i n d s p e e d a t a r e f e r e n c e h e i g h t . T h i s a l s o l e a d s t o f i n d i n g c h a r a c t e r i s t i c s o f s e n s i b l e h e a t a n d m o i s t u r e t r a n s f e r . T h e l i t e r a t u r e i n d i c a t e s t h a t t h e r e i s a s u r f a c e l a y e r a b o v e a n a t u r a l s u r f a c e i n t h e d o w n w i n d d i r e c t i o n . T h e t h i c k n e s s o f t h i s s u r f a c e l a y e r d e p e n d s o n t h e f e t c h d i s t a n c e f r o m t h e e d g e o f t h e s u r f a c e . F o r a g r i c u l t u r a l c r o p s , R o s e n b e r g e t a l . ( 1 9 8 3 ) s u g g e s t e d a r a t i o o f t h i c k n e s s t o f e t c h o f a b o u t 1 : 1 0 0 , i . e . t h e s u r f a c e l a y e r w o u l d b e 2 m t h i c k a t a d i s t a n c e o f 2 0 0 m f r o m t h e e d g e o f a f i e l d i n t h e d o w n w i n d d i r e c t i o n . I n t h i s s t u d y , t r a n s f e r p r o c e s s e s d i s c u s s e d w e r e m a i n l y w i t h i n t h e s u r f a c e l a y e r . S u b m o d e l A B O V E w i t h a s s o c i a t e d s u b p r o g r a m s A B V I , R S H A B V , I T E R , I N V E X C a n d H C O N V s i m u l a t e d t h e w i n d p r o f i l e a n d a e r o d y n a m i c r e s i s t a n c e s 0 : a 8 1 0 : T r o f . n 4 o 8 i 9 2 1 1 0 . t a i , d 6 a 1 r - 4 e 1 : v 6 a t w s - u g g n u o A l n d o 6 ) . 1 1 : . 0 r e a n t a w a n g w i D o h 0 . : d d c i t M 8 1 n e a n , g f n r o i u s 0 o e : H e n l a i L 2 1 ( f o t r s e p a m E i d 0 T . : 6 e n t i a l h u t m a i w s s 5 A a e 1 0 : . / 0 4 8 e r u g i F 0 r : 8 1 4 0 : 1 2 / 0 1 8 4 ( t u x / m ) e o u o g p b u l O A O M — O U O ‘ ] p i o m u m o q i e N 5 1 5 2 t o h e a t a n d m o i s t u r e t r a n s f e r a b o v e t h e s w a t h . 4 . 3 . 1 V i r t u a l T e m p e r a t u r e P a s t r e s e a r c h h a s s h o w n t h a t i n t h e a t m o s p h e r e n e a r t h e e a r t h ' s s u r f a c e , a n a i r p a r c e l m o v i n g u p w a r d s i s c o o l e d a t a r a t e o f a b o u t 0 . 0 1 ° K / m d u e t o a d e c r e a s e i n t h e a t m o s p h e r i c p r e s s u r e , i f t h e p r o c e s s i s a d i a b a t i c ( R o s e n b e r g e t a l . , 1 9 8 3 ) . T h i s r a t e i s c a l l e d t h e ' d r y a d i a b a t i c l a p s e r a t e ' . T h i s p r e s s u r e d e c r e a s e l e a d s t o a n e x p a n s i o n i n t h e a i r , a n d t h u s , a r e d u c t i o n i n t h e a i r d e n s i t y . I n a d d i t i o n , a i r h u m i d i t y a l s o a f f e c t s t h e a i r d e n s i t y . G o u d r i a a n ( 1 9 7 7 ) m e n t i o n e d t h a t a n i n c r e a s e i n w a t e r v a p o r p r e s s u r e o f 1 P a w a s e q u i v a l e n t t o a t e m p e r a t u r e i n c r e a s e o f 0 . 0 0 1 ° K . T a k i n g t h e s e t w o f a c t o r s i n t o a c c o u n t , G o u d r i a a n ( 1 9 7 7 ) p r e s e n t e d a n e q u a t i o n t o c a l c u l a t e ' v i r t u a l t e m p e r a t u r e ' o f t h e a i r a s : T p : T + 0 . 0 0 1 P v + 0 . 0 1 2 [ 6 3 ] w h e r e z = h e i g h t , m , T p = v i r t u a l t e m p e r a t u r e o f t h e a i r a t z , ° K , T = a b s o l u t e t e m p e r a t u r e o f t h e a i r a t z , ° K , a n d P v = w a t e r v a p o r p r e s s u r e o f t h e a i r a t 2 , P a . I n f u n c t i o n R S H A B V , t h e c o n c e p t o f t h e v i r t u a l t e m p e r a t u r e w a s a l s o u s e d t o c a l c u l a t e t h e d i f f e r e n c e i n v i r t u a l t e m p e r a t u r e b e t w e e n t h e r e f e r e n c e h e i g h t z r a n d t h e s w a t h t o p z c : A T p = A T + 0 . 0 0 1 A P V + 0 . 0 1 A z h [ 6 4 ] w h e r e A T V v i r t u a l t e m p e r a t u r e d i f f e r e n c e f r o m z r t o 2 0 , ° K , A T d r y - b u l b t e m p e r a t u r e d i f f e r e n c e f r o m z r t o z c , ° C , A P V w a t e r v a p o r p r e s s u r e d i f f e r e n c e f r o m z r t o z C , P a , S 3 a n d A z h = h e i g h t d i f f e r e n c e , z r - z c , m . I n t h e m o d e l , v a r i a b l e s D T P , D D B , D P V a n d D Z H s t o o d f o r A T P , A T , A P v a n d A 2 “ , r e s p e c t i v e l y . 4 . 3 . 2 Z e r o P l a n e D i s p l a c e m e n t a n d R o u g h n e s s P a r a m e t e r A s w a t h ( o r c a H O P y ) i s n o t a s o l i d b u t a l o o s e s t r u c t u r e . T h u s , w i n d s p e e d i s g e n e r a l l y n o t z e r o a t t h e t o p o f a c a n o p y . I n m i c r o c l i m a t o l o g y , a c a n o p y ( s w a t h ) i s a b s t r a c t e d a s i f t h e r e e x i s t e d a v i r t u a l s o l i d p l a n e a t a h e i g h t d w i t h a ' r o u g h n e s s p a r a m e t e r ' z o . A t t h e h e i g h t o f d + z o , t h e v i r t u a l e x t r a p o l a t i o n o f t h e w i n d p r o f i l e a b o v e t h e c a n o p y ( s w a t h ) r e a c h e d z e r o . T h i s a b s t r a c t e d h e i g h t d i s c a l l e d t h e ' z e r o p l a n e d i s p l a c e m e n t ' , w h i c h i s l o w e r t h a n t h e t o p h e i g h t o f a s w a t h ( o r c a n o p y ) . T h e z e r o p l a n e d i s p l a c e m e n t d a n d t h e r o u g h n e s s p a r a m e t e r 2 0 a r e o f t e n e s t i m a t e d w i t h e m p i r i c a l e x p r e s s i o n s . I n t h i s m o d e l , d a n d 2 0 w e r e c a l c u l a t e d a s ( G o u d r i a a n , 1 9 7 7 a n d R o s e n b e r g e t a l . , 1 9 8 3 ) : 0 . 9 7 9 l o g 1 o z c - 0 . 1 5 4 [ 6 5 ] l o g 1 o d 0 3 9 7 1 6 3 1 0 2 , , - 0 . 8 8 3 [ 6 6 ] 1 0 3 1 0 2 0 w h e r e z c = s w a t h ( c a n o p y ) h e i g h t , m . F o r a 0 . 1 m - t h i c k s w a t h , t h e e s t i m a t e d d w a s 0 . 0 7 4 m a n d 2 0 0 . 0 1 3 m . T h u s , z e r o w i n d s p e e d w a s a s s u m e d a t 0 . 0 8 7 m h e i g h t , w h i c h w a s 0 . 0 1 3 m b e l o w t h e s w a t h t o p . I n s u b m o d e l A B O V E , v a r i a b l e s Z R , Z C , D a n d 2 0 d e n o t e d z r , z c , d a n d 2 0 , r e s p e c t i v e l y . 0 a n d 2 0 w e r e e s t i m a t e d i n s u b r o u t i n e A B V I w h i c h i n i t i a l i z e d t h e a t m o s p h e r e a b o v e t h e s w a t h . 5 4 4 . 3 . 3 R a t e E q u a t i o n s o f M o m e n t u m , H e a t a n d M o i s t u r e T r a n s f e r A n a l o g o u s r a t e e q u a t i o n s h a v e b e e n e s t a b l i s h e d f o r m o m e n t u m , s e n s i b l e h e a t a n d m o i s t u r e t r a n s f e r i n t h e l a y e r w i t h i n t h e f i r s t f e w m e t e r s a b o v e v e g e t a t i o n c o v e r s . T = - m e 3 u / B z H = - p c p K h 3 T p / B z E = - x w a p w / a z w h e r e T = s h e a r s t r e s s , o r m o m e n t u m f l u x d e n s i t y , N / m z , H = s e n s i b l e h e a t f l u x , W / m z , E = m o i s t u r e f l u x , k g / m z s , p = a i r d e n s i t y a t c o n s t a n t p r e s s u r e , k g / m ’ , c p : a i r s p e c i f i c h e a t a t c o n s t a n t p r e s s u r e , J / k g ° K , a u / a z = w i n d s p e e d g r a d i e n t , 1 / s , a n / a z = v i r t u a l t e m p e r a t u r e g r a d i e n t , ° K / m , a p w / a z = w a t e r v a p o r d e n s i t y g r a d i e n t , k g w a t e r / m “ , K m = m o m e n t u m t r a n s f e r c o e f f i c i e n t , m z / s , K h : s e n s i b l e h e a t t r a n s f e r c o e f f i c i e n t , m z / s , a n d K H : m o i s t u r e t r a n s f e r c o e f f i c i e n t , m z / s . [ 6 7 ] [ 6 8 ] [ 6 9 ] N o t e t h a t T i s i n t h e h o r i z o n t a l d i r e c t i o n , o p p o s i t e t o h o r i z o n t a l w i n d s p e e d u , w h i l e t h e o t h e r t w o f l u x e s a r e v e r t i c a l . T o a v o i d c o n f u s i o n , t h e n e g a t i v e s i g n i n E q u a t i o n [ 6 7 ] w i l l b e o m i t t e d i n t h e d i s c u s s i o n . T h e d i f f e r e n t i a l f o r m s o f r a t e e q u a t i o n s a b o v e a r e r e w r i t t e n i n d i f f e r e n c e f o r m s : T = p A u / r m H : - p c p A T p / r h E 3 " p r / r w o f t e n [ 7 0 ] [ 7 1 ] [ 7 2 ] 5 5 w h e r e A u = w i n d s p e e d d i f f e r e n c e , m / s , A T p = v i r t u a l t e m p e r a t u r e d i f f e r e n c e , ° K , p r : w a t e r v a p o r d e n s i t y d i f f e r e n c e , k g w a t e r / m 3 , r m : r e s i s t a n c e t o m o m e n t u m t r a n s f e r , s / m , r h = r e s i s t a n c e t o s e n s i b l e h e a t t r a n s f e r , s / m , a n d r w : r e s i s t a n c e t o m o i s t u r e t r a n s f e r , s / m . T h e r e s i s t a n c e s r m , r h a n d r w h a v e b e e n d e f i n e d a s : _ I ‘ I ‘ m — C a l - 2 ° d Z / K m [ 7 3 ] r h = [ Z r d z / K h . [ 7 4 ] C I ‘ " : C r d z / K " [ 7 5 ] C I n t h e m o d e l , v a r i a b l e s R S H R a n d R S V R d e n o t e d r h a n d r w r e s p e c t i v e l y . 4 . 3 . 4 M i x i n g L e n g t h a n d F r i c t i o n S p e e d T h e t r a n s f e r c o e f f i c i e n t s K m , K n a n d K " h a v e b e e n d e f i n e d a s a f u n c t i o n o f a m i x i n g l e n g t h 1 m i x a n d a f r i c t i o n s p e e d u * i n l i t e r a t u r e . H e r e , 1 m i x a n d u * a r e t w o i m p o r t a n t p a r a m e t e r s w h i c h c h a r a c t e r i z e t h e m o m e n t u m e x c h a n g e i n t h e a i r . T h e m i x i n g l e n g t h 1 m i x i s r e l a t e d t o t h e f r e e p a t h l e n g t h o f a i r p a r c e l s , a n d t h e f r i c t i o n s p e e d u * c h a r a c t e r i z e s t h e s t r e n g t h o f m o m e n t u m t r a n s f e r . I n t h e l i t e r a t u r e , 1 m i x a n d u * a r e a s s u m e d t o b e : 1 , 1 , , = k ( z - d ) [ 7 6 ] u * = / T / p [ 7 7 ] w h e r e k = v o n K a r m a n c o n s t a n t , a b o u t 0 . 4 1 , d i m e n s i o n l e s s , a n d z - d a d j u s t e d h e i g h t f r o m t h e z e r o p l a n e d i s p l a c e m e n t , m . 5 6 4 . 3 . 5 A t m o s p h e r e S t a b i l i t y D u e t o t u r b u l e n c e , i r r e g u l a r t o p o g r a p h y 0 r b u o y a n c y , s o m e a i r p a r c e l s a r e p u s h e d u p w a r d s i n t h e a t m o s p h e r e , a s m e n t i o n e d a b o v e . I f t h e v i r t u a l t e m p e r a t u r e o f t h e a i r i s c o n s t a n t w i t h h e i g h t , t h e r e w o u l d b e n o h e a t e x c h a n g e b e t w e e n a n u p w a r d m o v i n g a i r p a r c e l a n d t h e s u r r o u n d i n g a i r . A l s o t h e d e n s i t y o f t h e a i r p a r c e l w o u l d b e t h e s a m e a s i t s s u r r o u n d i n g s , s o t h e a i r p a r c e l w o u l d c o n t i n u e t o m o v e u p w a r d s a t i t s i n i t i a l v e r t i c a l v e l o c i t y . T h i s i s c a l l e d a ' n e u t r a l ' c o n d i t i o n . T h e r e a r e t w o o t h e r c o n d i t i o n s i n t h e s u r f a c e l a y e r o f t h e a t m o s p h e r e . F i r s t , t h e v i r t u a l t e m p e r a t u r e o f t h e a i r i n c r e a s e s w i t h h e i g h t , s o a r i s i n g a i r p a r c e l w i l l b e c o o l e d b y i t s s u r r o u n d i n g s , a n d t h u s b e c o m e m o r e d e n s e . T h e u p w a r d v e l o c i t y o f t h e p a r c e l w i l l b e g r a d u a l l y r e d u c e d . T h i s i s c a l l e d a ' s t a b l e ' c o n d i t i o n , o r ' i n v e r s i o n ' . 0 n t h e o t h e r h a n d , w h e n t h e v i r t u a l t e m p e r a t u r e o f t h e a i r d e c r e a s e s , a n a i r p a r c e l w i l l b e w a r m e d b y i t s s u r r o u n d i n g a i r , a n d b e c o m e l i g h t e r . I n t h i s c a s e , v e r t i c a l m o v e m e n t o f t h e a i r p a r c e l w i l l b e a c c e l e r a t e d . T h i s i s a n ' u n s t a b l e ' c o n d i t i o n , o r ' l a p s e ' . u n s t a b l e c o n d i t i o n s o f t h e a t m o s p h e r e n e a r t h e g r o u n d o c c u r u s u a l l y f r o m a f t e r n o o n t o e a r l y i n t h e m o r n i n g w h e n t h e g r o u n d ( o r v e g e t a t i o n ) s u r f a c e i s m u c h w a r m e r t h a n t h e a i r a b o v e . I n v e r s i o n o c c u r s i n t h e m o r n i n g w h e n t h e a i r i s w a r m e d b y s o l a r r a d i a t i o n f i r s t w h i l e t h e v e g e t a t i o n s u r f a c e i s s t i l l c o o l . N e u t r a l c o n d i t i o n s e x i s t o n l y f o r v e r y s h o r t p e r i o d s d u r i n g t r a n s i t i o n . I n m i c r o c l i m a t o l o g y , t h e f o l l o w i n g a s s u m p t i o n s h a v e b e e n e s t a b l i s h e d b y M o n i n a n d O b u k h o v ( 1 9 5 4 ) : K m = 1 m 1 x u * / 0 m [ 7 8 ] l l m m i i x x u “ * * / / ¢ ¢ h w [ [ 7 8 9 0 ] ] 5 7 K h K w w h e r e 1 m i x = m i x i n g l e n g t h , m , u * = f r i c t i o n s p e e d , m / s , 0 m 2 s t a b i l i t y f a c t o r f o r m o m e n t u m t r a n s f e r , d i m e n s i o n l e s s , 0 h a s t a b i l i t y f a c t o r f o r s e n s i b l e h e a t t r a n s f e r , d i m e n s i o n l e s s , a n d 9 w : s t a b i l i t y f a c t o r f o r m o i s t u r e t r a n s f e r , d i m e n s i o n l e s s . T h o m ( 1 9 7 5 ) m e n t i o n e d t h a t t h e e f f e c t i v e m i x i n g l e n g t h 1 m i x / 9 m w a s s m a l l e r t h a n k ( z - d ) , i . e . 0 m > 1 , u n d e r s t a b l e c o n d i t i o n s ; a n d l a r g e r t h a n k ( z - d ) , i . e . ° m < 1 2 u n d e r u n s t a b l e c o n d i t i o n s . F r o m E q u a t i o n s [ 6 7 ] , [ 7 7 ] a n d [ 7 8 ] , t h e f o l l o w i n g w a s w r i t t e n : B u / a z : ¢ m u * / k ( z - d ) [ 8 1 ] I n t e g r a t i n g t h i s e q u a t i o n w i t h r e s p e c t t o h e i g h t g i v e s t h e w i n d p r o f i l e a b o v e a c a n o p y ( o r s w a t h ) . T h e s h a p e o f t h e w i n d p r o f i l e d e p e n d s u p o n t h e f o r m o f 0 m . L i k e w i s e , M a n / a z - ¢ t h / k ( z - d ) - ¢ w p : / k ( z - d ) a p p / 8 2 w h e r e a T P 4 0 w T h e r e f o r e , p r o f i l e s o f w i n d s p e e d , t e m p e r a t u r e a n d w a t e r v a p o r d e n s i t y H / p c p u * E / u * a r e a n a l o g o u s i n t h e a t m o s p h e r i c s u r f a c e l a y e r , a s m e n t i o n e d i n t h e l i t e r a t u r e . I n t h e s u r f a c e l a y e r n e a r t h e g r o u n d , H a n d E a r e o f t e n a s s u m e d t o b e c o n s t a n t . S o , t h e s u r f a c e l a y e r i s a l s o c a l l e d t h e ' c o n s t a n t f l u x l a y e r ' . A c c o r d i n g t o t h i s , H a n d E c a n b e e s t i m a t e d i f t e m p e r a t u r e a n d w a t e r v a p o r d e n s i t y a t t w o h e i g h t l e v e l s a r e k n o w n , e . g . a t t h e r e f e r e n c e h e i g h t z r a n d t h e s w a t h t o p 2 0 . 5 8 I n t e g r a t i o n o f E q u a t i o n [ 8 1 ] f r o m d + z o t o z r a n d f r o m 2 0 t o z r , a n d E q u a t i o n [ 7 4 ] f r o m z c t o 2 , g a v e : [ 1 * : k U P / I m [ 8 2 ] U 0 = u r I m c / I m [ 8 3 ] r h : I h / k l f ” = I h I m / k z u l m [ 8 4 ] w h e r e I m : [ Z r O m d z / ( z — d ) [ 8 5 ] d + z o I = c 0 d z / ( z - d ) [ 8 6 ] m c ( I i - 2 ° m I n = I : P 0 h d z / ( z - d ) [ 8 7 ] C I n f u n c t i o n s u b p r o g r a m R S H A B V , v a r i a b l e s U S T A R , U R , U C , I N T M , I N T H a n d I N T M C d e n o t e d u ’ , u p , u c , I m , I n a n d I m c , r e s p e c t i v e l y . S t a b i l i t y f a c t o r s 9 m : 6 h a n d 0 " a r e o f t e n e x p r e s s e d i n t e r m s o f a s t a b i l i t y p a r a m e t e r c o r R i c h a r d s o n n u m b e r R i : 8 m , h , w = 9 m , h , w ( C ) = 8 m , h , m ( R i ) T h e s t a b i l i t y p a r a m e t e r c i s d e f i n e d a s : c : ( z - d ) / L [ 8 8 ] w h e r e z - d : a d j u s t e d h e i g h t , m , a n d L = M o n i n - O b u k h o v s t a b i l i t y l e n g t h , m . L = z l - d = - p c p ( u * ) 3 T p / k g n [ 8 9 ] w h e r e z l v i r t u a l h e i g h t o f d i s t a n c e L a b o v e d , m , d = z e r o p l a n e d i s p l a c e m e n t , m , p c p = v o l u m e t r i c h e a t c a p a c i t y o f t h e a i r , J / m 3 ° K , u * = f r i c t i o n s p e e d , m / s , T p = m e a n v i r t u a l t e m p e r a t u r e , ° K , k = v o n K a r m a n c o n s t a n t , d i m e n s i o n l e s s , g : g r a v i t y a c c e l e r a t i o n , m / s z , S 9 a n d H = u p w a r d s e n s i b l e h e a t f l u x , W / m z . T h e M o n i n - O b u k h o v l e n g t h L i s a n a b s t r a c t e d d i s t a n c e a b o v e t h e z e r o p l a n e d i s p l a c e m e n t . A p o s i t i v e L ( o r c ) i n d i c a t e s a s t a b l e c o n d i t i o n 2 w h i l e a n e g a t i v e L ( o r c ) r e p r e s e n t s a n u n s t a b l e c o n d i t i o n . W h e n L i s i n f i n i t e , i . e . z e r o g , t h e a i r i s n e u t r a l . O n t h e o t h e r e n d , t h e s u r f a c e i s e x t r e m e l y s t a b l e a s L + 0 , i . e . c + a u T h e R i c h a r d s o n n u m b e r R i i s t h e r a t i o o f t h e b u o y a n c y a c c e l e r a t i o n g l . 1 x ( 3 T p / a z ) / T p t o t h e f r i c t i o n d e c e l e r a t i o n l m 1 x ( a u / a z ) 2 : R i = 3 ( a n / a z ) / T p ( a u / a z ) 2 [ 9 0 ] A n e g a t i v e , z e r o o r p o s i t i v e R 1 r e p r e s e n t s a n u n s t a b l e , n e u t r a l o r s t a b l e c o n d i t i o n o f t h e a i r , r e s p e c t i v e l y . T h e r e l a t i o n s h i p b e t w e e n R 1 a n d c i s : R 1 = r a h / 0 , 3 [ 9 1 ] T h e d i f f e r e n c e f o r m s o f L a n d R 1 a r e : L = T p ( u * ) 3 r h / g k A T p = T p i h ( u , . ) 2 / 3 A T P ( I , , ) 2 [ 9 2 ] R 1 = g A z h A T p / T p ( u , . . u c ) z = A c h I h / ( I m - I m c ) 2 [ 9 3 ] w h e r e A z h : z r - Z c A c h : A z h / L T p : ( T p r + T p c ) / 2 A T p = T p r - T p c W h e r e , s u b s c r i p t s r a n d c s t o o d f o r t h e r e f e r e n c e h e i g h t a n d t h e c a n o p y ( s w a t h ) t o p , r e s p e c t i v e l y . I n s u b m o d e l A B O V E , v a r i a b l e R I , L , D Z H , D T P , T P R , T P C , T P a n d K A R M A N r e p r e s e n t e d R i , L , A z h , A T P , T p r , T p c , T p a n d k , r e s p e c t i v e l y . 6 0 4 . 3 . 6 N e u t r a l C o n d i t i o n s A z e r o A T “ , R i , c o r a n i n f i n i t e L c h a r a c t e r i z e s n e u t r a l c o n d i t i o n s o f t h e a t m o s p h e r i c s u r f a c e l a y e r . T h e m o m e n t u m s t a b i l i t y f a c t o r m m i s c o m m o n l y a s s u m e d t o b e u n i t y ( o m z 1 ) u n d e r n e u t r a l c o n d i t i o n s . T h u s , K “ l = k ( z - d ) u * E q u a t i o n [ 8 1 ] b e c o m e s : a n = ( u * ) a z / k ( z - d ) [ 9 4 ] I n t e g r a t i n g f r o m d + z o t o z g i v e s : 4 ( 2 ) = ( u * / k ) 1 n [ ( z - d ) / z o ] [ 9 5 ] T h i s s h o w s t h e w i n d p r o f i l e a b o v e a c a n o p y u n d e r n e u t r a l c o n d i t i o n s i s l o g a r i t h m i c . W h e n w i n d s p e e d u p ( a t r e f e r e n c e h e i g h t z r ) i s k n o w n , t h e f r i c t i o n s p e e d u * a n d w i n d s p e e d a t a n y h e i g h t , e s p e c i a l l y t h e w i n d s p e e d a t t h e s w a t h t o p u c , c a n b e f o u n d : u * = k u r / B m [ 9 6 ] u c : p r a m / a m = u r ( 1 - B h / B m ) [ 9 7 ] w h e r e 3 m : l n [ ( z r - d ) / z o ] i . e . I m = B m B h = l n [ ( z , - d ) / ( z c - d ) ] i . e . I h = B h B m c = l n [ ( z c - d ) / z o ] i . e . I m c = B I n c I n t h e m o d e l , a n e u t r a l c o n d i t i o n w a s a s s u m e d i f t h e a b s o l u t e v a l u e o f A T P w a s l e s s o r e q u a l t o 0 . 0 0 1 . I n s u b r o u t i n e A B V I a n d R S H A B V , v a r i a b l e s B M , B H a n d B M C s t o o d f o r B m , B h a n d B m c , r e s p e c t i v e l y . I n t h e l i t e r a t u r e , N b a n d K H h a v e b e e n a s s u m e d t o b e e q u a l t o K m u n d e r n e u t r a l c o n d i t i o n s . H o w e v e r , r e c e n t r e s e a r c h h a s s h o w n s o m e d e v i a t i o n f r o m t h a t a s s u m p t i o n . B u s i n g e r e t a l . ( 1 9 7 1 ) f o u n d K h c o u l d o n 1 + 5 . 0 o n 1 + 5 . 0 D y e r ( 1 9 7 3 ) 6 1 b e 1 . 3 5 t i m e s a s l a r g e a s K m u n d e r f i e l d c o n d i t i o n s . I n t h i s s t u d y , i t w a s e x p r e s s e d a s : K h = K " = K m / f 1 . 8 . ¢ h : O w = f ¢ m w h e r e f = 0 . 7 4 ~ 1 . 0 F r o m t h i s , r h a n d r w w e r e f o u n d a s : r h = f r m : t h / k u * : t h B m / k z u r [ 9 8 ] 4 . 3 . 7 S t a b l e C o n d i t i o n s A p o s i t i v e A T P , R i , c o r L c h a r a c t e r i z e s i n v e r s i o n i n t h e a t m o s p h e r i c s u r f a c e l a y e r . I n t h e l i t e r a t u r e , m o s t e m p i r i c a l e q u a t i o n s o f O n a n d 9 h u n d e r s t a b l e c o n d i t i o n s h a v e t h e f o r m : 0 m : 1 + b ; 0 h : f + b ; w h e r e b ( 4 . 7 ~ 5 . 2 ) a n d f ( 0 . 4 7 ~ 1 . 0 ) a r e c o n s t a n t . F o r e x a m p l e , 9 m 1 + 4 . 7 o n 0 . 7 4 + 4 . 7 B u s i n g e r e t a l . ( 1 9 7 1 ) I t c a n b e s h o w n t h a t u n d e r s t a b l e c o n d i t i o n s : I , s a n + b A c I n = t h + b A c h I m c = B 1 1 1 c 4 ' P A C c H e n c e u * = k u p / ( B m + b A C ) [ 9 9 ] u ( z ) : ( u * / k ) { l n [ ( z - d ) / z o ] + b ( z - d - z o ) / L } [ 1 0 0 ] u 0 = u , [ 1 - ( B h + h A c h ) / ( B m + b A t ) ] [ 1 0 1 ] r h = ( t h + b A § h ) / k u * [ 1 0 2 ] 6 2 w h e r e A ; : A z / L A z : ( z r - d - z o ) A c e = A z c / L A z C = ( z c - d - z o ) A C h = A z h / L A z h : ( z r - z c ) / L E q u a t i o n [ 1 0 0 ] s h o w s t h a t u n d e r s t a b l e c o n d i t i o n s t h e w i n d p r o f i l e w i t h h e i g h t i s n o t s t r i c t l y l o g a r i t h m i c . I t h a s b e e n c a l l e d a ' l o g - l i n e a r ' p r o f i l e . H o w e v e r , G o u d r i a a n ( 1 9 7 7 ) i n d i c a t e d t h a t t h e d e v i a t i o n o f a l o g - l i n e a r p r o f i l e f r o m l o g a r i t h m i c s h a p e w a s v e r y s m a l l . T o d e t e r m i n e u * , L s h o u l d b e f o u n d . T h e d i f f e r e n c e f o r m o f L w a s r e w r i t t e n a s : L = [ T p ( u , . ) z / g A T p ] [ r B h + b A ; h ] / [ B m + b A t ] z [ 1 0 3 ] T h i s g a v e a q u a d r a t i c e q u a t i o n o f L : A L Z + B L + c = 0 [ 1 0 4 ] w h e r e A = ( 3 “ , ) 2 B = 2 m e A z - r B t h ( u , . ) 2 / g A T p c ( b A z ) 2 - p r A z h ( u , . ) 2 / g A T p T h e s o l u t i o n o f E q u a t i o n [ 1 0 4 ] d e p e n d e d u p o n : B 2 - 4 A C = [ T p ( u , . ) z / g A T p ] z + [ 4 m e T p ( u p ) Z / g A T p ] ( B m A z h - t h A z ) T h e f e l l o w i n g p o i n t s w e r e d r a w n f r o m E q u a t i o n [ 1 0 4 ] . ( 1 ) B z - 4 A C > 0 , s i n c e B m A z h > B h A z f o r z r > z c . ( 2 ) E q u a t i o n [ 1 0 4 ] h a d o n l y o n e p o s i t i v e s o l u t i o n : L = ( - B + / § 7 : E K C ) / 2 A i f a n d o n l y i f C < 0 , i . e . A T p < A T P o r = T p 1 u , ) 2 A z h / b g ( A z ) Z [ 1 0 5 ] w h e r e T p c r = c r i t i c a l v i r t u a l t e m p e r a t u r e d i f f e r e n c e , ° K . F o r e x a m p l e , i f z r : 0 . 6 m , 2 0 : 0 . 1 m , d : 0 . 0 7 4 m , 2 0 : 0 . 0 1 3 m , T p = 3 0 0 ° K , 6 3 u r = 1 . 0 m / s a n d b = 5 , t h e n A T p c r : 1 1 . 6 2 ° K . I n r e a l f i e l d s , t h i s r a r e l y o c c u r s . ( 3 ) L + 0 w h e n C + 0 , i . e . A T p + A T p c , . T h i s m e a n t c + a n P h y s i c a l l y , t h i s m e a n t t h e s u r f a c e l a y e r w a s e x t r e m e l y s t a b l e , i . e . f u l l i n v e r s i o n w a s e s t a b l i s h e d . E d d y m o t i o n w a s c o m p l e t e l y i n h i b i t e d , a n d c o n v e c t i o n c e a s e s . ( 4 ) R i a p p r o a c h e d a c r i t i c a l v a l u e R i c r a s L + 0 R i c , = 1 / b = 0 . 1 9 ~ 0 . 2 1 [ 1 0 6 ] T h i s w a s t h e d i r e c t r e s u l t o f E q u a t i o n s [ 9 3 ] . [ 1 0 3 ] a n d [ 1 0 5 ] . W e b b ( 1 9 7 0 ) , B u s i n g e r e t a l . ( 1 9 7 1 ) , O b u k h o v ( 1 9 7 1 ) , T h o m ( 1 9 7 5 ) a n d G o u d r i a a n ( 1 9 7 7 ) s h o w e d t h e s a m e c r i t i c a l R i v a l u e f r o m v a r i o u s a s p e c t s . T h e a n a l y s i s p r e s e n t e d b y t h e a u t h o r w a s a n o t h e r p r o o f a n d g i v e a n e w A T p c r w h i c h w a s e a s i e r t o e x a m i n e t h a n R i c r . O b u k h o v ( 1 9 7 1 ) s e t K h = 0 . 0 f o r R i > R i c r , w h i c h m e a n t r h + ¢ n G o u d r i a a n ( 1 9 7 7 ) t r e a t e d s u c h a c a s e 1 0 1 0 a s n o t r a n s f e r b y s e t t i n g r h t o b e e q u a l t o s / m . T h i s m o d e l a d a p t e d t h e s a m e p r o c e d u r e . 4 . 3 . 8 u n s t a b l e C o n d i t i o n s U n s t a b l e c o n d i t i o n s i n t h e a t m o s p h e r i c s u r f a c e l a y e r a r e r e p r e s e n t e d b y a n e g a t i v e c , R 1 o r L . U n d e r u n s t a b l e c o n d i t i o n s , m o s t s u g g e s t e d f o r m s o f 0 “ , 0 b a n d o n a r e n o n l i n e a r a n d d i f f i c u l t t o a n a l y t i c a l l y i n t e g r a t e : 9 = p H - q t ) r w h e r e p , q a n d r a r e c o n s t a n t . F o r i n s t a n c e , 0 . 7 4 ( 1 - 9 C ) " 1 / 2 B u s i n g e r e t a l . ( 1 9 7 1 ) c m ( 1 - 1 5 c ) “ / “ c h o m ( 1 — 1 6 g ) “ / ” 0 h ( 1 - 1 6 c ) ‘ ] / 2 D y e r ( 1 9 7 3 ) 6 4 A p r o c e d u r e b y G o u d r i a a n ( 1 9 7 7 ) w a s a p p l i e d i n t h e m o d e l t o g e t n u m e r i c a l s o l u t i o n s f o r i n t e g r a t i o n . S t a b i l i t y f a c t o r s w e r e a p p r o x i m a t e d w i t h a f o u r t h - o r d e r T a y l o r ' s s e r i e s : 0 = c , + C 2 ( z - d ) + c 3 ( z - d ) 2 + c u m - a ) 3 + 0 5 ( 2 — 0 ) “ [ 1 0 7 ] W h e r e C 1 ~ C 5 w e r e c o e f f i c i e n t s t o b e d e t e r m i n e d , a n d z - d w a s t h e a d j u s t e d h e i g h t . I f v a l u e s o f 0 a t f i v e d i f f e r e n t h e i g h t l e v e l s w e r e k n o w n , t h e n t h e s e c o e f f i c i e n t s c o u l d b e d e t e r m i n e d b y s o l v i n g a s y s t e m o f f i v e l i n e a r e q u a t i o n s . F o r t h i s p u r p o s e , t h e d i s t a n c e b e t w e e n 2 , a n d z c w a s d i v i d e d i n t o f o u r e v e n i n c r e m e n t s i n t h e m o d e l . F i v e a d j u s t e d h e i g h t s , i . e . z c - d , z , - d a n d t h e o t h e r s f r o m t h r e e d i v i s i o n p o i n t s , w e r e f o r m e d a n d u s e d t o d e t e r m i n e t h e c o e f f i c i e n t s o f t h e T a y l o r s e r i e s . U n d e r u n s t a b l e c o n d i t i o n s : I m = c m ] ; m + s z A z + C m 3 [ ( z r - d ) z - z o z ] / 2 + C m U [ ( z r ' d ) 3 ' z o a ] / 3 + C m 5 [ ( z p - d ) “ - z o “ ] / 4 [ 1 0 8 ] I n = c h 1 3 h + C h Z A z h + c h 3 [ ( z , - d ) ’ - ( z c - d ) z ] / 2 + C h 1 1 [ ( z p - d ) 3 - ( z c - d ) 3 ] / 3 + C h 5 [ ( z , ~ - d ) " - ( z c - d ) " ] / 4 [ 1 0 9 ] I I n c : C n 1 B n c + C m g A z c + C m 3 [ ( z c - d ) 2 - z o z ] / 2 + c m A [ ( z c - d ) 3 - z o ’ ] / 3 + c m 5 [ ( z c - d ) “ - z o “ ] / 4 [ 1 1 0 ] w h e r e C m 1 ~ C n 5 , a n d C h 1 ~ C h 5 w e r e c o e f f i c i e n t s i n t h e T a y l o r s e r i e s f o r e m a n d 0 h , r e s p e c t i v e l y . R e c a l l t h a t L w a s r e l a t e d t o I n a n d I m w h i c h w e r e d e p e n d e n t u p o n 9 h a n d 9 m : t h u s , c a n d L . L = T p ( u , . ) z i h / g A T p ( I m ) z [ 1 1 1 ] A n i t e r a t i o n t e c h n i q u e w a s u s e d i n t h e m o d e l . A v a l u e o f L ( - 0 . 0 0 1 ) w a s f i r s t a s s i g n e d a n d u s e d t o d e t e r m i n e t w o s e t s o f c o e f f i c i e n t s i n t h e T a y l o r s e r i e s o f 9 m a n d o n . T h e n , I m a n d I n w e r e c a l c u l a t e d . F r o m E q u a t i o n [ 1 1 1 ] , a n e w L w a s f o u n d . T h e p r o c e s s w a s r e p e a t e d u n t i l t h e 6 5 a b s o l u t e d i f f e r e n c e b e t w e e n t w o L v a l u e s r e a c h e d a p r e s e t c o n v e r g e n c e t o l e r a n c e ( 0 . 0 1 ) . I n t h e m o d e l , s u b r o u t i n e I T E R a n d I N V E X C c a r r i e d o u t t h e i t e r a t i o n a n d d e t e r m i n a t i o n o f t h e c o e f f i c i e n t s o f t h e T a y l o r ' s s e r i e s , r e s p e c t i v e l y . W h e n i n s t a b i l i t y w a s m o d e r a t e ( - 1 . 0 < R i < - 0 . 0 0 1 ) , r h w a s c a l c u l a t e d w i t h E q u a t i o n [ 8 4 ] . T h o m ( 1 9 7 5 ) i n d i c a t e d t h a t t h e w i n d p r o f i l e w a s v e r y c l o s e t o t h e l o g - l i n e a r s h a p e w h e n R i w a s n o t l e s s t h a n - 0 . 1 . U n d e r h e a v i l y u n s t a b l e c o n d i t i o n s ( R i g - 1 . 0 ) , h o w e v e r , T h o m ( 1 9 7 5 ) a n d G o u d r i a a n ( 1 9 7 7 ) p o i n t e d o u t t h a t b u o y a n c y f o r c e s r a t h e r t h a n f r i c t i o n f o r c e s d o m i n a t e d s e n s i b l e h e a t t r a n s f e r , w h i c h b e c a m e f r e e c o n v e c t i o n i n s t e a d o f f o r c e d c o n v e c t i o n . I n t h i s c a s e , r h w a s c a l c u l a t e d b a s e d o n a p r o c e d u r e b y H o l m a n ( 1 9 8 1 ) , s e e S e c t i o n 4 . 5 a n d f u n c t i o n s u b p r o g r a m H C O N V f o r d e t a i l . T a b l e 3 l i s t s s o m e a e r o d y n a m i c p a r a m e t e r s e s t i m a t e d i n t h e m o d e l . T h e d i s c u s s i o n a b o v e d e a l t w i t h n o n z e r o w i n d s p e e d a t 2 , . W h e n u r w a s z e r o , f o r c e d c o n v e c t i o n c e a s e s . I f A T p = 0 , t h e s u r f a c e l a y e r w a s n e u t r a l w i t h n o h e a t t r a n s f e r a t a l l . T h i s w a s r e p r e s e n t e d b y a v e r y l a r g e r h i n t h e m o d e l a s w e l l a s a z e r o A T P . I f A T p > 0 , t h e s u r f a c e l a y e r b e c a m e e x t r e m e l y s t a b l e , i . e . h e a t t r a n s f e r w a s s u p p r e s s e d , w h i c h w a s a l s o r e p r e s e n t e d b y a v e r y l a r g e r h . I f A T p < 0 , t h e s u r f a c e l a y e r w a s h i g h l y u n s t a b l e s i n c e L + 0 a n d R i + - m , w h i c h l e d t o f r e e c o n v e c t i o n a s R i < - 1 . 0 . 4 . 3 . 9 R e s i s t a n c e t o M o i s t u r e T r a n s f e r R e s i s t a n c e t o m o i s t u r e t r a n s f e r i n t h e a t m o s p h e r i c s u r f a c e l a y e r d e p e n d s h e a v i l y u p o n t h e t u r b u l e n c e s t a t u s i n t h e l a y e r . T h e r e s i s t a n c e 6 6 T a b l e 3 A e r o d y n a m i c p a r a m e t e r s i n t h e a t m o s p h e r i c s u r f a c e l a y e r w i t h z p = 0 . 6 m , z c = 0 . 1 m , T p = 3 0 0 ° K a n d u r : 1 . 0 m / s R i L t u * u r ( m ) ( K ) ( m / s ) ( m / s ) ( s / m ) - 1 - 0 . 1 7 5 - 3 5 2 3 0 . 1 8 8 0 . 2 4 1 1 2 4 8 - 0 . 1 - 1 . 6 9 3 ' 3 7 8 4 0 . 1 3 2 0 . 2 1 3 3 7 . 5 2 - 0 . 0 1 - 1 6 . 1 7 - 0 3 9 9 0 . 1 1 5 0 1 9 3 5 9 4 7 - 0 . 0 0 1 - 1 6 7 0 - 0 0 4 0 0 . 1 1 2 0 . 1 8 8 6 4 . 9 4 - 0 0 0 0 1 - 1 6 7 0 - 0 0 0 4 0 1 1 1 0 . 1 8 8 6 5 6 0 0 0 0 . 1 1 1 0 . 1 8 8 6 5 . 7 3 0 . 0 0 0 1 1 6 6 9 . 0 . 0 0 4 0 . 1 1 1 0 . 1 8 8 6 5 . 7 4 0 . 0 0 1 1 6 6 . 2 0 . 0 4 0 0 . 1 1 1 0 . 1 8 7 6 6 . 7 2 0 . 0 1 1 5 . 8 6 0 . 4 1 0 0 . 1 0 7 0 . 1 8 1 7 2 . 1 6 0 . 1 0 . 8 3 5 4 . 7 9 9 0 . 0 6 1 0 . 1 1 4 2 4 0 . 8 0 . 1 5 0 . 2 7 8 7 . 9 0 0 0 . 0 3 2 0 . 0 7 2 9 1 9 . 5 0 . 1 7 5 0 . 1 1 9 9 . 6 7 1 0 . 0 1 6 0 . 0 4 9 3 5 9 0 . 0 . 2 0 1 1 . 6 2 0 0 w a s c o m m o n l y a s s u m e d t o b e t h e s a m e a s t h e r e s i s t a n c e t o h e a t t r a n s f e r . T h i s w a s a p p r o p r i a t e w h e n b o t h A T p a n d A p v h a d t h e s a m e s i g n , i . e . b o t h s e n s i b l e a n d m o i s t u r e f l u x e s w e r e i n t h e s a m e d i r e c t i o n . H o w e v e r , t w o s p e c i a l c a s e s n e e d e d t o b e f u r t h e r d i s c u s s e d . F i r s t , A T p > O b u t A p v < 0 , i , e . a d o w n w a r d s e n s i b l e h e a t f l u x b u t a u p w a r d m o i s t u r e f l u x . I n t h i s c a s e , t h e s u r f a c e l a y e r w a s t h e r m a l l y s t a b l e a n d f o r c e d c o n v e c t i o n w a s r e d u c e d . S i n c e u p w a r d - m o v e m e n t o f a i r p a r c e l s w a s l i m i t e d , u p w a r d m o i s t u r e t r a n s f e r w a s a l s o r e d u c e d . T h u s , t h e e q u a l i t y o f b o t h r h a n d r w w a s r e a s o n a b l e i n t h i s c a s e . S e c o n d l y , A T p < 0 b u t A p v > 0 , i . e . a n u p w a r d s e n s i b l e h e a t f l u x w i t h a d o w n w a r d m o i s t u r e f l u x . T h e s u r f a c e l a y e r w a s t h e r m a l l y u n s t a b l e i n t h i s c a s e . F o r c e d c o n v e c t i o n w a s e n h a n c e d d u e t o b u o y a n c y e f f e c t s o f r i s i n g a i r p a r c e l s . T h e r e s i s t a n c e t o s e n s i b l e h e a t t r a n s f e r b e c a m e s m a l l e r . H o w e v e r , d o w n w a r d m o i s t u r e t r a n s f e r i n t h i s c a s e s h o u l d b e c o m p e n s a t e d r a t h e r t h a n e n h a n c e d b e c a u s e a i r p a r c e l s w i t h 6 7 l o w e r w a t e r v a p o r d e n s i t i e s i n s i d e w o u l d p i c k u p s o m e m o i s t u r e f r o m t h e s u r r o u n d i n g s o f t h e i r u p w a r d p a t h . I n t h i s c a s e , s e t t i n g r w v e q u a l t o r h w a s o b v i o u s l y u n r e a s o n a b l e . S i n c e t h e r e w a s a l a c k o f i n f o r m a t i o n a b o u t r e a l i s t i c t r e a t m e n t s o n t h i s s p e c i a l c a s e , r w w a s a s s u m e d t o e q u a l a v i r t u a l r h a s i f t h e r e e x i s t e d - A T p ( > 0 ) i n t h e s u r f a c e l a y e r . A p o s i t i v e v a l u e o f - A T p w o u l d i n d u c e a r e l a t i v e l y l a r g e r e s i s t a n c e . T o d e t e r m i n e t h e s t a b i l i t y o f t h e a t m o s p h e r i c s u r f a c e l a y e r , t e m p e r a t u r e a t t h e t o p o f a s w a t h ( c a n o p y ) s h o u l d b e k n o w n i n a d v a n c e . H o w e v e r , t h e s w a t h t o p t e m p e r a t u r e w a s o n e o f t h e s t a t e v a r i a b l e s t o b e d e t e r m i n e d i n t h e m o d e l . I t e r a t i o n t e c h n i q u e s o r o t h e r m e t h o d s s h o u l d b e a p p l i e d . I n t h i s m o d e l , t h e t h e r m a l s t a b i l i t y a n d r e s i s t a n c e a b o v e t h e s w a t h w e r e e s t i m a t e d f r o m t h e t e m p e r a t u r e a n d w a t e r v a p o r d e n s i t y a t e a c h t i m e s t e p a n d u s e d t o d e t e r m i n e t e m p e r a t u r e a n d w a t e r v a p o r d e n s i t y a t t h e n e x t t i m e s t e p . 4 . 4 E x c h a n g e s i n s i d e t h e S w a t h I n s i d e a s w a t h ( o r c a n o p y ) , e x c h a n g e s o f m o m e n t u m , s e n s i b l e h e a t a n d m o i s t u r e a r e a l s o a f f e c t e d b y w i n d s p e e d i n a i r s p a c e s . S t r e n g t h s o f t h e s e e x c h a n g e s a r e s m a l l e r t h a n t h o s e a b o v e t h e s w a t h ( o r c a n o p y ) d u e t o r e d u c e d w i n d s p e e d . G o u d r i a a n ( 1 9 7 7 ) p o i n t e d o u t t h a t t h e r e h a d n o t b e e n a w e l l - v e r i f i e d t h e o r y f o r e x c h a n g e s i n s i d e c a n o p i e s , l i k e t h e s i m i l a r i t y t h e o r y a b o v e c a n o p i e s . H o w e v e r , s o m e w o r k i n g h y p o t h e s e s f o r e x c h a n g e s a b o v e c a n o p i e s c o u l d b e a p p r o x i m a t e l y u s e d . S u b m o d e l I N S I D E w i t h a s s o c i a t e d s u b p r o g r a m s R S H I N S , P H I H M a n d H C O N V s i m u l a t e d t h e w i n d p r o f i l e a n d c a l c u l a t e d c o r r e s p o n d i n g r e s i s t a n c e s t o s e n s i b l e h e a t a n d m o i s t u r e t r a n s f e r w i t h i n t h e s w a t h . 6 8 4 . 4 . 1 N e u t r a l C o n d i t i o n s T h e h y p o t h e s i s o f K m b e i n g t h e p r o d u c t o f 1 m i x a n d u * h a s b e e n a d a p t e d b y m a n y r e s e a r c h e r s , e . g . T h o m ( 1 9 7 5 ) a n d G o u d r i a a n ( 1 9 7 7 ) . I n t h i s m o d e l , t h e p r o c e d u r e b y G o u d r i a a n ( 1 9 7 7 ) w a s e m p l o y e d w i t h s o m e m o d i f i c a t i o n . D i f f e r i n g f r o m t h e a i r a b o v e c a n o p i e s , t h e t r a v e l o f t h e a i r i n s i d e c a n o p i e s w a s r e s t r i c t e d b y l e a v e s a n d s t e m s . T h e m i x i n g l e n g t h 1 m i x i n s i d e c a n o p i e s w a s r e l a t e d t o t h e m e a n f r e e s p a c e b e t w e e n l e a v e s . G o u d r i a a n ( 1 9 7 7 ) p r o p o s e d a n e q u a t i o n t o c a l c u l a t e 1 m i x f o r n a r r o w a n d l o n g l e a v e s : 1 “ , , = / 4 § 7 ? E ; [ 1 1 2 ] w h e r e 1 m i x = m i x i n g l e n g t h , m , w = m e a n w i d t h o f l e a v e s , m , a n d L a = v o l u m e t r i c d e n s i t y o f l e a f a r e a , m 2 l e a f / m 3 . T h e v o l u m e t r i c d e n s i t y o f l e a f a r e a f o r u n i f o r m l y d i s t r i b u t e d l e a v e s i s : L a : A L / A z = L A I / z c [ 1 1 3 ] w h e r e A 2 = e l e m e n t o f h e i g h t , m , a n d A L e l e m e n t o f t h e l e a f a r e a i n d e x i n 2 . W h e n L A I = 3 . 0 , z c = 0 . 1 m a n d w = 0 . 0 0 5 m , 1 m i x = ° - 0 0 9 m . T h e m e a n e d d y v e l o c i t y i n s i d e c a n o p i e s w a s a s s u m e d t o b e p r o p o r t i o n a l t o l o c a l w i n d s p e e d : u * = i w u [ 1 1 4 ] w h e r e i " = r e l a t i v e t u r b u l e n c e i n t e n s i t y , d i m e n s i o n l e s s , l o c a l w i n d s p e e d , m / s . a n d u T h e r e l a t i v e t u r b u l e n c e i n t e n s i t y i w r a n g e d f r o m 0 . 3 a t t h e t o p t o 0 . 8 6 9 a t t h e s o i l s u r f a c e f o r a c o r n c a n o p y ( G o u d r i a a n , 1 9 7 7 ) . I n t h i s m o d e l , i w w a s a s s u m e d a s 0 . 5 5 i n v i e w o f a t h i n s w a t h . V a r i a b l e s W I a n d X I M L d e n o t e d i w a n d 1 m i x i n s u b m o d e l I N S I D E . M o m e n t u m l o s s i n s i d e s w a t h s ( c a n o p i e s ) w a s d u e t o d r a g f o r c e s b e t w e e n l e a v e s a n d t h e a i r f l o w . S h e a r s t r e s s i n s i d e a c a n o p y w a s o f t e n c h a r a c t e r i z e d w i t h : T = K m a u / B z [ 1 1 5 ] 8 1 / 8 2 = p C d e u z [ 1 1 6 ] w h e r e T = s h e a r s t r e s s , N / m z , p = a i r d e n s i t y , k g / m ’ , d r a g c o e f f i c i e n t , d i m e n s i o n l e s s , O a . 1 1 K m = t r a n s f e r c o e f f i c i e n t , m z / s , a n d v o l u m e t r i c d e n s i t y o f l e a f a r e a , m 2 l e a f / m 3 . i " a . 1 1 C d w a s a b u l k d r a g c o e f f i c i e n t o v e r a n e l e m e n t t h i c k n e s s o f p l a n t s ( l e a v e s a n d s t e m s ) , w h i c h w a s d i f f e r e n t f r o m a d r a g c o e f f i c i e n t o f a n i n d i v i d u a l l e a f ( o r s t e m ) . T h o m ( 1 9 6 8 ) f o u n d t h e l a t t e r w a s a f u n c t i o n o f w i n d s p e e d . H o w e v e r , C d w a s a s s u m e d a s c o n s t a n t b y C o w a n ( 1 9 6 8 ) a n d G o u d r i a a n ( 1 9 7 7 ) . T h o m ' s d a t a ( 1 9 6 8 ) s h o w e d t h a t w h e n w i n d s p e e d w a s b e l o w 0 . 2 m / s t h e d r a g c o e f f i c i e n t o f i n d i v i d u a l l e a v e s r o s e r a p i d l y . S i m i l a r t e n d e n c y w a s f o u n d b y u s e o f E q u a t i o n [ 1 1 6 ] t o c a l c u l a t e t h e b u l k d r a g c o e f f i c i e n t o f c a n o p i e s w i t h w i n d d a t a a t t h e t o p o f c a n o p i e s . I n t h i s m o d e l , a v a l u e o f 0 . 5 w a s u s e d . F r o m E q u a t i o n s [ 1 1 5 ] a n d [ 1 1 6 ] , a s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n w a s f o r m e d : u ( a z u / a z z ) + ( a c / 3 2 ) 2 - 2 a 2 u 2 = 0 [ 1 1 7 ] w h e r e a e x t i n c t i o n c o e f f i c i e n t , 1 / m . a 2 C d e / Z i w l m i x [ 1 1 8 ] T h e r a n g e o f a z c h a s b e e n r e p o r t e d f r o m 2 t o 4 . F o r i n s t a n c e , i f 7 0 L A I = 3 . 0 , 2 0 : 0 . 1 m , C d = 0 . 5 , i w = 0 . 5 , t h e n a z c = 4 . 0 3 . T h e s o l u t i o n o f E q u a t i o n [ 1 1 7 ] , w i n d s p e e d u a t h e i g h t z , w a s : 0 = u z e ' a ( z 2 ' 2 ) [ 1 1 9 ] w h e r e 2 2 = h e i g h t o f t h e t o p o f a l a y e r , m , a n d u 2 : w i n d s p e e d a t 2 2 . T h i s s h o w e d t h a t t h e w i n d p r o f i l e w a s e x p o n e n t i a l l y d e c a y i n g i n s i d e a c a n o p y ( s w a t h ) . T h o m ( 1 9 7 5 ) i n d i c a t e d t h a t r e a l i s t i c c a n o p y w i n d p r o f i l e s c o u l d b e g e n e r a t e d o n t h e b a s i s o f a n y r e a s o n a b l e a s s u m p t i o n a b o u t K m . F o r e x a m p l e , a n a s s u m p t i o n o f c o n s t a n t K . b y T h o m ( 1 9 7 5 ) l e d t o a w i n d p r o f i l e v e r y c l o s e t o e x p o n e n t i a l . I t s h o u l d b e p o i n t e d o u t t h a t t h e e x p o n e n t i a l p r o f i l e d i d n o t l e a d t o z e r o w i n d s p e e d a t t h e s o i l s u r f a c e a s e x p e c t e d . H o w e v e r , e s t i m a t e d w i n d s p e e d w i t h E q u a t i o n [ 1 1 9 ] a t t h e g r o u n d l e v e l w a s v e r y s m a l l , a n d t h e e r r o r i n t r o d u c e d w a s i m m a t e r i a l . 4 . 4 . 2 u n n e u t r a l C o n d i t i o n s B e c a u s e o f l a c k o f i n f o r m a t i o n , m a n y r e s e a r c h e r s h a v e i g n o r e d d i s c u s s i o n o n t h e t h e r m a l s t a b i l i t y i n s i d e c a n o p i e s . G o u d r i a a n ( 1 9 7 7 ) a s s u m e d u n d e r u n n e u t r a l c o n d i t i o n s K m a n d K h t o b e a s : K m 1 m i x i n / ¢ m [ 1 2 0 ] K b l m i x i w u / 9 h [ 1 2 1 ] W h e r e 9 m a n d 9 h w e r e t h e s a m e a s t h o s e e x p r e s s i o n s a b o v e c a n o p i e s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . T h e d i f f e r e n t i a l e q u a t i o n b e c a m e : ( 3 / 3 2 ) [ ( u / ¢ m ) 8 u / a z ] - 2 a 2 u z = 0 [ 1 2 2 ] T h e s o l u t i o n w o u l d b e v e r y c o m p l i c a t e d b e c a u s e 0 m w a s a l s o a f u n c t i o n o f 2 . T o s i m p l i f y t h e s o l u t i o n , G o u d r i a a n ( 1 9 7 7 ) a s s u m e d a n a v e r a g e 7 1 v a l u e o f 9 m o v e r a h e i g h t r a n g e , a n d h e n c e o b t a i n e d a n a p p r o x i m a t e s o l u t i o n , l i k e E q u a t i o n [ 1 1 9 ] . T h e e x t i n c t i o n c o e f f i c i e n t b e c a m e : a 2 = C d e O m a v / 2 i w l m 1 x [ 1 2 3 ] w h e r e ¢ m a v = a v e r a g e 0 m . G o u d r i a a n ( 1 9 7 7 ) a l s o r e d e f i n e d t h e R i c h a r d s o n n u m b e r : 1 1 1 = [ g ( a n / a z ) / T p ] ( l m x / u a v ) z [ 1 2 4 ] w h e r e “ a v = m e a n w i n d s p e e d o v e r a h e i g h t r a n g e , m / s . T h i s w a s b a s e d u p o n t h e f a c t t h a t t h e f r i c t i o n d e c e l e r a t i o n i n s i d e c a n 0 p i e s w a s d u e t o c o l l i s i o n s w i t h p l a n t e l e m e n t s r a t h e r t h a n v i s c o u s e f f o r t s . R e a r r a n g i n g E q u a t i o n [ 1 2 4 ] i n a d i f f e r e n t f a r m g a v e : R 1 = [ g A T p A z / T p ( u a v ) 2 ] ( 1 m 1 x / A z ) [ 1 2 5 ] N o t e t h a t t h e t e r m i n t h e b r a c k e t s w a s s i m i l a r t o t h e d i f f e r e n c e f o r m o f R 1 a b o v e c a n o p i e s , s e e E q u a t i o n [ 9 3 ] . I f t h i s t e r m w a s c o m p a r a b l e w i t h R 1 a b o v e c a n o p i e s , t h e r a t i o o f R 1 i n s i d e c a n o p i e s t o R 1 a b o v e c a n o p i e s w o u l d b e ( l m i x / A z ) 2 , w h i c h w a s n o r m a l l y s m a l l e r t h a n u n i t y . S i n c e w i n d s p e e d “ a v i n s i d e a s w a t h w a s v e r y v e r y s m a l l , a s m a l l v a l u e o f A T P w o u l d m a k e R i i n s i d e a s w a t h c o m p a r a b l e t o t h a t a b o v e t h e s w a t h . T h i s i m p l i e d t h a t i n s i d e t h e s w a t h , t h e r m a l s t a b i l i t y p l a y e d a n i m p o r t a n t r o l e o n e x c h a n g e s o f s e n s i b l e h e a t a n d m o i s t u r e . I n t h i s m o d e l , t h e r m a l s t a b i l i t y w a s e s t i m a t e d b e t w e e n e a c h l a y e r o f t h e s w a t h . T h i s w a s d u e t o t h e f a c t t h a t t e m p e r a t u r e a t t h e c e n t e r o f a s w a t h m i g h t b e h i g h e r o r l o w e r t h a n t h o s e a t t h e t o p a n d b o t t o m o f t h e s w a t h a t s o m e t i m e d u r i n g t h e d r y i n g p r o c e s s . T h i s k i n d o f v e r t i c a l t e m p e r a t u r e p r o f i l e h a s b e e n r e p o r t e d i n m a n y c a s e s f o r c r o p c a n o p i e s . T h u s , d i f f e r e n t p a t t e r n s o f t h e r m a l s t a b i l i t y s h o u l d b e d i s t i n g u i s h e d a t d i f f e r e n t l e v e l s o f a c a n o p y a t a g i v e n t i m e . T h e m e a n w i n d s p e e d o v e r a h e i g h t r a n g e f r o m z 1 t o 2 2 w a s 7 2 c a l c u l a t e d a s : “ a v = ( u g / a A z ) [ 1 - e a A z ] [ 1 2 6 ] w h e r e A z : 2 2 - 2 1 T h e r e s i s t a n c e t o s e n s i b l e h e a t t r a n s f e r r h u n d e r m o d e r a t e u n n e u t r a l c o n d i t i o n s w a s c a l c u l a t e d a s : r h = P h a v A Z / l m i x i w u a v [ 1 2 7 ] F o r f u l l i n v e r s i o n o r f r e e c o n v e c t i o n , t h e c r i t e r i a u s e d w e r e t h e s a m e a s t h o s e a b o v e t h e s w a t h . I n t h i s m o d e l , e m a v a n d o h a v w e r e e s t i m a t e d i n s u b r o u t i n e P H I H M b a s e d u p o n t h e r e l a t i o n s h i p g i v e n i n E q u a t i o n [ 9 1 ] . T h e r e s i s t a n c e t o m o i s t u r e t r a n s f e r r " w a s t r e a t e d b a s e d u p o n t h e s i g n s o f A T P a n d A p v i n t h e s a m e w a y a s t h a t a b o v e t h e s w a t h a f o r e m e n t i o n e d . I n t h e m o d e l , v a r i a b l e s R S H a n d R S V s t o r e d d a t a o f r h a n d r " , r e s p e c t i v e l y . F i g u r e 5 s h o w s a s i m u l a t e d w i n d p r o f i l e w i t h i n a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t o f 1 9 8 4 . 4 . 4 . 3 R e s i s t a n c e a t t h e S o i l S u r f a c e T h e r e s i s t a n c e t o s e n s i b l e h e a t t r a n s f e r b e t w e e n t h e s o i l s u r f a c e a n d t h e b o t t o m l a y e r o f t h e s w a t h c o n s i s t e d o f t w o c o m p o n e n t s . O n e r e s i s t a n c e c o m p o n e n t ( R G H ) w a s t h e a e r o d y n a m i c ' r e s i s t a n c e i n t h e b o t t o m l a y e r w h i c h w a s d e t e r m i n e d b y t h e t h e r m a l s t a b i l i t y o f t h e b o t t o m l a y e r . T h e o t h e r c o m p o n e n t ( R B S ) w a s a r e s i s t a n c e n e a r t h e s o i l s u r f a c e w h i c h w a s a f f e c t e d b y t h e b o u n d a r y l a y e r n e a r t h e s u r f a c e . I n t h e s o i l b o u n d a r y l a y e r , a i r f l o w w a s l a m i n a r a n d t h e c h a r a c t e r i s t i c d i m e n s i o n , z t e W H r n m t W o ? fl ° fl 7 fl i d H b - ] ] - - . - 1 0 0 8 0 4 0 4 . . . . . . . 2 0 0 0 0 0 0 , g n i s n a L t s a E n i h t a w s a n . i 4 h 8 t 9 i w 1 , 6 e l 1 - i 4 f o 1 r p t s d u n g i u w A d n e o t a n l a u g m i i h s c i A M . 5 e r u g i F ( e / u i ) p e e d s p u g M . 0 0 " 4 0 ° " 0 . 1 4 0 . 0 W 1 2 : 0 0 0 : 1 2 : 0 1 8 : 0 : 0 V 2 1 2 0 1 8 : 0 8 1 4 - 8 1 5 1 5 / / T l m e ( H o u r a n d D a t e ] 0 7 3 7 4 w a s t h e a v e r a g e s o i l c l o d s i z e . A v a l u e o f 0 . 0 5 m w a s a s s u m e d i n t h e m o d e l . T h e r e s i s t a n c e t o s e n s i b l e h e a t t r a n s f e r n e a r t h e s o i l s u r f a c e w a s e s t i m a t e d i n H C O N V , w h i c h w i l l b e d i s c u s s e d i n d e p t h i n t h e n e x t s e c t i o n . T h e r e s i s t a n c e t o m o i s t u r e t r a n s f e r ( R S V G ) h a d t h r e e c o m p o n e n t s ( R V G , R V S a n d R 5 6 ) . T w o o f w h i c h c o r r e s p o n d e d t o t h e t w o c o m p o n e n t s o f r e s i s t a n c e t o s e n s i b l e h e a t t r a n s f e r . T h e t h i r d c o m p o n e n t ( R S G ) r e p r e s e n t s t h e r e s i s t a n c e t o m o i s t u r e r e m o v a l f r o m t h e s o i l j u s t b e l o w t h e s o i l s u r f a c e . G o u d r i a a n ( 1 9 7 7 ) a s s u m e d a z e r o v a l u e f o r t h i s c o m p o n e n t i f t h e s o i l w a s w e t , o r 1 0 7 s / m f o r d r y s o i l . T h o m p s o n ( 1 9 8 1 ) u s e d a v a l u e o f 1 0 0 s / m f o r w e t s o i l a n d 1 0 0 0 s / m f o r d r y s o i l . I n t h i s m o d e l , t h e v a r i a b l e R S G w a s s e t t o 1 0 s / m f o r a t y p i c a l s o i l s u r f a c e i n E a s t L a n s i n g , M i c h i g a n ; a n d 1 0 0 0 s / m f o r t h e s o i l i n F r e s n o , C a l i f o r n i a . T h e r e s i s t a n c e s a t t h e s o i l s u r f a c e w e r e e s t i m a t e d i n f u n c t i o n s u b p r o g r a m R E S G N D , a c o m p o n e n t o f t h e s o i l s u b m o d e l w h i c h w i l l b e d i s c u s s e d i n S e c t i o n 4 . 7 . 4 . 5 E x c h a n g e s o n L e a f S u r f a c e s L e a f s u r f a c e s p l a y a m a j o r r o l e i n c o n n e c t i n g p l a n t s t o t h e i r e n v i r o n m e n t . T h i s i s e s p e c i a l l y i m p o r t a n t f o r c u t p l a n t s b e c a u s e t h e i r c o n n e c t i o n w i t h s o i l t h r o u g h r o o t s i s b r o k e n . E x c h a n g e s o f e n e r g y a n d m a s s b e t w e e n p l a n t s a n d t h e s u r r o u n d i n g a i r t a k e p l a c e o n s u r f a c e s o f l e a v e s a n d s t e m s . E x c h a n g e s o f w a t e r v a p o r a n d o t h e r g a s e s a r e n o r m a l l y t h r o u g h s t o m a t a . W a x a n d c u t i c l e s o n p l a n t s u r f a c e s p r e v e n t u n n e c e s s a r y e x c h a n g e s o f v a p o r a n d g a s e s . M o s t s t o m a t a a r e l o c a t e d o n l e a f s u r f a c e s s i n c e l e a v e s h a v e l a r g e r s u r f a c e a r e a a n d h i g h e r s t o m a t a d e n s i t y t h a n 7 5 s t e m s . L e a v e s a r e a l s o l e s s w a x y a n d c o v e r e d b y t h i n n e r c u t i c l e s . T h u s , t h e a m o u n t s o f e n e r g y a n d m a s s e x c h a n g e d o n l e a v e s a r e t h e d o m i n a n t c o m p o n e n t s o f t o t a l e x c h a n g e s o f a p l a n t . T h e r e f o r e , t h e s t a t u s o f t e m p e r a t u r e a n d t h e m o i s t u r e c o n t e n t o f p l a n t s c a n b e a p p r o x i m a t e l y r e p r e s e n t e d b y t h a t o f l e a v e s , r e s p e c t i v e l y . S u b m o d e l D B L E A F a n d a s s o c i a t e d s u b p r o g r a m s R E S I S T , H C O N V , R B V M C a n d D F M C L s i m u l a t e d t h e e x c h a n g e s o f s e n s i b l e h e a t , l a t e n t h e a t a n d m o i s t u r e o n l e a f s u r f a c e s b a s e d u p o n p h y s i c a l l a w s o f t r a n s f e r a n d c o n s e r v a t i o n o f e n e r g y a n d m a s s . 4 . 5 . 1 S e n s i b l e H e a t T r a n s f e r o n L e a f S u r f a c e s S e n s i b l e h e a t t r a n s f e r b e t w e e n o n e s i d e o f a n i n d i v i d u a l l e a f a n d i t s s u r r o u n d i n g a i r h a s b e e n c l a s s i c a l l y t r e a t e d a s c o n v e c t i v e h e a t t r a n s f e r o f a t h i n f l a t p l a t e i n a n a i r f l o w . F o r c o n v e c t i v e h e a t t r a n s f e r , t h e h e a t f l u x w a s e x p r e s s e d a s : Q = h ( T 1 - T 1 ) [ 1 2 8 ] w h e r e Q = h e a t f l u x , W / m z T 1 l e a f s u r f a c e t e m p e r a t u r e , ° C , T a a i r t e m p e r a t u r e , ° C , a n d h = c o n v e c t i o n h e a t - t r a n s f e r c o e f f i c i e n t , W / m 2 ° C L T h e r e s i s t a n c e t o h e a t t r a n s f e r w a s : P h 3 p e p / h w h e r e r h = r e s i s t a n c e t o h e a t t r a n s f e r , s / m , a n d p o p = v o l u m e t r i c h e a t c a p a c i t y o f t h e f l u i d , J / m 3 C . T h e v o l u m e t r i c h e a t c a p a c i t y o f t h e a i r w a s a b o u t 1 2 4 0 J / m 3 ° C a n d v a r i e d w i t h t e m p e r a t u r e a n d w a t e r v a p o r c o n t e n t o f t h e a i r . 7 6 I n t h e m o d e l , D B A , D B L , H L A , R B H a n d F X H L A d e n o t e d T a , T 1 , h , r h a n d 0 , r e s p e c t i v e l y . T h e c a l c u l a t i o n o f H L A a n d R B H w e r e c a r r i e d o u t i n H C O N V a n d R E S I S T , r e s p e c t i v e l y ; a n d F X H L A i n D B L E A F a n d R A T E . C o n v e c t i v e h e a t t r a n s f e r i s r e l a t e d t o f l o w c h a r a c t e r i s t i c s a n d t h e r m a l p r o p e r t i e s o f t h e f l u i d . W h e n t h e a i r f l o w s o v e r a f l a t p l a t e , a b o u n d a r y l a y e r i s d e v e l o p e d i n t h e - f l o w n e a r t h e p l a t e s u r f a c e . I n t h e b o u n d a r y l a y e r , t h e i n f l u e n c e o f v i s c o u s f o r c e s i s s i g n i f i c a n t . T h e f l o w i n t h e b o u n d a r y l a y e r c a n b e l a m i n a r o r t u r b u l e n t , d e t e r m i n e d b y t h e R e y n o l d s n u m b e r R e u X / v [ 1 2 9 ] w h e r e u = f r e e - s t e a m v e l o c i t y , m / s , x = c h a r a c t e r i s t i c d i m e n s i o n o f t h e p l a t e , m , a n d v = k i n e m a t i c v i s c o s i t y o f t h e f l u i d , m z / s . W h e n R e < 5 x 1 0 5 , t h e f l o w i s l a m i n a r ; o t h e r w i s e , t u r b u l e n t ( H o l m a n , 1 9 8 1 ) . F o r c a n o p i e s , t h e c h a r a c t e r i s t i c d i m e n s i o n X c o u l d b e t h e a r i t h m e t i c m e a n * o f t h e s u m o f l e a f w i d t h a n d l e n g t h . T h e m e a n w i n d s p e e d i n a c a n o p y c o u l d b e t a k e n a s t h e f r e e - s t r e a m v e l o c i t y . F o r a n a l f a l f a s w a t h , X = 0 . 0 1 2 5 m ( a s s u m e d l e a v e s o f 0 . 0 0 5 m w i d t h a n d 0 . 0 2 m l e n g t h ) , a n d \ 1 = 1 6 x 1 0 ' 6 m z / s a t 3 0 ° C . T o r e a c h t h e c r i t i c a l R e ( 5 x 1 0 5 ) , u s h o u l d b e 6 4 0 m / s w h i c h w a s i m p o s s i b l e . T h u s , i n r e a l f i e l d s , t h e a i r f l o w i n t h e b o u n d a r y l a y e r o v e r a s i n g l e l e a f w o u l d b e l a m i n a r . I f t h e w i n d s p e e d w a s 0 . 1 m / s , t h e n R e : 7 8 . 1 . F o r l a m i n a r b o u n d a r y l a y e r s o v e r p l a t e s w i t h c o n s t a n t t e m p e r a t u r e , t h e r e l a t i o n s h i p b e t w e e n h e a t t r a n s f e r a n d f l o w c h a r a c t e r i s t i c s h a s b e e n o f t e n e x p r e s s e d a s : N u = a 1 R e ) ‘ / 2 ( P r ) ‘ / 3 [ 1 3 0 ] 7 7 w h e r e N u : a v e r a g e N u s s e l t n u m b e r o v e r l e n g t h X , d i m e n s i o n l e s s , R e : R e y n o l d s n u m b e r , d i m e n s i o n l e s s , P r : P r a n d t l n u m b e r , d i m e n s i o n l e s s , a n d a = e m p i r i c a l c o n s t a n t . E q u a t i o n [ 1 3 0 ] h a s b e e n d e r i v e d b a s e d u p o n t h e a n a l o g y o f f l u i d f r i c t i o n a n d h e a t t r a n s f e r . T h e v a l u e o f a w a s f o u n d t o b e a b o u t 1 . 1 i n f i e l d c o n d i t i o n s w h i c h w a s 1 . 6 5 t i m e s a s l a r g e a s t h a t f o r s m o o t h p l a t e s ( 0 . 6 6 8 ) o f t h e s a m e d i m e n s i o n ( G o u d r i a a n , 1 9 7 7 ) . T h i s e n l a r g e m e n t w a s d u e t o r o u g h n e s s o f r e a l l e a v e s . T h e P r a n d t l n u m b e r P r c h a r a c t e r i z e s t h e r e l a t i v e m a g n i t u d e o f d i f f u s i o n o f m o m e n t u m a n d h e a t , a n d h e n c e , r e l a t e s t h e v e l o c i t y a n d t e m p e r a t u r e f i e l d s i n t h e f l u i d . P r = o l d [ 1 3 1 ] k i n e m a t i c v i s c o s i t y , m Z / s , w h e r e 0 t h e r m a l d i f f u s i v i t y , m z l s . a n d a T h e P r a n d t l n u m b e r P r f o r t h e a i r i s a b o u t 0 . 7 a t 3 0 ° C . T h e N u s s e l t n u m b e r N u c h a r a c t e r i z e s t h e c o n v e c t i v e e f f e c t w i t h r e s p e c t t o t h e c o n d u c t i v e e f f e c t i n t h e f l o w . A v a l u e o f u n i t y f o r N u i n d i c a t e s p u r e c o n d u c t i o n . N u = h X / k [ 1 3 2 ] w h e r e h = c o n v e c t i o n h e a t - t r a n s f e r c o e f f i c i e n t , W / m 2 ° C h X = c h a r a c t e r i s t i c d i m e n s i o n , m , a n d k = f l u i d t h e r m a l c o n d u c t i v i t y , W / m ° C . F o r R e : 7 8 . 1 , P r = 0 . 7 , X = 0 . 0 1 2 5 m , k : 2 6 . 4 x 1 0 ' 3 W / m ° C a n d a : 1 . 1 , t h e n N u : 8 . 6 4 a n d h : 1 8 . 3 W / m 2 ° C . H e a t t r a n s f e r d u e t o v i s c o u s f o r c e s i n a f l u i d f l o w i s f o r c e d c o n v e c t i o n . O n t h e o t h e r h a n d , n a t u r a l ( o r f r e e ) c o n v e c t i o n m a y a l s o w h e r e N u N P G c u r r = c ( G r a n d m r P = : : = ) G N P e u m r r m s a a p s s n i e h d r l o t i t f l c n n n u u u m m m c b b b o e e e n r r r s , , , t a l d d d i i i m m m e e e n n n s s s i i i o o o n n n l l l e e e s s s s s s , , , a n t s . [ 1 3 3 ] 7 8 o c c u r , w h i c h i s c a u s e d b y b u o y a n c y f o r c e s i m p o s e d o n t h e f l u i d d u e t o d e n s i t y c h a n g e s i n a h e a t e x c h a n g e p r o c e s s . F o r f r e e c o n v e c t i o n , t h e f o l l o w i n g e x p r e s s i o n h a s b e e n e s t a b l i s h e d : T h e G r a s h o f n u m b e r m e a s u r e s t h e r e l a t i v e i m p o r t a n c e o f t h e b u o y a n c y f e r c e s w i t h r e s p e c t t o t h e v i s c o u s f o r c e s . G r = g a x ’ A r / v [ 1 3 4 ] w h e r e B = v o l u m e t r i c c o e f f i c i e n t o f e x p a n s i o n , 1 / ° K , g = a c c e l e r a t i o n d u e t o g r a v i t y , m / s , A T = t e m p e r a t u r e d i f f e r e n c e b e t w e e n t h e s u r f a c e a n d f l o w , ° K , X : c h a r a c t e r i s t i c d i m e n s i o n , m , a n d u : k i n e m a t i c v i s c o s i t y , m z l s . F o r a i r , 8 i s t h e r e c i p r o c a l o f t h e a b s o l u t e t e m p e r a t u r e o f t h e a i r . W h e n h e a t e d s u r f a c e s f a c e u p o r c o l d s u r f a c e s f a c e d o w n , f r e e c o n v e c t i o n i s e n h a n c e d ; o r i t i s r e d u c e d i f h e a t e d s u r f a c e s f a c e d o w n o r c o l d s u r f a c e s f a c e u p . I f t h e p l a t e i s i n c l i n e d , f r e e c o n v e c t i o n i s a l s o d e p e n d e n t u p o n t h e i n c l i n a t i o n a n g l e . V a l u e s o f c a n d m i n E q u a t i o n [ 1 3 3 ] r e f l e c t t h e s e g e o m e t r i c a n d o t h e r f l u i d c o n d i t i o n s . P r o c e d u r e s a n d t y p i c a l v a l u e s o f c a n d m g i v e n b y H o l m a n ( 1 9 8 1 ) w e r e e m p l o y e d i n s u b p r o g r a m H C O N V . F o r a s i n g l e l e a f , o n e s i d e c o u l d b e c o n s i d e r e d a s f a c i n g u p a n d t h e o t h e r - f a c i n g d o w n . H o w e v e r , t h e o r i e n t a t i o n o f a l e a f i n a s w a t h c o u l d b e a n y a n g l e f r o m h o r i z o n t a l t o v e r t i c a l . 7 9 H o l m a n ( 1 9 8 1 ) i n d i c a t e d t h a t p u r e c o n d u c t i o n w a s o b s e r v e d w h e n G r < 1 7 0 0 . F o r a i r a t 3 0 ° C ( 3 0 3 ° x ) , 8 : 0 . 0 0 3 3 1 / ° K , 3 : 9 . 8 1 m / s z , v = 1 6 x 1 0 ' 6 m z / s a n d P r = 0 . 7 . S u p p o s e A T = 2 ° K a n d X : 0 . 0 1 2 5 m , t h e n G r = 4 9 4 w h i c h i m p l i e d p u r e c o n d u c t i o n , i . e . h = k / X : 2 . 1 1 W / m 2 ° C . I f , A T = 7 ° K , t h e n G r = 1 7 2 9 , c = 0 . 5 4 , m = 0 . 2 5 , R e = 3 . 1 8 a n d h = 6 . 7 2 W / m Z ° C . W i t h i n t h e s w a t h , w i n d s p e e d w a s s m a l l a s s h o w n i n S e c t i o n 4 . 4 , s o t h e e f f e c t o f f o r c e d c o n v e c t i o n m i g h t b e i n s i g n i f i c a n t . A c o m b i n a t i o n o f f o r c e d a n d f r e e c o n v e c t i o n o n e a c h s i d e o f a s i n g l e l e a f w a s l i k e l y t o o c c u r . I n H C O N V , t w o c o n v e c t i o n h e a t - t r a n s f e r c o e f f i c i e n t s w e r e e s t i m a t e d . O n e ( H C O N V F ) w a s f o r f o r c e d c o n v e c t i o n , a n d t h e o t h e r ( H C O N V N ) w a s f o r n a t u r a l c o n v e c t i o n . A s a n a p p r o x i m a t i o n , t h e c o m b i n e d h e a t - t r a n s f e r c o e f f i c i e n t ( H C O N V ) w a s o b t a i n e d b y a d d i n g H C O N V F a n d H C O N V N . T h e r e l a t i v e i m p o r t a n c e b e t w e e n H C O N V F a n d H C O N V N w a s d e t e r m i n e d b a s e d u p o n t h e i r m a g n i t u d e s . T h i s a d d i t i v e a p p r o x i m a t i o n m i g h t o v e r e s t i m a t e t h e h e a t t r a n s f e r c o e f f i c i e n t ; h o w e v e r , i t w o u l d n o t a f f e c t r e s u l t s m u c h s i n c e t h e m a g n i t u d e o f t h e c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t w a s n o t v e r y l a r g e . F o r t h e e x a m p l e g i v e n a b o v e , t h e c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t a n d t h e r e s i s t a n c e t o h e a t t r a n s f e r f o r e a c h s i d e o f a l e a f w a s a b o u t 2 2 W / m 2 ° C : a n d 5 6 s / m , r e s p e c t i v e l y . A s a w h o l e , t h e r e s i s t a n c e t o h e a t t r a n s f e r o f a s i n g l e l e a f w a s t h e s u m o f r e s i s t a n c e s f r o m t w o s i d e s i n p a r a l l e l . 1 / r h 1 = 1 / r h u + 1 / r h d [ 1 3 5 ] w h e r e ” b l = l e a f r e s i s t a n c e , s / m , P h u r e s i s t a n c e a t t h e u p w a r d - f a c i n g s i d e , s / m , a n d ” n o r e s i s t a n c e a t t h e d o w n w a r d - f a c i n g s i d e , s / m . I n t h e e x a m p l e a b o v e , r h 1 = 2 8 s / m a n d h 1 : 4 4 W / m 2 ° f o r o n e l e a f . 8 0 T h e o v e r a l l r e s i s t a n c e t o h e a t t r a n s f e r o f a l a y e r o f l e a v e s w a s t h e c o m b i n a t i o n o f r e s i s t a n c e s i n p a r a l l e l f r o m a l l l e a v e s o f t h e l a y e r . E q u a t i o n [ 1 3 5 ] w a s a l s o v a l i d f o r a l a y e r o f l e a v e s . A s d i s c u s s e d e a r l i e r , t h e d i s t r i b u t i o n o f l e a f o r i e n t a t i o n i n a l a y e r w a s a s s u m e d t o b e u n i f o r m ( s p h e r i c a l ) . W h e n h o r i z o n t a l w i n d f l o w s t h r o u g h a l a y e r o f l e a v e s , t h e w i n d c o m p o n e n t s p a r a l l e l t o e a c h l e a f s u r f a c e w e r e d i f f e r e n t d u e t o v a r i o u s o r i e n t a t i o n a n g l e s . I n t h e m o d e l , a m e a n i n c l i n a t i o n a n g l e ( 4 5 ° ) a n d a m e a n w i n d c o m p o n e n t ( 2 u / n ) w e r e u s e d t o e s t i m a t e a v e r a g e v a l u e s o f h a n d r h f o r l e a v e s i n a l a y e r . N a t e t h a t t h e d i s c u s s i o n a b o v e w a s w i t h r e s p e c t t o a u n i t l e a f a r e a . T h e o v e r a l l h e a t - t r a n s f e r c o e f f i c i e n t h a n d t h e r e s i s t a n c e r h c o u l d b e a l s o e x p r e s s e d w i t h r e s p e c t t o a u n i t l a n d a r e a : h h 1 8 L ” h r h l / B L W h e r e 3 L w a s t h e e l e m e n t l e a f a r e a i n d e x i n a l a y e r ( 1 1 1 2 l e a f / m 2 ) . T h e s u b s c r i p t l s t o o d f o r l e a v e s . 4 . 5 . 2 M o i s t u r e T r a n s f e r o n L e a f S u r f a c e s T h e r a t e e q u a t i o n f o r m o i s t u r e ( w a t e r v a p o r ) t r a n s f e r o n a l e a f s u r f a c e d u e t o m o l e c u l a r d i f f u s i o n w a s s i m i l a r t o t h e r a t e e q u a t i o n f o r s e n s i b l e h e a t t r a n s f e r : 3 = P ( O w l ' P w a ) = ( O w l - O w a ) / ’ w . [ 1 3 6 ] w h e r e E = m o i s t u r e f l u x , k g w a t e r / m z s , O w l : w a t e r v a p o r d e n s i t y o n t h e l e a f s u r f a c e , k g w a t e r / m 3 , O w a = w a t e r v a p o r d e n s i t y o f t h e a i r , k g w a t e r / m 3 , P : p e r m e a b i l i t y c o e f f i c i e n t o r m a s s - t r a n s f e r c o e f f i c i e n t , m / s , 8 1 a n d r w = r e s i s t a n c e t o m o i s t u r e t r a n s f e r , s / m . M o i s t u r e t r a n s f e r o n a l e a f s u r f a c e s h o u l d o v e r c o m e t w o r e s i s t a n c e s . O n e w a s t h e r e s i s t a n c e i n t h e b o u n d a r y l a y e r o v e r t h e s u r f a c e ; t h e o t h e r w a s t h e r e s i s t a n c e o f t h e s u r f a c e , i n c l u d i n g a s e r i e s o f r e s i s t a n c e s d u e t o s t o m a t a , s u b - s t o m a t a l c a v i t i e s a n d c e l l w a l l s i n p a r a l l e l w i t h a r e s i s t a n c e o f c u t i c l e s . T h e f o r m e r w a s a n e x t e r n a l r e s i s t a n c e a f f e c t e d m a i n l y b y t h e a i r f l o w c h a r a c t e r i s t i c s ; h o w e v e r , t h e l a t t e r w a s a r e s i s t a n c e d e t e r m i n e d b y i n t e r n a l f a c t o r s o f t h e p l a n t . T h e b o u n d a r y r e s i s t a n c e ’ v b a n d t h e s u r f a c e r e s i s t a n c e v a w e r e l i n k e d i n s e r i e s f o r e a c h s i d e o f a l e a f . T h e r e s i s t a n c e s o f b o t h s i d e s i n p a r a l l e l f o r m e d t h e r e s i s t a n c e f o r t h e l e a f . A s t o a l a y e r o f l e a v e s , t h e o v e r a l l r e s i s t a n c e t o m o i s t u r e t r a n s f e r r e s u l t e d f r o m p a r a l l e l r e s i s t a n c e s o f a l l l e a v e s i n t h e l a y e r . I n t h e m o d e l , t h e o v e r a l l r e s i s t a n c e r v w a s t r e a t e d t h e s a m e a s t h e r e s i s t a n c e t o h e a t t r a n s f e r d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n . C o n v e r s i o n o f r v f r o m u n i t l e a f a r e a t o u n i t l a n d a r e a w a s a l s o m a d e . T r a n s f e r o f w a t e r v a p o r a l s o c a r r i e d l a t e n t h e a t t r a n s f e r . T h e l a t e n t h e a t f l u x X E ( W / m z ) w a s t h e l a t e n t h e a t o f v a p o r i z a t i o n l t i m e s t h e m o i s t u r e f l u x E . T h e l a t e n t h e a t - t r a n s f e r c o e f f i c i e n t h v w a s A t i m e s t h e r e c i p r o c a l o f t h e r e s i s t a n c e r v . I n t h e m o d e l , R B V , H W , F V L A , V L A a n d F X V L A d e n o t e d r v , A , E , h v a n d 1 E , r e s p e c t i v e l y . 4 . 5 . 2 . 1 R e s i s t a n c e t o M b i s t u r e T r a n s f e r i n t h e B o u n d a r y L a y e r A c c o r d i n g t o t h e t h e o r y o f c o n v e c t i o n , t h e r e i s a n a n a l o g y b e t w e e n m o m e n t u m t r a n s f e r , s e n s i b l e h e a t t r a n s f e r a n d m a s s t r a n s f e r i n t h e b o u n d a r y l a y e r o v e r f l a t p l a t e s . F o r s o l v i n g p r o b l e m s i n c o n v e c t i v e 8 2 m a s s t r a n s f e r , t h e f o l l o w i n g e q u a t i o n h a s b e e n d e r i v e d : p c p P / h h b = ( P r / S c ) 2 / 3 = ( D / o ) 2 / 3 [ 1 3 7 ] w h e r e p c p = v o l u m e t r i c h e a t c a p a c i t y o f a i r , J / m 3 ° C L P = m a s s - t r a n s f e r c o e f f i c i e n t , m / s , h h b = h e a t - t r a n s f e r c o e f f i c i e n t , W / m 2 ° C , P r = P r a n d t l n u m b e r , S o : S c h m i d t n u m b e r , 0 = m a s s d i f f u s i v i t y , m z / s , a n d a = t h e r m a l d i f f u s i v i t y , m Z / s . T h e S c h m i d t n u m b e r S c , s i m i l a r t o t h e P r a n d t l n u m b e r P r , r e l a t e s m o m e n t u m a n d m a s s d i f f u s i o n o f t h e f l u i d . $ 0 = v / D W h e r e v i s t h e k i n e m a t i c v i s c o s i t y o f t h e f l u i d i n m z / s . B a s e d o n t h e s e e q u a t i o n s , t h e r e s i s t a n c e t o m a s s t r a n s f e r i n t h e b o u n d a r y l a y e r o f a f l u i d f l o w i s : r v b = ( a / D ) 2 / 3 r h b [ 1 3 8 ] F o r d i f f u s i o n o f w a t e r v a p o r i n t h e a i r a t 3 0 ° C , 0 : 2 5 . 7 x 1 0 " 6 m z l s , 0 : 2 2 . 8 x 1 0 ' 6 n Z / s , a n d t h u s , r v b = 0 . 9 2 r h b . N o t e t h a t k , P r , 0 , o a n d u v a r i e d w i t h t e m p e r a t u r e . I n t h e m o d e l , t h e s e p r o p e r t i e s w e r e e s t i m a t e d w i t h a n i n t e r p o l a t i o n r o u t i n e ( A P L 2 ) f r o m t a b u l a t e d d a t a p r o v i d e d b y M o n t e i t h ( 1 9 7 3 ) . I n t h i s s u b m o d e l , K , P R , D V , D H a n d D M d e n o t e d k , P r , 0 , o a n d v , r e s p e c t i v e l y . R e s i s t a n c e s ” v b a n d ’ h b w e r e c a l c u l a t e d i n s u b r o u t i n e R E S I S T . 8 3 4 . 5 . 2 . 2 S u r f a c e R e s i s t a n c e t o M o i s t u r e T r a n s f e r F o r l i v i n g p l a n t s , t h e s u r f a c e r e s i s t a n c e i s i n f l u e n c e d b y l i g h t i n t e n s i t y , l e a f t e m p e r a t u r e a n d l e a f w a t e r s t r e s s . S o m e i n f o r m a t i o n a b o u t t h e i n d i v i d u a l e f f e c t o f e a c h o f t h e s e f a c t o r s w a s r e p o r t e d ( M i l t h o r p e a n d M o o r b y , 1 9 7 4 , a n d N o r m a n , 1 9 7 9 ) ; b u t t h e r e w a s a l a c k o f i n f o r m a t i o n a b o u t t h e c o m b i n e d e f f e c t s o f t h e s e f a c t o r s . F o r c u t p l a n t s , t h e s u r f a c e r e s i s t a n c e f o l l o w s t h e p a t t e r n o f t h a t o f l i v i n g p l a n t s b e f o r e s t o m a t a a r e c l o s e d . H o w e v e r , a f t e r s t o m a t a a r e ‘ c o m p l e t e l y c l o s e d , t h e i n f l u e n c e o f l i g h t i n t e n s i t y w o u l d b e l e s s , e v e n n e g l i g i b l e . T h i s s h o u l d b e r e a s o n a b l e w i t h i n d e e p l a y e r s o f s w a t h s w h e r e p e n e t r a t i o n o f s u n l i g h t i s l o w , e s p e c i a l l y i f l e a f a r e a i n d e x i s l a r g e . I t w a s f o u n d t h a t l e a f s u r f a c e r e s i s t a n c e w a s a p p r o x i m a t e l y c o n s t a n t o v e r a w i d e r a n g e o f l e a f t e m p e r a t u r e s ( 1 0 ° t o 4 0 ° C ) f o r l i v i n g p l a n t s ( M i l t h o r p e a n d M o o r b y , 1 9 7 4 , a n d N o r m a n , 1 9 7 9 ) . I t w a s a s s u m e d t h a t t h e t e m p e r a t u r e i n f l u e n c e o n t h e s u r f a c e r e s i s t a n c e w a s n e g l i g i b l e f o r a l f a l f a s w a t h s i f l e a f t e m p e r a t u r e w a s w i t h i n t h e r a n g e a f o r e m e n t i o n e d . T h e r e f o r e , t h e d o m i n a n t f a c t o r a f f e c t i n g t h e s u r f a c e r e s i s t a n c e s h o u l d b e w a t e r s t r e s s . M i l t h o r p e a n d M o o r b y ( 1 9 7 4 ) p o s t u l a t e d a n e x p o n e n t i a l f u n c t i o n f o r t h e s u r f a c e r e s i s t a n c e w i t h r e s p e c t t o r e l a t i v e w a t e r c o n t e n t ( R W C ) o f l e a v e s : r v s = 1 0 9 - D R V C 1 1 3 9 ] w h e r e ’ v s = s u r f a c e r e s i s t a n c e , s / m , R W C = r e l a t i v e w a t e r c o n t e n t , d i m e n s i o n l e s s , a a n d b = e m p i r i c a l c o n s t a n t , d i m e n s i o n l e s s . F o r c o t t o n l e a v e s , a : 5 . 9 a n d b : 4 . 2 ( M i l t h o r p e a n d M o o r b y , 1 9 7 4 ) . I n 8 4 t h i s m o d e l , a a n d b f o r a l f a l f a l e a v e s w e r e p a r a m e t e r s t o b e d e t e r m i n e d . I t w a s e x p e c t e d t h a t t h e v a l u e s o f a a n d b f o r a l f a l f a w o u l d b e s m a l l e r t h a n t h o s e f o r c o t t o n . I n s u b p r o g r a m R B V M C , A a n d B d e n o t e d a a n d b , r e s p e c t i v e l y . T h e m i n i m u m s t o m a t a l r e s i s t a n c e i s l e s s t h a n 5 0 s / m ; t h e r e s i s t a n c e o f s u b - s t o m a t a l c a v i t i e s o r c e l l w a l l s i s a b o u t 2 0 s / m . T h e c u t i c u l a r r e s i s t a n c e i s 2 0 0 0 - 8 0 0 0 s / m f o r m e s o p h y t e s , a n d 5 0 0 0 - 8 0 0 0 s / m f o r x e r o p h y t e s ( M i l t h o r p e a n d M o o r b y , 1 9 7 4 ) . F o r m a n y c r o p s , t h e m i n i m u m s u r f a c e r e s i s t a n c e i s a b o u t 1 0 0 s / m ( M o n t e i t h , 1 9 7 3 , M i l t h o r p e a n d M o o r b y , 1 9 7 4 , G o u d r i a a n , 1 9 7 7 , a n d T h o m p s o n , 1 9 8 1 ) . T h e r e l a t i v e w a t e r c o n t e n t R W C i s t h e r a t i o o f t h e a c t u a l w a t e r c o n t e n t a t a g i v e n t i m e t o t h e m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n . 1 1 1 1 0 = M / M f [ 1 4 0 ] w h e r e M = m o i s t u r e c o n t e n t ( d r y b a s i s ) , k g w a t e r / k g d r y m a t t e r , a n d M y = M a t f u l l h y d r a t i o n , k g w a t e r / k g d r y m a t t e r . T h e l e a f m o i s t u r e c o n t e n t v a r i e s w i t h p l a n t s p e c i e s , t h e a g e o f p l a n t s , a n d t h e h i s t o r y o f w a t e r s t r e s s . B r a d f o r d a n d H s i a o ( 1 9 8 2 ) p o i n t e d o u t t h a t h e r b a c e o u s c r o p s h a d t h e a b i l i t y o f o s m o t i c a d j u s t m e n t t o w i t h s t a n d w a t e r s t r e s s t o a c e r t a i n e x t e n t . T h e y a l s o m e n t i o n e d t h a t t h e w i l t i n g p o i n t o f R W C , c o r r e s p o n d i n g t o p r o n o u n c e d s t o m a t a c l o s u r e , f o r t h e v a s t m a j o r i t y o f l e a v e s w a s i n t h e r a n g e o f 0 . 7 5 t o 0 . 8 5 . G o u d r i a a n ( 1 9 7 7 ) a l s o i n d i c a t e d t h a t w h e n R W C w a s a b o v e 0 . 9 ( m a x i m u m 1 . 0 ) , t h e s u r f a c e r e s i s t a n c e t o m o i s t u r e l o s s e s w a s n e a r c o n s t a n t . T h e r e w a s a l a c k o f i n f o r m a t i o n o f v a l u e s o f t h e m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n f o r a l f a l f a . V a n E l d e r e n e t a l . ( 1 9 7 2 ) u s e d a r a n g e o f 7 . 0 ~ 9 . 0 f o r t h e m a x i m u m m o i s t u r e c o n t e n t o f p l a n t s w h e n t h e y s t u d i e d w a t e r u p t a k i n g o f g r a s s s w a r d s . I f t h e r a n g e w e r e u s e d a s t h e a l f a l f a 8 5 m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n , a q u e s t i o n w o u l d a r i s e . F i e l d o b s e r v a t i o n s s h o w e d t h a t t h e m o i s t u r e c o n t e n t o f f r e s h l y c u t a l f a l f a w a s a b o u t 3 . 0 ~ 4 . 0 ( d r y b a s e s ) w h i c h w a s m u c h l o w e r t h a n 7 5 % t o 8 5 1 o f t h e m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n ( 7 . 0 t o 9 . 0 ) a s a s s u m e d . H o w e v e r , n o i n d i c a t i o n o f w i l t i n g w a s f o u n d f o r m a j o r i t y o f f r e s h l y c u t a l f a l f a » T h u s , t h e m o i s t u r e c o n t e n t o f f r e s h l y c u t a l f a l f a s h o u l d b e a b o v e 8 5 % t o 9 0 % o f i t s M f v a l u e . A r e a s o n a b l e a s s u m p t i o n w a s t h a t l i v i n g a l f a l f a m i g h t h a v e a n a d j u s t a b l e m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n w h i c h m i g h t d e p e n d u p o n t h e v a r i e t y , g r o w t h s t a g e a n d w a t e r s t r e s s o f t h e p l a n t . T h e r e f o r e , t h e a c t u a l m o i s t u r e c o n t e n t o f f r e s h l e a v e s s h o u l d b e c l o s e t o M y a t t h a t s t a g e o f g r o w t h u n d e r t h a t p a r t i c u l a r c o n d i t i o n s . I n t h i s m o d e l , t h e i n i t i a l m o i s t u r e c o n t e n t o f t h e a l f a l f a s w a t h w a s c o n s i d e r e d t o b e e q u i v a l e n t t o t h e m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n . I n t h e m o d e l , E N T M C , F M C a n d R W S d e n o t e d M , H f a n d ” v s : r e s p e c t i v e l y . R W S w a s e s t i m a t e d i n f u n c t i o n s u b p r o g r a m R B V M C . T o s p e e d u p d r y i n g , f i e l d c o n d i t i o n i n g h a s b e e n w i d e l y a p p l i e d i n h a y p r o d u c t i o n . M e c h a n i c a l c o n d i t i o n i n g b r e a k s c u t i c l e s o n a l f a l f a s u r f a c e s t o i n c r e a s e t h e e x p o s u r e a r e a o f a l f a l f a t o t h e s u r r o u n d i n g a i r . C h e m i c a l c o n d i t i o n i n g i n c r e a s e s t h e p e r m e a b i l i t y o f w a t e r v a p o r t h r o u g h c u t i c l e s . B o t h r e s u l t i n a r e d u c t i o n o f t h e s u r f a c e r e s i s t a n c e t o m o i s t u r e l o s s e s . T h i s c a n b e r e f l e c t e d b y i n t r o d u c i n g a r e d u c t i o n f a c t o r w h e n e s t i m a t i n g t h e s u r f a c e r e s i s t a n c e . P r e c i s e d a t a o f t h e e f f e c t o f c o n d i t i o n i n g a r e n o t y e t a v a i l a b l e . S m i t h ( 1 9 8 5 ) u s e d a v a l u e o f 0 . 5 f o r t h e r e d u c t i o n f a c t o r . I n t h i s m o d e l , t h e r e d u c t i o n f a c t o r , d e n o t e d a s C M E C H , w a s s e t t o 0 . 5 . T h e f o r e g o i n g d i s c u s s i o n d e a l t w i t h t h e d r y i n g p r o c e s s w h e r e l e a v e s l o s e m o i s t u r e . W h e n r e w e t t i n g ( r a i n o r d e w ) o c c u r s t h e s u r f a c e 8 6 r e s i s t a n c e h a s n o e f f e c t o n t h e c o n d e n s a t i o n o f w a t e r o n l e a f s u r f a c e s . I n t h i s c a s e , t h e s u r f a c e r e s i s t a n c e c o m p o n e n t w a s s e t t o z e r o i n t h e m o d e l . 4 . 5 . 3 R e s p i r a t i o n o n L e a f S u r f a c e s R e s p i r a t i o n i s a b a s i c f u n c t i o n o f p l a n t l i f e . I t i s a p r o c e s s o f c a r b o h y d r a t e o x i d a t i o n g o v e r n e d b y : C 5 H 1 2 0 5 + 6 0 2 + 6 C 0 2 + 6 H 2 0 + h e a t [ 1 4 1 ] R e s p i r a t i o n r e l e a s e s 1 0 . 7 2 x 1 0 ° J h e a t f o r e a c h k i l o g r a m o f c a r b o n d i o x i d e g e n e r a t e d . R e s p i r a t i o n c o n s i s t s o f m a i n t e n a n c e r e s p i r a t i o n a n d g r o w t h r e s p i r a t i o n f o r g r o w i n g p l a n t s . F o r c u t a l f a l f a , t h e g r o w t h r e s p i r a t i o n e v e n t u a l l y c e a s e s ; h o w e v e r , t h e m a i n t e n a n c e r e s p i r a t i o n c o n t i n u e s e v e n a f t e r d r i e d h a y i s i n s t o r a g e . T h e m a i n t e n a n c e r e s p i r a t i o n i s o f t e n r e f e r r e d t o a s ' d a r k r e s p i r a t i o n ' , s i n c e i t i s m e a s u r e d u n d e r d a r k c o n d i t i o n s . D a r k r e s p i r a t i o n i s a f u n c t i o n o f t e m p e r a t u r e a n d w a t e r s t r e s s o f t h e p l a n t . W o o d a n d P a r k e r ( 1 9 7 1 ) p r o p o s e d a n e m p i r i c a l f u n c t i o n : r s p = 0 . 4 2 5 x 1 0 ‘ 5 [ 3 . 6 6 1 1 / ( 1 + h ) - 1 1 x [ 1 4 2 ] w h e r e r s p = r e s p i r a t i o n r a t e , k g c a r b o n d i o x i d e / s - k g d r y m a t t e r , M = m o i s t u r e c o n t e n t ( d r y b a s i s ) , k g w a t e r / k g d r y m a t t e r , a n d K = f u n c t i o n o f t e m p e r a t u r e . 1 . 0 ( 2 5 ° < T < 4 5 ° C ) K : { 0 . 1 7 8 e 0 ' 0 6 9 T ( 5 ° < T < 2 5 ° ) T h e r e s p i r a t i o n h e a t f l u x f o r a l a y e r o f p l a n t s p e r u n i t l a n d a r e a w a s : 8 7 a s p = 4 . 5 5 6 K [ 3 . 6 6 M / ( 1 + M ) - 1 ] 3 m [ 1 4 3 ] w h e r e R s p = r e s p i r a t i o n h e a t f l u x , W / m z , a n d 3 m = a r e a d e n s i t y o f a l a y e r , k g d r y m a t t e r / m 2 . F o r e x a m p l e , K = 1 ( a i r a t 3 0 ° C ) , M = 4 . 0 ( f r e s h l y c u t ) , 3 m = 0 . 1 3 k g d r y m a t t e r / m 2 , r s p = 0 . 8 2 x 1 0 ‘ 6 k g c a r b o n d i o x i d e / s - k g d r y m a t t e r a n d R s p = 1 . 1 4 W / m z . I n t h e m o d e l , R S P a n d D D d e n o t e d R s p a n d 3 m , r e s p e c t i v e l y . T h e r e s p i r a t i o n h e a t f l u x w a s a n e n e r g y s o u r c e i n t h e d r y i n g p r o c e s s . A l t h o u g h r e s p i r a t i o n w a s n e g l i g i b l e d u r i n g t h e d a y t i m e w h e n s t r o n g s o l a r e n e r g y w a s a v a i l a b l e , i t p l a y e d a r o l e t o c o m p e n s a t e s o m e t h e r m a l r a d i a t i o n l o s s e s a t n i g h t w h e n s u n l i g h t w a s a b s e n t . R e s p i r a t i o n a l s o c a u s e d a l o s s i n d r y m a t t e r a n d a g a i n i n m o i s t u r e . G e n e r a t i n g 1 . 0 k g c a r b o n d i o x i d e l e a d s t o 0 . 6 8 k g d r y m a t t e r l o s s a n d 0 . 4 1 k g m o i s t u r e g a i n . H o w e v e r , r e s p i r a t i o n w a s s o s m a l l f o r o u t a l f a l f a , a s s h o w n , t h a t t h e l o s s a n d t h e g a i n w e r e n e g l e c t e d . W h e n M < 0 . 3 7 6 , E q u a t i o n [ 1 4 2 ] g a v e n e g a t i v e r e s p i r a t i o n . W o o d a n d P a r k e r ( 1 9 7 1 ) i n d i c a t e d t h a t r e s p i r a t i o n w a s s l i g h t w h e n M w a s b e l o w 0 . 3 7 6 . I n t h e m o d e l , r e s p i r a t i o n w a s s e t t o z e r o i n s u c h c i r c u m s t a n c e s . 4 . 5 . 4 E n e r g y B a l a n c e o n L e a f S u r f a c e s T h e e n e r g y b a l a n c e o n a l a y e r o f l e a v e s w a s d e s c r i b e d a s : R n + a s p + 0 1 ‘ - A E - Q = s [ 1 4 4 ] w h e r e a n = n e t a b s o r b e d r a d i a t i o n f l u x , W / m z , r e s p i r a t i o n h e a t f l u x , W / m z , R s p 8 8 O r = h e a t f l u x d u e t o r a i n , W / m z , A E : l a t e n t h e a t f l u x , W / m z , Q = s e n s i b l e h e a t f l u x , W / m z , a n d S = s e n s i b l e h e a t s t o r a g e f l u x , W / m z . T h e s i g n s o f t h e s e t e r m s i n t h e e q u a t i o n i n d i c a t e t h a t R n , R s p a n d O r w e r e a s s u m e d t o b e i n p u t s t o w a r d l e a f s u r f a c e s , a n d A E a n d 0 w e r e l o s s e s f r o m l e a f s u r f a c e s . I t w a s a s s u m e d t h a t S w a s p o s i t i v e i f l e a f t e m p e r a t u r e i n c r e a s e d ; o t h e r w i s e , n e g a t i v e . R n w a s t h e s u m o f t h e a b s o r b e d s h o r t - w a v e a n d l o n g - w a v e r a d i a t i o n f l u x e s , d e s c r i b e d i n S e c t i o n s 4 . 1 a n d 4 . 2 , r e s p e c t i v e l y . D u r i n g t h e d a y t i m e , R n w a s u s u a l l y p o s i t i v e b e c a u s e s o l a r r a d i a t i o n w a s p r e d o m i n a n t . R n w a s n e g a t i v e a t n i g h t w h e n s u n l i g h t w a s a b s e n t a n d t h e r e w e r e l o s s e s o f l o n g - w a v e r a d i a t i o n . S i m u l a t e d d a t a s h o w e d t h a t R n c o u l d b e f r o m - 2 0 t o 2 0 0 W / m 2 i n t h e t o p l a y e r o f a s w a t h . R a p a n d o r w e r e j u s t d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n s . Q r w a s n e g l i g i b l e . T h e h e a t s t o r a g e f l u x S w a s e x p r e s s e d a s : S : G E T / 8 t = C A T / A t w h e r e 3 T / 3 t t i m e d e r i v a t i v e o f t e m p e r a t u r e , ° C / s , A T : i n c r e m e n t o f t e m p e r a t u r e , ° C , A t = i n c r e m e n t o f t i m e , s e c , a n d C = h e a t c a p a c i t y o f l e a v e s i n t h e l a y e r , J / ° C - m z , C = 0 3 m w h e r e c = s p e c i f i c h e a t o f a l f a l f a , J / k g ° C , a r e a d e n s i t y o f t h e l a y e r , k g d r y m a t t e r / m 2 . a n d a m T h e s p e c i f i c h e a t o f a l f a l f a d r y m a t t e r i s 1 1 7 2 . 3 J / ° C - k g d r y m a t t e r , w h i c h i s a b o u t 2 8 % o f t h a t o f w a t e r ( M o h s e n i n , 1 9 8 0 ) . T h e 8 9 s p e c i f i c h e a t o f a l f a l f a i n c r e a s e s a s t h e m o i s t u r e c o n t e n t o f a l f a l f a i n c r e a s e s . S i e b e l ( 1 8 9 2 ) p r o p o s e d a n a p p r o a c h t o e s t i m a t e s p e c i f i c h e a t f o r f o o d m a t e r i a l s : 0 : ( c w M + c s ) / ( 1 + M ) [ 1 4 5 ] o v e r a l l s p e c i f i c h e a t o f t h e m a t e r i a l , J / k g ° C , w h e r e c = c w = s p e c i f i c h e a t o f w a t e r , 4 1 8 6 . 8 J / k g ° C , c s : s p e c i f i c h e a t o f d r y m a t t e r , J / k g ° C , a n d M = m o i s t u r e c o n t e n t o f t h e m a t e r i a l , k g w a t e r / k g d r y m a t t e r . T h i s e q u a t i o n w a s v a l i d f o r a w i d e r a n g e o f m o i s t u r e c o n t e n t s , e x c e p t v e r y l o w m o i s t u r e c o n t e n t s . A t v e r y l o w m o i s t u r e l e v e l s , t h e e f f e c t o f b o n d w a t e r w h i c h h a s a h i g h e r s p e c i f i c h e a t t h a n f r e e w a t e r w i l l c a u s e a p p r e c i a b l e e r r o r s ( M o h s e n i n , 1 9 8 0 ) . T h e s p e c i f i c h e a t o f a l f a l f a w a s a p p r o x i m a t e d i n t h i s m o d e l a s : c : 4 1 8 6 . 8 ( M + O . 2 8 ) / ( 1 + M ) [ 1 4 6 ] s l - l p p o s e t h a t i n a s w a t h l a y e r , M = 4 . 0 k g w a t e r / k g d r y m a t t e r ( f r e s h l y c u t ; a l f a l f a ) , 8 M = 0 . 1 3 k g d r y m a t t e r / m 2 ( m e d i u m d e n s i t y ) a n d A T = 3 C i n 1 h o u r ( f r o m m e a s u r e d d a t a ) , t h e n c = 3 5 8 4 J / k g ° C , C = 4 7 6 . 7 J / ° C - m a n d S = 0 . 4 W / m z . T h i s s h o w e d t h a t t h e h e a t s t o r a g e t e r m w a s n e g l i g i b l e i n t h e e n e r g y b a l a n c e o n l e a v e s . T h e r e f o r e , E q u a t i o n [ 1 4 4 ] w a s s i m p l i f i e d t o : R e - X E - Q = 0 [ 1 4 7 ] w h e r e R e = e n e r g y s o u r c e , W / m z R e : R n + R s p T h e e n e r g y ' s o u r c e ' t e r m m i g h t b e p o s i t i v e ( s o u r c e ) o r n e g a t i v e ( s i n k ) . R e w a s d e n o t e d a s E N E R G Y i n s u b m o d e l D B L E A F . 9 0 4 . 5 . 4 . 1 T h e L a t e n t H e a t F l u x R e c a l l t h a t : 4 5 = 4 ( l e ‘ 9 w a ) / r v T h e w a t e r v a p o r d e n s i t y i n t h e a i r w a s a p p r o x i m a t e d a s : 9 w = D a e e v / P a t m [ 1 4 8 ] w h e r e p H = w a t e r v a p o r d e n s i t y , k g w a t e r / m 3 , p a = d r y a i r d e n s i t y , k g d r y a i r / m 3 , P a t m = a t m o s p h e r i c p r e s s u r e , 1 0 1 3 2 5 P a , e v = w a t e r v a p o r p r e s s u r e , P a , a n d e = r a t i o o f m o l e c u l a r w e i g h t o f w a t e r t o d r y a i r , 0 . 6 2 2 . T h e n , A E = 4 0 a 5 ( e v l ' e v a ) / P a t m r v = p c p ( e v 1 - e v a ) / y r v [ 1 4 9 ] w h e r e l = l a t e n t h e a t o f v a p o r i z a t i o n , ~ 2 . 4 5 x 1 0 ° J / k g w a t e r , p o p = v o l u m e t r i c h e a t c a p a c i t y o f t h e a i r , J / m ° ° C , e v l : w a t e r v a p o r p r e s s u r e i n l e a v e s , P a , e v a = w a t e r v a p o r p r e s s u r e i n t h e a i r , P a , r v = r e s i s t a n c e t o m o i s t u r e t r a n s f e r , s / m , a n d y : p s y c h r o m e t r i c c o n s t a n t , ~ 0 . 6 7 P a / ° C . Y = P a t m c p / E A T h e w a t e r v a p o r p r e s s u r e e v w a s e x p r e s s e d a s : [ 1 5 0 ] e v = r e s w h e r e e 3 : s a t u r a t i o n v a p o r p r e s s u r e a t a g i v e n t e m p e r a t u r e , P a , a n d r = r e l a t i v e h u m i d i t y , d e c i m a l . I f s u b s c r i p t s a a n d l s t o o d f o r a i r a n d l e a v e s , r e s p e c t i v e l y , t h e n , e v a = ” a e s a a n d e v l = ” 1 3 5 1 T h e r e l a t i v e h u m i d i t y i n l e a v e s r 1 w a s o f t e n r e f e r r e d t o a s t h e 9 1 e q u i l i b r i u m r e l a t i v e h u m i d i t y r e . 4 . 5 . 4 . 2 E q u i l i b r i u m R e l a t i v e H u m i d i t y P a s t r e s e a r c h o n d r y i n g h a s s h o w n t h a t a n y b i o l o g i c a l m a t e r i a l t e n d s t o r e a c h e q u i l i b r i u m w i t h s u r r o u n d i n g a i r . A t e q u i l i b r i u m , t h e i n t e r n a l v a p o r d e n s i t y ( o r p r e s s u r e ) o f t h e m a t e r i a l e q u a l s t h a t o f t h e a i r , r e s p e c t i v e l y . O t h e r w i s e , m o i s t u r e t r a n s f e r w i l l o c c u r . A t e q u i l i b r i u m , t h e m o i s t u r e c o n t e n t o f t h e m a t e r i a l i s c a l l e d t h e e q u i l i b r i u m m o i s t u r e c o n t e n t , a n d a c c o r d i n g l y t h e r e l a t i v e h u m i d i t y i n s i d e t h e m a t e r i a l i s c a l l e d e q u i l i b r i u m r e l a t i v e h u m i d i t y . T h e e q u i l i b r i u m r e l a t i v e h u m i d i t y i s a f u n c t i o n o f t h e t e m p e r a t u r e a n d m o i s t u r e c o n t e n t o f t h e m a t e r i a l . l i n k ( 1 9 3 5 ) , D e x t e r ( 1 9 4 7 ) , D e x t e r e t a l . ( 1 9 4 7 ) , H e n d e r s o n a n d P e r r y ( 1 9 5 5 ) , a n d S h e p h e r d ( 1 9 5 8 ) p r o v i d e d s o m e e x p e r i m e n t a l d a t a o f t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y o f a l f a l f a . B a k k e r ( 1 9 6 1 ) d i d e x t e n s i v e e x p e r i m e n t s o n a d s o r p t i o n a n d d e s o r p t i o n o f a l f a l f a . H i s t e s t s c o v e r e d t e m p e r a t u r e f r o m 4 . 4 t o 4 8 . 9 C , m o i s t u r e c o n t e n t f r o m 0 . 0 4 7 t o & 2 . 2 3 ( d r y b a s i s ) , r e l a t i v e h u m i d i t y f r o m 0 . 1 1 4 t o 1 . 0 , a n d f o u r m a t u r i t y l e v e l s ( p r e b l o o m , 1 0 $ b l o o m , f u l l b l o o m a n d m a t u r e ) . H e n d e r s o n ( 1 9 5 2 ) s u g g e s t e d a n e q u a t i o n f o r e q u i l i b r i u m r e l a t i v e h u m i d i t y i n b i o l o g i c a l m a t e r i a l s : n r e s 1 - e ‘ a T “ [ 1 5 1 ] w h e r e M = m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , T = t e m p e r a t u r e , ° C , a a n d n = e m p i r i c a l c o n s t a n t . T h o m p s o n ( 1 9 8 1 ) p r o p o s e d a n o t h e r m o d e l b a s e d o n S h e p h e r d ' s d a t a ( 1 9 5 8 ) . 9 2 r e = [ 1 - e ' ( a + b T ) " ] n [ 1 5 2 ] W h e r e a , b a n d n w e r e e m p i r i c a l c o n s t a n t s . A n o n l i n e a r r e g r e s s i o n t e c h n i q u e ( N O N L I N E A R p r o c e d u r e , S P S S 8 . 3 , 1 9 8 2 ) w a s u s e d i n t h i s s t u d y t o f i t B a k k e r ' s d a t a w i t h b o t h H e n d e r s o n ' s a n d T h o m p s o n ' s a p p r o a c h e s . B a k k e r ( 1 9 6 1 ) f o u n d t h a t H e n d e r s o n ' s e q u a t i o n d i d n o t f i t h i s e x p e r i m e n t a l d a t a w e l l . R e g r e s s i o n r e s u l t s s h o w e d t h a t T h o m p s o n ' s e q u a t i o n w o r k s b e t t e r t h a n H e n d e r s o n ' s . T h e c o e f f i c i e n t o f d e t e r m i n a t i o n w a s 0 . 9 5 ( d e s o r p t i o n ) o r 0 . 9 8 ( a d s o r p t i o n ) f o r m a t u r e a l f a l f a w h e n T h o m p s o n ' s e q u a t i o n w a s a p p l i e d ( T a b l e 4 ) , i n c o m p a r i s o n w i t h t h a t o f 0 . 6 9 ( d e s o r p t i o n ) o r 0 . 7 3 ( a d s o r p t i o n ) f o r H e n d e r s o n ' s e q u a t i o n . T a b l e 4 E m p i r i c a l c o n s t a n t s i n T h o m p s o n ' s e q u a t i o n o f e q u i l i b r i u m r e l a t i v e h u m i d i t y b a s e d u p o n B a k k e r ' s d a t a ( 1 9 6 1 ) T y p e M a t u r i t y a b n r 2 D e s o r p t i o n M a t u r e 8 . 5 8 6 0 . 0 3 8 2 2 . 0 7 7 0 . 9 5 A d s o r p t i o n M a t u r e 1 1 . 8 6 0 . 0 2 1 4 2 . 4 0 6 0 . 9 8 F u l l B l o o m 1 0 . 6 1 0 . 0 8 0 7 2 . 7 3 3 0 . 9 7 1 0 % B l o o m 9 . 9 8 2 0 . 0 8 4 2 2 . 9 3 2 0 . 9 7 P r e b l o o m 8 . 6 1 4 0 . 0 7 5 4 2 . 8 4 5 0 . 9 7 r 2 - t h e c o e f f i c i e n t o f d e t e r m i n a t i o n o f r e g r e s s i o n a n a l y s i s T h e p h e n o m e n o n o f r e a c h i n g d i f f e r e n t e q u i l i b r i u m r e l a t i v e h u m i d i t y d u r i n g d e s o r p t i o n a n d a d s o r p t i o n a t t h e s a m e t e m p e r a t u r e a n d m o i s t u r e c o n t e n t o f a m a t e r i a l i s k n o w n a s t h e h y s t e r e s i s e f f e c t . L i t e r a t u r e h a s s h o w n t h a t t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y f o r a d s o r p t i o n i s h i g h e r t h a n t h a t f o r d e s o r p t i o n . T h i s i s p r e s u m a b l y a t t r i b u t e d . t o w a t e r v a p o r d e n s i t y ( o r p r e s s u r e ) i n c r e a s i n g l o c a l l y n e a r t h e s u r f a c e 9 3 w h e n a m a t e r i a l a b s o r b s m o i s t u r e . B a k k e r ( 1 9 6 1 ) d i d n o t g i v e d a t a o f d e s o r p t i o n o f a l f a l f a f o r t h r e e o f f o u r m a t u r i t y l e v e l s . H o w e v e r , h e a s s u m e d t h a t t h e d i f f e r e n c e i n e q u i l i b r i u m r e l a t i v e h u m i d i t y b e t w e e n a d s o r p t i o n a n d d e s o r p t i o n f o r m a t u r e a l f a l f a a t a g i v e n t e m p e r a t u r e a n d m o i s t u r e c o n t e n t c o u l d b e a p p l i e d t o o t h e r m a t u r i t y s t a g e s . H i s a p p r o a c h w a s a d a p t e d i n t h e m o d e l . I n s u b m o d e l D B L E A F , E O R H w a s u s e d t o d e n o t e r e a n d e s t i m a t e d i n f u n c t i o n s u b p r o g r a m R H E . I n t e r p o l a t i o n w a s u s e d t o e s t i m a t e t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y w h e n b l o o m l e v e l s w e r e o t h e r t h a n 0 % , 1 0 1 o r 1 0 0 % . T a b l e 5 l i s t s s o m e e s t i m a t e d v a l u e s o f t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y r e f o r d e s o r p t i o n o f m a t u r e a l f a l f a a t v a r i o u s t e m p e r a t u r e a n d m o i s t u r e c o n t e n t s . T h e t a b l e s h o w s t h a t t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y i n c r e a s e d a s e i t h e r t e m p e r a t u r e o r t h e m o i s t u r e c o n t e n t o f a l f a l f a i n c r e a s e d . T h e e q u i l i b r i u m r e l a t i v e h u m i d i t y o f m a t u r e a l f a l f a w a s n e a r u n i t y w h e n t h e m o i s t u r e c o n t e n t w a s a b o v e 0 . 5 , d e p e n d i n g u p o n a l f a l f a t e m p e r a t u r e . T a b l e 5 E s t i m a t e d e q u i l i b r i u m r e l a t i v e h u m i d i t y o f m a t u r e a l f a l f a i n d e s o r p t i o n a t v a r i o u s t e m p e r a t u r e a n d m o i s t u r e c o n t e n t s T e m p e r a t u r e ( ° C ) 0 . 0 5 0 . 1 0 0 . 5 0 0 . 6 0 0 . 6 2 0 . 6 5 0 . 6 8 1 0 0 . 1 2 0 . 3 4 0 . 9 8 0 . 9 9 0 9 9 0 9 9 1 0 0 2 0 0 . 1 3 0 3 6 0 . 9 8 0 . 9 9 0 9 9 1 0 0 1 0 0 3 0 0 . 1 4 0 . 3 7 0 . 9 8 0 9 9 1 0 0 1 0 0 1 0 0 9 4 4 . 5 . 4 . 3 M o d i f i e d M o n t e i t h - P e n m a n E q u a t i o n R e c a l l t h a t : R e - A E - 0 3 0 0 E = p c p ( e v 1 - e v a ) / y r v Q 2 O C P ( T 1 ' T a ) / r h e v 3 F e e s T h e s e c o n d a n d t h i r d e q u a t i o n s w e r e i n t h e s a m e f o r m s i n c e ( e v l - e v a ) / k h a d t h e d i m e n s i o n o f t e m p e r a t u r e . W h e n l e a f a n d a i r t e m p e r a t u r e s w e r e c l o s e , t h e f o l l o w i n g f i r s t - o r d e r T a y l o r ' s - s e r i e s e x p a n s i o n w a s v a l i d : e 3 1 : e 3 3 + s ( T 1 - T a ) [ 1 5 3 ] w h e r e T 1 = l e a f t e m p e r a t u r e , ° C , T a = a i r t e m p e r a t u r e , ° C , e 3 1 = s a t u r a t i o n v a p o r p r e s s u r e a t l e a f t e m p e r a t u r e , P a , e 3 3 = s a t u r a t i o n v a p o r p r e s s u r e a t a i r t e m p e r a t u r e , P a , a n d s = d e r i v a t i v e o f e 3 w i t h r e s p e c t t o T , P a / ° C . s = a e s / a T [ 1 5 4 ] T h e q u a n t i t y 3 w a s k n o w n a s t h e s l o p e o f t h e c u r v e o f s a t u r a t i o n - p r e s s u r e - t e m p e r a t u r e a t a g i v e n t e m p e r a t u r e . T h e f u n c t i o n f o r m o f t h e s a t u r a t i o n - p r e s s u r e - t e m p e r a t u r e c u r v e w a s a d a p t e d f r o m A S A E D a t a 2 7 1 . 2 ( 1 9 8 4 ) . F o r a i r f r o m 1 0 ° t o 4 0 ° C , 5 w a s 0 . 8 t o 3 . 9 P a / ° C , r e s p e c t i v e l y ; a n d f o r a i r a t 3 0 ° C , 3 : 2 . 4 4 P a / ° C . W h e n u s e d i n E q u a t i o n [ 1 5 3 ] , s w a s e s t i m a t e d a t t h e a r i t h m e t i c m e a n o f t h e l e a f a n d a i r t e m p e r a t u r e s . B a s e d u p o n t h e f i r s t f i v e e q u a t i o n s l i s t e d i n t h i s s e c t i o n , T h o m p s o n ( 1 9 8 1 ) o b t a i n e d t h e f o l l o w i n g e q u a t i o n s : X E : ( r e s R e + 6 ) / ( v * + r e s ) [ 1 5 5 ] 9 5 Q : ( 7 * R e - 6 ) / ( y * + r e s ) [ 1 5 6 ] w h e r e X E = l a t e n t h e a t f l u x , W / m z , O = s e n s i b l e h e a t f l u x , W / m z , R e 2 e n e r g y s o u r c e , W / m z , r e = e q u i l i b r i u m r e l a t i v e h u m i d i t y , d e c i m a l , 7 * = a p p a r e n t p s y c h r o m e t r i c c o n s t a n t , P a / ° C , a n d 6 = d r i v i n g f o r c e , W P a / m 2 ° C . T h e a p p a r e n t p s y c h r o m e t r i c c o n s t a n t w a s : 7 * : Y ( P v / I ' h ) [ 1 5 7 ] w h e r e r v = r e s i s t a n c e t o m o i s t u r e t r a n s f e r , s / m , r e s i s t a n c e t o h e a t t r a n s f e r , s / m . 1 ‘ h T h e d r i v i n g f o r c e , d e n o t e d a s D R V i n t h e m o d e l , w a s : 6 = p c p ( r e e s a - e v a ) / r h [ 1 5 8 ] w h e r e p c p = v o l u m e t r i c h e a t c a p a c i t y o f t h e a i r , J / m 3 ° C , r e = e q u i l i b r i u m r e l a t i v e h u m i d i t y o f t h e m a t e r i a l , d e c i m a l , e 8 a : s a t u r a t i o n v a p o r p r e s s u r e a t a i r t e m p e r a t u r e , P a , a n d e v a : v a p o r p r e s s u r e o f t h e a i r , P a . G o u d r i a a n ( 1 9 7 7 ) n a m e d t h e t e r m 6 a s ' d r i v i n g f o r c e ' t o r e f l e c t a v i r t u a l v a p o r p r e s s u r e d e f i c i t ( r e e s a - e v a ) . T h e d r i v i n g f o r c e 6 w a s p o s i t i v e a s l o n g a s t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y o f a l f a l f a w a s h i g h e r t h a n t h e r e l a t i v e h u m i d i t y o f t h e a i r . T h i s w a s t y p i c a l f o r f i e l d d r y i n g i f t h e a l f a l f a m o i s t u r e c o n t e n t w a s n o t v e r y l o w . E v e n i f 6 w a s n e g a t i v e , X E m i g h t s t i l l b e p o s i t i v e a s l o n g a s r e s R e w a s l a r g e r t h a n 6 i n m a g n i t u d e . T h i s e x p l a i n e d w h y a l f a l f a c o u l d b e d r i e d t o a v e r y l o w m o i s t u r e c o n t e n t w i t h a n e q u i l i b r i u m r e l a t i v e h u m i d i t y e v e n b e l o w t h e r e l a t i v e h u m i d i t y o f t h e a i r i n t h e f i e l d e n v i r o n m e n t . I f R 8 w a s n e g a t i v e , t h e n 0 w a s n e g a t i v e w h i c h o c c u r r e d n o r m a l l y 9 6 f r o m s u n s e t t o n e x t s u n r i s e . W h e n 0 < 0 , l e a f t e m p e r a t u r e w a s l o w e r t h a n a i r t e m p e r a t u r e . T h i s m i g h t e x i s t e v e n a f t e r s u n r i s e i f s h o r t - w a v e r a d i a t i o n w a s l o w . L e a f t e m p e r a t u r e i s h i g h e r t h a n a i r t e m p e r a t u r e w h e n 0 > 0 , w h i c h e x i s t e d d u r i n g m o s t o f d a y t i m e a n d p e r h a p s f o r a s h o r t p e r i o d a f t e r s u n s e t . W h e n R e > 0 , X E w a s p o s i t i v e s i n c e 6 > 0 , a c c o r d i n g t o E q u a t i o n [ 1 5 5 ] . O n t h e o t h e r h a n d , X E m i g h t s t i l l b e p o s i t i v e e v e n w h e n R e < 0 . T h i s i m p l i e d t h a t t h e d u r a t i o n f o r X E > 0 w a s l o n g e r t h a n t h a t f o r O > 0 , w i t h i n a d i u r n a l t i m e c y c l e . D u r i n g m o s t o f d a y t i m e , b o t h X E a n d 0 w e r e p o s i t i v e , i . e . l e a f t e m p e r a t u r e w a s h i g h e r t h a n t h a t o f t h e a i r a n d l e a v e s l o s e m o i s t u r e . A t n i g h t , b o t h X E a n d 0 w e r e n e g a t i v e , i . e . l e a f t e m p e r a t u r e w a s b e l o w t h a t o f t h e a i r a n d r e w e t t i n g o c c u r r e d . N e a r s u n s e t o r s u n r i s e , t h e s i t u a t i o n w a s c o m p l i c a t e d . Q a n d X E m i g h t b e e i t h e r b o t h p o s i t i v e o r n e g a t i v e ; o r o n e m i g h t b e p o s i t i v e w h i l e t h e o t h e r w a s n e g a t i v e . W h e n a l f a l f a w a s f r e s h l y c u t , r e s > 7 * a n d 6 w a s q u i t e l a r g e , t h e p e r i o d o f l o s i n g l e a f m o i s t u r e ( X E > 0 ) w a s l o n g e s t . I t w a s a l s o p o s s i b l e f o r l e a f t e m p e r a t u r e t o b e b e l o w a i r t e m p e r a t u r e i f t h e r e w a s s u c h a l a r g e v a p o r p r e s s u r e d e f i c i t i n t h e a i r t h a t O < 0 . T h i s m i g h t o c c u r o n l y a t t h e b e g i n n i n g o f f i e l d d r y i n g w h e n 7 * w a s s m a l l . A s t h e m o i s t u r e c o n t e n t o f a l f a l f a d e c r e a s e d , 6 a n d r e d e c r e a s e d , b u t 7 * i n c r e a s e d r a p i d l y . H e n c e , t h e d u r a t i o n o f O > 0 i n c r e a s e d w h i l e t h a t o f X E > 0 d e c r e a s e d . A t a c e r t a i n m o i s t u r e c o n t e n t a n d b e l o w , t h e p e r i o d o f O > 0 b e c a m e l o n g e r t h a n t h a t o f X E > 0 . F o r l i v i n g p l a n t s a n d o p e n w a t e r s u r f a c e s , t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y w a s u n i t y . T h e n , E q u a t i o n s [ 1 5 5 ] . [ 1 5 6 ] a n d [ 1 5 7 ] b e c a m e : X E = ( s R e + 6 ) / ( 7 * + s ) [ 1 5 9 ] Q = ( 7 * R e - 6 ) / ( 7 * + s ) [ 1 6 0 ] 9 7 6 = p c p ( e s a - e v a ) / r h [ 1 6 1 ] E q u a t i o n [ 1 5 9 ] h a s b e e n w e l l k n o w n a s t h e M o n t e i t h - P e n m a n e q u a t i o n w h i c h h a s b e e n s u c c e s s f u l l y a p p l i e d t o m a n y c r o p s a n d n a t u r a l w a t e r s u r f a c e s f o r a v a r i e t y o f c l i m a t e s t h r o u g h o u t t h e W o r l d . E q u a t i o n [ 1 5 5 ] w a s a m o d i f i c a t i o n o f t h e M o n t e i t h - P e n m a n e q u a t i o n . 4 . 5 . 4 . 4 L e a f T e m p e r a t u r e L e a f t e m p e r a t u r e w a s e s t i m a t e d f r o m E q u a t i o n [ 1 7 4 ) a s : T 1 : T a - t h / p c p [ 1 6 2 ] w h e r e T 1 = l e a f t e m p e r a t u r e , ° C , T a = a i r t e m p e r a t u r e , ° C , r h = b o u n d a r y r e s i s t a n c e t o h e a t t r a n s f e r , s / m , p c p = v o l u m e t r i c h e a t c a p a c i t y o f t h e a i r , J / m 3 ° C , a n d O = s e n s i b l e h e a t f l u x , W / m z . S o l u t i o n s s h o u l d b e o b t a i n e d w i t h i t e r a t i o n t e c h n i q u e s , s i n c e t h e M o n t e i t h - P e n m a n e q u a t i o n i s a n i m p l i c i t f o r m u l a . R e , 6 , 7 * , r e a n d s a r e f u n c t i o n s o f t e m p e r a t u r e a n d / o r t h e m o i s t u r e c o n t e n t o f t h e m a t e r i a l . R e i s n o t i n d e p e n d e n t o f l e a f t e m p e r a t u r e a n d m o i s t u r e s t a t u s s i n c e l o n g - w a v e r a d i a t i o n a n d r e s p i r a t i o n a r e f u n c t i o n s o f l e a f t e m p e r a t u r e . A b s o r p t i o n , r e f l e c t i o n a n d t r a n s m i s s i o n o f s h o r t - w a v e r a d i a t i o n a r e a f f e c t e d b y l e a f m o i s t u r e c o n t e n t . H o w e v e r , t h e e f f e c t o f l e a f m o i s t u r e o n s h o r t - w a v e r a d i a t i o n i s o n l y m i n o r i n c o m p a r i s o n w i t h t h e l e a f a r e a i n d e x , s o l a r a n g l e a n d s k y c o v e r . S h o r t - w a v e r a d i a t i o n c a n b e a s s u m e d t o b e n e a r l y i n d e p e n d e n t o f l e a f t e m p e r a t u r e a n d m o i s t u r e . L o n g - w a v e r a d i a t i o n i s p r o p o r t i o n a l t o t h e f o u r t h p o w e r o f t h e a b s o l u t e t e m p e r a t u r e o f a b o d y , w h i c h c h a n g e s s l i g h t l y w h e n a p p a r e n t 9 8 t e m p e r a t u r e c h a n g e s . F o r e x a m p l e , w h e n t e m p e r a t u r e c h a n g e s f r o m 3 0 ° C b y 5 a n d 1 0 ° C , l o n g - w a v e r a d i a t i o n c h a n g e s b y 6 a n d 1 4 % , r e s p e c t i v e l y . T h e m a g n i t u d e o f r e s p i r a t i o n h e a t i s v e r y s m a l l a s s h o w n e a r l i e r , s o i t s e f f e c t o n t h e d e t e r m i n a t i o n o f l e a f t e m p e r a t u r e i s n e g l i g i b l e . I n a d d i t i o n , i t e r a t i o n i s n e c e s s a r y b e c a u s e r e c h a n g e s s i g n i f i c a n t l y b e t w e e n a d s o r p t i o n a n d d e s o r p t i o n . I n t h i s m o d e l , l e a f t e m p e r a t u r e T 1 ( d e n o t e d a s D B L ) i n e a c h s w a t h l a y e r w a s e s t i m a t e d i n s u b r o u t i n e D B L E A F w i t h i n a l o o p t i l l a c o n v e r g e n c e r e q u i r e m e n t w a s m e t . I n t h e l o o p , t h e l o n g - w a v e r a d i a t i o n p r o f i l e i n s i d e t h e s w a t h w a s r e e s t i m a t e d o n l y i f t h e d i f f e r e n c e i n t e m p e r a t u r e b e t w e e n t w o s u c c e s s i v e e s t i m a t e s w a s g r e a t e r t h a n 7 ° C . E s s e n t i a l l y , t h e l e a f t e m p e r a t u r e e s t i m a t e d f r o m E q u a t i o n [ 1 6 2 ] w a s l e a f - s u r f a c e t e m p e r a t u r e . A c o n d u c t i v e h e a t t r a n s f e r e x i s t s i n s i d e a l e a f i f i t s s u r f a c e t e m p e r a t u r e d i f f e r s f r o m t h a t o f i t s i n t e r i o r . U n e v e n t e m p e r a t u r e s a r e o f t e n n e g l e c t e d b e c a u s e a l e a f c a n b e t r e a t e d a s a ' l u m p e d - c a p a c i t y ' s y s t e m , i . e . w i t h u n i f o r m t e m p e r a t u r e t h r o u g h o u t t h e l e a f b o d y . T h i s c a n b e c h e c k e d b y e s t i m a t i n g a c h a r a c t e r i s t i c t i m e o r a B i o t n u m b e r : T : ( A x ) 2 / o [ 1 6 3 ] B i = h ( A x ) / k < O . 1 ~ 0 . 2 [ 1 6 4 ] w h e r e T = c h a r a c t e r i s t i c t i m e , 3 , B i = B i o t n u m b e r , d i m e n s i o n l e s s , h = c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t , W / m z , k = h e a t c o n d u c t i v i t y o f t h e b o d y , W / m ° C , o = t h e r m a l d i f f u s i v i t y o f t h e b o d y , m z / s , a n d A x = c h a r a c t e r i s t i c d i m e n s i o n , m . T h e c h a r a c t e r i s t i c t i m e i n d i c a t e s h o w f a s t h e a t c a n d i f f u s e 9 9 t h r o u g h o u t a b o d y . I f T i s v e r y s m a l l , t h e c o n d u c t i o n h e a t t r a n s f e r c a n b e n e g l e c t e d . F o r a l f a l f a s t e m s , 0 1 : 9 8 . 3 x 1 0 ' 9 m Z / s ( M o h s e n i n , 1 9 8 0 ) . A l e a f w a s c o n s i d e r e d a s a f l a t p l a t e a n d i s o t h e r m a l i n t w o p l a n e d i m e n s i o n s , b u t n o t t h r o u g h i t s t h i c k n e s s . T h e c h a r a c t e r i s t i c d i m e n s i o n i s o n e h a l f o f i t s t h i c k n e s s , a s s u m e d t o b e 0 . 0 0 0 5 m . T h u s , T = 0 . 6 3 3 . S h o u l d a t e m p e r a t u r e p r o f i l e a l o n g t h e l e a f t h i c k n e s s b e s t u d i e d , a t i m e s t e p o f l e s s t h a n 0 . 6 3 5 w o u l d b e r e q u i r e d . T h i s i s o b v i o u s l y i m p r a c t i c a l f o r t h i s s t u d y . C o m p a r e d w i t h r e l a t i v e l y s l o w c h a n g e s i n f i e l d e n v i r o n m e n t d u r i n g a d i u r n a l c y c l e , t h e c o n d u c t i v e p r o c e s s i n l e a v e s w a s n e g l i g i b l e , i . e . i t w a s a s s u m e d t h a t l e a v e s c o u l d r e s p o n d t o e n v i r o n m e n t a l c h a n g e s i m m e d i a t e l y . T h e B i o t n u m b e r i s t h e r a t i o o f r e s i s t a n c e t o h e a t t r a n s f e r o f c o n d u c t i o n t o t h a t o f c o n v e c t i o n . A s m a l l B i o t n u m b e r i n d i c a t e s t h a t c o n d u c t i v e t r a n s f e r i n s i d e a b o d y i s f a s t e r t h a n c o n v e c t i v e t r a n s f e r a t i t s s u r f a c e . I f B i < 0 . 1 ~ 0 . 2 , n o n - u n i f o r m i t y o f b o d y t e m p e r a t u r e i s n e g l i g i b l e . A c c o r d i n g t o t h e a u t h o r ' s m e a s u r e m e n t o n 1 0 s a m p l e s o f a l f a l f a p l a n t s , t h e d r y m a t t e r d e n s i t y o f a l f a l f a l e a v e s w a s 7 4 . 5 k g / m ’ . S u p p o s e t h a t h = 2 0 " / 1 1 1 2 a n d 1 1 : 1 . 0 ( d r y b a s i s ) , t h e n o c = 0 . 3 9 9 x 1 0 ‘ J / m 3 ° C , k = o p c = 0 . 0 3 9 W / m ° C , a n d B i = 0 . 1 3 . T h i s a l s o s h o w s t h a t a ' l u m p e d - c a p a c i t y ' a s s u m p t i o n w a s r e a s o n a b l e . 4 . 6 R e w e t t i n g a n d R e d r y i n g L e a v e s g a i n m o i s t u r e f r o m d e w o r r a i n . T h i s p r o c e s s i s a l s o r e f e r r e d t o a s r e w e t t i n g . T h i s s e c t i o n f o c u s e s m a i n l y o n m o d e l i n g o f r e w e t t i n g d u e t o d e w a n d r a i n . D i s c u s s i o n o n c h a n g e s i n t h e m o i s t u r e c o n t e n t o f a l f a l f a i n d r y i n g , r e w e t t i n g a n d r e d r y i n g 1 0 0 i s a l s o p r e s e n t e d . T h e c a l c u l a t i o n o f a l f a l f a m o i s t u r e c o n t e n t s w a s c a r r i e d o u t b y s u b p r o g r a m D F M C L . 4 . 6 . 1 R a i n I n t e r c e p t i o n R a i n f a l l i s p a r t l y c a u g h t b y l e a v e s . S i n c e r a i n i s a b u l k f l o w o f w a t e r , i t d o e s n o t f o l l o w t h e r a t e e q u a t i o n d e r i v e d f r o m d i f f u s i o n t h e o r y . S m i t h ( 1 9 8 5 ) p r o p o s e d a m o d e l : 1 , . = R 1 1 - e ' ° " - ) [ 1 6 5 ] w h e r e I r = i n t e r c e p t e d r a i n , k g w a t e r / m 2 , R = r a i n i n c i d e n t , k g w a t e r / m 2 , L : d o w n w a r d l e a f a r e a i n d e x , 1 1 1 2 l e a f / m 2 , a n d a = e x t i n c t i o n c o e f f i c i e n t , m z l m 2 l e a f . T h i s e q u a t i o n w a s a s s u m e d t o b e v a l i d f o r b o t h a w h o l e c a n o p y a n d a l a y e r o f l e a v e s w h e n a n a p p r o p r i a t e L v a l u e w a s u s e d . T h e e x t i n c t i o n c o e f f i c i e n t a w a s a f u n c t i o n o f t h e i n t e r c e p t i o n c a p a c i t y I c o f t h e c a n o p y ( o r l a y e r ) , a n d e x i s t i n g i n t e r c e p t e d r a i n f a l l ( T h o m p s o n , 1 9 8 1 a n d S m i t h , 1 9 8 5 ) . T h e y r e p o r t e d t h a t I c w a s a b o u t 0 . 1 - 0 . 2 k g w a t e r p e r u n i t d r y m a t t e r . T h e i r m o d e l f o r a w a s : o : 1 0 . 1 5 f o r { 0 . S I C L < I < I C L 0 . 0 o t h e r w i s e T h u s , t h e i n t e r c e p t e d r a i n f o r a l a y e r w a s m i n [ 0 . 5 I c a L , R ( 1 - e ‘ ° ' " 3 L ) ] I r < 0 . 5 1 C 3 L I " : { 0 1 B L . f o r m i n [ I c 3 L , 0 . 5 1 0 3 L + ( R - 0 . 5 I C B L ) ( 1 - e ' ° 5 ) ] o t h e r w i s e U n i n t e r c e p t e d r a i n f a l l , i f a n y , f l o w e d t o t h e n e x t l a y e r . R a i n w a s a s s u m e d t o f a l l e v e n l y d u r i n g t h e t i m e i n t e r v a l o f l O l r e c o r d i n g . F o r e x a m p l e , w h e n d a t a f r o m t h e l o c a l w e a t h e r s e r v i c e w e r e u s e d , t h e t i m e i n t e r v a l w a s 3 h o u r s . A n y p r e c i p i t a t i o n r e c o r d e d w a s d i v i d e d b y 1 0 8 0 0 s e c o n d s , a n d t h e n m u l t i p l i e d b y a t i m e s t e p u s e d f o r i n t e g r a t i o n t o g e t t h e a v a i l a b l e r a i n f a l l d u r i n g t h e t i m e s t e p . C a l c u l a t e d I r w a s d i v i d e d b y t h e t i m e s t e p t o o b t a i n a r a i n i n t e r c e p t i o n f l u x I , d e n o t e d b y R A I N a n d e s t i m a t e d i n s u b r o u t i n e F R A I N . R a i n i n t e r c e p t i o n w a s a l s o a c c o m p a n i e d w i t h e n e r g y e x c h a n g e . H o w e v e r , t h e m a g n i t u d e o f t h e e n e r g y f l u x w a s s m a l l a n d n e g l i g i b l e . 4 . 6 . 2 D e w W h e n l e a f t e m p e r a t u r e i s b e l o w t h e d e w p o i n t o f i t s s u r r o u n d i n g a i r , w a t e r v a p o r i n t h e a i r c o n d e n s e s o n l e a f s u r f a c e s w h i c h i s c a l l e d d e w . D e w d e p o s i t i o n i s t h e c o u n t e r p r o c e s s o f m o i s t u r e e v a p o r a t i o n o n l e a f s u r f a c e s . I t h a s b e e n o b s e r v e d t h a t d e w d e p o s i t i o n o n l e a v e s h a s t w o f o r m s : f i l m a n d d r o p w i s e , a s c l a s s i f i e d i n c o n d e n s a t i o n t h e o r y . H o w e v e r , t h e r e w a s a l a c k o f i n f o r m a t i o n a b o u t d i r e c t a p p l i c a t i o n o f e x i s t i n g t h e o r y o n c o n d e n s a t i o n t o d e w f o r m u l a t i o n o n p l a n t s . N o s p e c i a l p r o v i s i o n w a s m a d e i n t h e p r e s e n t m o d e l . I t w a s a s s u m e d t h a t d e w f o r m a t i o n w a s g o v e r n e d b y t h e s a m e r a t e e q u a t i o n d e s c r i b e d i n E q u a t i o n [ 1 3 6 ] . H o w e v e r , t h e s u r f a c e r e s i s t a n c e ‘ h a d n o e f f e c t o n d e w d e p o s i t i o n a t a l l . A s a f o r e m e n t i o n e d , ” v s w a s s e t t o z e r o , w h e n E w a s n e g a t i v e , w h i c h i m p l i e d t h e m o i s t u r e t r a n s f e r f l u x w a s f r o m t h e a i r t o l e a v e s . 1 0 2 4 . 6 . 3 R e w e t t i n g V a n E l d e r e n e t a l . ( 1 9 7 2 ) s t u d i e d m o i s t u r e u p t a k e i n g r a s s s w a r d s . T h e y c l a s s i f i e d m o i s t u r e o f p l a n t s i n t o t h r e e t y p e s . T h e f i r s t w a s ' i n t e r n a l m o i s t u r e ' , e i t h e r o r i g i n a l l y h e l d o r a b s o r b e d f r o m s u r f a c e s b y p l a n t s . T h e s e c o n d t y p e w a s ' s u r f a c e m o i s t u r e ' w h i c h w a s t i g h t l y h e l d o n l e a f s u r f a c e s w h e n d e w o r r a i n o c c u r r e d . T h e t h i r d t y p e w a s ' r u n a b l e m o i s t u r e ' , t h a t w a s l e s s a s s o c i a t e d w i t h p l a n t s w h e n d e w o r r a i n w a s p r e s e n t . T h e i r a p p r o a c h w a s a d a p t e d i n t h e m o d e l . W h e n c o n d e n s a t i o n o r r a i n i n t e r c e p t i o n o c c u r r e d , w a t e r f i r s t f o r m e d a t h i n f i l m o n l e a f s u r f a c e s . T h e f i l m t h i c k n e s s i n c r e a s e d t o a c e r t a i n l i m i t . W i t h i n t h e t h i n f i l m , t h e c l o s e r t o t h e l e a f s u r f a c e w a t e r m o l e c u l e s w e r e , t h e t i g h t e r t h e y w e r e h e l d b y t h e s u r f a c e . D r o p s m i g h t b e f o r m e d a s w e l l . W a t e r d r o p s w e r e l e s s a s s o c i a t e d w i t h t h e s u r f a c e t h a n t h e f i l m . W h e n f u r t h e r d e p o s i t i o n o f w a t e r e x c e e d e d t h e h o l d i n g c a p a c i t y o f t h e l e a f s u r f a c e , t h e w a t e r d r o p s r a n o f f t h e s u r f a c e . T h e w a t e r d r o p s a n d s o m e w a t e r i n t h e f i l m w h i c h w e r e l e s s a s s o c i a t e d w i t h t h e s u r f a c e b e c a m e r u n a b l e m o i s t u r e . W h e n t h e f i l m a n d / o r d r o p s e x i s t e d o n a l e a f s u r f a c e , t h e l e a f i n t e r i o r a b s o r b e d m o i s t u r e f r o m t h e s u r f a c e w a t e r . A f t e r t h e w a t e r n e a r e s t t o t h e s u r f a c e w a s a b s o r b e d , o t h e r w a t e r w a s a t t r a c t e d t o t h e s u r f a c e a n d b e c a m e f i r m l y h e l d b y t h e s u r f a c e . W h e n r e d r y i n g o c c u r r e d , t h e w a t e r l e s s a s s o c i a t e d w i t h t h e s u r f a c e w a s e v a p o r a t e d f i r s t , t h e n , t h e w a t e r f i r m l y h e l d b y t h e s u r f a c e . U n t i l a l l s u r f a c e w a t e r w a s c o m p l e t e l y d r i e d , n o i n t e r n a l m o i s t u r e c o u l d b e d r a w n f r o m t h e p l a n t . S u b r o u t i n e D F M C L h a n d l e d t h e e s t i m a t i o n o f m o i s t u r e c o n t e n t s . 1 0 3 4 . 6 . 4 I n t e r n a l M o i s t u r e I n t e r n a l m o i s t u r e w a s r e p r e s e n t e d b y t h e i n t e r n a l m o i s t u r e c o n t e n t ( d r y b a s i s ) . T h e i n t e r n a l m o i s t u r e c o n t e n t w a s d e n o t e d a s E N T M C i n t h e m o d e l . B e f o r e c u t a l f a l f a d i e d , i t c o u l d b e c o m p l e t e l y r e c o v e r e d f r o m w a t e r s t r e s s , i f s u f f i c i e n t m o i s t u r e w a s p r o v i d e d f o r a l o n g p e r i o d . T h e r e f o r e , t h e i n t e r n a l m o i s t u r e c o n t e n t h a d a m a x i m u m . I n t h i s m o d e l , t h e m a x i m u m i n t e r n a l m o i s t u r e c o n t e n t , d e n o t e d a s X M A X M C , w a s s e t t o 7 . 0 ( d r y b a s i s ) a c c o r d i n g t o V a n E l d e r e n e t a l . ( 1 9 7 2 ) . W h e n d e w w a s p r e s e n t , c u t a l f a l f a a b s o r b e d s o m e m o i s t u r e f r o m l e a f s u r f a c e s t o t h e p l a n t i n t e r i o r . V a n E l d e r e n e t a l . ( 1 9 7 2 ) s h o w e d t h a t t h e m a x i m u m a b s o r p t i o n r a t e o f r y e g r a s s w a s p r o p o r t i o n a l t o t h e d e f i c i t o f i n t e r n a l m o i s t u r e f r o m i t s m a x i m u m . T h e p r o p o r t i o n a l c o e f f i c i e n t w a s c a l l e d t h e s p e c i f i c m o i s t u r e f l u x o f i n t e r n a l m o i s t u r e . T h e y f o u n d a v a l u e o f 1 1 . 1 x 1 0 ’ 6 k g w a t e r / s f o r e a c h k i l o g r a m o f w a t e r d e f i c i t . I n t h e m o d e l , t h i s v a l u e w a s p r e s u m e d t o b e v a l i d f o r a l f a l f a . T h u s , f o r a b s o r p t i o n : a h / a t - . - h u m a n - 1 1 ) ; ) » : [ 1 6 6 ] w h e r e a M / a t = r a t e o f c h a n g e i n i n t e r n a l m o i s t u r e , k g w a t e r / s - m z , M = i n t e r n a l m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , R M = s p e c i f i c m o i s t u r e f l u x o f M , k g w a t e r / s - k g w a t e r , M m a x = m a x i m u m M , k g w a t e r / k g d r y m a t t e r , a n d 3 m : a r e a d e n s i t y , k g d r y m a t t e r / m 2 . I n t h e m o d e l , R E N T M C , E N T M C , S M F E , X M A X M C a n d D D s t o r e d d a t a o f a M / a t , H , R M , “ m a x a n d 3 1 1 1 . A p o s i t i v e a M / B t i n d i c a t e d a n i n c r e a s e i n M , w h i c h r e s u l t e d f r o m a p o s i t i v e a v a i l a b l e m o i s t u r e f l u x . T h e a v a i l a b l e m o i s t u r e f l u x f o r a 1 0 4 l a y e r c o n s i s t e d o f a n y d e w , n e g a t i v e c o n v e c t i v e m o i s t u r e f l u x , r a i n i n t e r c e p t i o n o r e x c e s s m o i s t u r e f l u x f r o m a n u p p e r l a y e r . E a : D e w / a t + 1 , . - E + E e x [ 1 6 7 ] w h e r e E a = a v a i l a b l e m o i s t u r e f l u x , k g w a t e r / s - m z , D e w : e x i s t i n g d e w , k g w a t e r / m 2 , a t = t i m e s t e p o f i n t e g r a t i o n , s e c , I r : r a i n i n t e r c e p t i o n f l u x , k g w a t e r / s - m z , c o n v e c t i v e m o i s t u r e f l u x , k g w a t e r / s - m z , [ ' 1 I I a n d S e x = e x c e s s m o i s t u r e f l u x f r o m u p p e r l a y e r , k g w a t e r / s - m z . I n t h e m o d e l , R M O I S T , D E W , O T a n d E X C E S S s t o o d f o r E a , D e w , 3 t a n d E e x : r e s p e c t i v e l y . A I n t h i s m o d e l , i t w a s a s s u m e d t h a t a s l o n g a s t h e r e w a s m o i s t u r e e i t h e r e x i s t i n g o n o r m o v i n g t o w a r d l e a f s u r f a c e s , t h e i n t e r n a l m o i s t u r e i n c r e a s e d . T h i s m e a n t t h a t t h e a l f a l f a i n t e r i o r h a d t h e h i g h e s t p r i o r i t y t o a b s o r b m o i s t u r e f r o m i t s s u r f a c e . T h e o p p o s i t e p r o c e s s o f a d s o r p t i o n i s d e s o r p t i o n . I t w a s a s s u m e d t h a t t h e i n t e r n a l m o i s t u r e c o u l d b e l o s t o n l y i f t h e r e w a s n o d e w , r a i n , n o r e x c e s s f l u x f r o m u p p e r l a y e r s . I n a d d i t i o n , t h e c o n v e c t i v e m o i s t u r e f l u x s h o u l d b e p o s i t i v e , i . e . f r o m l e a v e s t o w a r d t h e a i r . U n d e r t h e s e c o n d i t i o n s , t h e d e s o r p t i o n w a s t o t a l l y c o n t r o l l e d b y t h e d i f f u s i o n m e c h a n i s m . I n o v e n d r y i n g , t h e i n t e r n a l m o i s t u r e c o n t e n t c a n a p p r o x i m a t e l y d r o p t o z e r o . I n t h e f i e l d , t h e i n t e r n a l m o i s t u r e c o u l d b e r e d u c e d t o a v e r y l o w l e v e l d e s p i t e t h e a i r h u m i d i t y ; b u t r a r e l y d o w n t o z e r o . I n t h i s m o d e l , a v a l u e o f 0 . 0 5 ( d r y b a s i s ) w a s a s s u m e d a s t h e m i n i m u m i n t e r n a l m o i s t u r e c o n t e n t , n a m e d X M I N M C . F o r d e s o r p t i o n , t h e r a t e o f c h a n g e i n i n t e r n a l m o i s t u r e w a s : 1 0 5 a u / a t = m a x i - E , 3 m ( M m i n - M ) / 3 t ) } [ 1 6 8 ] I f t h e m a g n i t u d e o f t h e c o n v e c t i v e m o i s t u r e f l u x E w a s g r e a t e r t h a n t h e m a x i m u m p o s s i b l e f l u x a m ( M - M m i n ) / a t , t h e s i z e o f t h e t i m e s t e p a t w a s r e d u c e d t o s u c h t h a t d u r i n g o n e t i m e s t e p t h e i n t e r n a l m o i s t u r e c o n t e n t c o u l d n o t d r o p b e l o w “ m i n - I n a d d i t i o n t o t h i s , i f t h e i n t e r n a l m o i s t u r e c o n t e n t r e a c h e d i t s m i n i m u m , t h e s u r f a c e r e s i s t a n c e r v s w a s s e t t o a v e r y l a r g e v a l u e t o p r e v e n t f u r t h e r d e c r e a s e i n t h e i n t e r n a l m o i s t u r e c o n t e n t . 4 . 6 . 5 S u r f a c e M o i s t u r e S u r f a c e m o i s t u r e w a s r e p r e s e n t e d b y a s u r f a c e m o i s t u r e c o n t e n t M s ( d r y b a s i s ) , d e n o t e d a s S F C M C . T h e s u r f a c e m o i s t u r e e x i s t s i n f i l m s o n l e a f s u r f a c e s , a n d i s q u i t e f i r m l y h e l d b y l e a v e s . V a n E l d e r e n e t a l . ( 1 9 7 2 ) f o u n d t h a t t h e s u r f a c e m o i s t u r e c o n t e n t c o u l d r e a c h a m a x i m u m o f 1 . 5 ( d r y b a s i s ) , a n d t h e s p e c i f i c m o i s t u r e f l u x f o r s u r f a c e m o i s t u r e w a s 0 . 2 5 : : 1 0 ' 3 k g w a t e r / s p e r k g o f w a t e r d e f i c i t f r o m i t s m a x i m u m . F o r a b s o r p t i o n , t h e r a t e o f c h a n g e i n s u r f a c e m o i s t u r e w a s : O M S / 3 t = m i n { E a , R 3 ( M s m a x - M s ) a m } [ 1 6 9 ] w h e r e E M S / 8 t = r a t e o f c h a n g e i n s u r f a c e m o i s t u r e , k g w a t e r / s - m z , a v a i l a b l e m o i s t u r e f l u x , k g w a t e r / s - m z , E a = M 8 = s u r f a c e m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , M s m a x : m a x i m u m M 3 , k g w a t e r / k g d r y m a t t e r , R 5 = s p e c i f i c m o i s t u r e f l u x , k g w a t e r / s - k g w a t e r , a n d 8 m = a r e a d e n s i t y , k g d r y m a t t e r / m 2 . I n t h e m o d e l , R S F C M C , S F C M C , S F C M A X a n d S M F S d e n o t e d E M S / 3 t , M s , M s m a x a n d R S , r e s p e c t i v e l y . 1 0 6 T h e a v a i l a b l e m o i s t u r e f l u x R M O I S T w a s r e d u c e d s i n c e s o m e w a s a b s o r b e d b y t h e i n t e r i o r , w h i c h h a d f i r s t p r i o r i t y . I n t h e m o d e l , t h e s u r f a c e m o i s t u r e h a d s e c o n d p r i o r i t y t o i n c r e a s e a n d d e c r e a s e . T h i s m e a n t t h a t w h e n r e w e t t i n g o c c u r r e d , t h e s u r f a c e m o i s t u r e m i g h t i n c r e a s e i f t h e r e w a s e x t r a m o i s t u r e a v a i l a b l e b e s i d e s t h a t a b s o r b e d b y t h e l e a f i n t e r i o r . T h e e x t r a m o i s t u r e c a m e f r o m e i t h e r e x i s t i n g r u n a b l e m o i s t u r e , r a i n , o r c o n v e c t i v e f l u x . T h e s u r f a c e m o i s t u r e m i g h t d e c r e a s e d u r i n g r e w e t t i n g , i f t h e r e w a s l i t t l e o r n o r u n a b l e m o i s t u r e , r a i n , n o r n e g a t i v e c o n v e c t i v e f l u x , b u t a l a r g e d e f i c i t i n i n t e r n a l m o i s t u r e . T h e s e c o n d p r i o r i t y a l s o m e a n t t h a t t h e s u r f a c e m o i s t u r e c o u l d n o t b e d r i e d a s l o n g a s a n y r u n a b l e m o i s t u r e e x i s t e d ; b u t s h o u l d b e d r i e d o u t f i r s t b e f o r e a n y d e c r e a s e o f i n t e r n a l m o i s t u r e c o u l d o c c u r . F o r d e s o r p t i o n , t h e r a t e o f c h a n g e i n s u r f a c e m o i s t u r e w a s : B M S / a t = m a x i E a , - M s a m / 3 t } [ 1 7 0 ] W h e r e t h e a v a i l a b l e m o i s t u r e f l u x s h o u l d b e n e g a t i v e . T h e s u r f a c e m o i s t u r e c o u l d d i m i n i s h c o m p l e t e l y s i n c e i t w a s l i k e ' f r e e ' w a t e r . E v a p o r a t i o n o f t h e s u r f a c e m o i s t u r e w a s n o t c o n t r o l l e d b y t h e s u r f a c e r e s i s t a n c e . F o r d r y l e a v e s , t h e s u r f a c e m o i s t u r e c o n t e n t w a s z e r o . 4 . 6 . 6 R u n a b l e M o i s t u r e R u n a b l e m o i s t u r e w a s r e p r e s e n t e d b y t h e r u n a b l e m o i s t u r e c o n t e n t , d e n o t e d a s R U N M C i n t h e m o d e l . R u n a b l e m o i s t u r e e x i s t s i n d r o p s a n d / o r f i l m s o n l e a f s u r f a c e s . V a n E l d e r e n e t a l . ( 1 9 7 2 ) f o u n d t h a t t h e m a x i m u m r u n a b l e m o i s t u r e c o n t e n t w a s 1 . 0 ( d r y b a s i s ) , a n d t h e s p e c i f i c r u n n i n g - o f f f l u x f o r r u n a b l e m o i s t u r e w a s 0 . 2 0 8 ° 1 0 " 3 k g w a t e r / s p e r k g w a t e r a v a i l a b l e , r e s p e c t i v e l y . 1 0 7 R u n a b l e m o i s t u r e w a s a s s u m e d t o h a v e t h e l e a s t p r i o r i t y o f i n c r e a s e , b u t t h e f i r s t p r i o r i t y t o d e c r e a s e . T h e o n l y c h a n c e t o i n c r e a s e r u n a b l e m o i s t u r e w a s d u e t o r a i n o r n e g a t i v e c o n v e c t i v e m o i s t u r e f l u x ; a n d t h e i n t e r n a l m o i s t u r e c o n t e n t r e a c h e d i t s m a x i m u m . F o r a d s o r p t i o n , t h e r a t e o f c h a n g e i n r u n a b l e m o i s t u r e w a s : a m p / a t = m i n { E a , ( M r m a x - M r ) 8 m / 8 t ] [ 1 7 1 ] w h e r e a M r / B t = r a t e o f c h a n g e i n r u n a b l e m o i s t u r e , k g w a t e r / s - m z , E a = a v a i l a b l e m o i s t u r e f l u x , k g w a t e r / s - m z , M r = r u n a b l e m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , “ r m a x = m a x i m u m M r , k g w a t e r / k g d r y m a t t e r , a m = a r e a d e n s i t y , k g d r y m a t t e r / m 2 , a n d a t = t i m e s t e p , s e c . E a w a s t h e r e m a i n i n g m o i s t u r e f l u x a f t e r a p o r t i o n w a s a b s o r b e d b y t h e l e a f i n t e r i o r w i t h a p o r t i o n f i r m l y h e l d b y t h e s u r f a c e . I n t h e m o d e l , R R U N M C , R U N M C a n d R U N M A X s t o o d f o r a M r / a t , M r a n d “ r m a x v r e s p e c t i v e l y . I t w a s f u r t h e r a s s u m e d t h a t t h e r e w a s a r u n - o f f f l u x , a s l o n g a s r u n a b l e m o i s t u r e e x i s t e d . E r : R e r a m [ 1 7 2 ] w h e r e E r r u n - o f f f l u x , k g w a t e r / s - m z , a n d R r s p e c i f i c r u n n i n g - o f f f l u x , k g w a t e r / s - k g w a t e r , T h u s , i f t h e r e w a s r u n a b l e m o i s t u r e o n l e a f s u r f a c e s , a n y a v a i l a b l e m o i s t u r e f l u x t o w a r d t h e s u r f a c e w o u l d b e r e d u c e d d u e t o t h e r u n - o f f f l u x . T h e r u n - o f f f l u x w a s a p o r t i o n o f t h e e x c e s s f l u x a v a i l a b l e t o t h e n e x t l a y e r i n t h e s w a t h . O t h e r s o u r c e s c a m e f r o m e x c e s s r a i n o r a n y e x t r a a v a i l a b l e m o i s t u r e a f t e r i n c r e a s e s i n t h e s e t h r e e t y p e s o f m o i s t u r e . B e s i d e s t h e r u n - o f f f l u x a n d c o n v e r s i o n t o i n t e r n a l o r 1 0 8 s u r f a c e m o i s t u r e , r u n a b l e m o i s t u r e c o u l d b e e a s i l y d r i e d b y e v a p o r a t i o n , s i n c e i t w a s f r e e w a t e r . F o r d e s o r p t i o n , t h e r a t e o f c h a n g e i n r u n a b l e m o i s t u r e w a s : a m p / a t = - M r 3 m / 3 t [ 1 7 3 ] w h e r e a M r / a t = r a t e o f c h a n g e i n r u n a b l e m o i s t u r e , k g w a t e r / s - m z , T h i s e q u a t i o n i m p l i e d t h a t r u n a b l e m o i s t u r e c o u l d b e d r i e d a t a n y r a t e , a s l o n g a s r e q u i r e d e n e r g y w a s a v a i l a b l e . 4 . 6 . 7 D e t e r m i n a t i o n o f M o i s t u r e C o n t e n t s D e w a n d i n t e r c e p t e d r a i n w e r e c o n s i d e r e d a s t h e c o m b i n a t i o n o f t h e s u r f a c e a n d r u n a b l e m o i s t u r e . I n t h e m o d e l , t h e f o l l o w i n g e x p r e s s i o n s w e r e u s e d : b e H = ( M 3 + M r ) 3 m [ 1 7 4 ] w h e r e n e w = d e w o n l e a f s u r f a c e s , k g w a t e r / m 2 , a n d B D e w / a t = r a t e o f d e w f o r m a t i o n , k g w a t e r / s - m e 2 . I n t h e m o d e l , D E W a n d R D E W d e n o t e d D e w a n d a n e w / a t , r e s p e c t i v e l y . T h e d u r a t i o n o f d e w t d e w v d e n o t e d a s T D W , w a s a c c u m u l a t e d a s l o n g a s t h e r e w a s d e w i n a l a y e r . W h e n r a t e s o f c h a n g e i n t h e t h r e e t y p e s o f m o i s t u r e w e r e d e t e r m i n e d , e a c h m o i s t u r e a f t e r o n e t i m e s t e p w a s e s t i m a t e d . T h e o v e r a l l m o i s t u r e c o n t e n t o f a l f a l f a , M C ( k g w a t e r / k g d r y m a t t e r ) , w a s : M C : M + M s + M r [ 1 7 6 ] S i n c e e v a p o r a t i o n o f d e w w a s d i f f e r e n t f r o m d r y i n g o f i n t e r n a l m o i s t u r e , i t w a s n e c e s s a r y t o s e p a r a t e t h e s e t w o p r o c e s s e s . I n t h e m o d e l , t h e t i m e s t e p o f i n t e g r a t i o n w a s a d j u s t e d s o t h a t i n o n e s t e p B r e 1 + 1 E E a E a a a l , a a a i , , , + i i i - E = = : a m m m a o o o , i i i i s s s t t t + u u u r r r l E e e e a f f f , l l l i u u u x x x = 0 u n n o i i w a h n e d ( i = t t s o i t h t f e o d e t t i e e h h a h i i t t l h h y l l e a a r y y , e e r r , g 1 k , , a 2 w g g k k a t w w n e a a d r t t / e e 3 m r r ) z / / s m m , z z s s , . [ 1 7 7 ] 1 0 9 o n l y d e w m i g h t d r y c o m p l e t e l y . T h e n , f r o m t h e n e x t s t e p , i n t e r n a l m o i s t u r e b e g a n t o d r y . F i g u r e 6 s h o w s a s i m u l a t e d m o i s t u r e p r o f i l e i n s i d e a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 t o 1 6 o f 1 9 8 4 . 4 . 7 B a l a n c e o f M o i s t u r e a n d E n e r g y i n A i r S p a c e s C o n s e r v a t i o n o f e n e r g y a n d m a s s i s a u n i v e r s a l l a w . F r o m a m a s s a n d e n e r g y b a l a n c e , t e m p e r a t u r e a n d h u m i d i t y s t a t u s i n a i r s p a c e s w i t h i n a s w a t h c a n b e e s t i m a t e d . S u b m o d e l M A T R I X s i m u l a t e d e x c h a n g e s o f w a t e r v a p o r a n d s e n s i b l e h e a t a n d e s t i m a t e d t e m p e r a t u r e a n d h u m i d i t y s t a t u s o f a i r s p a c e s i n s i d e a s w a t h . T h e n a m e o f ' m a t r i x ' i m p l i e d t h a t a m a t r i x m e t h o d w a s u s e d t o s o l v e a s y s t e m o f s i m u l t a n e o u s f i n i t e d i f f e r e n c e e q u a t i o n s . 4 . 7 . 1 M o i s t u r e B a l a n c e i n A i r S p a c e o f a S w a t h L a y e r M o i s t u r e b a l a n c e i n t h e a i r s p a c e o f a s w a t h l a y e r w a s w r i t t e n a s : E l a , i w a s t h e m o i s t u r e f l u x f r o m l e a v e s t o w a r d t h e a i r s p a c e i n s i d e t h e i t h l a y e r . E a a , i a n d E a a , i + 1 w e r e t h e w a t e r v a p o r f l u x e s b e t w e e n a i r s p a c e s . 6 3 3 , 1 w a s f r o m t h e i t h l a y e r t o t h e a d j a c e n t l a y e r a b o v e w h i l e E a a , i + 1 w a s f r o m t h e a d j a c e n t l a y e r b e l o w i n t o t h e l a y e r . A t t h e t o p o f t h e s w a t h , E a a , i w a s t o w a r d t h e a t m o s p h e r e . A t t h e b o t t o m o f 0 I : 8 1 a 0 n I : i 2 h 1 t i . w 4 8 t 9 1 n e t , n 6 o 1 c - 4 1 e r u t t s 0 I : 6 6 ) 0 I 1 e s u : 0 / t i g o u 8 a m A r e y a L p o T r e e y r D a n f o l y a 0 d a n a L I : f a L e l d d m h a 8 1 n l g a i o e c t t r t M h i i o u M B 0 , f I : o g o 2 1 H e i n ( l s i n f a o L e r p t 0 m I s : 6 i d a e E T t a n l i u m h i t s a w A s . 6 e r u g i F 5 0 I 1 : 0 / 8 0 I : 8 1 4 0 : 1 - 0 . 2 1 - 9 - - 2 / 0 0 0 0 . . . . 8 1 2 2 0 1 m0 3 . 0 - ' P V 0 . 0 4 3 . 0 - 2 . 0 - 1 . 0 - i u e i u o o 9 . 1 m . 0 4 3 . 0 — ] 3 . 0 - fl 3 2 o w a b o J a A v u i o m s S w a t h O v e r a l l 1 1 0 1 1 1 t h e s w a t h , E a a , i + 1 w a s f r o m t h e s o i l s u r f a c e i n t o t h e s w a t h . A m o i s t u r e s t o r a g e t e r m w a s i g n o r e d i n E q u a t i o n [ 1 7 7 ] . T h e c h a r a c t e r i s t i c t i m e f o r m o i s t u r e d i f f u s i o n w a s : r = ( A x ) 2 / D W h e r e 0 w a s t h e d i f f u s i v i t y o f w a t e r v a p o r i n a i r , ~ 2 5 . 6 x 1 0 m z / s ( H o l m a n , 1 9 8 1 ) . T h e c h a r a c t e r i s t i c d i m e n s i o n f o r t h e a i r s p a c e o f a s w a t h l a y e r w a s t h e d i m e n s i o n o f a i r g a p s a l o n g t h e v e r t i c a l d i r e c t i o n . S u p p o s e a 0 . 1 m t h i c k s w a t h w i t h a p o r o s i t y o f 0 . 1 w a s d i v i d e d i n t o t h r e e l a y e r s , t h e n A x = 0 . 0 0 1 7 m a n d T = 0 . 1 1 s . T h u s , a u n i f o r m d i s t r i b u t i o n o f w a t e r v a p o r i n t h e a i r s p a c e o f a s w a t h l a y e r c o u l d b e r e a s o n a b l y a s s u m e d . F l u x e s i n E q u a t i o n [ 1 7 7 ] w e r e a p p r o x i m a t e l y e x p r e s s e d a s : E 1 a , i ( O w l , i - O w a , i > / P v , i E a a , i = ( D w a , i - O w a , i - 1 ) / P s v , i ( i = 1 , 2 a n d 3 ) E a a , i + 1 = ( D w a , i + 1 - P w a , i ) / P s v , i + 1 v a p o r d e n s i t y i n l e a v e s o f t h e i t h l a y e r , k g w a t e r / m 3 , w h e r e 9 w l , i v a p o r d e n s i t y i n t h e a i r s p a c e o f t h e l a y e r , k g w a t e r / m 3 , P w a , i = P s v , i : r e s i s t a n c e t o m o i s t u r e t r a n s f e r i n a i r s p a c e , s / m , a n d r v , 1 = r e s i s t a n c e t o m o i s t u r e t r a n s f e r o n l e a f s u r f a c e s , s / m . F o r t h e t o p l a y e r o f t h e s w a t h ( i = 1 ) , P w a O w a s O w r : t h e v a p o r d e n s i t y a t t h e r e f e r e n c e h e i g h t . F o r t h e b o t t o m l a y e r o f t h e s w a t h ( i = 3 ) , D w a 4 a n d P s v 4 w e r e t h e v a p o r d e n s i t y a n d r e s i s t a n c e t o m o i s t u r e t r a n s f e r a t t h e s o i l s u r f a c e , i . e . o n e a n d ” s v g o r e s p e c t i v e l y . A s y s t e m o f s i m u l t a n e o u s e q u a t i o n s w a s c o n s t r u c t e d : A 1 1 p w a 1 + A 1 2 0 w 3 2 : B 1 ( t o p l a y e r ) A 2 1 p w a 1 + A g z p w a z + A 2 3 p w a 3 : B 2 ( m i d d l e l a y e r ) } [ 1 7 8 ] A 3 2 0 w 3 2 + A 3 3 9 w a 3 : B 3 ( b o t t o m l a y e r ) 1 1 2 w i t h 1 / I ‘ s v ’ 1 + 1 / I ‘ s v ’ i + 1 + 1 / r v ’ i A i , i A i , i - 1 ' 1 / r s v , i - 1 / r s v , 1 + 1 ( i = 1 , 2 a n d 3 ) A i , i + 1 B 1 0 w l 1 / r v 1 + O w r / r s v l 9 w 1 2 / r v 2 B 2 B 3 w h e r e P w r : w a t e r v a p o r d e n s i t y a t r e f e r e n c e h e i g h t , k g w a t e r / m 3 , 0 w l 3 / P v 3 + O w g / P s v g a n d O w g = w a t e r v a p o r d e n s i t y a t t h e s o i l s u r f a c e , k g w a t e r / m 3 . T h i s s y s t e m o f s i m u l t a n e o u s e q u a t i o n s w a s s o l v e d i n s u b r o u t i n e M A T R I X . A r r a y s R V A , R V L , R S V , R B V , a n d v a r i a b l e R V R a n d R V S s t o r e d P w a , i : O w l , i 9 r s v , i v r v , i v D w r a n d O w s v r e s p e c t i v e l y . A c c o m p a n i e d w i t h a m o i s t u r e b a l a n c e , t h e r e w a s a l a t e n t h e a t b a l a n c e . W h e n e a c h t e r m i n E q u a t i o n [ 1 7 7 ] w a s m u l t i p l i e d b y c o r r e s p o n d i n g l a t e n t h e a t o f v a p o r i z a t i o n , t h a t e q u a t i o n b e c a m e a l a t e n t h e a t b a l a n c e e q u a t i o n . T h e l a t e n t h e a t o f v a p o r i z a t i o n c h a n g e d s l i g h t l y a s t e m p e r a t u r e c h a n g e d ; t h e r e f o r e , i t w a s a s s u m e d t h a t t h e l a t e n t h e a t b a l a n c e a u t o m a t i c a l l y h e l d a s l o n g a s t h e m o i s t u r e b a l a n c e h e l d . 4 . 7 . 2 S e n s i b l e H e a t B a l a n c e i n A i r S p a c e s T o t a l e n e r g y i n t h e a i r s p a c e o f a s w a t h l a y e r c o n s i s t e d o f s e n s i b l e , l a t e n t a n d r a d i a t i o n h e a t . I t w a s a s s u m e d e a r l i e r t h a t r a d i a t i o n e n e r g y e x c h a n g e d o n l y o n l e a f a n d s o i l s u r f a c e s . L a t e n t h e a t w a s a s s u m e d t o b e b a l a n c e d a l o n g w i t h w a t e r v a p o r b a l a n c e i n a i r s p a c e s . T h u s , t o t a l e n e r g y w o u l d b e b a l a n c e d i f s e n s i b l e h e a t w a s b a l a n c e d . 1 1 3 F o r s e n s i b l e h e a t t r a n s f e r i n a i r s p a c e o f t h e s a m p l e s w a t h l a y e r d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n , t h e c h a r a c t e r i s t i c t i m e T w a s 0 . 1 2 s , a s c a l c u l a t e d f r o m E q u a t i o n [ 1 6 3 ] w i t h t h e t h e r m a l d i f f u s i v i t y o f t h e a i r a t 3 0 ° C o f 2 2 . 8 1 : 1 0 ’ 6 W / m 2 ° C . T h e r e f o r e , a h e a t s t o r a g e t e r m w a s e x c l u d e d f r o m t h e s e n s i b l e h e a t b a l a n c e , a n d u n i f o r m l y d i s t r i b u t e d t e m p e r a t u r e w a s a s s u m e d w i t h i n t h e a i r s p a c e o f e a c h l a y e r . A s e n s i b l e h e a t b a l a n c e e q u a t i o n w a s w r i t t e n a s : Q a a , i + 1 - 0 3 3 , 1 + 0 1 3 , 1 = 0 ( i = 1 . 2 a n d 3 ) [ 1 7 9 ] h e a t f l u x o u t o f t h e i t h l a y e r , W / m z , w h e r e 0 3 3 , 1 h e a t f l u x i n t o t h e i t h l a y e r , W / m z , Q a a , i + 1 h e a t f l u x f r o m l e a v e s t o a i r s p a c e , W / m z . a n d Q l a , i F l u x e s i n E q u a t i o n [ 1 7 9 ] w e r e a p p r o x i m a t e l y e x p r e s s e d a s : Q a a , i + 1 = h a a , i + 1 ( T a , i + 1 ‘ T a , i ) Q a a , i = h a a , i ( T a , i - T a , i - 1 ) } ( i = 1 . 2 a n d 3 ) Q l a , i = h l a , i ( T l , i ‘ T a , i ) h e a t - t r a n s f e r c o e f f i c i e n t b e t w e e n l a y e r s , W / m 2 ° C , w h e r e h a a , i h l a , i : h e a t - t r a n s f e r c o e f f i c i e n t f r o m l e a v e s t o a i r , W / m Z C . T a , 1 = a i r t e m p e r a t u r e i n t h e i t h l a y e r , ° C , a n d T 1 , 1 : l e a f t e m p e r a t u r e i n t h e i t h l a y e r , ° C . T a u , Q a a 4 a n d h a a 4 w e r e t h e t e m p e r a t u r e , h e a t f l u x a n d h e a t - t r a n s f e r c o e f f i c i e n t a t t h e s o i l s u r f a c e , r e s p e c t i v e l y . T a o w a s t h e a i r t e m p e r a t u r e a t t h e r e f e r e n c e h e i g h t . S i m i l a r t o t h e w a t e r v a p o r b a l a n c e , a s y s t e m o f s i m u l t a n e o u s e q u a t i o n s w a s f o r m u l a t e d : C 1 1 T a 1 + C 1 2 T a 2 = D 1 } [ 1 8 0 ] 1 1 D N C 2 1 T a 1 + C 2 2 T a 2 + C 2 3 T a 3 C 3 2 T a z + C 3 3 T a 3 : D 3 1 1 4 w i t h h a a , i + h a a , i + 1 + h l a , i ‘ h a a , 1 } ( 1 : 1 , 2 a n d 3 ) C i , i C i , i - 1 C i , i + 1 = - h a a , i + 1 D 1 = h 1 a 1 T l 1 + h a a 1 T r D 2 = h l a 2 T 1 2 w h e r e T r = a i r t e m p e r a t u r e a t r e f e r e n c e h e i g h t , ° C , t e m p e r a t u r e a t s o i l s u r f a c e , ° C . a n d T 8 T h i s s e t o f s i m u l t a n e o u s e q u a t i o n s w a s a l s o s o l v e d i n s u b r o u t i n e M A T R I X , w h e r e a r r a y s D B A , D B L , H A A , H L A , a n d v a r i a b l e s B E R a n d 0 8 0 d e n o t e d T a , 1 , T 1 , 1 , h a a , i : h l a , i a T r a n d T B ' r e s p e c t i v e l y . F i g u r e s 7 a n d 8 s h o w s i m u l a t e d p r o f i l e s o f t e m p e r a t u r e a n d w a t e r v a p o r d e n s i t y w i t h i n a s w a t h i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 t o 1 6 o f 1 9 8 4 . 4 . 8 E x c h a n g e s i n S o i l S o i l i s a r e s e r v o i r o f b o t h h e a t a n d w a t e r . T h i s s e c t i o n d i s c u s s e s m o d e l i n g o f h e a t e x c h a n g e s a t t h e s o i l s u r f a c e a n d i n s i d e t h e s o i l . W a t e r s t a t u s o f t h e s o i l w i l l b e d i s c u s s e d b r i e f l y . S u b m o d e l D B S O I L a s s o c i a t e d w i t h s u b p r o g r a m G R O U N D s i m u l a t e d e x c h a n g e s i n s i d e s o i l a n d a t t h e s o i l s u r f a c e , r e s p e c t i v e l y . S u b r o u t i n e S O I L I i n i t i a l i z e d a s o i l t e m p e r a t u r e p r o f i l e a t t h e b e g i n n i n g o f a s i m u l a t i o n . S e n s i b l e h e a t f l u x e s b e t w e e n a d j a c e n t l a y e r s o f s o i l w e r e c a l c u l a t e d i n s u b r o u t i n e D F D B S . 0 : 8 1 t - s a E n i 0 : 2 h 1 t a w s a r o f . 4 e 8 r 9 u 1 t a 0 f : 6 6 0 1 h t a w w r e c a : , 0 r 6 8 e e 1 t p - m 4 a e 1 D t r e t f 0 e y r s r : d y a i u u 8 n S L a g a L S 1 a u f A e m l v e o r i o o b a r i A } ° 5 l n o t u S d d i t 0 o o w : H e o l i n M B o 2 ( f a l 1 e b e o g r i p h a m c i d i 0 T e M : t a 6 , l g u n m i i s s n 5 a 0 1 : / 0 8 A L . 7 e r u 0 g : i 8 F 1 r 1 . W W 4 0 : 1 d : - n 0 0 a 5 3 ] 2 / 0 0 3 8 1 1 ( 9 ° ) a i n l o i e d u i a i 1 1 5 W H - e . . c 4 s J 0 : 2 1 a 8 p 9 s 1 r , i 6 s a q r m H " f / X a 1 W + 0 : 6 — $ ~ e 4 h 1 t t n s i u g y u t A i - 6 s n H 0 1 n o : 0 e d n e a ¥ t r g a o i \ H \ p h \ \ * \ \ ‘ X ” _ D a c 0 v i : d M 8 1 n r ‘ q ” : m “ a e , t g a n a : i “ r w " ’ H u s / / H : . P 0 o f n o a / L ’ e e 1 / 2 ( c l t I f r a i s e r e y a L e f f a - m y r o E a u r r i V - i : L S l 0 p n m l 6 d h t a w S e v o ? p S b a r i A a t t — T P i w o B o 5 l ‘ 0 1 e : o o i m e h t t a a l w u s m i a b fl 0 / s N A o 8 f — \ k \ W W 0 : 8 1 s A { fi 0 i 1 : 4 1 0 2 / . 8 1 0 . 8 e r u g i F 3 3 5 3 3 L 5 . 1 x / \ Z . 3 ( J 1 0 , u J / J a i o m 5 7 ) £ 4 1 3 q u J O d D A 8 3 3 8 3 t - — — I — - ? _ § — - r — - 1 r — w — - 1 3 6 1 ; ] 3 1 8 : 0 i . I : 1 . I I 3 . I I 1 1 6 l l 7 u . 8 . 1 E n e r g y B a l a n c e a t S o i l S u r f a c e T h e s o i l s u r f a c e i s a c o n v e c t i v e b o u n d a r y f o r t h e s o i l u n d e r n e a t h . E x c h a n g e s o f s e n s i b l e , l a t e n t a n d r a d i a t i o n h e a t t a k e p l a c e o n t h e s o i l s u r f a c e . T h e s o i l s u r f a c e w a s a s s u m e d t o b e a f l a t p l a t e w i t h a n i n f i n i t e s i m a l t h i c k n e s s ( H o l m a n , 1 9 8 1 ) , a n d a b a l a n c e e q u a t i o n w a s w r i t t e n a s : R n - Q - A E = G [ 1 8 1 ] w h e r e R n : n e t a b s o r b e d r a d i a t i o n f l u x , W / m z , Q = u p w a r d s e n s i b l e h e a t f l u x , W / m z , A B = u p w a r d l a t e n t h e a t f l u x , W / m z , a n d G : d o w n w a r d h e a t f l u x i n t o s o i l , W / m z . E a c h t e r m w a s e x p r e s s e d a s : Q : h a g ( T g - T 3 3 ) A B = “ 3 8 ( 9 W 8 - p w a B ) G = h 3 1 ( T g ‘ T 3 1 ) w h e r e h a g = h e a t - t r a n s f e r c o e f f i c i e n t a t t h e s o i l s u r f a c e , W / m 2 ° C L l a t e n t h e a t t r a n s f e r c o e f f i c i e n t , W m / k g w a t e r , < 9 1 o n u h 3 1 : h e a t - t r a n s f e r c o e f f i c i e n t i n s i d e t h e s o i l , W / m 2 ° C h t e m p e r a t u r e a t t h e s o i l s u r f a c e , ° C , v - i o n 0 1 T 3 3 = t e m p e r a t u r e i n t h e b o t t o m l a y e r o f t h e s w a t h , ° C , t e m p e r a t u r e o f t h e t o p l a y e r o f t h e s o i l , ° C , v - J u I : o w e : v a p o r d e n s i t y a t t h e s o i l s u r f a c e , k g w a t e r / m 3 , a n d 9 3 3 = v a p o r d e n s i t y i n t h e b o t t o m l a y e r o f t h e s w a t h , k g w a t e r / m 3 . T h e l a t e n t h e a t t r a n s f e r c o e f f i c i e n t b e t w e e n t h e s o i l s u r f a c e a n d t h e b o t t o m l a y e r o f t h e s w a t h w a s : 1 1 8 w h e r e A l a t e n t h e a t o f v a p o r i z a t i o n , J / k g w a t e r , r e s i s t a n c e t o m o i s t u r e t r a n s f e r a t t h e s o i l s u r f a c e , s / m . a n d r s v g T h e c o n d u c t i v e h e a t - t r a n s f e r c o e f f i c i e n t i n t h e t o p l a y e r o f s o i l w a s 3 w h e r e k s = t h e r m a l c o n d u c t i v i t y o f t h e s o i l , W / m ° C , a n d A 2 = t h i c k n e s s o f t h e t o p l a y e r o f t h e s o i l , m . T h e t e m p e r a t u r e a t t h e s o i l s u r f a c e w a s , t h u s , e s t i m a t e d a s : T 8 = [ ( R n - X E ) + h a g T a 3 + h 3 1 T 3 1 1 / ( h a g + h 3 1 ) [ 1 8 2 ] T h i s i s a n i m p l i c i t f o r m u l a , s i n c e t h e l a t e n t h e a t f l u x d e p e n d s o n t e m p e r a t u r e . I n t h e m o d e l , T 8 w a s e s t i m a t e d i n s u b r o u t i n e G R O U N D t h r o u g h a n i t e r a t i o n p r o c e d u r e . I n t h e s u b m o d e l , R V A ( 3 ) , R V G , D B A ( 3 ) , D B G , 0 8 8 ( 1 ) , H A A ( 4 ) a n d V A A ( H ) d e n o t e d 0 w a 3 a O W S ’ T 3 3 , T 3 , T 3 1 , h a s a n d v 3 8 , r e s p e c t i v e l y . T h e e s t i m a t i o n o f h a s a n d V 3 8 w a s d i s c u s s e d i n S e c t i o n 4 . “ . 3 . I n t h i s m o d e l , h a g a n d V 3 8 w e r e e s t i m a t e d i n f u n c t i o n s u b p r o g r a m R S H G N D . u . 8 . 2 R e l a t i v e H u m i d i t y a t S o i l S u r f a c e V a p o r d e n s i t y a t t h e s o i l s u r f a c e w a s a n i m p o r t a n t p a r a m e t e r f o r e s t i m a t i o n o f s o i l s u r f a c e t e m p e r a t u r e a n d t h e v a p o r e x c h a n g e b e t w e e n t h e s o i l s u r f a c e a n d t h e s w a t h . W a t e r v a p o r d e n s i t y i n a i r i s a f u n c t i o n o f t e m p e r a t u r e a n d r e l a t i v e h u m i d i t y o f t h e a i r . A i r i s o f t e n t r e a t e d a s a m i x t u r e o f i d e a l g a s e s o f d r y a i r a n d w a t e r v a p o r . A c c o r d i n g t o t h e i d e a l g a s l a w , 0 w = e v / R w T = r a e s / R W T [ 1 8 3 ] w h e r e p w = v a p o r d e n s i t y i n t h e a i r , k g w a t e r / m 3 , 1 1 9 T = a b s o l u t e t e m p e r a t u r e o f t h e a i r , ° K , v a p o r p r e s s u r e i n t h e a i r , P a , ( D < u e s = s a t u r a t i o n v a p o r p r e s s u r e a t T , P a , r e l a t i v e h u m i d i t y o f t h e a i r , d e c i m a l , a n d R w = g a s c o n s t a n t o f w a t e r v a p o r , 4 6 1 J / ° K - k g w a t e r . W h e n T i s k n o w n , e s c a n b e e s t i m a t e d , a n d t h e n p " c a n b e p r e d i c t e d f r o m k n o w n T a n d r a . I n t h e s o i l , t h e r e l a t i v e h u m i d i t y c a n b e e s t i m a t e d f r o m t h e o s m o t i c p o t e n t i a l o f t h e s o i l . r 3 : e - a n / R T [ 1 8 4 ] w h e r e r s = r e l a t i v e h u m i d i t y i n s o i l , d e c i m a l , V m o l e c u l a r v o l u m e o f w a t e r , 0 . 0 1 8 8 m 3 / k g m o l e , m u = o s m o t i c p o t e n t i a l i n t h e s o i l , P a , R u n i v e r s a l g a s c o n s t a n t , 8 3 1 ” J / ° K - k g m o l e , a n d T a b s o l u t e t e m p e r a t u r e o f t h e s o i l , ° K . I n u n s a t u r a t e d s o i l , t o t a l w a t e r p o t e n t i a l 0 i s t h e s u m o f t h e o s m o t i c p o t e n t i a l o n a n d t h e w a t e r p o t e n t i a l d u e t o c a p i l l a r y e f f e c t s w e , b o t h a r e n e g a t i v e . I f t h e t o t a l w a t e r p o t e n t i a l o f a s o i l i s - 1 . 5 x 1 0 6 P a , t h e n t h e s o i l i s s a i d t o b e a t t h e w i l t i n g p o i n t . 0 n t h e o t h e r h a n d , a s o i l i s a t f i e l d c a p a c i t y i f t h e t o t a l w a t e r p o t e n t i a l i s - 3 3 x 1 0 3 P a . I t c a n b e s h o w n t h a t f o r s u c h a w i d e r a n g e o f t h e t o t a l w a t e r p o t e n t i a l o f a s o i l , t h e r e l a t i v e h u m i d i t y i n t h e s o i l c h a n g e s v e r y s l i g h t l y . S u p p o s e t h e o s m o t i c p o t e n t i a l o f a s o i l v a r i e s f r o m - 1 . 5 x 1 0 6 t o - 3 3 x 1 0 3 P a , t h e n t h e r e l a t i v e h u m i d i t y o f t h e s o i l c h a n g e s J u s t f r o m 0 . 9 8 9 t o 1 . 0 0 . T h i s i n d i c a t e s t h a t t h e r e l a t i v e h u m i d i t y o f t h e s o i l i s v e r y c l o s e t o u n i t y ; t h e r e f o r e , t h e w a t e r v a p o r d e n s i t y a t a s o i l s u r f a c e c a n b e a s s u m e d t o b e n e a r t h e s a t u r a t i o n v a p o r d e n s i t y a t t h e s u r f a c e t e m p e r a t u r e . 1 2 0 I n t h e m o d e l , t h e r e l a t i v e h u m i d i t y a t t h e s o i l s u r f a c e w a s s e t a t a - c o n s t a n t v a l u e o f 0 . 9 9 . B a s e d u p o n t h e f o r e g o i n g d i s c u s s i o n , n e i t h e r w a t e r m o v e m e n t n o r m o i s t u r e t r a n s f e r i n s o i l w a s m o d e l e d i n t h i s s t u d y . “ . 8 . 3 S o i l T e m p e r a t u r e P r o f i l e A t e m p e r a t u r e p r o f i l e i n a s o i l i s t h e r e s u l t o f a c o n d u c t i v e h e a t - t r a n s f e r p r o c e s s i n t h e s o i l . T o m o d e l t h i s p r o c e s s , a h e a t b a l a n c e e q u a t i o n w a s f o r m e d a s : Q s , i - Q 3 ’ 1 + 1 = 3 3 . 1 ( 1 : 1 t o 1 0 ) [ 1 8 5 ] w h e r e 0 3 , 1 = h e a t f l u x i n t o t h e i t h l a y e r , W / m z , Q s , 1 + 1 = h e a t f l u x o u t o f t h e i t h l a y e r , W / m z , a n d 3 3 , 1 = h e a t s t o r a g e f l u x i n t h e i t h l a y e r , W / m z . I n t h e m o d e l , d o w n w a r d h e a t f l u x e s i n t h e s o i l h a d a p o s i t i v e s i g n . I n s o i l , t h e s t o r a g e t e r m c o u l d n o t b e n e g l e c t e d , b e c a u s e t h e s o i l h a d a l a r g e h e a t c a p a c i t y . I n t h i s m o d e l , a s o i l o f 0 . 5 m i n d e p t h w a s d i v i d e d i n t o 1 0 l a y e r s . E a c h l a y e r w a s 0 . 0 5 m d e e p . T h e t h e r m a l c o n d u c t i v i t y k a n d v o l u m e t r i c h e a t c a p a c i t y b e o f t h e s o i l w e r e a s s u m e d t o b e 1 . 3 H / m ° C a n d 2 x 1 0 6 J / m 3 ° C , r e s p e c t i v e l y ( G o u d r i a a n , 1 9 7 7 ) . H e n c e , 0 1 : 0 . 6 ‘ 5 x 1 0 ' 6 m z / s , a n d 1 : 3 8 4 6 3 . A t t h e t o p l a y e r o f t h e s o i l , i t s c e n t e r w a s 0 . 0 2 5 m b e l o w t h e s o i l s u r f a c e . T h e c h a r a c t e r i s t i c t i m e f o r t h e t o p l a y e r w a s 9 6 1 3 . T h e s e s h o w t h a t t h e c o n d u c t i v e h e a t t r a n s f e r i n s i d e t h e s o i l w a s a s l o w p r o c e s s , i n c o m p a r i s o n w i t h t h o s e i n s w a t h e s . S i n c e t h e h e a t c a p a c i t y i n s o i l w a s v e r y l a r g e , i t w a s n e c e s s a r y t o i n c l u d e a s t o r a g e t e r m i n t h e h e a t b a l a n c e e q u a t i o n [ 1 8 5 ] . A l t h o u g h h e a t s t o r a g e w a s n o l o n g e r n e g l i g i b l e f o r s l o w h e a t 1 2 1 t r a n s f e r i n s a i l , a ' l u m p e d - c a p a c i t y ' a p p r o a c h w a s s t i l l a p p l i c a b l e . S i n c e w i n d s p e e d w a s n e a r z e r o a t t h e s o i l s u r f a c e , a v e r y s m a l l c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t w a s e x p e c t e d a t t h e s o i l s u r f a c e . T h u s , i t w a s p o s s i b l e t o m e e t t h e B i o t n u m b e r c r i t e r i o n . S u p p o s e a t t h e s o i l s u r f a c e , T = 3 0 ° C = 3 0 3 ° K , A T : 1 0 ° C , X : 1 m , A z = 0 . 0 2 5 m a n d k s : 1 . 3 W / m ° C , t h e n G r = 1 . 2 6 x 1 0 9 , G r P r = 0 . 8 8 5 x 1 0 9 , 0 : 0 . 1 5 , m = 0 . 3 3 , h : 3 . 8 W / m Z ‘ C a n d B i = 0 . 0 7 < 0 . 1 . T h e r e f o r e , t h e l u m p e d - c a p a c i t y a s s u m p t i o n w a s a l s o u s e d i n m o d e l i n g t h e c o n d u c t i v e h e a t t r a n s f e r i n t h e s o i l . T h e o n e - d i m e n s i o n a l h e a t c o n d u c t i o n i s g o v e r n e d b y F o u r i e r ' s l a w : Q = - k 8 T / a z w h e r e Q = h e a t f l u x , W / m z , k = t h e r m a l c o n d u c t i v i t y , W / m ° C , a n d a r / a z = t e m p e r a t u r e g r a d i e n t a c r o s s 8 2 , ° C / m . A b a c k w a r d f i n i t e d i f f e r e n c e f o r m u l a w a s e m p l o y e d t o e x p r e s s e a c h t e r m i n E q u a t i o n [ 1 8 6 ] a s : Q s , i = k s ( T s , i - 1 ‘ T s , i ) / A Z Q s , i + 1 = k s ( T s , 1 - T s , 1 + 1 ) / A z } ( i = 1 t o 1 0 ) 8 3 , 1 = p c A z ( T 3 , 1 - T x , 1 ) / A t w h e r e k 3 = t h e r m a l c o n d u c t i v i t y o f t h e s o i l , W / m ° C , A t = t i m e s t e p , 3 , A 2 = t h i c k n e s s o f t h e s o i l l a y e r , m , T x , 1 = p r e s e n t s o i l t e m p e r a t u r e o f t h e l a y e r , ° C , a n d T s , 1 : f u t u r e s o i l t e m p e r a t u r e o f t h e l a y e r , ° C . T s , i - 1 ’ T s , i a n d T s , i + 1 w e r e s o i l t e m p e r a t u r e s i n t h r e e a d j a c e n t l a y e r s a f t e r o n e t i m e s t e p A t f r o m a g i v e n t i m e . U s i n g b a c k w a r d f i n i t e d i f f e r e n c e f o r m u l a e h a s t h e a d v a n t a g e o f r e l e a s i n g a s t a b i l i t y c o n s t r a i n t o n t i m e s t e p s f o r n u m e r i c a l i n t e g r a t i o n . S h o u l d f o r w a r d 1 2 2 d i f f e r e n c e f o r m u l a e b e a p p l i e d , t h e s t a b i l i t y c o n s t r a i n t r e q u i r e s t i m e s t e p s m u c h s m a l l e r t h a n t h e c h a r a c t e r i s t i c t i m e T ( H o l m a n , 1 9 8 1 ) . B a c k w a r d f i n i t e d i f f e r e n c e s a r e i m p l i c i t f o r m u l a e w h i c h m a y b e s o l v e d w i t h v a r i o u s m e t h o d s , s u c h a s m a t r i x , i t e r a t i o n a n d r e l a x a t i o n m e t h o d s . I n t h i s m o d e l , a m a t r i x m e t h o d w a s u s e d . A _ s e t o f s i m u l t a n e o u s e q u a t i o n s w a s f o r m u l a t e d : G 1 1 T s 1 + G 1 2 T s 2 = F 1 0 1 , 1 - 1 T 3 ’ 1 _ 1 + 0 1 , 1 T 3 , 1 + G i , 1 + 1 T s , i + 1 : F 1 } ( 1 = 2 t o 9 ) [ 1 8 6 ] G 9 1 0 T s 9 + G 1 0 1 0 1 3 1 0 = F 1 0 w i t h 0 1 , 1 _ 1 : - a A t / ( A z ) 2 6 1 , 1 : 1 + 2 a A t / ( A z ) 2 ( 1 : 2 t o 9 ) 0 1 ’ 1 “ : — c h t / ( A z ) z F 1 = T x , 1 G 1 1 = 1 + 3 e A t / ( A z ) 2 r 1 = T x 1 + Z T g a A t ) / ( A z ) 2 0 1 0 1 0 = 1 + 3 a A t / ( A z ) 2 5 1 0 = T X 1 O + Z T d a A t / ( A z ) 2 w h e r e T 3 = s o i l s u r f a c e t e m p e r a t u r e , ° C , T a = s o i l t e m p e r a t u r e a t 0 . 5 m d e p t h , ° C , A t = t i m e s t e p , 3 , A 2 : t h i c k n e s s o f a s o i l l a y e r , m , a n d a = t h e r m a l d i f f u s i v i t y o f t h e s o i l , m z / s . A s m e n t i o n e d i n C h a p t e r 3 , s o i l t e m p e r a t u r e b e l o w 0 . 5 m w a s a s s u m e d t o b e a c o n s t a n t d u r i n g d i u r n a l c y c l e s . T h i s s e t o f e q u a t i o n s w a s s o l v e d i n s u b r o u t i n e D B S O I L . A r r a y s 0 8 8 , D X S , D E P , a n d v a r i a b l e s A L F , D T , 0 8 0 a n d a r r a y e l e m e n t D B S ( 1 1 ) w e r e 1 2 3 u s e d t o d e n o t e T s , i ’ T x , i : A 2 , a , A t , T 8 a n d T d , r e s p e c t i v e l y . F i g u r e 9 s h o w s a s o i l t e m p e r a t u r e p r o f i l e s i m u l a t e d f o r t h e f i e l d i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 n t o 1 6 f 1 9 8 4 . 4 . 9 M o d e l S t r u c t u r e T h e p r e c e d i n g s e c t i o n s o f t h i s C h a p t e r d e s c r i b e d e a c h s u b m o d e l i n d i v i d u a l l y . I n t h i s s e c t i o n , t h e i n t e g r a t i o n o f a l l s u b m o d e l s i n t o a c o m p l e t e s i m u l a t i o n m o d e l i s d i s c u s s e d . T h e w h o l e m o d e l c o n s i s t e d o f a n i n i t i a l i z a t i o n s e c t i o n , a s i m u l a t o r s e c t i o n a n d a n o u t p u t s e c t i o n . T h e b l o c k d i a g r a m o f t h e m o d e l i s s h o w n i n F i g u r e 1 0 . A l i s t o f F O R T R A N c o d e f o r t h e A L F D R Y m o d e l i s g i v e n i n A p p e n d i x A . 5 . 9 . 1 I n i t i a l i z a t i o n S e c t i o n I n t h e i n i t i a l i z a t i o n s e c t i o n , d a t a o f t h e r m a l p r o p e r t i e s o f a i r , w a t e r v a p o r , s o i l a n d o t h e r i m p o r t a n t c o n s t a n t s w e r e i n i t i a l i z e d i n s u b p r o g r a m s A I R P A R a n d M I S P A R , a c c o r d i n g l y . D a t a o f s k y c o v e r a n d p r e c i p i t a t i o n f r o m l o c a l w e a t h e r s e r v i c e a n d m i c r o c l i m a t o l o g i c a l d a t a m e a s u r e d i n f i e l d s w e r e r e a d f r o m s p e c i f i e d w e a t h e r . d a t a f i l e s i n s u b p r o g r a m s D A Y W T R a n d M I C H T R , r e s p e c t i v e l y . T h e l a t i t u d e , l o n g i t u d e , a l t i t u d e a n d t i m e - z o n e m e r i d i a n o f t h e l o c a t i o n b e i n g s t u d i e d w a s r e a d t o p r o v i d e i n f o r m a t i o n f o r s i m u l a t i o n o f s h o r t — w a v e r a d i a t i o n . T e m p o r a l d a t a o f t h e y e a r , d a t e a n d t i m e f o r b e g i n n i n g a n d f i n i s h i n g a f i e l d d r y i n g p r o c e s s w e r e r e q u i r e d . S p a t i a l i n f o r m a t i o n o f t h e s w a t h , s u c h a s t h e d r y - m a t t e r a r e a d e n s i t y , i n i t i a l m o i s t u r e c o n t e n t , r a t i o o f l e a f a r e a i n d e x t o y i e l d , m a t u r i t y a n d c u t t i n g , r m u t e i e — o W — 0 : n 2 i 1 0 : 6 h t a w s a h t a e n r . 6 e " 0 1 d 8 : 0 n 9 u 1 e t e , r 6 a 1 u r D t - a fl r 1 r : e 8 y 1 0 a L h t 6 fi w M ' d e p t n m s a e u t g u r l A u i 0 o : H o n s o 2 ( f n 1 o a g e e i m l h i c i f i 0 T o M : 6 r p , g d n e i t s a n l a 5 u L 0 1 m : / 0 i t s s 8 a A E . 9 0 e : r 8 u 1 g i F 4 0 1 : 2 / 1 E 8 ( o n ) a J n i o J a d w a l : 1 g ] g ] 1 0 3 0 2 0 1 0 3 0 2 ° ] 1 0 3 0 m ] 1 0 3 0 2 0 3 r d L a y e r W 4 t h L a y e r A A A v f f A A . I 5 t h L a y e r W 1 J J » 1 1 0 1 . b s 4 L 1 ; 1 L 1 1 J J » 1 7 t h L a y e r 8 t h L a y e r 9 t h L a y e r 1 0 t h L a y e r 1 0 r r U 1 1 8 0 1 2 4 n o i t a z i l a i t i n I n o i t c e S r o n t o a i l t u c m e i S S t u p t u O n o i t c e S s s e c o r P t s a F s s e c o r P w o l S ~ F i g u r e 1 0 . B l o c k d i a g r a m o f t h e A L F D R Y s i m u l a t i o n m o d e l . ( : B E G I N N I N G : ) V I N P U T 1 [ E E A T R E R I N F O R M A T I O N ] [ S N A T H * I N I T I A L I Z A T I O R ] [ S O I L I N I T I A L I Z A T I O R ] 1 ; V [ L E A P T E M P E R A T U R E C A L C U L A T I O N ] [ S N A T R S T A T U S I N T E G R A T I O N ] “ C o n v e r g e d . “ O . = y e s [ E A S T T I M E I N C R E M E N T ] S l o w s t e p r e a c h e d 9 [ S O I L T E M P E R A T U R E I N T E G R A T I O N J [ w E A T R E R U P D A T E } [ R E S E T S L O N P R O O E S S ‘ E L O E E ] 1 % O u t p u t ? 4 [ @ P U T J C o m p l e t e d ? \ \ — “ ° _ ~ y e s S T O P 1 2 6 w a s r e a d . T O c o n t r o l a s i m u l a t i o n , t h e S i z e s o f t i m e s t e p s f o r i n t e g r a t i o n a n d o u t p u t i n t e r v a l s w e r e a l s o n e e d e d . T h e r e f e r e n c e h e i g h t o f m i c r o c l i m a t o l o g i c a l m e a s u r e m e n t s w a s a n i m p o r t a n t p a r a m e t e r . A t t h e i n i t i a l i z a t i o n s t a g e , i n i t i a l v a l u e s O f s t a t e v a r i a b l e s , s u c h a s t e m p e r a t u r e a n d v a p o r d e n s i t y o f l e a v e s a n d a i r s p a c e s w i t h i n t h e s w a t h , a n d s o i l t e m p e r a t u r e w e r e e s t i m a t e d i n s u b p r o g r a m S W A T H I a n d S O I L I , r e s p e c t i v e l y . I t w a s a s s u m e d t h a t i n i t i a l p r o f i l e s o f t e m p e r a t u r e a n d r e l a t i v e h u m i d i t y w e r e l i n e a r a l o n g t h e v e r t i c a l d i r e c t i o n . S o , d a t a a t t w o p o s i t i o n s , t h e s o i l s u r f a c e a n d r e f e r e n c e h e i g h t , w e r e r e q u i r e d . T h e s e d a t a w e r e o b t a i n e d b y u s i n g i n t e r p o l a t i o n t e c h n i q u e s f r o m m e a s u r e d d a t a . I t w a s a s s u m e d t h a t i n i t i a l l e a f t e m p e r a t u r e w a s t h e s a m e a s t h e a i r t e m p e r a t u r e i n e a c h l a y e r o f t h e s w a t h , r e s p e c t i v e l y . I n a d d i t i o n , v a p o r d e n s i t y , p r e s s u r e a n d r e l a t i v e h u m i d i t y o f l e a v e s w e r e a s s u m e d t o b e s a t u r a t e d a t l e a f t e m p e r a t u r e . S i n c e h e a t a n d v a p o r t r a n s f e r p r o c e s s e s i n s i d e a s w a t h w e r e f a s t p r o c e s s e s , t e m p e r a t u r e a n d v a p o r s t a t u s w o u l d b e a d j u s t e d t o p r o p e r s t a t e s b y t h e m o d e l a s h o r t t i m e a f t e r i n i t i a l i z a t i o n . I t w a s f u r t h e r a s s u m e d t h a t t h e r e w a s n o s u r f a c e a n d r u n a b l e m o i s t u r e , i . e . n o d e w , a n d t h a t a l f a l f a m o i s t u r e c o n t e n t w a s u n i f o r m t h r o u g h o u t t h e w h o l e s w a t h a t i n i t i a l t i m e . T h e r e w a s l a c k o f i n f o r m a t i o n o n d i u r n a l p a t t e r n s o f t e m p e r a t u r e o f l o c a l s o i l s . T o i n i t i a l i z e a s o i l t e m p e r a t u r e p r o f i l e , d a t a g i v e n b y C a r s o n a n d M o s e s ( 1 9 6 3 ) w e r e a d a p t e d . T h e y m e a s u r e d s o i l t e m p e r a t u r e a t d e p t h s o f 0 . 0 1 , 0 . 1 , 0 . 2 , 0 . 5 a n d 1 . 0 m , a n d a t 2 : 0 0 , 6 : 0 0 , 1 0 : 0 0 , 1 H : O O , 1 8 : 0 0 a n d 2 2 : 0 0 i n A r g o n n e , I l l i n o i s o n a t y p i c a l s u m m e r d a y , J u l y 2 7 , 1 9 6 3 . I n t h e m o d e l , t h e s e t e m p e r a t u r e d a t a w e r e s t o r e d i n a t w o - d i m e n s i o n a l a r r a y . A n i n t e r p o l a t i o n m e t h o d w a s 1 2 7 u s e d t o g e t a t e n t a t i v e s o i l t e m p e r a t u r e p r o f i l e f o r a g i v e n i n i t i a l t i m e . N o s p e c i a l p r o v i s i o n w a s m a d e t o d i s t i n g u i s h v a r i a t i o n o f t h e t e m p e r a t u r e p r o f i l e o n d i f f e r e n t d a t e s . T h e s o i l t e m p e r a t u r e i n t h e t o p l a y e r w a s s e t t o t h e l o c a l s o i l t e m p e r a t u r e m e a s u r e d j u s t u n d e r n e a t h t h e s o i l s u r f a c e . S o i l t e m p e r a t u r e i n o t h e r l a y e r s w e r e a d j u s t e d p r o p o r t i o n a l l y . P r o f i l e s o f w i n d , s h o r t - a n d l o n g - w a v e r a d i a t i o n a n d r e s i s t a n c e s t o h e a t a n d m o i s t u r e t r a n s f e r w i t h i n t h e s w a t h w e r e a l s o e s t i m a t e d a t i n i t i a l t i m e . I n i t i a l v a l u e s o f v a r i o u s f l u x e s o f m o i s t u r e a n d h e a t , a n d o f r a t e s o f c h a n g e s i n s t a t e v a r i a b l e s w e r e d e t e r m i n e d , a c c o r d i n g l y . u . 9 . 2 S i m u l a t o r S e c t i o n T h e s i m u l a t o r s e c t i o n w a s t h e m a i n b o d y o f t h e m o d e l . I t e s t i m a t e d r a t e s o f c h a n g e a n d f l u x e s f o r v a r i o u s s t a t e v a r i a b l e s , a n d i n t e g r a t e d s t e p b y s t e p t o s i m u l a t e t h e s t a t u s o f t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m . T w o k e y p o i n t s n e e d t o b e e m p h a s i z e d : 1 ) h a n d l i n g ' f a s t ' a n d ' s l o w ' p r o c e s s e s i n a s i m u l a t i o n m o d e l , a n d 2 ) c h o o s i n g p r o p e r s i z e ( s ) o f t i m e s t e p ( s ) f o r i n t e g r a t i o n . A s d i s c u s s e d p r e v i o u s l y , s e n s i b l e a n d l a t e n t h e a t t r a n s f e r a t l e a f s u r f a c e s , a n d s e n s i b l e h e a t a n d v a p o r t r a n s f e r i n a i r s p a c e s o f a s w a t h w e r e ' f a s t ' p r o c e s s e s w i t h t i m e c o n s t a n t s o f l e s s t h a n o n e s e c o n d . O n t h e o t h e r h a n d , s e n s i b l e h e a t t r a n s f e r i n s i d e t h e s o i l w a s a ' s l o w ' p r o c e s s w i t h a t i m e c o n s t a n t o f o n e t h o u s a n d s e c o n d s t o o v e r a n h o u r . A p p l i c a t i o n o f o n e s i n g l e t i m e s t e p w o u l d n o t b e a p p r o p r i a t e f o r b o t h k i n d s o f p r o c e s s e s . I f a t i m e s t e p w a s v e r y s m a l l , 1 2 8 s a y o n e o r s e v e r a l s e c o n d s , m u c h c o m p u t e r t i m e w o u l d b e w a s t e d i n u n n e c e s s a r y e s t i m a t i o n s o f s o i l t e m p e r a t u r e w h i c h w a s i n s e n s i t i v e t o e n v i r o n m e n t a l c h a n g e s . O n t h e o t h e r h a n d , a c c u r a c y i n e s t i m a t i n g s w a t h s t a t u s m i g h t b e r e d u c e d h e a v i l y i f a t i m e s t e p w a s v e r y l a r g e , s a y a b o u t o n e h o u r o r l o n g e r . T h e a p p r o a c h i n t h i s m o d e l w a s t o u s e d i f f e r e n t t i m e s t e p s f o r d i f f e r e n t p r o c e s s e s . F a s t ( o r s o f t ) p r o c e s s e s w e r e i n t e g r a t e d w i t h a s m a l l t i m e s t e p w h i l e s l o w ( o r s t i f f ) p r o c e s s e s w e r e i n t e g r a t e d w i t h a l a r g e t i m e s t e p . D u r i n g m o s t t i m e o f a s i m u l a t i o n r u n , e s t i m a t i o n o f s o i l t e m p e r a t u r e w a s b y p a s s e d w h i l e e s t i m a t i o n o f t e m p e r a t u r e a n d m o i s t u r e s t a t u s w i t h i n t h e s w a t h w a s e x e c u t i n g . S o i l t e m p e r a t u r e p r o f i l e s w e r e e s t i m a t e d o n l y w h e n t h e t i m e o f i n t e g r a t i o n o f f a s t p r o c e s s e s w a s a c c u m u l a t e d u p t o t h e s l o w t i m e s t e p . U p d a t i n g o f w e a t h e r c o n d i t i o n s , s u c h a s s o l a r i r r a d i a n c e , s k y c o v e r f r a c t i o n , r a i n , w i n d s p e e d , t e m p e r a t u r e a n d r e l a t i v e h u m i d i t y a t t h e r e f e r e n c e h e i g h t , a n d p r o f i l e s o f w i n d , s h o r t - a n d l o n g - w a v e r a d i a t i o n w i t h i n t h e s w a t h , w a s a l s o b y p a s s e d b e c a u s e w e a t h e r i n f o r m a t i o n w a s m e a s u r e d a t i n t e r v a l s o f o n e h o u r o r l o n g e r . T h e a l g o r i t h m f o r b y p a s s i n g t h e s l o w p r o c e s s a n d w e a t h e r u p d a t i n g i s a l s o s h o w n i n F i g u r e 1 0 . T o d e t e r m i n e t h e s i z e s o f t h e t w o t i m e s t e p s , t h e f r e q u e n c y o f m e a s u r e m e n t o f w e a t h e r a n d m i c r o c l i m a t e d a t a w e r e c o n s i d e r e d . L i t e r a t u r e h a s i n d i c a t e d t h a t t h e r e a r e v a r i o u s f l u c t u a t i o n s o f b o t h h i g h a n d l o w f r e q u e n c i e s i n t h e b o u n d a r y l a y e r o f t h e a t m o s p h e r e ( R o s e n b e r g e t a l . , 1 9 8 3 ) . M i c r o c l i m a t o l o g i c a l d a t a a r e n o r m a l l y d e t e r m i n e d b y a v e r a g i n g m e a s u r e m e n t s o v e r a n i n t e r v a l o f a h a l f t o o n e h o u r ( M o n t e i t h , 1 9 7 3 ; G o u d r i a a n , 1 9 7 7 ; a n d R o s e n b e r g e t a l . , 1 9 8 3 ) . W i t h s u c h a l o w t i m e r e s o l u t i o n o n m i c r o c l i m a t o l o g i c a l d a t a , a s t e p s i z e i n t h e o r d e r o f s e c o n d s w a s n o t m e a n i n g f u l . T e s t r u n s o f t h e m o d e l w i t h 1 2 9 v a r i o u s s t e p s i z e s w e r e m a d e f o r a t y p i c a l d r y i n g t e s t : T r i a l 5 ( 0 . 4 4 8 k g d r y m a t t e r / m 2 , a t E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 t o 1 6 o f 1 9 8 4 ) . S i m u l a t i o n r e s u l t s s h o w e d t h a t c h a n g e s i n s i m u l a t i o n a c c u r a c y w e r e i n s i g n i f i c a n t w h e n t h e f a s t t i m e s t e p w a s i n c r e a s e d f r o m 6 3 u p t o 6 0 s a n d 6 0 0 5 w h i l e t h e s l o w t i m e s t e p w a s 1 h o u r . T h e t o t a l s u m o f s q u a r e s ( S S T O ) f o r T r i a l 5 w a s 1 4 . 0 6 ; a n d t h e s u m o f s q u a r e s d u e t o e r r o r ( S S E ) w a s 1 . 3 2 , 1 . 3 0 a n d 1 . 0 5 f o r t h e f a s t s t e p o f 6 , 6 0 a n d 6 0 0 s , r e s p e c t i v e l y . T h e c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) w a s 0 . 9 1 , 0 . 9 1 a n d 0 . 9 2 , r e s p e c t i v e l y . F i g u r e 1 1 s h o w s a c o m p a r i s o n b e t w e e n m e a s u r e d a n d p r e d i c t e d d r y i n g c u r v e s f o r T r i a l 5 w i t h t h e f a s t s t e p o f 6 , 6 0 a n d 6 0 0 s , r e s p e c t i v e l y . T h e p r e d i c t e d d r y i n g c u r v e s w i t h t h e f a s t s t e p o f 6 a n d 6 0 s w e r e a l m o s t i d e n t i c a l , w h i c h s u g g e s t e d t h a t t h e f a s t t i m e s t e p d i d n o t n e e d t o b e s m a l l e r t h a n 1 m i n u t e . T h e i n c r e a s e i n S S E w a s p r e s u m a b l y a t t r i b u t e d t o a c c u m u l a t e d n u m e r i c a l e r r o r s i n s i m u l a t i o n r u n s w i t h s m a l l s t e p s . T h e s i z e s o f t i m e s t e p s i n f l u e n c e e f f e c t i v e u t i l i z a t i o n o f c o m p u t e r s , a n d c o m p u t a t i o n a l c o s t s . F o r s i m u l a t i o n o f T r i a l 5 , a 4 9 - h o u r d r y i n g p r o c e s s , o n e r u n w i t h t w o s t e p s o f 6 0 0 s a n d 1 h o u r r e q u i r e d a b o u t 1 8 m i n u t e s o n a C O M P A Q - 2 8 6 m i c r o c o m p u t e r w i t h a n 8 0 2 8 ? m a t h c o p r o c e s s o r r u n n i n g a t 1 0 m H z . W h e n t h e s m a l l s t e p w a s r e d u c e d t o 6 0 o r 6 s , i t t o o k a b o u t 1 . 5 o r 1 2 . 4 h o u r s , r e s p e c t i v e l y , f o r e x e c u t i o n o n t h e s a m e m i c r o c o m p u t e r . O n t h e m a i n f r a m e c o m p u t e r C D C 7 5 0 o f M i c h i g a n S t a t e U n i v e r s i t y , i t c o s t $ 2 . 9 5 a t a n e c o n o m i c r a t e f o r o n e s i m u l a t i o n r u n o f t h e m o d e l f o r T r i a l 5 w i t h t h e t w o s t e p s o f 6 0 0 s a n d 1 h o u r . S h o u l d t h e f a s t s t e p b e r e d u c e d t o 6 0 o r 6 s , t h e c o m p u t i n g c o s t w o u l d b e e x p e c t e d t o i n c r e a s e t o a b o u t $ 1 5 o r $ 1 2 4 , r e s p e c t i v e l y . R u n n i n g t h e s i m u l a t i o n w i t h a v e r y s m a l l s t e p s i z e w a s n o t p r a c t i c a l . 0 : 8 1 fl 0 , : 2 1 - — ) e h t i w s e . v s r u 0 c 0 6 g ) : i o 0 r 0 n r 6 y s 0 0 6 = - 6 T = A ) T ( 6 e 0 A 1 6 d e 0 ( i o d T : = d T e A t t a e e p c i d t r e a p t r 0 d 6 d , e 6 t c f 4 8 / 4 1 / 8 n o n a g i h ( e D c s c r d 0 i P i d n t M d e t e : r d s a r 8 a n P 1 f . c a g i n d d e r a i e — r u s r u s d n a L t s a E n i 5 l a i r T 4 0 . 5 P 0 o \ a n e a — \ : H T m 2 1 ( r n h e e e 1 m w i t f 0 T e o i : b 6 p n e o t s s i r w 5 a o 0 p l 1 1 : m S / 0 o 8 C a . 1 1 0 , : e 8 r 1 u g i F 4 0 1 : 2 / 0 1 8 . 0 ( ° q ' p ) q u a l u o o a l n l s i o w m o m s a O O J a A v 2 . 0 — 1 3 0 1 3 1 4 . 9 . 3 O u t p u t S e c t i o n T h e m o d e l w a s a b l e t o p r i n t b o t h a s u m m a r y a n d a d e t a i l e d r e p o r t u p o n u s e r - s p e c i f i e d t i m e i n t e r v a l s s i m u l a t e d . T h e s u m m a r y c o n t a i n e d i n f o r m a t i o n o n t h e y e a r , d a t e , t i m e , p r e d i c t e d m o i s t u r e c o n t e n t a v e r a g e d o v e r t h e w h o l e s w a t h , m e a s u r e d m o i s t u r e c o n t e n t ( i f a v a i l a b l e ) , a n d r e s i d u a l . I t a l s o g a v e s t a t i s t i c a l i n f o r m a t i o n , s u c h a s t h e t o t a l s u m o f s q u a r e s ( S S T O ) , t h e s u m o f s q u a r e s d u e t o r e g r e s s i o n ( S S R ) a n d t h e s u m o f s q u a r e s d u e t o e r r o r ( S S E ) , t h e c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) , t h e m e a n s u m o f s q u a r e s o f r e s i d u a l s ( M S E ) a n d F - s t a t i s t i c s a f t e r s i m u l a t i n g a w h o l e c a s e s e t o f f i e l d d r y i n g . T h e r e p o r t g a v e d e t a i l e d i n f o r m a t i o n a b o u t w e a t h e r c o n d i t i o n s a t t h e r e f e r e n c e h e i g h t , p r o f i l e s o f t e m p e r a t u r e , m o i s t u r e a n d v a r i o u s f l u x e s a s s o c i a t e d w i t h l e a v e s i n e a c h l a y e r o f t h e s w a t h , p r o f i l e s o f t e m p e r a t u r e , h u m i d i t y a n d v a r i o u s f l u x e s a s s o c i a t e d w i t h a i r s p a c e i n e a c h l a y e r o f t h e s w a t h , p r o f i l e s o f s h o r t - a n d l o n g - w a v e r a d i a t i o n i n t h e s w a t h , a n d t h e t e m p e r a t u r e p r o f i l e i n s i d e t h e s o i l . 4 . 9 . 4 P s y c h r o m e t r i c L i b r a r y I n t h e m o d e l , d e t e r m i n a t i o n o f p s y c h r o m e t r i c p a r a m e t e r s i n t h e m o i s t a i r w a s o f t e n r e q u i r e d . T h e t a s k w a s c a r r i e d o u t b y c a l l i n g a s e r i e s o f f u n c t i o n s u b p r o g r a m s . T h e f o r m u l a e o f t h o s e f u n c t i o n s w e r e o b t a i n e d f r o m A S A E D a t a 2 7 1 . 2 ( 1 9 8 4 ) . T h e s e f u n c t i o n s u b p r o g r a m s w e r e g r o u p e d i n a p s y c h r o m e t r i c p a c k a g e . A l i s t o f F O R T R A N c o d e f o r t h e p s y c h r o m e t r i c p a c k a g e i s s h o w n i n A p p e n d i x B . C H A P T E R 5 F I E L D E X P E R I M E N T E x p e r i m e n t s w e r e c o n d u c t e d t o c o l l e c t e s s e n t i a l i n f o r m a t i o n o n t h e p r o c e s s o f a l f a l f a d r y i n g i n t h e f i e l d e n v i r o n m e n t . I n f o r m a t i o n o b t a i n e d w a s u s e d t o e s t i m a t e u n k n o w n p a r a m e t e r s o f t h e m o d e l , s u c h a s t h e e m p i r i c a l c o n s t a n t s a a n d b i n E q u a t i o n [ 1 3 9 ] r e l a t i n g l e a f s u r f a c e r e s i s t a n c e t o l e a f t e m p e r a t u r e a n d m o i s t u r e c o n t e n t , a n d t o v a l i d a t e t h e m o d e l . E x p e r i m e n t s w e r e c a r r i e d o u t o n f a r m s i n E a s t L a n s i n g , M i c h i g a n i n 1 9 8 4 , a n d i n F r e s n o , C a l i f o r n i a i n 1 9 8 5 . 5 . 1 M e a s u r e m e n t o f S w a t h D r y i n g P e r i o d i c w e i g h i n g i s a c o n v e n i e n t m e t h o d t o m o n i t o r t h e p r o g r e s s o f d r y i n g o f b i o l o g i c a l m a t e r i a l s . A l f a l f a s w a t h s a r e u s u a l l y h u n d r e d s o f m e t e r s i n l e n g t h , s o t a k i n g m e a s u r e m e n t s o n a w h o l e s w a t h i s i m p r a c t i c a l a n d i m p o s s i b l e . S e l e c t i n g a c e r t a i n a m o u n t o f a l f a l f a f r o m a r e a l s w a t h , a n d t h e n w e i g h i n g t h e s a m p l e m a t e r i a l a t t i m e i n t e r v a l s c a n p r o v i d e r e p r e s e n t a t i v e i n f o r m a t i o n a b o u t t h e d r y i n g r a t e o f a w h o l e s w a t h a n d e v e n a l l s w a t h s u n d e r s i m i l a r c o n d i t i o n s . I n t h i s s t u d y , a p r o c e d u r e o f w e i g h i n g t r a y s w i t h a l f a l f a s a m p l e s w a s e m p l o y e d . T h i s m e t h o d h a s b e e n a p p l i e d i n h a y r e s e a r c h a t t h e U S D A D a i r y F o r a g e R e s e a r c h C e n t e r a t M i c h i g a n S t a t e U n i v e r s i t y s i n c e 1 9 7 7 ( R o t z a n d C h e n , 1 9 8 5 ) . T r a y s w e r e m a d e o f m e t a l w i r e m e s h . T h e 1 3 2 1 3 3 d i m e n s i o n s o f e a c h t r a y w e r e O . 5 x 0 . 5 x 0 . 1 m . T h e e m p t y w e i g h t o f e a c h t r a y w a s p r e d e t e r m i n e d . A f t e r a l f a l f a w a s c u t , t r a y s w e r e r a n d o m l y p u t a t s e v e r a l p o s i t i o n s ( c e n t e r a n d e d g e ) a l o n g a f r e s h s w a t h . T h e a l f a l f a b e n e a t h a t r a y w a s s e p a r a t e d f r o m o t h e r a l f a l f a i n t h e s w a t h , a n d p l a c e d i n t o t h e t r a y ( F i g u r e 1 2 ) . T h e t r a y l o a d e d w i t h a l f a l f a w a s t h e n w e i g h e d a n d t h e g r o s s w e i g h t w a s r e c o r d e d . S o m e t r a y s w e r e l o a d e d w i t h t h e e x a c t a m o u n t o f a l f a l f a t a k e n f r o m a s w a t h , w h i c h r e p r e s e n t t h e d e n s i t y o f n a t u r a l s w a t h s . O t h e r t r a y s w e r e l o a d e d w i t h m o r e o r l e s s a l f a l f a t o f o r m a r t i f i c i a l ' h i g h e r ' o r ' l o w e r ' d e n s i t i e s t h a n t h a t o f t h e a c t u a l s w a t h s . I n E a s t L a n s i n g , M i c h i g a n , t h r e e s a m p l e s w e r e t a k e n f o r e a c h d e n s i t y t r e a t m e n t ; a n d i n F r e s n o , C a l i f o r n i a , t h e n u m b e r o f s a m p l e s w a s t w o . D r y i n g t e s t s w e r e c o m b i n e d w i t h o t h e r t e s t s t o d e t e r m i n e t h e e f f e c t i v e n e s s o f c h e m i c a l c o n d i t i o n i n g . F o r t h i s w o r k , s a m p l e s w e r e o n l y t a k e n f r o m s w a t h s o f c o n t r o l t r e a t m e n t s ( m e c h a n i c a l , b u t n o c h e m i c a l c o n d i t i o n i n g ) . A l l t r a y s w e r e a r r a n g e d a s i f t h e y w e r e i n a r e a l s w a t h t o d r y . T h i s w a s d o n e b y p u t t i n g t r a y s a s i d e r e a l s w a t h s w i t h t w o o r t h r e e t r a y s i n a r o w . T h e t r a y s w e r e s u r r o u n d e d w i t h a l f a l f a t o a v o i d e d g e e f f e c t s ( F i g u r e 1 3 ) . T h e g r o s s w e i g h t s o f a l l s a m p l e t r a y s w e r e m e a s u r e d a t 2 h i n t e r v a l s . S o m e m e a s u r e m e n t s w e r e t a k e n t h r o u g h o u t n i g h t s d u r i n g t h e h a r v e s t s e a s o n o f 1 9 8 4 . A K T R O M d i g i t a l b a l a n c e w i t h a n a c c u r a c y o f 1 0 . 0 0 2 k g i n t h e r a n g e o f O ~ 6 k g w a s u s e d . W h e n s w a t h s w e r e b a l e d , s a m p l e m a t e r i a l s w e r e r e m o v e d f r o m t h e i r t r a y s a n d p l a c e d i n p a p e r b a g s w i t h k n o w n w e i g h t s . S a m p l e b a g s w e r e o v e n - d r i e d a t a t e m p e r a t u r e o f e i t h e r 6 5 ° C f o r 7 2 h o u r s o r 1 0 4 ° C f o r 2 4 h o u r s d e p e n d i n g u p o n w h e t h e r n u t r i t i o n a l a n a l y s i s w a s p l a n n e d , a c c o r d i n g . 1 , 1 . . . e 0 4 a s a w m 2 9 : 6 G . 3 3 5 3 0 : . . . o n « m e — E b 2 . 3 6 5 « s o 3 3 b . . 9 5 . ] . x . L B F » 1 3 5 t o A S A E S t a n d a r d 8 3 5 8 . 1 ( A S A E , 1 9 8 4 ) . T h e d r y - m a t t e r w e i g h t o f e a c h s a m p l e w a s t h e f i n a l g r o s s w e i g h t o f e a c h s a m p l e b a g m i n u s t h e b a g ' s w e i g h t , r e s p e c t i v e l y . T h e w e t w e i g h t o f a n a l f a l f a s a m p l e w a s t h e g r o s s w e i g h t o f t h e c o r r e s p o n d i n g s a m p l e a n d t r a y m i n u s t h e t r a y ' s e m p t y w e i g h t . T h e m o i s t u r e w e i g h t o f a s a m p l e w a s o b t a i n e d b y s u b t r a c t i n g t h e s a m p l e ' s d r y - m a t t e r w e i g h t f r o m i t s w e t w e i g h t . T h i s w a s ' b a s e d u p o n t h e a s s u m p t i o n o f n o l o s s i n d r y m a t t e r . T h e n , t h e m o i s t u r e c o n t e n t ( d r y b a s i s ) o f a s a m p l e w a s e s t i m a t e d b y d i v i d i n g t h e s a m p l e ' s m o i s t u r e w e i g h t b y i t s d r y - m a t t e r w e i g h t . T h i s c a l c u l a t i o n w a s d o n e t o e a c h m e a s u r e m e n t o f e a c h s a m p l e t o p r o v i d e a s e r i e s o f m o i s t u r e c o n t e n t d a t a f o r e a c h t e s t . A m e a n m o i s t u r e c o n t e n t w a s o b t a i n e d b y a v e r a g i n g m o i s t u r e c o n t e n t m e a s u r e m e n t s o v e r a l l s a m p l e s o f a g i v e n d e n s i t y a t e a c h m e a s u r e m e n t t i m e . T h e s a m e t r a y s a m p l e s w e r e u s e d t o c a l c u l a t e t h e m e a n t r a y d r y - m a t t e r w e i g h t . T h e m e a n a r e a d e n s i t y o f a s a m p l e d s w a t h w a s t h e a v e r a g e d t r a y d r y - m a t t e r w e i g h t m u l t i p l i e d b y 4 , s i n c e e a c h t r a y h a d a h o r i z o n t a l a r e a o f 0 . 2 5 m 2 . M e a s u r e d d a t a o f s w a t h d r y i n g w e r e s t o r e d i n a f i l e n a m e d D R Y C A S E w h i c h i s g i v e n i n A p p e n d i x C . A t y p i c a l d r y i n g c u r v e f o r T r i a l 5 ( E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 ) w a s s h o w n i n F i g u r e 1 1 o f S e c t i o n 4 . 9 . 2 . 5 . 2 M i c r o m e t e o r o l o g i c a l M e a s u r e m e n t s M i c r o m e t e o r o l o g i c a l d a t a w e r e a c q u i r e d w i t h t w o p o r t a b l e m i c r o p r o c e s s o r - b a s e d w e a t h e r s t a t i o n s . O n e u n i t w a s u s e d t o m e a s u r e 1 3 6 m i c r o c l i m a t o l o g i c a l d a t a a t a r e f e r e n c e h e i g h t a b o u t 0 . 6 m a b o v e t h e g r o u n d . T h e o t h e r u n i t w a s u s e d t o m e a s u r e p s y c h r o m e t r i c d a t a i n s i d e s w a t h s . E a c h u n i t c o n s i s t e d o f a m i c r o l o g g e r a n d a s s o c i a t e d s e n s o r s . T h o s e s e n s o r s i n c l u d e d t e m p e r a t u r e p r o b e s , r e l a t i v e h u m i d i t y p r o b e s , p y r a n o m e t e r s , a n e n o m e t e r s a n d s o i l m o i s t u r e b l o c k s . F i g u r e 1 4 s h o w s o n e w e a t h e r s t a t i o n a n d i t s c o m p o n e n t s u s e d i n t h i s s t u d y . 5 . 2 . 1 M i c r o l o g g e r s T h e d a t a l o g g e r s u s e d i n t h i s s t u d y w e r e C R — 2 1 u n i t s m a n u f a c t u r e d b y C a m p b e l l S c i e n t i f i c , I n c . E a c h C R - 2 1 w a s b a t t e r y - p o w e r e d a n d e q u i p p e d w i t h a m i c r o p r o c e s s o r , a r e a l t i m e c l o c k , a s e r i a l d a t a i n t e r f a c e , a n d a p r o g r a m m a b l e a n a l o g - t o - d i g i t a l c o n v e r t e r . T h i s m i n i a t u r e , n i n e - c h a n n e l , c o m p u t e r i z e d d a t a r e c o r d e r h a n d l e d u p t o s e v e n a n a l o g i n p u t s a n d t w o p u l s e i n p u t s . I t p r o v i d e d D C a n d A C e x c i t a t i o n v o l t a g e s t o m e e t r e q u i r e m e n t s o f c e r t a i n t r a n s d u c e r s . T h e C R - 2 1 ' s w e r e f a c t o r y - p r o g r a m m e d t o c o n v e r t v o l t a g e s i g n a l s o f c e r t a i n t y p e s o f s e n s o r s , i n t o d i g i t a l s i g n a l s o f m e a s u r e d q u a n t i t i e s ( t e m p e r a t u r e , h u m i d i t y , s o l a r i r r a d i a n c e , w i n d s p e e d , e t c . ) . T h e C R - 2 1 s c a n n e d e a c h c h a n n e l o n c e e v e r y 1 0 s e c o n d s , a n d e a c h s c a n t o o k a b o u t 0 . 3 t o 2 s e c o n d s d e p e n d i n g o n a u s e r - p r o g r a m m e d i n p u t t a b l e . T h e C R - 2 1 d u m p e d s t o r e d d a t a t o c a s s e t t e t a p e s f o r s t o r a g e a t a u s e r - s p e c i f i e d i n t e r v a l . I n t h i s s t u d y , t h e C R - 2 1 ' s w e r e p r o g r a m m e d t o o u t p u t a v e r a g e d h o u r l y d a t a . T h e m i c r o l o g g e r h a d a n a m p l i f i e r a c c u r a c y o f £ 0 . 5 1 o f r e a d i n g o r 2 t i m e s t h e r e s o l u t i o n o f a m e a s u r e d q u a n t i t y , w h i c h e v e r w a s l a r g e r . T h e r e s o l u t i o n w a s $ 0 . 0 0 5 m V i n t h e r a n g e o f - 2 t o 2 5 m v , a n d : 1 m V i n t h e F i g u r e 1 4 . A m i c r o l o g g e r w i t h a s s o c i a t e d s e n s o r s a n d a c a s s e t t e r e c o r d e r . F i g u r e 1 5 . A s o i l m o i s t u r e b l o c k . 1 3 8 r a n g e o f - 0 . 2 t o 2 . 5 V , a c c o r d i n g t o m a n u f a c t u r e r ' s s p e c i f i c a t i o n s . T h e C R - 2 1 r e q u i r e d a w o r k i n g e n v i r o n m e n t o f - 2 5 t o 5 0 C , a n d 0 . 0 t o 0 . 9 5 r e l a t i v e h u m i d i t y . I n t h i s s t u d y , e a c h C R - 2 1 u n i t w a s s e t i n a m e t a l b o x . I t w a s f o u n d t h a t t h e C R - 2 1 ' s s t o p p e d w o r k i n g w h e n t h e m e t a l b o x e s w e r e e x p o s e d t o v e r y h i g h s o l a r r a d i a t i o n . T h e r e f o r e , t h e b o x e s w e r e c o v e r e d w i t h c u t a l f a l f a t o s h a d e t h e i n s t r u m e n t . 5 . 2 . 2 T e m p e r a t u r e P r o b e s T e m p e r a t u r e p r o b e s u s e d w e r e C a m p b e l l M o d e l 1 0 1 t e m p e r a t u r e s e n s o r s m a n u f a c t u r e d b y F e n v a l E l e c t r o n i c s . E a c h s e n s o r w a s a t h e r m i s t o r ( 0 0 . 2 x 2 5 . 4 m m ) s e t t i n g i n a w a t e r - r e s i s t a n t p r o b e w i t h a 2 4 9 k 9 p i c k o f f r e s i s t o r . A c c o r d i n g t o m a n u f a c t u r e r ' s s p e c i f i c a t i o n s , t h e g r e a t e s t s y s t e m e r r o r f o r a M o d e l 1 0 1 t e m p e r a t u r e s e n s o r u s e d w i t h a C R - 2 1 w a s 1 0 . 6 ° C i n t h e r a n g e o f - 3 0 ° t o 4 7 ° C . 5 . 2 . 3 R e l a t i v e H u m i d i t y P r o b e s T h e r e l a t i v e h u m i d i t y p r o b e s u s e d i n t h i s s t u d y w e r e C a m p b e l l M o d e l 2 0 1 . E a c h c o n t a i n e d a r e l a t i v e h u m i d i t y s e n s o r w i t h : 1 1 c a l i b r a t i o n , m a d e b y P h y s - C h e m i c a l R e s e a r c h C o r p . , a n d a M o d e l 1 0 1 t e m p e r a t u r e s e n s o r . I n m o i s t a i r , r e l a t i v e h u m i d i t y s e n s o r s c h a n g e d t h e i r r e s i s t a n c e a s h u m i d i t y c h a n g e d . T h e r e a d i n g o f r e l a t i v e h u m i d i t y w a s l i n e a r i n t h e r a n g e o f 0 . 0 t o 0 . 9 7 w i t h a n e r r o r o f : 3 1 , a c c o r d i n g t o m a n u f a c t u r e r ' s s p e c i f i c a t i o n s . T e m p e r a t u r e o v e r n o r m a l r a n g e s d i d n o t a f f e c t t h e r e l a t i v e h u m i d i t y s e n s o r s . 1 3 9 T h i s k i n d o f s e n s o r i s s u s c e p t i b l e t o o r g a n i c s o l v e n t s , i r o n i c - l a d e n l i q u i d s , o i l v a p o r , s u l f u r , w a t e r c o n d e n s a t i o n a n d i m m e r s i o n . I t w a s f o u n d i n t h i s s t u d y t h a t a f e w r e a d i n g s w e r e a b o v e 1 . 0 a t n i g h t , w h i c h w a s c a u s e d b y d e w f o r m a t i o n . T h e s e r e a d i n g s w e r e t h e n c o r r e c t e d t o 1 . 0 , a c c o r d i n g l y . 5 . 2 . 4 P y r a n o m e t e r L I - 2 0 6 S p y r a n o m e t e r s m a n u f a c t u r e d b y L I - C O R , I n c . w e r e u s e d . E a c h w a s a s i l i c o n d e v i c e w h i c h p r o d u c e d m i l l i v o l t s w h e n e x p o s e d t o s u n l i g h t . E a c h p y r a n o m e t e r w a s i n d i v i d u a l l y c a l i b r a t e d i n t h e f a c t o r y a n d a c o n v e r s i o n c o e f f i c i e n t w a s p r o v i d e d . T h e c o n v e r s i o n c o e f f i c i e n t s o f t w o p y r a n o m e t e r s u s e d i n t h i s s t u d y w e r e 1 1 9 . 8 a n d 1 3 9 . 2 W / m z / m v , r e s p e c t i v e l y . T h e m e a s u r e m e n t r a n g e o f t h e p y r a n o m e t e r w a s 0 t o 1 3 0 0 W / m z , a c c o r d i n g t o m a n u f a c t u r e r ' s s p e c i f i c a t i o n s . A f e w r e a d i n g s w e r e f o u n d t o b e s l i g h t l y n e g a t i v e a t n i g h t , a n d a c c o r d i n g l y , c o r r e c t e d t o b e z e r o . T h e m e a s u r e m e n t a c c u r a c y w a s $ 5 . 9 9 a n d 1 6 . 9 6 W / m z , r e s p e c t i v e l y . 5 . 2 . 5 A n e n o m e t e r T h e a n e n o m e t e r , m a d e b y W E A T H E R t r o n i c s I n c . , w a s a t h r e e - c u p t y p e . I t o u t p u t t e d p u l s e s w h e n a D C v o l t a g e w a s p r o v i d e d . T h e C R - 2 1 c o u n t e d t h e p u l s e s a n d c o n v e r t e d p u l s e c o u n t s t o w i n d s p e e d i n m / s . 1 4 0 5 . 2 . 6 S o i l M o i s t u r e S e n s o r s S o i l m o i s t u r e s e n s o r s u s e d w e r e C a m p b e l l M o d e l 2 2 1 m a n u f a c t u r e d b y D E L M H O R S T ( F i g u r e 1 5 ) . E a c h s e n s o r w a s a c y l i n d r i c a l b l o c k c o m p o s e d o f g y p s u m c a s t a r o u n d t w o c o n c e n t r i c e l e c t r o d e s . E l e c t r i c a l c u r r e n t f l o w w a s c o n f i n e d i n t h e i n t e r i o r o f t h e b l o c k , s o t h e e f f e c t o f s o i l c o n d u c t i v i t y w a s r e d u c e d . T h e g y p s u m o n t h e o u t e r e l e c t r o d e a s s o c i a t e d w i t h t h e s o i l c r e a t e d a b u f f e r a g a i n s t s o i l s a l t s w h i c h m i g h t i n f l u e n c e t h e b l o c k ' s c o n d u c t i v i t y . T h e s e n s o r s c h a n g e d t h e i r r e s i s t a n c e a s s o i l m o i s t u r e c h a n g e d . T h e s e n s o r s w e r e e x c i t e d b y a n A C v o l t a g e w i t h z e r o D C c o m p o n e n t . T h e C R - 2 1 ' s p r o v i d e d a s q u a r e - w a v e A C e x c i t a t i o n o f 4 V a n d 7 0 0 H z . T h e C R - 2 1 ' s o u t p u t t e d v o l t a g e d a t a c o r r e s p o n d e d t o t h e r e s i s t a n c e o f s o i l m o i s t u r e b l o c k s . I t w a s n e c e s s a r y t o c o n v e r t t h e v o l t a g e o u t p u t t o s o i l m o i s t u r e p a r a m e t e r s . T h e m a n u f a c t u r e r p r o v i d e d c a l i b r a t e d d a t a r e l a t i n g s o i l w a t e r p o t e n t i a l v s . v o l t a g e o u t p u t o f t h e 0 8 - 2 1 . A n a p p r o x i m a t i o n m e t h o d o f t h i r d - o r d e r s p l i n e s w e r e a p p l i e d t o t h e n o n l i n e a r f u n c t i o n o f s o i l w a t e r p o t e n t i a l v s . v o l t a g e o u t p u t o f t h e C R - 2 1 . E a c h s e g m e n t o f t h e f u n c t i o n w a s a p p r o x i m a t e d w i t h a t h i r d - o r d e r p o l y n o m i a l . E a c h p a i r o f a d j a c e n t p o l y n o m i a l s h a d c o m m o n f i r s t - a n d s e c o n d - o r d e r d e r i v a t i v e s a t t h e i r i n t e r c e p t i o n p o i n t . A t t h e t w o e n d p o i n t s o f t h e f u n c t i o n , t h e s e c o n d - o r d e r d e r i v a t i v e s g o f c o r r e s p o n d i n g p o l y n o m i a l s w e r e a s s u m e d t o b e z e r o . F i g u r e 1 6 s h o w s t h e a p p r o x i m a t i o n c u r v e . n o i t s a t m n i i 0 o . P 2 fl x o r n p o p i A t a e r n b i i l l p a 8 ) S C s r . 1 t . l s o o — V k c o ( l b s e 6 k r u I . 1 c t o s i l o B m e l i o r s - u 4 I . r t o 1 f s i e o v M l i r u c n o 2 i o r . S t 1 a r f b o i l a c t u e 0 r . p t 1 u h T . 6 O 1 e r 1 u g 2 i — F 8 I . 0 R C 6 . 0 5 1 _ - 0 . 0 ( c d . 0 1 ) I n n u e l o d J 9 1 0 M n o s l f i ' 9 . < 9 0 2 l l l 1 N , J 1 1 4 1 1 4 2 5 . 2 . 7 M e a s u r e m e n t P r o c e d u r e O n e w e a t h e r s t a t i o n w a s s e t u p t o m e a s u r e t e m p e r a t u r e , r e l a t i v e h u m i d i t y a n d w i n d s p e e d a t a r e f e r e n c e h e i g h t , s o l a r i r r a d i a n c e a t t h e t o p o f s w a t h s , r e l a t i v e h u m i d i t y i n s i d e a s w a t h , a n d s o i l m o i s t u r e . T h e r e f e r e n c e h e i g h t w a s t h e h e i g h t o f t h e r o t a t i n g c u p s o f t h e a n e n o m e t e r a b o v e t h e g r o u n d . F i g u r e s 1 3 a n d 1 7 s h o w t h e s t a t i o n i n t h e f i e l d . T h e o t h e r w e a t h e r s t a t i o n w a s u s e d t o m e a s u r e t e m p e r a t u r e a n d r e l a t i v e h u m i d i t y a t t h e t o p a n d b o t t o m o f a s w a t h , s o i l t e m p e r a t u r e a n d m o i s t u r e b e n e a t h t h e s w a t h . O n e h u m i d i t y p r o b e w a s p u t o n t h e t o p o f a s w a t h . T h e p r o b e w a s c o v e r e d w i t h a v e r y t h i n l a y e r o f a l f a l f a t o a v o i d d i r e c t s u n s h i n e . F i g u r e 1 8 s h o w s t h e p o s i t i o n o f t h e h u m i d i t y p r o b e . A n o t h e r h u m i d i t y p r o b e w a s p l a c e d o n t h e g r o u n d i n s i d e t h e s w a t h . S i n c e t h e d i a m e t e r o f t h e p r o b e s h e l t e r w a s a b o u t 3 c m , t h e s e n s o r m e a s u r e d t e m p e r a t u r e a n d r e l a t i v e h u m i d i t y a t a h e i g h t o f a b o u t 1 . 5 c m . S o i l m o i s t u r e b l o c k s w e r e p l a c e d i n t h e s o i l a b o u t 1 . 5 c m b e l o w t h e s u r f a c e . O n e w a s i n t h e o p e n f i e l d , a n d t h e o t h e r w a s u n d e r t h e s w a t h . A t e m p e r a t u r e p r o b e w a s a l s o p l a c e d i n t h e s o i l v e r y c l o s e t o t h e s o i l s u r f a c e u n d e r n e a t h t h e s w a t h . B o t h w e a t h e r s t a t i o n s w e r e p r o g r a m m e d t o d u m p a v e r a g e d d a t a o f a l l s e n s o r s a t a n h o u r l y i n t e r v a l t o a c a s s e t t e . D a t a s t o r e d o n c a s s e t t e t a p e s w e r e u p l o a d e d t o t h e m a i n f r a m e c o m p u t e r , C D C 7 5 0 , o f M i c h i g a n S t a t e U n i v e r s i t y a n d s t o r e d i n d a t a f i l e s . A p p e n d i x D l i s t s t h e c o n t e n t o f d a t a f i l e W T R M I C . F i g u r e s 1 9 , 2 0 a n d 2 1 s h o w s o m e m e a s u r e d m i c r o m e t e o r o l o g i c a l d a t a o f s o l a r i r r a d i a n c e , w i n d s p e e d a t t h e r e f e r e n c e h e i g h t , a n d t e m p e r a t u r e a n d r e l a t i v e h u m i d i t y a t f o u r l o c a t i o n s o f t h e s w a t h . F i g u r e 1 7 . A m i c r o w e a t h e r u n i t i n t h e f i e l d . F i g u r e 1 8 . A r e l a t i v e h u m i d i t y p r o b e i n t h e t o p l a y e r o f a s w a t h . 0 1 ' : 8 1 6 . . 0 4 8 0 " 9 f I : 2 1 ‘ — ' B o 1 t , h 6 g 1 i - e 4 1 r h 0 : 4 9 K " fl — r ” 9 6 e t c s n u e g r u e A 6 f 0 1 e n a ) T : r o \ [ “ / 0 e 8 t a n a a D t g a i h " 0 d c 7 : e i d 8 n e M ' 1 p a s , g r d n n i ' i s u 0 ” o w n T : ‘ 2 H 1 ( e d l n a t s ‘ e e a ' m 1 9 c E n 0 i “ a n ? : M 6 T i i E H \ ‘ F , d a d r e r r d i u b ‘ ' 1 5 s 0 1 r a ‘ : a e / 0 l m ‘ 8 o S m . 9 1 e r u g i F 0 r : 8 1 1 : 0 : 1 2 / 1 8 1 0 0 0 , i 8 0 0 1 6 0 0 1 4 0 0 1 2 0 0 1 l ( E t a / . 1 7 1 1 ) 9 3 0 0 1 0 0 1 1 : J D : O S " “ 1 O 1 - . O . . - . . , 1 1 n . 1 ( s / u Q b e a d s p u n , N ‘ B 1 - - + 0 1 1 1 4 0 : 8 1 h n t i a 0 w : 2 1 0 : 6 6 0 1 : 0 e t s d e e r h u t s a f e o m m o h t t t a . o w 4 b 8 s 9 d e 1 n h a t , 6 p h 1 o — t h t m o e c a f a 0 a t 4 a D e 1 e n h r t w S e v p t o t r : d t e s u 8 n d u T o S 1 B a h t l i t n g a u u A o , o e h r b a a w S t a S u h c n 0 o t a o w w a f : H r 2 S o n w r i l ( 1 s u a A s e b 1 0 4 1 1 0 0 4 4 e m g a i l h e i c i v o i 0 l o s M : b 6 , a e h g e t n r i u w s t n o 5 a a l 0 e r L 1 : e b / 0 p t 8 m s d e n a T a E . 0 2 e r u g i F 0 : 8 1 4 0 : 1 2 / 1 8 ( 3 . ) a J n l O J a d u l a i ” - 1 2 0 4 1 0 r . 7 1 4 5 0 : 8 1 0 r : 2 1 0 I : 6 h t a w S e v o b a r i A p m o t t 6 0 : 0 1 ) / e 8 t e c a f 0 r : a D d a u 8 n T o S 1 h B t h a t w a S w S l i o S w o l e b a r u o 0 : 2 H 1 ( e m 0 i : 6 T 5 0 U 1 : / 0 8 0 r : 8 1 , - - 1 0 0 0 5 ) 1 0 0 7 5 2 0 . . . . , 1 0 0 . W - 0 4 : 1 1 - - 5 0 5 0 e 0 0 5 0 / 2 7 5 2 0 0 5 7 2 1 0 8 . . . . . . . . . 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ( I o w l o e p ) £ 1 1 l w n H 9 1 9 1 0 1 9 2 1 . 1 F i g u r e 2 1 . R e l a t i v e h u m i t i d i e s a b o v e a s w a t h , a t t h e t o p a n d b o t t o m o f t h e s w a t h , a n d b e l o w t h e s o i l s u r f a c e u n d e r n e a t h t h e s w a t h m e a s u r e d a t E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 - 1 6 , 1 9 8 4 . 1 4 6 1 4 7 5 . 3 L o c a l C l i m a t o l o g i c a l I n f b r m a t i o n I n f o r m a t i o n o f s k y c o v e r f r a c t i o n , p r e c i p i t a t i o n a n d o t h e r w e a t h e r d a t a w e r e o b t a i n e d f r o m t h e l o c a l s t a t i o n o f t h e N a t i o n a l W e a t h e r S e r v i c e . T h e l o c a l w e a t h e r d a t a w e r e m e a s u r e d a t t h e L a n s i n g A i r p o r t , 1 1 k m N o r t h - W e s t o f t h e f i e l d S i t e . D a t a f i l e W T R D A Y i s l i s t e d i n A p p e n d i x E . F o r d r y i n g t e s t s i n F r e s n o , C a l i f o r n i a , a z e r o s k y c o v e r f r a c t i o n w a s a s s u m e d s i n c e t h e s k y w a s v e r y c l e a r d u r i n g t h e t e s t s . I ' I I I I I I I 1 1 I ; I I 1 I ' I ‘ I I 1 1 01 1 2 1 1 1 1 1 1 1 . “ ! ' ‘ . l n . _ ~ . fi v . ¢ - , “ — I 5 . 1 — , . . s . A _ I . . ’ W _ M ~ ' . - . N . n - - I I o . » . ) . > « . ~ , ' ~ - . M . . . 4 ‘ ~ . > . . _ ‘ , . . . 4 . m ~ . _ . . . . , > , . 3 . . “ . . . . . , W . . - 4 — . ~ . « , - a M . 4 . r - n V - . 1 - w o N . % . g — . y . . I “ ~A u . A g - l H 4 . . - ‘ m I c ‘ J “ w — — » . d . n b - a . _ L H w * . “ _ U ' ‘ " . - — ' , } J 4 . ~ I . . ¢ . . . - A . . . ' Y ‘ T Y . . " w . I . - L . ‘ . . . fl : H . . . . “ - 3 I ’ . , 2 1 ” , " m 3 m u 2 : m 2 ' K " 7 W 1 . 3 4 - w $ 1 1 ,‘II‘, 1 1 3 1 1 1 1 1 3 1 ": 3I I ,1 1 'I‘ 1 ' 5 1 111;“ " 1 1 1 ' : : « « 5 a . " - 9 ” m a ‘ c “ ' IIII’I‘I‘I“ 1 . ~ > u A , . “ . fi m : 9 . 0 1 “ ‘ - 2 . - . o » . 6 — — . . . - , ” - a u - ¢ . g - v - . — - m . h . m « . . . w u ~ . . . L ‘ . 4 - . U — ' : a . ' 1 m . o ‘ - : . - _ - - . . : ' w A - 4 . w . 5 C n , u u y 5 s ' . - " . M . _ 1 . — v ” ' 3 U - I - . - . ‘ . _ V _ - . . d r : : . m ” ’ ‘ - w W ” , . " W - " m H . . . « m , r - a fi . s < h . r “ ’ M ‘ I 5 I I I 5 ' . I 1 I ' 1 , I , ; " I ' - 1 _ I I I I 1 1 1 . 1 ,I 1 " I 1 " I . M 1 I " « 1 , 1 I 1 ’ I : 1 . . |1 1 1 . 1 :1 . . . 1 1 1 1 ; [ I ‘ I I I. ' I I 1 I . a k fi w . . _ ' , . ‘ 1 : n t : . ‘ ° ' . . ‘ 7 " 3 5 . : m m - M . - - ~ . - - - ‘ . ~ . . ~ 4 . . . . - V - . 1 0 . . . . v - v o . . - w - 4 s . . . . . - — - s o . « A “ - u o , . m - , . “ . « I _ L t - ‘ u 7 3 ? » h m ” . M g m . . . , . : ~ t “ — - v : 3 3 ; v a — c ‘ l A . _ | ~ 4 ” . W q . ‘ - 0 » — o 0 3 1 * « ~ v a u - . _ - s . o 3 _ u . ‘ u v - - ~ — 1 { . 3 I 1 . I . I I I I ' I T 3 - 1 0 1 , ' 1 I I I . . I ‘ 1 I ‘ I I 1 1 , - } . I I I ' 1 ‘ I I I ' I 1 ' 1 ” , I 1 I I I I 1 ‘ 1 ' 1 1 1 : 4 ( 1 ' ; - 1 : 1 1 ' I 1 I ‘ I I I ' I i I ' I ' I ‘ I . I I . I I l ‘ I I I I ‘ I I ‘ I I J 1 1 1 1 1 1 ' , ‘ 1 | 1 I . ‘ , 1 9 1 1 1 1 W 1 1 I . ' 1 0 ! 1 ‘ I ‘ 1 ‘ I 1 1 I “ 1 1 1 1 1 1 1 1 1 1 . . . . 1 1 1 . 4 i I i 1 1 1 ‘ . , 1 I I ? “ I I 1 ° 1 1 § I 1 I | X I 1 . 1 1 H J - I I 1 I I I I I I I I I I I I I I I I u ” H I ? I ’ 4 ' ; , I I I I I I E ‘ T I r ' I 1 1 : 1 I 1 » : . 1 7 1 “ I I I — ‘ : I b 1 . . ‘ ' I I I I , I . ' 3 1 “ . I . I . ' I I I ' I . 4 1 " M W I ' I ‘ I . ; . " I L ' 1 1 ‘ I I I I F I “ 2 ' I H “ I I , . I I I I I I 1 1 1 1 1 . 1 1 1 } 1 . 1 1 1 . 1 9 3 ” . I I I I . I I _ " I 1 ' m I I I I I ! " . I I I ‘ " N I " I I I 1 I ' I I I 1 1 1 1 ; : { 1 1 1 1 ‘ 1 . 1 1 " . u 1 1 m 1 1 1 “ . W 1 ‘ 1 1 1 " I ‘ 1 ' I 1 1 1 ; I I I 1 1 I I I 1 1 ' I 1 I I I 1 1 1 1 I ‘ 1 1 1 ’ 1 1 1 1 I ' 1 1 I I ' I ' q m m p 9 1 ‘ I v 1 1 5 1 ’ S H 3 . 1 1 1 1 1 1 1 1 1 ' : I . . - 1 ) . ! ” I } I ‘ I I ‘ 5 ' 1 1 1 % “ ! 5 ' # 4 1 1 1 , 1 7 1 ’ I ? I M I ‘ I I ' I I I I I S I I ‘ I ' 1 I I I i 1 I I I 1 I ’ , 1 1 1 I 1 1 1 ” 1 1 1 1 1 1 1 1 1 ; £ 1 1 1 I I F ‘ I I I n 1 1 ‘ 1 1 1 1 1 ' 1 ‘ 3 . ‘ I 5 1 1 * ] ; ' 3 1 : } ; " I I / . I I I ‘ I , I 1 ' . ' . . | ’ . l . : ' 1 . : ; ; ‘ I I I I ' I ' I I I I I I 1 ' I : I 1 ' 1 | . ‘ I I I I ' 1 I I 1 1 I ” . 1 1 1 ' 1 1 ' 7 \ 1 “ 1 : 1 1 “ . 1 . : . . ‘ . 1 1 ' 1 . I 1 1 | 1 1 I | ‘ I ' I ' m I l ‘ . , 1 I | I 1 \ . - n I 1 1 1 1 ' ” I , ‘ « 1 1 1 1 1 3 1 1 1 1 1 2 : 1 1 « " 1 . 1 . . ‘ I I . ‘ I I I 1 1 1 I 1 1 I I I I I I I I 1 1 1 ' I I I I J 1 1 1 q I I 1 1 1 1 1 I I I I ' 1 I 1 1 . 1 1 1 1 . - w ‘ I I I ' I I I I I I ‘ 1 1 I I I ' H I I I I W I I F I I I I I I “ 1 : 1 3 1 ! I I I . I I I J I I ' I I I ' I I I l ‘ n 1 I 1 1 1 4 1 1 1 3 1 1 1 1 1 1 ; I I I I I I I I I I 1 I I ‘ I I ' I ‘ I ‘ “ ‘ i H I 1 I 1 I ; I . — $ 5 . : “ I 1 ' i j 1 : 3 1 . " W I I ’ 2 1 I 1 1 1 1 ! 1 1 , 1 ; . . 1 ? 1 1 ; x I 1 1 I I 1 1 : 1 , . u . 7 1 1 1 - 1 1 ' 1 1 ‘ ! ’ _ 1 1 1 , 1 I - ’ : I 1 , 5 ' 1 1 2 1 I 1 I I J I - I I ‘ 1 1 1 ( 9 1 . ‘ I ' 1 ' I u \ ' ; £ I h l \ 1 ' ‘ 1 - ~ ‘ i I ' I 1 . 1 J h 1 é 1 . - , 1 1 1 ' M : I n S u i | 1 ’ I I 1 5 " " $ 1 ; I n t l e l i . I , . 1 E 1 1 1 1 . 1 1 1 1 . I I ‘ I I I I - h I I I : " I 1 > , 1 [ 1 : 1 ” I I I I I 3 1 : 1 1 ! ' 1 ’ : 1 I 1 I 1 ‘ ” 1 I I z I I ' I I ' I I I I I I I I ‘ ? ‘ I " I I I I I I I u I I I { 1 1 1 1 1 1 1 1 1 % I I 1 , 1 1 , 1 2 1 , , 3 1 “ ” ‘ 1 1 ‘ 1 , I 1 1 I - I ' l b n i I I J " 1 I ' r 1 : 1 1 . ” . 1 : ' H I . 1 1 I | . { ‘ 1 1 1 “ ” ! ( 1 1 1 ; . I I I P H I ‘ I I I I ‘ I I I 1 / 2 1 1 I I I 1 - 1 I ’ m m : w “ 4 n . , . . - ' . . : 1 1 - I I : « 1 1 1 1 ' 1 1 1 1 : , 1 1 ‘ 1 1 l 1 1 . 1 1 - ‘ - ‘ 7 5 " . I ' I I I ' s 1 . 1 1 ; " “ . 1 1 3 , " 4 1 . 1 1 1 1 1 1 ’ 1 1 I 1 1 I 1 V I I I ; 1 " 3 ' 1 1 ? I I I ‘ { 3 1 4 1 1 2 1 1 1 1 I 1 3 1 5 6 1 5 1 1 3 1 1 1 1 I I i ' f I ’ I I 1 1 , 1 1 1 “ I ' F I ' i ’ ? I 1 1 p d fl 1 1 1 I i . 1 1 1 5 1 : 1 1 : 1 1 I ' I I I I I I I I : I f I I ' I I i I I ' " : 1 “ ? I “ ! I . 2 | : L . I ' " ' I I ‘ I I I E . - I . , . 1 1 I I F 1 I 1 ’ \ 1 . ' ; E I I ' 1 ' 1 ' . 1 ' 1 1 5 1 1 1 1 ‘ 1 1 1 1 1 1 . I " ’ I , . : J ‘ . l 1 1 1 L I B R A R Y M i c h i g a n S t a t e U n i v e r s i t y C H A P T E R 6 P A R A M E T E R E S T I M A T I O N T h i s c h a p t e r d i s c u s s e s s t a t i s t i c a l e s t i m a t i o n o f t w o p a r a m e t e r s o f t h e s i m u l a t i o n m o d e l . T e c h n i q u e s u s e d w e r e d e r i v a t i v e - f r e e o p t i m i z a t i o n a n d n o n l i n e a r r e g r e s s i o n . T h e s t a t i s t i c a l p a r a m e t e r s e s t i m a t e d w e r e t w o c o e f f i c i e n t s w h i c h r e l a t e l e a f s u r f a c e r e s i s t a n c e t o t h e m o i s t u r e c o n t e n t o f l e a v e s i n E q u a t i o n [ 1 3 9 ] . T h e s e p a r a m e t e r s , h o w e v e r , w e r e e s t i m a t e d b y r e l a t i n g t h e m t o t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t i n l i e u o f t h e l e a f m o i s t u r e c o n t e n t . 6 . 1 T h e C o m p l e x i t y o f t h e P r o b l e m A s d i s c u s s e d i n S e c t i o n n . 5 , l e a f s u r f a c e r e s i s t a n c e t o m o i s t u r e r e m o v a l i s a n i n t e r n a l p r o p e r t y o f l e a v e s . I n f o r m a t i o n o n t h i s r e s i s t a n c e r e q u i r e d m i c r o s c o p i c s t u d y b e y o n d t h e s c o p e o f t h i s s t u d y . O n e a l t e r n a t i v e w a s t o c h a r a c t e r i z e t h i s r e s i s t a n c e t h r o u g h m a c r o s c o p i c b e h a v i o r o f l e a v e s . M o n t e i t h e t a l . ( 1 9 6 5 ) e x t e n d e d t h e c o n c e p t o f l e a f r e s i s t a n c e t o a b u l k c a n o p y r e s i s t a n c e w h i c h b e h a v e d l i k e t h e l e a f r e s i s t a n c e , a n d c o u l d b e d e r i v e d f r o m m i c r o m e t e o r o l o g i c a l o b s e r v a t i o n s . R o s e n b e r g e t a l . ( 1 9 8 3 ) r e v i e w e d e x t e n s i v e l y o t h e r s ' r e s e a r c h w i t h a s i m i l a r t h o u g h t . M o n t e i t h e t a l . ( 1 9 6 5 ) f o u n d t h a t m e a s u r e d l e a f r e s i s t a n c e w a s l i n e a r l y c o r r e l a t e d t o b u l k c a n o p y r e s i s t a n c e , e s t i m a t e d f r o m a n e n e r g y b a l a n c e w i t h m i c r o m e t e o r o l o g i c a l d a t a i n a b a r l e y f i e l d . 1 4 8 1 4 9 T h e i r a p p r o a c h w a s a d a p t e d i n t h i s s t u d y , h o w e v e r , t h e p r o c e d u r e w a s r e v e r s e d s u c h t h a t l e a f r e s i s t a n c e d e s c r i b e d i n E q u a t i o n [ 1 3 9 ] w a s c h a r a c t e r i z e d f r o m m a c r o s c o p i c d r y i n g o f s w a t h s . E s t i m a t i o n o f l e a f r e s i s t a n c e t o m o i s t u r e r e m o v a l f o r d r y i n g p l a n t s m a y b e e a s i e r t h a n f o r l i v i n g p l a n t s , b e c a u s e t h e l e a f r e s i s t a n c e o f d r y i n g p l a n t s w a s p r e s u m a b l y u n i n f l u e n c e d b y l i g h t i n t e n s i t y a n d t e m p e r a t u r e , b u t m e r e l y a f f e c t e d b y t h e w a t e r s t r e s s i n l e a v e s . T h e e s t i m a t i o n o f p a r a m e t e r s i n t h i s s t u d y w a s a c o m p l i c a t e d p r o b l e m . F i r s t , t h e m e a s u r e d q u a n t i t y w a s t h e a v e r a g e m o i s t u r e c o n t e n t o f t h e s w a t h r a t h e r t h a n t h e l e a f r e s i s t a n c e i t s e l f , s i n c e d e t e r m i n a t i o n o f m o i s t u r e c o n t e n t s w a s o f p r a c t i c a l i n t e r e s t , a n d e a s i e r t h a n m e a s u r e m e n t o f l e a f r e s i s t a n c e . T h u s , t h e p a r a m e t e r s o f t h i s s t u d y w e r e n o t e s t i m a t e d d i r e c t l y w i t h E q u a t i o n [ 1 3 9 ] , b u t t h r o u g h a n i n d i r e c t f u n c t i o n b e t w e e n l e a f r e s i s t a n c e a n d t h e a v e r a g e m o i s t u r e c o n t e n t o f t h e s w a t h . S e c o n d l y , t h e p r o b l e m w a s n o n l i n e a r , n o t o n l y d u e t o t h e n o n l i n e a r i t y o f E q u a t i o n [ 1 3 9 ] , b u t a l s o d u e t o t h e c o m p l e x i t y o f t h e f u n c t i o n a l r e l a t i o n s h i p b e t w e e n t h e a v e r a g e m o i s t u r e c o n t e n t o f t h e s w a t h a n d l e a f r e s i s t a n c e . A c t u a l l y , t h i s r e l a t i o n s h i p w a s u n a b l e t o b e d e s c r i b e d w i t h a n e x p l i c i t a l g e b r a i c f o r m . A n i m p l i c i t f u n c t i o n w a s : A M C = f ( r 1 , . . . ) = r ( 1 o a ‘ b R “ ° , . . . ) = r ( 1 o a ' b ” / ” o , . . . ) [ 1 8 7 ] w h e r e A M C a v e r a g e s w a t h m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , r 1 = l e a f r e s i s t a n c e t o m o i s t u r e t r a n s f e r , s / m , R W C = r e l a t i v e w a t e r c o n t e n t o f l e a v e s , d e c i m a l , M : l e a f i n t e r n a l m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r , M o : i n i t i a l m o i s t u r e o f l e a v e s , k g w a t e r / k g d r y m a t t e r , a a n d b = s t a t i s t i c a l p a r a m e t e r s t o b e e s t i m a t e d , d i m e n s i o n l e s s . 1 5 0 T h e a v e r a g e m o i s t u r e c o n t e n t o f t h e s w a t h w a s d e f i n e d a s t h e d e p e n d e n t v a r i a b l e ( s t a t i s t i c s t e r m ) , o r s y s t e m o u t p u t ( s y s t e m s c i e n c e t e r m ) . T h e q u a n t i t i e s , e x c e p t c o e f f i c i e n t s a a n d b , i n c l u d e d i n t h e p a r e n t h e s e s o f E q u a t i o n [ 1 8 6 ] w e r e i n d e p e n d e n t v a r i a b l e s ( s t a t i s t i c s t e r m ) . T h e s e i n d e p e n d e n t v a r i a b l e s w e r e a c o m b i n a t i o n o f t h e s y s t e m i n p u t ( t h e i n i t i a l s w a t h m o i s t u r e c o n t e n t ) , e n v i r o n m e n t a l v a r i a b l e s ( s o l a r r a d i a t i o n , a i r t e m p e r a t u r e a n d h u m i d i t y , w i n d s p e e d a t t h e r e f e r e n c e h e i g h t ) , s t a t e v a r i a b l e s ( l e a f r e s i s t a n c e , t e m p e r a t u r e a n d m o i s t u r e c o n t e n t s , a e r o d y n a m i c r e s i s t a n c e , a i r t e m p e r a t u r e a n d h u m i d i t y a b o v e a n d w i t h i n t h e s w a t h , s o i l t e m p e r a t u r e a n d m o i s t u r e , e t c . ) , a n d s y s t e m p a r a m e t e r s ( a l f a l f a m a t u r i t y , s w a t h d e n s i t y , l e a f a r e a i n d e x , c o n d i t i o n i n g f a c t o r , e t c . ) f r o m t h e v i e w p o i n t o f s y s t e m s c i e n c e . T h e c o e f f i c i e n t s a a n d b w e r e c a l l e d s t a t i s t i c a l p a r a m e t e r s i n o r d e r t o b e d i s t i n g u i s h e d f r o m l e a f r e s i s t a n c e , a s t a t e v a r i a b l e , a n d o t h e r s y s t e m p a r a m e t e r s . T h e f u n c t i o n i n E q u a t i o n [ 1 8 6 ] w a s c a l l e d a r e g r e s s i o n f u n c t i o n , r e s p o n s e f u n c t i o n o r p r e d i c t i o n f u n c t i o n . I n a d d i t i o n , a l l s t a t e v a r i a b l e s w e r e n o t e v e n l y d i s t r i b u t e d t h r o u g h o u t t h e s w a t h . A s d i s c u s s e d i n C h a p t e r s 3 a n d fl , s t a t e v a r i a b l e s v a r i e d f r o m t i m e t o t i m e , a n d f r o m l a y e r t o l a y e r a l o n g t h e v e r t i c a l d i r e c t i o n i n t h e s w a t h . T h i r d l y , t h e a v e r a g e m o i s t u r e c o n t e n t o f t h e s w a t h a t a p a r t i c u l a r t i m e w a s n o t d e t e r m i n e d b y t h o s e i n d e p e n d e n t v a r i a b l e s a f o r e m e n t i o n e d a t t h a t i n s t a n t a n e o u s t i m e , b u t b y t h e p a s t h i s t o r y o f a l l t h e s e v a r i a b l e s s i n c e t h e b e g i n n i n g o f d r y i n g ( e v e n b e f o r e c u t t i n g ) . T h i s m e a n s t h a t e a c h q u a n t i t y o n t h e r i g h t s i d e o f E q u a t i o n [ 1 8 6 ] r e p r e s e n t e d a s e r i e s o f v a l u e s o f t h a t q u a n t i t y f r o m t h e t i m e o f c u t t i n g o n . O n e m a y s a y t h a t t h e a v e r a g e m o i s t u r e c o n t e n t o f t h e s w a t h w a s a f u n c t i o n o f t h e 1 5 1 h i s t o r y o f i t s e l f . L i k e w i s e , l e a f r e s i s t a n c e w a s a l s o a f u n c t i o n o f i t s o w n p a s t h i s t o r y , a l t h o u g h i t w a s c a l c u l a t e d f r o m i n s t a n t a n e o u s m o i s t u r e c o n t e n t s o f l e a v e s t h r o u g h E q u a t i o n [ 1 3 9 ] . F o u r t h l y , t h e l e a f r e s i s t a n c e w a s a c o n t i n u o u s f u n c t i o n b u t w i t h d i s c o n t i n u i t y a t t h e t r a n s i t i o n b e t w e e n a d s o r p t i o n a n d d e s o r p t i o n . T h i s i m p l i e d t h a t f i r s t - o r d e r p a r t i a l d e r i v a t i v e s o f t h e l e a f r e s i s t a n c e w i t h r e s p e c t t o t h e s t a t i s t i c a l p a r a m e t e r s a a n d b w e r e p i e c e w i s e l y c o n t i n u o u s w i t h d i s c o n t i n u i t y o f t h e f i r s t k i n d . T h i s k i n d o f s t e p w i s e c h a n g e i n l e a f r e s i s t a n c e m i g h t c a u s e o s c i l l a t i o n d u r i n g s i m u l a t i o n . F i f t h l y , t h e r e w e r e o n l y d i s c r e t e d a t a a v a i l a b l e f o r t h e d y n a m i c p r o c e s s . T h e d i s c r e t e n e s s e x i s t e d i n a l l d a t a , b o t h f i e l d m e a s u r e d a n d s i m u l a t e d . M e a s u r e d d a t a w e r e t a k e n a t 1 - o r 2 - h o u r i n t e r v a l s . P r e d i c t e d d a t a w e r e i n t e g r a t e d a t G O O - s e c o n d o r 1 - h o u r i n t e r v a l s . D i s c r e t e d a t a c o u l d r e p r e s e n t p a r t i a l b u t i n c o m p l e t e i n f o r m a t i o n a b o u t t h e h i s t o r y o f t h e p r o c e s s . I n a d d i t i o n t o t h o s e r e a s o n s a b o v e , t h e r e w e r e o t h e r f a c t o r s w h i c h c a u s e d d i f f i c u l t y i n a p p l y i n g t e c h n i q u e s o f p a r a m e t e r e s t i m a t i o n . B e c k a n d A r n o l d ( 1 9 7 7 ) o u t l i n e d b a s i c s t a t i s t i c a l a s s u m p t i o n s c o m m o n l y u s e d i n , r e g r e s s i o n a n a l y s i s . T h o s e a s s u m p t i o n s a n d t h e d e v i a t i o n o f t h i s m o d e l a r e d i s c u s s e d a s f o l l o w s . 1 ) R e g r e s s i o n m o d e l s , o r r e g r e s s i o n f u n c t i o n s , s h o u l d h a v e c o r r e c t f o r m s , n o t e m p i r i c a l a p p r o x i m a t i o n s o r b e s t g u e s s e s . T h e m o d e l o f t h i s s t u d y w a s a m a t h e m a t i c a l l y s i m p l i f i e d a p p r o x i m a t i o n o f a c o m p l i c a t e d f i e l d d r y i n g p h e n o m e n o n . T h e m o d e l w a s c o m p r i s e d o f s e v e r a l s u b m o d e l s a n d m a n y s u b - s u b m o d e l s , e a c h o f w h i c h w a s a m a t h e m a t i c a l a p p r o x i m a t i o n o f a s p e c i f i c p h y s i c a l o r p h y s i o l o g i c a l p r o c e s s . I n t h e m o d e l s t u d i e d , s o m e e m p i r i c a l f u n c t i o n s w e r e a d a p t e d f r o m p a s t r e s e a r c h . 1 5 2 T h o s e n u m e r i c a l t e c h n i q u e s u s e d i n t h e m o d e l w e r e a l s o a p p r o x i m a t i o n s o f a n a l y t i c a l m e t h o d s . G e n e r a l l y s p e a k i n g , t h e r e a r e a l w a y s s o m e a p p r o x i m a t i o n s w h e n n a t u r a l p h e n o m e n a a r e m o d e l e d . 2 ) E r r o r s a b o u t t h e q u a n t i t i e s s t u d i e d a r e e i t h e r m e a s u r e m e n t e r r o r s o r m o d e l e r r o r s , b u t n o t b o t h . I n t h i s s t u d y , t h e r e w e r e e r r o r s b o t h i n t h e m e a s u r e m e n t s a n d t h e m o d e l . M e a S u r e m e n t e r r o r s w e r e d u e t o i n s t r u m e n t s , i n a p p r o p r i a t e o p e r a t i o n s , a n d o t h e r r a n d o m f a c t o r s . E r r o r s i n t h e m o d e l w e r e c a u s e d b y a p p r o x i m a t i o n , n u m e r i c a l t e c h n i q u e s , t r u n c a t i n g a n d r o u n d - o f f i n i m p l e m e n t a t i o n o n d i g i t a l c o m p u t e r s . F o r e x a m p l e , H o l m a n ( 1 9 8 1 ) p o i n t e d o u t t h a t t h e r e m a y b e : 2 0 } o r m o r e u n c e r t a i n t y i n c a l c u l a t i n g c o n v e c t i o n h e a t t r a n s f e r c o e f f i c i e n t s a n d £ 1 0 1 i n s u r f a c e r a d i a t i o n p r o p e r t i e s . M e a s u r e m e n t e r r o r s o f t e m p e r a t u r e , r e l a t i v e h u m i d i t y , s o l a r i r r a d i a n c e , w i n d s p e e d , a n d s o i l m o i s t u r e m a y b e n e g l i g i b l e i n t h e s t u d y . 3 ) E r r o r s a r e r a n d o m w i t h z e r o m e a n , c o n s t a n t v a r i a n c e a n d n o r m a l d i s t r i b u t i o n . G e n e r a l l y , r a n d o m e r r o r s e x i s t i n p r o p e r l y c o n d u c t e d m e a s u r e m e n t s a n d c a r e f u l l y c a l i b r a t e d i n s t r u m e n t s . M o d e l e r r o r s d u e t o a p p r o x i m a t i o n , n u m e r i c a l m e t h o d s a n d t r u n c a t i n g o r r o u n d - o f f w e r e p r o b a b l y s y s t e m a t i c r a t h e r t h a n r a n d o m , b u t t h e o c c u r r e n c e , d i r e c t i o n a n d m a g n i t u d e o f t h e s e e r r o r s s e e m u n p r e d i c t a b l e . I n t h i s s t u d y , a l l e r r o r s w e r e a s s u m e d t o b e p s e u d o - r a n d o m . 9 ) E r r o r s a r e u n c o r r e l a t e d . T h i s m i g h t n o t b e t r u e f o r t h e m o d e l s t u d i e d , ‘ s i n c e a n y e r r o r a t a t i m e s t e p o n a s i m u l a t i o n r u n m i g h t b e c a r r i e d o v e r t o t h e n e x t s t e p . I n a d d i t i o n , B e c k a n d A r n o l d ( 1 9 7 7 ) p o i n t e d o u t t h a t a p p l i c a t i o n o f m o d e r n d i g i t a l d a t a a c q u i s i t i o n i n s t r u m e n t s m a y i n d u c e s o m e c o r r e l a t e d e r r o r s i n t i m e s e r i e s . F o r t h i s s t u d y , i t w a s d e s i r a b l e t o f o r m a n a u t o c o r r e l a t e d 1 5 3 e r r o r m o d e l . H o w e v e r , p r e l i m i n a r y t e s t s s h o w e d t h a t s o m e s t a t i s t i c a l s e n s i t i v i t i e s w e r e l o s t w h e n s u c h a n e r r o r m o d e l w a s f o r m e d . F u r t h e r m o r e , w h e n t h e m o d e l i s u s e d t o p r e d i c t a d r y i n g p r o c e s s w i t h o u t m e a s u r e m e n t s , a n a u t o c o r r e l a t e d m o d e l w i l l b e i n c o n v e n i e n t b e c a u s e i t n e e d s m e a s u r e d d a t a t o a d j u s t s i m u l a t i o n o u t c o m e s . T h e r e f o r e , a u n c o r r e l a t e d e r r o r m o d e l w a s a s s u m e d i n t h i s s t u d y . 5 ) I n d e p e n d e n t v a r i a b l e s a r e n o t s t o c h a s t i c . I n t h i s s t u d y , o n l y a f e w v a r i a b l e s w e r e m e a s u r e d d i r e c t l y , s u c h a s s o i l t e m p e r a t u r e , a i r t e m p e r a t u r e , r e l a t i v e h u m i d i t y , w i n d s p e e d a n d s o l a r i r r a d i a n c e a t t h e r e f e r e n c e h e i g h t . S o m e q u a n t i t i e s , s u c h a s v a p o r p r e s s u r e , d e n s i t y a n d o t h e r p s y c h r o m e t r i c p r o p e r t i e s o f t h e a i r a b o v e t h e s w a t h , a n d s o i l m o i s t u r e , w e r e c a l c u l a t e d f r o m t h o s e m e a s u r e d v a r i a b l e s . E v e n t h e d e p e n d e n t v a r i a b l e , t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t , w a s i n d i r e c t l y c a l c u l a t e d f r o m w e i g h t m e a s u r e m e n t s o f s a m p l e t r a y s a n d s a m p l e b a g s . M a n y q u a n t i t i e s w e r e e s t i m a t e d , s u c h a s p s y c h r o m e t r i c p r o p e r t i e s o f t h e a i r . s p a c e s i n s i d e t h e s w a t h , a l f a l f a m o i s t u r e c o n t e n t s o f s w a t h l a y e r s , t e m p e r a t u r e i n s o i l l a y e r s , a n d r a d i a t i o n p r o f i l e s i n s i d e t h e s w a t h . T h u s , e r r o r s e x i s t e d i n a l l q u a n t i t i e s . A c c o r d i n g t o t h e t h e o r y o f p a r a m e t e r e s t i m a t i o n , t h e n u m b e r o f d e p e n d e n t v a r i a b l e s s h o u l d i n c l u d e a l l s t o c h a s t i c q u a n t i t i e s . T h i s w i l l r e q u i r e h u g e c o m p u t e r m e m o r y a n d h i g h c o m p u t i n g c o s t s . I n t h i s s t u d y , n o n - s t o c h a s t i c i n d e p e n d e n t v a r i a b l e s w e r e a s s u m e d , s i n c e t h o s e q u a n t i t i e s m e a s u r e d w i t h a c c u r a t e s e n s o r s w e r e e r r o r l e s s a s a s s u m e d . 6 ) T h e p a r a m e t e r s b e i n g e s t i m a t e d a r e d e t e r m i n i s t i c , i . e . n o n - s t o c h a s t i c . L e a f r e s i s t a n c e o f a l f a l f a , i f d i r e c t l y m e a s u r e d , m i g h t s h o w r a n d o m p a t t e r n s . F o r e x a m p l e , l e a f r e s i s t a n c e , c h a r a c t e r i z e d b y t h e t w o p a r a m e t e r s , m i g h t v a r y a s t h e v a r i e t y , m a t u r i t y l e v e l , c u t t i n g , 1 5 4 l o c a t i o n , r e w e t t i n g / r e d r y i n g , a n d h i s t o r y o f w a t e r s t r e s s v a r i e d . I n t h i s s t u d y , e m p h a s e s w e r e p u t o n t h e d a t e o f c u t t i n g . I t w a s a s s u m e d t h a t a l f a l f a c u t o n t h e s a m e d a t e h a d t h e s a m e m a t u r i t y l e v e l a n d h i s t o r y o f w a t e r s t r e s s , a n d a s i m i l a r d r y i n g - r e w e t t i n g - r e d r y i n g p a t t e r n . T h e v a r i e t y f a c t o r w a s c o m b i n e d w i t h t h e l o c a t i o n a n d y e a r ( E a s t L a n s i n g , M i c h i g a n , 1 9 8 ” o r F r e s n o , C a l i f o r n i a , 1 9 8 5 ) . N o s p e c i a l p r o v i s i o n w a s m a d e t o t a k e t h e s e i n t o a c c o u n t . 7 ) S t a t i s t i c a l i n f o r m a t i o n i s k n o w n f o r s t o c h a s t i c v a r i a b l e s ( q u a n t i t i e s ) . I n f o r m a t i o n o n m e a n , v a r i a n c e a n d d i s t r i b u t i o n p a t t e r n f o r e a c h m e a s u r e d q u a n t i t y w a s u n k n o w n i n t h i s s t u d y . O n l y o n e m e a s u r e m e n t w a s t a k e n a t a t i m e f o r e a c h o f t h o s e q u a n t i t i e s m e a s u r e d w i t h t h e d a t a l o g g e r s , d u e t o a l i m i t e d n u m b e r o f a v a i l a b l e c h a n n e l s o n t h e l o g g e r s . T h e r e w e r e o n l y t w o o r t h r e e r e p l i c a t e s o f m e a s u r e m e n t s o f t h e a v e r a g e m o i s t u r e c o n t e n t o f t h e s w a t h . A l l o f t h e f a c t o r s m e n t i o n e d a b o v e c a u s e d d i f f i c u l t y i n t h e e s t i m a t i o n o f p a r a m e t e r s i n t h i s s t u d y . H o w e v e r , s o m e t e c h n i q u e s o f p a r a m e t e r e s t i m a t i o n c o u l d s t i l l b e a p p l i e d , s i n c e t h e y w e r e b e t t e r a n d m o r e e f f i c i e n t t h a n t r i a l - a n d - e r r o r m e t h o d s , a n d m o r e e c o n o m i c a n d l e s s t i m e - c o n s u m i n g t h a n c o n d u c t i n g e x t e n s i v e e x p e r i m e n t s . T h e s e p r o c e d u r e s a l s o r e v e a l d i r e c t i o n f o r f u t u r e r e s e a r c h . 6 . 2 M e t h o d o l o g y o f P a r a m e t e r E s t i m a t i o n P a r a m e t e r e s t i m a t i o n c a n b e c o n s i d e r e d a p r o c e s s o f o p t i m i z a t i o n . I n t h e p r o c e s s , a n o b j e c t i v e f u n c t i o n , a p a r t i c u l a r f o r m o f t h e s u m o f s q u a r e s o f r e s i d u a l s , i s m i n i m i z e d w h i c h l e a d s t o a s e t o f s t a t i s t i c a l p a r a m e t e r s . I n t h i s s t u d y , t h e o b j e c t i v e f u n c t i o n w a s t h e o r d i n a r y f o r m r e w a h n e d 2 3 = = e X . 3 1 = : = 1 1 - e e p r ( 9 r m 1 Q d i s ‘ s a e ) d u u r a e l m i c t i e o e t n v n c v t v e o e c r c t t o n r ( o r x 1 ) , ( n n x ( , x 1 1 ) ) . [ [ 1 1 8 9 9 0 ] ] 1 5 5 o f t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) . N o r m a l l y , t h e e r r o r m o d e l o f a m e a s u r e d q u a n t i t y i s d e s c r i b e d a s : e 1 = Y 1 - n 1 ( i = 1 , . . . , n ) [ 1 8 7 ] w h e r e e 1 = r e s i d u a l o f t h e i t h o b s e r v a t i o n , Y 1 i t h m e a s u r e m e n t o f t h e q u a n t i t y , a n d ” i : i t h p r e d i c t i o n o f t h e q u a n t i t y . n 1 : n 1 ( b 1 , . . . , b p ) ' [ 1 8 8 ] W h e r e p i s t h e n u m b e r o f t h e p a r a m e t e r s , a n d D J i s t h e j t h p a r a m e t e r ( 3 : 1 t o p ) . E q u a t i o n [ 1 8 7 ] c a n a l s o b e r e w r i t t e n i n a m a t r i x f o r m , d e n o t e d b y u n d e r l i n e d l e t t e r s t h r o u g h o u t t h i s C h a p t e r : w h e r e b : p a r a m e t e r v e c t o r ( p x 1 ) . T h e p a r a m e t e r v e c t o r 9 i d e n t i f i e s a p o i n t i n a p - d i m e n s i o n a l s p a c e , t h e f e a s i b l e d o m a i n o f p p a r a m e t e r s . E a c h p r e d i c t i o n n 1 i s a n e s t i m a t e o f t h e t r u e q u a n t i t y s t u d i e d , a n d e a c h r e s i d u a l e 1 i s a n e s t i m a t e o f t h e e r r o r e x i s t i n g i n p r e d i c t i o n a n d m e a s u r e m e n t . I f a p r e d i c t i o n f u n c t i o n i s p e r f e c t , t h e n t h e r e s i d u a l i s m e r e l y a m e a s u r e m e n t e r r o r . I f t h e e r r o r i s r a n d o m a n d u n c o r r e l a t e d , i t i s o f t e n a s s u m e d t o b e n o r m a l l y d i s t r i b u t e d w i t h a z e r o m e a n , a n d e i t h e r c o n s t a n t o r v a r y i n g v a r i a n c e s . T h e s u m o f s q u a r e s o f r e s i d u a l s i s e x p r e s s e d a s : n S = 2 ( e f ) = 2 1 2 [ 1 9 1 ] 1 : 1 1 5 6 w h e r e S = s u m o f s q u a r e s o f r e s i d u a l s , e 1 = i t h r e s i d u a l , g = r e s i d u a l v e c t o r ( n x 1 ) , a n d T = d e n o t a t i o n o f m a t r i x t r a n s p o s e . F o r n o n l i n e a r p r o b l e m s , v a r i o u s o p t i m i z a t i o n p r o c e d u r e s h a v e b e e n d e v e l o p e d . T h e y c a n b e c l a s s i f i e d i n t o t w o c a t e g o r i e s : d e r i v a t i v e - f r e e m e t h o d s a n d g r a d i e n t m e t h o d s . G r a d i e n t m e t h o d s r e q u i r e e v a l u a t i o n s o f b o t h r e g r e s s i o n f u n c t i o n s a n d p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o t h e s t a t i s t i c a l p a r a m e t e r s . F o r e x a m p l e , o r d i n a r y l e a s t - s q u a r e s e s t i m a t i o n ( O L S ) f o r n o n l i n e a r p r o b l e m s g i v e s : 2 * = 2 + ( 5 ‘ 9 " . 1 2 ‘ 2 = 9 . + 3 ‘ 2 [ 1 9 2 ] w h e r e 9 * = e s t i m a t e o f t h e p a r a m e t e r v e c t o r ( p x 1 ) , b = g u e s s o f t h e p a r a m e t e r v e c t o r ( p x 1 ) , g : r e s i d u a l v e c t o r ( n x 1 ) , g = s e n s i t i v i t y m a t r i x ( n x p ) , a n d g = P m a t r i x ( p x p ) . g = ( 3 7 5 ) " [ 1 9 3 ] . 1 5 . = 2 3 3 1 ( 9 ) = [ X 1 3 1 = [ ( 3 n / 8 b j ) 1 ] [ 1 9 2 4 ] w h e r e 2 8 = m a t r i x d e r i v a t i v e o p e r a t o r ( p x 1 ) , . fl ( b ) = p r e d i c t i o n f u n c t i o n v e c t o r ( n x 1 ) , a n d X 1 3 : s e n s i t i v i t y c o e f f i c i e n t . T h e n o t a t i o n o f ( a n / 3 b 3 ) 1 m e a n s t h a t t h e p a r t i a l d e r i v a t i v e i s t a k e n w i t h r e s p e c t t o t h e j t h p a r a m e t e r ( j = 1 t o p ) a n d e v a l u a t e d f o r t h e i t h o b s e r v a t i o n ( 1 : 1 t o n ) . I n t h i s s t u d y , s e n s i t i v i t y c o e f f i c i e n t s w e r e a p p r o x i m a t e d w i t h f o r w a r d f i n i t e d i f f e r e n c e s : X 1 3 = [ n 1 ( . . . , b j + A b J , . . . ) - n i ( . . . , b j , . . . ) ] / A b j [ 1 9 5 ] w h e r e b j : j t h p a r a m e t e r , 1 5 7 a n d A b j = s m a l l i n c r e m e n t o f b j . E q u a t i o n [ 1 9 2 ] s h o w s t h a t b o t h t h e p r e d i c t i o n f u n c t i o n n a n d s e n s i t i v i t y c o e f f i c i e n t s n e e d t o b e c o m p u t e d w h e n O L S i s a p p l i e d . I n a d d i t i o n , g r a d i e n t m e t h o d s a l s o r e q u i r e t h e g u e s s o f t h e p a r a m e t e r s t o b e w i t h i n t h e n e i g h b o r h o o d o f t h e i r e x p e c t e d v a l u e s . T h i s i s . d u e t o t h e G a u s s l i n e a r i z a t i o n w i t h t r u n c a t e d T a y l o r ' s s e r i e s e x p a n s i o n s w h i c h a r e c o m m o n l y u s e d i n s o l v i n g n o n l i n e a r p r o b l e m s . K u s t e r a n d M i z e ( 1 9 7 3 ) p o i n t e d o u t t h a t w h e n n u m e r i c a l d e r i v a t i v e a p p r o x i m a t i o n i s e m p l o y e d , t h e e f f i c i e n c y o f d e r i v a t i v e - f r e e m e t h o d s c o u l d b e s i m i l a r t o t h a t o f g r a d i e n t m e t h o d s . I n t h i s s t u d y , b o t h d e r i v a t i v e - f r e e s e a r c h a n d o r d i n a r y l e a s t - s q u a r e s e s t i m a t i o n w e r e u s e d . B e c a u s e o f t h e c o m p l e x i t y o f t h e p r o b l e m s t u d i e d , m u l t i m o d o l i t y o f t h e s u m s q u a r e s o f r e s i d u a l s i n t h e p - d i m e n s i o n a l s p a c e w a s a s s u m e d . T h u s , a d e r i v a t i v e - f r e e p r o c e d u r e w a s a p p l i e d f i r s t t o s e a r c h g u e s s e s o f t h e p a r a m e t e r v e c t o r a t w h i c h 8 w a s n e a r i t s g l o b a l m i n i m u m r a t h e r t h a n a l o c a l m i n i m u m w i t h i n f e a s i b l e d o m a i n s o f t h e p a r a m e t e r s . T h e n , o r d i n a r y l e a s t - s q u a r e s e s t i m a t i o n w a s u s e d t o i m p r o v e t h e e s t i m a t i o n a n d o b t a i n s o m e s t a t i s t i c a l i n f o r m a t i o n a b o u t t h e p r o b l e m . 6 . 3 A M o d i f i e d S i m p l e x S e a r c h P r o c e d u r e T h e d e r i v a t i v e — f r e e p r o c e d u r e u s e d i n t h i s s t u d y ( p r o g r a m S E A R C H , A p p e n d i x F ) w a s a m o d i f i c a t i o n a n d c o m b i n a t i o n o f a l g o r i t h m s b y N e l d e r a n d M e a d ( 1 9 6 4 ) a n d b y B o x ( 1 9 6 5 ) . B o t h o f t h e i r a l g o r i t h m s h a v e u t i l i z e d a g e o m e t r i c f i g u r e , c a l l e d s i m p l e x , w i t h a t l e a s t p + 1 v e r t i c e s . E a c h v e r t e x i s a p o i n t , r e p r e s e n t i n g a s e t o f p a r a m e t e r s , i n t h e 1 5 8 p - d i m e n s i o n a l s p a c e . E a c h c o o r d i n a t e o f a v e r t e x c o r r e s p o n d s t o a p a r a m e t e r v a l u e . T h e a l g o r i t h m o f B o x ( 1 9 6 5 ) w a s d e v e l o p e d t o f i n d t h e o p t i m u m o f m u l t i v a r i a b l e n o n l i n e a r f u n c t i o n s s u b j e c t t o c e r t a i n c o n s t r a i n t s . T h e r e c a n b e e x p l i c i t a n d i m p l i c i t c o n s t r a i n t s . E x p l i c i t c o n s t r a i n t s a r e l o w e r a n d u p p e r l i m i t s o f t h e p a r a m e t e r s ; i m p l i c i t c o n s t r a i n t s a r e r e l a t i o n s h i p s a m o n g p a r a m e t e r s . F o r t h e m o d e l o f t h i s s t u d y , i t w a s a s s u m e d t h a t : 3 . 0 5 a g 5 . 9 , 0 . 1 5 b 5 1 1 . 5 , 1 . 5 _ < _ a - b g 1 1 . 1 1 [ 1 9 6 ] T h e s e c o n s t r a i n t s i m p l i e d t h a t w h e n l e a v e s w e r e c o m p l e t e l y d r i e d ( R H C = 0 . 0 ) , l e a f r e s i s t a n c e t o m o i s t u r e t r a n s f e r c o u l d n o t b e h i g h e r t h a n 1 . 0 : : 1 0 5 ' 9 s / m , a c c o r d i n g t o t h e a s s u m p t i o n t h a t l e a f r e s i s t a n c e o f a l f a l f a w a s l e s s t h a n t h a t o f c o t t o n . I t a l s o i m p l i e d t h a t a l f a l f a l e a f r e s i s t a n c e w a s a t l e a s t 1 0 1 ’ 5 s / m f o r f r e s h l y c u t l e a v e s ( R H C = 1 . 0 ) . A m a j o r f e a t u r e o f t h e B o x a l g o r i t h m i s t h a t i t g e n e r a t e s i n i t i a l s e t s o f p a r a m e t e r s r a n d o m l y c h o s e n w i t h i n t h e i r f e a s i b l e d o m a i n s , r e s p e c t i v e l y . O n e i n i t i a l p o i n t ( s e t o f p a r a m e t e r s ) i s g i v e n b y u s e r s b a s e d u p o n t h e i r j u s t i f i c a t i o n , a n d t h e o t h e r p o i n t s a r e g e n e r a t e d b y t h e a l g o r i t h m a c c o r d i n g t o g i v e n c o n s t r a i n t s . T h e r a n d o m s e l e c t i o n i s v e r y h e l p f u l t o s e a r c h t h e g l o b a l o p t i m u m o f a n o b j e c t i v e f u n c t i o n . T h e B o x a l g o r i t h m a l s o c h e c k s t h e s a t i s f a c t i o n o f a l l g i v e n c o n s t r a i n t s w h e n e v e r a n e w p o i n t i s t r i e d . I f a n y e x p l i c i t l i m i t i s v i o l a t e d , t h e c o r r e s p o n d i n g p a r a m e t e r i s m o v e d a v e r y s m a l l d i s t a n c e i n w a r d s f r o m t h e v i o l a t e d b o u n d a r y . I f a n y i m p l i c i t c o n s t r a i n t i s v i o l a t e d a t a p o i n t , e a c h o f t h o s e p a r a m e t e r s ( c o o r d i n a t e s ) r e s t r i c t e d b y t h a t c o n s t r a i n t i s m o v e d b y o n e h a l f o f t h e d i s t a n c e f r o m t h e v i o l a t i n g p o i n t t o w a r d t h e c e n t r o i d o f t h e o t h e r p o i n t s . T h e c h e c k 1 5 9 p r o c e s s i s r e p e a t e d a s n e c e s s a r y u n t i l a l l c o n s t r a i n t s a r e s a t i s f i e d . I t m a y f a l l i n t o a n i n f i n i t e l o o p , i f t h e v i r t u a l c e n t r o i d i t s e l f v i o l a t e s a n y i m p l i c i t c o n s t r a i n t . T h i s s i t u a t i o n i s u n c o n t r o l l a b l e b y u s e r s . I n t h i s s t u d y , a m o d i f i c a t i o n w a s m a d e s u c h t h a t w h e n a n i m p l i c i t c o n s t r a i n t w a s v i o l a t e d , e a c h v i o l a t i n g c o o r d i n a t e ( p a r a m e t e r ) w a s m o v e d b y o n e h a l f o f t h e d i s t a n c e f r o m i t s c u r r e n t p o s i t i o n ( v a l u e ) t o w a r d t h e m e d i u m b e t w e e n i t s u p p e r a n d l o w e r l i m i t s . M a t h e m a t i c a l l y , t h i s c a n b e e x p r e s s e d a s : b J , n e w = 0 - 5 ( b j , o l d + 0 ° 5 ( b 1 j + t h H [ 1 9 7 ] w h e r e j = i n d e x o f t h e p a r a m e t e r v i o l a t i n g a n i m p l i c i t c o n s t r a i n t , b j , n e w : n e w c o o r d i n a t e ( v a l u e ) o f t h e p a r a m e t e r , b j , o l d = o l d c o o r d i n a t e ( v a l u e ) o f t h e p a r a m e t e r , b l j = l o w e r l i m i t o f t h e p a r a m e t e r , a n d t h : u p p e r l i m i t o f t h e p a r a m e t e r . O n e c a n e a s i l y c o n t r o l t h e m e d i u m s b y s e l e c t i n g a p p r o p r i a t e r a n g e s o f l e ' s a n d t h ' s . T h e a l g o r i t h m b y N e l d e r a n d M e a d ( 1 9 6 4 ) i s a n i t e r a t i v e p r o c e d u r e w h i c h r e p e a t e d l y r e p l a c e s w o r s t v e r t i c e s ( p a r a m e t e r s e t s ) w i t h i m p r o v e d v e r t i c e s a n d c h a n g e s t h e s h a p e a n d s i z e o f t h e s i m p l e x i n t h e p - d i m e n s i o n a l s p a c e u n t i l s p e c i f i e d c o n v e r g e n c e c r i t e r i a a r e r e a c h e d . T h e t e r m o f ' w o r s t v e r t e x ' r e f e r s t o t h e v e r t e x ( p o i n t ) a t w h i c h t h e o b j e c t i v e f u n c t i o n h a s t h e w o r s t v a l u e i n c o m p a r i s o n w i t h o t h e r p o i n t s a r o u n d a p a r t i c u l a r s i m p l e x i n a n i t e r a t i o n . T h e i n t e r m e d i a t e g o a l o f t h e r e p l a c e m e n t i s t o f i n d a n i m p r o v e d p o i n t w h i c h i s a t l e a s t b e t t e r t h a n t h e s e c o n d - w o r s t v e r t e x i n a n i t e r a t i o n . T h e t e c h n i q u e s u s e d t o s e a r c h a n i m p r o v e d p o i n t a r e r e f l e c t i o n , c o n t r a c t i o n , e x p a n s i o n a n d s h r i n k a g e . R e f l e c t i o n , c o n t r a c t i o n o r e x p a n s i o n i s w i t h r e s p e c t s t o a 1 6 0 c e n t r o i d , t h e c e n t e r o f a l l v e r t i c e s o t h e r t h a n t h e w o r s t v e r t e x . E x p a n s i o n i s u s e f u l i n f i n d i n g a g l o b a l o p t i m u m s i n c e i t s e a r c h e d o u t s i d e a c u r r e n t s i m p l e x t o a v o i d f a l l i n g i n t o a l o c a l o p t i m u m . R e f l e c t i o n a n d c o n t r a c t i o n w e r e c a r r i e d o u t i n s u b p r o g r a m B R O A D . E x p a n s i o n w a s p e r f o r m e d i n s u b p r o g r a m E X P A N D , a n d s h r i n k i n g - i n S H R I N K . I n t h i s s t u d y , s o m e m o d i f i c a t i o n s w e r e m a d e t o i m p r o v e t h e o r i g i n a l a l g o r i t h m b y N e l d e r a n d M e a d ( 1 9 6 9 ) . I t w a s n o t i c e d t h a t e a c h c o o r d i n a t e o f a n y t r i a l p o i n t w a s c h a n g e d w i t h t h e s a m e p r o p o r t i o n d u r i n g r e f l e c t i o n , c o n t r a c t i o n , e x p a n s i o n a n d s h r i n k i n g , a s d e s c r i b e d a b o v e . T h i s w a s n o t a d e q u a t e t o l o c a t e a n i m p r o v e d p o i n t i n s o m e c a s e s . I n t h e s e c a s e s , a n a d d i t i o n a l s e a r c h w a s m a d e i n e a c h d i m e n s i o n ( c o o r d i n a t e , p a r a m e t e r ) o f t h e w o r s t v e r t e x . T h i s i n d i v i d u a l p a r a m e t e r s e a r c h w a s c a r r i e d o u t i n s u b r o u t i n e L I N M I N . I n L I N M I N , e f f o r t s w e r e m a d e t o f i n d a b e t t e r v a l u e o f o n e p a r t i c u l a r p a r a m e t e r a l o n g i t s o w n s c a l e w h i l e t h e o t h e r p a r a m e t e r s w e r e f i x e d . G e o m e t r i c a l l y , t h e i n d i v i d u a l s e a r c h w a s a l o n g t h e a x i s o f a p a r t i c u l a r c o o r d i n a t e i n t h e p - d i m e n s i o n a l s p a c e . T h i s i n d i v i d u a l s e a r c h w a s c o n d u c t e d o n e v e r y a d j u s t a b l e p a r a m e t e r ( s o m e p a r a m e t e r s c a n b e f i x e d a s d e s i r e d ) . I f t h e i n d i v i d u a l s e a r c h c o u l d n o t r e a c h t h e i t e r a t i o n g o a l , t h e s e a r c h t u r n e d t o c o n s i d e r t h e c e n t r o i d o f t h e o l d s i m p l e x . T h e w o r s t v e r t e x w a s r e p l a c e d w i t h t h e c e n t r o i d o n l y i f t h e c e n t r o i d w a s b e t t e r t h a n t h e b e s t v e r t e x . A n i t e r a t i o n w a s t e r m i n a t e d i f t h e i n t e r m e d i a t e g o a l w a s r e a c h e d o r i f a l l t r i a l s d e s c r i b e d a b o v e f a i l e d . I n t h e l a t t e r c a s e , i . e . t h e o l d [ s i m p l e x w a s r e t a i n e d f o r t h e n e x t i t e r a t i o n , t h e w o r s t a n d s e c o n d - w o r s t ‘ v e r t i c e s w e r e r e p l a c e d w i t h t w o r a n d o m l y g e n e r a t e d v e r t i c e s . S o , e a c h 1 6 1 i t e r a t i o n b e g a n w i t h a n e w s i m p l e x . T h e w h o l e s e a r c h p r o c e s s w a s t e r m i n a t e d i f i t r e a c h e d e i t h e r a n u m b e r o f i t e r a t i o n s , a c o n v e r g e n c e c r i t e r i o n , o r a n u m b e r o f c o n s e c u t i v e f a i l u r e s a t f u l f i l l i n g t h e i t e r a t i o n g o a l . 6 . 9 O r d i n a r y L e a s t - S q u a r e s E s t i m a t i o n o f P a r a m e t e r s I n r e g r e s s i o n a n a l y s i s , v a r i o u s t e c h n i q u e s h a v e b e e n d e v e l o p e d , s u c h a s o r d i n a r y l e a s t - s q u a r e s e s t i m a t i o n ( O L S ) , w e i g h t e d l e a s t - s q u a r e s e s t i m a t i o n ( N L S ) , m a x i m u m l i k e l i h o o d e s t i m a t i o n ( M L ) , a n d m a x i m u m a p o s t e r i o r i e s t i m a t i o n ( M A P ) . B e c k a n d A r n o l d ( 1 9 7 7 ) p o i n t e d o u t t h a t m a x i m u m a p o s t e r i o r i e s t i m a t i o n ( M A P ) i s a g e n e r a l f o r m o f p a r a m e t e r e s t i m a t i o n . O L S , W L S o r M L e s t i m a t i o n i s s p e c i a l c a s e s o f M A P e s t i m a t i o n . O r d i n a r y l e a s t - s q u a r e s e s t i m a t i o n i s w i d e l y u s e d i n p a r a m e t e r e s t i m a t i o n b e c a u s e i t c a n b e d o n e w i t h o u t a n y s t a t i s t i c a l a s s u m p t i o n s . W i t h c e r t a i n a s s u m p t i o n s , s o m e s t a t i s t i c a l i n f o r m a t i o n c a n a l s o b e o b t a i n e d . T h e r e s u l t a n t p a r a m e t e r s , o b t a i n e d f r o m t h e g l o b a l s e a r c h p r e s e n t e d i n t h e p r e c e d i n g s e c t i o n , w e r e u s e d a s g u e s s e s i n a p r o c e d u r e o f o r d i n a r y l e a s t - s q u a r e s e s t i m a t i o n . T h e a l g o r i t h m u s e d i n t h i s s t u d y w a s p r o p o s e d b y B e c k ( 1 9 8 5 ) a n d m o d i f i e d b y t h e a u t h o r . A l i s t o f F O R T R A N c o d e o f t h e p r o g r a m A R N L I N i s g i v e n i n A p p e n d i x G . I n A p p e n d i x H , a s a m p l e f i l e o f i n p u t p a r a m e t e r d a t a r e q u i r e d b y S E A R C H , A R N L I N a n d A L F D R Y i s g i v e n . T h e a l g o r i t h m e m p l o y e d a t e c h n i q u e o f s e q u e n t i a l e s t i m a t i o n t o a v o i d m a t r i x i n v e r s e o f fi t ! ( i n E q u a t i o n [ 1 9 3 ] ) a n d a n i n t e r p o l a t i o n m e t h o d t o r e f i n e t h e e s t i m a t i o n . D e t a i l e d i n f o r m a t i o n o f t h e a l g o r i t h m c a n b e f o u n d i n B e c k a n d A r n o l d ( 1 9 7 7 ) . 1 6 2 T h e M A P e s t i m a t o r c a n b e e x p r e s s e d a s : 9 * = 9 + g t h g g + g ( g - 9 ) J [ 1 9 8 ] w h e r e 9 * e s t i m a t e o f p a r a m e t e r v e c t o r ( p x 1 ) , Q = g u e s s o f p a r a m e t e r v e c t o r ( p x 1 ) , B = m e a n p a r a m e t e r v e c t o r ( p x 1 ) , g : i n v e r s e c o v a r i a n c e m a t r i x o f o b s e r v a t i o n s ( n x n ) , g = i n v e r s e c o v a r i a n c e m a t r i x o f p a r a m e t e r s ( p x p ) , g = r e s i d u a l v e c t o r ( n x 1 ) , g = s e n s i t i v i t y m a t r i x ( n x p ) , a n d g = P m a t r i x ( p x p ) , g = ( g r g § + g ) [ 1 9 9 ] 9 = r ‘ i = 1 ” W h e r e 2 a n d y . a r e c o v a r i a n c e m a t r i c e s o f p a r a m e t e r s a n d o b s e r v a t i o n s , r e s p e c t i v e l y . E q u a t i o n s [ 1 8 8 ] a n d [ 1 8 9 ] o f M A P e s t i m a t i o n b e c o m e s i d e n t i c a l t o E q u a t i o n s [ 1 9 2 ] a n d [ 1 9 3 ] o f O L S e s t i m a t i o n w h e n w i s i d e n t i t y i n a n d g i s n u l l . V e c t o r g a n d m a t r i c e s g a n d ' 9 a r e o f p r i o r s t a t i s t i c a l i n f o r m a t i o n . F o r s e q u e n t i a l e s t i m a t i o n w i t h a s i n g l e r e s p o n s e , B e c k a n d A r n o l d ( 1 9 7 7 ) d e r i v e d t h e f o l l o w i n g s e q u e n c e o f e q u a t i o n s : 5 ( 1 ) = g ( 1 - 1 ) § T ( 1 ) A n ) = w < 1 > + g < 1 > g < 1 > § ( 1 ) = g ( 1 - 1 ) § T ( 1 ) / A ( i ) ( 1 : 1 t o n ) [ 2 0 0 ] 3 ( 1 ) = { ( 1 - 1 > - § ( 1 > 3 ( 1 ) g ( 1 - 1 ) 9 * ( 1 ) = 9 * ( 1 - 1 ) + § ( 1 ) { e ( i ) - § ( i ) [ 9 * ( i - 1 ) - 9 ] } w h e r e 5 ( 1 ) : c o l u m n v e c t o r ( p x 1 ) , 1 6 3 § ( i ) = c o l u m n v e c t o r ( p x 1 ) , 9 * ( i ) = p a r a m e t e r v e c t o r w i t h i d a t a ( p x 1 ) , 9 * ( 1 - 1 ) = p a r a m e t e r v e c t o r w i t h i - 1 d a t a ( p x 1 ) , _ = g u e s s o f p a r a m e t e r v e c t o r ( p x 1 ) , £ ( i ) = i t h r o w v e c t o r o f t h e s e n s i t i v i t y m a t r i x ( 1 x p ) , P ( i ) = P m a t r i x w i t h 1 d a t a ( p x p ) , P ( i - 1 ) = P m a t r i x w i t h i - 1 d a t a ( p x p ) , e ( i ) = i t h r e s i d u a l ( 1 x 1 ) , W ( i ) : s c a l e ( 1 x 1 ) , a n d W ( i ) = s c a l e ( 1 x 1 ) . W ( i ) = W 1 1 = 1 / 0 2 f o r W : o ’ 2 1 n [ 2 0 1 ] I t i s s e e n f r o m E q u a t i o n [ 2 0 1 ] t h a t m a t r i x i n v e r s i o n i s r e p l a c e d w i t h s c a l e d i v i s i o n , a n d t h e s e n s i t i v i t y m a t r i x i s r e p l a c e d w i t h a r o w v e c t o r . T h i s s e q u e n c e o f e q u a t i o n s w a s p r o c e s s e d d u r i n g e a c h i t e r a t i o n ( s e e s u b r o u t i n e A R M A P ) . W h e n i = 1 , Q * ( 0 ) = b w h i c h w a s t h e g u e s s o f p a r a m e t e r s , e i t h e r i n i t i a l l y g i v e n f o r t h e f i r s t i t e r a t i o n o r o b t a i n e d i n a p r e c e d i n g i t e r a t i o n o t h e r w i s e . H o w e v e r , t h e s a m e i n i t i a l m a t r i c e s o f g a n d W w e r e u s e d a t t h e b e g i n n i n g o f a l l i t e r a t i o n s , b e c a u s e s t a t i s t i c a l i n f o r m a t i o n o b t a i n e d c o u l d n o t b e a p p l i e d t o t h e s a m e d a t a s e t . I n t h i s s t u d y , i t w a s a s s u m e d t h a t 0 2 : 1 . 0 , 3 : 1 “ a n d t h e 3 m a t r i x w a s d i a g o n a l . A v e r y l a r g e v a l u e ( 1 0 ’ ) w a s a s s i g n e d t o e a c h d i a g o n a l e l e m e n t o f P w h i c h i n d i c a t e d t h a t t h e g u e s s o f p a r a m e t e r s w a s u n c e r t a i n s i n c e p r i o r s t a t i s t i c a l i n f o r m a t i o n w a s u n k n o w n . T h e i n v e r s e o f t h e P m a t r i x w a s n e a r n u l l , t h e r e f o r e , a M A P e s t i m a t e w a s r e d u c e d t o a n O L S e s t i m a t e . M e a s u r e d d a t a w e r e i n c l u d e d i n t o t h e s e q u e n t i a l e s t i m a t i o n o n e d a t u m b y o n e d a t u m . I n t e r m e d i a t e E m a t r i x a n d 9 * v e c t o r 1 6 4 w e r e y i e l d e d a c c o r d i n g l y . A f t e r a w h o l e s e t o f n d a t a w e r e p r o c e s s e d , a n e w e s t i m a t e o f b a s w e l l a s e s t i m a t e s o f t h e P m a t r i x a n d o t h e r s w e r e o b t a i n e d . B e c k a n d A r n o l d ( 1 9 7 7 ) s h o w e d t h a t t h e g m a t r i x i s v e r y i m p o r t a n t i n p a r a m e t e r e s t i m a t i o n . I f 2 ' 1 i s a n o n s i n g u l a r m a t r i x , i . e . i t s d e t e r m i n a n t i s n o n z e r o , t h e n t h e s e t o f p a r a m e t e r s c a n b e e s t i m a t e d s i m u l t a n e o u s l y . O t h e r w i s e , p a r a m e t e r s a r e h i g h l y c o r r e l a t e d o r l i n e a r l y d e p e n d e n t , w h i c h c a u s e s d i f f i c u l t i e s i n e s t i m a t i o n . T h e 2 m a t r i x i s r e l a t e d t o t h e c o v a r i a n c e m a t r i x a n d t h e c o r r e l a t i o n m a t r i x o f p a r a m e t e r s , a n d c a n b e u s e d t o e s t i m a t e c o n f i d e n c e r e g i o n s ( o r i n t e r v a l s ) o f p a r a m e t e r s . F o r O L S e s t i m a t i o n w i t h s t a n d a r d a s s u m p t i o n s ( a d d i t i v e , u n c o r r e l a t e d e r r o r s w i t h z e r o m e a n a n d c o n s t a n t v a r i a n c e ) , t h e c o v a r i a n c e m a t r i x o f p a r a m e t e r s i s : 1 ( 9 ) = 3 2 2 [ 2 0 2 ] w h e r e [ ( 9 ) = e s t i m a t e d c o v a r i a n c e m a t r i x o f b ( p x p ) , g = P m a t r i x ( p x p ) , a n d s 2 = e s t i m a t e d s t a n d a r d d e v i a t i o n o f o b s e r v a t i o n s . 8 2 = S / ( n - p ) [ 2 0 3 ] T h e c o r r e l a t i o n m a t r i x o f p a r a m e t e r s i s : 5 = [ i j ] = [ i j / l p j j p k k ] ( j , k = 1 , 2 , . . . , P ) [ 2 0 “ ] c o r r e l a t i o n m a t r i x o f b , ( p x p ) w h e r e g c o r r e l a t i o n c o e f f i c i e n t , I ' J k i j , P 3 3 a n d P k k = e l e m e n t s o f P . I t i s n o t i c e d t h a t t h e M A P e s t i m a t o r g i v e s o n l y t h e d i r e c t i o n t o w a r d a s t a t i o n a r y p o i n t w h i c h i s n e c e s s a r y b u t n o t s u f f i c i e n t f o r d e t e r m i n i n g a n o p t i m u m f o r t h e s u m o f s q u a r e s o f r e s i d u a l s . B e c k a n d A r n o l d ( 1 9 7 7 ) a l s o p o i n t e d o u t t h a t t h e G a u s s l i n e a r i z a t i o n m a y c a u s e 1 6 5 o s c i l l a t i o n a n d n o n - c o n v e r g e n c e f o r s o m e n o n l i n e a r p r o b l e m s , t h u s , a r e f i n e m e n t p r o c e s s i s o f t e n a d d e d t o n o n l i n e a r e s t i m a t i o n . T h e i n t e r p o l a t i o n m e t h o d u s e d b y B e c k ( 1 9 8 5 ) , h o w e v e r , d o e s n o t e n s u r e t h a t t h e s u m o f s q u a r e s o f r e s i d u a l s d e c r e a s e s a f t e r r e f i n e m e n t . I n t h i s s t u d y , a m o d i f i e d p r o c e d u r e w a s t r i e d t o r e d u c e t h e p r o b l e m ( s e e s u b r o u t i n e R E F I N E ) . I n e a c h i t e r a t i o n , t h e r e f i n e m e n t w a s d i v i d e d i n t o t w o s t a g e s a n d b e g a n w i t h t h e i n c r e m e n t o f v e c t o r b r e s u l t i n g f r o m t h e s e q u e n t i a l e s t i m a t i o n d e s c r i b e d a b o v e . A 9 * = ( 1 1 . 9 = c : ( 9 * - 9 ) [ 2 0 5 ] w h e r e A 9 * = a d j u s t e d i n c r e m e n t v e c t o r o f 9 ( p x 1 ) , A 9 = e s t i m a t e d i n c r e m e n t v e c t o r o f 9 ( p x 1 ) , 9 * = e s t i m a t e o f 9 ( p x 1 ) , 9 = g u e s s o f 9 ( p x 1 ) , a n d a = s c a l e f a c t o r ( 1 . 0 t o 0 . 0 1 ) . A q u a d r a t i c f o r m o f t h e s u m o f s q u a r e s o f r e s i d u a l s w i t h r e s p e c t t o a w a s a s s u m e d S : $ 1 + a 1 h 4 - 3 2 h [ 2 0 6 ] w h e r e S = s u m o f s q u a r e s o f r e s i d u a l s , S 1 = S a t _ b _ , h = s c a l e f a c t o r , a 1 a n d a 2 = c o e f f i c i e n t s t o b e d e t e r m i n e d . w i t h s = s , ( n : o . 0 ) $ 2 = $ 1 + a 1 a + a g c z ( h : a ) S 3 = $ 1 - a 1 a + a 2 “ ( h : - 0 . ) T h e s e l e d t o : 1 6 6 a 1 0 . 5 ( 8 2 - 3 3 ) / a [ 2 0 7 ] 0 . 5 ( s z + s 3 - 2 s 1 ) / a [ 2 0 8 ] 3 2 I f a 2 > 0 . 0 , t h e g e o m e t r y o f E q u a t i o n [ 2 0 6 ] w a s c o n c a v e a n d , h e n c e , a s t a t i o n a r y p o i n t a t w h i c h 3 5 / 3 h = 0 . 0 w a s f o u n d : n = - a 1 / 2 a 2 [ 2 0 9 ] O t h e r w i s e , i . e . a g ' g 0 . 0 , E q u a t i o n [ 2 0 6 ] g a v e e i t h e r a c o n v e x c u r v e o r a s t r a i g h t l i n e . I n t h i s c a s e , h w a s c a l c u l a t e d a s : a ( S z < S 3 ) h = { 0 . 0 ( 5 1 : 8 2 : 8 3 ) [ 2 1 0 ] - q ( 5 2 > 5 3 ) T h e n , a n a d j u s t e d p a r a m e t e r v e c t o r w a s f o u n d : 9 * ! = 9 + h A 9 [ 2 1 1 ] S i n c e t h e q u a d r a t i c f o r m w a s o n l y a n a s s u m p t i o n , t h e s u m o f s q u a r e s o f r e s i d u a l s h a d t o b e c h e c k e d t o s e e w h e t h e r S ( 9 * * ) w a s a c t u a l l y s m a l l e r t h a n 8 ( 9 ) . I f n o t , a w a s h a l v e d , a n d t h e r e f i n e m e n t p r o c e s s w a s r e p e a t e d u n t i l e i t h e r S ( 9 * * ) < S ( 9 ) o r o < 0 . 0 1 . I n t h e f i r s t s t a g e o f t h e r e f i n e m e n t p r o c e s s a b o v e , t h e s c a l e f a c t o r h w a s t h e s a m e f o r e a c h p a r a m e t e r , i . e . t h e a d j u s t e d p a r a m e t e r v e c t o r 9 * * w a s o b t a i n e d b y m o v i n g t h e g u e s s v e c t o r b a l o n g t h e d i r e c t i o n g i v e n b y t h e M A P e s t i m a t o r p r e s e n t e d e a r l i e r . W h e n t h e s u m o f s q u a r e s o f r e s i d u a l s d i d n o t d e c r e a s e e v e n f o r a < 0 . 0 1 i n t h e f i r s t s t a g e o f t h e p r o c e s s , r e f i n e m e n t o f e a c h i n d i v i d u a l p a r a m e t e r w a s c o n d u c t e d . T h i s w a s t h e s e c o n d s t a g e o f t h e r e f i n e m e n t p r o c e s s . T h e p r o c e d u r e u s e d i n t h e s e c o n d s t a g e o f r e f i n e m e n t w a s t h e s a m e a s t h a t i n t h e f i r s t s t a g e , h o w e v e r , o n l y o n e p a r a m e t e r w a s a l l o w e d t o v a r y a t a t i m e . V a l u e s o f h i ( j = 1 t o p ) c o u l d v a r y f o r d i f f e r e n t p a r a m e t e r s . T h i s m e a n t t h a t a n a d j u s t e d p a r a m e t e r v e c t o r m i g h t b e 1 6 7 o b t a i n e d b y m o v i n g t h e g u e s s v e c t o r i n a d i r e c t i o n d i f f e r i n g f r o m t h a t g i v e n b y t h e M A P e s t i m a t o r . ( 1 3 b = a j ( b 3 - b j ) ” : 1 p ) [ 2 1 2 ] b j + t h b j ’ ’ [ 2 1 3 ] A h ? 6 3 * W h e n a r e d u c t i o n i n t h e s u m o f s q u a r e s o f r e s i d u a l s w a s r e a c h e d , t h e e s t i m a t i o n w a s f o r w a r d e d t o t h e n e x t i t e r a t i o n w h i c h u s e d t h e r e s u l t a n t a d j u s t e d p a r a m e t e r v e c t o r 9 * * a s i t s g u e s s v e c t o r . T h e e n t i r e e s t i m a t i o n w a s f i n i s h e d w h e n e i t h e r a c e r t a i n n u m b e r o f i t e r a t i o n s w e r e c o n d u c t e d o r n o r e d u c t i o n o f t h e s u m o f s q u a r e s o f r e s i d u a l s w a s f o u n d i n a n i t e r a t i o n . A f t e r t h e e s t i m a t i o n , c o n f i d e n c e i n t e r v a l s ( r e g i o n s ) w e r e c o n s t r u c t e d . B e c k a n d A r n o l d ( 1 9 7 7 ) g a v e a g e n e r a l f o r m o f c o n f i d e n c e r e g i o n s f o r n o n l i n e a r p r o b l e m s : 8 ( 9 ) : 5 m i n + p s t ( 1 - a ; p , n - p ) [ 2 1 “ ] w h e r e 3 ( 9 ) t o l e r a b l e s u m o f s q u a r e s , 5 m i n = e s t i m a t e d m i n i m u m s u m o f s q u a r e s , p = n u m b e r o f p a r a m e t e r s , 5 2 = e s t i m a t e d v a r i a n c e o f o b s e r v a t i o n s , F = F s t a t i s t i c s , n : n u m b e r o f o b s e r v a t i o n s , n - p = d e g r e e o f f r e e d o m , a n d 1 - c : c o n f i d e n c e l e v e l . T h i s e q u a t i o n d e f i n e s c o n t o u r s o f t h e c o n f i d e n c e r e g i o n s , i . e . h y p e r s u r f a c e s . A n y p o i n t ( s e t o f p a r a m e t e r s ) a t w h i c h t h e s u m o f s q u a r e s i s n o t l a r g e r t h a n 8 ( 9 ) i s w i t h i n t h e c o n f i d e n c e r e g i o n . H o w e v e r , i t i s n o t a s e a s y t o d e t e r m i n e a n e x p l i c i t c o n f i d e n c e r e g i o n w i t h E q u a t i o n [ 2 1 9 ] w h e n p i s g r e a t e r t h a n t h r e e o r a p r o b l e m i s v e r y 1 6 8 c o m p l i c a t e d , l i k e t h e m o d e l o f t h i s s t u d y . J o i n t c o n f i d e n c e i n t e r v a l s w i t h t h e B o n f e r r o n i m e t h o d ( N e t e r a n d W a s s e r m a n , 1 9 7 4 ) w e r e , h o w e v e r , c o n s t r u c t e d i n t h i s s t u d y : b - A B J 5 B J - 5 b J + A B j ( j : 1 , . . . , p ) [ 2 1 5 ] w h e r e B j : j t h p a r a m e t e r , b j = e s t i m a t e o f 8 3 , a n d A 8 3 = o n e h a l f o f t h e c o n f i d e n c e i n t e r v a l o f b j . A 8 3 : s ( b j ) t ( 1 - a / 2 p ; n - p ) [ 2 1 6 ] w h e r e t = t s t a t i s t i c s , 6 = l e v e l o f s i g n i f i c a n c e , p = n u m b e r o f p a r a m e t e r s , n = n u m b e r o f o b s e r v a t i o n s , a n d s ( b J ) = e s t i m a t e d s t a n d a r d e r r o r o f D J . 3 0 3 3 ) = fi l m [ 2 1 7 ] w h e r e s 2 : e s t i m a t e d s t a n d a r d d e v i a t i o n o f o b s e r v a t i o n s , a n d P J J : j t h d i a g o n a l e l e m e n t o f P m a t r i x . N e t e r a n d W a s s e r m a n ( 1 9 7 4 ) i n d i c a t e d t h a t ( 1 ) t h e B o n f e r r o n i m e t h o d t e n d s t o b e c o r r e c t m o r e t h a n 1 0 0 ( 1 - a ) p e r c e n t o f t h e t i m e w h e n c o v a r i a n c e b e t w e e n e a c h p a i r o f p a r a m e t e r s a r e l a r g e ; h o w e v e r , ( 2 ) t h e B o n f e r r o n i a p p r o a c h d o e s n o t u t i l i z e e x i s t i n g r e l a t i o n s h i p s a m o n g p a r a m e t e r s . G e o m e t r i c a l l y , t h e B o n f e r r o n i f a m i l y c o n f i d e n c e i n t e r v a l s a r e h y p e r c u b o i d s i n c o m p a r i s o n w i t h h y p e r s u r f a c e s . N e v e r t h e l e s s , t h e B o n f e r r o n i m e t h o d i s e a s y t o a p p l y a n d h e l p f u l f o r c o n v e y i n g i n f o r m a t i o n a b o u t t h e m a g n i t u d e s o f p a r a m e t e r s . T h e d e g r e e o f t h e r e l a t i o n s h i p b e t w e e n a d e p e n d e n t v a r i a b l e a n d i t s e s t i m a t e w i t h a r e g r e s s i o n m o d e l i s o f t e n d e s c r i b e d w i t h t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n . I < ‘ N I 1 6 9 r 2 = 1 - [ ( n - 1 ) / ( n - p ) ] S S E / S S T O [ 2 1 8 ] w h e r e r 2 = a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n , n = n u m b e r o f o b s e r v a t i o n s , p n u m b e r o f p a r a m e t e r s , S S E : s u m o f s q u a r e s o f r e s i d u a l s , a n d S S T O = t o t a l s u m o f s q u a r e s . n s s m = { ( y 1 - § ) 2 = { T i - n ? [ 2 1 9 ] 1 : 1 w h e r e y 1 = i t h o b s e r v a t i o n , = m e a n o f n o b s e r v a t i o n s , o b s e r v a t i o n v e c t o r ( n x 1 ) . a n d F o r a g i v e n s e t o f d a t a ( i . e . k n o w n 5 a n d n ) a n d a p a r t i c u l a r m o d e l ( i . e . k n o w n p ) , t h e s m a l l e r t h e s u m o f s q u a r e s o f r e s i d u a l s , t h e h i g h e r t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n . S o , r 2 i s a d e s c r i p t i v e m e a s u r e w h i c h i n t e r p r e t s h o w m u c h t h e t o t a l v a r i a t i o n o f a d e p e n d e n t v a r i a b l e i s r e d u c e d b y e m p l o y i n g a s e t o f p a r a m e t e r s i n t o t h a t m o d e l . 6 . 5 R e s u l t s o f P a r a m e t e r E s t i m a t i o n A s m e n t i o n e d i n S e c t i o n 6 . 1 , t h e s t a t i s t i c a l p a r a m e t e r s o f t h e m o d e l s t u d i e d , m i g h t v a r y f r o m t r i a l t o t r i a l , a c c o r d i n g t o t h e h i s t o r y o f w a t e r s t r e s s t h a t t h e a l f a l f a r e c e i v e d b e f o r e c u t t i n g , a n d o t h e r f a c t o r s o f t h e e n v i r o n m e n t a n d t h e p l a n t . I n t h i s s t u d y , a l f a l f a c u t o n t h e s a m e d a y a t t h e s a m e s i t e w a s a s s u m e d t o b e a r a s i m i l a r h i s t o r y o f w a t e r s t r e s s , a s i m i l a r m a t u r i t y l e v e l i n a s i m i l a r e n v i r o n m e n t . B a s e d u p o n t h e s e , m e a s u r e d d r y i n g d a t a w a s d i v i d e d i n t o n i n e g r o u p s ( T a b l e 6 ) . E a c h g r o u p c o n s i s t e d o f s e v e r a l t r i a l s w i t h a s i m i l a r i n i t i a l m o i s t u r e 1 7 0 c o n t e n t b u t d i f f e r e n t s w a t h a r e a d e n s i t i e s c u t o n t h e s a m e d a t e . T r i a l s o f G r o u p 1 w e r e c o n d u c t e d d u r i n g t h e s e c o n d c u t t i n g i n E a s t L a n s i n g , M i c h i g a n o n J u l y 2 3 , 1 9 8 4 . G r o u p s 2 t o 4 w e r e m o w n d u r i n g t h e t h i r d c u t t i n g i n t h e s a m e a r e a i n A u g u s t o f 1 9 8 4 . T h e r e m a i n i n g g r o u p s w e r e t e s t e d d u r i n g t h e f o u r t h c u t t i n g i n F r e s n o , C a l i f o r n i a i n 1 9 8 5 . D u e t o l a c k o f i n f o r m a t i o n , t h e r a t i o o f l e a f a r e a i n d e x t o y i e l d f o r t h e f O u r t h c u t t i n g o f a l f a l f a i n F r e s n o , C a l i f o r n i a w a s a s S u m e d t o b e t h e s a m e a s t h a t o f t h e t h i r d c u t t i n g i n M i c h i g a n ( T a b l e 2 ) . I t w a s a l s o t h o u g h t t h a t f i r s t - d a y ' s d r y i n g d i f f e r e d f r o m s u c c e s s i v e - d a y s ' d r y i n g d u e t o r e w e t t i n g a n d r e d r y i n g . T h u s , t w o p a i r s o f t h e p a r a m e t e r s w e r e t e n t a t i v e l y d e s i g n a t e d . O n e p a i r ( a a n d b ) w a s f o r f i r s t - d a y ' s d r y i n g a n d t h e o t h e r p a i r ( a 2 a n d b 2 ) w a s f o r s u c c e s s i v e - d a y s ' d r y i n g . P a r a m e t e r e s t i m a t i o n w a s c o n d u c t e d f o r n i n e t r i a l s ( o n e f r o m e a c h g r o u p ) . P r e l i m i n a r y r e s u l t s s h o w e d t h a t t h e i n t r o d u c t i o n o f t h e s e c o n d p a i r o f p a r a m e t e r s w a s u n n e c e s s a r y . I t d i d n o t r e d u c e t h e s u m o f s q u a r e s o f r e s i d u a l s m u c h , a n d i t d i d c a u s e d i f f i c u l t y w i t h e s t i m a t i o n . T h e s e n s i t i v i t y c o e f f i c i e n t s o f a 2 a n d h e w e r e v e r y s m a l l i n m a g n i t u d e ( i n t h e o r d e r o f t h o u s a n d t h s o f t h o s e o f a ) . T h e r e w e r e t w o c o l u m n s o f n e a r n u l l i n t h e s e n s i t i v i t y m a t r i c e s . T h i s l e d t o i l l - c o n d i t i o n e d P m a t r i c e s a n d c o r r e l a t i o n m a t r i c e s . H e n c e , t h e s e c o n d p a i r ( a 2 a n d b 2 ) w a s r e m o v e d f r o m t h e e s t i m a t i o n . T h e e s t i m a t e d p a r a m e t e r s w i t h 9 6 % j o i n t c o n f i d e n c e i n t e r v a l s a r e s h o w n i n T a b l e 7 , a n d e l e m e n t s o f t h e P m a t r i c e s a n d e s t i m a t e d c o r r e l a t i o n c o e f f i c i e n t s a r e l i s t e d i n T a b l e 8 . I t i s o b s e r v e d f r o m T a b l e 7 t h a t t h e e s t i m a t e d p a r a m e t e r s v a r i e d a m o n g t r i a l s . T h i s m i g h t b e a t t r i b u t e d t o 1 ) i n h e r e n t c h a r a c t e r i s t i c s o f t h e a l f a l f a p l a n t , 2 ) t h e c o r r e l a t i o n b e t w e e n t h e p a r a m e t e r s a a n d b , T a b l e 6 G r o u p i n g d r y i n g t r i a l s b a s e d u p o n t h e d a t e o f c u t t i n g 1 7 1 G r o u p T r i a l 1 7 / 2 3 - 7 7 2 3 / 8 4 8 / 1 4 - 8 / 1 6 / 8 4 8 / 2 1 - 8 / 2 2 / 8 4 8 / 3 0 - 8 / 3 1 / 8 4 7 / 3 0 - 7 / 3 1 / 8 5 7 / 3 0 - 8 / 0 1 / 8 5 7 / 3 0 - 8 / 0 2 / 8 5 7 / 3 0 - 7 / 3 1 / 8 5 7 / 3 0 - 8 / 0 1 / 8 5 7 / 3 0 - 8 / 0 2 / 8 5 7 / 3 0 - 8 / 0 1 / 8 5 7 / 3 1 - 8 / 0 1 / 8 5 7 / 3 1 - 8 / 0 2 / 8 5 7 / 3 1 - 8 / 0 2 / 8 5 7 / 3 1 - 8 / 0 1 / 8 5 7 / 3 1 - 8 / 0 2 / 8 5 7 / 3 1 - 8 / 0 2 / 8 5 8 / 0 2 - 8 / 0 3 / 8 5 8 / 0 3 - 8 / 0 4 / 8 5 8 / 0 5 - 8 / 0 6 / 8 5 3 . 7 7 1 9 3 . 5 4 7 1 3 . 6 8 4 2 4 . 3 1 2 1 3 . 4 6 3 6 3 . 4 0 2 1 3 . 6 7 0 0 3 . 5 8 8 9 3 . 3 5 0 6 4 . 0 1 7 7 2 . 6 4 5 3 2 . 6 7 3 9 3 . 6 7 4 5 4 . 4 9 5 1 3 . 6 8 5 0 4 . 0 0 0 0 3 . 9 7 5 0 3 . 9 3 0 0 4 . 2 2 0 0 4 . 1 1 0 0 3 . 0 4 5 0 3 . 3 4 0 0 3 . 3 8 5 0 2 . 6 6 0 0 2 . 8 2 5 0 2 . 9 5 5 0 3 . 1 5 5 0 3 . 2 1 5 0 3 . 4 6 0 0 2 . 9 3 5 0 3 . 1 4 5 0 3 . 1 7 0 0 3 . 1 1 0 0 3 . 2 5 0 0 3 . 4 6 5 0 3 . 0 1 5 0 3 . 1 7 0 0 3 . 3 7 0 0 5 % 8 % 5 % 4 5 % 3 5 % 4 0 1 1 0 0 % 6 . 6 2 0 6 . 6 2 0 6 . 6 2 0 6 . 6 2 0 6 . 6 2 0 6 . 6 2 0 I n i t i a l m o i s t u r e c o n t e n t ( M o ) - k g w a t e r / k g d r y m a t t e r S w a t h a r e a d e n s i t y ( D ) - k g d r y m a t t e r / m 2 L e a f a r e a i n d e x / y i e l d ( L / Y ) - m 2 l e a f / k g d r y m a t t e r 1 7 2 T a b l e 7 E s t i m a t e d p a r a m e t e r s ( a a n d b ) w i t h 9 6 % c o n f i d e n c e i n t e r v a l s , t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) a n d t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) f o r n i n e s e l e c t e d t r i a l s G r o u p T r i a l a b S S E r 2 1 4 4 . 5 3 9 0 . 3 1 5 1 . 4 6 4 0 . 3 9 9 0 . 0 1 0 0 . 9 9 8 2 5 5 . 0 9 4 0 . 1 1 8 2 . 2 3 7 0 . 3 0 6 0 . 7 0 3 0 . 9 4 8 3 9 4 . 8 3 4 0 . 3 9 0 1 . 8 3 1 0 . 9 6 8 0 . 2 0 8 0 . 9 8 1 4 1 3 4 . 8 7 9 0 . 1 5 2 1 . 7 6 0 0 . 3 4 1 0 . 1 2 2 0 . 9 7 9 5 2 5 4 . 4 2 7 0 . 1 8 0 1 . 1 0 2 0 . 3 3 4 0 . 2 7 8 0 . 9 8 2 6 3 2 3 . 9 6 3 0 . 0 7 5 0 . 3 2 0 0 . 1 5 2 0 . 2 0 9 0 . 9 7 3 7 3 5 4 2 3 8 0 . 2 1 8 0 . 8 4 3 0 . 4 6 2 0 . 4 4 0 0 . 9 3 8 8 3 8 4 . 4 9 5 0 . 2 3 5 2 . 5 6 0 0 . 2 7 5 0 . 2 3 2 0 . 9 7 0 9 4 4 4 . 2 3 2 0 . 4 1 1 2 . 4 5 6 0 . 5 4 8 0 . 0 4 2 0 . 9 9 2 T a b l e 8 E l e m e n t s o f t h e P m a t r i x ( P a a , P a b a n d P b ) , t h e c o r r e l a t i o n c o e f f i c i e n t ( r a b ) , t h e s t a n d a r d d e v i a t i o n 3 2 ) a n d t h e n u m b e r o f o b s e r v a t i o n s ( n ) f o r t h e n i n e s e l e c t e d t r i a l s G r o u p T r i a l P a a P a b P b b F a b s n 1 4 1 . 6 0 7 8 1 . 9 3 4 2 2 . 5 7 7 3 0 . 9 5 8 0 . 0 0 3 5 2 5 0 . 0 7 7 1 0 . 1 6 5 9 0 . 5 1 5 7 0 . 8 3 2 0 . 0 2 9 2 6 3 9 1 . 5 0 1 9 3 . 6 8 9 9 9 . 2 6 5 3 0 . 9 8 9 0 . 0 1 5 1 6 4 1 3 0 . 3 8 5 1 0 . 8 1 3 6 1 . 9 3 7 3 0 . 9 4 2 0 . 0 0 9 1 6 5 2 5 0 . 2 1 6 4 0 . 3 7 9 7 0 . 7 4 1 2 0 . 9 4 8 0 . 0 2 1 1 5 6 3 2 0 . 0 4 4 9 0 . 0 2 1 3 0 . 1 8 5 0 0 . 2 3 4 0 . 0 1 7 1 4 7 3 5 0 . 1 2 1 6 0 . 2 0 6 2 0 . 5 4 8 5 0 . 7 9 8 0 . 0 4 9 1 1 8 3 8 0 . 2 6 9 6 - 0 . 1 0 9 3 0 . 3 6 7 8 - 0 . 3 4 7 0 . 0 2 6 1 1 9 4 4 2 . 2 9 0 3 - 0 . 8 9 8 9 4 . 0 6 7 7 - 0 . 2 9 5 0 . 0 0 9 1 0 a n d 3 ) o t h e r f a c t o r s . I n T a b l e 8 , f i v e o u t o f n i n e v a l u e s o f ” a b w e r e b e l o w 0 . 9 w h i c h i s a c r i t i c a l v a l u e c o m m o n l y u s e d f o r d e c l a r a t i o n o f h i g h c o r r e l a t i o n s ( B e c k a n d A r n o l d , 1 9 7 7 ) . A l t h o u g h ’ a b w a s m o s t l y h i g h e r t h a n 0 . 9 f o r t h e d a t a o f E a s t L a n s i n g , M i c h i g a n , i t s p r e a d w i d e l y f o r t h o s e o f F r e s n o , C a l i f o r n i a . T h e s e s h o w e d t h a t a a n d b w e r e i n t e r c o r r e l a t e d t o a n e x t e n t . H o w e v e r , t h e i n c o n s i s t e n c y o f t h e h i g h v a l u e o f " a b s u g g e s t e d o t h e r i n f l u e n c e s o n t h e v a r i a t i o n o f t h e 1 7 3 p a r a m e t e r s . D u r i n g t h e s e q u e n t i a l e s t i m a t i o n , i n t e r m e d i a t e r e s u l t s s h o w e d t h a t t h e p a r a m e t e r s a a n d b v a r i e d f r o m t i m e t o t i m e a s e a c h c a s e o f o b s e r v e d d a t a w a s i n c l u d e d . T h i s m e a n t t h a t t h e s e t w o p a r a m e t e r s w e r e p r o b a b l y a f u n c t i o n o f t i m e o r t h e m o i s t u r e c o n t e n t o f t h e a l f a l f a p l a n t . A l s o a s e x p e c t e d , t h e p a r a m e t e r s v a r i e d a m o n g t r i a l s . A t p r e s e n t , t h e v a r i a t i o n p a t t e r n o f t h e p a r a m e t e r s w a s n o t c l e a r l y e x p o s e d w i t h t h e l i m i t e d e x p e r i m e n t s o f t h i s s t u d y . T h o r o u g h p h y s i o l o g i c a l s t u d y o f t h e a l f a l f a p l a n t b e f o r e a n d a f t e r c u t t i n g i s r e q u i r e d , w h i c h i s b e y o n d t h e s c o p e o f t h i s s t u d y . T h e a c c u r a c y a n d d e t a i l o f t h e m e a s u r e d d a t a a n d o t h e r i n f o r m a t i o n u s e d i n t h i s s t u d y m i g h t a l s o a f f e c t t h e e s t i m a t i o n r e s u l t s . F o r e x a m p l e , s o m e m e a s u r e m e n t s w e r e t a k e n d u r i n g n i g h t p e r i o d s i n E a s t L a n s i n g , M i c h i g a n . T h e s e o b s e r v a t i o n s g a v e i n f o r m a t i o n o n t h e p r o g r e s s o f s w a t h r e w e t t i n g . B y u s i n g t h e t e c h n i q u e o f s e q u e n t i a l e s t i m a t i o n , t h e s e d a t a w e r e h e l p f u l t o a d j u s t t h e e s t i m a t e s o f a a n d b a c c o r d i n g l y . I n f o r m a t i o n o n t h e l o c a l w e a t h e r a n d p l a n t c h a r a c t e r i s t i c s f o r F r e s n o , C a l i f o r n i a w a s n o t s u f f i c i e n t . A s m e n t i o n e d e a r l i e r , s o m e c h a r a c t e r i s t i c s w e r e a s s u m e d o r a s s i g n e d t o b e t h e s a m e a s t h o s e o f E a s t L a n s i n g , M i c h i g a n . T h e e s t i m a t e d 9 6 % j o i n t c o n f i d e n c e i n t e r v a l s w e r e q u i t e s m a l l . M o s t o f t h e m r a n g e d f r o m 4 % t o 6 0 % o f t h e m a g n i t u d e o f t h e r e s p e c t i v e p a r a m e t e r . T h e 9 6 % c o n f i d e n c e i n t e r v a l f o r p a r a m e t e r a w a s s m a l l e r t h a n t h a t o f p a r a m e t e r b , a s s e e n i n T a b l e 7 . T h i s w a s a t t r i b u t e d m o s t l y t o t h e d i f f e r e n c e i n t h e m a i n d i a g o n a l e l e m e n t s o f t h e P m a t r i x , w h i c h i n d i c a t e d t h a t t h e e s t i m a t e o f a w a s m o r e c e r t a i n t h a n t h a t o f b , s i n c e t h e c o r r e l a t i o n b e t w e e n t h e s e t w o p a r a m e t e r s a f f e c t e d t h e e s t i m a t i o n . N e t e r a n d W a s s e r m a n ( 1 9 7 4 ) p o i n t e d o u t t h a t h i g h c o r r e l a t i o n s o f t e n c a u s e d i f f i c u l t i e s i n p a r a m e t e r e s t i m a t i o n a n d m a y l e a d t o e r r a t i c 1 7 4 i n f e r e n c e s o n e s t i m a t e d p a r a m e t e r s , h o w e v e r , t h e h i g h c o r r e l a t i o n s a r e u s u a l l y n o t a p r o b l e m w h e n i n f e r e n c e s o n t h e r e s p o n s e f u n c t i o n s o r p r e d i c t i o n s o f f u t u r e o b s e r v a t i o n s a r e m a d e w i t h i n t h e r a n g e o f t h e o b s e r v a t i o n s u s e d t o d e t e r m i n e t h e p a r a m e t e r s . F o r t h i s s t u d y , t h e i n i t i a l m o i s t u r e c o n t e n t o f a l f a l f a a n d t h e a r e a d e n s i t y o f t h e s w a t h v a r i e d o v e r w i d e r a n g e s n o r m a l l y e n c o u n t e r e d i n h a y p r o d u c t i o n . I t w a s e x p e c t e d t h a t p r e d i c t i o n s m a d e b y u s i n g t h e m o d e l w i t h t h e s e e s t i m a t e d p a r a m e t e r s w o u l d n o t l e a d t o e r r o n e o u s o u t c o m e s . T h i s w i l l b e s h o w n d u r i n g m o d e l v a l i d a t i o n i n t h e n e x t c h a p t e r . . S i n c e t h e r e s u l t s o f p a r a m e t e r e s t i m a t i o n s h o w e d d e v i a t i o n s b e t w e e n v a r i o u s t r i a l s , i t w a s n e c e s s a r y t o t e s t w h e t h e r t h o s e d i f f e r e n c e s w e r e s i g n i f i c a n t . A n F - t e s t s h o w e d t h a t t h e 4 t r i a l s ( T r i a l s 4 , 5 , 9 a n d 1 3 ) s e l e c t e d f r o m t h e E a s t L a n s i n g g r o u p m o s t l i k e l y h a d a c o m m o n v a r i a n c e w i t h a 9 5 % c o n f i d e n c e . S i m i l a r h y p o t h e s e s w e r e c o n c l u d e d f o r t h e 5 t r i a l s ( T r i a l s 2 5 , 3 2 , 3 5 , 3 8 a n d 4 4 ) s e l e c t e d f r o m t h e F r e s n o g r o u p , a n d f o r a l l t h e 9 t r i a l s . A n o t h e r F - t e s t i n d i c a t e d t h a t t h e 4 E a s t L a n s i n g t r i a l s c o u l d b e p o o l e d t o g e t h e r t o h a v e o n e c o m m o n p a i r o f t h e p a r a m e t e r s a a n d b w i t h a 9 5 % c o n f i d e n c e . H o w e v e r , a h y p o t h e s i s t h a t t h e 5 F r e s n o t r i a l s h a d s e p a r a t e p a i r s o f t h e p a r a m e t e r s a a n d b c o u l d n o t b e r e j e c t e d a t t h e s a m e c o n f i d e n c e l e v e l . C o n s e q u e n t l y , t h e 9 t r i a l s f r o m b o t h l o c a t i o n s w o u l d n o t h a v e a c o m m o n p a i r o f t h e p a r a m e t e r s a a n d b . F i g u r e s 2 2 t o 3 0 s h o w s i m u l a t e d d r y i n g c u r v e s f o r t h e n i n e t r i a l s . I n A p p e n d i x I , a s a m p l e o u t p u t o f A R N L I N f o r T r i a l 5 i s g i v e n . ' m / d e v h t i w 4 l a i r T r o . f 3 d i ( P r a u e o s 9 M . D 4 g 8 k 9 1 4 6 n 5 a g r e s b O - — - \ d e t \ c i d e r . i h c i M 0 : y t i . s g n n e D s n a a e L r A t s h a E t a w S 4 l a i r T 0 e r : v s 5 ) r 1 e u c t . a g 4 D 6 n i 4 y . r 1 d : n b a d e d v n r H b 3 o 5 . d 4 n = e a a m i d r e e 0 T t t r : c e 2 i m 1 d a e r r a P p . 2 2 e r u g i F ( ° q ° p ) i u e i u o o a m i s g o w 8 5 0 J 8 A v m o m s 4 . 0 4 3 . 0 - 4 1 . 0 ~ I 1 8 : 0 7 / 2 3 1 7 5 a m / . M . D 4 g 8 k 9 h t i w 5 l a i r T r o f s e v I r : 0 { u 1 c 1 8 n 4 a 4 g . i h c i M 0 : y t i . s g n n e i D s 8 e . t g 7 a n 3 i 2 D y . r 2 d = d b 0 T : 8 n d 1 e d a v n r a r e s 4 u b 9 \ — 0 o o 0 l . : H n d 5 h e E r 0 i m T l P : d a 6 e r d e t 2 1 ( n = a a c e i d d r m e e i t t c e r a P p . 3 2 e 5 0 1 : 0 / r 8 u g i F a a e L r A t s a t a w S 5 l a i l r T — - 0 0 . . 3 2 f 0 : 8 1 4 0 1 : 2 / 0 1 8 . 0 ( ' q ' p ) i u a q u o o a a n s g o w 8 5 0 J 8 A V L n o M S 0 1 6 1 7 6 0 : 2 1 h t i w 9 l a i r T r o 0 f : 5 s e v ) r e u t c a . D g 1 n 3 i 8 d . y n r 1 d : 2 a b 2 d r e d 8 u v n r a o e H s 4 0 : 9 d e v r ( e s b 3 o 8 e . b d 4 O n = m i a a - T 0 : 8 1 1 r 2 e 3 d r e e t t c e i m d a e r r a P p . 4 2 e r u g i F 4 0 . 2 3 . ( ' q ' p ) i u a i u o g e m i s g o w e fi o J e A v m o m s 0 . 0 T r i a l 9 S w a t h A r e a D e n s i t y : 0 . 4 5 9 9 k g D . M . / m ‘ E a s t L a n s i n g . M i c h i g a n 1 9 8 4 \ \ P r e d i c t e d - - \ 1 7 7 ' m / . M . D g k 4 8 4 9 4 1 4 5 n . a g : i y h t c i i s n e M D . a e g n i s r n A a 0 : 8 1 h t i w 3 1 0 l : a 2 i 1 r T r o f s e v r 0 I : 6 ) u e c l t . a 0 g D n 6 i 7 y . d r 1 n d : b a d e d r v n 1 r a u 0 e 3 r o : / s 9 0 b 7 8 H ( o 8 L . h t t a s w a S E 3 1 l a i r T e d 4 n = m a a I i T d r e e t t c e i m d a e r r a P p . 5 2 e r u g i F 0 l : 8 1 [ 0 0 r 3 : 2 / 1 8 0 . 0 1 0 . N : 3 . ( ° q ' p ) i u e q u o g s i n i s g o w a b o J e A v m o m s 1 7 8 0 : 2 h t i 1 w 0 : 6 5 2 l a i r T r o f 1 s 0 e I o : v { 0 r s u e c 4 t f r \ u e — 0 o s 7 ' m / . M . D g k 2 2 3 9 n e D a e r A 8 9 . 1 : a y i t n i r s o i l a C a g 2 . D n 0 i 1 0 y . I : r d 1 8 d : n 1 b a d e d v n r a I : H b 2 2 1 o 4 ( . d 4 n = e a a m l d r e e , e o t n c s i d h e d t r e a F w S 5 2 l a i r T - 0 . 4 r 0 T t t I P : c e 6 i m d a e r r a P p 1 0 . 6 I 3 2 e r u g i F : / 0 7 0 : 8 1 ( ' q ‘ p ) q u e i u o o e m i s g o w e fi o J e A v L n o M S 3 . 0 - 2 . 0 - 0 . 0 1 2 m 7 / 3 0 1 7 9 \ . — _ d e . t - c _ i d _ e r d e v r _ P I 0 0 s 0 : 8 1 fl 0 I : 2 1 h t i w 2 3 l a 0 I i : 6 2 r T r o f : e 0 v s e r t a D u c . g 0 0 n 2 I i : 3 d 8 1 n y . r 0 1 a d = 0 b / r d 8 u 0 o e d v n r a I : 2 1 H e ( s 3 b 9 o 3 e . m d 3 i n = 0 T a a I : 6 0 I d r e e t t c e i m d a e r r a : P p 0 . 7 2 e r 0 I u : 8 1 g i F 0 1 : 3 2 / 1 7 ] 0 . 0 * n r / . M . D g k 1 6 5 4 s 8 b _ O e _ . 9 1 1 : a y i t n i r s o n f e i D a e r l a C . o A n s h e t r a F w S 2 3 l a i r T \ - 0 . 3 ( - q ' p ) i u e i u o g a m i s g o w s l i m e / ( v u i o M s 2 . 0 - 4 1 8 0 5 8 9 1 a i n r o f i l a C . o n s e r F r T r h t i w 5 3 l a i r T r o f s e v r u ) c r e . t g 3 a n 4 i 8 D y . d r 0 d = b n d a e d v n r r a e u s 8 o b 3 H o 2 ( . d 4 n = 3 a a e 0 0 r d r : m 0 8 e e t t c e i m d a e r r r a P p f 0 I : 8 1 . 8 2 e r u g i F 4 0 . 1 2 0 : 0 2 / 1 8 ( ' q ' p ) i u a i u o o a m i s g o w a b o J e A v m o m s O J ) T r i a l 3 5 S w a t h A r e a D e n s i t y : 1 . 4 9 6 k g D . M . / m ' fi 6 : 0 1 2 : 0 1 8 : 0 1 8 1 ’ m / . M . D g k 8 7 5 . 5 1 8 9 : 1 y t a i i s n n r e o D f i l a a e r A . o h n t s a e w r S F 4 4 C e d e v r s b O - - \ h 0 t k i T 2 w 1 4 4 l a I i D r A S r T r o f s e v r u c T 6 g . n 5 i 4 y . r 2 d = 6 b l d fi C 0 e d k / ( v n 8 r a _ e _ s 2 , b 3 o 2 . d 4 n = a a d r e e t t c e i m d a l e r a i - r T d e t c i d e r P r a P p . 0 3 e r u 5 g 0 0 i r k F 2 / 1 8 4 - 0 . 2 ‘ 0 . 0 j 1 8 2 1 Q , . \ \ ( ' q ° p ) i u e i u o o e m i s g o w 9 6 0 1 9 1 0 9 1 1 1 0 1 1 1 1 3 r 1 8 : 0 C H A P T E R 7 M O D E L V A L I D A T I O N A N D S E N S I T I V I T Y A N A L Y S I S T h r o u g h p a r a m e t e r e s t i m a t i o n , t h e m o d e l w a s f i t t e d t o a p o r t i o n o f o b s e r v e d d r y i n g d a t a . T h i s w a s i l l u s t r a t e d b y l o w v a l u e s f o r t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) a n d h i g h v a l u e s f o r t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) l i s t e d i n T a b l e 7 , a n d s h o w n i n F i g u r e s 2 2 t o 3 0 . F u r t h e r t e s t s o n t h e m o d e l w e r e n e e d e d t o s e e h o w w e l l i t p r e d i c t e d o t h e r o b s e r v a t i o n s , a n d t o f i n d h o w s e n s i t i v e i t w a s t o d i s t u r b a n c e s i n i m p o r t a n t p a r a m e t e r s a n d v a r i a b l e s . I n t h i s c h a p t e r , v a l i d a t i o n a n d s e n s i t i v i t y a n a l y s i s o f t h e m o d e l a r e p r e s e n t e d . 7 . 1 M o d e l V a l i d a t i o n V a l i d a t i o n o f t h e s t u d i e d m o d e l w a s p e r f o r m e d t h r o u g h s i m u l a t i n g f i e l d d r y i n g o f t h e r e m a i n i n g 3 1 t r i a l s ( 3 8 1 o b s e r v a t i o n s ) n o t u s e d i n p a r a m e t e r e s t i m a t i o n . I n f e r e n c e s w e r e m a d e b a s e d u p o n t h e s i m u l a t i o n r e s u l t s f r o m a l l t r i a l s a n d o b s e r v a t i o n s . T h e t r i a l s w e r e d i s t i n g u i s h e d i n s e v e r a l d i f f e r e n t w a y s : b y c u t t i n g d a t e , b y l o c a t i o n a n d b y t h e e n t i r e t y . T h e e n t i r e e x p e r i m e n t o f t h i s s t u d y c o n s i s t e d o f 2 l o c a t i o n g r o u p s , 9 c u t t i n g d a t e g r o u p s , 4 0 t r i a l s a n d 5 0 5 o b s e r v a t i o n s . T h e E a s t L a n s i n g g r o u p i n c l u d e d 4 d a t e g r o u p s ( G r o u p s 1 t o 4 ) , 1 5 t r i a l s ( T r i a l s 1 t o 1 5 ) a n d 1 9 0 o b s e r v a t i o n s ; a n d t h e F r e s n o g r o u p - 5 d a t e g r o u p s ( G r o u p 5 t o 9 ) , 2 5 t r i a l s ( T r i a l s 2 1 t o 4 5 ) a n d 3 1 5 o b s e r v a t i o n s . 1 8 3 1 8 4 T h e m o d e l ' s p e r f o r m a n c e w a s t e s t e d t h r o u g h 4 t r e a t m e n t s . T h r e e t r e a t m e n t s d e a l t w i t h t h e s e l e c t i o n o f t h e p a r a m e t e r s a a n d b b a s e d u p o n t h e c u t t i n g d a t e , t h e l o c a t i o n o r t h e w h o l e e x p e r i m e n t . D u r i n g t h e s e t h r e e t r e a t m e n t s w h e n v a r i o u s v a l u e s o f t h e p a r a m e t e r s a a n d b w e r e u s e d , t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) a n d t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) w e r e e x a m i n e d f o r e a c h i n d i v i d u a l t r i a l . T h e o v e r a l l p e r f o r m a n c e f o r t h e p o p u l a t i o n o f e a c h d a t e g r o u p , e a c h l o c a t i o n g r o u p a n d t h e w h o l e e x p e r i m e n t w a s a l s o s u m m a r i z e d . T h e f o u r t h t r e a t m e n t w a s t h e u s e o f w e a t h e r d a t a f r o m t h e l o c a l w e a t h e r s e r v i c e i n s t e a d o f t h o s e f r o m f i e l d m e a s u r e m e n t s . T r i a l 5 w a s s e l e c t e d t o t e s t t h e m o d e l ' s p r e d i c t a b i l i t y w i t h t h e u s e o f d a t a f r o m t h e l o c a l w e a t h e r s e r v i c e . T a b l e 9 c i t e s m i n i m u m v a l u e s o f r 2 f o r v a r i o u s s a m p l e s i z e s f r o m W a n g ( 1 9 7 7 ) . F o r a p r e f e r r e d c o n f i d e n c e l e v e l ( a ) a n d a k n o w n s a m p l e s i z e ( n ) , a r e g r e s s i o n m o d e l w i t h t w o p a r a m e t e r s c a n b e d e c l a r e d t o b e m e a n i n g f u l , i f i t r e s u l t s i n a n r 2 a b o v e a c o r r e s p o n d i n g v a l u e g i v e n i n t h e t a b l e . A s d i s c u s s e d e a r l i e r i n t h e p r e c e d i n g c h a p t e r , i t i s p r e f e r r e d t o h a v e r 2 a s c l o s e t o u n i t y a s p o s s i b l e . N o t e t h a t t h e c o e f f i c i e n t o f d e t e r m i n a t i o n u s e d i n c o n s t r u c t i o n o f T a b l e 9 i s t h e o r d i n a r y d e f i n i t i o n o f r 2 : r 2 = 1 - S S E / S S T O [ 2 2 0 ] w h e r e r 2 : c o e f f i c i e n t o f d e t e r m i n a t i o n , S S E = s u m o f s q u a r e s o f r e s i d u a l s , a n d S S T O t o t a l s u m o f s q u a r e s . T h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n d e f i n e d i n E q u a t i o n [ 2 1 8 ] i s a l w a y s s m a l l e r t h a n t h e o r d i n a r y c o e f f i c i e n t o f d e t e r m i n a t i o n u n l e s s S S E i s z e r o , i . e . r 2 i s u n i t y - a p e r f e c t m o d e l . I n f e r e n c e s w o u l d b e 1 8 5 T a b l e 9 M i n i m u m v a l u e s o f t h e c o e f f i c i e n t o f d e t e r m i n a t i o n ( 9 2 ) f o r d e c l a r a t i o n o f r e g r e s s i o n m o d e l s w i t h t w o p a r a m e t e r s n - 2 0 : 5 1 0 : 1 1 I n - 2 0 : 5 1 0 : 1 1 I n - 2 : 5 1 : 1 1 1 0 . 9 9 7 1 . 0 0 0 | 1 6 0 . 4 6 8 0 . 5 9 0 I 3 5 0 . 3 2 5 0 . 4 1 8 2 0 . 9 5 0 0 . 9 9 0 I 1 7 0 . 4 5 6 0 . 5 7 5 I 4 0 0 . 3 0 4 0 . 3 9 3 3 0 . 8 7 8 0 . 9 5 9 I 1 8 0 . 4 4 4 0 . 5 6 1 I 4 5 0 . 2 8 8 0 . 3 7 2 4 0 . 8 1 1 0 . 9 1 7 I 1 9 0 . 4 3 3 0 . 5 4 9 | 5 0 0 . 2 7 3 0 . 3 5 4 5 0 . 7 5 4 0 . 8 7 4 I 2 0 0 . 4 2 3 0 . 5 3 7 | 6 0 0 . 2 5 0 0 . 3 2 5 6 0 . 7 0 7 0 . 8 3 4 I 2 1 0 . 4 1 3 0 . 5 2 6 I 7 0 0 . 2 3 2 0 . 3 0 2 7 0 . 6 6 6 0 . 7 9 8 I 2 2 0 . 4 0 4 0 . 5 1 5 | 8 0 0 . 2 1 7 0 . 2 8 3 8 0 . 6 3 2 0 . 7 6 5 I 2 3 0 . 3 9 6 0 . 5 0 5 I 9 0 0 . 2 0 5 0 . 2 6 7 9 0 . 6 0 2 0 . 7 3 5 | 2 4 0 . 3 8 8 0 . 4 9 6 I 1 0 0 0 . 1 9 5 0 . 2 5 4 1 0 0 . 5 7 6 0 . 7 0 8 I 2 5 0 . 3 8 1 0 . 4 8 7 I 1 2 5 0 . 1 7 4 0 . 2 2 8 1 1 0 . 5 5 3 0 . 6 8 4 | 2 6 0 . 3 7 4 0 . 4 7 8 I 1 5 0 0 . 1 5 9 0 . 2 0 8 1 2 0 . 5 3 2 0 . 6 6 1 I 2 7 0 . 3 6 7 0 . 4 7 0 | 2 0 0 0 . 1 3 8 0 . 1 8 1 1 3 0 . 5 1 4 0 . 6 4 1 I 2 8 0 . 3 6 1 0 . 4 6 3 I 3 0 0 0 . 1 1 3 0 . 1 4 8 1 4 0 . 4 9 7 0 . 6 2 3 I 2 9 0 . 3 5 5 0 . 4 5 6 | 4 0 0 0 . 0 9 8 0 . 1 2 8 1 5 0 . 4 8 2 0 . 6 0 6 I 3 0 0 . 3 4 9 I 1 0 0 0 0 . 0 6 2 0 . 0 8 1 I F r o m W a n g ( 1 9 7 7 ) c o n s e r v a t i v e i f t h e y a r e m a d e b y c o m p a r i n g v a l u e s o f t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n w i t h t h o s e o f T a b l e 9 . 7 . 1 . 1 V a l i d a t i o n b y N i n e C u t t i n g D a t e G r o u p s T h e p e r f o r m a n c e o f t h e m o d e l i n p r e d i c t i n g d r y i n g o f a l f a l f a c u t o n t h e s a m e d a t e u n d e r v a r i o u s c o n d i t i o n s o f t h e s w a t h a n d t h e p l a n t w a s e v a l u a t e d f i r s t . E a c h t r i a l o f t h e n i n e d a t e g r o u p s w a s s i m u l a t e d b y u s e o f o n e r e s p e c t i v e s e t o f e s t i m a t e d p a r a m e t e r s ( T a b l e 7 ) a l o n g w i t h o t h e r s w a t h a n d p l a n t c o n d i t i o n s ( T a b l e 6 i n C h a p t e r 6 ) . A s a s a m p l e , c o n t e n t ( b o t h m e a s u r e d a n d d a t a o f t h e a v e r a g e s w a t h m o i s t u r e s i m u l a t e d ) a n d r e s i d u a l s f o r T r i a l 5 a r e g i v e n i n A p p e n d i x J . A s u m m a r y o f r e s i d u a l s o f t h e t o t a l s u m o f s q u a r e s ( S S T O ) , t h e s u m o f s q u a r e s ( S S E ) a n d t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) f o r t h e 1 8 6 p o p u l a t i o n o f e a c h t r i a l , d a t e g r o u p , l o c a t i o n a n d t h e w h o l e e x p e r i m e n t i s s h o w n i n t h e l e f t t h r e e c o l u m n s o f T a b l e 1 0 . U n d e r t h i s t r e a t m e n t , t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) r a n g e d f r o m 0 . 6 5 7 t o 0 . 9 9 8 a m o n g t h e 4 0 t r i a l s w i t h s a m p l e p o p u l a t i o n s f r o m 4 t o 2 6 , a n d f r o m 0 . 8 0 9 t o 0 . 9 7 3 a m o n g t h e n i n e d a t e g r o u p s w i t h v a r i o u s s a m p l e s i z e s f r o m 1 7 t o 1 0 6 . I t w a s s e e n t h a t m o s t o f t h e s e r 2 v a l u e s w e r e m u c h h i g h e r t h a n t h o s e r e q u i r e d f o r t h e 9 9 % c o n f i d e n c e ( T a b l e 9 ) . A s f o r i n d i v i d u a l t r i a l s , o n l y T r i a l 2 r e s u l t e d i n a n r 2 v a l u e ( 0 . 7 6 0 ) s l i g h t l y b e l o w 0 . 8 1 1 f o r 9 5 % c o n f i d e n c e ; h o w e v e r , t h e o r d i n a r y r 2 o f T r i a l 2 w a s 0 . 8 4 w h i c h w a s a b o v e 0 . 8 1 1 . T r i a l s 6 a n d 8 a l s o h a d r e l a t i v e l y l o w r 2 i n c o m p a r i s o n w i t h o t h e r t r i a l s i n D a t e g r o u p s 2 a n d 3 , r e s p e c t i v e l y . I t s h o u l d b e n o t e d t h a t t h e s e t h r e e t r i a l s w e r e c o n d u c t e d w i t h e x t r e m e l y l o w s w a t h d e n s i t i e s , 0 . 2 6 2 0 t o 0 . 2 7 7 6 k g d r y m a t t e r / m 2 . F o r t r i a l s i n F r e s n o , C a l i f o r n i a , T r i a l s 3 0 , 3 3 , 3 6 , 3 9 , 4 2 a n d 4 5 s h o w e d r e l a t i v e l y l o w r 2 v a l u e s a n d h a d t h e h i g h e s t s w a t h d e n s i t y , 2 . 2 4 5 t o 2 . 5 8 4 k g d r y m a t t e r / m 2 . F o r t h e r e m a i n i n g t r i a l s w i t h t h e s w a t h d e n s i t y b e t w e e n 0 . 3 2 9 2 a n d 1 . 9 3 9 2 k g d r y m a t t e r / m 2 , t h e p e r f o r m a n c e o f t h e m o d e l w a s g o o d a n d r e s u l t e d i n r 2 v a l u e s f r o m 0 . 8 5 0 t o 0 . 9 9 8 w i t h a m a j o r i t y a b o v e 0 . 9 0 . I n e v e r y d a t e g r o u p , e x c e p t G r o u p 7 , t h e r e w a s o n e o r t w o t r i a l ( s ) r e s u l t i n g i n t h e s a m e o r b e t t e r r 2 t h a n t h e r e s p e c t i v e o n e f r o m t h o s e n i n e s e l e c t e d t r i a l s . T h e r 2 v a l u e s f o r t h e E a s t L a n s i n g g r o u p ( 0 . 9 6 0 ) , t h e F r e s n o g r o u p ( 0 . 9 2 1 ) a n d t h e e n t i r e e x p e r i m e n t a s a w h o l e ( 0 . 9 3 7 ) , w e r e s u f f i c i e n t l y h i g h . T h e s e r e s u l t s p r e s e n t e d a b o v e s h o w t h a t a l l o b s e r v e d d r y i n g p r o c e s s e s w e r e c l o s e l y s i m u l a t e d b y u s i n g t h e m o d e l w i t h t h e n i n e s e t s o f e s t i m a t e d p a r a m e t e r s . T a b l e 1 0 S u m m a r y o f t h e t o t a l s u m o f s q u a r e s . 1 1 3 ' 7 s q u a r e s o f r e s i d u a l s ( S S E ) d e t e r m i n a t i o n ( r ‘ ) f r o m t h r e e t r e a t m e n t s o f v a l i d a t i o n ( S S T O ) , t h e s u m o f a n d t h e a d j u s t e d c o e f f i c i e n t o f D a t e L o c a t i o n E n t i r e t y G r o u p T r i a l n 8 5 7 0 $ 5 8 r ‘ 3 5 8 r 1 5 5 8 r ‘ ' ' 1 4 4 . 3 1 0 . 3 0 0 . 8 9 4 0 . 4 7 0 . 8 3 7 0 . 3 8 0 . 8 6 9 2 4 4 . 3 7 0 . 7 0 0 . 7 6 0 0 . 9 0 0 . 6 9 1 0 . 7 7 0 . 7 3 5 3 4 3 . 3 7 0 . 0 2 0 . 9 9 1 0 . 0 2 0 . 9 9 0 0 . 0 2 0 . 9 9 0 4 5 4 . 8 1 0 . 0 1 0 . 9 9 8 0 . 0 4 0 . 9 9 0 0 . 0 3 0 . 9 9 3 1 1 7 1 9 . 1 5 1 . 0 3 0 . 9 4 2 1 . 4 3 0 . 9 2 0 1 . 2 0 0 . 9 3 3 5 2 6 1 4 . 0 6 0 . 7 0 0 . 9 4 8 1 . 1 6 0 . 9 1 4 1 . 4 4 0 . 8 9 3 6 2 6 1 2 . 8 7 1 . 6 0 0 . 8 7 0 0 . 9 7 0 . 9 2 2 0 . 8 6 0 . 9 3 0 7 1 9 1 4 . 7 9 0 . 7 0 0 . 9 5 0 0 . 7 9 0 . 9 4 3 0 . 9 3 0 . 9 3 3 2 7 1 4 2 . 7 7 3 . 0 1 0 . 9 2 9 2 . 9 2 0 . 9 3 1 3 . 2 3 0 . 9 2 3 8 1 6 1 2 . 9 2 1 . 5 8 0 . 8 6 9 1 . 6 8 0 . 8 6 1 1 . 2 6 0 . 8 9 5 9 1 6 1 1 . 4 2 0 . 2 1 0 . 9 8 1 0 . 2 1 0 . 9 8 0 0 . 2 2 0 . 9 7 9 1 0 1 0 8 . 7 5 0 . 1 5 0 . 9 8 0 0 . 1 6 0 . 9 7 9 0 . 1 3 0 . 9 8 3 1 1 ° 1 0 1 1 . 0 3 0 . 1 7 0 . 9 8 2 0 . 1 8 0 . 9 8 2 0 . 2 1 0 . 9 7 8 3 5 2 4 7 . 4 6 2 . 1 1 0 . 9 5 5 2 . 2 3 0 . 9 5 2 1 . 8 2 0 . 9 6 1 1 2 1 6 6 . 3 6 0 . 4 6 0 . 9 2 3 0 . 2 1 0 . 9 6 4 0 . 1 6 0 . 9 7 2 1 3 1 6 6 . 2 2 0 . 1 2 0 . 9 7 9 0 . 2 2 0 . 9 6 1 0 . 3 1 0 . 9 4 7 1 4 9 7 . 4 9 0 . 5 4 0 . 9 1 8 0 . 2 0 0 . 9 7 0 0 . 1 4 0 . 9 7 9 1 5 9 7 . 3 1 0 . 2 7 0 . 9 5 8 0 . 8 4 0 . 8 6 9 1 . 0 7 0 . 8 3 2 4 5 0 4 7 . 8 0 1 . 3 8 0 . 9 7 0 1 . 4 7 0 . 9 6 9 1 . 6 8 0 . 9 6 4 e . L a n s i n g 1 9 0 1 9 0 . 7 0 7 . 5 3 0 . 9 6 0 8 . 0 6 0 . 9 5 8 7 . 9 3 0 . 9 5 8 2 1 9 7 . 7 3 0 . 4 1 0 . 9 3 9 0 . 6 1 0 . 9 1 0 0 . 2 8 0 . 9 5 9 2 2 1 5 1 5 . 5 4 0 . 3 8 0 . 9 7 4 0 . 8 2 0 . 9 4 3 0 . 4 1 0 . 9 7 2 2 3 2 1 2 0 . 4 5 0 . 8 1 0 . 9 5 9 1 . 0 7 0 . 9 4 5 1 . 1 2 0 . 9 4 2 2 4 9 9 . 0 1 0 . 2 8 0 . 9 6 5 0 . 4 8 0 . 9 3 9 0 . 3 5 0 . 9 5 5 2 5 1 5 1 6 . 5 6 0 . 2 8 0 . 9 8 2 0 . 5 2 0 . 9 6 6 0 . 4 2 0 . 9 7 3 2 6 2 1 2 3 . 3 5 0 . 7 6 0 . 9 6 6 1 . 1 6 0 . 9 4 8 0 . 8 9 0 . 9 6 0 - 2 7 1 6 2 0 . 6 8 0 . 2 4 0 . 9 8 8 0 . 3 4 0 . 9 8 3 0 . 2 5 0 . 9 8 7 5 1 0 6 1 1 8 . 3 0 3 . 1 6 0 . 9 7 3 5 . 0 0 0 . 9 5 7 3 . 7 2 0 . 9 6 8 2 8 1 0 6 . 3 9 0 . 8 5 0 . 8 5 0 0 . 2 0 0 . 9 6 6 0 . 0 4 0 . 9 9 2 2 9 1 5 1 0 . 1 9 0 . 2 2 0 . 9 7 7 0 . 1 9 0 . 9 8 0 0 . 1 0 0 . 9 8 9 3 0 1 6 1 0 . 4 5 3 . 3 4 0 . 6 5 7 2 . 3 9 0 . 7 5 5 3 . 1 3 0 . 6 7 9 3 1 9 5 . 1 5 0 . 5 2 0 . 8 8 5 0 . 8 6 0 . 8 1 0 0 . 4 8 0 . 8 9 4 3 2 1 4 8 . 4 9 0 . 2 1 0 . 9 7 3 0 . 3 5 0 . 9 5 6 0 . 2 7 0 . 9 6 6 3 3 2 0 1 2 . 9 0 3 . 2 5 0 . 7 3 4 3 . 0 1 0 . 7 5 3 3 . 6 1 0 . 7 0 4 6 8 4 5 5 . 2 6 8 . 4 0 0 . 8 4 6 7 . 0 0 0 . 8 7 5 7 . 6 3 0 . 8 6 0 3 4 9 5 . 7 7 0 . 5 3 0 . 8 9 6 0 . 8 8 0 . 8 2 5 0 . 3 7 0 . 9 2 6 3 5 1 1 7 . 8 8 0 . 4 4 0 . 9 3 8 0 . 4 9 0 . 9 3 1 0 . 7 4 0 . 8 9 6 3 6 1 6 1 3 . 6 1 4 . 1 7 0 . 6 7 2 4 . 5 2 0 . 6 4 4 4 . 5 1 0 . 6 4 5 7 3 6 2 7 . 7 0 5 . 1 4 0 . 8 0 9 5 . 8 9 0 . 8 0 1 5 . 6 2 0 . 7 9 0 3 7 6 5 . 3 8 0 . 0 3 0 . 9 9 4 0 . 0 2 0 . 9 9 6 0 . 0 7 0 . 9 8 4 3 8 1 1 8 . 6 6 0 . 2 3 0 . 9 7 0 0 . 2 5 0 . 9 6 8 0 . 4 5 0 . 9 4 3 3 9 1 2 8 . 8 1 1 . 8 9 0 . 7 6 4 2 . 0 4 0 . 7 4 5 2 . 6 1 0 . 6 7 4 8 2 9 2 3 . 1 3 2 . 1 5 0 . 9 0 4 2 . 3 1 0 . 8 9 6 3 . 1 3 0 . 8 6 0 4 0 9 6 . 6 6 0 . 2 5 0 . 9 5 8 1 . 7 1 0 . 7 0 7 2 . 3 5 0 . 5 9 7 4 1 1 0 7 . 1 4 0 . 1 6 0 . 9 7 5 1 . 7 7 0 . 7 2 1 2 . 7 8 0 . 5 6 2 4 2 1 1 7 . 8 6 1 . 0 5 0 . 8 5 2 3 . 4 9 0 . 5 0 6 4 . 2 8 0 . 3 9 5 4 3 9 6 . 3 8 0 . 4 7 0 . 9 1 6 0 . 7 5 0 . 8 6 5 1 . 3 2 0 . 7 6 3 4 4 1 0 6 . 4 5 0 . 0 4 0 . 9 9 2 0 . 9 1 0 . 8 4 2 1 . 4 9 0 . 7 4 0 4 5 1 1 7 . 0 8 1 . 1 1 0 . 8 2 7 3 . 0 1 0 . 5 2 8 3 . 5 5 0 . 4 4 3 9 6 0 4 5 . 4 2 3 . 0 7 0 . 9 3 1 1 1 . 0 4 0 . 7 3 9 1 5 . 7 7 0 . 6 4 7 F r e s n o 3 1 5 2 7 7 . 5 0 2 1 . 9 2 0 . 9 2 1 3 1 . 8 5 0 . 8 8 5 3 5 . 8 9 0 : 8 1 0 O v e r a l l 5 0 5 4 6 9 . 1 2 2 9 . 4 5 0 . 9 3 7 3 9 . 9 1 0 . 9 1 4 4 3 . 8 2 0 . 9 0 6 1 8 8 7 . 1 . 2 V a l i d a t i o n b y T w o L o c a t i o n G r o u p s I t i s n o t c o n v e n i e n t t o u s e t h e m o d e l w i t h a s e r i e s o f s t a t i s t i c a l p a r a m e t e r s e t s f o r p r e d i c t i n g f u t u r e d r y i n g o a s e s . T h e r e f o r e , t h e s e c o n d t r e a t m e n t w a s t o c h e c k t h e m o d e l ' s p e r f o r m a n c e w i t h t w o s e t s o f s t a t i s t i c a l p a r a m e t e r s f o r t w o l o c a t i o n s o f t h i s s t u d y . T h e s e t o f p a r a m e t e r s f o r t h e E a s t L a n s i n g g r o u p w a s o b t a i n e d b y a n o t h e r r u n o f p a r a m e t e r e s t i m a t i o n w i t h t h e J o i n e d p o p u l a t i o n ( 6 3 o b s e r v a t i o n s ) f r o m T r i a l s 4 , 5 , 9 a n d 1 3 . T h i s w a s d u e t o t h e i n f e r e n c e a b o u t t h e c o m m o n p a r a m e t e r s f o r t h e E a s t L a n s i n g g r o u p ( s e e t h e p r e c e d i n g c h a p t e r ) . E v e n t h o u g h t h e i n f e r e n c e d i d n o t f a v o r a c o m m o n p a i r o f p a r a m e t e r s f o r t h e F r e s n o g r o u p , a s d e s c r i b e d i n t h e p r e v i o u s c h a p t e r , a s e t o f p a r a m e t e r s a a n d b w a s a l s o o b t a i n e d f r o m r e g r e s s i o n o f t h e c o m b i n e d p o p u l a t i o n o f T r i a l s 2 5 , 3 2 , 3 5 , 3 8 a n d 4 4 ( 6 1 o b s e r v a t i o n s ) . T a b l e 1 1 g i v e s v a l u e s o f t h e s e t w o s e t s o f p a r a m e t e r s . S i m u l a t i o n r e s u l t s f o r e a c h l o c a t i o n g r o u p a r e s u m m a r i z e d i n t h e m i d d l e t w o c o l u m n s o f T a b l e 1 0 . T h e r e s u l t a n t S S E ( 8 . 0 6 ) a n d r 2 ( 0 . 9 5 8 ) f r o m e m p l o y i n g o n e s i n g l e s e t o f p a r a m e t e r s f o r t h e w h o l e E a s t L a n s i n g g r o u p w e r e a l m o s t t h e s a m e a s t h o s e ( S S E = 7 . 5 3 a n d r 2 = 0 . 9 6 0 ) o b t a i n e d u n d e r t h e p r e v i o u s t r e a t m e n t w i t h f o u r s e t s o f p a r a m e t e r s . S i m i l a r t e n d e n c y w a s f o u n d f o r D a t e g r o u p s 1 t o 4 u n d e r t h i s t r e a t m e n t . F o r T r i a l s 1 , 2 , S , 1 3 a n d 1 5 , S S E w a s i n c r e a s e d s o m e w h a t , s o r 2 d e c r e a s e d . T r i a l s 6 a n d 1 H , i n v e r s e l y , r e s u l t e d i n d e c r e a s e s i n S S E a n d i n c r e a s e s i n r 2 . F o r t h e F r e s n o g r o u p a s a w h o l e , S S E w a s m a r k e d l y i n c r e a s e d f r o m 2 1 . 9 2 t o 3 1 . 8 5 , a n d t h u s , r 2 w a s r e d u c e d f r o m 0 . 9 2 1 t o 0 . 8 8 5 , b u t r 2 w a s s t i l l m u c h h i g h e r t h a n t h e m i n i m u m r e q u i r e d ( T a b l e 9 ) . I n c r e a s e s i n S S E a n d d e c r e a s e s i n r 2 w e r e o b s e r v e d f r o m a l l d a t e g r o u p s o f t h e F r e s n o 1 8 9 T a b l e 1 1 T h e s e t s o f t h e s t a t i s t i c a l p a r a m e t e r s ( a a n d b ) u s e d d u r i n g m o d e l v a l i d a t i o n T r e a t m e n t a b 1 4 . 5 3 9 1 . 4 6 4 2 5 . 0 9 4 2 2 3 7 3 4 . 8 3 4 1 8 3 1 C u t t i n g 4 4 . 8 7 9 1 . 7 6 0 D a t e 5 4 . 4 2 7 1 . 1 0 2 6 3 . 9 6 3 ’ 0 . 3 2 0 7 4 . 2 3 8 0 8 4 3 8 4 . 9 7 5 2 5 6 0 9 4 . 2 3 2 2 . 4 5 6 L o c a t i o n E a s t L a n s i n g 4 . 8 3 6 1 . 8 2 2 F r e s n o 4 . 7 3 0 1 . 8 8 3 E n t i r e t y 4 . 8 2 3 1 . 8 4 9 P o p u l a t i o n g r o u p , e s p e c i a l l y G r o u p 9 . I n c r e a s e s i n S S E w e r e s u b s t a n t i a l f o r a l l t r i a l s i n D a t e g r o u p 9 . T h e r 2 v a l u e o f T r i a l 4 2 ( 0 . 5 0 6 ) w a s b e l o w t h e m i n i m u m 0 . 6 0 2 ( T a b l e 9 ) . T h e s e n o t i c e a b l e d i f f e r e n c e s i n t h e s i m u l a t i o n r e s u l t s b e t w e e n t h e t w o l o c a t i o n g r o u p s m a y b e a t t r i b u t e d t o t h r e e r e a s o n s . F i r s t , t h e s w a t h d e n s i t y f o r t h e F r e s n o g r o u p v a r i e d o v e r a v e r y w i d e r a n g e f r o m 0 . 4 8 2 t o 2 . 5 8 4 k g d r y m a t t e r / m 2 , i n c o m p a r i s o n w i t h t h a t f r o m 0 . 2 6 2 t o 0 . 5 4 4 k g d r y m a t t e r / m z f o r t h e E a s t L a n s i n g g r o u p . S e c o n d l y , i n f o r m a t i o n n e e d e d b y t h e m o d e l w a s l e s s a c c u r a t e f o r t h e F r e s n o g r o u p t h a n t h e E a s t L a n s i n g g r o u p . T h i r d l y , t h e r e w a s n o o v e r n i g h t m e a s u r e m e n t o f t r a y w e i g h t s d u r i n g t h e t e s t p e r i o d i n F r e s n o , C a l i f o r n i a . A l l f i v e s e t s o f p a r a m e t e r s f o r t r i a l s o f t h i s l o c a t i o n w e r e e s t i m a t e d w i t h o u t a d j u s t m e n t t o r e w e t t i n g i n f o r m a t i o n . T h i s a f f e c t e d t h e p e r f o r m a n c e o f t h e m o d e l p r e d i c t i n g d r y i n g a l f a l f a i n t h a t s p e c i f i c a r e a . N e v e r t h e l e s s , a l l 2 5 d a t e g r o u p s , a n d t h e F r e s n o g r o u p a s a w h o l e s t i l l h a d r 2 v a l u e s f r o m 0 . 7 3 9 t o 0 . 9 5 7 w h i c h w e r e 1 9 0 s u f f i c i e n t l y h i g h e r t h a n t h o s e r e s p e c t i v e m i n i m u m v a l u e s ( T a b l e 9 ) . F o r t h e e n t i r e e x p e r i m e n t p o p u l a t i o n , S S E i n c r e a s e d f r o m 2 9 . 4 5 t o 3 9 . 9 1 , r 2 d e c r e a s e d f r o m 0 . 9 3 7 t o 0 . 9 1 4 . T h i s v a l u e w a s s t i l l g o o d d u e t o t h e l a r g e S S T O ( 4 6 9 . 1 2 ) a n d t h e l a r g e s a m p l e s i z e ( 5 0 5 ) . T h e r e s u l t s o f t h e s e c o n d t r e a t m e n t i n d i c a t e d t h a t i t w o u l d b e p o s s i b l e t o a p p l y o n e s e t o f s t a t i s t i c a l p a r a m e t e r s t o t h e m o d e l f o r m a k i n g r e a s o n a b l y a c c u r a t e p r e d i c t i o n s o f f i e l d d r y i n g o f a l f a l f a a t a p a r t i c u l a r l o c a t i o n . 7 . 1 . 3 V a l i d a t i o n b y t h e E n t i r e E x p e r i m e n t a s a W h o l e F u r t h e r m o r e , t h e t h i r d t r e a t m e n t w a s t o u s e a s i n g l e s e t o f s t a t i s t i c a l p a r a m e t e r s f o r a l l o b s e r v a t i o n s o f t h i s s t u d y . T h i s s e t o f p a r a m e t e r s ( T a b l e 1 1 ) w a s o b t a i n e d f r o m a r e g r e s s i o n o f a l l 9 s e l e c t e d t r i a l s ( 1 2 4 o b s e r v a t i o n s ) . S i m u l a t i o n r e s u l t s a r e s u m m a r i z e d i n t h e r i g h t t w o c o l u m n s o f T a b l e 1 0 . R e g a r d i n g t h e e n t i r e e x p e r i m e n t a s a w h o l e , S S E w a s s l i g h t l y i n c r e a s e d t o 4 3 . 8 2 , a n d r 2 c h a n g e d s l i g h t l y t o 0 . 9 0 6 i n c o m p a r i s o n w i t h t h o s e f r o m t h e s e c o n d t r e a t m e n t . F o r t h e E a s t L a n s i n g g r o u p a s a w h o l e , S S E ( 7 . 9 3 ) w a s n e a r l y u n c h a n g e d f r o m t h o s e ( 7 . 5 3 a n d 8 . 0 6 ) o f t h e f i r s t a n d s e c o n d t r e a t m e n t s , r e s p e c t i v e l y ; a n d r 2 r e m a i n e d a r o u n d 0 . 9 6 . F o r t h e F r e s n o g r o u p , S S E w a s s l i g h t l y i n c r e a s e d t o 3 5 . 8 9 f r o m 3 1 . 8 5 o f t h e s e c o n d t r e a t m e n t , s o r z c h a n g e d a l i t t l e f r o m 0 . 8 8 5 t o 0 . 8 7 0 . A s t o i n d i v i d u a l d a t e g r o u p s , G r o u p s 3 a n d 5 h a d r e d u c e d S S E a n d i m p r o v e d r 2 f r o m t h o s e o f t h e s e c o n d t r e a t m e n t , r e s p e c t i v e l y . G r o u p 3 h a d t h e l o w e s t S S E a n d t h e h i g h e s t r 2 u n d e r t h i s t r e a t m e n t c o m p a r e d t o t h e o t h e r t w o t r e a t m e n t s . G r o u p 9 h a d a n i n c r e a s e i n S S E a n d a 1 9 1 d e c r e a s e i n r 2 . I n c r e a s e s i n S S E f o r G r o u p s 2 , 4 a n d 9 w e r e q u i t e s i g n i f i c a n t . A m o n g t h e f o r t y t r i a l s , T r i a l s 6 , 8 , 1 0 , 1 0 , 1 2 , 1 4 , 2 1 , 2 8 a n d 2 9 h a d t h e l o w e s t S S E a n d t h e h i g h e s t r 2 u n d e r t h i s t r e a t m e n t r e l a t i v e t o t h e o t h e r t w o . V a l u e s o f S S E a n d r 2 f o r T r i a l s 2 a n d 1 5 o f t h e E a s t L a n s i n g g r o u p , a n d T r i a l s 3 3 a n d 4 0 t o 4 5 o f t h e F r e s n o g r o u p w e r e t h e w o r s t u n d e r t h e t h i r d t r e a t m e n t . T h e r 2 v a l u e s o f T r i a l s 2 ( 0 . 7 3 5 ) , 4 0 ( 0 . 5 9 7 ) , 4 1 ( 0 . 5 6 2 ) , 4 2 ( 0 . 3 9 5 ) a n d 4 5 ( 0 . 4 4 3 ) w e r e b e l o w t h e m i n i m u m v a l u e s g i v e n i n T a b l e 9 , r e s p e c t i v e l y . F o r t h e r e m a i n i n g t r i a l s , r e s u l t s u n d e r t h e t h i r d t r e a t m e n t w e r e b e t w e e n t h o s e f r o m t h e o t h e r t w o t r e a t m e n t s d i s c u s s e d a b o v e . I t s h o u l d b e p o i n t e d o u t t h a t t h e t w o f i e l d e x p e r i m e n t s o f t h i s s t u d y a r e 4 0 0 0 k m a p a r t w i t h d i f f e r e n t c l i m a t e s , s o i l s , p a t t e r n s o f a l f a l f a g r o w t h a n d h a y p r o d u c t i o n . T h e p e r f o r m a n c e o f t h e m o d e l u s i n g o n e s e t o f p a r a m e t e r s t o s i m u l a t e f i e l d d r y i n g c a s e s a t b o t h l o c a t i o n s w a s g o o d . F i g u r e s 3 1 t o 3 3 s h o w s c a t t e r d i a g r a m s o f o b s e r v e d v e r s u s p r e d i c t e d d r y i n g d a t a . T h e r e l a t i o n s h i p b e t w e e n f i e l d m e a s u r e d a n d m o d e l p r e d i c t e d d a t a w a s d e s c r i b e d w i t h a l i n e a r f u n c t i o n : c h z a - o - B M C P [ 2 2 1 ] w h e r e M C l l : m e a s u r e d a v e r a g e s w a t h m o i s t u r e c o n t e n t , M C p : p r e d i c t e d a v e r a g e s w a t h m o i s t u r e c o n t e n t , a : i n t e r c e p t i o n o f t h e l i n e , a n d B = s l o p e o f t h e l i n e . T a b l e 1 2 l i s t s v a l u e s o f a a n d 8 f o r t h e E a s t L a n s i n g g r o u p , t h e F r e s n o g r o u p a n d t h e t h e e n t i r e e x p e r i m e n t f r o m t h e r e s u l t s o f t h e t h r e e t r e a t m e n t s . I f t h e m o d e l w a s p e r f e c t , o a n d B w o u l d b e z e r o a n d u n i t y , 0 1 . 5 h d t l . i e a w i i f n t r n o e f f t o i ) . b 0 . n l r o a . d 4 ( c e C g e n , t a r o n n u r e t s s e t i a r n o F o m C r d h e n t v a e a o 0 r n w r s s a . u 3 t r g e e i s g t h i a e c o r m i M e a M v r a a , e p g g d n a e l i t a s r a c n e l i a 0 v u t L F . A m s 2 i i t s t s h a a t s t E u s a s n w i r f S e o v s s n d d o t e e i e t r t s c u i 0 s d e i F . a n n 1 d e i o e M n c r P . 1 3 e / r u g i F 0 . 4 0 A 0 4 - 0 0 0 . . . - 4 2 0 1 ( ' q ' p ) i u a i u o g a m i s g o w G E D J O A V m o m s p a i n s o e w I q u ) 5 5 . 0 - A — - R e g r e s s i o n L i n e w i t h 9 5 % C o n f i d e n c e B e l t 0 E a s t L a n s i n g D a t a x F r e s n o D o h a 1 9 2 0 . 5 h d t l i e w i . a i f n t r n O e f f t o i n l ) . b . 0 o a d c e C 4 ( . g e n , t o r a n e u r n t s s e t i a r n o F o m C h e n r d t v a e a o 0 r 3 u 3 t w n s s a r g e e i s g t h i a e c o r m i M e a M v r a a e p , g g d n a e l i t a s r a c n e l i a D v E A Z u t L m s i i t s t s h a a s t E t u s a s n w r f i S e o v s s n d d t o e e e i t c r s t u i O s d I a o n i 1 e w o d e M c t r P . 2 3 e r u g i F t l e B e c n e d i f n o C d n a a t a e n D i l g n a n i t o s a i n D s a s o L e n r s t g s e e a r R E F 0 x — — 0 . 4 - 0 . 5 - 0 . 1 0 . 0 0 . 0 ( ' q ' p ) i u e i u o o e m i s g o w 9 5 0 1 9 7 0 9 1 1 4 3 0 4 1 3 p e d n s o e w i 0 . 1 n 1 9 3 0 I . 5 - ) . b 0 . r . d 4 ( t n e t n o C e 0 r T . u 3 t s i o t l e M B e c n e d i f n o C % 5 9 h t i w a t e a n D i L g n a n i t o s a i n D s a s L o e n r t s g s e e a t R E F — 4 o ) - e g a r e 0 v r . A 2 h t a w S d e t c 0 i 1 . 1 d e r P / 0 . 0 - 0 4 - . . 1 2 0 1 0 ° ( ° q ' p ) i u e i u o o s i n i s g o w e fi o J e / ( v q l e S p e m s o e w I q 1 0 4 . 0 - 1 3 . 0 - 4 F i g u r e 3 3 . M e a s u r e d v e r s u s s i m u l a t e d a v e r a g e s w a t h m o i s t u r e c o n t e n t w i t h o n e s e t o f s t a t i s t i c a l p a r a m e t e r s o v e r a r a n g e o f f i e l d c o n d i t i o n s i n E a s t L a n s i n g , M i c h i g a n a n d F r e s n o , C a l i f o r n i a . 1 9 4 1 9 5 T a b l e 1 2 V a l u e s o f t h e i n t e r c e p t ( a ) a n d s l o p e ( B ) o f E q u a t i o n [ 2 2 1 ] o b t a i n e d f r o m s c a t t e r d i a g r a m s o f t h r e e t r e a t m e n t s E a s t L a n s i n g F r e s n o E n t i r e t y T r e a t m e n t - - - - - - - - - - - - - - - - - - - - - - - - - - - - — — — — — — — = - - - - 8 a B a 8 a B y C u t D a t e 1 . 0 1 3 7 - 0 . 0 9 3 0 0 . 9 8 8 4 - 0 . 0 5 8 3 0 . 9 9 8 6 - 0 . 0 7 2 9 B y L o c a t i o n 0 . 9 8 4 0 0 . 0 0 0 1 0 . 9 4 5 3 0 . 1 0 8 1 0 . 9 6 1 8 0 . 0 6 4 4 B y E n t i r e t y 0 . 9 2 8 6 0 . 2 6 4 1 0 . 9 8 4 2 - 0 . 0 3 6 7 0 . 9 4 0 3 0 . 1 1 2 6 . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ‘ - - - - - - — - - - - - - - - - - - r e s p e c t i v e l y . I t w a s s e e n t h a t t h o s e v a l u e s o f a . a n d B i n T a b l e 1 2 w e r e c l o s e t o t h e p r e f e r r e d v a l u e s . I n F i g u r e s 3 1 t o 3 3 , 9 5 % c o n f i d e n c e p r e d i c t i o n i n t e r v a l s a r e a l s o d r a w n . I t c a n b e s e e n t h a t m o s t o f t h e o b s e r v e d d r y i n g d a t a p o i n t s f e l l i n t o t h e s e c o n f i d e n c e p r e d i c t i o n b e l t s . I n t h e r a n g e o f l o w s w a t h m o i s t u r e c o n t e n t s , d a t a p o i n t s d i v e r g e d , a n d a f e w p o i n t s w e r e o u t s i d e t h e c o n f i d e n c e b e l t . T h i s w a s d u e t o a c c u m u l a t e d e r r o r s o f t h e m o d e l . 7 . 1 . 4 V a l i d a t i o n b y u s e o f D i f f e r e n t H e a t h e r D a t a A s d i s c u s s e d i n C h a p t e r 4 , t h e m o d e l w a s d e v e l o p e d t o u s e a f e w w e a t h e r v a r i a b l e s a s e n v i r o n m e n t a l i n p u t s . T h e y w e r e a i r t e m p e r a t u r e a n d r e l a t i v e h u m i d i t y , w i n d s p e e d a t a r e f e r e n c e h e i g h t a b o v e t h e s w a t h , s o l a r i r r a d i a n c e a t t h e s w a t h t o p , p r e c i p i t a t i o n a n d t h e s k y c o v e r f r a c t i o n . T h e s e w e a t h e r d a t a c o u l d b e o b t a i n e d f r o m m i c r o c l i m a t i c m e a s u r e m e n t o r t h r o u g h t h e l o c a l w e a t h e r s e r v i c e . I n a l l d i s c u s s i o n s s o f a r , t h e f i r s t f o u r q u a n t i t i e s w e r e d i r e c t l y m e a s u r e d a t a h e i g h t a b o u t 0 . 6 m a b o v e t h e g r o u n d , a n d t h e o t h e r t w o w e r e f r o m t h e l o c a l w e a t h e r s e r v i c e . T h e f o u r t h t r e a t m e n t w a s t o c h e c k w h e t h e r t h e m o d e l c o u l d w o r k 1 9 6 n o r m a l l y i f a l l w e a t h e r d a t a w e r e f r o m t h e l o c a l w e a t h e r s e r v i c e . T h e m o d e l e s t i m a t e d s o l a r i r r a d i a n c e s i n c e i t w a s n o t m e a s u r e d a t t h e l o c a l w e a t h e r s t a t i o n . T h i s t e s t w a s c o n d u c t e d t o a l l 4 d a t e g r o u p s , 1 5 t r i a l s a n d 1 9 0 o b s e r v a t i o n s o f t h e E a s t L a n s i n g g r o u p . T h e s e t o f s t a t i s t i c a l p a r a m e t e r s a a n d b u s e d w a s t h e s a m e s e t a p p l i e d u n d e r t r e a t m e n t 2 . S i m u l a t i o n r e s u l t s a r e s u m m a r i z e d i n T a b l e 1 3 . T a b l e 1 3 S u m m a r y o f t h e s u m o f s q u a r e s o f r e s i d u a l s ( S S E ) a n d t h e a d j u s t e d c o e f f i c i e n t o f d e t e r m i n a t i o n ( r 2 ) f o r t h e E a s t L a n s i n g g r o u p b y u s e o f l o c a l w e a t h e r s e r v i c e d a t a G r o u p T r i a l S S E r 2 | G r o u p T r i a l S S E r 2 1 ‘ _ — 0 . 6 9 ‘ — _ — 0 . 7 5 9 — — I 5 0 . 8 3 0 . 9 3 8 2 1 . 2 2 0 . 5 8 3 I 6 1 . 9 9 0 . 8 3 9 3 0 . 0 7 0 . 9 6 7 I 7 0 . 6 9 0 . 9 5 1 4 0 . 0 8 0 . 9 7 7 : 1 2 . 0 6 0 . 8 8 5 I 2 3 . 5 1 0 . 9 1 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | . . - - - - - - - - - - . . - - - - - - - - - - - - - - - - - - - - - - - - 8 1 . 8 5 0 . 8 4 7 | 1 2 0 . 3 4 0 . 9 4 3 9 0 . 2 5 0 . 9 7 6 I 1 3 0 . 5 1 0 . 9 1 2 1 0 0 . 2 6 0 . 9 6 7 1 1 4 0 . 2 1 0 . 9 6 8 1 1 0 . 1 6 0 . 9 8 4 : 1 5 1 . 0 3 0 . 8 3 9 3 2 . 5 2 0 . 9 4 6 I 4 2 . 0 9 0 9 5 6 T h e s e r e s u l t s w e r e v e r y c l o s e t o t h o s e u n d e r t h e s e c o n d t r e a t m e n t l i s t e d i n T a b l e 1 0 . T r i a l s 1 a n d 2 h a d r e l a t i v e l y l o w r 2 v a l u e s , t h e y w e r e b e l o w t h e m i n i m u m s f o r 9 5 % c o n f i d e n c e g i v e n i n T a b l e 9 . T h e r e s u l t s f o r t h e E a s t L a n s i n g g r o u p a s a w h o l e w e r e a l m o s t t h e s a m e a s t h o s e o f t h e s e c o n d t r e a t m e n t . 1 9 7 I n s u m m a r y , t h r o u g h t h e s e f o u r t r e a t m e n t t e s t s , i t w a s p r o v e d t h a t t h e m o d e l h a d t h e a b i l i t y t o p r e d i c t f i e l d d r y i n g o f a l f a l f a u s i n g t y p e o f w e a t h e r i n f o r m a t i o n . v a r i o u s c i r c u m s t a n c e s . 7 . 2 S e n s i t i v i t y A n a l y s i s T h r o u g h s e n s i t i v i t y a n a l y s i s , e f f e c t s o f v a r i o u s p a r a m e t e r s a n d v a r i a b l e s c a n b e i d e n t i f i e d q u a n t i t a t i v e l y . S e n s i t i v i t y a n a l y s i s o f t h e m o d e l s t u d i e d w a s d o n e b y e x a m i n i n g s e n s i t i v i t y c o e f f i c i e n t s o f t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t ( t h e d e p e n d e n t v a r i a b l e a n d s y s t e m o u t p u t ) , w i t h r e s p e c t t o t e n i m p o r t a n t p a r a m e t e r s a n d v a r i a b l e s o f t h e m o d e l . T h e y w e r e t h e t w o s t a t i s t i c a l p a r a m e t e r s i n E q u a t i o n [ 1 3 9 ] , t h e a s s u m e d i n i t i a l m o i s t u r e c o n t e n t ( M C O ) , a n d s e v e n s y s t e m p a r a m e t e r s . T h e y i n c l u d e d t h e s w a t h a r e a d e n s i t y ( D E N S ) , t h e r a t i o o f l e a f a r e a i n d e x t o y i e l d ( L A I / Y L D ) , t h e c r o p m a t u r i t y ( B L O O M ) , t h e c o n d i t i o n i n g f a c t o r ( C M E ) , l e a f m o i s t u r e c o n t e n t a t h y d r a t i o n ( F M C ) , r e s i s t a n c e t o m o i s t u r e t r a n s f e r a t t h e s o i l s u r f a c e ( R V S ) a n d t h e r a t i o o f s w a t h t h i c k n e s s t o a r e a d e n s i t y ( Z C / D E N S ) . S i n c e d i f f e r e n t p a r a m e t e r s h a v e v a r i o u s d i m e n s i o n s ( u n i t s ) a n d m a g n i t u d e s , i t i s b e s t t o f o r m n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s w h i c h a r e d i m e n s i o n l e s s i n o r d e r t o c o m p a r e t h e r e l a t i v e i m p o r t a n c e o f e a c h p a r a m e t e r . I n t h i s s t u d y , n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s w e r e c o n s t r u c t e d a s : i j : x i j A B j / n j : ( a n / a j ) 1 A B J / n 1 [ 2 2 2 ] ( i : 1 , . . . , n ; j = 1 , . . . , q ) w h e r e i j : n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t , d i m e n s i o n l e s s , x i j = o r i g i n a l s e n s i t i v i t y c o e f f i c i e n t , d i m e n s i o n a l , 1 9 8 A 8 3 i n c r e m e n t o f t h e j t h p a r a m e t e r , d i m e n s i o n a l , i t h p r e d i c t e d m o i s t u r e c o n t e n t , k g w a t e r / k g d r y m a t t e r . a n d n 1 H e r e , 8 3 i s o n e o f t h e t e n p a r a m e t e r s t o b e e x a m i n e d ( q = 1 0 ) . T h e v a l u e o f B j w a s 1 0 % o f 8 3 ( o r 0 . 1 i f 8 3 : 0 ) . T h e q u a n t i t y 1 0 0 X i j w a s t h e p e r c e n t a g e o f c h a n g e i n n 1 d u e t o a n i n c r e m e n t A B j i n B j - N o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s w e r e c a l c u l a t e d i n p r o g r a m s A R N L I N a n d A L F D R Y ( s e e A p p e n d i c e s I , a n d J ) . I n A R N L I N , n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s f o r t w o s t a t i s t i c a l p a r a m e t e r s a a n d b w e r e d e n o t e d w i t h B X ( 1 ) a n d B X ( 2 ) . T h e o t h e r e i g h t n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s w e r e d e n o t e d w i t h S M C O , S D E N , S L / Y , S B L M , S C M E , S F M C , S R V S a n d S Z / D , r e s p e c t i v e l y , i n A L F D R Y . I t s h o u l d b e p o i n t e d o u t t h a t t h e s e s e n s i t i v i t y c o e f f i c i e n t s , l i k e t h e s t u d i e d m o d e l i t s e l f , w e r e n o n l i n e a r f u n c t i o n s o f a l l o f t h e t e n s e l e c t e d p a r a m e t e r s a n d v a r i a b l e s a s w e l l a s o t h e r s . T h e s e s e n s i t i v i t y c o e f f i c i e n t s w e r e c o m p l i c a t e d f u n c t i o n s o f t i m e a n d t h e h i s t o r y o f t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t . T h e s e s e n s i t i v i t y c o e f f i c i e n t s d e s c r i b e d t h e r e s p o n s e o f t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t t o s m a l l d i s t u r b a n c e s o f t h e s e p a r a m e t e r s a n d v a r i a b l e s , r e s p e c t i v e l y , u n d e r g i v e n c o n d i t i o n s . H o w e v e r , t h e y m a y n o t b e i n d i c a t i v e o f t h e d i r e c t i o n s o f c h a n g e s i n t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t w i t h l a r g e c h a n g e s o f t h e s e p a r a m e t e r s . F i g u r e s 3 4 t o 3 6 s h o w t y p i c a l c u r v e s o f t h e t e n n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s v e r s u s t i m e f o r T r i a l 5 t e s t e d i n E a s t L a n s i n g , M i c h i g a n o n A u g u s t 1 4 t o 1 6 o f 1 9 8 4 . I n F i g u r e 3 4 , c u r v e s o f B X ( 1 ) a n d B X ( 2 ) a r e p l o t t e d . C u r v e s o f t h e o t h e r e i g h t s e n s i t i v i t y c o e f f i c i e n t s a r e s h o w n i n F i g u r e s 3 5 a n d 3 6 , r e s p e c t i v e l y . I t w a s n o t i c e d t h a t t h e s e e i g h t n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s f o r t h e 0 : 8 1 0 : 2 1 0 : 6 l a c i t s i t a t s o w t . f 5 o l s a 6 t i 0 1 n r T : e T 0 i { 8 e c r t i o a f f D f e ) 0 2 o : d c ( T 8 X n 1 y B a t i d r n v u i a 0 o f : H t i ) s 1 2 1 ( n ( e X s B e m d s i e r 0 T T : 6 z e i t l e a m m a r r o a N p 5 0 1 . r : 4 0 3 / 8 e r u g 0 i : F 8 1 ) 1 ( X B e — e 4 4 5 0 . . 1 1 . 1 3 . : 1 U 9 1 0 ! 9 0 : ) 6 4 1 4 1 1 1 8 4 9 5 9 6 2 1 1 0 1 4 1 4 0 1 1 2 . 0 5 0 . 5 1 0 . 0 1 0 . 2 , 0 . 1 1 0 . 0 1 H B X ( 2 ) 1 2 : 0 8 / 1 4 F r T — l 1 9 9 O C M N l E Y / D L S S S a — a - H m . O e - J fl fl E ’ O 0 h p : t o 1 8 a r 1 w c s ‘ d 0 I : 2 1 0 I : , n ) a O C ) M Y S / ( L S e ( r u d t l s e 6 i i o y m o l t a 6 i x 0 1 t e I : { i d 0 n n 8 e i i t a f a A o e D r 0 s a A V T : d t 8 n f n 1 e a m V a i e c l . V r i u f f . f o 5 . r V : H 0 o e 2 1 o o l ( c i a A V t i A ' A f A v . ~ 4 1 0 e y a r t r T m i i v e r 0 T i h o r t f : 6 t i s , ) n ) M e L N s E B D S 5 d S ( 0 ( r 1 e : / 0 8 z y i y t l t i a i r m s u r n t o e a 0 N d m r : 8 . 1 5 3 e r 4 u 0 1 : 2 / g i 2 F 0 1 8 . 0 - . . — 0 - 3 . O l _ . 2 . 0 . 0 4 - 0 . 0 2 - 1 1 1 1 9 1 0 4 1 9 0 0 6 4 1 4 1 1 1 8 1 1 9 5 p e z u o q u N 2 0 0 f , a ) e S 0 l V 1 : 8 R S , 1 ( ) E M C e S c ( n E C M D 0 1 a : . M [ C S H ' F S - 2 8 1 1 1 1 - — + — 1 1 1 2 r t 5 1 o s t i l c s a a e i f r r T 0 g e 1 : n c r 6 i o a n f f o r i u ) t s D i / d Z l 6 0 1 l i n S : 0 ! o o ( E e c s t e t , y a h ) i D C s t M n 0 f e F f : d o d S 8 1 n ( a s o t n t n o r e i s u i t s 0 o c e a I : i r n H 2 f d k ( 1 f y c e h i e o h m c l t l i y h u 0 t t T f [ : i a 6 v t w i a s t i f t s o n 5 n e 0 1 e t o fi n i s : / 0 L o t E d c a 0 r : 8 1 e r z e i r e l u h a t t m s r i d o o n N m a . 6 3 1 1 1 4 1 1 1 M - 7 4 1 - r 1 2 L 0 4 : 1 e / 2 1 0 2 1 0 0 2 u 2 0 2 2 1 E . . . . . 0 g 0 0 0 0 0 0 0 0 0 0 . i . 3 0 0 0 . . - 0 F 0 0 0 . — i u e g o w s o o 1 0 , 1 1 1 1 g s u e s p e z u a u u o N 2 0 1 2 0 2 s a m e T r i a l 5 v a r i e d a m o n g t h e t h r e e v a l i d a t i o n t r e a t m e n t s . T h i s w a s e v i d e n c e o f d e p e n d e n c e o f s e n s i t i v i t y c o e f f i c i e n t s t o t h e s t a t i s t i c a l p a r a m e t e r s a a n d b . T h e d a t a u s e d i n F i g u r e s 3 5 a n d 3 6 w e r e t a k e n f r o m c o r r e s p o n d i n g o u t p u t o f T r i a l 5 i n A p p e n d i x J . 7 . 2 . 1 E f f e c t s o f T w o S t a t i s t i c a l P a r a m e t e r s A c c o r d i n g t o E q u a t i o n [ 1 3 9 ] , i t w a s e x p e c t e d t h a t p a r a m e t e r a h a d a p o s i t i v e e f f e c t o n a v e r a g e s w a t h m o i s t u r e c o n t e n t . T h i s w a s p r o v e d b y a l l p o s i t i v e v a l u e s o f B X ( 1 ) f o r T r i a l 5 a n d o t h e r t r i a l s . P a r a m e t e r a m a i n l y d e t e r m i n e d t h e m a g n i t u d e o f l e a f r e s i s t a n c e . A n i n c r e m e n t i n a c a u s e d a n i n c r e a s e i n l e a f r e s i s t a n c e , w h i c h r e s u l t e d i n s l o w i n g f i e l d d r y i n g , i . e . l o n g e r t i m e t o d r y a l f a l f a t o a c e r t a i n e x t e n t , o r a h i g h e r s w a t h m o i s t u r e c o n t e n t f o r a g i v e n p e r i o d o f t i m e . B X ( 1 ) g r a d u a l l y i n c r e a s e d a s d r y i n g p r o g r e s s e d , e s p e c i a l l y i n t h e d a y t i m e . T h i s m e a n t t h a t t h e s w a t h m o i s t u r e c o n t e n t w a s m o r e a n d m o r e a f f e c t e d b y p a r a m e t e r a a t l o w a l f a l f a m o i s t u r e c o n t e n t s a s s e e n f r o m E q u a t i o n [ 1 3 9 ] . A t n i g h t , B X ( 1 ) w a s r e d u c e d a c c o r d i n g l y b e c a u s e l e a f r e s i s t a n c e h a d n o i n f l u e n c e a f t e r d e w s t a r t e d . B u t , B x ( 1 ) d i d n o t r e t u r n t o z e r o a s s h o w n i n F i g u r e 3 4 , s i n c e t h e i n f l u e n c e h i s t o r y o f B X ( 1 ) s t i l l r e m a i n e d . T h e m a g n i t u d e o f B X ( 1 ) w a s u p t o 1 . 6 6 o c c u r r i n g a t t h e e n d o f d r y i n g i n t h e t y p i c a l t r i a l , i . e . a 1 0 % i n c r e a s e i n p a r a m e t e r a ( 5 . 0 9 4 ) c o u l d c a u s e 1 6 6 1 p r e d i c t i o n v a r i a t i o n o n t h e f i n a l s w a t h m o i s t u r e c o n t e n t f o r T r i a l 5 . T h i s w a s v e r y s i g n i f i c a n t . I n o t h e r w o r d s , t h e s y s t e m o u t p u t w a s v e r y s e n s i t i v e t o t h e a c c u r a c y o f p a r a m e t e r a . P a r a m e t e r D h a d a n e g a t i v e e f f e c t o n l e a f r e s i s t a n c e , i . e . a h i g h 2 0 3 v a l u e o f b w o u l d l e a d t o a l o w l e a f r e s i s t a n c e a c c o r d i n g t o E q u a t i o n [ 1 3 9 ] . H e n c e , a n i n c r e m e n t o f b c o u l d r e s u l t i n a l o w s w a t h m o i s t u r e c o n t e n t . T h i s c a n b e s e e n f r o m n e g a t i v e v a l u e s o f t h e B X ( 2 ) c u r v e i n F i g u r e 3 4 . T h e m a g n i t u d e o f B X ( 2 ) i n c r e a s e d a s d r y i n g t i m e p r o g r e s s e d ; h o w e v e r , B X ( 2 ) d e c r e a s e d n e a r t h e e n d o f f i e l d d r y i n g ( F i g u r e 3 4 ) . T h i s d e c r e a s e i n t h e i n f l u e n c e o f b w a s d u e t o t h e r a t i o o f M t o M 0 i n E q u a t i o n [ 1 3 9 ] w h i c h b e c a m e s m a l l e r a n d s m a l l e r a s d r y i n g a d v a n c e d . B X ( 2 ) a l s o d e c r e a s e d t e m p o r a r i l y a t n i g h t b e c a u s e o f d e w f o r m u l a t i o n . T h e r e w e r e t w o i n s t a n c e s o f p o s i t i v e B X ( 2 ) o c c u r r i n g a t t h e m o r n i n g o f t h e t h i r d d a y a c c o r d i n g t o F i g u r e 3 4 . T h i s o s c i l l a t i o n m a y b e a t t r i b u t e d t o t w o r e a s o n s . F i r s t , t h e i n c r e a s e o f b l o w e r e d l e a f r e s i s t a n c e , w h i c h r e d u c e d l e a f a n d s w a t h m o i s t u r e c o n t e n t . W h e n s i m u l a t e d l e a f m o i s t u r e c o n t e n t w a s l o w e n o u g h t h a t i t s e q u i l i b r i u m r e l a t i v e h u m i d i t y w a s b e l o w t h a t o f t h e s u r r o u n d i n g a i r , r e w e t t i n g t o o k p l a c e . S e c o n d l y , a s d i s c u s s e d i n S e c t i o n 7 . 1 , t h e d i s c o n t i n u i t y o f l e a f r e s i s t a n c e a t t r a n s i t i o n b e t w e e n a d s o r p t i o n a n d d e s o r p t i o n m i g h t i n d u c e s u c h a n o s c i l l a t i o n . T h e m a g n i t u d e o f B X ( 2 ) w a s a b o u t o n e t e n t h o f t h a t o f B X ( 1 ) w h i c h i n d i c a t e d t h e m o d e l w a s l e s s s e n s i t i v e t o p a r a m e t e r b t h a n p a r a m e t e r a . T h e i n t e r c o r r e l a t i o n b e t w e e n a a n d b i s a l s o s e e n i n F i g u r e 3 4 . T h e t w o c u r v e s , e x c l u d i n g t w o p o s i t i v e p o i n t s o f B X ( 2 ) , w e r e n e a r s y m m e t r i c w i t h r e s p e c t t o t h e h o r i z o n t a l a x i s i f v e r t i c a l c o o r d i n a t e s o f o n e c u r v e w a s m u l t i p l i e d b y a s c a l e f a c t o r . R e c a l l t h a t F a b f o r T r i a l 5 i n T a b l e 8 w a s 0 . 8 3 2 . T h i s q u i t e h i g h v a l u e o f r a b w a s v i s u a l l y c o n f i r m e d w i t h F i g u r e 3 4 . 2 0 4 7 . 2 . 2 T h e E f f e c t o f I n i t i a l M o i s t u r e C o n t e n t A p o s i t i v e e f f e c t o n t h e s w a t h m o i s t u r e c o n t e n t d u e t o a n i n c r e m e n t o f t h e i n i t i a l m o i s t u r e c o n t e n t o f l e a v e s w a s e x p e c t e d . T h e h i g h e r t h e i n i t i a l m o i s t u r e c o n t e n t , t h e m o r e m o i s t u r e t o b e r e m o v e d , t h u s a l o n g e r d r y i n g t i m e i s r e q u i r e d o r a h i g h e r s w a t h m o i s t u r e c o n t e n t i s o b s e r v e d a f t e r a c e r t a i n p e r i o d o f t i m e . I n t h e m o d e l s t u d i e d , t h e m o i s t u r e c o n t e n t o f l e a v e s a t f u l l h y d r a t i o n w a s a s s u m e d t o b e t h e s a m e a s t h e i n i t i a l m o i s t u r e c o n t e n t o f l e a v e s d u e t o i n t e r n a l a d j u s t m e n t o f l e a v e s . T h u s t h e i n c r e m e n t o f M C O h a d a n i n f l u e n c e o n l e a f r e s i s t a n c e i n t h e m o d e l w h i c h w a s a l s o i n f a v o r o f i n c r e a s i n g l e a f r e s i s t a n c e a n d s w a t h m o i s t u r e c o n t e n t . P o s i t i v e v a l u e s o n t h e c u r v e o f S M C O i n F i g u r e 3 5 r e p r e s e n t e d t h e e f f e c t o f i n i t i a l m o i s t u r e c o n t e n t . T h e m a g n i t u d e o f S M C O w a s 0 . 0 1 7 , i . e . a m a x i m u m 1 . 7 1 d e v i a t i o n i n s i m u l a t e d s w a t h m o i s t u r e c o n t e n t c o r r e s p o n d e d t o a 1 0 1 i n c r e a s e i n N O D . T h i s w a s s m a l l . O n e n e g a t i v e S M C O o c c u r r i n g a t t h e b e g i n n i n g o f d r y i n g w a s f o u n d i n F i g u r e 3 5 . S i m i l a r c i r c u m s t a n c e s w e r e o b s e r v e d f r o m s i m u l a t i o n r e s u l t s o f o t h e r t r i a l s . I n a f e w c a s e s , n e g a t i v e S M C O l a s t e d f o r a p e r i o d o f t i m e . T h e s e w e r e u n e x p e c t e d a n d p r e s u m a b l y d u e t o s o m e n u m e r i c a l i n s t a b i l i t y o f t h e m o d e l i n t h o s e c a s e s . 7 . 2 . 3 T h e E f f e c t o f S w a t h A r e a D e n s i t y T h e e f f e c t o f s w a t h a r e a d e n s i t y ( D E N S ) o n t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t w a s c o m p l i c a t e d b e c a u s e D E N S d e t e r m i n e d n o t o n l y t h e a m o u n t o f m o i s t u r e i n a l f a l f a p l a n t s t o b e d r i e d b u t a l s o t h e l e a f a r e a 2 0 5 i n d e x a n d t h e s w a t h t h i c k n e s s i n t h e m o d e l . T h e l a t t e r t w o f a c t o r s a f f e c t e d p r o f i l e s o f r a d i a t i o n a n d w i n d s p e e d i n s i d e t h e s w a t h . T h e o v e r a l l e f f e c t d u e t o c h a n g e i n D E N S w a s c h a r a c t e r i z e d b y t h e d o m i n a n t f a c t o r . V a r i o u s p a t t e r n s o f t h e n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t S D E N o f D E N S w e r e o b s e r v e d f r o m s i m u l a t i o n r e s u l t s . W h e n t h e s w a t h a r e a d e n s i t y w a s l o w , l i k e t h o s e o f t h e E a s t L a n s i n g g r o u p , a n d l e a v e s w i l t e d , a n i n c r e a s e i n D E N S m i g h t l e a d t o i n c r e a s i n g i n t e r c e p t i o n o f s o l a r r a d i a t i o n d u e t o a n i n c r e a s e i n t h e l e a f a r e a i n d e x , h e n c e , s p e e d i n g u p d r y i n g ( n e g a t i v e S D E N ) , e s p e c i a l l y a t t h e l a t e s t a g e o f d r y i n g . T h i s i s s e e n f r o m F i g u r e 3 5 . H o w e v e r , a t t h e b e g i n n i n g o f d r y i n g w h e n l e a v e s a r e n o t w i l t e d , t h e i n c r e m e n t o f t h e l e a f a r e a i n d e x w o u l d n o t i n c r e a s e l i g h t i n t e r c e p t i o n b u t d e c r e a s e w i n d s p e e d a n d l i g h t p e n e t r a t i o n , a s w e l l a s i n c r e a s e t h e a m o u n t o f m o i s t u r e , s o s w a t h d r y i n g w a s r e d u c e d ( p o s i t i v e S D E N ) a s s h o w n i n F i g u r e 3 5 . O n t h e o t h e r h a n d , w h e n t h e s w a t h w a s v e r y d e n s e , l i k e t h o s e o f t h e F r e s n o g r o u p , l i g h t p e n e t r a t i o n a n d w i n d s p e e d w e r e l i m i t e d a n d r a d i a t i o n i n t e r c e p t i o n w a s s a t u r a t e d b y o v e r l a p p i n g l e a v e s w i t h i n t h e s w a t h , a n i n c r e m e n t o f t h e s w a t h d e n s i t y r e s u l t e d i n f u r t h e r r e d u c t i o n o f l i g h t p e n e t r a t i o n a n d i n t e r c e p t i o n i n d e e p l a y e r s . I n t h i s c a s e , t h e w h o l e p r o c e s s o f d r y i n g s l o w e d d o w n ( p o s i t i v e S D E N ) a s s e e n i n A p p e n d i x J . F o r T r i a l 5 , t h e m a g n i t u d e o f n e g a t i v e S D E N i n c r e a s e d a s d r y i n g p r o g r e s s e d i n t h e d a y t i m e , a n d c o n t i n u e d r i s i n g u p 2 t o 4 a m , t h e n l o w e r e d t i l l t h e n e x t m o r n i n g . T h i s w a s p r o b a b l y d u e t o t h e h e a t c a p a c i t y o f t h e s w a t h . S t a r t i n g f r o m s u n r i s e , a l f a l f a p l a n t s g r a d u a l l y a b s o r b e d s o l a r e n e r g y , s p e n t s o m e f o r d r y i n g a n d s t o r e d s o m e t o r a i s e s w a t h t e m p e r a t u r e . S t o r e d e n e r g y s u p p o r t e d d r y i n g i n t h e e v e n i n g a n d a t 2 0 6 e a r l y n i g h t u n t i l d e w o c c u r r e d . A n i n c r e m e n t o f t h e s w a t h d e n s i t y i n c r e a s e d t h e s w a t h h e a t c a p a c i t y a n d , t h u s , h e l p e d t o e x t e n d t h e d u r a t i o n o f d r y i n g . T h e m a g n i t u d e o f S D E N S w a s l e s s t h a n 0 . 0 3 2 w h i c h w a s n o t v e r y n o t i c e a b l e . T h e r e w a s a p o s i t i v e S D E N f o u n d n e a r t h e e n d o f d r y i n g . T h i s a b n o r m a l o s c i l l a t i o n c a s e m i g h t b e c a u s e d b y t h e s a m e r e a s o n s d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n . 7 . 2 . 4 T h e E f f e c t o f t h e R a t i o o f L e a f A r e a I n d e x t o Y i e l d T h e r a t i o o f l e a f a r e a i n d e x t o y i e l d ( L A I / Y L D ) d e t e r m i n e d t h e l e a f a r e a i n d e x f o r a g i v e n s w a t h d e n s i t y . A s m e n t i o n e d i n t h e p r e c e d i n g s e c t i o n , a n i n c r e m e n t o f t h e l e a f a r e a i n d e x h a d i n f l u e n c e s o n l i g h t i n t e r c e p t i o n , p e n e t r a t i o n a n d w i n d s p e e d w h i c h a f f e c t e d m o i s t u r e r e m o v a l o f t h e s w a t h . T h e d o m i n a n t f a c t o r c h a r a c t e r i z e d t h e b e h a v i o r o f t h e s e n s i t i v i t y c o e f f i c i e n t S L / Y o f L A I / Y L D . F o r T r i a l 5 , t h e e f f e c t o f L A I / Y L D w a s i n f a v o r o f f a s t d r y i n g w h i c h w a s r e p r e s e n t e d b y n e g a t i v e S L / D i n F i g u r e 3 5 . T h e m a g n i t u d e o f S L / Y i n c r e a s e d a s t h e s w a t h g r a d u a l l y d r i e d . A t n i g h t , t h e n e g a t i v e e f f e c t o f l e a f a r e a i n d e x t e m p o r a r i l y d r o p p e d b e c a u s e a n i n c r e a s e i n l e a f s u r f a c e a r e a e n h a n c e d t h e r m a l r a d i a t i o n l o s s e s f r o m p l a n t s . A t t h e l a t e s t a g e o f d r y i n g d u e t o l e a f w i l t i n g , t h e i n f l u e n c e o f l e a f a r e a i n d e x w a s r e d u c e d . T h e m a x i m u m m a g n i t u d e o f S L / Y w a s a b o u t 1 7 . 7 % w h i c h w a s q u i t e s i g n i f i c a n t . 2 0 7 7 . 2 . 5 T h e E f f e c t o f A l f a l f a M a t u r i t y T h e m a t u r i t y o f a l f a l f a w a s r a t e d a s p e r c e n t a g e o f c r o p i n f u l l b l o o m ( B L O O M ) . I n t h e m o d e l s t u d i e d , t h e m a t u r i t y a f f e c t e d e q u i l i b r i u m r e l a t i v e h u m i d i t y o f l e a v e s w h i c h d e t e r m i n e d w h e t h e r m o i s t u r e t r a n s f e r o n l e a f s u r f a c e w a s d e s o r p t i o n o r a d s o r p t i o n . T h e e f f e c t o f e q u i l i b r i u m r e l a t i v e h u m i d i t y w a s n o t i c e a b l e o n l y w h e n l e a f m o i s t u r e c o n t e n t w a s l o w . T h e r e f o r e , a t t h e e a r l y s t a g e o f d r y i n g , n o r m a l i z e d s e n s i t i v i t y c o e f f i c i e n t s S B L M o f t h e b l o o m l e v e l w e r e z e r o . A c c o r d i n g t o B a k k e r ' s d a t a ( 1 9 6 1 ) , e q u i l i b r i u m r e l a t i v e h u m i d i t y i n c r e a s e d a s t h e b l o o m l e v e l i n c r e a s e d . T h u s , a n i n c r e m e n t o f t h e b l o o m l e v e l f a v o r e d f a s t d r y i n g ( n e g a t i v e S B L M ) . T h e m a g n i t u d e o f n e g a t i v e S B L M w a s l e s s t h a n 0 . 0 2 w h i c h w a s u n i m p o r t a n t . T w o p o s i t i v e v a l u e s o f S B L M w e r e f o u n d n e a r t h e e n d o f d r y i n g . T h i s o s c i l l a t i o n m a y b e c a u s e d b y t h e s i m i l a r f a c t o r s m e n t i o n e d i n S e c t i o n 7 . 2 . 2 . 7 . 2 . 6 T h e E f f e c t o f t h e C o n d i t i o n i n g F a c t o r T h e c o n d i t i o n i n g f a c t o r ( C H E ) d e t e r m i n e d t h e r e d u c t i o n i n l e a f r e s i s t a n c e d u e t o m e c h a n i c a l ( o r c h e m i c a l ) t r e a t m e n t d u r i n g c u t t i n g . C M E w a s a s s i g n e d a v a l u e o f 0 . 5 i n t h e m o d e l . O b v i o u s l y , a n i n c r e a s e i n C M E w o u l d c a u s e a n i n c r e a s e i n l e a f r e s i s t a n c e , t h u s , s l o w i n g a d r y i n g p r o c e s s . S C M E g r a d u a l l y i n c r e a s e d a s d r y i n g p r o c e e d e d , a n d d r o p p e d a t n i g h t d u e t o d e w a s s h o w n i n F i g u r e 3 6 . T h e m a g n i t u d e o f S C M E w a s 1 4 % f o r T r i a l 5 . 2 0 8 7 . 2 . 7 T h e E f f e c t o f L e a f M o i s t u r e C o n t e n t a t F u l l H y d r a t i o n T h e l e a f m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n ( F M C o r M y ) w a s u s e d i n E q u a t i o n [ 1 3 9 ] . A s e x p r e s s e d i n t h a t e q u a t i o n , a n i n c r e a s e i n M f w o u l d r e S u l t i n i n c r e a s i n g l e a f r e s i s t a n c e , t h u s , d e l a y i n g f i e l d d r y i n g , a s s h o w n b y p o s i t i v e S F M C i n F i g u r e 3 6 . S i m i l a r t o o t h e r p a r a m e t e r s d i r e c t l y r e l a t e d t o l e a f r e s i s t a n c e i n E q u a t i o n [ 1 3 9 ] d i s c u s s e d a b o v e , t h e e f f e c t o f F M C i n c r e a s e d a s f i e l d d r y i n g a d v a n c e d ( l e a f r e s i s t a n c e i n c r e a s e d ) i n t h e d a y t i m e , a n d r e d u c e d a t n i g h t d u e t o d e w ( w h e n l e a f r e s i s t a n c e h a d n o e f f e c t o n m o i s t u r e t r a n s f e r ) . T h e m a g n i t u d e o f S F M C w a s 2 1 % w h o s e s i g n i f i c a n c e w a s h i g h , s e c o n d t o p a r a m e t e r a f o r T r i a l 5 . 7 . 2 . 8 T h e E f f e c t o f S o i l S u r f a c e R e s i s t a n c e t o M o i s t u r e T r a n s f e r T h e r e s i s t a n c e t o m o i s t u r e t r a n s f e r a t t h e s o i l s u r f a c e ( R V S ) a f f e c t e d m o i s t u r e t r a n s f e r b e t w e e n t h e s o i l s u r f a c e a n d t h e b o t t o m l a y e r o f t h e s w a t h . W h e n m o i s t u r e c o n t e n t i n t h e b o t t o m l a y e r w a s h i g h , a n i n c r e m e n t o f R V S w o u l d r e d u c e m o i s t u r e t r a n s f e r f r o m t h e b o t t o m l a y e r t o s o i l , t h u s , d e l a y i n g d r y i n g . O n t h e o t h e r h a n d , w h e n s w a t h m o i s t u r e c o n t e n t w a s l a w , a n i n c r e a s e i n R V S c o u l d p r e v e n t a b s o r p t i o n o f m o i s t u r e f r o m s o i l , h e n c e , e n h a n c i n g d r y i n g . T h i s w a s s e e n f r o m t h e S R V S c u r v e o f F i g u r e 3 6 . T h e m a g n i t u d e o f S R V S w a s e x p e c t e d t o b e s m a l l s i n c e t h e w i n d s p e e d a t t h e s o i l s u r f a c e i n s i d e t h e s w a t h w a s z e r o a n d , t h u s , n o s t r o n g t r a n s f e r a c t i v i t y t o o k p l a c e . F o r T r i a l 5 , t h e m a x i m u m S R V S w a s o n l y 0 . 1 2 % . I t w a s a s s u m e d t h a t R V S f o r t h e E a s t L a n s i n g g r o u p w a s 1 0 s / m 2 0 9 a n d 1 0 0 0 s / m f o r t h e F r e s n o g r o u p i n t h i s s t u d y . S u c h a l a r g e d i f f e r e n c e i n R V S f o r t h e t w o l o c a t i o n s d i d n o t i n d u c e l a r g e v a r i a t i o n i n t h e m a g n i t u d e o f S R V S . T h i s w a s f u r t h e r e v i d e n c e t h a t t h e m o d e l w a s n o t s e n s i t i v e t o c h a n g e s i n R V S . 7 . 2 . 9 T h e E f f e c t o f t h e R a t i o o f S w a t h T h i c k n e s s t o D e n s i t y T h e r a t i o o f s w a t h t h i c k n e s s t o d e n s i t y ( Z C / D E N S ) w a s a n i n d i c a t i o n o f s w a t h p o r o s i t y . F o r a g i v e n s w a t h d e n s i t y , a h i g h s w a t h t h i c k n e s s 2 0 m e a n t a r e l a t i v e l y l o o s e s w a t h . A i r m o v e m e n t i n s i d e a l o o s e s w a t h w o u l d e n h a n c e h e a t a n d m o i s t u r e t r a n s f e r . T h e c u r v e o f S Z / D i n F i g u r e 3 6 s h o w e d t h a t a n i n c r e m e n t o f Z C / D E N S p r o m o t e d m o i s t u r e l o s s e s f r o m a l f a l f a p l a n t s ( n e g a t i v e 5 2 / 0 ) a t t h e e a r l y s t a g e o f d r y i n g . H o w e v e r , a i r m o v e m e n t i n s i d e t h e s w a t h a l s o e n h a n c e d p o s s i b l e m o i s t u r e t r a n s f e r f r o m t h e s u r r o u n d i n g a i r t o l e a v e s . T h i s b e c a m e s i g n i f i c a n t w h e n a l f a l f a m o i s t u r e c o n t e n t w a s l o w a t t h e l a t e s t a g e o f d r y i n g ( F i g u r e 3 6 ) . T h i s s u g g e s t e d t h a t r a k i n g i n h a y p r o d u c t i o n i s u s e f u l n o t o n l y d u e t o t u r n i n g a l f a l f a p l a n t s b u t a l s o b y i n c r e a s i n g s w a t h c o m p a c t n e s s . T h e m a g n i t u d e o f 5 2 / 0 w a s o n l y 2 % f o r T r i a l 5 . I n t h e m o d e l , s w a t h t h i c k n e s s w a s a s s u m e d t o b e p r o p o r t i o n a l t o s w a t h d e n s i t y . A c t u a l l y , t h e r a t i o o f s w a t h t h i c k n e s s t o d e n s i t y s h o u l d d e c r e a s e a s d e n s i t y i n c r e a s e s b e c a u s e o f g r a v i t a t i o n a l f o r c e s . A s f i e l d d r y i n g p r o c e e d e d , e s p e c i a l l y a f t e r l e a v e s w i l t e d , t h e s w a t h b e c a m e c o m p a c t e d . T h e s e w e r e n o t y e t t a k e n i n t o a c c o u n t i n t h e m o d e l . I n s u m m a r y , t h r o u g h t h e s e n s i t i v i t y a n a l y s i s , i t w a s f o u n d t h a t ( 1 ) S e n s i t i v i t y c o e f f i c i e n t s o f t h e m o d e l w e r e c o m p l i c a t e d n o n l i n e a r f u n c t i o n s o f t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t , t w o s t a t i s t i c a l 2 1 0 p a r a m e t e r s a n d o t h e r v a r i a b l e s a n d p a r a m e t e r s ; ( 2 ) T h e s e n s i t i v i t y c o e f f i c i e n t s o f t h e m o d e l v a r i e d f r o m t r i a l t o t r i a l a n d f r o m t i m e ( o b s e r v a t i o n ) t o t i m e ( o b s e r v a t i o n ) ; ( 3 ) T h e m o d e l w a s m o s t s e n s i t i v e t o c h a n g e s i n p a r a m e t e r a ; a n d ( 4 ) O t h e r i m p o r t a n t p a r a m e t e r s w e r e p a r a m e t e r b , l e a f m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n , t h e r a t i o o f l e a f a r e a i n d e x t o y i e l d , t h e c o n d i t i o n i n g f a c t o r a n d t h e s w a t h a r e a d e n s i t y . I n A p p e n d i x K , a s a m p l e o u t p u t o f p r o f i l e s o f s h o r t - w a v e a n d l o n g - w a v e r a d i a t i o n , s o i l t e m p e r a t u r e , l e a f t e m p e r a t u r e a n d m o i s t u r e c o n t e n t , a i r - s p a c e t e m p e r a t u r e a n d h u m i d i t y i n t h r e e l a y e r s o f t h e s w a t h f o r T r i a l 5 i s g i v e n . C H A P T E R 8 C O N C L U S I O N S A N D R E C O M M E N D A T I O N S T h e f o l l o w i n g c o n c l u s i o n s a n d r e c o m m e n d a t i o n s c a n b e d r a w n f r o m t h i s s t u d y . 8 . 1 C o n c l u s i o n s M o d e l i n g o f t h e s o i l - s w a t h - a t m o s p h e r e c o n t i n u u m u n d e r c o n d i t i o n s o f f i e l d d r y i n g a l f a l f a i s a p p r o p r i a t e . T h r o u g h t h i s m o d e l i n g , t h e p h y s i c a l a n d p h y s i o l o g i c a l p r o c e s s e s o f f i e l d d r y i n g c a n b e c l o s e l y s t u d i e d . U n d e r f i e l d d r y i n g c o n d i t i o n s , p l a n t s c a n b e p r i m a r i l y r e p r e s e n t e d b y l e a v e s , s i n c e t r a n s f e r o f h e a t a n d m o i s t u r e t a k e s p l a c e m o s t l y o n l e a f s u r f a c e s . T h e M o n t e i t h - P e n m a n e q u a t i o n c a n b e a p p l i e d t o d r y i n g p l a n t s w i t h m o d i f i c a t i o n t o c o n s i d e r t h e e f f e c t s o f t h e e q u i l i b r i u m r e l a t i v e h u m i d i t y o f p l a n t l e a v e s , a n d l e a f r e s i s t a n c e t o m o i s t u r e t r a n s f e r u n d e r s e v e r e w a t e r s t r e s s . L e a f r e s i s t a n c e t o m o i s t u r e t r a n s f e r c a n b e a p p r o x i m a t e d w i t h t h e e x p o n e n t i a l f u n c t i o n f o r m o f E q u a t i o n [ 1 3 9 ] . W i t h o u t d e t a i l e d p h y s i o l o g i c a l i n f o r m a t i o n o n l e a f r e s i s t a n c e t o m o i s t u r e t r a n s f e r u n d e r s e v e r e w a t e r s t r e s s w i t h i n v a r i o u s s w a t h l a y e r s , t h i s r e s i s t a n c e c a n b e a p p r o x i m a t e d f r o m m a c r o s c o p i c 2 1 1 1 0 . 2 1 2 i n f o r m a t i o n a b o u t i t s o v e r a l l e f f e c t o n t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t . O v e r n i g h t m e a s u r e m e n t s o f r e w e t t i n g p r o c e s s e s a r e i m p o r t a n t a n d w o r t h w h i l e t o t h e d e v e l o p m e n t o f a c o m p r e h e n s i v e m o d e l o f f i e l d d r y i n g . S e q u e n t i a l p a r a m e t e r e s t i m a t i o n i s e f f e c t i v e i n m o d e l i n g c o m p l i c a t e d ' b i o l o g i c a l s y s t e m s . D e r i v a t i v e - f r e e t e c h n i q u e s o f o p t i m i z a t i o n a r e a l s o v e r y h e l p f u l t o d e t e r m i n e p r o p e r v a l u e s f o r p a r a m e t e r s i n c o m p l i c a t e d n o n l i n e a r p r o b l e m s . T h e s i m u l a t i o n m o d e l A L F D R Y w a s s h o w n t o b e v a l i d b y c o m p a r i n g i t s p r e d i c t i o n s o f t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t w i t h f i e l d o b s e r v e d d r y i n g d a t a u n d e r v a r i o u s c o n d i t i o n s i n E a s t L a n s i n g , M i c h i g a n a n d F r e s n o , C a l i f o r n i a . 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T h e s i m u l a t i o n m o d e l c a n p r e d i c t t h e a v e r a g e s w a t h m o i s t u r e c o n t e n t o f f i e l d d r y i n g a l f a l f a o v e r a w i d e r a n g e o f s w a t h d e n s i t i e s ( 0 . 2 6 2 t o 2 . 5 5 k g d r y m a t t e r / m 2 ) , i n i t i a l m o i s t u r e c o n t e n t s ( 2 . 6 5 t o 4 . 5 0 k g w a t e r / k g d r y m a t t e r ) , m a t u r i t y l e v e l s ( 0 t o 1 0 0 % b l o o m ) a n d m e t e o r o l o g i c a l c o n d i t i o n s w i t h r e a s o n a b l e a c c u r a c y . T h e m o d e l w a s m o s t s e n s i t i v e t o c h a n g e s i n t h e s t a t i s t i c a l p a r a m e t e r a o f E q u a t i o n [ 1 3 9 ] . O t h e r i m p o r t a n t q u a n t i t i e s f o r t h e m o d e l a r e 2 1 3 t h e s t a t i s t i c a l p a r a m e t e r b i n E q u a t i o n [ 1 3 9 ] , t h e l e a f m o i s t u r e c o n t e n t a t f u l l h y d r a t i o n , t h e m e c h a n i c a l / c h e m i c a l c o n d i t i o n i n g f a c t o r , t h e l e a f a r e a i n d e x a n d t h e s w a t h d e n s i t y . 1 1 . T h e m o d e l r e q u i r e d i n p u t w e a t h e r d a t a o f a i r t e m p e r a t u r e , r e l a t i v e h u m i d i t y , w i n d s p e e d , s o l a r i r r a d i a n c e a t a r e f e r e n c e h e i g h t a b o v e a s w a t h , p r e c i p i t a t i o n a n d s k y c o v e r f r a c t i o n t o m a k e p r e d i c t i o n s o f f i e l d d r y i n g . T h i s w e a t h e r i n f o r m a t i o n m a y b e m e a s u r e d a t t h e s i t e o r o b t a i n e d f r o m t h e l o c a l w e a t h e r s e r v i c e . 1 2 . A s e t o f t w o s t a t i s t i c a l p a r a m e t e r s i n E q u a t i o n [ 1 3 9 ] i s n e e d e d f o r t h e m o d e l . O b t a i n i n g o n e s e t o f t h e s e t w o s t a t i s t i c a l p a r a m e t e r s f o r a p a r t i c u l a r l o c a t i o n u n d e r n o r m a l c o n d i t i o n s o f h a y p r o d u c t i o n i s s u f f i c i e n t t o m a k e r e a s o n a b l e p r e d i c t i o n s . 8 . 2 R e c o m m e n d a t i o n s 1 . C o n t i n u o u s t e s t i n g t h e m o d e l a t v a r i o u s l o c a t i o n s i n o r d e r t o e x p a n d i t s p r e d i c t i o n a b i l i t y i s n e e d e d . 2 . F u r t h e r r e s e a r c h o n l e a f r e s i s t a n c e t o m o i s t u r e t r a n s f e r , e s p e c i a l l y u n d e r a h i s t o r y o f s e v e r e w a t e r s t r e s s i s e s s e n t i a l . 3 . T h o r o u g h i n v e s t i g a t i o n o n e x c h a n g e s w i t h i n p l a n t c a n o p i e s ( s w a t h s ) i s r e c o m m e n d e d . 4 . 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H , , 8 D ) n m , Y U V P u ( m O K , x : , x x 1 N W z N O ( : I A R : S , m “ 1 1 n J , T D , D ) O 1 O 1 : z : : 0 M H C A L 1 N 0 I N 0 ) M W n m a 3 3 $ , , ) I K , + N ) + 0 Z m B H m m M , m / o o o B S . c m 3 H Y , I I v w + , : , I R I D : , / ( / ( ( U ( o O G 5 0 / V S , o v 5 , N S / D m / ‘ N N ( N C : N N N B I I , I N / / / , ( , D / / ( H / ' H , ( ' / m N / H / s W m 6 m R L n ( V T o ( D S A R / Q J 0 A I S / , ~ R P N ) R B B V R A V D / 1 w B N ( ( ‘ N ( H L V A H ‘ S s m 3 a / ) U L H ‘ U R Z M N O , M D N t n D G a m ( B u M H I L / R o p Y / W G I T V I S N V S B S 7 C e / E C A R R R I M D G C R D R R D S R W I u a 0 m / S $ I m § 0 m M m m / R fl , 1 fl W / m M fi W / m M / S s 5 S S S S S S s S / M / m R M W / m / ” 9 1 1 u s m W m n N n m n N m o O o m o O n n c I m " " m m 1 n T m e 1 s R : a I a 0 a I e a 0 0 6 m 0 m W W ( m c m 0 N W m m W P a d W m m m W 0 m W m m m m m 0 0 2 g u A U P , ) : J N ) I , N , H ( W ( H S X , I X , 7 , ) 2 ) N A K - , 6 , , H L 5 ( H T ) S ) ( r P 1 r C B 6 I + , , , I V H ) H , , ( a 6 ( ) ) D l Z , 3 H 0 6 0 3 - H B ( , , m X N ( , ) / 2 ) 0 , ( I , N M 2 F 0 0 ( 0 9 2 1 1 1 1 3 Y ) , J , - ( ( T H , 6 H , 0 I 3 B X R P 1 ) 0 r , B 1 5 3 , ) , r I P , l 1 , , 1 H ) S fl ‘ C 7 m a 3 m 1 1 1 0 ) ) 1 ) 1 ' 1 o I L ( 6 P J D R , Z , N G 6 , - D , I ( ) L m H , P S R D , , , D F ' ( O I S e , u ) 0 ? 1 , + , , . 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I ( 1 0 V V L o I P L R G M A o m I L U A E R n 1 . u ' R 1 n r E n 1 1 a m x 1 A m A n a T R T I A 1 a a n a 0 U n n ¢ - A m A P N 3 E E n C B B H W m m n W 0 a a W u C m E A H a N a 9 D ‘ 6 ( ” l I ( V H u 6 R ) m m 0 D I C c m R P c C [ c 0 c W m 1 0 M n 3 N 0 B S E 0 0 2 g u A ' — + + - — — — — — O ‘ J , 3 ) ' 0 ( J , 1 ) E L S E O ( J , 2 ) ' V I ( I , J ) ' O ( J , 1 ) O ( J , 3 ) - V I ( I , J ) ' ( 1 . 0 § O ( J , 1 ) ) E N D I ? 1 0 W I N " : 3 “ ) I ? W I W I M “ m 2 1 1 " l e n ' I O O O ' J + I D ( I ) " ( J ) ' O ( J , 3 ) x ‘ I I J ) - ( m ( B I R I I Y ‘ I ) ! m ( I ) I T ( I ) I Y ( I ) l m ( I ) l m l m s l m ( I ) l + n , I . I . ) - B ‘ r M I ) ) “ ( J ) ' V I ( I , J ) I P ( I D ( I ) . m . l ) X ( I , J ) - 0 . 0 W ! “ “ U U U U U U U U U U U U U U U U U 0 0 0 + + + + + + + + + + + + + + + + + + 4 . 2 1 8 N O I T A n I T I N I E L B L ‘ A ) / E ) T 3 J A ( D m A Y P W L U L - o ! , I N A o O V L , I A I ) O N c 0 E I . ' 1 ) ‘ ) S 0 1 T 1 L 0 S , ) w * A P B 1 ) A . 2 . ) T s V 5 m J D , O ) 9 m ( 0 Y 0 S 1 A ) u ) S C C - . 8 B T L . L s u N . A n I I a . m P N L - r L L ' A , o T 0 8 ) T P 1 ( P g D 1 C L , , i H L I s m , P w , L R R , , . X o I . s m ) 7 1 O B c n , 2 . ( m 1 s c r , A N r 1 R u m T ( m S H S c . ) T a . B . ) 1 ! n , 1 3 N , G m 2 . T 6 C u L E ‘ m R 2 2 C . V 2 m o T L - ( 1 1 H 1 L ) I , , n T L T W 1 , O 0 H m ) P J 5 R , J m c C 1 A A ( S 5 1 s . A . - m a T ) a n 1 C a . o I D 1 . - C R m u 0 0 o 4 U O ( N 0 ) x 1 A I ) M 2 I I I H 1 o 1 - E 3 L m T R ) R - H S 1 A - “ A T I I I m ) . A I o 1 I N I , 1 - J m u ¢ ' ‘ 0 ‘ 0 o a L a D . D D L n ( A V I 0 , I 3 I 1 C N - R T 1 1 ? . M A 1 6 ( O d P 0 . 8 - I 1 7 R 0 N R J G ( B J R 0 ( J N u D D B I R A A C 3 ' m m . B m l s n . 5 n m : A m S m m ‘ 0 R O - H 1 V m 1 - m B u L m - ) + 3 H I n O . 6 R I ( T . a . D B E I a 2 e . a - ? n 8 T - m L - D A 5 L . ‘ B - - N A 9 m m m a o . a 1 l R O P 8 1 ( I I m . H s M A G 9 - r C l L 3 O 6 8 m 5 m F m m I m D ( l c C P n m m 1 H I n H N ? I 2 m l - I o - 9 1 A - P 1 a n L L l L — r A 1 E l L L 1 L L L 1 a A - o H u L L S 5 W C S L L L L L i I s I I i I u u 1 ( a 1 o A A ? L a a ' A A A O A A 3 3 D T 1 I a m c H L L L L L 1 a o ‘ 1 A A A A A o H l I m 0 m - R R L L L L L 9 r . m ( : ) N n t C C E m n C C C C C o D c I c I 7 m [ u A C n C C C C . ” 6 1 : 6 0 ) ) J ( L D , ) A T J A ( D O L D ) O C H L L A F N I A R E 1 V u — — 0 — - 3 g u — — - — — - ( L 1 r w — — A C C c C c c 6 , . ) n 5 2 n ( a I Q m J o s M H J , I 5 A ) ) 5 S 2 1 C S , N L 4 A / / E m 6 L 3 C ¢ S L , m E 6 3 R s U , ) , 5 ) 2 3 C . 5 9 8 2 R 9 . 0 , T T 0 . 2 A A , 0 C 0 , . 0 W 5 , m ( . 5 . U 2 a 7 0 I 1 1 x fl ( 1 n C . 2 ‘ , ’ 0 1 I t s , m 6 , T , 1 L ) . L 7 s 0 ) n . 4 5 A A , 5 2 m 8 9 ) 2 , o D . 1 c m 5 m ¢ , 2 m , x C 3 1 N A A 3 , 8 7 , ) ¢ 9 ) 5 m 7 7 m , . . N N m 2 A 6 2 A 5 2 . m § 3 , A 2 I m I 9 / / 0 D 1 m ( P ) ) D P 5 a , s L 4 4 I I m ) A , , ! R , , 5 2 , R 1 1 . ( - - E 7 m J J ) 2 1 B 1 H M H ) ) ) A ) s W ( , , : m o 9 1 1 8 1 m , ) 7 1 2 . , 1 4 , u 2 r ) , W Z C I 1 m L m . X 1 L C m ) D X s o s o - 1 m 6 ) m I L J ) n l s 1 N / m A , K 1 5 J 1 , J 0 w L J 0 - J c m H ( , / 1 N 0 S ‘ N H ? 2 e S m ? T m ‘ 3 ' T P n 0 u 0 m C m a n m ) ) . r n 7 1 a / S S U l S 1 L S S ) ) ) ) T 1 2 v ” a l m P w 0 0 1 s Y l - I a L L 1 6 , A U 1 I L L 1 1 a : u S L . W Y s 0 x 7 A : c l 8 m , m m m A A - 0 1 - ( D w “ - 9 A A 1 A - - m I 1 9 ( m P ‘ ‘ 3 p ) ) ) ) m - w n 1 - " ( ( 1 1 s I . 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W R A B ) W T J H 6 , ) J o m : I D K A , C V , P C S , W , ) I ) I N A w ) ) N T E ( P w w 1 M ( I L P S : P ( R , ) C ) J ( A J H ) D ( ) F ) Y 1 D S s R L s C * , N V N F P H ' w R U E m Y , C R , ( P ) S A ) R A D M N Y N D , I , Y ; c C ) . H , H , H v ( ( L 1 , V P C " S H J . ) I I . A — X ? m R S G , 0 m ) R P E H m m R S R U m , , N U E R ) V H R V , T E ( L H R , , , ) ) L I ) . ‘ W G G U R N N P H I ( 2 1 e g a l l K P V . , S P Y Y . , S ) , A A , R C S R L B D D O B U O P I U H R R ) I O I — J R . R ‘ , . , ‘ L G D ‘ I H ) T D H I O V ( 0 ( N H H W U R H I ' P 0 P O W H H ) , , , S N N a V N R D I M L S C L ) ) , R 8 Y I R R - A , Y , C V w v P , , ( R O F J R K D A , ) 1 ) ) , ) I l R U U ( I S P O C J C a R I D T y P ( ( U m H S r L z D : ( N I P A R . 0 D . N W , J D N 0 N N , ) c E 0 H 1 ( ( R N J m ( ) L R 1 L ) A - R p v R » 5 L . m Q , , . V ‘ W v N w ( H A L A , d . , , D . I ) I W 2 , S F A 1 1 H Y , I m N c , D D H R Z B O L m E v 1 U A S P R R o - R R N R H C r , 1 R T a c , m D 3 I m / B ) ) / / R G S A L H I a - S A A M L m ‘ D . . R m c N N s H D 1 I U I / s R ( ( 1 o ? 8 L m n Y A A . ( m R V E U . D m L ( / ( 9 M V 1 p L . B ) , A I H a J N D w S N N u - / ) s ' L o H n L L L L L A N E A B A A c m D m ' r A 1 G C 8 A A I 9 ( C n A A A A A o a C C C C C n R o L L L L L 7 L L V R , N . R R P A R L o D R R ( n I 1 / ‘ O E E f / / / fl / / . L A a x ( I C 5 0 - 1 T I e 1 P ‘ . F I - . J m M S N P T E H u N u n N I I P c 0 M A T I R ' U o A N ? 2 - ( H m - L - U Z A o L o o o 1 1 : 6 0 0 2 g u D P U — U A m c P I L I R A R O m 1 o r o n E R A A E I O u m m m R o m A A R D n c " C I B P R m o m m m m ' m N S U E S ' + A C 2 C ) ) J ( " H I ) J ( P ) ) J ( L D , ) J ( 0 L D - ) ) 0 c H / ) u m — 0 C . N R . E H 1 T ( ) l - R ) J - s ‘ Y ) ( H A R R G D I J A H T , L E D 0 . N A L H 1 A T L - C L I P ‘ 4 ) U D 1 ) A R P I Y 0 | N . 2 ( A } I ‘ 4 l A ‘ 1 . - D G Z 2 l I m ' . G A R C l . - E E 1 C 1 S P e . V P , B D , J ( g a P t . y z d I Y A D J - 4 2 + ' 8 . I O P Y . A D J H P T ) 1 G N J . ( 1 R ) O H O c C — H ( p x 3 ( J L A N D ( , - C 1 ) H - ) J ‘ ) ! 2 f P S . H ) l I A D I E L O . R H R B x 7 D E 0 0 I F I . U T Y . I R T 8 L 9 L m L A A ‘ M L O I ‘ L D . P - - ‘ 1 a S E N E - J I H u N 2 P s . 1 2 D D ( A U ( P D O b K D S A F I P I I m m J [ H ( [ U D J ( N ( E L L ? I - I T R Y H N a A W : l 1 A L U J o T o Y R ’ R m I C A 1 C o ' M ( - ( A U H P h I D O C E P F n 1 o m S m J A H H I I N F E I : 6 0 0 2 g u 9 2 3 A 9 1 C C ( ) U U + + + 2 2 0 S E R ) L S / R ) . . 1 5 o / s 1 J 1 H ' A R B ( 1 R . ) I 0 A P A 5 ( P E M 0 ) ) H R S U O I R 1 L n T I 2 ) S s A C / / 3 N E - n n 1 H H I 0 / T . . A o o 1 H Y V ( ( T s s N ( ( I I ' V I E I ' A ) S R ' Y z R U U ! n T o R s r I 0 ( ‘ I s s V E ' A E T I o R E P R o S O I F P s U D F R O U I F I ) ) ) ) I I W D P m ) r ' V F R I E I O I I I I , , , , J J J J , J T ' R E c A v A L V ( R r m I ( ( ( ( 0 5 E 0 P P A m R m M R s W T A V A H 1 I R E m n a ) ) I I . , J J ( ( W N S I N A R R S S - - A 5 A S , s O R A R m E T I ) ) H R R 1 V I H A J J 5 5 - $ S P A A A m n m T H - ( ( - - - - J T m ) 6 1 e g a P y n d ) f l I , J A ( N S I 7 l - ) 8 ) - u 9 J J o 1 ( L - w m 1 1 u 3 D s 7 A - I : o o 6 u n 0 0 2 g u A H N ) ) ) ) A S I J J J - J 4 D s A - - - - N A ( ( ( a ( 7 C m m R R A H V A n I - A S 5 o A S A o H R R n V c 4 7 Y P m H P a , ) N ( , , ) ) , , , , , K K , ) ) K K - ~ C o N K , K K , K ) 0 0 ) ) ) ) ) , : : . o A R m S : C : ( - P , ) V I 0 0 T A . m n s , ( ‘ 0 R C I : ( ( N , , V I o R S ) , ( U r n , S 1 ( U U , I , , ) ‘ C S S § ) , N N X R ) S n 1 n N N - K ) Z , , X + G ( ( S F V R E s A N s , P ( : K V I S ) ) N ( F H S L , L I G S O : , K K ( - C , , . X n W H , 0 A V A S R ( 0 ) E T , F ) L R , R G ( R K , ) u : : ( a , S 2 X , s N ) K O 0 s E , R G . W G L ) D F ) ( K : , , , ) N : ) A H S H I , 5 0 N N 0 ) 1 D L S N R 1 . V O , N N ( R N + H ( R C V R S N K V A G ) ' + G X ) ( ( , ( , I x F ‘ N S R , X S , , ' m n ( N : B L F K : E ( S X V F , z O E s A . R K O I ) 2 D A K ' R ) , 0 n ( , D A R ( K A ) S S . < A E S 1 ) D A 9 N 1 U ) V - s R : K R , , ' V S , . ‘ L ) ) ) , P : S O J , P 0 ( + R m N C 0 ) A ) ) , K K E m ( ( R , L ) K H : : , 0 , , N G K c ) , ) G . P C 1 P H ( K : 0 O : S N m ) ) N U ( K ) X , ( 0 S A 1 C ) N S 0 N : , , u ( V M F H ( ( ) H , A 0 1 ( 0 N N , R C . R ( + S N L ) , ( ( N H , S R K V ) K M { H L A ( N , ) H S X I B : N N R N ( H G 1 ( F A O D R ( , U , : ( V L L U H U + S , ( R S S 5 X ) O S P V ) D R , , , , B ) R , N G ( F ( , & S S x N S 1 R n D 1 V ) A , ) ) ) , , ( G - ) D , , E , * S V N ) A ) I K K S ) ) ) C R V o N I , M R , ( ' : K K H : , R K ( , ) N , x 1 H C ) H ( G 1 ; ) . J V N 0 O ) : : ( S S 5 X I ( N ( A + o X N ( K 0 O 0 1 H S , , , V S 1 ' A R G 0 ( P : ( H E ) N : , , P , ) H e D L L X E N G ' X : : ( ( A ( ( R ( , S € ) O E F R H H L K ( 0 N N K ) , X N S N P H g a P D , F S , N D F O C 0 0 G A L P , m ( H ) F , , , ) I D R , P 1 , ( V V C H A w ) R ( N , n ) I , ) ) ) ) S R R F R N C 9 R G S S o N , S N K , ) N R S S N H S ( 5 H N , V ( A z ( C M ( S , , : C C J ) K ( 0 , A ) : H N V ( S P h A L , ) ) 1 N ) m D , ) P S R N D A A 0 K D S S , I H H V ) K K : ( , ( : x N K 1 O R ' t S R K : : A Z D . ) X 0 I ) N S Z 0 2 E M X o N , , ) ' D O 1 X X ' R I E ) E , K H o : 0 0 R , 0 0 R 0 F F F 0 , , ) , L * H N : ( ' K / , , / R R , m . N ) C / N S : ( 0 N P N / / 7 m ( N N . X ( N ( / / 0 H ( m ( S G ( ( 1 O / / I n ( , C A ( V ) V ( L G H L H D ' 7 R R D G I M S L N U A L L F A V N B m L O S B B R R u ' S C R D R D R w I D M C R H A O I A S m ‘ n I ) 0 / / S S / / S S S S 5 s S [ / / m / ( D n H — - ) u c 9 1 1 1 : m m 1 m s u u m m n E Y m V A n m - o s - o 1 a e a o A D 6 c m m 0 m c H S m 0 c u 0 0 2 g u A + & + + + + + + + + + + + + + + + + ~ ~ ~ ~ \ . . . 1 u n n a n a u u n a I I R E T U R N 0 0 0 0 0 0 0 0 0 0 0 I ) ! I } ! [ N I X : 2 2 2 6 M , L - O i I O S / S . T B 9 - / 0 1 E . m 1 , , 4 ) 1 R 6 0 F G 0 1 K , 1 ' 0 J M O 1 - ' / O B . 1 m , E I ' V A 9 m , . 0 0 . E , 3 M 6 , H 3 I s 3 5 1 R L 8 B C O 0 - r 0 . , P R 0 0 n . 2 C , . S , I 5 1 s N H 5 , . 5 4 , - 8 A R P 0 2 2 N O , R . O m 5 T , 0 I 8 . L C S 0 1 ' / 1 H 1 . O 0 I $ P ' / c n s ‘ , W R ) F J , m W m I ( m 9 1 ( W 5 , ) 2 , 1 ’ ) " I m W N O , ? S . / 2 2 - T R T 0 0 , x A 5 , O N , 1 5 , 0 T / P E 5 . 0 2 . S m - J R W I , E V T A ) m I J T 8 6 A u , 1 E 1 , 1 L 2 M 2 P 6 , I D 3 R 1 1 . V R 0 / 1 - A 6 I P ( 9 n ( I ) ) 6 U . 6 R , R . e , S 0 0 , + , 0 g 0 N 1 1 . C O A / , 8 1 . L D Y I ) 1 C 6 , E , I a 3 x 1 B 5 0 1 E G 0 ( P , . / C M H 9 , , . 5 6 R . 1 0 m . ! O R c 1 ( P ' ( D ) Y 1 N 2 R y n , 3 A 0 1 ? , P . , s 0 5 A O D / , 5 , R P 4 / 6 5 O 2 , T ‘ S . , . 5 2 . / I B 0 3 G ) ) E a I , 3 H I - ( T 5 . v N D . D , T M A . E P 1 s ) Y ) T ) “ I R l 7 S 2 9 3 C 1 , . n f , / 1 I G C / - , 4 5 , 2 1 A M / 6 1 u 8 5 I A , I D H C ' m , W - - , ‘ S ) P , T 1 3 D 0 0 4 0 E , 0 W 0 E A I . 6 H 1 . H c E 1 . , . ) Y 0 . . I D 0 8 5 / i D , V 7 / 0 , I 3 / E C , O K , , m 6 , 1 ( O I m U + K V I X Y K A ( K ( H l B ( 2 I ( 1 . D ( K S 6 D G 1 . , ) 1 ’ ) 1 1 I 0 5 , O ( . 8 ? D 1 H ' Y T L N ( + D A I A R ( - K L I 0 I N E - ( B P I I 1 P I M I R ” X w I 1 E W R ( ” . 8 A C / S R S / D , 0 / ‘ 0 1 , B 9 1 L N N I ’ V R L 3 C A . . B R 2 E P E H m m E I 1 fl N I A R Z l m T 1 T A E A A W A A A A A A A U 1 A F T T T T T T T Y O S D A E A A A A A T T T A L R S D R D D m U D D D D “ D I B + + N A U E D S : 6 0 0 2 g u A C C 1 C C C , 5 - 8 1 5 , , , 5 5 5 - " - 8 8 8 5 7 2 , 4 ) 1 1 ( / . 1 4 , . . . . 1 0 2 , 2 5 2 1 6 / / , / / , 8 4 2 , , 5 9 5 5 5 5 5 , 0 - - - - 6 - - 5 5 T . . 8 8 8 8 3 - . 8 “ S 0 6 4 3 B 4 , 9 0 0 , W 4 7 7 2 4 3 8 ‘ 8 7 3 , 5 . 0 , . . 2 . . 4 . ) ) 1 . . 2 0 1 . 2 1 2 2 1 1 , 2 7 , 1 , 5 0 1 1 ( ( 3 . , 5 5 , 5 , 5 . S 5 , , , , - 5 - - - - 5 - , T 3 8 8 . - 8 3 / ‘ 9 T B B S 0 2 9 5 S I I 4 3 2 8 2 5 3 1 9 2 7 6 2 7 6 . 8 0 , 0 . ‘ 2 L 7 . . . 9 0 3 M R . . - . . 1 1 2 1 2 . . 2 2 , P 5 / ) , 2 4 5 0 5 8 , , 5 5 8 0 , 5 5 1 ) 5 - 7 , , , , , , / 2 , 1 1 , - 8 - - 3 . 5 5 - - 1 1 2 3 2 . 8 9 0 3 8 8 ( 1 0 0 0 7 4 7 6 4 0 1 ( 2 3 . . . . 3 0 8 3 T 6 6 , 6 - ‘ 2 3 T , . - 2 5 3 S I L S 2 . I 9 2 . 6 0 1 1 . . 2 1 6 6 3 , , . . 6 . 9 3 , , 5 4 7 , , 1 , , 9 2 8 7 1 1 K L 5 , 1 2 2 5 5 5 5 9 . 0 C , 0 0 , , - 5 A - - - - 8 7 6 e g a P , D . . 5 5 . 2 B E 8 8 3 - . . ) , V - - 8 E 0 3 0 , , 7 6 , , , 1 ) , 1 3 2 2 8 8 3 6 7 2 5 5 7 8 1 . 1 , 9 . 9 1 . . 3 7 3 4 . 6 ( 2 2 1 1 . 2 1 2 1 9 0 0 W 8 . 2 ' 6 , ( , , 7 V . , 5 0 0 1 2 . . t , . . 1 . , I T / 0 5 5 5 , . . 1 5 5 8 , 1 . 5 y r S - 1 . - - S - , , H W . 0 5 - 5 I A 8 3 B 8 8 I 0 4 , , 9 8 / ” , T 5 - 8 ‘ L 4 1 / L P 9 5 2 4 A 6 9 I d , H f 9 A B V 2 2 2 5 2 5 E C . 0 0 V 2 4 S T 8 P 3 D 0 3 0 4 . , 5 0 . . . 7 7 . , l 0 / / a P u 1 . 1 ” / 1 8 . - . 1 . 2 . 2 1 2 W S 5 R H 0 / . / fl . I V 1 , E 1 / 1 m T 1 5 T / S / R / T T I P 1 T I T P A C H S S m T S A S I H 1 m / [ / T I I S L S / L , 1 K L A A L L I I V I W R W P K T M M D M P H C E A A A A A A A A A V T T T T T T T T A E T A A A A A A A A A R m m S m m D D D D D D “ D D W . 6 8 1 4 7 8 9 1 1 1 : 6 0 0 2 g u A 0 0 0 0 0 0 0 U 0 0 0 U P R - P R A I D I ' L N I H B B R ( D W I O N L R S S ) H A - M O L E C U L A R V E I G H ‘ O F A I R ( K G A I R / 1 0 L ) N W - M O L E C U L A R 9 8 1 0 0 O F W A T E R ( K G m T B R / I ‘ D L ) W - I D L B C U L A R V O L U M E 0 ? W A T E R ( C U B I C H H A T E R / 1 0 L ) V A - m m m u m O F A I R ( C U B I C H A I R / 1 0 L ) C W - S P E C I F I C H E A T O F W A T E R ( J / K G - C ) E N D H I W S P A R A M E T E R S A N D D A T A a n o c x D A T A M I S P A R R A R A H E ' I E R ( n - 3 , n - 1 0 ) R E A L L A I , H C O c a n n o n R A B / P I c a m o n R I P S / B P S c a m o n / 9 ¢ A m / u 1 , n n n s , v o , m . , n c o , m c , R s r c c a n n o n / n T / z R , z c , n , z o c a m o n / E n x s n 1 / c n , n 1 , x n u . c a m o n / A B V / D B R , R H R , R V R , P V R , U R , C V , R C P R , P S G , R I R , R 9 £ R , R S V R c a m o n / G N D / C I D D , D B G , R H G , P V G , R V G , 0 6 , R C P G , P S O G , R S H G , R S V G , R V S c a n o n / R A n m / R n 1 m n ) , R m m , E m c a n o n / R A D S H / R N S R ( N ) , R n s m , R s c c a n o n m u n / c o n n , m c p , n e p ( n ) , n n s m + 1 ) , o ( n o 1 ) , c o m , A L R , V r c a m o n / S ' I ‘ A B / A 1 , B l , A 2 , 8 2 , P I , 0 1 , R 1 , P 2 , 0 2 , R 2 c a n n o n / A L F A / A L F I , A L P 2 c a n o n m m m u , m m 1 n , m m m x c a n o n M I S T / m m ) , S E O I C ( N ) , R q u C ( N ) , R ( m r E ( n ) , E m E S S ( n ) , R E m u C ( n ) , R S I m 0 ( n ) , R R U M 1 C ( n ) , m , m , s - a s c a n n o n fl ' B P / I C , ’ I ' X L ( 4 0 ) , P L ( 4 0 ) S A V E C U 0 0 0 - - — - J I H E $ 1 2 1 : o r n A m I . D E P E M ) o n T H E 5 1 2 8 o r D A T A F I L E ° e r n A y ' P A R M E I ‘ E R ( n - 1 5 4 , L - 1 8 ) I n m C E R L Y ( 2 , 2 ) , L D ( 2 , L , 3 ) I O G I C A L o x I N T E G E R m , m , R A , o u , o r , n R m o w n / I o r L / m , m , P A . o u , o r , n R c o n m n M N / I Y D ( N , 2 ) , W I ‘ R ( N , 6 ) m a s o n fl I A I E S / r m A r , J n A x , n o u n c a n n o n / P A R I N / I Y R , H D A Y I . H R I , H R F , H D A Y P , L B A P , J D A Y I , J D A Y F , + U I ' P , X H C , S I H U L , D I ' P A S T , U 1 ' S I D W C - - — - ' 1 H B s u n o r 1 0 c o R R E S R o n n S n o n — 2 : 0 0 T o 2 5 : 0 0 A T 3 1 4 I N I ‘ E R V A L S c a n n o n ” m o / m u m ) , n A L ( 1 0 ) , R H L ( 1 0 ) , C V L ( 1 0 ) , P C L ( 1 0 ) , H A L ( 1 0 ) , L I . S A V E n A T A 1 . 1 1 . 1 0 1 , I m , a n , L ¥ , L n , 1 1 , . 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( 1 ' 1 1 " 1 ( 1 ( . 1 0 1 1 1 0 3 I ( 1 ' . - ' 1 m 1 ” - 1 ( n ' 1 0 - 0 u 1 0 ’ 1 1 ' ( 1 m 1 ' 1 ' 1 1 ' 1 1 ( 1 ( ' 0 1 ' . 1 1 0 c ( ' - ( 1 m ’ ' ( ' 1 ' c u ( 1 ( - 0 1 ' n ' 1 0 1 n 1 z ' - 1 ' ( 1 ' 1 1 ' 1 1 ' 1 3 1 1 0 1 3 ’ n 1 1 1 n 1 1 ' 3 ( ( 0 1 1 c I m I 1 ( N 0 A 0 0 - 1 ' 1 ' 1 ' ( t ' “ n l 0 0 - ' 3 ' 0 ' 1 n 1 1 3 1 ' 1 1 m m ' - . m 1 I ' 1 3 I a A ' 1 ' 0 1 3 1 h x A 3 k ' 1 0 1 n L 1 ' 1 0 c 1 v 1 t ' 3 1 0 ' 1 1 ( 1 1 1 0 1 x 1 1 1 ' U 3 ' 1 ( ( 3 1 n 1 n 1 ' V 1 ( ' 1 3 1 1 ' 0 3 r 1 1 n + 0 1 3 - 1 0 1 1 1 1 3 n n - 1 3 1 0 0 - 1 ' - n 1 1 n 1 n n 1 1 1 1 0 r 1 1 ( 1 1 1 1 1 9 1 1 1 1 - 1 1 1 1 ' 1 0 ' ( 1 ' r 0 3 u 1 1 0 1 x ' ' 0 1 1 ' 1 ‘ 3 1 1 3 1 1 1 1 8 + 1 1 0 1 t ' 1 3 0 0 - ' 3 1 ( 1 A 1 1 1 1 1 1 2 0 1 ' r u u u - - u - 1 1 r ' 1 1 ' ( 1 r r ' ( ( ( 1 0 1 ‘ 1 1 1 1 0 ' ( r 3 1 1 ! 1 1 1 I 1 1 1 . 1 n 1 ‘ V . 1 1 1 ' o 3 1 1 - ' 1 1 3 3 5 0 1 1 1 1 1 1 . 0 1 3 0 V ' 0 0 1 0 1 3 - - 3 1 3 0 3 ' 1 1 1 1 1 1 0 o 1 0 1 1 T 1 “ 1 1 1 1 0 1 1 1 1 1 ’ 1 1 n 1 1 3 0 n - 1 1 1 1 1 1 x 3 1 3 1 1 1 0 ‘ 5 1 1 1 3 - 1 o 1 1 1 0 1 0 0 1 1 1 1 ' 1 1 1 r 1 1 1 1 3 1 ' 1 ' 0 1 1 1 1 3 3 3 1 1 1 1 1 1 3 3 : 1 1 1 . 1 0 - - - 1 1 1 3 1 1 1 0 r 3 1 1 3 1 1 1 3 1 1 1 0 0 5 5 1 1 1 ) 0 1 1 0 5 3 H 1 1 n 3 3 1 n 3 1 3 1 1 1 1 1 3 1 n 3 3 1 3 n 1 3 3 3 1 3 0 ‘ 3 ' — " o 1 : o 1 o 3 S o - — - — — 1 1 — — — n 1 n — 1 — — — — — — — — — — — ) o n 1 1 1 n + ( 3 - 1 1 1 — 0 9 5 1 1 0 9 0 0 ! 5 2 2 4 U H 3 U U U U L L , , ) 5 ) , 2 0 ) ( 1 5 2 R ( 2 V . L S ( 1 A B R , 5 W D ) . R , F 5 H 2 ) , S 1 0 ) R 1 1 . S 1 ( 1 R 2 H L I 1 ( 5 T C R A W L , P , R ) , R U 5 E S C , ) N ( A H R , ) N ( L B D , ) N ( A B D , C ) 2 S , ( 0 I ) E P ) a . V N H , S o 1 1 R L , P , ( ) ) V C E T R s 2 ( 1 3 1 1 0 C R L 5 5 I 2 2 3 , , 1 ( 1 V S 1 ) 1 2 L R B E A H E S ( B V R , ) N ( L V P . ) N ( A V P 1 1 3 1 1 1 1 L T C , 0 1 m , ) 3 1 1 R S S , B , 2 ( U ) n I A A E V F m , 5 L A I ( ) L ) R V 2 1 ) H 2 R ) ) P 2 1 ! T 1 R V ) ) L L ) ( A A 2 H 5 e 0 E ) 3 J , R > A I L D 1 , I P ) 1 3 P L L ) L L V R a T C R , a 1 g ) 1 ) R , 0 1 W 3 R O D L L . I I , P B , A H I L a P ! . 7 5 1 2 . . 1 1 0 1 ” . 1 ) 0 1 1 A 1 3 . 1 1 . 9 M 1 . 6 6 1 1 3 M . , ) 1 I A L L , W , R ) ) ) ) 5 ) 2 . L 1 U O ‘ , , L , V H C ) 1 L P D R I ! D I , R R ' S ) L R “ , R m , ( D H ( 2 L T L L H H R F 1 1 ‘ R R L V R A 2 5 , R R R , ) 1 0 I C 1 C I I C 1 1 , 1 1 R T R R 1 H B R N ! ' 1 I , J I 1 m I R R , m D 0 , , , E H I 0 L R R 3 1 M U R , R R L I ' R , - U 0 1 l ) L 1 J ' C I 1 a , , U R 1 H 1 1 3 U ‘ X / S R F 1 0 E I L I A R L J . 1 ( 1 R 1 m R D U R . . 0 1 1 B , W O U B R R R B B £ 0 | A ' D D H D D 3 0 1 3 3 1 i ) O ' D B ) U U “ / } ( ( X 1 R I 1 5 5 W . 1 0 0 , D ' L ) I 3 N A 1 " 1 0 1 1 0 . ( 1 1 I V / 5 . I ( ' | I R H H ( I ( ( ' R Y U 1 I I D D 1 ( I 4 1 0 A I A B Z T E R R 3 - . 3 1 1 1 3 E ( ’ 3 ( Q Q U L ( H 0 A V D B B A G L D 2 ( 1 m r 1 1 8 ' - - 3 ) 1 1 ) B N / M L J U R I L D . P A ( D D A P P P H 9 0 1 1 1 1 1 e 1 l 1 1 ) 1 3 1 3 N I L P B R A B A C T 1 E A P A A U 1 1 1 1 . 3 I A m 1 J + 1 1 1 0 l 1 N 1 1 fl 5 1 1 ' m E 1 0 0 - R 1 1 ( 1 1 - I 1 ( . 1 1 1 3 ( - a 1 1 1 1 J U 1 m 3 “ 3 3 J 1 1 1 - T 1 1 R U 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 I H A - D R P N ' A - - L 1 1 E D 1 H R T R R - A S - - R M 0 P R R - 1 3 I ? 1 I R U B H A E R B H P R R F 0 A L A - 1 1 1 ( L D D R U S D R U - R I - P C T R A 1 0 1 E L R C S N E A E V V 1 1 R 1 0 0 1 1 L 1 1 1 1 1 ? O 1 1 0 6 1 R $ E M R W R P R I P R I ) I I W D 3 0 M 0 0 0 0 B U C C 0 I A 1 1 0 2 g u A B W S R R 5 1 + « I H — C C 5 0 1 0 0 ) ) ) 2 1 , 2 1 1 - 1 ) 1 . 0 4 1 . 3 . 0 1 . 3 ( 1 1 1 . 0 . 0 1 0 . 1 1 . ) ) 1 T G . ) 0 1 . o 4 , = 2 K 1 1 . 1 u - 1 1 ) ~ 1 1 1 3 1 1 - . 0 ) 1 1 1 2 1 - ) 3 1 , 2 , 1 e ¢ 1 ) - 2 n . T 0 . 3 1 3 . 1 l L 1 3 . 1 . 1 , , I . 1 m 1 J 1 A . 1 ( 1 R J 1 0 1 . . . 1 1 3 1 I 1 1 1 1 - 1 , 0 D 1 1 N 1 3 1 W 1 1 1 , , 1 0 1 ( ) 0 1 1 A 0 1 1 1 . 2 S , B 1 1 1 g > 3 ) . 1 , ) a m . , P 1 x 1 1 1 1 . ¢ 1 1 ) 1 J ) A ) ( 1 , J 1 1 1 1 ) 1 3 ( , , , 3 ) 1 1 D 1 0 . 1 R 1 1 1 y m 1 . 1 - 1 1 m 0 0 ) J 1 1 . 1 1 ) J 1 , 1 , , . o 1 1 1 n . ) - ) 0 ( 1 1 1 . 0 0 1 1 1 0 1 - , J ( ) J 0 ) 2 1 . x J Y J ) . 3 , 3 . + 1 1 1 1 1 . ) ) ” 1 1 ) ) 1 3 S , 3 0 L 1 ( 1 1 3 2 , 4 D , 1 7 1 1 0 - 1 u 1 , 1 w 3 ) ( 1 1 8 0 L 1 . 1 l 1 1 ( 1 1 . 0 1 . . . ) , 1 , 1 N . . Y . . ) ( 1 0 1 3 + I , 1 J I 1 1 ) 1 0 . 1 1 3 3 1 1 R 1 1 1 1 3 . . 0 B 1 1 3 . - 0 , + n 1 1 . 0 n ) x , 0 , o . , 0 . 1 1 . B . 3 ‘ 1 e 3 1 1 1 ( m 0 N ( , I 3 3 1 1 . 1 1 R l 7 1 1 x 1 1 1 1 3 1 1 0 1 . 1 1 n Y 1 m 1 3 + 1 t . 1 1 1 1 ( . 1 1 1 u W 1 “ 1 H 1 1 1 - m 1 1 Y . 1 1 1 w 1 . . 1 1 1 1 1 ¢ 1 1 1 1 - 4 - 1 3 1 . - 1 1 R N 3 1 1 a 3 1 - I R 1 1 1 1 n 1 1 + 1 1 4 1 1 , 1 1 . o 1 \ 3 1 1 m ) 9 l w w 1 1 1 . ( 1 Y - 0 1 D 1 1 ) x - J 0 1 1 1 - 1 1 1 1 1 1 1 1 1 1 1 ) 1 1 ) 1 1 1 1 3 ( - m a 1 1 1 0 1 1 1 1 1 . 1 1 1 1 - - J 3 3 1 ( I 1 1 1 a 1 1 3 1 3 3 1 1 1 1 1 1 5 1 x 0 1 0 1 1 1 3 3 1 1 1 1 ( J - m 1 ( 1 3 3 ( O 1 1 9 P I 0 1 J 1 1 1 : 3 1 1 3 1 1 ) 3 - J 0 1 0 1 1 ( 5 3 1 - 0 0 - 3 9 . 1 1 n n ( 1 ' L 3 B 3 . 1 3 1 0 1 1 3 0 a 1 1 1 1 3 - 1 1 ( 1 ( . 1 1 1 1 1 1 1 . J 1 1 1 1 P 1 1 1 1 P 3 3 3 3 1 D 0 L u u 1 ? 5 5 0 1 I 1 1 I I 0 1 1 1 1 1 1 1 1 1 . 0 0 J 1 m 1 ( m 3 1 5 0 0 0 - S “ 3 m F ” n 1 1 1 l n 1 0 1 1 . . 1 1 3 1 3 3 1 1 3 1 1 1 1 - 1 1 6 3 3 3 3 3 3 1 1 0 J 1 1 1 m 1 0 m 1 0 2 g 1 1 1 — — 1 — — — — — - 1 r w u — m 3 5 6 c c c 1 c C C 0 0 0 0 C 0 0 0 0 2 2 5 4 B s m , l D , H W 2 ) ( , ) , ) , n N , x e o N N ) g N v ( ( o m ) N L I w N , , ( O E ) ) s N s m w s ( P D R N , s s U l , , , , e r ) , R ) m N A H A D ' I J D ) I o , ) ) A N N n N , , O R y z R n 1 2 - U X , r D o A m M e O c I x 2 ( T 1 ’ I m U s m m H m o a A P o Y , G ( ' T m N ' o m 0 m Z . : m P C ) V o A I L 1 c , , , ) ) ) z w N s c ( n n ) N ( A , , , H H , ) ) , R ) 1 ) , N N ) + N N ) ( ( N : : N N M m o O ( R D s ( ( L , T L R B ) , , D S m D ) N , , ( m ) ) ) N R 1 ) n N A ( + N : , M . N N ( O 1 ) A 1 E ( ( v N ) ) ) B D u , ( Y D v , c , ) , ) R K r ) N ) N I S G N I s ( 1 N ( V m F 2 U ) H ) S 1 ( U ) ( D S A A D m n P , + S . , N ) , C V m V ( ) N S E L c M I n c E I A R m ( , M N : , T , n M ‘ m , ) O s ( H m I fl , ) ) N ( Y 9 . n , s r m T G A - o D N N ( ) s , ( , ( Y c C X : D E ' L K N a N x N T S y V s S ( N I , P x s P w P T 1 N G . - m m m , - o 0 H N A ( c I w 0 N A . D 6 ( m d n , ( m u ) x m a ‘ I - . m S D u o L z f D u E T 8 , m s 1 s R 1 E l , m ) , - V L C + / / D A 0 a / N ) ) H / N . n V . E 1 m A , , A u R s 0 m o u . N m . 1 n 0 m M m m S N ( V L r H . a m ( c N o , 1 u u ( . m n m v R u V u S o E A 7 Y R s s s R R D s 8 / M / / / T - T L T N E . 9 E U u N w s G x r 1 s o z - I - s 1 S m c m z - m 1 m 1 m n S W N o m - U N S D U H o ? - L 1 n r L U m ' u m m A H o 1 x m N m o s m C S n : H o I S B - N r 6 0 0 2 g u A 3 3 m c I m m o F H + — a — — — — - — — — — c c , , ) ) . N N 1 ( ( V C B A , ) N , . 1 , R F m , ( , , s ) , ) A 1 ) 0 H x ) N + ) S 1 N N M s X ( ( ( L N , F . H H V c m , m B S ‘ m m ) v R R E , , , N , m x , r ( ) ) v , ) ) A ) 0 N N 1 L c x 1 ( I A m M ' ( + V n L X N s E . V M ~ D 1 ( ‘ x , A , M E x r c ) N V ) , V , p , ( , ) F N , m A N ) ) , ) L ( { P A ) 1 m m 0 ( - s , C + 1 l N V L N ( n ( 0 6 Y , , ( , & s L P M ) m ) ) I M m V N N X C H ) x o c N m A ( ( , ' , 1 r C H ) , E ) + a S R , N N ) m 0 P E ( ) ( , , A N 1 m 1 H ( R ( + m ) L ) H V A m ) u C E D I . S m N , X H n N N ( s m p ) R O F a 1 c X , N s . L , 9 ) n D ( F m ) x . l , , , y 0 D , 9 + D ) 1 . x m . ) ) , ( N ) N N N m ( ) m L ( ( . G y M L M N n s A H N n P n I V i D ( v D d , , D x R N Z ) S A O a , ) m ) 1 a u . n I D L E I F M I 1 R ) x X , m m N ) T F A . 0 m 0 m I 8 D F r r n ) + ) / I a N N ( ( I / X m N N N O ) / u A U V / X A M ( V L I S L n E R 7 R R D R D F a G . . 0 N n 1 r u / 1 8 9 1 / m / c N I m ) - 1 ( I 1 . + ( m m L A 1 u m m m 1 : 6 0 o n g u A z o o s u m c c 5 c I L 1 m 1 - C 6 1 m P r N - o U R n H - T R O H S A R E V + + + + + + + c I r ( 1 m e x . a o . 0 ) T H E N n o 6 2 I - 1 , N m u r a l s - D u ! ) R ' I ' R Y - R ' ( 1 - a x m — o . 4 ' D L ( I ) ) ) I F ( R ‘ I ‘ R Y . L E . m u ! ) M N “ m u r m u r a r s e M I N D - M I N N W H R - W ) ‘ ( l - E X P ( - 0 . 1 5 ' D L ( I ) ) ) , * W U ” E N D I P n - R - R A I N u ) R I S E R A I N u ) - o . o 2 m m 6 1 c o m m u t e : c — - — - — — — n n N n L L c o m ' 1 0 T H E G R O U N D R A I N G - R R E T U R N E N D H U I ) U S U B W U T I N B S W A D ‘ H I C ) P m m m n ( N - J ) R E A L C ( N ) , M N ) , u m ) , V ( N ) , A ( 2 ' N , 2 ' N ) , A B ( 2 ' N , 2 ' N + 1 ) n a n . H ( 2 ' N ) , P ( 2 ' N ) , m u , n c o H C L c o m m n m a s m r , m a y , n o u n o o m o N / A H V / D a m m n , n v n , m u n , c v , n c m , P s c n m n m n s u n n s v n + + + + C D m 1 m ( 1 ) - o . o R m - ( l - c u n / ( n c m ) m n - s m u u . 0 , N ( 1 ) ) - m a x 1 ( 1 . o a — 9 , A n s m 1 ) ) ) 1 r ( A B S ( s u u z - C ( I ) - 1 . 0 ) . L ' r . 1 . o r . — 3 0 ) T H E N D - A H A X M I . 0 3 - 9 , A H I N 1 ( 1 . 0 , 1 - R ( I ) ) ) t u n — ( q 1 ) - m . ( 1 ) + o . 5 - N ( 1 ) ) / n V ( I ) - ( C ( I ) ' R ( I ) ' D L ( I ) + 0 . 5 ) / D u s e u v - 1 - x x - s u u a n 2 U V - S I G N ( 1 . 0 , U V ) / A M A X 1 ( 1 . 0 3 - 9 , A B S ( U V ) ) 0 ( 1 ) - ( 1 + s n u 2 ) - u v w n - u — s m m - u v m m 2 2 6 x 1 ) ) ( ( ( ( 1 I 8 3 ( ( ( ( o o ) ) n n a o u v a 0 ( n 8 1 x 1 ) x 0 0 s 3 a 8 1 1 H a 0 u 1 a w u x ( a ' ' 0 a s l x o ( ( 3 a x s a s ) d x v X 1 1 0 a a n a s u ) ) — u E m n a u u m a 0 0 1 ' ' I r a 8 x 1 H x ( ( ( a a D n 1 x ) a s o o o n J N V a s a s a l — ) ) a N a a a - v O S u a a x a I o a a x a . u x ( L s ’ ‘ s 8 a o I o x a N ( ( a 0 a s a ) s I 1 1 a d v o o / ( ( N ) u a a a 1 G ) ) + + a x u N V u n ( N N N a ( ‘ x ' N O o ’ ’ x O x c ' ( s ( N ( ( ) I x I x ( t a t O a x x a L m S c s 1 4 ) ) a L u ) ) a N n S v 0 n N + L v q 1 ' ' ( V n . ( o ( I N s z ) l z a M ’ w n n s N a r I l I ’ ' a G a H ( ( a V v S m ( m r r N a u N a ' ) a d ) v o ) ) + V a ( ( - O a u a H a S ( a X s ( O O 9 o H u ' ( a x s 3 ) ’ x N ' ‘ ' ' A 1 A d ( o d ( o - a ( a V a ( s E I 1 ) 0 q ( 1 n 9 D A a a ' d ) 1 . I s t u x H M x - u O 5 I N ( ( - a ) ' ( s ) ) a n n x a 9 N 1 I d O s N a + O a O s a m n / u . 1 s S 5 L d + ’ d u N — ' ‘ o a n ' ' N 1 z 5 N ) o ' o ' . u m ) m N l 1 d . 0 0 x w ) . - ( ( ( s O J d ( ) u s ’ c ' ' x ( s I I + o ( ( 0 d o N a v H e n n N ' . ’ 1 H ( ) o o s 1 ) ) ( 0 ( ( x I 0 x S l O o s E m K m o 1 ‘ N . A a ‘ ’ 1 o ) L u a fl m ’ O m 1 q x ) x ) H m n 1 n s 5 s I z 1 ' o 0 x 1 s ) X v ( 0 5 ) 1 s a a ) n 1 ’ a L s s L a a a ' M s P I x - m ‘ t I G - v v a m a ’ n 0 H S ’ ' H x r + n 0 ' u ) ) u x s 9 a x V ‘ u u s ( + n T 1 o ’ x ' o o x + x x o a a s ( - - s T O w 1 S s S ) ) I s x u x 1 ) ( - a a a s I ( 3 1 a H N n n a a u r a ' ) l a ’ ‘ N V ’ u n ' m ‘ I 0 H - V w s ) a ( x H a s u x q a s ' n I + + ( ( o M - a t O d N a s P n v q s u a ) - x - N ' O a - - - I ) ) - a s o - o G N 1 m N z O 1 N I M a n n u n ( T s s u a a n u a I I s r x a O ) ' L z - ' O ( v - a x M l s - I m I x ( ‘ 5 a a a u 1 ' n n 0 - m ' I 1 I m O ( + O / 1 1 1 s ( N d r O 1 m u s s a O - a O 1 s s x I 1 1 I N n m t I ’ ) - t S d I H 1 1 t ’ 1 x n I 1 a G ( a - i u C l 5 ) J 1 1 m ) u x d r 1 u T q m t ) t - n v a 8 m 1 1 1 t ) m a 1 1 u H o t n - - 1 v v a q E J n a x m s s m a m a a s v I n a o u o v a v o o l v 1 v a x n o — — v o a ) ) u x s V n 1 ) 1 : a n a 1 1 l u H I v ) s P X s N : : s N u 1 T s n o : v : n o a N : s ; a s n v - - - a - - ' 9 0 0 8 5 . - s — - — — — — - — - — — a s a a — - — — - a s a n — — — — — — — — — - — — — + — + — — - — - — — — — — - — — — — — - — - — - “ — — I a — — — — - ? I t a a o o 3 a a 3 3 m L s a / a ( + ( ( N O I L N I G V H " w a s 1 0 3 8 8 ( X ( ( S x x - s s ) - - d ) ) x d a a x m a a a ( u . ( ( ( X ( N O L L H I G V I I N S ( ) ( 2 t A N 5 + u ) 0 x ( A 1 ) X + 3 1 n ) ( 1 0 . M N 0 ( P ) ( ( ( ( N N 1 1 1 1 0 0 n . ( ( N N ) 0 I ( n 0 ) D 1 + I . 1 3 ) x + s ( ( — a d ( z ) n X ( C O ) X s n ) u a n x ( + N ( d d d 3 p A ) ) ( r r 1 u 3 ( N x ( ( N S C ( - . 1 r ' ’ / fl a ' t t ( . 1 + + ( I u 9 ( 5 S 8 O ) I N O W H I n + ) L ( H S L O . X n L ( ) 1 1 r + ( t D / ( x ’ v ) ) x c ) ( S I x S - V N ( ) e H ) r . u u ) ’ n ' ' x V ( H ( I I x H s I a ) ( / / u I 0 o - N G O G ) ) s . O 1 T a ) 0 V H V — v x t I — ) ' ' ( ( ' ' v ( ) u u u ) C N ) t / t - ) - - v U S c ) X . 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J 0 a 1 c O ’ ’ 9 - - ) ) ) ) ) ) ‘ O L ' o ' 1 - X a C a q a n 1 I 0 O L 9 v t L V ) a ) ) J a m n N A N t I 1 v v [ O - 1 o ) c ) n 1 O q 1 n v v a o K I ) - 0 t x v v V v O r v v S : 0 6 0 d J ' A x p ; [ t 3 9 0 0 8 fi n v I t L 3 6 I - O H x n S v V u - ) 1 1 8 O N 0 1 n O N n O ) 1 ) o o o O n a o - x X a v 0 o 0 0 O o a H V ! n s X s n v o r ) S v 5 : n 5 0 O o — — — — — — — — 2 2 7 ( N 1 T S - 0 M S N I a u n o a s 3 H 1 N 0 » N O L m v 1 0 v u a u u n n n o a J E N - — — — — — - 3 J I O N B l n N I J N O O C I - a I a N a ( 1 ) 1 5 - ( t - 1 ) q s - ( 1 ) N 5 N u a s u a $ 0 0 1 0 0 0 “ M R V S R , M R , R Y K S R A E L C I E N L N O I T C N U F ‘ I W I ' A ' I ' S N O I T A I D m m A m R R A r ' s a 1 1 : 6 0 0 2 g u A 7 2 e g a P n a r g R , G S I B L / / m A N L o J 0 3 S . 1 P / 9 0 I c A O I B 0 . , V T ) , , / L M C R A Y A I K T L e A 0 6 2 . 1 0 m T A P , A ) V 4 0 1 * p 2 2 D S M 5 . . 2 A l , E R I R L W Y O , , , " ‘ " A C I D D ' J 3 A , 8 ' N 0 8 9 . 1 9 m 5 . . ) ) ) T T T L L L W S U A , I T L I A S A T T R , E U S , I , T R A N R O A A 3 0 w 0 A O O A D E N . 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W " A u s e v - A N A x u c , o o s a z / X ) a r - e x m - I / r ) U U U U U 0 0 0 U 2 2 9 Y P m A C Y E P H O T N A N C I H S R E Y A ) L D ) N H T A U O W G S N E O Y H I D T Y I ( U E , ) U M E T T ) , H I ) T W 0 S N E ( D A D G M I E N D B I C A I H ) I ( L O V U L R G C N I O E E T : B I A A F ) E , ) S A I L B E H A A R T I M I U 0 ( Q e N N R N W E O O E O ( L ” N D l I I H T T T : , R A A ) A E I 1 I ) I ' I T A I N D ( D D D I A % A A 2 A L R R 7 ) ( A , A I R R R ) E A O I D I T A ) I D U A R L A M R E , ” , L E , L m D H T D , B , 2 3 e g a P ) ) ) ) N ) N ( ( A G L L ) ( N A A 1 T 1 ) I H M ( l - W ( E V R R L ) N m T L E D E L E I T E ) ( ) L L ) K L D D N O H ' H M - ' E I S T T U ) ) 2 N E ) ( S ( ( ( K L - D D L ’ ) 2 ) S 1 ( m ‘ L - A I ) ‘ I ( . y x d f ( E ) ) A M A 3 ( 0 L - 1 - ' - O 7 M I , U A ) U ( D + ) ) ) ( ) 1 ( D I + P ( B A ( ( 1 B ( - L E p P m ) ' - » ' - ( I x 1 ) I U U ) S L I T E R , I I c E A T f A - L A L ( - D D X L ) m N L P D N M l 2 ' ) a , E ( A ) A A - A E ( J I E D ) G o X ( a ' , I 1 - L M L 3 L n D O : * ' ( ( - ) ' ) U M O 1 P + ) ( A ( ) I E + R s O O M ) R fl ) 1 N I A L - J L I S - D - . 1 2 - J - - L N A B R I D T I 0 M 7 P E , ) , , P S O o fl ' ( ( + . U ) ) U ) , , , E E N N 8 - ' J 1 1 E - U a - 2 B E A 0 T J L 1 1 1 1 0 S I - - W E W A ) m 9 ) m ) l , fl + + U ) A U B . a ) A - + , , - + - ( - ) ) - 1 l L - I I I N L G N - W 0 - “ I I 1 I L I L I L I I 0 L N N E I T I R I E N U . ’ L L F E 1 l I ( ( ( ¢ I l ( ( , ( I . , , I ( ) ) F L R S R D R N M E ( ' 1 + l - 1 5 - B B 1 1 m L ‘ J K A A ‘ N U B D 3 T - L L L L 6 T 0 L T 0 I 2 U L N N F A E A T n N U N A A A N L A ( L ( ( M R ( ( L E M S E E E C I E E T I : F B O E O P B B A O U A N n A B U E E W a L A A A 6 m I A A D C C D O 0 0 2 g u A , , L L , ) ) ) N N E E a C S m ( R R W D T E m S ) M K S E I , ( N ) V ( 0 A B 1 H R ( C S E S A , K T . 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C ‘ ) D T N G A C . E ' P L R P E T H G I E H S S E N I Y ‘ P G U O M A R C ) E M H A T T N E E R ) T W F A H Y G A , . I A 1 T . ) U £ L E R D ( Y ) I I Z A 1 H C A N O E D N J , ) 2 ' I 2 E D A 1 N 5 / / L , O " E - / m Y H 1 K I N C I M . Z 1 D P I S T A A m I 6 N E 1 O l " T ( D P J N , O M N ) 3 E H 0 F X ) 5 D C D , A , 1 H A T O ( U m ) 1 I N H Z E ¢ 1 e T M ( . ( L O O ( H D 6 F g F 9 Y , O m I 3 3 I ( c , O I O N A a P ) R O + X A P A Y A D N I I M N Y N M A E — M ' Y C A I T C H Z m 8 T 5 C D . D G s 5 H ' 0 A E 8 1 T fl 2 ) . R I ‘ m , D £ ' N P E A 1 , D D ! L G G A , 0 O ) T . H D 1 P A P O + r N I D - - N S ( C . y T A £ 3 ' R 5 4 0 E B Z I , ) C C . ) I - A I ) I F I I z 2 F O O 6 E I ' A ) L F H C z Z . 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D 5 ) o 0 N 0 - E ) 5 J . t ' I 0 0 5 9 7 E M I 1 ( ‘ I ) I m S 1 ‘ o 1 . N l A P S 0 5 5 3 6 H 9 , ( I T U B - 0 P ( N T R 3 , W I 1 ' 1 G I I a 1 B I 4 6 9 1 6 T Y 0 ' A / I ) P C 1 8 2 6 0 9 1 ‘ A G E . . . 0 1 . . S 0 B N m . S x S N / . R E 1 X F T A . E 0 ( I - Y 0 / A N . A m ) M 1 R - 0 0 0 — . , ( O E P o - - r c I N . I . - ( ( E 0 P / 0 1 ' E m [ [ E D m A E C / O L R E B s S I W 9 1 : T D A A N N I a I N - J N I E x A - . S I U W I T E C I u m n + - R U N P I O N B ( B S E D s A W ( I E D R ! G / - 0 N 5 M . N a P - A P B R A Y ( - B ‘ O Y 1 E A A B B 2 U - A n E L N E o A ? N 1 U E E A A P I 1 V T T ' - X T O V ( E E S E E 0 T T V T T I I G o I u M U I m A : A A A D A D E A I 1 E P N A A A o I A T e 6 S A E R D n D D W G m I W I m S E E R S W D D I D D I 0 0 2 g u A N D E W 1 E S W ( 3 / R L , O U U U U U U U U C 9 3 5 8 5 8 3 1 2 3 2 S A u N A ’ s S u A N O I L I S a ' ( I I ) L a n m ’ r s s ' u N ' m N o o m m m N m D A m W 3 8 m 0 - 1 5 A l W . 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' m n q A u / a - I t u u A / o o l o m a d E o o o l 0 2 G t s 2 u a U s . d 0 S I m o v m ' E ' c S / O ( ' 3 - I m ( z U m Z m a a s s 0 S M 9 Z ' m u a z l . 8 4 0 I ( ( D - a - S m m ' ' ' ' ' m m T m i E 2 ' l m / / / / / / o w o I h 1 - I P z ( C z a : 6 m — o - O n 0 G O O 2 t ( ' 0 0 1 a s m P I m ! - 3 - U T 8 2 8 0 ) - - ( ( - ( - ( t ( O u ( ( 4 4 - - - m o n - ' a I “ 0 2 4 2 ) n 5 9 s t ( S C I C S ( ( - S ( m n C s u a S m 0 a u - T - m - m : ( m o ( ( a a a - 3 4 ) z - a - E H u m m m A O n - - - 1 - - r o - T ) m M N R R m m ’ s o R ’ m ) 1 ’ ” ’ ) H a 3 o b N m 6 - - - d - 4 “ 8 5 0 I I O 0 ! 0 ! 2 G 0 Z z 9 0 0 : 6 m : o m ! v v W m m m N m w w v N m o u V S ' m m m a a a m 1 m . m H m m Y 3 a n s 0 9 2 3 3 U U U U ( 1 N 3 N I K K I - 3 8 3 U U U U ( N E W S R T O H T E ) R C G C A - I A L M F S F E H T . R . N N E I R O E I G E S R D T I L . 0 A 9 T W ( N O I M ) ) I C C N Y R I ( ( E A “ I T M C I T T C F A P C C I O P P E E F L O + R J J P , S ) ) E E C ) E S 3 V D , 6 U R ( O E B B ) ) O U U C H ) O S S C ( ( ( ( T , L E T 2 N z 5 A T R E N R B E ( V I * E N ' E T O E G E , N A L D A I E P L S S 3 M O N P A I F ) E A , . R N R T I ( ) N P U W m N R A T ' 0 l ) S R " € ) I A A C E A 6 U , O 5 , T P R I B R E E E E N ) + 5 ) e I E C , - g R P , / N T ) ) ) 2 z ( z E T D H ( N C X ( Q T L S ) ( D E F D / T T T T a , E T , 3 4 t C 2 I m A F R r P P L S 2 N , , ) l 2 A u E I E O o O O I G F S H H H H I ‘ . y x d , 2 R U , P T l a ' / D ‘ B R 2 I R H Y T I I 5 ' z T E , , E X ( ( ( ) E E ' E E S V 1 ) ' F N B B C R R R I N 2 5 A A o B O I U U U R I ( ( * - + I I " L T T T E + ( C N ) ) € D ' ? 2 R A C 2 ) ) B ) ) 1 4 T A r I + H c A A A I R R R C I ( I I A ) ( E A 1 ) I H E , 0 ( ( ( ' , 6 2 T E V E E E A T V ) 5 . I 1 5 1 5 , z - V W P P P P R N 5 , 1 1 z z z ( I t , T N R I - fl M M M A F I ( 1 - - - - 1 S ( ) A O I E E E I W ( E ( 1 R f E X 3 7 A O ) ) ) ) S B 4 E O T T T T G 1 - - 2 ) * W A 8 / 9 1 2 I I 1 2 4 5 E U A E ( H W - 6 ) . I M , , E n , , A - U C - - - - - - - m I ( 1 m E U N L I B B B B B L 2 ( T U ( ' - N " m N I I I I I G ) N N R T ( ( ( ( I I R ( C L I 1 V T A A ” A A A A L N T C : A E E ) ) N N A D E O R G I ! ! 0 A I T 6 m S R R ) I R m C I ( I 0 M E T ) l l I o m 0 2 g u A E W E m N I + W 1 c c 2 ) R C P T U ) 1 0 E L B P T D 0 . o A - A R ‘ I S W > L T S Y L B E I B A T T A ( N O S R E V N I < U I E N N O N I I m U T a C Y E T I D N I S L ( W L C R E E L N V ( I + ) 3 O ) T C ) A ) R E C E B E L E A T ' S 2 R , 2 U A E A R D P ' M P + N 0 M ) R U N D R ) ) P B ) , 2 3 ( I ‘ O + 2 ) N ) " ) S I 2 ) ) E ) H H ( C - V I T 5 P ) ) V + C , . ( 1 A 0 R E D I U 5 , ( ) 1 H 5 R , 1 i ) 0 N A / ) ) 2 " ) C B ( ( / 0 P , V A 9 ( E 1 I U C R o V S ( . ) I A A 3 N - A Z - C C 0 N ) . ( ) S C P T R R ) , o 1 ) E T T C - e H D ' U N 0 ' I ' ' I 0 N R ) ) N . A . , P 0 T 5 m P 5 S P I P 5 D E O , ( 2 I S U C o C . ( , , z g A o T ! R L M T 0 D R , * 1 L B H a P ) H U W P 4 L C I L m I I N C ) N ‘ A . A I B ) V E U ' / R L N A I , I o I N 2 I N ( ' , ( - , A ( A Q ) 1 T T L Z - / Z 0 R P ' ' 2 H / D ' J U G 0 . ) ’ T 3 L " Z Z I Y R T K B ( 0 T B o D e . T C ' P E Z / O U + ( l z N K I M D V A M F z R I A O I + , + R ) h R N R , A P A T ) c 9 R T , , . E A ' , , , 5 O T H D H I ( , 1 L 4 K M - D H B ( V 0 T , T , M B ( r ( E T H A D B A . I , C P N C P ( ( / M D O C H 0 ‘ 0 C 0 T A , D T 0 / 1 H I I E I C N ) d ’ T Z 2 C T / - I I T fl B E . m A ) I A / . N . : S ( 1 / T H R R fl B N + E R U H 1 O ( ' I ‘ ' I I 1 H x P V - P d Z m O $ C . - fl m I R ) R ( N t ( P ) I ( / R P G U H . 0 . B ' ’ ( 0 I . . ? l 2 T 2 ' I R I I I L z ' H - C P T R Z A ' D I - 5 A A 7 U . 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( 1 B I H N I I C C . 0 X H ' ( 1 E V N D R I T / T T I X E - I V B A N A I A - O H N R v E ' T , H I L / / Z S / / 0 I l - O A A - n M H - - P A N N N m R C U P ( I I I I I I H 1 M - ” C B L R 5 R D L F R R 1 I - V V P H P R L O O 1 P T E F 1 N E A I A U - S I V A I S c A E A C I M I - L m N N 1 T U 3 S I E S “ C I ( ( : L L D T - R R S P P H T P R R D I C A H H o N S c L A E I E O O W 6 E 0 E E n U E E R R R ) c W O O I I 0 + 2 g u A C C c c C c C M E E M S U N O I T A R U , , 5 A A ( ( H R R / E S T 2 B B T N / ‘ A Z U U U U U J - 0 L - T R I A L R E P E A T I C O N T I N U E T R I A L - L « 1 + 1 C A L L I N V E X C ( z s , A H , P 1 , o 1 , R 1 , T R I A L , I N I H , 0 4 ) C A L L I N V E X C ( z s , A H , P 2 , o z , R 2 , T R I A L , I N I H , C H ) W ' I N ’ I H ' U R 2 / ( 9 . 8 1 - 1 7 1 1 » I N T a n ) I P u A B s u - T ' R I A L / L n . c T . E P S . A N D . J . L T . 6 ) ( m o 1 C — M I L ( A B S ( I - T ' R I A L / L ) . L E . E P S ) c U U U U U U U U U U U U U U U U U U > B § § § n 2 3 4 N O I T e l I 1 D w o n N P @ O U ” C G ( T N M N R E I C A N A I L U B I C A F F E M A C R L A U T F R U . S D - I ) “ F R U S D I ( D X K " W ( R ' ' K D 2 R X ' 3 O 3 O F 2 . w T ( 5 O / 3 o 2 P W 2 " 4 U 3 n 3 " m . R . ) X e N ; P M 0 ) ' g O 0 G K I H ) ) a I ' C 6 U " I L " I P T 8 R C K ' B A R 0 A I m 1 X S t F . P F N S ’ P ‘ D . 8 R O ) O - y G . E P L 0 I R ( G r T A G . G C T S . T 0 T d L ' F ' . G ( 0 A L C t . F A U S 0 l R R C 2 D B R “ . O 5 A P U I N . A 1 U . - S S R S G E 0 . F I N O ‘ V § T ' I 1 R 1 ( ’ 7 ( W W S N F K E 8 L 8 F I . L E m V . E M ( m m 9 I l S ' F D E E H F 1 9 R § 1 I L R I - U ' I ( E B D P R ( 1 — ' N E S F 1 B S W ( I : L — — 6 0 0 2 g u A — — E - — — - — — - - C C C C B T A , ) P C P C E E P I N P 3 P O E E I N 3 T O N 3 N 3 I I O L l O D . T I P T O D C u / I N I T 0 r " 8 C C N E E s S E m 5 E O V R - fl ' I u 9 N H , I N ' O 5 O R C E 3 ) A 6 ) ) I C A I 3 0 2 P ‘ V C N ) V 1 9 U N L O S ) B I u 1 r ( . ) I A B . 3 0 / H 3 0 N R R C U I H - S W ' ‘ , 9 C ' . 5 A U ‘ 8 N s 4 E I R I L T m O L I H . ) S S J I 5 0 O 0 - A ) U P T L 2 . “ S ) ) I T m ( 3 0 5 S S S O L A I ( B S S R , 6 C B B S R L I R ) P 6 R P ' 4 8 5 O I R N L L I , 8 , . S 3 3 ) 2 “ I A P A O I N N I ) m ) . 9 T 0 0 A I ' O O ( 1 E 2 " E I I S F V P 1 0 , R T N S S O F R S 0 ( . A E N N O ) I ‘ 0 2 . ) 0 E E O L Y E E M T E Y I M H u . I 0 5 R 3 3 . O 1 P " V S 0 ) ) L U 8 ( 4 S 1 I ’ D I I M D D I ( V E I , I 0 1 R 1 — G ' , 8 1 ) E L 0 ' P ‘ I ‘ ( N / N O 3 e 5 8 O " E I 8 ( ( ' S , P g ' 1 D 0 1 P I . I S L I . I 1 I I U m ) C E 1 ‘ , ' X H N T ' a G R 1 . N X , T I E N I , ' P m E ) E S H O N ) 1 C S ) R J ( I S X E / , 6 T I Q 0 0 R , S A A I F 1 B 4 V 9 ) L s . N H " ) O E T I N U I F N O V N P B - ' 1 H R ( 1 E B T 1 I U R ) P N T H m ) y S L D G 8 L ( m U I . P . T H N C B T R ) , R / 1 ’ ) O I , n W M R ( C I M H T T B S E ' C 8 n I 6 3 ) L E 0 L X ( I . ‘ ' 1 O I ) A N R T S 5 8 , , 1 5 1 - R H V L ' T . ) “ I T A I T 7 L C - F 2 0 T T ‘ ) E A 7 I r T N 0 L v 7 E ( ( C 1 D , I D , 2 F ' A P . M . z S S S S 0 3 C T < ) A H I 1 / 0 / I T , + ) I I E . N 0 T O R S $ w O x 7 . C P . I C P 2 A O I M S T L o 1 m 0 & a T S / L L 0 1 R 0 z x 1 ' T A S . ) . ' s I N E R Y A A / . T T . . I - . R L O L . . S I I I o O 1 o T . ' M B R R . m B I R , , ' 5 T 0 R N . T G O A 0 E 1 m G K R F P A L w ( P P . L C 8 O - L V R U P I , R L 1 ( S ( o L 1 8 [ / T T C P . ' . H E N R . U T 9 ’ ' ' ‘ " - " , R ( ( C E P L P R O L U U R T ( 1 9 ( 1 I I ( T ' 5 G . x L ' a S C R F 1 8 ( N m . - A n n A O ' 1 R N ( G 2 Z x . R I E C T T T G U E A O R P S 2 S - m . 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( / J ) ' I a S U S I u ” I m e U / - O T £ u ’ I ' a m w E w T U m 9 s X ' 1 8 ( I A o 0 d S M ) - ) a m x m E E t 1 I N ) C a 3 o I L V E ' ' 1 ' S ! v 0 m " m 3 / ( 0 S m o u ) V ( ( C ' N o u M ‘ c 0 I I m t m - T ' a C C m N H 1 a 1 T I a o H - d a - ' a 0 0 . . a v - - - - a - A m E ( E 1 N a S ( u O - N I - s - d ( ' ' S L - ' a ( - ( ( . ) I ( u U a m V ) 1 V l ' n I O 1 a I ) m ( M I ' I ( / ' m m M o I N u ) d C o : 1 ) I 1 I I s T I ‘ - C a I I - a ( a S I m I ) ’ ( a S a m N - ( I ( I - N m C - s ) s - 1 L W I - S n m l / / ' a C O I I u . T J x S a I + d I ) a m n m V E u I k 1 a V ) ) ( - d ( ) v - E I I m v m T T I ' I m - 1 V m ( I m m N C J O - o C V - a - I N s m m u 1 a m a m O V N o O a d s a E O T I a m m N v m m I N E V I I - U I a n I m m H L r d J m M I N I — N n u S a a ) m 1 1 I T I C a I x I 1 T u a N E u 1 C m A I m a T U . m E M m - - N P I m s C a S a L u T ) m d s A a I s a ) ) ( V ) ' R a ' a n ) ) I I ’ C V o 9 O C n s 0 0 a a — — 8 — + 5 0 V — I : t 1 ! - — — — — a — a + N 1 1 — — — — o — n o : a : a a a 1 1 2 3 9 ( C ' C ) V U C ( ( O — 1 d X 3 - I 1 - m ( ( W ( I " Z ) V + ( P ' I ) V h D H ' O ' O S H ' N I m - O ( I ‘ c w u u a - a n — U - I m ( ( ‘ L L U ‘ I ' Z ’ V + ( I ' I ’ V 1 3 m ' 0 ' 0 5 1 I N n fl - 3 U U U U U U U U U U U U U U U S E I I S m ' ) 1 ) 3 0 0 7 7 0 0 7 7 . . 0 0 - , , D I ( A a B ) s r w ) H / Q T E N E S ( w R s ) W E R B F ‘ I S ' N I I , m ‘ 4 A A R H c m T Y , ” E R W T n a m I fl i S a I I U , i I , ) S ' E ) I m ) S n I I ( R m v m m V [ n 3 c I c o m r z o T G p A u K , L W / 2 E R R R U E E c ) , o T T ) I ( P A A R O W H 1 H C M P . / D H r n N & G w D u L N c , T , A m . o m D / s O I K , X T ( ‘ m m u 0 m , c C H , 2 T 3 x I I M 1 C E o ) ) A A A 0 ) o I m m . 5 A ( ( ( 1 G v m H N ) u v M 1 a , 1 E U m e H A L B v 0 ) m ) n T F 0 P ( 1 c g W B m A ‘ P , U 1 n m + a 0 A D . S 1 E A m S ) u ” 5 1 D , 1 m P ‘ T n , n 0 N u m H ? s R ” . 1 N I , P I : S ) o R n ) T N ( A n . w 3 o o A a S , 0 I J I I 0 M y a / P ) / E E / B m o s m S s W N ( ( 1 m m S H ' C w m ' n ) . a u r ” , W m m ' r T n C A o I ) E n n B u ' / m B ' v , ) I I n F H a T A a n S N d m u J . m ( R R y s u ) n M - I f r m e ) I I m E a S , o I d 7 8 9 t A ( 1 n A D ” . £ a D I F I N m u - x w U h a V m I A ‘ B s u m u 1 W . s - a - S s 0 a D I C R c , O O , ” ( a S H I B ( I . H v n ) P a u - P u L B W X " ' U M ” R . 1 / w . A a - - A I - E - - E R W n s R T U 0 . O m £ M 2 ' U P F n m 3 ( : - A ' Y L n L - r I c I . - W 1 m 5 I A s E C R m I 1 2 ( . p ) S A l 2 3 m N - L m ) m w m u D M S a s . 0 C Y D W ( M A a I a I R . " C C w C ? ( n a y e u s D 1 I ? m H ? r m 2 B m ( I ' 1 m e a V m 1 ? J - E - ' - A W r 1 I A B S B A ( S ( m 5 m n u w V n I . m a I m a n m R U L E E H ' ‘ r V I T ” C A ( S V W S W I : L E E W W 8 P W P a w m B a o A L P D B 1 6 M W 1 W W E R m 3 a I 1 s R s 0 0 2 g u A — w m — — z v o — c c R T D I - — R I c m E S E R N E M M m C fl — U c — C , , ) ) , m N ) v ( N s C , ) n M A , N F ) , m , M , , ) N S ( P R I H , 1 ) x S H J ) N , S S ) + I N N S , r A M M m ( ) ( , P X L I E , € C E S - W ‘ M X M H A , I J ‘ P ) , S ) D ) , L ) N V I m m I 1 S P V B R N D B , R , , P m U I S , , 4 R L ) H . J N C A I H W ( T , ( c X Y N w P m - ) N ( P A u + € M L N , D u N A ) v D V , A R , J 1 M ( , F , I P } 3 F S C D , B ) ) N M P ) I 1 ) , D ) V L W 1 ( 2 V { n A U ) D T 1 N D , C X m W C ) ( ( E R ( I S ( L F P L T P w ) , E , , C 1 J 5 ) S I C E P S L S ) 1 ) ) W m , M I m , X ( ( , V N I 1 I R N C + m B N , , ) Y D L P J U ‘ F T N I a N C ( , C I R 9 A ) ) M x R I S H M D D ) ( L B D ( B D ) ) ) ) . m n ) " 4 ) 5 , 2 ( A + A + , u H L ( A m J M , 1 ) N ( , - U , , P M ) ) 1 O , r A J 2 r M H C F R ) m R 1 ( N O E , ( , M S R ‘ , N D s , M ) 1 T P C P O n m M ) s W n P L , I L ( C K 1 ) ( ) H R n R x Z V , , H ( I ' I H D N X ) N 1 , C , N , ( S ( ) I ) M I ) S ) 1 P + m ) ) u , C , C 0 u - o l . H . m m ( 0 o X a s P 5 B m m : ( ( X - 1 B 1 - - l 1 N - N ( n n 8 I m I - 1 5 ) ) 4 I H L N , X C C 1 + L ‘ I ) ) U m “ m a N I O 5 5 , , ( + F m O , w U S X ) I N , I W I I , [ E m e . A I S A “ , I . a , S h N O N s L I , , M 2 ( H M 1 ‘ ) ) n n g o 1 J ( a M ( A 3 A M ( . ) C ( M c L ) ( Y S K O O Y A H l D : , , A ) N P D R R s s P I ) , , m 7 N I P E r r l W X S , , P I A , , m ) J S D P P A ( C L S ) , F D , ) D m ( " - fl - ) c ) o ) I ) H . , N A ) w s s , M ) 6 , J ) ) X 1 U , , T e N N 1 N I T O 0 N F X L 5 : c , S ( ( ) Z S . n ) R 1 E N . l x n u 1 m M ( , A m , 6 n p A w ' ' 6 - ' I D H U I R 1 H N m 2 m , V S A u c n , ) ( I L ‘ I H fl N Y , l 1 ( 1 K . m I J W . ( C I 5 D C ( 6 - Z m P ) , N I . A / K p , O H , , , / 1 X 9 V P ) u N m ) n r . S / 1 P ) + S ) D m m F L , ' 0 I N J C ) n N ( a n o I H N m - N P I 1 A m r s V m , m m 0 E I E A . ( M fl o u I S . ( / / / 6 B v R . M s m N S ( N / N E C L m ( E . R R P A T R I A V I s A l S m m ) I m , R m " ) 1 C Y M - s W ) ) W m I W m , w R R n D R P A ) ” m P ' I / A 1 W E M M [ [ M [ [ fl M M m . I - E E H v P W n m / 9 m m A a i I , T l u 1 m - - N R E W ( ! u " u n n n n O I o m c c n m u m W R m N ‘ m E M I o O o o o ? a S A A H o o a S S H e l 5 0 0 2 g u A m I S U R A m - m M m s n - - n m V R - t m . C o N A E O a O o a a a o A A I i v o m 3 - F ' A L m n E m V m - - . e m W D A R B N R U 8 S T P R m c C m m c c c m o c S m m M H " u s D m p P + + + + ¢ u D m x m q W P - p I p p m A ( C 0 U U U U C C C C U U U 2 4 0 N O I T A N I M I L E . E U R T . - E C A P S D N S A ) T U S ) , ) E O U P ) ) R E A E G C C ) ) ) I L R A , ) A R P Y B R A S B D E ( I A S I N N R S ) ( L K A ) , ( V I 1 B R 0 + A , . R , ) 0 N I A W E E I N A E O R ( . ) C S C E A R ) I A Y N A T E ( T A . A C I S C I P E S ( P A P I A 1 E E , . S B Y A I V B S L S H U 0 ) , D Y T Q R L T R R ) ) T T R E . ) ) ( 5 I I V ( V S R / ) ) ) / ) N N I I I 4 ( R ( S I I ? A E S 1 1 I ( ) ( A A S R A E 0 I ( ( L N + ( V N L L N A V F L O E P ) A E P E E ( ) B ( A A O I I E B V ? N D 0 E O ) E D E P D O N D O C I L V N D O R ) 1 ) I / R I D I / ( N - E ) ) A ( E ( ' H H / O ) I I T ( . ' M ) R V R A ) I 2 + M E T 1 R E L ) O A I O S R / L N . R O + R ) S E ) - , B ( ‘ P ) U P R U I ) H ( C + I N N P I H I L ) 0 D X 5 I R ¢ ) I ( H A / T I ( A N T O N + A ( T ) E I L + V A 1 ( A ” U . A e V ) + S R A V L R N H ) V V ( A ( ) L A 0 . R . W E M E 1 T J ( S R D - ( - g R I T N I R E H E ) n T ) I S V E ( + E D A J / ) I . ) a + ( E P + R ) T N R P A G . ) 1 R T 1 ( I L + S m ( i I P ) A A I A R ( ( ( S T / M ) L E + M A B N / S E L R R R ) I ) E V V I N L l I L E A ) E I ( I E H D I C Y D . C E P I ( A f ( M N H I T L ( C m T + ' V V + S S J O S S ( ( ) H . C O V ) . V + ( I ) ) ¢ A M S C S r ) C E A P J T R a - R R A V N A R y S m ) ) ) t R ‘ 2 2 I d / : ( ( I ( R L R A F A R R V R ! J J E B / / A a S ( D . . ( A ' 2 ( D , R O V O O ( ( V A N F / S / I 1 E ) R 1 I N R A A I 0 0 ~ P R S P 1 R R C N — m E A A L , N P R + - - - / A - V R R , V f R A 1 A - A A V O O R / X . I . I E £ ) R C ) A I N ) ) ) A S - - - - 1 C H H ( R S 0 N M R S I A 1 l V N ) A A A . 1 V E R a R N R , 2 S H , A R I S O P E - - - - 1 - K L L R E U - - - ) E ) E - V B I K R x ) ) ) ) . 2 2 R 7 ) ) ) I l + 1 , , , , U ) I , , , - - T V A I I ) I U J P 1 O ) R S L J R - E H - W 2 I + - - - - I - - S I A R ( A I - + - ) ) + A m T N W A . C 8 L 2 L - I - I K R + M R x N K L L C - N - 1 N E I ( ( E B 4 x x R K R x ( O n ' 9 2 ( ( , , , - 2 I - x I ( ( , , , z M A z I 1 I N R i I ( D I D , , , ( ¢ ( u P R - : 2 2 2 ) ) ) ) - - H . 1 E E 1 2 2 N - I 4 - B B E - E E T 2 - + + A K m R L u - E V R x N I B T ( . H 3 S R V V S R ( ( ( - R J ( L A A A A A A x A A N L ( ( ( L x [ P K D N N P R ( S P L L ( ( u ( 1 B B B I B O - B - A o B E - E E A A A N m C A x A A K A m A J C n 1 A : 6 0 0 2 g u O ( - R - L L I ( R C A A - - - C c - L I 1 2 3 V a n C C M R R R v P I A 4 C C c C C C C c C C 2 4 ' , , , ) ) ) , N N ( N ) ( V ( N , B , C ) A S S R A , ( N , ) P ( A , E m V N R , ) , s H ) 1 N ) S N + x ) ( X N ( R R V , S I S A S C X N R V I E S N H F r ( A ( ( B , W . , L R H ) , T m , ) , S R C , R R X E T ) A V R ( E c R H N , E P A ) , , S N D L ( S C , N M E m L N ( 1 V X , E c N R I A ( ) N + O R V . E R , ( R P W N ) R a ( , R ) V S N ) N ( N , A ) I m E . N S L X , ‘ I m , ) A + R , . ( ' R E O ) . u ) N - A . C P , O A C ’ C A P S R I X I D N , ( ) ) A N ( A I z 1 N G S ( S P N N A U , V s m C , Z A N P C ( C . A , . C N , C ) P R N L ( I P R . , , ( P V ) P S H T A N ! I ( ) ) 8 C N , R R I A R I x C ) P M 9 V ¢ , N R N m ~ . ) m , C R , ( ) N I H C R fl s , ( U C R R , ) . N C , N m I m , ) u ( N I ) m , A V U R B M . , N N ( ( ( H A L H C R ) o m s R C A , , M 1 , , V ) C x , U + ) ) R R A N x C G C R V W E R I m B Z Y Y ) A T I L I ( S V T N E S E R R D / O I I 3 5 m N u I C ) V ) N m ( V L W A I 0 T , , A A N R N 4 o I C P N N N V S N m S R R E , m U , ( P , , ¢ ) , , R , ) , ) C N ~ ( ( n ( , ) H C ) m . m , L A e N X R S A x E m ) s , C S I s g , F P L v R D a I , P R R ) R V w , ) , R P 2 V D C R C m E , , N H O D , m R . V C , V C T I L A u , U N ( t I C P N ( m N N P , S r ( 0 o B , I N I I P R + O ) N P I I A ( V V S R R E / , , s P m A ) m s , , , R , P S N L 1 N L ) ) ) C ) , m N ) N N ( ( ) , N ) , ) , ) N ) N , R f E N A , + R u N N E E . T ) N ( N ( H D , 0 ( u m P N N ( C ) T 1 A + H ) ) 2 y I I ( ( 1 L A u S I ( W W R N A , I L v , z N A N N - + V m s R E O F m 1 , P R P h C Z ) I R , a c V m o I . ( B 0 . R ( ( O V V d N T 8 R C I X A C L m 1 H R S D , x X A S l T S S , A D C , E f - A P I / + ) N E n P R I A B 0 R , B / I u R l 2 E N R / S N / B / D 0 C / n m D A x l / / a ) , L / N ( D ( ) m 2 R ) N / w R D T ’ l , P t r T ( E X / 2 / A ) R ( ( N m / m S D / o A N Y A N E A V U T D P ) J — V N E E 7 R X L D F R A C C A N 2 s O o R R R N N + ( 0 I U ) R A I I M ( A V - R I S L A N L B P U U 1 I M m P I A M - N E - u - . m m - - T m 0 O ( L B E B / [ / / m J / R / fl / / / m I X I 4 T N 3 A n 9 I A T E A D T B T J 2 1 1 I , ” N N E 1 N I E T A C N N N N N T N - D N + o O T 1 - 4 P L N N N m m R T M A L I O O O O O O O J O E A O z x n C N ( o O o ( 1 U U A R L A G H m M m M m N E - J D C C E R - - o O E 1 T O R A G E D N M M I N V J P x L ” A n C m m m : E R A E E R I O O O I O A E J a a a o 0 6 R T R W c W c c P R C C c C C W 0 C S R 0 N U m E S I 0 2 g u A + + o 1 4 4 c C - l I 1 U n , E - 1 I - U U U U D O + + + + + + + H U U U U U U B U U U ' U U U 2 4 1 , ) I ) ( ) D ) I I ‘ ( ) D ) ) ) ) ) I ) ) ( ) ) ) C I I I ( C H ) ' ) S ' P m H O H ( ( H C C I I M N Q U E ‘ - T R I x R S R S N * R W * A B - m , 0 . X I ) A N ) 4 U I ' - + m ) I + I 0 R ( U E + , C ) ) ( N E S T I C I C H ) ( 1 S N E T ) 0 , ( T I U S I I C R O R / ( ( m U ) 1 . S N R C W P R 0 X R — ) I E M S , . A M ( T ' , X U I ) 0 I , . H R I S D X R ) x A ( . A N I O A T C ( R A M I ) I ) T C I O L ' + W I O M 1 I N I U ( N M . B ) ( N C ( ( M o U R I A R ) F € I I E ( R R ) C 1 , , I I M D H N o 0 ) ( ) ) ( N S 1 + ( + S ) C — ' C ) A . . C ( N ) 2 A u I ( P ) N ) I I I B m ( a , . o 0 n 1 I I L I T 2 C S ( ( A R ( N H 0 T V ‘ H ( N S R I A / D C Q - P 1 I U n . L . ) I I A E C - P I I T x X R N , . N , 1 C O H S A A R U 0 T . I / S m a P R N I A N N + R . O y S M ( ) R ' J N A 0 0 R X O s . , P . C I I + - + . A A ) R fl m ) o 0 S A ! ( - W - I a ) 0 I J ) o R H ) ) ( A . ( ( . a O C I ( I . I n - ( ‘ A R R I I P M C ( . o 1 ' O m ( C I C I P H a 0 0 ( ( C T I ( C S U . X ) I R R v T I A I C S ~ I O M X I N ' U S S . . - R ( x H C I S G m R A N T . . I C ! I . M E s E £ I T 1 0 ) - D H A A N N M - T ) ) O U C P N R S R B E S H N - R R I X U , - E - - I C I ) R - O A R R R R R ? D I ) E ) , ( n n J - . n ( I ) A - - N I 1 - I C a ) N R o . C ( I - ) ) E ( ( N 0 I E A I m o I I C ) H u w 6 5 e g a P l y z d f l a 7 8 9 1 I I ( 1 N I L T E ( O E ( m O M I K n o H . 8 P N 2 N C E N I P U m N ” S m m I O E O m S R E U H T I 1 . K C L O R o R A R R I R M 5 + O m C ( L I M 1 — 1 c c R N O E 1 1 : 6 0 0 2 g u A , . , ) ) ) N N N ( ( , ( V C ) , ) , A B A , N H R P ) , m W S R , , ( S W 1 ) s , S , ) N + A ) ) x B N B N u M N m ( ( m n I L E L r C F z X V , , I N - E O , R l c B a S s F ) I T A R D , , R N , , S , , ) ) u L ( ) ) ) m 1 N , V M R R ) ' ‘ N X I + ( N N ) A F M 1 W ( N P Q . I I R ( ( m , P L E + I H A N s A A , R O V O B L N ) S ( A L " { U , D , , x N E G R ) F ) A r ) I , ) , L , ( A V m ) N N V . N , C F N ) ( , ) M K ) ( A , ) ( P ) L N N ‘ 5 m 1 C ) ( . N C x N V N ( 1 A ( 5 + L R s ‘ . , ( C R N A fl , M L \ . , ) ) I U ) N ( £ u N w m x H R R , ) , , ( C A R m R , L S V r , R , , ) N X s N M N R A I N ( ( U F A u A U m ( N , A ( ) ( ) N , ) ) 1 A m ( N ( ) H V P R E ) ) , N N x N , ) ) P ) , , R 5 ) 1 m 5 e g a N N fl , I N ( + W N I C ( ( ) I + A ( a B P 0 N S ( 1 E N X m R L R A C L B U S P ( ( F X B V D R V R O s E S P . , A , D F D I H S ) s , R R , , 2 P A , ' ) ) , ) ) N , , m ) , N ) P S ( N N ) V ) N L ) N N , I ) N J ( A S N I ( M . N M P . N L ( 0 ( X L A m ( fl ) ( N ( y Z r N ) C C m 1 E V I A ( C I S m I I - M , R W S P A H 1 N , R ) R B ) d L R U X X R m R N / 1 ) I , , u D f L ) , S ) + F m F H / B x A / F m E R N ) / N N m d u / E N m S I . L V S A L R Y A I N ( N ( I m ( A T A A V ( P A M ' / 7 m A R R R R I D I R A O 8 9 1 O E / . M I T C ? E M W / M A / / m N N R N 1 I A L m I M N E O O M O O o 1 S R A T M V N m o E A E a O O A O W T P R m m c C S C C c A R S : 6 0 0 2 g u A U U U U + + + + + + + + + c J - O c R E P E A T C — — — — — — R E N m C , R S P O I C a R R U M C — ( R C H A T E R / S E C - S Q . n L A N D ) C 1 C O N T I N U E c J - J + 1 C U P O - m ' O O 1 1 I - 1 , N O C T - n o m / m ' I P ( I . E 0 . 1 ) m m O I S ' I b R A I N u r t ‘ v u u ) E L S E m I O I S I h R A I N ( I ) - M . A ( I ) + E x c z s s u - I ) E N O I P I P ( D E W ( I ) . G I ‘ . 0 . 0 ) T H E N n e w 1 5 P R E S E N T - R u m n u r s n m ' m u p a u m c u ) R N O I S T » R I « > I S T - R U N O P P ( I ) m u r m ' m u m m m fl - W U ) ” ( 0 ( 1 ) c R E q u I ) - ( R m a R N ( O E L ( I ) , 1 . 0 ) - R m a m ( n a 1 . u ) , E R N ( I ) ) ) / C + R N S ( I ) - 2 - O m u ) W I W W I W R M C ( I ) I P I O E N ( I ) + R N O I S T - O T . I E . — A L P I ) T H E N c — — — — — T C T A L O E N E V A P O R A T E D T C I N E A I R s A E S O R E m m r P L A N r S w m r u l c m , m l ( m , m ( l ) m l fl ) ) E l m - n o n v m R S T Q I C I I ) - S P Q I C ( I ) - w r R R U N I C ( I ) - — R U N 4 C ( I ) - w r E X C E S S ( I ) - R U N O P H I ) C E L S E A V A I I I - R I ‘ D I S ' N R U M C H P M C O E C R E A S E O R I m R E A S E I N S U R P A C E N C R S E O ‘ I C H P A H I N M A V A I L , + S M F S ' A M A X 1 ( 0 . 0 , S E ’ 0 4 A X - S m ( l ) ) ' w ( 1 ) ) c O E C R E A S E O R I N C R E A S E I N R U N N A E I E E C R m r s ' l m e I s ' I L R S P o m I ) R R u m q I ) - A N I m ( R N C I S T , A H A x I ( o . o , R U M I A x - n g I ) ) - O m - ) E X C E s s u ) - R U N O P P ( I ) + A E A X I ( 0 . 0 , R E O I S T — R R U N ~ I C ( I ) ) E N D I P E I S E c — — — — — D E N I S A B S E N T R U N O T P I I ) - o . o I P ( R ! ' D I S T . L B . 0 . 0 ) T H E N C - — - — - — - — - D R Y I N G R s m I C ( I ) - o . o A m m a n - 0 . 0 E X C E S S ( I ) - 0 . 0 O E c h - ( E c h u r m m r w m I P ( D E M ‘ H C + R K ) I S T ‘ U P . L T . - A L F 1 ) T H E N c I N T E R N A L N C R E w m s ' 1 0 M I N I M U M m - A H I N 1 ( U T , A N A x I ( A L n , m e m e / m o w n ) R E N I E C I I ) - — O E N I H C / U T E m R E c h u ) - R m 1 5 T B N D I F E L S E C — — — — ' — R E V E I ' I ‘ I M 3 C 2 4 2 ( ' I ' ( T S u ’ m m d ( ' I ' T ) . N O N I L V I ' T ’ ( T ) d a ( S a I s m 0 6 9 4 ( ( ( ( I I - I L ' I - ' K - I 7 G I I h . z g ' ( ( V N ' ' ’ ( ( ( N N ’ N ) ) ( ) A O N ’ V ' E V 1 m N ( E I V ' S A M s E ( E I ' ‘ H ( s ' + ( ( S u a ’ ( N N N X m E ' ’ A O S m A ' N u n N ) ) ) J ) H H N ’ m s S ’ T s E O ( l m E E a ’ E N ' ' E u E I N m s ' I I O O ’ ' ' ) ' ( ( ( V ( T S O ( u A E H ( I N N 1 I ’ N S I ’ N + ) ) A E + + S E o N ' I ’ ) N V N X ) m N E E O S V ) T S J ) ( D 1 0 0 8 ! ) 3 1 1 1 ; . I N O V A O ‘ V V u O T ’ ) O V A ' ' ( ' H S E ' A ( ( N . R E O O ( ' N N ) ( I E ' ' S N ( ) ) V m ' ' S O ( ) I ( V d 1 s ( O ' s + T O H N u N O E I n N N A K + ) E O r s u ) ' ' J N ' O x n I N s E ’ V ( ( ’ a ( V N N ( m V n ) ' N H ) ) I a ' E ’ A ’ S O S ) ’ O H + ( ’ T ( S u N I ( O O A a ’ A N d E ) ( m N N S E E a d ) ' ' V N J ) E E ‘ O ) N ' ’ O ' ' a N S ( N N A H I O O A u ( m ) d O J ( x O N ) ) X ' V N M d A I I n u T S ' a m A u O ( ) a a E N N u V 3 T N ’ . x N N ) ’ O u S V d ' ' I O ’ ' ' N A ’ ' d ' ( ( + O ( ( ( ( ( ' u a ( . — a ( N ' T O ' . V S I N ( ‘ o ' a ! “ ( ( N N N ' N u N N N O ‘ O N ) ) ) ( r ) O ) ) O E I ) T V N V M ' d N N N O N ) T m - T A O ) V O S I E m s M I I z . + s ( N z a ( H 3 E N z ? u d V N N O ' s m x ’ . N m ‘ H ' ' O I X x D E E E O u u m P N ) 9 ( ' ( 1 I t C c S ( U 5 ' + ) a V c o / ( ( + ‘ a J a m / I A A O O I ' ! I . O 8 u S o ' - ‘ ' ' s - N S N N ( / N N M a O O N o N x m N ) ) N § T S / / E I J ' 3 N fl z o n ( ( ' 0 I C I d “ G O c - ' ) o A l A V V ) fl O O O A H V L ' I ’ - — I - ' n W O 1 N E O fl E E O V D O a D ' ° O N a I T s V A I T i V V N E J E d w s I X ' l ' d o / I V ‘ ( B 8 I I ' G I H 1 ' - s - I U t I I a E d C N ( e I ( M ( s S I - - a ' I N — n T O V T - a E m N / / I / M / / / m I O R e N N N N N N N N N o / O N I M l ) N n - N - T T ) 9 I 1 u I ( M9 I a 0 I O a N I 5 0 V I S a ' m I - I ( “ d a J J - I I a C E ' O I m O 8 S N - ' n I H a J N N m I ' N 0 z ! I . O ) x O O 0 O I I I - I N m a O O O O O C O O T H N H N N a N H H S V m M ! V H H N N H H H A I 9 I I m : n m m m I O O . 2 E m E D O O O O : D D S O ! ( C C N I - I E O O O O C m O O O N N 0 a t I J M 6 — m D I N S s I A ‘ I ' I V ' E O N C O s a u n a / I I l I I l ’ s 0 : c : t t c ( ' 0 ' ' ‘ ' t N ' s 9 9 9 9 9 m I t I t I 1 I I ) ‘ ' ’ ' ' ' ' o ( O L s ' H S ( D ' B L 5 ' c ' ' 8 ‘ ‘ O I a a 1 s I ’ ' t ' t - O L ‘ ' 9 I ' I + R r a t ' c I C ' a r I ' ' ) a I O ‘ G ' o ' s t z ‘ I O a z a ' O ’ ' ’ z t ’ ( 0 0 ( n d a m 9 O O e Z c L $ S ( C / ( I O ' S ' H I ' ' ' ' ( L ' S r o d I v c ' s z a ' o 8 m O S 3 N d ' a ' a I a S 9 z ' I ' s 1 ' ' ' t ( a ) O z ' ' s s ° 0 6 0 : I I W “ D I 9 ' L ' z ' o u a ' t : ( ' V O I ' a O d / r U - N ) I ’ O ( ( c ' 9 a I / / H s I M V ( I / N I ( I I N s S / ' I O s / ( a C . ' ' I I M L S ’ ’ ( s t N m S ’ M S ! I I ! ( a ( ( I s s ' - - a . + A t ( P J S u m ! < s - ' ) O ) n ' T I T a O ( I n T I I I t - O ’ . ( V O M I ' I ( - d / M - T R I ' ~ ' ' m v D N V £ N I z T ( O I L W O O I ' u I m ’ I ' ' - O ' ( ' O - - T ' 9 I I ( - I ( t I a o s D S I ' ( O a T ' o H I o ( a t u V T ( ( G ' o L - M R / d J s T ) / - a S n 9 ( W a W N ) . S I E N T - N ) ) n I ' I I ' I / O 6 J I O 1 I L t H ' w - S I O m I V Z r - T I M m C ' I l ' m ) ) S m m m ) I a . 1 W W L W 2 I M I I ' N u - I O 5 I N c I O O u O - V E E N ( m m - m N " m a M I m O V V V V V V V V N d - C O m : N J . m A I I I I I I I I I a N O I O o I I o u V N N N N N N N N n a M 9 m m I u s O Oo Oo o O O o Oo Oo O I 0 o : a n C N I s o o o o o o = = = = = = a 9 o v a z I t t z s - — - — — - I S 2 4 3 + + + + + + + U U U U I + I - I Z O N I M 1 3 3 6 3 8 — — O - I W h i t - U H ( 0 ' Z ' J fl ' I I H M I ( ( I R H ' O ‘ O ) I M ' O ' 9 8 1 I N I H V - I I H U H U U U U U U U U U U U , ) N ) M S H E S , C X N ' ) E E R m E , ) F m F I D E ‘ N , M ) L 1 V + X N I ( P ) ) 2 6 v 1 P ) . N E H T ) V 0 R . + ) o z , G V O I S N ' I N I P F O A , A V ) P m , ) u N t m L , H ) I T X + F N ) + V 3 T ( 1 A 7 ) S m O ) ) A N v P ) ) 2 p I I O ) ) 0 P I T . T ( r + 2 T I L Z - . p 2 6 D x E , C T , / T I B R + S o , W A A M V H ) X L M E 5 - T S Z S C D ( X D R 1 - , a . c ) ‘ T D * 1 C T E . T R ) P F P C L N 7 / o n 0 . n , C ( V D 6 M , 3 P L , 1 t ) ) 7 ) { v , ) 2 V P V 0 . A ( A O D 6 O I E . / G ) 6 m , V + D I A + P , C A D ) T C ) ) 1 N ) + A 0 ) N + ( y H I ) , ' ) A 1 P H 2 . ' T ‘ B , . 1 n A m z R d , f G 1 B l T , V , V M P S L P o m m I ) o I S I G ( ' H H - , A u B - z D ( v I P ( M H v n P A U D I L 1 H U D E A ' - - - s - V A L ) . L ( L T 8 S , z ( X a D 2 + D T ) ) 1 1 T N o S 0 , $ R Q ( N E N I C s F K T R - ) Z 3 ( H S S I C I 0 ( V ) 0 m O A / B P B V A N 0 0 « S A 0 N T N S X , M I . 2 G , O E / U N ( . N - V ( T a A o E R G G I ( I C D ( D 7 R ( H m . . S 1 D + P V T H ' H a P P ( . . T I L I H 3 8 B E R T S R ( S C N o ( C 1 - B ( 0 z o O C N S S 0 / / - N 2 D D V + + z - P - ( c a ( 2 N R R A R 9 D w D o R R R 5 I B A P - 2 C L 1 - s 1 . V P I 5 C E L T - - P m - - L S m - B D ) U H L R ( F E S / 1 - - c - D J - I J R - L P G - T E O M R - H A - G P R a ) ) N o - - H P A . . o - T o D O H A V H V P A - R M E D N M x s . D 0 R A s C T a E P P X . R M R G G C ( E v v c S P P V - - 1 2 M w - G - ( C + ( U T A N L T 0 0 - X S T - - - M s S S E S T - I A - P - R C R R S D R S c M E A A U I A C s 2 1 P P L I P C C A T P R 2 2 H P H T 1 X O T C C R L T N - i S I E S A - S E L A E O E E m O I V A N V T R A T + ) ( . . R U ) ) ) L 3 3 X A F S ) L . C L R 0 V ) R I 1 R , 1 O R , , , R M P G O , ) L A T o U + H z 2 C B ) D N L S F E : , n G T P M , ) ( A I . . ) ) N M ) A a , D T ( P N ( ( T ) J J L M I M I , l , S ' , I R G X H ( H A A ) ) . I N o G L R l E . E 5 V . H G R 0 D E O c ) ( , V V S E D T 1 R P . N H N n 0 I m B 0 S + 0 A , X D . P , P ( S 0 , , M , , s ) 1 ) , C 5 . ) ) E m 0 ) - ) F 1 E L N m V P ) G , 1 0 H ( , J , G O u m M m T V . ) - P N ( A E - H G 0 T N , ( N I , P P 0 N H 0 A ) ( m L P . ) T O ( R ( G V M G E . S R 0 ) - P O ( N A D 0 N L A N M E ) , 2 G 3 v N L . 2 , , ( L H V C ( I 0 O . N , T D 5 P T M A , E ( o A I . V ( N 1 ) , 3 H 1 G B R ' I A M T . C N S R S L A V c P , X m 1 T , S , R o - A A H N - R D G , ) , ( , N 2 , P I X L i . B E M u n R ) ) , ) ( D S ’ G I . C , N C ) T I E A G H R ) D H T ) ) P ' N N r R - P I V T ) u O N ( N S N C , B N m N r s N A C - R I ( G E T P , m A G / V , B C W A ( , s H I ( A S I D V , , I d G / c ‘ E ( A W E - T L O L C O T E , , ( , T I a I z 0 D E V M ' , D C V L N T ) R 0 / D ( V N s E T ( v T U . / D A n E D E I ) ' C O . C A D , S M , 1 T N N O R / , L D I ) , O l ' , 0 R C L ( - M J S V L R P R T T + T ' I O I / I A A L u . C I / ) A ' B 1 T - ( 0 T C S ) S G L - ) , P / S N N P P - s E I V P N / N C C C 0 E ( ( J ) N H A I ( ' I S N R T 1 m T - O M O N - E N ‘ N - C c O A N I X G ( ' G G L R N 1 a ( E s D G O I L L N P S R I — 1 D c A S S D R E ( C V V A ( R R T ' A D S O L N / P S E / T / / X M C N T R V 1 D X ( ( R M S A I L A T m A R ( N N G P - A ( S A C O N T N N I P - H R - - G H S A E - J B E L / m J / / - E - A P A R , L “ R O R E s T S G A M - s A ( m P D A H E A H A P H H G T - A S s C I T I A ( - - N - V A I / / / N N N N N N N / / fi 1 P R E P P R ' P B E I H S Y U F E U I T D D T S s c I U O U N A X x a I D E C R A E O T D R A M L F E m A L M T O O O O n O M M M T M T M E - T T T T T T T V O O O O O O A O + R G — c c C c c c F R D R B P R C C C N E U E A S W C S C + U C C C m T R R R J H R R D I P R C N O E S W + , : , ) ) ) N N N ( ( C A W 1 A S , . M B 7 V T F ) R , , , R V R T ) + L 2 , , ) 2 N , G P N N ( N ( ) , V L ( W n 2 S A W T L U 1 n R , E B § , ' , , 2 O R R ) , , s ? G O N N m 1 H ) L ) ( , O O S N ( W 1 R , 1 H C 1 8 1 , + ) 1 , ) N ( m A B o M E A D N ) ) n R G E V , l ( 1 , ( C O , L ) A l + - S T ) W N A N H T P P 2 ' C , C R c ( ( D E N V S A s G R P , P 0 ' S C , I ( Z - C , T P , S 2 C ) ) ) ) N ( A B I 0 6 e g 1 G : V 6 R 0 0 2 g u A 9 5 e g a P J y f H . 7 o 8 E 9 . 1 O 1 ( : P 6 I 0 0 2 g u A U U U U U U + + + + U U U U 2 4 4 , , , ) ) ) N N , N ( ( ( V C A B A ) , N ) , H R F , C S ( N R , , A ( ) , I V S , H S 1 S ) R F R ) + X N ) , V H 5 s ) N N ( N P S G S ! H H x E , ‘ ? ( , V H A N N R T I ( A E E C , S B , H G L ' X " A V W , R I R E S , D , , B R T E ) N L D , 5 H R R , ( R ) , X C , L N S I ) I N ‘ H “ W S E H ) ) ) F M N ) N 5 I N 1 , A ) V S , N V I ( L N ( R + G R + E ' T R P ( A W N V H , F 1 X , R T M A M E H T N B , V D O R A X E C S I D A R ) C , , L N , A F S , N A , S A A L E , R T , ( X S I C ( I E N N S S N N S R N ) ) C , , L , R ( ( ( N P P ) ) C V , M P ( U ) ) V E T N ) H I C , C ( 1 P ) ( S H A P C , B L C R M R R + C S N A ) N ( , ) E i H V P U m ( N N R H , , M A R , , N C , ( ( , S C , ) L R ) ) X A C I M T N , C ) R ( R R T m , A P ) A F N N A E U ( T ) N , M R ( ( , + x V , A T N C H N 0 E C P S R I A F O E C A P S ) R I I ) ( A I ) A ) E I ( P E ) V ) V O ( , ) V , C S U ( L R E ) 1 S R , ( C ( I , L F ) V C R R ) L ( 0 ‘ ) 1 , , N T P , , ‘ I ‘ S V V A M R N + R 1 N U , ) + ) , A ( L ) ) N V ( X C ' R 3 S ( A H 3 , ( ) , ( E I 2 1 L I T ) A N N N S M X V E I E / R N R ) N V + 6 e g ( ( ( , S ( P P ( S N T T , S S ) / V ) E ) ( N S ) H P ( O P O I R A P 1 R C N M ) ) ) I R 5 E 5 D B D S 1 A C L V T B . , , , E ( / G ( ' l ) ) U x V R D , O ) D - I U ( ' x H I 1 A N X R ) R C [ V ) V A a / V P , , S N T R S S ) I ( m O P T ) R V D P M E R P - + R I , S , , N ( R R 1 ( 2 f . ) E / A D — I , N C / , O , ( F ) ) D , V A H ) ( R , ) ) N N , ) ) ( D ) R C M ) P A ) ) V ( R - R S ' A ) 0 ) N ( ( N I I V E N ) ( u I V L A N 1 N ( M N A N , A R E ) R / y I S s - - . V D ( A ( ( A . V u N - ) M T R ' V V D 1 ( ) r d B s z ) u q R E , L D E X I , , , ( r N ) ( I — N A I ( D O I L D 1 R ( C X E z 5 D S H N R V m 1 L P ( ) I R — A R , N I + f m P M R / ( S R 3 I H ) ) + , m / s F B T H ) 1 V ( A I ( V A l 1 a - - " 1 ) 2 ' S ( R I N X E E T ( N ( ( X C ' / / S E A N V ( - V S E / / - , R A V O / N N N ) D C T S ( L R T A N E V R S G A A ( A V U N R H R . V I S L L E N w O ( V A N R 2 V D A I R G F ' C , V ( , - F G E 1 - H S E E E A M I / / ( / / / A Y R R R D R I E A M s C R 7 8 9 m M I V , ( - m ) 1 I N T N S E U - U ) ( H C T L N L S I I 1 1 T 2 X ( D T N S T R N : S O o O O A L M ' T T m T M N E o U O R A T T T M B F M I O m O O O ' ) N N N N / I - 1 - I G ) ) - ( U A ( U C I E U ( I E H A ( ' ) - R ' I ) A - 1 ) - N 1 1 0 M A N 1 A N V P ( ( 9 L A I 9 R ? V I L 0 A A 1 ( V T 1 X R T 0 A L A F W N F R N C M n C I R A E ' B P R W O C A U R S 6 0 0 2 g u A 1 6 e g a P J y r d f u 1 8 9 O O O ) C c C C M + V A V E I O o O T F W I C n C T 3 1 A M R ‘ — C 1 9 1 C , C T V T , S F X L F $ V R , C V A , 2 , C , S ) R J N e , , l M ( m C ) , u N H X 5 C E ( R m H D , , ) 1 + S R , X , C E m ) C N 5 H , m P ( ( ) , L O , 1 C C , ' ) 1 U ) M P S C N R H ( R , + S H X N , ) ( S B ' N G I E A ( U , R A , ) , S E D V C R U T , ) R V , ) N R R u M , ( , m ( N C ) ) A P s T N V M R E W T E A s ( P ( D R L , S , X m E C E P P D N D C , , , R R R H V ) , , , ) L ) N ) ) C , N I D ( 0 N E ( N w - N P 1 A ( ( D ) V A N , M S R E D D I N , R ( X D T C I 3 S / - P n R T R N N D / C N E / E E ( A T D M D . A N A R I R C E P I ‘ E / / / T E N N N 1 G 1 1 : 6 0 0 2 g u A ? M O I O O 0 ' A M T T E U R M T T V E A X O T P ( R A B U R S O O A C C S F I N ! ) ( ) 0 ! ) U 4 ‘ + 4 } + + + v a N ) , m 5 5 ( M + 1 ) , E X S N S ( M ) , E X S H C , n n N C , n s m , m s P ( N ) , E X R A I N ( N ) , R A I N ( N ) , m n . R , n V L R C O T T O N A m / C O N D , V N C P , D E P ( M ) , D E S ( M + T ) , o ( M + 1 ) , C O N D 2 , A L P , V T C O T T O N / R A T E / R D E A ( N ) , R D E L ( N ) , R R V A ( N ) , R M C L ( N ) , R D E N ( N ) , R ' T ' O T ( N ) , R D E S ( M ) C I T M M O N / C M > / C T O D , D E C , R H C , P V C , R V G , U C , R C P C , P S C C , R 5 N C , R S V C , R V S S A V E L - M T I D O 2 I - 1 , M s t m - D a s u ) P ( I ) - U I " A L F / D € P ( I ) D O 3 J - 1 , L A G ( I , J ) - 0 . 0 C O N T I N U E C O N T I N U E S E T U P m E E ' F I C I E N I ‘ M A T R I X A G U J ) ‘ 1 * P ( 1 ) ' ( 0 ( 1 ) + 0 ( 2 ) ) A G U J P ‘ P U P O U ) A C ( T , L ) - s t u ) + P ( T ) - o ( 1 ) - D a a R - M — I D O 5 T - 2 , R ” ( I I I ‘ I P ' P U P O U ) “ ( 1 : 1 ) ' 1 * P ( I ) ' ( O ( I ) * Q ( I + 1 ) ) A C ( I , I + 1 ) - P ( T ) - o ( 1 + 1 ) A C ( 1 , L ) - o x s u ) C O N T I N U E ” ( H I K ) F P ( H ) . O ( H ) A G ( H . H ) - 1 + P ( H ) ' ( 0 ( H ) * O ( L ) ) A G ( H , L ) - D x 5 ( M ) + P ( M ) - 0 ( L ) - D E S ( L ) S O L V E N I ' T H M A T R I X M E T H O D C A L L C A U S S ( M , A C ) n o 6 I - 1 , M D E s ( I ) - A C ( I , L ) C O N T I N U E R E I U R N E N D D - H H . “ ( U m - ( m - D E S U D ' O U ) N U U U U U + + + + + + + R E T U R N E M ) L - N + 1 1 9 0 C 2 4 5 , ' ‘ ' , R ' A T L N O E l m : , : ' N A N O F I , I L A ) S C V ) T A R I ' R , E P ' C 2 A F ' ( , . R L , , H , . 0 A U U fl ' I ' ) 9 A m , ' D , N ' A ‘ A , ) . ' : E fl 3 T l I R ' T - , P , Y ) A D N J W H , M I E Y W R R A S U , D ‘ T J ) , I S N D I P ( A , D L E T E C . S I N 6 R , , A P T R I E ) I M D ‘ ‘ N D A ' m 1 I ' P T W R W T U p , ’ N L S S ' B , ( , 6 S D V I I N ' , I , H X ' x I ) , ) N I ) A 3 D P ' ' . S F R 4 I I , O R . E “ ( ' ' A N I I O A . A a 0 ) X , N R ( 6 V H ' v 1 m W ' U W , ’ ' , , P ' N S I D L ' fl ‘ T P , ‘ ) L W , 4 A ' W A I M X I E X 1 ' A Y F , C , , I R ' , C E , ( L A E , I ' m ' , 2 , H C ' ' X , W P A . I 3 T ' F R Y S . T X ( Y 0 S . Z C ' T T G N , ‘ ' 0 F S I , A , , ) F ' 1 ' ( L ) H I 1 C ' w C P , I R R M R V I 3 ? V S H , , , s P X S E 2 R ' ‘ , , , 0 N N R ' ' L , V ) R ( , X I P P 3 F R ' A ! 1 S . ' R E E 1 N O , R I m , , R E ( N B . , I R ' R P ' 7 , V 0 2 ) X A A R D J E , A I U O , C I L L 1 N . R I F V m P ' I / J ) N A N A 7 C M ' V S 7 O A ( , ' C B ' , A ' l R , F F P L ) N M D 3 E A , V , U , , B I ‘ R ' . , X W , A E , ' , 7 U ' X 2 E C D ( X ' C S V 4 R N . ' F L F A C D H 0 H U H . R , , A R ' ' V ' L V L , ' , I , F , , D S 1 T ' ) ) U E R , V V 4 I ' ' I S T , N C R R , ( Y , A 5 I ( R ‘ N S ' ' X . ' ' H , U U 7 W I , , A C L G ' A ' F 7 A R ' R A 0 A T R H A , , D D D D ' , P R U , 1 ' X P E X E ' R ~ J C , ' A ) , E D , A I 4 R F I B N N L ) , ( P N F D N I 2 I , , I V . ' “ C ’ ) I A ) Y , ' V . “ M E ' ' S ' 4 , R Y 7 A I , 9 ( L I , , R , ' ' ' M . A N I I , ( T 2 A I , L E P U S V I F Y N T R I ) A O I , R S D E , , S P A S ) ) T I / T , , O 4 I N I ) A R 2 ( Y ) 6 R D I 3 ( H 1 K C T L S ( 5 H ' S M T T e N B B C 4 T , , L g 7 : I D H I E R S a 1 r Y V ) R I X A , P 0 N O A I ' O R , , , 0 L O ) ) D ) G S I , N N T 1 y N D D ( n C 9 L , fl , E , , ' . T u R ) 7 R N 0 1 S C d I Y 7 Z T P O U . 8 A . E T P R ' A I L ( ) D ' 0 H ) T ' , , R , C , D P 0 N I ) 6 N ' 2 ° ' ' ' , ' R A . ) ) m T , R . A X V 6 , ) , - ) ) 1 w V R , Y R . V V E m P , ' 2 ' ' , H I I E " ' J S E 3 L ' A c ) E A I T Y W X E ) N ' I , S 1 T , 0 C . ( T T A R G / Y ’ R F A 0 l A I / m e , F R D L I P D R 1 , I I P A , ) I t T O R H R S / R 0 0 I 7 ' L H A m E 2 , p 0 0 ' / ' ' I ' ' 4 A ' ' l / L / Z A ) . T N P P , 0 ( R ) 2 1 1 . Q ) C A ) ) ) a O I / . - / ( C T 1 ? 6 A 9 1 R I 2 E 7 N 0 T R T N I L R ' M N / T E , ) 9 ‘ T ‘ , 0 8 1 G 0 ( G 2 , , N , 1 P / S l / ) A / 9 7 A D , G A A , R l C N U ) ' . J ' , , , 8 N 9 R Q 7 , V 0 9 P ' 7 R D A P P C ( S S H - / M E V A 9 I . I , A 1 O M ( O 8 J V 8 E R I [ / I M E / / o C A U I O / M 6 ( V 8 - ( E C ( O , I O , ( 9 7 ' , I , V 9 S , ( E - N M R C T ( T ( I ( E W P O I ( V O 1 U X ¢ M T T 3 c I A m I T A T T ( T E E I I w T O o ‘ T E . E ( ( 1 E A S D T V m A C N ( c 1 O V T L N M R C I P M O : m A A R I E E A A m C E P I I I N L R T N 7 T I I m N I ' R L O I R R R R I N T R T I O 1 O A P R D R O O U W W R N E E R m - 7 n - T - - I A ( T H A 0 E E ' ' A ( A I T R T ' M E 6 m m W W M m S D P P C I C V F H T V m T I m 0 6 0 P 2 ‘ g U 8 2 9 0 u — 0 9 0 0 1 A C 9 C C 9 7 9 I 6 ) u w O ) ) ) I ( ) I ( . A H A D ) V X F - ) ' 0 9 . 6 1 0 - . 1 ) ) 0 + ) I ( I ( ( A A T A I V T P I X ( ) E C ) ' H X T I 3 I ) ) ) ) 1 1 ) ) ' T N ) ( 1 ) Z — - ) ) ) ) T I I T ) ) ( ( ( 1 1 A ( A A P T ) L 8 + - E 7 1 + ) ) D 1 ( 0 I ) z - . - a , u A ) D 0 ( m L I ( / . , , , ) ) ) , N N ( ( N ) ) N V C N ( N , w B A I S A , R P ( m , ) , , V , M R , , , A , I ) R R ) ) ) ) + N V , m N N M m , N S m N S ( G : ) : ) N , N fl ( X A R N V 0 H , 0 ' ( ( H M D P I G S ( , m : D L R T R R R ) ' P , o S R , E . , { I S , V I R L , B , M , D ) ) N S S D ) ( ) P ' ) ) , I R ( X N T 1 m N ’ , S 1 ) N , U A F I A R R , + + N V L , T N N L , R + : , 0 ( E V ) I ( ( I ) ) ( A V I x H N N U A A R C I H G , , M E A , P A ( S ( : D ) ) V , V S , G P N Y I K ) 0 I , N N Q , ) F H S , ) ( ( N ) N , S P G ¢ S D P N A P ) ( ) I , F L C I ( X P , A D T N N N , + H V A C ) F R , S ( N I E : ) . T L , P R 1 T ( L A ) 0 N R A ) ) I ( : R N G N U O N m H ) ‘ 1 R I ) C , + , , A N , U ( ) N ( , X + X N , ( I ‘ ) , Y E C ) , H F V S I E N I ) N G ( P K W S ( R ) 1 , ( C B Y G G ) ) E ) ) ( V V ¢ ( D R 0 A I . ( 1 C R 3 6 e R R O ) ) A A E V - - N ) N - O E R R E V D R D R ) ) - ) ( I ( — I I ) T A - 1 ) g — 1 ( ( E ) ) I A V ( T P ) ( 1 - - ) ( A O ( I - ) ) 1 T 1 A J , , H U I O . H L S , D P E N + ) L S K B S ( E V , , N R , V W R E S , , ) ( N N L S , P , ) ) N ( ( I S ‘ 3 H ) W , N , N N N A A , K N , A M : X P H W ( ( , H R ) , C ) U ) A , , 0 P O I ) ) X ' S ( U N ( N N N V R G C L X F P N N N F U R ( ( , R D V T x H ( B F , : R , E , a ) A ( O P I 1 L V R ( C A ( , D ) ) D m , , ) R A , , P , R C R R 0 P 1 + A - + N ( ( I E L V V R - R ( m A A ( R R X G ( N B D ) ( V c I I ) E V L ( ( F V / I L R E ( H T D E U , ) , V ( V , I Y N , ) N ) O ) N ) \ 0 ( ( ) L R ) H H : P , N , ) N R N ( ( G D U , R N B P : ( D T D R V T ' - ' R ) A A D A D 1 N N R L L ‘ ) M A P D C N 0 S I ' T , m ( ( ) ) A R ( I R ) - ( M , - - ( I T I - ' T V ( - 6 D P ) - - I 6 H Y M T R R R W ( H N E ) S D V D F S E A R m ( T ) ) ) ' ) ( ( ( ) ) O I A V H B I N 5 . S , . L , , X X X , D E D A D 1 , m N E T - A 6 , ( ( ' N V H S , H U ( D , D V C I 1 1 1 ) 1 A A W T ( ( ) 1 T A A ( L H - ( L L P H T D K P T 5 3 3 C P P F I ) ) , + m m I , ' ( W 0 X D ( S A P ( C L ( - M N N ) N C D U N / / ( ( N ( H ( ' / m / , / m 1 I A A N V V A A X N A N + - ' . ( A - A A ( M , - - N A F , ) T S W A A ( V W S D M V A L 2 - 1 I ) ) V - O - - U I - ) - A N ) E W V V I L R S N / B R A P D A W m R m A G / R O I R 1 - - I T - ) R R R O Y R R D . ) V I 9 ( ) ) ( E ) I I u M 1 E E E E N M m E / fl / m W M [ / T n . . - T , 9 1 I A A A U L ( a U T T R T I H ( C O - L - - A N I T ) ) M I ( ( T V / / W E D N ( ( H R / H I 1 ( T 0 A A I N ( 1 A N L A s N U N E C E N N M A A 0 V l V I A 0 B I R O M A A - n 1 u S I ‘ G O O 1 A A 1 T V 1 A m ) ( m X 2 n X T T A T 2 I D T U u P R P M T N ‘ ) I I U O I A R L E T T N N N O O O T T T E P m F P N O V R A A T T T T R T : E T O ¢ M O T X O E W A E H O O O O O N 6 0 0 2 g u P E m I m P P R ! N R D C I C m I B R U S E P S P R C C C I C m C C m m W m W + + + + 1 0 A 2 2 0 0 U U U U U + + + + + + + 0 0 0 0 0 0 0 0 0 0 0 0 9 0 7 2 4 6 a ' 0 1 3 1 1 ’ o u ' ( v ’ N ' ( ' u N ' I / - z t I - ' I ' ' 0 ( ' ( 1 N I ( ( 3 ' ’ I ) ( 1 ) 0 1 ( s - s 1 . N s 1 a . ' m ' ’ a . I ( ) 0 ) ( - 1 l I ) 1 ' 3 ’ ’ : ) s m 1 e ' I 3 s ' ' a c ) < x a V x t ' I 3 ' a . ) . ’ s ‘ a ' ’ L = . m o . N U 9 9 0 ( c 9 3 = v 6 9 v u x . v = w m m S 0 ' 1 ' 5 ' 3 . 3 . 0 3 1 = N 3 3 1 o ) u I 3 X 3 m ' ’ 3 u O 0 I L 9 v I v 1 1 ’ I 8 ' N a m ’ P 1 1 o L n I I P 1 . 1 . . . o . u ' w 1 ( ‘ ' ' n 1 6 5 1 ’ o v u ’ ’ w 9 n 3 v 6 1 1 1 o ‘ 1 ) u 1 1 3 u 3 0 3 u m m m 1 % 0 : a [ m £ 9 8 : 6 6 ( ' I ’ . ( . m $ 1 1 . ' M n m ' I . " a 3 O n a S ( a N . ' ( n m n N . I o I L ( 1 m a o ' 1 I “ I ' ' a h d ) ( C m 3 ) m ' u 1 < e 1 u x o m ( I ( e ' ’ . . I 0 m N m I o 3 ' v s 1 I ( 1 . 1 ) 1 S 1 I » ' a o o 3 3 . s N u . M s ' 0 N x 1 5 I I ' 1 ‘ ) ( ) : . a ( = ' a ) a a C 1 N < N u 3 ' 1 v ~ ' m I . I m . N u H ( E m ( 3 N v < o . 1 a ’ x N / N ' I ' I S u ) ( s n m a A v x s L I c I 3 I 3 u m ’ I 1 ' ( N 5 m ( ' a S 3 s . ' ' L m . ' ( . w z ' ’ ’ ' ( u . ( I ( ( 1 A N 1 1 v « s 5 s ) 1 ' n a ( 1 W 1 ’ I m v C I 1 3 I m ' 3 u x a s ( 3 c ( . m o m m 3 1 1 m 1 - o 1 ) o ) I . A o 1 n a e I . ) z 1 ' I I d 3 s M ) ‘ ' 3 ’ c 1 . d ’ x ' v ' 3 1 4 ' ( 3 . n ( m x S 3 ' n ' r m ) . . V ’ ' = n o I u 3 ' N 1 v a n m s . a ( 1 1 / W 8 I x s a = S I . . 0 ' a S ’ 9 A s a N ( ’ s N I . : m ’ ) . N ’ ) ( I 1 z 1 I a ( u a C C N < a N I ( 3 I ' 3 . ( ' 3 1 N N 1 1 o ’ x ( 3 o ' I V I m 3 a w 1 1 ) : N v l H z ' 1 ( S 0 a ' I C 3 . 3 n 3 5 . ’ 3 . ) v ‘ 3 z 5 $ I N m I I c ¢ 1 ’ ‘ N 1 v o ) ’ 1 . = u . s . ‘ Q ( H 1 ) 3 ) V 3 . ( u I 1 1 a 0 ( ) . u m m 1 ' . x ' m I a 3 s ( ' t I a * I I V . 3 . ' 0 n . ' 3 n : m I w m 3 3 3 x ' C C w 3 n I a ’ . N ) ( 0 . 1 ( ; 1 ’ ( 3 ' u 1 ' ' 3 ' 1 . 1 . x ' x c 3 5 ( r o 3 ' 3 . N n 9 m . d O m h - G a ( I m I 1 1 z ' ' N - ' U V n 3 n s I G ' ) I . . o . 6 ' m 3 ) m ( ' m m x ’ 5 . v M N 3 0 x I U a ( 0 ' a 3 1 I u 1 N I d N m s s a . m o ( ( ' 3 m 6 3 . S I S O ’ 1 a 1 r 1 ' ' a A v 3 . . ( o I a ( A 1 3 1 ’ u 3 V ' ) 3 ( . ) ’ 3 m 1 a 3 ’ W 1 I ( 5 . v ) u m a z 1 c V < ' n ( . N n ' ) ( - a v 3 ) 3 ' ) C 0 : I A < 1 . I I 3 o ’ m I M X I u ' > s 1 ' x 3 N 3 3 I V 1 N ’ 1 1 u v ’ ( ) . I w ' 9 ' s s u 5 1 1 ) M . M n n 1 O v 9 ’ s E 1 a 3 o ' 9 O ( 3 x t ' I 3 3 L I M 1 1 0 ' . a ( m ’ ( ) O r ' ' . 1 1 L < O I . 3 m d o 3 n z 8 a O o 9 ' . 1 n m s n 1 1 ‘ . M 3 ( ' - ' o a ( 3 O x a 1 I 3 u N M n x ’ l 3 3 . ( I I ' 3 1 E a 0 ' . r N v I ( 1 ( ' o 1 1 0 I N v 3 9 ' ( S ’ 3 3 3 . ’ a ) x 1 S ( w ' ' u a 3 1 ( 3 ' 3 3 ) a I ’ ’ ) a ( 5 ' O 9 N I ( ' c 1 a ’ 1 3 A 3 s P V . U A c . t . ’ 0 . . 3 I I . I 3 3 ' a ( ' O x 9 1 2 a H u s o . l . G a x O ) . 1 m M 3 0 I ' ' ( a 3 ' J a ( 3 3 a I a 3 U ' x ( S x I ’ ’ v v ‘ r ( x ) 3 1 3 3 S 3 3 u ) O / t C u 1 L n . x . a ' ' 6 N n ’ 1 a 3 3 V 0 / I I 1 H 3 3 ’ O ’ s ’ 3 a X 3 3 3 t ' ) J / ’ ' ' z n E a L 0 3 3 P z a c 0 1 I a s U < ’ d 1 9 ' v ' m 4 m ' o r ) M 1 n 1 ' m m B 3 / 1 1 A a a L V z I . x ’ ' 1 v ' ( < U 1 J N V o n N ' t s ( 9 . ( . ' 1 ’ 6 1 3 s ’ c 3 . ( . 3 1 K a m ' . 1 1 ' m n I 3 ' . ’ v G V . n ’ 9 x . a 5 ' 1 1 . 3 1 1 3 I ' v z . 9 P £ ' a u c ' / L ' 1 O J x ( I . ’ ’ ' . . . . 0 9 x ’ z W ( . ( 1 ( 6 t a 5 ' I ’ ( . u ’ 3 3 S t ( c . v 1 n 3 t N ' . . . L ‘ ( I . . 0 a I O n . I 0 3 o N 6 o 0 - L I 3 m I z m t ' v v s 1 L u L I ) 6 s 3 a ( x A m o I N / x C ' 1 6 N I ' ' C o ) 6 3 . G ‘ ' c ) 6 z / ( ( ' ' L £ L u ' A ’ 0 5 s . 1 z z 6 c n c 5 N ' O / z z I ) N t 6 ’ z v . ’ n v ’ ' J ' - 3 3 ' ' z L ' ' A ' ’ C C A ’ A 3 ) L . A ' z A ’ C ’ ' A o ‘ A ’ ’ ' ’ A 6 ' 3 6 ( x ' ) o ) C A 6 ) I S L I 1 u I . - C V V u u ' 3 o L I u D 1 m ' ) L s A - O 8 6 1 I . 3 m ) A . s ) 3 3 ) 3 / o 1 ) . 3 3 o v ’ 3 / u ) I . n ’ . n 3 t H v N ) 5 I a ' I O n I I 3 u . n ’ V M ) A u H m a . . I . I m 3 L N N ’ A n . 3 3 N 3 . 3 I I 3 M . N . m s m H . I I u I I L M 1 1 I v N . ' H I ’ . . o I I m o N L . J n c m . u m I n L n + a U N N C I 3 u I o I m L . I u I 1 u . I m N m I Q N M M 9 M m n O m m 1 I N I 1 o n m . O N m I m m a n N v - u V m X 3 o - n 1 m O ' m I M n 3 m m 1 O I 3 I 1 1 o z o O I M M m m n o I m a m m m A » o z fi Z S 9 8 6 0 I C n 1 I 1 C 1 T 8 8 8 N v 6 6 6 l . 6 6 6 . 6 6 2 4 7 O N E W 8 + + + [ . 1 6 £ 1 6 I L + 4 - 4 - A P P E N D I X B F O R T R A N C o d e o f t h e P s y c h r o m e t r i c L i b r a r y 2 4 8 ) ) N N E E H H C C Y Y B B ( ( 2 H ) A H ) ) K + 1 ( / A + 1 ~ ( 1 6 r . R 6 0 ) 8 0 ) I 8 6 A ) 6 . 5 G K / J ( ) A H * 4 ) N T ) , H 7 8 T R S 1 S I + / ) H A ' 6 ) ) 0 . 5 7 ) O N ( 8 E E 1 C ( 1 S , E P , N E ( 1 N e g a P b i _ c i t a m h c y a p 7 0 9 1 , ) 1 R , N I 4 M B L Y D 5 " ( 2 ) + U P I ) ) 0 4 T O M 1 . P I , I O H R H 5 A 9 . w u 2 R N , ( , , A ( A 1 0 O H R ' 9 E E 0 B B ) ’ I ' T B ) . P P . , I ) ) A 6 v m , S A D B 0 p V P H 6 M E ) ( 0 ( L I 0 p E P D P V ( 0 V E E 1 P ( P E L X H C ( 1 ( c 2 0 T D ( A P 1 D I O ( E I . V V H ( B X B 1 ( 9 P , N C P n P P B T I A D ( - N 1 N N T O X X O C L S I A ( I o L A C - M N H R V A R O N A A 9 H L P R U ' P 0 H H U I E V L A E - w E H 5 - L I - B T - - 1 Y P A - P - T T P A - A 2 P I n U : 3 2 3 2 g u A P P C V E A D P H 7 R ’ E H E C P I s - E H I - B ' — P A P R D H Q V A A G Q I G R D N R z E R m n w E I m ’ P U ' N u N N U Y S E E r E E F M C C C 2 9 1 . N S S M M A A ) R ) O ) O R R H X G P G G I O O K A C D V I R R / . E P P J ( B R U M m ) E C I I N T O N / A m ) ) ) ) ) ) ) ) I . L L I ‘ J H C ‘ L L T A I I Y A A I G B B C C ‘ K U G ” I N N I I I I / K C J ( / ) R L ( B A N N N N N N N N N N N N N N N N N - E E E E E E E E E E E E E E E H I H H H H C O C C C C H I I C O G H H H I H H C C C O C C f ' G O E f R I Y Y 1 Y Y Y 1 Y Y Y a B B ! Y Y Y Y Y ! B B B B a I A R 8 B B B B B 3 B B R D R S A I ) ( N E D E E ) R O A Y R T C ) I ! P R H V K H A P R I / A a A A E I ) S 1 I U P ( I N / T G ) e E g G . W U ( Q Y O K M L A R E m ) N A ) I E / P N D R ( I ) ) ) E T A R H H C G E ) 3 a A V A U L — R K C G I N K C R ( 1 R P K E — - ) ( ( ( ( ( R Z “ S A R C I K ' C C L K E U H I S A I O / E I h A - R C E R P E E I E R F B T J D N i . P G U O O R C R ) A R R I N U 1 N E K T B E P U O V U U P A A C . ' S R T _ c i r C I E H N E N / U A D Y D Y P ( A S Y T T C . E R T R V S T T D R R A A J P ( ( P ( L L R S L B E C S I E O E R E R I C E ( P O R F H P T E I D E M E S S ' N R H I Y S T M V O A R O I A P U P I P C ‘ U A A . A t E I ‘ I I . 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( x . 0 C D P B B E / D . U D 0 P E E V 6 R 0 0 T R ' l . I $ - B / . H R D S 0 I / B S W E / D R S 7 P C S , , , 3 X - S V ( 3 C 0 . ' B V E ( ' V E P P 8 - P N P A A D G 7 E R S P 1 7 P R / 1 [ P R A R 9 V - O , , , , E X 2 P X P 2 - - - - . 1 P P ( T 1 P V N I R P R C X + E E l . A I B P R T N ' A H B D D B T R P O M B 9 fl D ' B U A E D A A S E S I U I A D B U I U I E A m E E 5 D H D T R C V T T A T A - ( P P S O T T - - D V - T T T T V R D 0 : C C C E U - A A A A P L N E O I B A V E C E A I A A V N R N A + N N N N [ A [ - V R R O R O H V S m ) B N ( m 5 P - s 3 P P P R I S D D m m E E R T P N I T P I S P R W T N S P 0 D T ) R D ) I 2 T A M U N U M U A E W S E P V E F E P 3 2 g u A ) 2 O O A ) T R ' E 8 T 0 . 0 E P 7 G 6 9 a 5 g 9 a 9 P 5 1 R / J ( ) R I ) . . 0 1 9 4 9 1 2 6 . 0 ( / ) H I ' A P / P V ' 7 7 5 5 1 . 0 - 1 R A + ) N O ) - > P E 0 0 9 . B A H 0 V T . . ) D 1 0 0 ) 0 P ( 8 1 0 1 ! D ' 9 5 F E 4 O , , 1 0 T B 6 B . 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N L T - o T . 7 L E - N E F D I A I 8 I E F O A . 9 T S R F D O C C T 1 I I A R N E E B D ( 7 N F 1 E I : 2 0 2 g u 0 A 9 ) 8 , 1 - J , ) J , L ( ) P ) N ) P ) 6 , 1 X E L ) N ) P , T N 1 P , . - ) N P J 1 E M a , 0 H O ) T C . A , - N 1 ) ‘ 1 J . - L ! - , 1 J K I . S J 1 - . ) , ( , - J ) J 1 , , J ) - , ) ) ( b J L P , ( L , ' 1 ( , - V L ( L ' B J . , , 5 ( J 1 , I O S T , T 1 . ) T N P E S R E E - F S . D J 8 , N 1 N , ( ( ' o V ) ' I S I 0 1 ( B , ) O 0 F R . P V ( 0 I S , L ) 8 , L ' ) C 1 T , E ) ' J ) ) n ( I ( , R O L , ) R ' S C I S 1 - I ) C D C R H ( Y N . R E 2 T H , L ) ( ' L , ' ) ' S ( ) X ( ) V I ) ) X ) ' ' , C T - E ( 1 ) L , I D H I T , ) P S T ( . , m T ) L K 1 I . m N S A , N T , , - I X ) b ' B 6 T N ( N 3 , 1 D ) I ' 2 O . I L ( . ) ) , ) 6 , ' ) ) ) H I O . L , : 5 5 3 X ( ' N . 6 ) . X D 1 , R L P O ) 3 . ) 1 1 I S : O N ) E . E . e E ' 1 , ( 4 X m ) O A I T o T . J ( N I ' g , ) E 6 1 3 1 A E S , O T a P 6 , X . ) 8 , - . T . A T R . 4 1 1 ( 1 ' S S T , e A E ( 1 I A ( Y I I N E S L ) ) R E . R . L . C H N R 0 2 O ( u N 1 ( 6 A ) ) S T E O 3 L N T u 0 1 I f E ( , 6 , ' 5 2 1 E n R T R E I T . ) E H . . E N L . H 6 3 2 ) . , m 1 C T D 2 I 0 A I o F , ) m 7 ) m L ( T ' h T , 1 1 5 , l L N 1 O L . V 3 S c L 3 , ( , 7 1 . 1 1 x ) t 0 1 L ' 7 A 8 , 3 ( ( ) 6 3 3 6 I ' I , A . 7 I A R C E . E P 1 7 ) S ) 0 . L ) ) ) 0 1 , T ) , U - N G X 0 - a 0 3 U ( ( 3 , A ( 0 A S ' R 0 O I ) L , A E 1 - . e . E L ' T 1 P ( 1 . ) , E s , 1 0 H E N ' ' E 0 6 . e 0 , , 0 0 0 A , O 0 ( A ( 1 1 8 U F . V v ( 1 T ' P T 1 C 8 S 1 U N . T m H 6 x ( I V P B E 0 - 1 o 1 T U , E 0 1 . D I I L L L . O T L E A L ( A J C 0 C S C ( . ( F . ( ‘ . ' ( s 1 E C O , E E ( S , O . T / E R N U T C O E T 9 F 7 - ( R ' , ' ' - w C T 3 8 ) . E T ' ) H E , O L + D T F D T 1 1 ’ , , T - ‘ ' P O 1 E A ( H 7 1 O I L C n L - ( P I A ( + I T I 9 L I T F ( P I - F E E I L U ' R ( T A C S 6 T P . - O O D R 1 L 2 2 9 N F D 2 F T - D C ( 1 A L E 0 L ( m T . I 1 m T D 1 A E ( I E ( I N t R W H E H L N I 0 0 R L S L O O L I I 5 I E T 6 ‘ I F ( 7 ! S C R A m S S D H - - I N I E T C L I R ‘ I L - L - E ( ' I 9 D C L W D T N D S K P C ( I L N T T 1 T 0 o R A O R m A I 5 s A N A E L N H T 0 T 1 F 3 F E F 1 I E E N I R - A I - C N C C F F s L I 1 1 I I E I C R E E I I - I D O R m 9 N ' 2 T E C " I I 1 N S a — — E R - A 6 8 1 6 C c C c C C C . 6 0 6 7 — — — — — — — : 2 0 2 g u C A L L B R O A D C A L L D A Y ' m m u m - . 1 7 ) ' C O N P I E X S E A R G I ‘ , I 3 , C I . O C K C R I T E ( O P , 1 7 ) ' m L E X S E A R C I ' , 1 3 , C I . O C 1 < C A L L F w s m s ) L 6 - 0 1 3 - 0 I P ( S ( 1 . 2 ) . G E . S 3 ) T H E N S H R I N K C R R E F L E C T T H E m m C A L L S H R I N K C A L I . D A r I H H R I T E P J 7 ) ' S N R I N A S E A R C N ' , 1 3 , C 1 0 C K C m I T E ( o F , 1 7 ) ' S H R I N R S E A R o I ' , 1 3 , C I O C I < C A L I . F L U S N ( 6 ) 1 3 - 0 E m I F I F ( L 6 . m . x L 1 ) T N E N C — — — - — — — — T U R N T C I m I V ' I D U A L P A R A M E T E R S E A R C H L - L 0 + 1 S ( L ) - 1 . 0 E 3 0 . I - o c — — — — - — — m n . E ( . I C . L E . N P ) 1 7 0 I F ( J . E o . N P ) C O T O 1 7 1 J a I + 1 I N K O I B . G I ‘ . 0 ) R O D E a I I P ( J P I X ( J ) . m . O ) C A L L L I M I N U ) _ ( 1 1 “ ) 1 7 0 1 7 1 C O N T I N U E c — — - — — - E m 9 4 1 1 1 . 3 C A L L D A r I H N R I T ' E a n ' I N D I V I D U A L S E A R O I ' , I 3 , C I D C K C 9 8 1 1 8 ( O P , 1 7 ) ' I N D I V I D U A L S E A R a I ' , I 3 , C I O C 1 < C A L L F w s m 6 ) E m I F I F ( S ( L 2 ) . C E . S J ) T H E N I n g o . I . T . S I ) T H m s u m - s o D O 1 7 6 0 4 , 1 0 E S V ( L 2 , . I ) - E C ( . I ) 1 7 6 C O N T I N U E D O 1 7 2 J - 1 , 8 S T A T ( I . 2 , J ) - S T A T ( L 0 , . I ) 1 7 2 C O N T I N U E D O 1 7 4 I - 1 , N C D O 1 7 5 . I - 1 , 6 V ( 1 . 2 , I , J ) - V ( L O , I , J ) C 2 7 1 ( ( r ‘ c 1 ( ) A T S ‘ E E + ' ( I r M ) s O a a - ) ( . N m V E s I E ( r ’ 8 ' 1 1 A S E - ( 1 0 A a 1 L + ‘ N 3 ( ( r ) 3 8 — ( r " I ' I ) A S H ) a m m e e - ( ( P N S ‘ I 1 S ( ( T 5 A 3 E 8 L ( ( ) 6 S 0 O ( 0 ' N 0 R z L ( ( ( 3 E 0 8 ( P ( 0 ' 8 T I E H l I N - 0 3 2 1 ( N 0 5 0 3 I ' 0 1 L ° 1 ' T 1 m J 1 . r O S ' H A 5 ' P I ( ( 0 0 3 - ' ' 1 I ( ( 1 ( 3 ' 5 T ‘ . ( ( ( . 0 E 5 D C C C 1 8 ' ( I I 1 1 ( 1 P d d P I 1 / 3 . ' ' 1 ' N N B ' - C ' N ( ’ 1 1 ( ( 1 1 1 1 5 1 I m ( ' I z ( S ' T 1 ) 8 I G J ' 4 8 I ( a 1 ' 4 q 3 0 a 8 ( ' ' 3 ' 5 1 0 ) A I ) a ' N A r 5 m S 3 a ) 1 n 3 1 T A n 3 L - x ' - 1 I ' 3 S N ( ) T 1 E E c ‘ 8 0 ' 0 I 1 E I 3 s ' R 3 M S 3 1 3 m ( 1 I I ' T 1 6 ) - 9 1 N - I C ( L N 3 9 ) 9 J M I 3 1 ' z ' I Z 8 C I S I 3 1 1 3 I 1 V N ' I 5 V d X S m 1 8 I 1 ' S ) d 8 0 E 1 - n 1 9 ‘ 8 X r N 1 - 4 1 - I I ' ( 1 ' . 8 1 1 I . 1 1 . o N 9 s m 1 ' 1 0 S 8 m 3 / 1 C 3 O 3 . 9 ( 1 1 u - ' 1 1 T 1 1 L 1 5 s r 0 ' ( 3 T 1 1 H ‘ 1 1 1 I ' ‘ 1 1 . T 3 m 3 . 1 I - $ 1 N ( I 6 0 a s I C - 8 3 N W 9 5 5 1 1 9 3 3 m 9 ) I C N 1 ” ’ 8 0 8 D ) I N I 0 C 3 3 C S ) C 1 3 a 3 0 ) s C O 3 O 5 0 ' 1 1 1 1 8 S 3 1 1 2 1 5 J 1 ' 8 1 9 - N L 3 d a ' 8 I 1 V 1 9 A 3 - 1 I 5 1 ( 1 1 8 I ' 1 u 1 n 1 1 1 1 N 3 1 ( l I 1 5 I 3 n a I 0 0 3 3 e 1 8 9 1 o N m 1 1 0 ' 1 8 0 . ' 1 3 S 8 I V 5 a N 8 } ! 1 ( 1 O 3 m 1 8 ' 1 0 : m 1 ! 1 m 3 ( ! ) 1 1 ) 5 : 3 - ! 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S E P V . a 9 r 3 0 I T I 1 fl T 3 . 5 1 1 3 1 n M 9 ' E 1 L ' n v v I I A ) m 3 9 1 R 3 I m z J 9 3 a 0 ) 1 1 N d ) ‘ ( m 1 H ( S ) s 1 6 S 1 1 ' S m 0 S 1 1 a X S 1 5 N L m A U 8 3 I E 0 3 ) 6 6 1 T W E ' I 1 3 3 1 ' m C m 1 C 1 m v 3 9 E 1 3 . 1 1 S V I m 6 1 J ' 5 R 9 A 3 - ) m A 1 1 1 ' - I q m I 1 3 0 O - O 3 C ' L 3 I 3 I 1 ' 1 1 I ( v 1 3 ' C m 3 I m z I N O V I 3 3 I m ) ' H w E S 1 S u . S < 1 - I I I 5 3 5 u I 1 8 1 3 O s v 1 1 3 I H ' ) a m 3 L 3 T 1 0 m 9 C ' I M 3 1 5 s . t — I — N 1 N S I V u s ' ' m L I H Z ' 1 I u 0 3 C 3 3 I 3 m D 1 m ' O o = — 1 1 8 m — — 8 I 3 3 - O 9 ~ — - — — — 8 5 1 — - — — 3 D — — — — 6 £ — V A E — — — - — ” a 5 . W 1 D 1 D 3 3 1 a 5 9 3 9 — _ — — — — 3 9 9 1 2 7 2 U U U 3 3 D 3 1 9 1 U D O U ( 1 N ! 9 1 8 1 1 1 3 8 3 N E H T ) ) J , 2 L ( V S B . T L . ) 3 ( N S E S H B T . N ) E ) H 2 T ( 2 S R O . ) 1 ( T T N N I I O O P P T T S D S E I O R I ' N E B B E E H H T E T C O O T E T H T T T C C E T E P A P S ) ) J , 1 L ( V S B - ) J , L ( V S B ( ' S E A N E R E N O R f E 1 S I H ) H T ) S 2 B e 5 . S B g T . . ) ) a D T T N 2 2 P . L . . 1 1 L E ) E ) 5 5 l U 2 J 5 5 . ) J H ) T 2 J ) h . ' R , 3 3 ) ) , T c r a I T 2 ) ) 2 A m 2 . - - ( . ) 1 L 2 2 L E L 2 S - 2 z 3 1 ( 5 L ) ( P ( L o T H T J T N C T I , I N I W 1 N O ) ) U W F X X E L L E P ) S S ) ) 0 L S ( H 2 J . , . V T T E E P S , Q L T S S V L I N P I ’ f I . T E M I N , L E ) ( G B E E ) ( . D . ) I - B H P L V ) ) L S N ) K C H O ( T N ( S J ( . A L J W h e S V h ( 5 T ( s ( U . V ( J c E N E T C V ( S B I ( . U T E ( P P - 5 A - 5 E L T S S ( R I J E ) ( - I X L L L N X S a E E A L E I B S 6 5 s 5 I R E E H L ( E ( 1 ) - I E A . c P S ( 8 3 s 3 D E B V - . H K O 3 H T N J U ) J F V V A P 2 T 2 2 B L ) ) ) , S ) 1 E E E F L N I E ) ) ( L A ( S - S S T , . I ( N J S E L L I . C 7 S R E I E D E 2 2 ( E ( ) S P T ' E . T T A 1 0 M S E ( B ( I L T 8 L N I L L ( I O S P 1 J 2 S E H A - A S N T L A E 7 F S 9 E E D L ( 5 E - E E N N 5 B , 1 S 5 N E A I 2 I R I - - 2 7 L N N P F 1 E E E I I L I A 2 J ( - L ( V E ( 3 S ( . S C D D ( X ) ) L 2 ( S S N I D N £ E L S O A L P G N E T T L - E . N L I ( C S I B S . C K T E R U - ( S T F E 9 m C C I - . S - U L U S A E I 0 L 6 P A T O C R I ( - P S F D - C C F K I O I I E E N E O R £ L N F S O 1 H — - — : 2 0 2 g u A T B S S B R D B I I E E L 1 R D { R N N U S — U - — - - c E S — - G — P - — + - — — — — - — — — — - — — - — - C C C C C C C 7 C - — 0 - 2 - N c ) 2 E H + T ( ) S ) / ) S ) ) 3 N E ) 3 ( E ( 2 ( S H ~ " S T S s ) B S S 2 ~ E ) ) . ) + ) ( 2 T 2 E z I S P R E U - ) 1 B S ) S - B J D E S ( 2 L ( V S S . S 1 1 N E D e E ) T A g D B H N T S E C U D E R a P ) J ( S + N ) E ) N E 2 O E C N 2 , - ( ( S " ' E 2 s E S H L p S ) V S ( E T E 1 A + V . - ( C ) S E ) ) 2 S N ) E G ) ) O . . 1 ) 2 - S C D ) 2 ( a B ( ) 3 ( S N ) ( S C S N S ' S A 2 I S E E 6 B . ( ( ) ) J J T S B H . - ) S , , S A - + T T 2 2 S ) J S R ) L 3 , E E ) ) ) L L ( ) J 2 J ( ( D l . ( 2 - - 2 ) . 2 S 2 U S S s ) B V , 2 5 U z " ) E 2 S ( J S J J H N 5 N I 1 L ) S ( ) D . ( , A V V f E , . T E E T ) - ‘ . H 2 - ) ) ) J A L S L Q . B ) B L R H C , 3 S h T L 1 J ) J J , D T E G T ( B ( - - 3 B ( S & 2 2 ( T T I ( 3 ) c . ( , 2 2 ) ( ( E R V - V ) ) 7 A ( - ( . ' E L L a . E . s U C . 1 r V o D I 3 S ) S E ' 1 . 3 S 2 . ( ( 2 S 2 S L S ) L - u 1 ) - ) E . a S , L B ( L B . o - ¢ D B J B E G 6 ) A T ( ( 5 S V S T V 2 1 ) ) ) S s s e B A ) 1 ( ( T - - V . - , - D B $ V S N A E s - . V U o L ) ) S H 2 ) 2 ) - I ) L L 2 ( 0 E . - y . D E E L S S E E I a . ) A e L . 2 2 2 L o . 1 S L E 2 2 3 B H 3 L I A M . ) 2 C ( ( S E 1 ) ) ( ( ( P S s D s B B 5 o J E o ( ) ( S ( A ( B A . 5 - J + . S S ( ( ( T R T L J I S S E L E L N - D N E ( . - 7 - I V - 2 I S - S ) G I I , F - - F E S S V L O ( E ( I L E R E ' E , 8 N ) 2 E 5 L I S 6 S J - I 6 S I S S E S . F E S E I - A 2 S A E I H 2 1 B T 9 L ( B S B ( 5 I 2 ( D r S - - ( $ B S B E I L D D E S E S I I C D C L 1 - A ( F N I A ) . S - 1 H T ( L 2 E E ( S L E S I E L N I E E I ( E E N p ( ) E T - ( 7 - 0 o S S A L L N I ( E S F E E E A S L A V ( I E E S B D S E E S D D I I ( ( I V L 2 P A E — S o - ( L S 2 S S - E E E T S R E I 1 - - E N S P U B E R - ’ — - — - + + 1 D - - B B C I R : E I 1 2 D I 1 E 0 R - - — - — — — — - - — — - 2 — - — g - — u - - - — — — — - — — — — - A C I c C C C 2 C + 2 1 7 3 ( ( N L T a ’ a V — L ( T 1 E 1 a A + 3 ( ' E ( O 1 O ' N ( ( ( ( 1 r 1 1 3 3 3 3 + - ( ( E 1 N ! 1 1 ( ’ ’ 3 T ( T T 1 3 E x T T 1 x 1 1 3 ( 1 ) A A 3 ( x 3 S S N 1 V v 1 L u E A L 3 E E + + + N n v ( ( ( 1 E - x I 8 H a ( ( x N X N E V - ( ' ' ' E O N ’ ( T T T A O E E E L N A L E T D ' L I ‘ I H T T T 1 1 1 A A A ( E N ( T 5 S S S o G 1 ( E T 1 E E E E N S 1 ' + A c X N E 1 1 1 1 ' 1 I 1 ( E ' ’ L A E A . . . 0 1 1 N E ' 3 E T s 9 9 S A L S 9 I V 3 ' a ) ( N 3 T ' ' ' E L ' T E ( ‘ 1 z o o o 1 E 6 E 5 A E o ( E A 1 ( x - - - 9 3 n 1 L ' ’ ( 1 0 o E 1 s 3 1 ( ( ( S a ° 0 1 x O N a 3 ' 0 t z N t ' N x x N N x 0 ' x ’ 0 ( - O N 0 I 0 v T ( ' o 3 ' ' u - 3 ' ’ 3 t ‘ ' 3 u x T I 0 o 0 a . x T a ' T T ( 1 ' h H 0 1 0 1 q ° v 1 X 3 1 ' x n N T 3 T T L T 3 T ' L t d V 1 3 1 1 ( - A ' ' N a t E O E I ( I ’ — M I H I h r N ( 5 I T 3 ' 0 A 0 A A 3 n 1 ' h J . 0 1 1 o x v ( . T 1 T ‘ 1 q T x X ' 3 E 3 T D ’ ) o ' 3 6 5 3 S E S 1 ( u 3 x . J A I - o 9 I ( D 0 I x ’ 1 1 1 ' 1 E 0 E 0 3 E S ( 0 L 9 ’ 1 H J c K e 1 G u T L c - 1 T E 1 + A 1 3 c 3 N I 3 T E N d m t 3 I H 0 ' H E 3 c 1 N 1 s - I 0 E 0 n D x ' + 1 1 E O 3 a E 5 n 0 N 0 - 1 ) ' 1 1 3 E o 1 A A - - N 3 3 1 0 1 - E E 0 I ' I 0 x . 1 1 3 ) 3 D 1 I 1 3 L 0 1 X £ ‘ 3 N 1 N 9 N 1 t E . 3 D ’ ' x 0 3 N Z S v c 3 s 3 3 t 1 H 1 1 N L L E - - 3 0 0 1 1 1 6 n N S N E 3 x 3 1 x 0 I 1 3 N 3 N N E P 0 ( 1 N L d I U o 0 O 1 3 3 N D x D 1 1 N I U 1 1 3 3 3 x u J - H V n 5 0 3 1 1 3 1 o 0 T u L D 0 1 1 1 1 1 T n T E D L E 3 1 N 3 1 3 I n n ! 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O T J ° 1 1 u a T ' H ( N 1 I W - P H S 9 T 5 o T N I q Q P a ' T O P ' a ° ' n x x V H a S fl ( ’ 3 ’ 1 1 fl S L 9 L ’ L ’ ' s H L T l 3 N + C ) N S 8 m 5 z 0 z I 9 E Y m D 1 T 1 W I 9 A L I - V I 1 S J ( L J - U N T J ( I 1 ' 0 3 - N ' - T 0 A d n N ( 1 I ( ( ( 5 3 T ( : T E ‘ 0 ( T 1 1 E E 1 o , S ' D S ' 1 1 X A N U 1 £ B 9 0 O I ) S T N T a 3 E E E E E 1 s E H G I E S J 8 8 ( ( G H 1 0 O G J S n N N S A n A 1 - n - E N S L H - - O I 1 [ fl - - 1 ' 1 8 I s T N I N N ' I G L £ 5 N L c N I + * + D P ( N ' ( ( 0 ( I G 1 1 1 L J D N W fl T a 1 3 I 0 1 N 3 a 3 L H T N T I I I I 1 N 1 1 I L 1 1 5 5 I N D Z fl ) J N - - 1 1 N N - 1 E N - - O D fl - - - T 8 N I 3 8 G a G S H D I J 6 1 ' s I 6 T G 1 1 I J I Z £ f I d 9 S H l J 1 1 ‘ C O B fl J N I o 1 o I 8 o C o 1 8 H N O V O I C C 0 J N O H H 3 T 1 n o 3 S n I 0 I o 5 3 H D O O 0 E I I I I I O N D D — 3 S S - E 5 0 ( £ 1 L 6 6 6 — — - — - V I 3 9 3 3 3 1 D 1 3 3 D 3 I 2 7 4 U U U U 0 N 3 N a n S H 3 J J J K K . ) I ( C I . S T U I O F O 3 / 0 0 0 9 P H Y A D , R L C O B F I ) E T T U I ) I ( U I ' ) E A ) ( G Y P A T ' I A , M U O X ' C 2 G N ' E A S C , A ( 1 ) H T T L ' P 9 ) 1 6 1 e g a P f ) D Y Y U M . ) A ) ' , A A N S B A ' N / ' ‘ , ) ' ) U h I f ) 5 f E D D S K I I W c ( T A B E A B . K 2 M , , C O M D 9 1 . L r P S 2 2 T S s I 1 h I T 8 S O N D E S C ( ) h a S S R R S s R R H c T C ' I I : A T S c N L ' O ' ) Y e - - - - - - - T T / - s S A 1 E s ) L , T A C H Y Y B B C X C x C D ( Y O D m n J ) K ( Y a ) ) ) ) ) ) ) ) 5 . 1 3 4 S 8 2 6 7 A ‘ ' I ( S S S O N c 1 ) ( X o x c 1 A c L I E , , , , , , , , D N E H R R R M 7 o E E ) 1 I D C W 7 U . L L L L L L L L E E E U R L 1 E V H T T E ' ' M C H D C 8 L L L L L L L L A H L N S E T T T / E C / I A E I E T 9 1 ( ( 1 ( ( ( ( L I S N E C C C ' X m I D I H 1 H A X D T N E I ) M I 1 F T T N R N D N ( T T T T T T T - I D A A A A A T E D I I ' ( c C ' V N A A A A A A ) T R R R L V O T - V T L L T D U A A E R U P R U H R ( ( 7 O T L T T T T T T I T U U L A A A L L 1 1 I Y — L L U S I M N U U L 1 C S S S S O H H H A R R R T A A A R R O - S S L L S S T D I T T C C : N ( E R C A D C R C C C N C C W D 0 E N N A I T K E O 0 2 E D S R B C T 1 I D R D B W D K R B I 0 2 N U H T E 1 — — — — N U N E S E S E — - — — — — — — — — — — — — — - - g 2 - - - — — u 0 — - - - — A 3 I L 5 K 1 / ) ) e J ) g , J a P f . h 1 , 1 L 1 . 1 I ( ( ) 1 ) ) I T ‘ ) ‘ I I ) ) ( S ) ( ( ) ) 1 l A S T ? 2 I I ’ — T ' S G N 6 ( ( I I E P ' I 1 2 2 ( ( ' N P ) S 5 E E O Y N N ) 6 G N E H N E H T ) ) 2 5 , ) L L ( S ( 1 N I H R . T L . S S . ) I ) ) ) ) ( I I I I A ( ( ( ( T V ' ) ) I I T W ' H ' ( N V / / 1 I 5 ( f J I ( O R O / ( c 3 r + S B . h ) N , , ( G ) G H ‘ H C 8 / " " / / 2 2 5 S a ) e J ) J c 1 $ ’ I ‘ 5 S N R ( T . S C ' I / ' ' - ' . I I " " s C - I / € / D ) ( ) T ‘ ' ) ) ( 2 N 0 , ) ) ) ) ) m ( ( ) ) - 1 ) ) 2 2 I A I 1 I N E ) Y 3 G H R ) R 5 S C s m ( C ( C o . Y E I I , U E ) I . ( ( T S 1 T 1 ' I ( ( T , 1 : , , 0 ' Y 2 N J E L V ' ) E I I A I I A ) " / 1 2 3 ‘ 5 S ) ) . 1 E ) c + + ( ( E B - 7 U ) ( ‘ 2 E G F I S " ' 0 I J O ( Y I ' ' G ' ' ' ' B C / C . , 1 , , , 1 3 N ) B I ( ) E Y ( E Y E B / ' S ‘ L 1 I 1 I I I 8 N J - 0 9 ( 1 I ) E 0 0 H N P - Y D I F S ( I I B I I I U S ( B 0 L L L L L 1 3 D ( T Y + N ( ) S N ) ) ) ) ) # E S N O 2 L L L L L S S B S 2 S 3 Y S + + ( ‘ A 1 I C T J 1 N D 0 . 0 . 0 0 1 O I L ) S ( ( ( E ( N ( S ( I ’ ‘ - ' ( ‘ ( ( ( I ) Y ‘ ‘ ' S S . 3 E R U N ( . 0 . . 0 . I C L G H S H S V G I V V V V 0 D I ' F F D I 1 ' ? E ( P - l 1 L 1 ! ( S B 7 O 1 C U L 0 - 0 0 - 0 3 F L F “ S N ( Y A L T N A ' - ' ‘ S O Y S B S 1 1 C C B 1 T - 2 ' - E - C B E S S S E N B S S S A 3 ‘ A B D I I T ' ( : E Y Y A B S S O N O Z 2 E I 1 S T F 1 0 2 R S S S S S S D I 0 C S R R S S R R I 0 2 g u A 3 0 3 1 0 2 0 3 0 1 E N D S U B R O U T I N E F U N C ( L L ) U U U U U U U U U U U U U U U U U U U 2 7 7 5 , D 1 N . L 1 , I A ) , , m J , 1 N ) I T A 3 , N G N , 1 H , , , ( H 1 C S H 2 H x N I = , 1 7 X ) , 6 , , , 1 2 L ) K T A N 1 P L ' 6 L I H , 1 A N , 5 1 , , ‘ ( 2 , H , , N 5 ' I I 5 ) ) P ) , L P fl ' V 3 6 1 P N 1 ) , , , , + R , P , 1 8 A ) H H , O ) ( , 1 I 8 3 > K , 6 ( I 3 ( L 1 , Z , , D , ' N D H B ) H m 6 , ( , ( A c ( 1 3 , T N L ) A , , E 1 2 T , D z ) 0 R ) ( ) 2 , X N , , 1 H O 1 L S ) 1 , I G < 0 , ( ( , 3 I 6 P G H 1 1 H B S ' E 1 J ( , 6 , I Y , . 3 1 R 3 , X A I , x ) I ' . E T B 5 . S 1 A , , fl , ) , L R A , ‘ 1 1 0 ) H ) A P ) I I 1 0 L H ( 6 , I H R , D ( 3 , 1 0 , C 3 G 2 , H . , ( 3 R P F L S E . 1 s G 6 A 0 B ' D N ' , q , , r I ( , 1 2 I , , r ) u ) S m . S R P R U E L m H L S , , , , E . , , 1 O , D s ( a 3 ) ) ) ) S A 1 6 , O 1 I E , 4 0 R 2 1 6 2 , 2 U H , , , ) S r e 1 D R O ) + , ( g ? - ) ' ) 6 , - I , ( H 1 1 I H ( 3 H R , , I , A 6 6 a A , E ' ( ( T ( P P t V , s O Y S , S S I ” > ( , ( S C , 1 S D , P , , P S 3 A , ) 3 H , 8 , 6 , 2 , I a R I T S U , ) . E - 1 , S r ) E ) L I N 1 N x , 2 P O H S ) , H , , E C , 1 ( , ' , I ( 6 S ) K m , E N S 6 C A , A . , K G x S K T L + ( E m O T O > P ( 6 T ( , D a , L I H C I v E O ( L S I , I H E ( E S V 3 I , C S / P S I - I S I D H T D D C E S I 1 A H , ) , ) E C I F H / ( L I I H N / / / / / / / C C , I P E C E E H D G E D L P O B E D G C F E 7 / R N R R N P I A D E D C P K 8 9 E P E / / / / / / / T T R / / m 1 M E E E R C N N N N N N N N 7 E A R M B N A M M M 1 T R A N U ‘ A M M M m M M M L O G A O D O O O O O O : A H O O O N G a O C O O O 6 P C D I C c C C C C C C C W W 0 0 2 g u A C C C 2 7 6 A P P E N D I X G F O R T R A N C o d e o f A R N L I N ' ' ) ) ) ' 6 A ) . , ) 9 1 3 F 1 ' . ‘ , ) 9 , 8 ) ' P A P ) ) N 1 ' A 3 , ' , , . ' , P 1 I 0 , 2 , I 1 ) - ' , P 3 P , 7 ' . N 3 A , 7 , . ( 2 F ' 7 ' ' 2 , . ) 5 . 1 - J . ) J . 1 L E 1 1 1 1 1 O v , ' S c 1 1 . . ) 1 1 I I P 1 , P , ) , P N 5 A 3 N , . , ) ) 1 ) E 1 1 A A M ( 8 r 1 . ' ) ) , ' 0 1 6 J ' , 1 1 ’ ( , 3 P , 1 1 1 . ) E , , A , , m 3 3 B B N . 1 S S m 1 ) S 5 5 ) F X ) J 7 . 2 1 I E 1 I 1 I U 1 2 I ) - J 1 A 3 , , U 1 , 1 A 6 ( A ( . P 2 ( 2 2 1 , 6 ' , ' e 1 1 . 1 2 . P P ) ) 2 g 6 , 0 0 ( 1 a ) P 1 1 2 6 N N F , , P l 1 , , , , 1 1 1 6 3 , 2 ‘ ' , , 1 - - 1 J J A 3 X J 1 P ) I L 7 ' I 0 S 1 I - 1 1 E ) ) ) ) S T . , S B ° F ' ' ' ) 1 ) ) ) I A ) . O ) ( J O ) I ( 1 I I 1 ‘ I 1 , A S m ' ” ' 1 1 1 I 1 - , 1 A A E B U A m ( ( N , 1 P , , 2 A ) I 1 1 I I 1 I 1 2 A 1 S ‘ ‘ B J H ' ( m ) , I L , , 1 1 I U , , , 2 I 1 A 2 1 1 1 1 ( ( A ) X 0 , N ‘ N , S ) + ( , c ° O S B . a J ( ( ( ( ) ) ( 1 ( 1 ° ' m , J ( I L E - E E I ' ) ( I 1 ' ' ' ' ' ( 0 0 , , 7 m . u N H H A H S 2 ( O P 0 , . ( 1 1 ) ) : P n ( 1 ) 1 1 S ) 0 P E N T T 1 L X J ) T n T ( 6 7 I L n H 6 ) : : . ' L 1 ) H 6 , ) B A + ) ) ) ) ) ) ) A A 6 ( - - a 0 ( ( . ( 1 5 5 0 ) s S , 1 - P - I 3 O 0 0 0 4 I M 1 s ( ( a 0 I ) 1 , , I B 3 ( 0 T A 7 ( ) H X E ) O 0 0 0 , M 8 A ‘ 7 S : m 7 , A ' * ) 1 L , , 4 - 3 I ( u F 0 F , + . 3 3 - E 0 ( T E - 1 I 1 n m S 6 , ) ) 1 2 3 1 , N . J . 1 E N 1 1 . ‘ ( 0 n r L ) H 1 S 9 3 9 1 1 8 8 B m . I ' 1 ) 1 s N ) P P I ' 8 3 1 3 S T ) - 1 - 0 n 3 ( , 1 E P . , T A A P T - z - X L X l 0 ’ 0 L 1 - D o O T 0 1 T T - l 1 - J 1 1 o E P w I 9 1 1 1 n O ' W P - U ( A A P 1 G N s A n 3 . ) I 1 - N N . I P 3 . L ( ( - 1 A F E M 1 R 1 . O o ( I 1 N J D N P P ( T . - - N o J P I A B E E m I J F L I 1 + E 3 U e - 1 ' 1 A I ( ( E A 2 P 1 A 1 - D 1 P 7 N - ( M S T T I I J K M I P m ( 1 I ( E ( N n O M m ( T B m 4 T D D E M P 0 I E I E x S L B N - - - I I I E - E R 1 1 6 E T E u L 3 N P 1 N E - u S m 1 1 - ? 1 1 I . u 3 R N A A D T I T ( R N 1 T I 1 1 1 D D B a D I B E ( S 1 B m D I C E 1 l S g I A 1 1 ( m m R I R 1 1 1 L - ? 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H I , I A m N 1 N E G ' I ' A I I ‘ D D I O ‘ I I ' O O A L ' H I - ) F O I ' , 7 C ' U I I ‘ ' S O ' T ‘ - - U D R G I ' H O s 1 E E S I , I , L R R R 1 ( O A F P N I , V H . n S E ) ‘ ) ) D ' ' E H I , P , G L ) m E 2 B ( A , , 2 T I A V I R S P ' : ) A T L O E D D , L 8 Y I H P , E E I M A F I J D O Y T B I I l l ‘ v ' e — g a P . 1 1 1 1 1 1 1 2 R ( F A ' C D I , , ‘ N F 0 , , m 7 " I ' I N U ' 0 L J , 0 ‘ h I S O B U S N I ' I ) ’ G w O A W M G ' ' T A A N - A S T N A s , T S U R ' ' S E , H e , I T ' I ) S I , T ) ( I P ) 6 6 , I O D 1 “ I V I 1 I S R , ' , J - , s ‘ ' E J ' R 1 a y S l c L A S X , , ‘ i 7 ‘ c a R m 1 I y E [ d R 8 H 1 H - ' ) O A ) 1 I ! A E ) ) ) A ' D C G G L G r T u S 5 P I , , M ‘ t E v , J J ‘ N N " ' X A A c F A I S L E L L s ( L A ' ' d N , v ) 1 I P m U I ' ' E ' ' L , H ( R ' N ? 0 F 8 S 0 , H B D ' G F ' - - D E I - 0 3 N A E A ' C M fl l E N P I E L D / N ’ K a I M I C T A F - ) ' D B E , 5 I ( ( ( D ) ' m 1 E O - . 1 ’ J ) L A L M E L 7 N I P D A ' E V A D I ( ) I L ) ) ) ) ) ) R t t E C I P I - ) ' ' . ' ' L R F I H I P J r 8 R E L u u V N ‘ , F J E N Z N P ' H F D E p l “ K K ' , , I , , ' 9 A A I o o l H I O I U A H X N , , , P , N U M - ( K R , A A A A A 1 C t a C I E R I R A ‘ m I I ' M A ' ' I L L L L I m m ' A N - D E 9 I ( ( P ( 9 H I ¢ P P P P W P J 2 S ( P P ( ( ( ( ‘ ( ( I D T ( T m 0 A ' R G A L L L L - - R L L 3 R A T D A A A A A P L I L L U m : G A A O R H A I U P - C C C C 0 - N N N D 1 E E E A I 1 . P P P R E 9 E D O D D D D D 2 N N A E J u B A - A A A A A m P X X o E P E E B E E E ) 4 O C C C D C C A P D 1 1 R O O O N O D ( m R N R R R R R R I 1 R N B U I P E S N 7 1 g u A U C " — 9 9 9 l , J 1 fl K , - U U U U U U 9 0 0 3 7 2 7 8 ’ . a ' . a ' u / ' M ' ( m ' ( u d I d N x r ' S ' u / ( d ’ d . ( I ' ( ( L N L x l - o . u d N ’ ( o ' 9 J ' I s ( ( ’ ( o ' ’ ’ ' s r u ' . d ( . ( ( ( ( ' v n . ’ t I ’ I r I m . ( l ( I ’ c m . 9 m m ) ) - l . . ’ u N ' ( ’ . ' x ( ’ ’ - ' m s r I I u ' r . . ' I t ( ( ( m 9 I t L I z r u ( ' ( ’ 9 m m L u m m A ’ ' - . m - - ’ ' l ) M 8 a - n ( z . ( . u ( ' ' ' A ’ a " m ' s o X 1 . r I v d a I ' ( ' ' . I d . ' ‘ ' 9 a I - t t t ’ . l . ' z ' t u l ' r . . ' - ' - ( . . ' - ( ( ' 1 o ) d ) ’ m . ’ x ' x ) o d u ’ I . s . x r n I I m . n x ( ( m m ’ r . s W ( ’ - . ( ( z u m s ' o ’ v ( . . s ' . ' ) ' S u . . l ) ’ ’ x . l ( m ’ . X ’ ’ t ’ ) . I ( ( ( P a ' u . s x u . ’ . ’ o ' ( m X r I o . 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V 1 1 N ) ) r r 6 2 3 ( ( 1 1 ’ ' 1 1 1 ' ’ 1 3 3 9 5 ' E z r 1 E r L w 1 0 £ S ' + ) N o 8 H H 1 r ) ) 0 L ) ) 3 3 “ L 0 N m 3 o 1 3 3 ) ) ( 1 ' 0 1 ) 0 ' 1 H z ' r 1 1 1 ( P ' 1 3 3 x ’ a 3 1 ) ' ' o o r o J r ( c I o 1 l 1 3 ) ) C . . s s 1 1 P s s ‘ ‘ o 1 3 a ( ) ) u . 1 N c ( ( 1 ) N M q 1 ' 1 m ) v 1 ) a o o - - o N 3 0 1 1 9 1 A C ' s - X a 3 3 t u 1 1 0 x v d — V ( ( 8 3 ‘ - - ( a ( - a ' ( r ( ' r 3 1 - B ( ' 1 ‘ v 3 - ) 1 r ) z 3 a 3 1 L ' I ( n ' c ( 0 0 ' - ' ( r D 3 d I d E o n 0 ( u C 1 1 1 r 3 ) 3 ' 5 0 3 m r 3 I r ' D u 3 x N ' v v ‘ N S r 3 D ’ x ) ) 3 1 N 1 ( 1 z t b I L 1 3 ’ - n C a - - 3 0 - c - 1 X ‘ - - 8 N 1 3 a N ' P M 1 3 ( I ) 1 r N D I t u 3 I E U I - C 3 ) 3 3 ) 5 3 3 0 - 3 8 1 1 a 1 1 N 8 3 ( D . 1 6 1 1 1 n n ) 3 1 1 ' r a N 3 ) 0 - 3 3 0 3 1 ) 1 ' c f 5 ‘ f I 1 ) a 1 N N 1 0 1 L ) 1 0 0 1 1 3 0 3 1 N 3 3 I ' G ‘ M 0 3 1 r H N 3 1 3 1 0 3 3 3 0 1 1 = A N 1 t v 1 1 N 5 3 1 3 u D 1 a o 1 1 1 o : 0 0 1 N O H 2 1 3 n o N N s H a 3 1 1 o o o m H 9 3 3 I 3 3 m I — — 1 o n o 1 ! - — 1 1 1 D S 3 - — - — — - - — — fi 0 — 1 a 1 n — — - — — - n 1 1 t : — - - 0 3 v 1 1 : 1 1 3 3 3 3 N 1 1 1 1 1 0 1 3 3 " ' l N N 1 3 ) 1 ) 1 N 3 1 3 1 1 2 8 4 ( 1 ) 1 3 1 5 / 1 . . 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U J J 1 ' ‘ 1 T N ' Q U I ' O 1 O 3 I 1 E S 3 S m S ’ L 8 M W 3 N 1 O O H U D 1 E 3 D O + S ) . . 3 0 m ( 1 E 1 ( - 3 ' 6 N - N 3 3 1 ( I V C ‘ V I L I b n v 9 ‘ 3 5 " " “ L 1 0 0 3 ) 1 0 3 L 1 0 5 0 ) n ' D n 1 1 1 1 1 v O o 3 - 3 ) - . U z H 1 N 1 x 1 N 3 I 1 1 1 1 1 x 3 I m S s ( ( £ 5 / H 0 8 ’ I " . d I . ' C N I ’ . X 5 5 ( 1 - 1 ) 3 . ( 1 ( ) 1 3 - + z . . ( 1 ) N 3 3 + 3 1 1 s - ( 1 3 5 ’ S 0 ' I — ) I X V H V ' G 3 N 3 I ' . . ’ 5 d 5 N 3 ' 3 . / . d 1 ’ N 0 N 0 . ( ( O ’ 0 3 1 ( 1 N 1 3 3 ° 0 3 . 1 1 ( ( ) ' ( 5 ( 0 I ‘ I 1 5 1 ) 1 1 v ( ( 1 ( 5 1 3 ) 1 — - 1 1 1 1 1 . S - I 3 1 1 3 ) ) ' ) 3 . N N N 0 ' 0 1 3 + ( ( 5 1 ) ) . : 1 1 5 N d 1 5 3 o I 3 3 x 0 3 u 3 ( 3 ' 1 N N ( 1 3 N - . 3 I ’ ’ . . 0 . 1 L 1 m c 0 0 3 ( 5 ) 5 N 0 / ) 5 ' 1 . - 0 / . . 3 1 1 1 1 ( 0 ' ) N 5 3 V A 3 l J I 9 I ' - ‘ 1 3 m 1 9 v ( 3 3 - 0 / 3 / ( ) ( ( ( ( 1 3 o 3 / . 3 1 ( 3 ( ' c 1 z 1 1 1 1 1 1 c ' ' s S N S Q 3 3 ( ’ . 1 ) t 1 I O ( — - 1 1 . 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O P 0 S I . 4 ' 8 1 9 2 9 3 . 0 1 0 + e ) ) S 2 5 0 ( 0 ( 8 X 8 5 I 4 P 1 J . 0 R 0 0 5 . 0 E H C 0 5 2 L . 0 E D O H M L B 0 0 X 0 3 A 2 . M 0 5 T 0 0 I N 0 D A 1 L . G Y . 0 I 0 / 0 H 6 I C 3 A 2 T 0 I L 0 N M S T , 4 0 D G 6 0 N 5 0 I 3 0 S 1 . . 1 N . 0 0 A 0 0 0 X e L 6 S I g a N 1 F T E 0 N P S D P A T 1 0 + e ) ' 2 T 8 S 5 0 8 . 4 A 2 R 2 R 8 5 8 ) 3 0 4 6 6 1 4 0 5 6 6 ( 4 0 0 1 3 2 . . . . . 8 0 0 0 0 0 ) 0 5 I 5 0 ( 0 0 P . . 0 0 g 2 ) 6 1 7 5 u 1 I 5 9 2 4 A 3 ( 9 0 2 7 H . . . . . n 3 1 4 3 2 u S ) 1 1 9 2 5 I 8 9 9 1 0 ( 0 1 7 3 0 C . . . . . 4 4 2 3 1 ) 3 6 0 6 2 2 8 8 0 2 6 ( 0 0 0 0 0 X . . . . . 8 0 0 0 0 0 - - - ' ) 3 0 4 6 6 1 4 0 5 6 6 ( 4 0 0 3 1 X . . . . . 8 0 0 0 0 0 ) 3 4 0 5 5 I 1 0 5 3 5 1 ( 0 0 1 0 0 0 0 . . 0 . R . . ) + + 0 0 0 0 0 9 d - - d 4 0 0 E D ) 2 2 0 1 1 3 0 2 0 ( 0 0 . 7 0 2 ) 1 7 6 5 t 0 1 ( 8 u 0 2 8 5 X T + + 2 5 2 0 1 1 5 9 4 2 1 . 4 F S 2 2 0 5 2 0 9 1 8 . . . . . I S e ) 8 5 e 1 ) 0 ( 3 9 2 0 7 3 0 0 2 P 1 . 0 N . J 0 ( 0 . 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H 0 A A 9 R E 2 0 7 0 6 6 7 9 1 0 . . . 3 4 6 2 ~ * R A L 1 1 3 S 4 4 4 A 0 0 0 S R 0 0 0 0 G 9 4 5 7 5 9 7 2 7 9 . . D . . . 7 7 A 6 9 0 4 0 R 6 3 1 3 0 0 0 3 : F 4 6 2 . L 9 0 7 1 Y T I A 9 7 0 5 7 1 V D . . . I A 0 3 7 I M E R 9 2 2 3 D I 6 5 5 5 N A 4 0 2 5 U 4 5 7 9 O D . . . . R 2 4 5 A 4 G 4 6 0 R 4 3 1 : Y T I V I T C U D N O C : : L T H A : I T L H 0 0 0 E 0 O g a P 5 1 a 1 r t H ’ " 0 0 0 , A 9 0 N T 0 0 0 0 H S N H T E 2 1 A 6 6 6 5 E 0 0 0 7 I 0 ! I 0 0 0 0 0 L O 1 1 S D 0 0 0 e S 1 ! L 9 0 0 0 A 0 0 I O 1 ! D 7 . . . T . . N 0 ' . 0 E 2 R I 2 U L 3 5 1 H H 0 9 9 9 9 N L G S I 1 . . . . I . I 0 2 0 0 0 O . D N 0 N I . S S . L E . E F D V . L 1 R V C D F N N A L A D U E L L H I F N E F I F C O I 5 N I F T . N T A 3 D T 0 X . _ 9 0 7 D . D D . . . N T 0 0 3 D 0 L N L 1 1 6 3 4 A 4 O 5 A 4 . E 1 7 E B 6 1 9 D 2 I 2 3 I e l A 0 D E N H 0 6 L E T D t 1 . O 2 5 6 . 6 . . . A A A D r o p e 2 . . H 1 1 1 0 0 0 I R 7 1 . - . C . Y T D A E . 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' 3 s 0 0 3 0 9 c o 1 1 m 1 3 3 L : 3 1 1 3 z o 3 0 0 0 0 1 3 1 a 1 o 5 3 - 1 0 0 0 0 0 9 L 0 V 9 3 1 p 1 0 s 9 9 9 ' : 3 3 0 0 0 a V 1 : I 3 o 1 1 3 3 5 s 9 0 1 0 ' 1 1 0 1 3 3 a : 5 3 1 3 G 1 s 9 3 1 6 6 6 s 0 0 0 0 u 0 A 0 O 5 0 s t ' : 1 L 1 9 3 0 C : 9 9 : 0 1 L 1 3 1 : 1 0 6 5 ' ‘ 1 0 0 1 O 9 o 0 1 9 0 ' ‘ ‘ ' 0 0 0 o 1 1 1 5 t : 0 1 0 0 3 0 O 0 v 0 ' 0 0 9 0 ' o 0 0 ' 3 s 1 5 5 ' t . 1 1 3 0 0 o o 0 3 0 1 c 3 0 L e . 3 5 3 3 x 0 o 3 6 3 9 1 I I ° . 1 1 1 0 0 3 5 : . 3 ' 1 0 0 0 V 9 5 1 9 1 1 n 0 1 1 1 1 I 1 5 g : 1 3 1 2 : 3 - . 0 s 1 0 3 3 1 : 3 1 1 3 : 9 9 1 1 1 . ! 3 ' 5 1 0 0 0 0 0 0 3 1 t u n — n 3 1 9 9 9 9 5 3 3 : 0 0 : 3 : : 1 1 ' 3 ' 3 3 ( 0 o o : 3 1 0 3 0 0 t C 6 0 o 0 6 : 3 s 0 5 £ 3 I 6 v c 1 3 L 9 9 0 0 1 1 1 1 3 3 1 3 1 £ 9 3 0 0 5 s 5 9 9 1 : " : ' 0 0 0 ' 3 3 0 1 L 1 0 9 1 : 0 3 3 0 3 1 6 L L L 1 5 o 1 0 L 0 0 ' 0 : 0 0 1 3 ‘ ' 3 5 1 ' 1 9 1 : 0 1 1 0 I ‘ 0 z s 9 s ' 3 3 o 1 3 5 o ‘ 3 3 o 1 0 ' ' ' 1 1 o 0 O 1 5 5 3 . 0 1 9 3 1 0 ' : : : 0 0 1 0 o o o d s 0 3 1 5 0 ' v 0 1 0 t ' 1 5 3 0 3 » 1 L 3 1 t I 0 5 3 3 1 1 C : 9 I 9 V 1 6 1 1 0 3 3 0 1 0 1 - 0 0 3 V 0 a L 1 0 : 1 L 3 ) 3 5 I 1 0 G 1 A Z : 1 ) 3 0 0 ' 3 3 3 3 G 6 3 8 ' 3 ‘ ' ° ' 0 6 V 1 3 a 3 3 0 3 3 ' ' 5 L 1 0 3 c 1 9 3 9 s S : L 0 n I 3 s : 9 1 3 1 3 1 3 0 1 o 3 0 O 0 1 ! 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R 6 6 6 8 B 1 1 8 R 9 4 3 B 3 1 1 9 E . . X . . . 7 R 1 8 4 O A . . . ~ - - . 0 F 0 0 0 3 O . . . S 0 3 6 B 5 8 4 A 1 N 5 6 8 1 H 7 5 1 5 7 0 . . V 4 4 4 . B 2 8 5 B 0 0 0 0 A 1 . - 3 S 1 4 6 N 5 + 5 3 7 8 5 9 1 R + + + W 5 9 4 3 e 0 4 8 4 1 R A 0 3 e e e O 7 2 3 7 0 2 6 . . Z L 1 4 6 4 2 8 D 6 8 3 4 0 9 0 0 9 V 1 7 5 6 6 5 N 8 6 0 4 . . . . 0 . 6 1 X . . . . . . W 6 8 7 8 O 8 7 5 4 T 2 8 0 0 2 6 . . F 6 6 5 0 0 O D 5 7 5 1 E 5 1 - 1 . . 0 . . . . N ' - 0 . H 2 2 2 : E C N A T O 7 1 9 8 D 5 5 3 0 . . . . T 4 5 3 9 E 3 1 2 7 T 6 2 5 5 E 7 4 2 9 N 3 2 1 . . D 0 0 0 R + + + : T 7 3 . 8 e e 4 3 2 1 5 5 A 7 8 e N Y 7 . R 7 0 7 L 6 9 1 I 3 9 4 P C 3 1 6 0 . O 2 F 8 0 5 0 U 3 8 2 5 A 3 C . 9 6 2 T . . . H 6 5 1 9 E 0 5 I 0 9 4 V 7 . X . . . 1 3 . . M S N A Y 1 3 4 7 P 9 R 0 A K 1 8 8 6 A H 7 T S . . . . C . L 8 F 3 6 6 9 X . E . N 3 7 5 D 2 1 T 1 . R 2 8 8 F 3 V A 2 . H 8 2 8 5 I P 3 6 4 1 D 1 7 0 1 C O 8 9 6 3 . . 0 . : A E 2 2 2 E - ' - F A 7 7 3 P 9 3 5 8 N 4 1 1 7 E . 7 L . . . W 7 7 7 3 H 8 . 8 0 0 T C U 8 7 4 0 . . . . I - F H 2 0 6 U 4 8 6 6 R 5 2 4 1 0 9 2 6 R I 7 6 0 1 1 3 5 5 T D . . . . 4 4 4 4 E L A F L S 7 9 7 E R 0 3 3 0 X . . . I 9 3 9 4 F 3 4 2 D . . . . F 3 8 7 9 H 5 6 A 7 F 5 8 7 8 D 7 4 2 U 0 1 R 3 D O 0 2 7 3 0 9 8 R . I 0 N 7 3 3 8 V . 5 1 L 0 V 9 . N : H T A A H A 0 G 8 1 1 5 5 1 S S V D 0 7 4 1 1 N 0 6 0 7 5 9 8 1 4 E X N 0 4 3 6 D 0 0 6 5 D 9 . . . . X F A 2 3 F 3 D 0 1 2 6 Y 0 9 1 3 A . 9 9 8 4 U L 0 0 0 O Y 0 6 6 4 K 0 6 6 6 0 8 6 4 4 R L V 0 0 0 K 1 0 2 8 S . . . . 1 3 0 3 S F F . . . S . . . . F 0 8 9 5 D 1 1 1 0 6 E 0 0 0 P 0 6 7 2 0 . Y 0 6 8 0 C A D 2 2 2 3 9 G 2 3 0 0 C A A 2 2 2 : E : C F 1 4 8 . 0 0 0 R A P H 4 6 0 4 . 1 L 5 8 9 1 . 1 + + E F S X . . . . Y T A 7 2 7 . e e e N P 3 2 0 8 N A N 0 7 7 7 I 3 5 1 I E 1 0 7 0 2 7 D 0 5 5 3 V D . . . . 5 4 1 R I 1 C D 0 3 4 0 T N 8 1 1 8 R 9 1 0 2 I A 6 3 9 I 6 6 7 0 S R 6 6 5 : 3 9 . . . D 3 3 3 A E R 0 1 9 3 N . . . A 0 I 3 L 0 4 4 8 1 D . . . . 4 4 4 S A 0 0 0 N F 3 I 0 1 5 Y T 2 4 . 4 L 4 1 6 F D . . . . I V 7 1 5 E F 0 2 2 5 D 5 5 4 M 9 I 8 E X . . . 7 7 8 N 0 0 0 R D E 1 . R I 0 0 0 S F 6 6 5 R 8 9 5 4 V I 1 U A 0 0 0 E D 1 I 7 6 2 3 T T R . . . X E V N - 0 0 0 A 8 4 9 6 C . S U I Y 0 0 7 6 U 8 1 9 1 U . I L T Y 1 1 6 7 K 9 0 9 9 O D . . . . D . O F A 7 8 8 C K 7 8 2 7 S 1 4 1 7 R A 0 3 1 4 N 9 . M H 0 0 0 L 6 9 1 E S 8 8 3 6 L 4 8 3 5 G R 1 9 0 5 O 5 . N 5 . E 0 0 0 Y H 6 5 5 F L 0 3 3 9 S . . . . 4 1 1 C 1 . U 7 N D 0 0 G 0 . . . . . X F . . S 6 1 1 0 8 2 1 4 1 H T : L I T . H A . 3 0 0 9 E 0 6 2 1 2 5 1 0 S I T . . . . 0 5 E . R 8 8 9 5 F 6 A R 1 N 5 : H T 1 0 0 0 0 O E - E ' 0 0 0 0 S H N , A 0 0 0 2 6 7 A A 6 6 6 5 N A E D 0 0 0 5 4 2 H 5 1 N 9 9 N 9 9 R 7 3 3 3 1 L D N 0 0 0 0 S 7 V 0 1 R 9 3 8 2 O 1 I I 6 3 1 2 1 I 0 0 0 A 0 0 . 4 1 . 0 I 7 2 7 9 S F . . . . I D 3 2 1 6 5 0 0 0 0 O D . . . . . N 3 7 . 9 2 . T D 0 8 9 2 e 7 6 3 8 L 2 N S S . 9 . 7 A 0 L 0 0 0 0 0 I 5 L 8 0 9 6 E g . . . . S . I E R - 8 5 . E 5 S . . . . a 0 7 9 2 0 X 8 L U 1 . . I 1 5 6 A 4 D 5 7 R P 7 8 9 3 U . D V 6 . F D A 1 8 0 3 8 2 2 3 4 W L 2 L 0 2 8 4 6 6 4 8 5 5 4 4 4 E E F C D 0 1 1 F 0 0 0 I 3 1 L M L M 1 8 9 9 0 . 1 . 0 R V . . 0 E . 0 0 4 2 1 S 1 I 8 8 8 8 5 N N 3 3 3 F . . . 1 I 3 F . . . . T . 0 0 0 A M ) 1 T 7 7 2 : Y 0 0 0 0 0 A 9 1 N 5 3 3 T r A E 6 4 8 N 7 9 L N L 0 2 2 8 A 8 1 1 0 0 O S A 7 5 0 t C 7 6 7 C T I H 0 . 9 E 1 E B 3 6 6 D 4 9 9 3 0 I E 8 8 C 8 , _ S 7 7 7 I V T 0 5 D 0 D D . . . . Y T N 0 9 5 2 0 E T S 7 7 7 t . . . T L I D . I 2 . O 1 0 0 0 . . . . R . A . . . r 0 0 0 6 . A B S N N . M 3 3 0 0 0 0 0 3 0 0 0 I G S , o p e r I S D A W . N 0 1 D E D I O . C 1 Y T E A D A M S D . D L N R I 0 2 A 2 0 2 5 1 1 R E E R P N R 9 9 9 R . . E . . . . . S D 1 1 1 V 7 7 7 8 1 ( 7 4 8 0 V R T E U 0 D 0 0 0 1 1 1 1 2 A P O 1 0 6 E E U E R 3 E . . . 5 5 3 5 5 V O N 7 6 4 1 9 2 8 9 5 A E 7 6 9 5 9 1 A D O . A H I 7 9 3 9 1 2 N 1 H 6 4 1 7 R H 2 V - T 0 0 1 3 R 1 . . . - P . . . . E . S 1 T A . . . O C 7 4 6 A 0 6 1 1 1 G S R 5 5 5 5 0 P P M 1 2 9 B 0 2 R V 1 1 1 : , 3 N O 2 2 2 2 2 M E A 7 9 3 D . . O E E E 1 M D E Y V 0 0 1 9 5 H L : 5 1 I T T A : R : S U X . . . 1 1 1 R U 2 2 S E A L S T U , T E A P A L E 5 R P R P : N R M R R R 1 2 3 1 2 3 D 0 F T 1 1 2 3 P 1 2 3 D R 1 3 7 E O E O N 1 A 8 R K E E E E E O N E 1 g u A Y T Y T G T D Y E C P T P Y T G Y A A Y M I A , M A A L L L L A E H E C E . A L L T T H H T M C 2 6 . 8 4 2 6 . 3 4 2 5 . 7 5 E M L U P 4 6 2 . 5 2 3 0 4 5 8 . 2 7 6 4 4 5 2 . 8 0 8 4 4 4 5 . 2 4 3 6 G N D 1 0 . 0 5 2 0 . 0 5 1 8 . 7 8 3 0 . 0 5 4 0 . 0 5 . 8 5 6 0 . 0 5 1 9 . 8 9 1 9 0 . 0 5 1 9 . 1 3 0 . 0 5 S O L A R 5 7 9 . 2 - 1 0 . 3 0 6 6 - . . . D B R 2 7 . 9 9 . . . . . N I N D 5 . 1 4 1 0 S N R N 1 2 1 . 1 1 9 8 4 . 8 2 0 5 6 . 4 9 0 9 M C - 3 5 . 5 4 4 1 2 . 6 3 9 6 . 5 5 4 0 . 0 0 0 0 . 2 4 8 1 0 . 0 3 2 0 0 . 0 1 2 0 D F S K Y U P 3 7 . 7 7 0 0 2 3 . 9 3 6 8 1 3 . 7 7 8 4 4 . 6 4 9 1 R S H 0 . 1 3 e + 0 2 0 . 1 4 é + 0 2 0 . 6 3 e + 0 1 0 . 1 5 e + 0 2 0 . 1 0 0 + 1 1 3 1 2 0 5 8 1 0 6 3 V 2 2 2 1 1 4 S 0 0 0 0 1 . . 4 0 8 1 1 . 9 D 4 0 9 E 3 7 4 9 5 4 4 E 8 8 2 0 0 2 R + + + + + N 0 0 0 e e e e e I 0 0 0 2 4 3 5 0 A 0 0 0 1 1 6 1 1 R . . . . . . . . X 0 0 0 0 0 0 0 0 F H 2 2 1 2 1 S 0 0 0 0 1 R + + + + + D E E R 2 8 8 . . 2 P 7 5 9 e e e e e D 0 0 5 R O . . . 0 9 8 S 3 4 6 2 4 3 5 0 E 4 4 5 O S . 8 0 0 1 R 5 5 5 1 1 6 1 1 E 9 1 4 S E 2 2 2 7 X . . . . . . . . A - - - F 0 0 0 0 0 0 0 0 R 4 1 3 E O . . . A S 9 3 0 N 9 5 6 3 8 5 4 1 R + + f W 4 1 4 0 0 3 2 R 4 3 A 8 9 e e e O 2 0 1 7 . 8 . 1 6 L 7 0 6 6 5 l D 2 4 1 7 0 9 7 0 7 7 7 V 4 8 3 . . . . 1 . 6 . . . X . . . T 9 0 0 0 5 . 6 F 5 5 O 0 0 E 4 2 ' 2 0 N - ' N 6 W 9 O 0 N 6 7 2 2 D 9 W 7 3 9 4 . O 9 0 8 4 T 5 D 8 4 2 9 E 4 7 A 5 5 3 e e e N 4 3 2 1 H 2 2 2 . . . . N - B 0 0 0 T 4 5 2 1 R + + + E 0 0 3 8 V 4 4 4 E 9 7 5 E 0 0 0 A R 0 4 3 9 7 L 1 2 E 0 4 1 1 2 H 3 1 V 0 . X . . . . . O 4 F 0 1 8 0 0 0 C . 8 6 3 P 2 2 7 5 U 6 5 4 2 0 A 1 6 4 Y 2 2 0 0 H 2 1 0 K 8 9 4 4 L 7 0 1 S . . . . X . . . F 2 7 5 5 6 6 F 0 4 0 D 4 2 1 6 D 0 - - N 8 6 4 1 2 N 1 W 1 5 6 3 4 I 4 O 7 1 2 8 . W 1 D 1 0 8 5 0 . : A F A 6 9 1 . . . . 5 L H 6 2 3 7 4 7 4 3 6 7 8 4 4 4 4 A S 6 3 8 F N . 5 . X . 4 5 L R 9 1 4 F 8 8 4 5 5 9 A W 4 4 1 3 2 1 0 6 5 R . 9 S . . . . 6 V 9 N 3 9 0 : N T D 0 4 6 6 0 9 9 R . 9 I 7 5 N 5 9 9 5 1 . 5 A A 6 3 8 0 W A 1 9 0 0 G 7 3 8 0 0 1 S S V 3 0 2 0 9 8 1 7 D . . . . A 2 5 0 5 R 9 7 5 5 1 3 - E X . . . . X A 2 2 F 0 F 2 2 6 3 U L 2 1 0 0 0 O L V 0 0 0 4 5 5 7 . . F F . S 0 6 6 0 0 0 R 1 E . . 2 E Y 3 C A 3 9 0 8 0 9 2 D . G C 2 2 2 A A 2 0 0 9 F 7 5 5 1 . 9 R A 0 0 0 P H 0 6 0 3 L 8 6 6 1 2 E F + + S X t . . . . A 7 4 4 1 2 6 D . . . A 0 8 0 N E e e F 6 e 8 3 0 7 9 7 7 2 2 R D 3 4 4 I A 1 0 8 1 N 1 1 0 0 D 0 4 5 2 R 0 3 5 5 R 9 8 9 3 5 0 . . 7 N . 4 4 4 0 7 8 0 0 0 R N 9 A L 7 6 0 D . . . . A 8 3 9 I 0 6 5 0 . . 5 A . I V 4 8 3 F 0 4 3 0 0 9 2 L 0 E X . N 0 0 0 . . D 7 8 7 1 . O 5 R 6 F 5 5 I 0 0 0 S N 1 W 1 O 7 D 7 . 2 5 4 : Y T I V I S S I M E R 1 D I 1 N A 7 R 8 8 6 5 1 S 5 U A 0 0 0 E I 1 7 7 4 1 T R . . . X A 7 9 3 6 1 0 3 9 D . . . . . U S 0 0 0 . I L . O F A 8 7 7 8 2 . M W 0 0 0 5 5 3 1 2 4 Y 0 5 5 7 U 7 K 5 4 6 2 O D . S 7 3 2 9 R A 2 R 3 0 0 5 4 2 1 0 0 0 0 0 0 0 0 6 6 6 5 9 9 9 9 . . . . 1 0 0 0 0 5 3 7 0 5 8 . . 5 . . . . , E 0 0 0 Y 3 1 4 L 8 1 4 6 G R 5 N D 0 0 0 G . . . S . . . . 4 I T . . . R 0 1 8 A 0 0 0 E 6 8 3 0 1 2 7 3 6 3 1 R N 1 , M W 0 0 0 A 2 0 6 N 9 5 2 E 0 0 D 0 A T A E E . : " L O 0 0 0 D 0 0 V 0 N R 6 2 9 W 5 I 0 A 0 0 0 0 0 F I 4 7 6 1 S T . . . . . . D 0 9 4 6 A 0 0 0 0 0 0 E L 9 3 7 0 N 1 2 0 9 0 1 8 R R S . . . . I . 2 . U L 5 6 A 4 3 4 6 0 9 8 8 0 3 D V U 2 1 L 5 2 2 2 2 4 4 8 D A M 9 T R . V . . 0 0 4 0 2 1 L M 1 S D F 3 4 4 0 0 0 E E 1 2 0 5 . 0 D N 3 3 3 I . . . A 0 M 0 0 N W L 2 3 3 A 9 3 2 0 2 , 0 G E B 7 2 4 D 4 9 3 9 0 Y I 9 F . : Y 0 e T T l . D D . . . T W 5 9 2 0 0 I V N . 3 3 , S 3 0 0 0 0 0 0 1 2 2 . . . . . I W . Y N O . T E A I I S D S I D . I A 7 7 8 D L A 8 M . . S 0 0 0 D V 6 R E P E E 6 D 2 . S 5 R 2 E Y A L N 9 9 9 R R . E . . . O 7 D 0 0 0 P 2 R V A O C 8 5 1 6 9 A 6 4 E E V R A E A 6 W H 6 E - D P . : E P M 5 8 6 5 6 E 6 , 8 O S 0 A 5 6 D 0 . . . . E N 3 O : R : R V 9 9 0 0 0 1 8 . . . 3 2 3 3 O M [ S U X S T U E A L , T A 0 0 1 U T R P D 0 F 1 : N R R M R R P 3 D 2 3 1 2 E O N 1 R K E E E E E O N P A Y T G E C P Y T T P Y T G A L Y I M A , M A A . ' A E H E C E L L L T T N M T R P E O Y T A L A u g 1 7 1 5 : 1 3 1 9 8 7 r e p o r t _ t r 1 a 1 5 P a g e 1 2 I D N G - W A V E R A D I A T I O N F I E L D I N S W A T " : A T M O S P H E R E E M I S S I V I T Y : 0 . 8 8 6 3 G R O U N D E M I S S I V I T Y : 0 . 9 5 0 0 D E A D E L E M A 0 . 8 8 6 3 0 . 8 5 2 4 0 . 8 5 2 7 0 . 8 5 3 0 E M L U P 4 8 6 . 3 9 6 7 4 8 4 . 4 1 7 1 4 7 7 . 9 4 1 0 4 6 3 . 8 1 2 8 T E M P E R A T U R E S A N D H E A T F L U X E S I N S O I L : S O I L N E A T C O N D U C T I V I T Y : 1 . 3 0 0 0 V O L U M E T R I C H E A T C A P A C I T Y : 0 . 2 0 0 0 3 + 0 7 . . . . . . . . . . . . l . . . . . . . . . . . . . . . O . . . . . . . . . . . . . . . . . 0 . . . . . 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I 5 L 4 9 2 H F 6 9 5 R 7 8 9 4 4 4 T E A E S 2 7 9 F X . . . L F 7 3 1 H 5 5 7 7 A - - U 0 8 8 . R 4 L . . 5 N O 4 V O 0 9 I 4 . R . V 1 . . 5 : H T 8 A A 3 7 0 0 8 W A 9 6 9 0 5 9 1 . S S V 9 9 0 6 2 . . . E X . . . . . X A F 5 2 0 0 6 2 2 F 0 U L 0 1 1 0 0 O L V 0 0 0 0 0 5 9 . 0 0 F F 5 3 3 4 . . . S 0 0 2 5 R 9 0 0 0 E 0 . . 3 E 7 Y C A 5 3 8 0 3 0 0 9 D . G C 2 2 2 A A 7 0 1 0 7 2 5 . 2 . 9 . R A 0 0 0 P H 5 5 8 0 8 3 6 1 8 2 t E F + + + S X . . . . 7 1 6 1 . N e e e F 4 9 7 0 : E C N A : F 9 2 4 Y L 5 2 1 T A 3 2 5 3 7 0 . . . 7 0 2 9 0 0 4 5 5 . E 9 6 2 4 4 4 R . . D 4 4 4 I A 8 8 6 1 T 6 5 2 N 0 I 7 4 0 C 8 7 7 D 0 V D . . . E 1 9 9 R 0 I A 1 4 6 : A 3 . 5 6 4 N . . . 0 9 . 8 R 8 A N 0 0 0 7 7 4 L 5 1 I 0 8 S R 0 0 0 Y T I V . . I 7 A . V 1 3 7 E F 0 1 0 1 I 0 0 6 L 2 E N 0 0 0 X . . . R D 5 6 5 M 2 . O 3 R F 5 I 0 5 5 0 S 0 L 4 8 3 F D . . . . S 5 5 5 1 1 9 7 S 4 I 0 0 E U A 0 1 8 4 3 T T R . . . X C 0 . S U 0 0 U . I L E V I T E R 4 2 8 7 D I 2 2 5 6 N A 6 0 9 8 D . O F A 0 6 C 5 K 4 0 3 O 4 D . . . . N 5 . 2 1 1 M W 0 0 0 L 4 0 8 E S 5 1 3 7 R A 5 6 3 2 O 0 . 4 N 2 2 E , 0 0 0 H Y 8 5 F 8 L 0 2 0 0 G 5 1 1 6 R C . . . 2 1 N U D 0 0 0 X G . . . F S . . . . 4 2 1 Y 0 4 3 4 U 0 7 8 2 7 3 8 0 5 7 . . . 5 2 1 1 1 6 2 1 : I . . T F . 5 S . 0 2 . R 5 5 7 E 7 1 0 4 L A 0 0 0 . H 7 T 2 . E 4 6 2 0 5 2 1 0 7 0 0 E 0 O E I A 2 4 R N : H 1 0 0 0 S H 0 6 6 6 5 3 9 9 9 9 N L 1 . . . . I I 1 5 0 3 . , N O W 0 0 0 A 0 5 3 A E 0 0 D 0 A 7 4 W 2 2 M E T L D 0 0 N 0 V 0 S 0 0 1 R 5 5 2 0 9 N 0 0 0 0 O 0 2 7 . D 4 T . . . . . . N 7 D 0 5 0 0 S . S . 5 . L 8 A E 0 0 0 I 0 0 0 4 L 5 9 2 7 0 0 9 I 0 0 0 0 0 0 A F . I 4 2 4 0 I E 0 5 2 E . 2 R R S . . . . . X 2 . . I 4 U L 3 5 A 9 U 1 4 2 D 5 5 1 4 0 4 7 0 U 1 . F . D V 4 6 L ' 1 2 2 2 L 2 7 5 9 5 5 L 6 8 . N 0 R . . . V T 0 0 3 0 1 E 5 5 5 5 1 F . C D 4 5 5 F S 0 0 0 I 1 8 8 8 5 N 3 3 N 3 I . . . F : a . . . T . 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O I Z C ‘ S I O O L I ' S I c v c s ' v t t t s t ’ C I 9 9 0 9 ° 0 1 1 9 L 6 ' 9 8 0 6 0 ‘ I - G C C L ' O I - L 9 9 9 ' z c - L Z I I ' S O - L Z I I ' S 9 - : x n w a L v a n 0 9 a t 9 £ ' 6 t L 6 ' 6 t c s ' o z 9 0 ‘ I : $ 9 ' I 8 9 9 ‘ 1 8 9 9 ° 1 8 a t [ 8 9 £ ° o z 5 5 ‘ s ! = a a n m v a a d u a L s 0 ' 0 9 0 ' 0 5 0 ' 0 9 0 ' 0 5 0 ° o 9 0 ' 0 9 0 ' 0 9 0 ' 0 9 0 ' 0 9 0 ' 0 = S S S N X D I H & 0 1 6 a L 9 s 9 c z t 0 H 9 : u a w i I t a b v d s r v t n ’ u o d a x ( . 9 6 1 c t = s r L I b n v 3 2 2 3 V 8 1 V 8 1 V H I 5 3 X fl 1 8 A S 8 3 1 1 3 O H V 3 8 0 1 5 1 0 1 1 1 1 “ ' H O I - L V 8 0 0 0 N V M 0 ’ M I S H 3 0 8 0 8 V A ' 8 H 3 1 ' D H 8 8 ' 8 9 - 8 8 L I ' 0 - 9 9 0 9 ' 0 0 9 1 0 ' 0 6 1 8 9 ' 8 9 0 ' 8 1 0 0 ° 0 8 9 ' L 0 8 9 L ' 6 9 9 L 6 8 ' 0 L O L 9 ° 0 9 1 ° 8 1 9 I 8 1 . 1 1 3 3 8 3 8 8 0 8 8 3 8 2 8 3 M » O H I H 8 A 8 8 0 0 8 V 1 0 5 H I I S H 0 1 3 1 8 D H 1 3 0 0 ' O H a n o n 3 . 1 V 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s . . . . . . u . . . . . . . . . 8 9 6 9 ' 8 1 9 8 8 L ' 8 1 0 L 9 8 ' I I 8 8 9 8 ' 6 6 8 9 9 ' 9 8 0 8 1 ' 9 8 9 9 9 ' 8 I L 8 9 ' 8 9 1 8 8 ' 8 8 L 9 0 8 ' 9 6 I L 9 0 8 ' 9 6 1 ’ X fl 1 8 1 V 3 H L L ' 8 1 6 8 ' 6 1 8 L ' 6 1 8 8 ' 0 8 L 9 ° 0 8 8 8 ' 0 8 6 6 ° 0 8 8 1 ° 1 8 9 9 ' 1 8 8 L ' 8 8 0 9 ° 9 8 ’ 3 8 0 1 1 1 8 3 8 9 1 3 1 9 0 ' 0 9 0 ' 0 9 0 ' 0 9 0 ' 0 9 0 ' 0 9 0 ' 0 9 0 ° 0 9 0 ' 0 9 0 ' 0 9 0 ' 0 ‘ S S 3 N X D I H J . 0 1 6 8 L 9 9 9 8 8 I O H ! ) : 8 3 1 1 V 1 L 0 + 3 0 0 0 8 ' 0 ‘ A J J O V d V D L V 3 H D I 8 L 3 H D 1 0 A 0 0 0 8 ° 1 : u u a n a n o a L V 3 H 1 1 0 5 H 9 ' 0 0 1 . 9 8 0 0 8 3 3 0 S 8 3 A V 1 0 1 : 1 1 0 5 H 1 5 3 X I I 1 8 1 V 3 8 0 H V 5 3 8 M V 8 3 8 H 3 & - 0 0 9 6 ° 0 0 9 ' 9 8 G N D 9 0 1 8 ' 9 1 - 8 8 9 8 ' 8 L 0 8 0 ° 0 8 9 0 6 9 6 ' 8 8 9 8 9 0 9 ' 8 9 8 6 L 8 9 ° I 6 9 9 L 6 9 ' L 9 0 0 9 6 ' 0 0 9 9 8 ' 0 1 8 ° 0 8 9 6 ' 6 8 8 9 6 9 0 ' 9 8 - 1 8 9 8 ' 8 1 - 8 6 1 0 ' 9 8 9 1 L L 9 ' 8 L 9 9 9 9 I ' L L I 9 0 9 9 ' 8 6 9 9 1 0 L ° 9 0 1 0 0 9 6 ' 0 L 9 9 8 ' 0 6 6 ' 0 8 0 8 ' 6 8 8 8 L 1 8 ' I 9 - 9 1 6 8 ' 9 8 - 9 9 9 9 ' 6 8 9 6 8 L 0 ' 8 9 9 8 8 0 1 ' 9 6 9 8 1 9 ' 0 6 9 8 9 6 0 ' 8 6 1 0 0 9 6 ' 0 9 9 9 8 ' 0 8 9 ° 0 8 1 9 ° 6 8 1 : 8 6 0 9 ' L L - 1 8 9 0 ' 0 6 9 L 8 9 9 ' 8 1 9 6 0 L 8 ° 8 9 L 8 9 9 ' 8 1 9 1 8 9 8 ° 0 6 0 ' 6 8 8 0 1 . 0 3 0 8 O S O V H 1 4 0 0 J S H 8 0 m o o 0 N 9 0 V 8 8 1 V 0 V 8 8 1 V 0 V 8 1 K ! 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M V 0 0 0 0 0 8 3 H J 0 0 5 0 0 3 H . 3 0 0 0 0 V 8 8 1 H 9 9 9 9 9 V 8 0 H A 0 0 0 S 8 0 0 L 8 9 6 9 H 9 1 6 6 6 6 H 1 8 8 0 0 0 V 0 0 0 ' S ' ' ' ' 1 1 0 0 0 V 1 9 8 8 9 8 I 9 L 8 . 1 3 6 ' ' ' 6 ' ' ' 0 0 0 0 H 0 8 0 0 0 8 0 0 ' ' ' 0 8 8 . 9 0 0 0 0 0 0 I 9 3 1 9 0 9 6 H 8 S 5 ' ' 1 9 . 9 1 V 0 0 0 3 8 0 5 I ° ' ' ' 3 ' 0 9 3 6 . 8 8 8 9 1 0 0 9 9 9 8 ' . 1 V 8 X 1 1 0 8 L 9 8 9 9 0 6 9 0 4 A 1 1 1 9 0 1 1 1 8 0 9 9 0 V 1 9 8 6 0 8 . 8 ' 0 A $ L 9 9 . 8 ' ° ' 0 0 0 A 3 9 1 9 8 1 5 9 8 8 L 8 L 1 1 H 6 . 0 8 5 ' ' ' 9 L 8 0 0 0 8 1 3 8 8 8 3 0 8 9 L 1 8 8 . 1 ! 8 8 8 8 ' ' ' 8 1 8 8 8 8 9 H H 8 8 8 0 0 H 0 0 0 1 8 ' ' ' ' 1 ' V H ‘ ' H 0 8 0 0 0 3 0 0 0 0 0 0 0 0 N s t u n 1 V 0 6 V 8 8 0 0 8 5 9 8 H 1 9 1 ' - L 0 9 X 0 9 9 0 1 6 8 0 3 9 9 9 0 D H J 8 8 9 ‘ 1 1 1 0 0 0 0 V 0 0 8 8 3 H 9 8 8 6 9 0 1 8 1 H O 9 ' 9 3 3 0 6 9 8 6 J . 8 6 0 0 0 6 ' ' ' . 0 ° ' 8 9 1 0 1 3 8 8 S ' 8 0 8 1 A L n o 8 9 9 I ' ' ' ' ' V 8 ' ' ' 1 1 0 0 8 8 9 9 8 ' 1 0 6 . 1 8 8 8 S 0 0 0 0 0 1 9 0 0 0 V 0 5 N 8 H ' 8 8 8 9 9 H u L d I 0 3 1 5 0 ° ' V 8 . ' 6 0 ' A e : ( . 8 6 G V 0 0 1 8 9 9 0 3 3 8 . J 1 8 G V H 8 8 8 . 0 V 0 8 8 9 1 8 8 S 0 H 1 0 I 8 0 0 0 6 6 6 A 6 6 6 6 6 1 8 3 . 0 3 8 6 6 S 8 8 8 8 ' 8 ' ' 3 ' ° 8 8 H 8 8 8 8 . ' ° ° 8 8 8 8 9 A 0 H 3 L L L 3 3 0 3 8 9 3 0 ' ° ° 0 0 0 1 1 1 1 8 V 8 0 H 9 8 1 A 8 1 3 0 8 0 0 0 0 8 V 8 1 8 6 6 V 3 V 0 0 ' V V 8 9 0 9 8 A 1 9 9 9 8 0 8 — 8 H 9 A 0 9 8 8 9 8 ! L L L D V 8 8 9 0 9 1 - V ° ' ' 1 8 0 3 ° ' ' ° ° S 0 D 9 1 6 H 0 9 8 1 0 9 8 8 9 ' 8 9 A 0 0 0 5 8 8 8 8 8 8 E 8 H 9 . 9 8 ‘ ’ 8 ' 8 6 6 ° ' ' ' 0 0 3 H 0 8 8 8 8 8 H X 3 3 3 . V 9 1 1 1 9 9 9 8 8 8 8 8 H 1 O H 3 V 8 ‘ 8 ‘ 8 . A 8 9 9 : ' ° ° 8 8 8 8 8 5 3 1 1 1 ’ ' ' ‘ 9 0 0 O 1 0 1 V H 1 . 9 3 3 V 8 V V I 0 0 0 S 1 ' 1 0 . S 0 X . 0 8 8 8 8 8 I I 8 8 8 0 0 8 8 8 8 8 8 8 8 O I 8 8 1 8 I H 8 0 8 1 u 1 z H 8 u 3 3 3 0 H 5 3 0 3 0 H I . V 8 8 X 3 3 3 3 1 A 1 9 1 1 8 V V V 0 b 3 1 1 1 . 1 9 1 1 9 1 A 0 3 D 8 1 8 A I H ’ H V V V H 3 3 D 3 1 n 1 1 L L H H L v ' ' A 1 1 5 H 3 2 3 B I B L I O G R A P H Y B I B L I O G R A P H Y A S A E . 1 9 8 “ . P s y c h r o m e t r i c d a t a , A S A E D a t a D 2 7 1 . 2 . S t a n d a r d s ( 3 1 t h e d . ) . S t . J o s e p h , M i c h i g a n . A S A E . 1 9 8 “ . M o i s t u r e m e a s u r e m e n t - f o r a g e s , A S A E S t a n d a r d $ 3 5 8 . 1 . S t a n d a r d s ( 3 1 t h e d . ) . S t . J o s e p h , M i c h i g a n . B a k k e r - A r k e m a , F . H . 1 9 6 1 . D e s o r p t i o n a n d a d s o r p t i o n i s o t h e r m s f o r a l f a l f a o f f o u r g r o w t h s t a g e s i n t h e t e m p e r a t u r e r a n g e o f “ 0 ° F t o 1 2 0 ° F . M . S . T h e s i s , D e p a r t m e n t o f A g r i c u l t u r a l E n g i n e e r i n g , M i c h i g a n S t a t e U n i v e r s i t y , E a s t L a n s i n g , M i c h i g a n . B e c k , J . V . 1 9 8 5 . M E 8 6 0 C l a s s h a n d o u t o f N L I N p r o g r a m . P e r s o n a l c o m m u n i c a t i o n . M e c h a n i c a l E n g i n e e r i n g D e p a r t m e n t , M i c h i g a n S t a t e U n i v e r s i t y , E a s t L a n s i n g , M i c h i g a n . B e c k , J . V . a n d K . J . A r n o l d . 1 9 7 7 . P a r a m e t e r E s t i m a t i o n i n E n g i n e e r i n g a n d S c i e n c e . J o h n W i l e y & S o n s , I n c . , N e w Y o r k , N e w Y o r k . B o x , M . J . 1 9 6 5 . A n e w m e t h o d o f c o n s t r a i n e d o p t i m i z a t i o n a n d a c o m p a r i s o n w i t h o t h e r m e t h o d s . T h e C o m p u t e r J o u r n a l 8 : 9 2 - 5 2 . B r a d f o r d , K . J . a n d T . C . H s i a o . 1 9 8 2 . P h y s i o l o g i c a l r e s p o n s e s t o m o d e r a t e w a t e r s t r e s s . I n : 0 . L . L a n g e e t a l . ( e d . ) . P h y s i o l o g i c a l P l a n t E c o l o g y I I W a t e r R e l a t i o n s a n d C a r b o n A s s i m i l a t i o n . S p r i n g e r - V e r l a g I n c . , N e w Y o r k , N e w Y o r k . B r u c k , I . C . M . a n d E . v a n E l d e r e n . 1 9 6 9 . F i e l d d r y i n g o f h a y a n d w h e a t . J . A g r i c . E n g n g . R e s . 1 H ( 2 ) : 1 0 5 - 1 1 6 . B r u t s a e r t , H . 1 9 7 5 . O n a d e r i v a b l e f o r m u l a f o r l o n g - w a v e r a d i a t i o n f r o m c l e a r s k i e s . w a t e r R e s o u r c e s R e s e a r c h 1 1 ( 5 ) : ? 9 2 - 7 9 9 . B u s i n g e r , J . A . , J . C . W y n g a a r d , I . I z u m i a n d E . F . B r a d l e y . 1 9 7 1 . F l u x - p r o f i l e r e l a t i o n s h i p s i n t h e a t m o s p h e r i c s u r f a c e l a y e r . J o u r n a l o f t h e A t m o s p h e r i c S c i e n c e s 2 8 : 1 8 1 - 1 8 8 . C a m p b e l l , 6 . 8 . 1 9 7 7 . A n I n t r o d u c t i o n t o E n v i r o n m e n t a l B i o p h y s i c s . S p r i n g e r - V e r l a g , I n c . N e w y o r k , N e w Y o r k . C a r l s o n , R . E . , D . N . Y a r g e r a n d R . H . S h a w . 1 9 6 3 . F a c t o r s a f f e c t i n g t h e s p e c t r a l p r o p e r t i e s o f l e a v e s w i t h s p e c i a l e m p h a s i s o n l e a f w a t e r s t a t u s . A g r o n o m y J o u r n a l 6 3 ( 3 ) 9 8 6 . 3 2 4 D y e r , A o o . n c J s i . t e c S a t n y a t n 9 d 6 3 l 7 . u 1 f : x 8 5 . - 7 H l 2 i a 1 c y . k e s r . . 1 9 Q 7 u 0 a . r t e x - F r l l u y g J r o a u d r i n e a n l t l a r o e f t R i o o y n a s l h i p M s e t e i o n r o l t o h g e y 3 2 5 C l a c k , B . J . a n d D . M c D o n a l d . 1 9 7 7 . T h e d r y i n g p a t t e r n o f g r a s s s w a t h i n t h e f i e l d . J o u r n a l o f t h e B r i t i s h G r a s s l a n d S o c i e t y 3 2 : 7 7 - 8 1 . C o u t u r i e r , D . E . a n d E . A . R i p l e y . 1 9 7 3 . R a i n f a l l i n t e r c e p t i o n i n m i x e d g r a s s p r a i r i e . C a n a d i a n J o u r n a l o f P l a n t S c i e n c e 5 3 : 6 5 9 - 6 6 3 . D a l e , J . A . , D . A . H o l t a n d R . M . P e a r t . 1 9 7 8 . A m o d e l o f a l f a l f a h a r v e s t a n d l o s s . A S A E P a p e r N o . 7 8 - 5 0 3 0 , A S A E , S t . J o s e p h , M i c h i g a n . . d e V r i e s , F . N . T . P . a n d H . H . v a n L a a r ( e d . ) 1 9 8 2 . S i m u l a t i o n o f P l a n t G r o w t h a n d C r o p P r o d u c t i o n . C e n t r e f o r A g r i c u l t u r a l P u b l i s h i n g a n d D o c u m e n t a t i o n ( P u d o c ) , N a g e n i n g e n , t h e N e t h e r l a n d s . d e N i t , C . T . a n d H . v a n K e u l e n . 1 9 7 2 . S i m u l a t i o n o f T r a n s p o r t P r o c e s s e s i n S o i l s . C e n t r e f o r A g r i c u l t u r a l P u b l i s h i n g a n d D o c u m e n t a t i o n ( P u d o c ) , W a g e n i n g e n , t h e N e t h e r l a n d s . D e x t e r , S . T . 1 9 9 7 . T h e m o i s t u r e c o n t e n t o f v a r i o u s h a y s i n e q u i l i b r i u m w i t h a t m o s p h e r e s a t v a r i o u s r e l a t i v e h u m i d i t y . J o u r n a l o f A m e r i c a n S o c i e t y o f A g r o n o m y 3 9 : 6 9 7 . D e x t e r , S . T . , 8 . 0 . S h e l d o n a n d D . I . N a l d r o n . 1 9 9 7 . T h e e q u i l i b r i u m m o i s t u r e c o n t e n t o f a l f a l f a h a y w h e n e x p o s e d t o a i r a t v a r i o u s r e l a t i v e h u m i d i t i e s . A g r i c u l t u r a l E n g i n e e r i n g 2 8 : 2 9 5 . D u f f i e , J . A . a n d N . A . B e c k m a n . 1 9 8 0 . S o l a r E n g i n e e r i n g o f T h e r m a l P r o c e s s e s . J o h n W i l e y & S o n s , I n c . , N e w Y o r k , N e w Y o r k . D y e r , J . A . a n d D . M . B r o w n . 1 9 7 7 . A c l i m a t i c s i m u l a t o r f o r t h e f i e l d - d r y i n g h a y . A g r i c . M e t e o r o l . 1 8 : 3 7 - 9 8 . E l d e r e n , E . v a n , J . d e F e i j t e r a n d S . P . J . H . v a n H a v e n . 1 9 7 2 . M o i s t u r e i n a g r a s s s w a r d . J . A g r i c . E n g n g . R e s . 1 7 : 2 0 9 - 2 1 8 . F u e s s , F . N . a n d M . B . T e s a r . 1 9 6 8 . P h o t o s y n t h e t i c e f f i c i e n c y , y i e l d a n d l e a f l o s s i n a l f a l f a . C r o p S c i e n c e 8 ( 2 ) : 1 5 9 - 1 6 3 . G o u d r i a a n , J . 1 9 7 7 . C r o p M i c r o m e t e o r o l o g y : A S i m u l a t i o n S t u d y . C e n t r e f o r A g r i c u l t u r a l P u b l i s h i n g a n d D o c u m e n t a t i o n ( P u d o c ) . W a g e n i n g e n , t h e N e t h e r l a n d s . H a n s o n , C . C . ( e d . ) . 1 9 7 2 . A l f a l f a S c i e n c e a n d T e c h n o l o g y . A m e r i c a n S o c i e t y o f A g r o n o m y , M a d i s o n , W i s c o n s i n . H a r r i s , C . E . a n d J . N . T u l l b e r g . 1 9 8 0 . P a t h w a y s o f w a t e r l o s s f r o m l e g u m e s a n d g r a s s e s c u t f o r c o n s e r v a t i o n . G r a s s a n d F o r a g e S c i e n c e 3 5 : 1 - 1 1 . 3 2 6 H a y h o e , H . N . a n d L . P . J a c k s o n . 1 9 7 “ . W e a t h e r e f f e c t s o n h a y d r y i n g H e n d e r s o n , S . M . 1 9 5 2 . A b a s i c c o n c e p t o f e q u i l i b r i u m m o i s t u r e . A g r i c u l t u r a l E n g i n e e r i n g 3 3 : 2 9 - 3 1 . H e n d e r s o n , S . M . a n d R . L . P e r r y . 1 9 5 5 . A g r i c u l t u r a l P r o c e s s E n g i n e e r i n g . J o h n W i l e y & S o n s , I n c . , N e w Y o r k , N e w Y o r k . H i l l , J . D . , I . J . R o s s a n d B . J . B a r f i e l d . 1 9 7 7 . T h e u s e o f v a p o r p r e s s u r e d e f i c i t t o p r e d i c t d r y i n g t i m e f o r a l f a l f a h a y . T r a n s a c t i o n s o f A S A E 2 0 ( 1 ) : 3 7 2 - 3 7 4 . H i l l , J . D . 1 9 7 6 . P r e d i c t i n g t h e n a t u r a l d r y i n g o f h a y . A g r i c u l t u r a l M e t e o r o l o g y 1 7 : 1 9 5 - 2 0 4 . H i l l e l , D . 1 9 7 7 . C o m p u t e r S i m u l a t i o n o f S o i l - W a t e r D y n a m i c s ; a C o m p e d i u m o f R e c e n t W o r k . I n t e r n a t i o n a l D e v e l o p m e n t R e s e a r c h C e n t e r , O t t a w a , C a n a d a . H e l m a n , J . P . 1 9 8 1 . H e a t T r a n s f e r ( 5 t h e d . ) M c G r a w - H i l l B o o k C o . , N e w Y o r k , N e w Y o r k . H o t t l e , H . C . 1 9 7 6 . A s i m u l a t i o n m o d e l f o r e s t i m a t i n g t h e t r a n s m i t t a n c e o f d i r e c t s o l a r r a d i a t i o n t h r o u g h c l e a r a t m o s p h e r e s . S o l a r E n e r g y 1 8 : 1 2 9 . 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A g r i c u l t u r a l E n g i n e e r i n g 1 S ( 6 ) : 1 9 8 - 2 0 1 . J o n e , T . N . a n d L . 0 . P a l m e r . 1 9 3 6 . N a t u r a l d r y i n g o f f o r a g e c r o p s . A g r i c u l t u r a l E n g i n e e r i n g 1 7 ( 1 0 ) : u 3 3 - u 3 u , u 3 7 . J o n e , T . N . 1 9 3 9 . N a t u r a l d r y i n g o f f o r a g e c r o p s . A g r i c u l t u r a l E n g i n e e r i n g 2 0 ( 3 ) : 1 1 5 - 1 1 6 . J o n e , T . N . , 0 . A . L e o n a r d a n d I . E . H a m b l i n . 1 9 A 1 . M e t h o d s o f f i e l d c u r i n g h a y . T e c h n i c a l B u l l e t i n 2 7 , M i s s i s s i p p i S t a t e C o l l e g e , , M i s s i s s i p p i A g r i c u l t u r a l E x p e r i m e n t S t a t i o n , S t a t e C o l l e g e , M i s s i s s i p p i . 3 2 7 K e m p , J . G . , G . C . M e s e n e r a n d W . S . R o a c h . 1 9 7 2 . D e v e l o p m e n t o f e m p i r i c a l f o r m u l a f o r d r y i n g o f h a y . T r a n s a c t i o n s o f A S A E 1 5 ( u ) : 7 2 3 - 7 2 5 . L a n g e , 0 . L . , P . S . N o b e l , C . B . O s m o n d a n d H . Z i e g l e r ( e d . ) . 1 9 8 2 . P h y s i o l o g i c a l P l a n t E c o l o g y I I W a t e r R e l a t i o n s a n d C a r b o n A s s i m i l a t i o n . S p r i n g e r - V e r l a g , I n c . , N e w Y o r k , N e w Y o r k . M i l l e r , D . A . 1 9 8 ” . F o r a g e C r o p s . M c G r a w H i l l B o o k C o . , N e w Y o r k , N e w Y o r k . M i l t h o r p e , F . L . a n d J . M o o r b y . 1 9 7 “ . A n I n t r o d u c t i o n t o C r o p P h y s i o l o g y . C a m b r i d g e U n i v e r s i t y P r e s s . L o n d o n , U K . M o n t e i t h , J . L . , G . S z e i c z a n d P . E . W a g g o n e r . 1 9 6 5 . T h e m e a s u r e m e n t a n d c o n t r o l o f s t o m a t a l r e s i s t a n c e i n t h e f i e l d . J o u r n a l o f A p p l i e d M e t e o r o l o g y 9 : 8 5 7 - 8 6 1 . M o n t e i t h , J . L . 1 9 7 3 . P r i n c i p l e s o f E n v i r o n m e n t a l P h y s i c s . A m e r i c a n E l s e r v i e r P u b l i s h i n g C o m p a n y , I n c . , N e w Y o r k , N e w Y o r k . M o n t e i t h , J . L . ( e d . ) . 1 9 7 5 . V e g e t a t i o n a n d t h e A t m o s p h e r e V o l . 1 : P r i n c i p l e s . A c a d e m i c P r e s s , N e w Y o r k , N e w Y o r k . M o n i n , A . S . a n d A . M . O b o h k o v . 1 9 5 4 . B a s i c l a w s o f t u r b u l e n t m i x i n g i n t h e g r o u n d l a y e r o f t h e a t m o s p h e r e . T r u d y G e o p h y s i c s I n s t i t u t e A N S S S R N o . 2 u ( 1 5 1 ) : 1 6 3 - 1 8 7 . N e l d e r , J . A . a n d R . M e a d . 1 9 6 4 . A s i m p l e x m e t h o d f o r f u n c t i o n m i n i m i z a t i o n . T h e C o m p u t e r J o u r n a l 7 : 3 0 8 - 3 1 3 . N e t e r , J . a n d W . W a s s e r m a n . 1 9 7 “ . A p p l i e d L i n e a r S t a t i s t i c a l M o d e l s . R i c h a r d D . 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