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Filmed as Xerox University Microfilms 300 North Z e«b Road Anri Arbor, M ichigan 48106 I I 76-12,411 CHAMBERS, Richard Lee, 1947INF0BMATICN CCNTENT OF GRAIN-SIZE FREQUENCY DISTRIBUTIONS IN ooas t a l deposits o n THE EASTERN SHORE OF LAKE MICHIGAN. Michigan State University, Ph.D., 1975 Geology Xerox University Microfilms , Ann Arbor, Michigan 40106 INFORMATION CONTENT OF GRAIN-SIZE FREQUENCY DISTRIBUTIONS IN COASTAL DEPOSITS ON THE EASTERN SHORE OF LAKE MICHIGAN By Richard Lee Chambers A DISSERTATION Submitted to Michigan S ta te U n iv e rs ity in p a r t i a l f u l f i l l m e n t o f the requirements f o r the degree o f DOCTOR OF PHILOSOPHY Department o f Geology 1975 ABSTRACT INFORMATION CONTENT OF GRAIN-SIZE FREQUENCY DISTRIBUTIONS IN COASTAL DEPOSITS ON THE EASTERN SHORE OF LAKE MICHIGAN by Richard Lee Chambers The problem o f d is c rim in a tin g sediments deposited by d i f f e r e n t sedimentary processes in ad jacen t environments from two rece n t deposit io n a l systems on the eastern coast o f Lake Michigan has been i n v e s t i ­ gated by m u l t iv a r ia t e s t a t i s t i c a l methods. These two systems are located a t 1) L i t t l e Sable Point and 2) the Sleeping Bear P o in t/ Manitou Passage a rea . environments. Each system has fo u r adjacent d ep o s itio n a l The L i t t l e Sable Point environments are: beach, a e o lia n , f l u v i a l , and o ffs h o re bar and trough; w h ile the environments in the Sleeping Bear Point/M anitou Passage area a re: shore (<20m in w ater d e p th ), t r a n s i t i o n a l and profundal (>75m>100m in water d ep th). beach, shoal and near­ (>20m<75m in w ater d e p th ), The sediment from the L i t t l e Sable Point area has a lim it e d g r a in - s iz e range (00 to 30) and is derived from a s in g le source area. The Sleeping Bear P oint/M anitou Passage sediments are derived from several source areas and are c h a ra c te rize d by g r a in -s iz e s ranging from lag g ravels to f i n e s i l t and c la y . Principal-components a n a ly s is was performed on frequencies of occurrence o f sediment w it h in g r a i n - s i z e in t e r v a l s in o rder to group R icha rd Lee Chambers the s iz e classes in to independent c lu s te rs which may r e f l e c t deposi­ tio n from d i f f e r e n t tra n s p o rt mechanisms. I t was found t h a t in both systems, the size classes were c lu s te re d m ainly around two fa c to rs which account f o r 70-90% o f the t o t a l g r a i n - s i z e va ria n c e. G e n e ra lly , s p e c if ic s iz e classes c lu s te re d around s p e c if ic fa c to rs and each c lu s t e r corresponds to one o f the log-normal sub-populations observed on t r a d i t i o n a l cumulative p r o b a b ilit y graphs o f g r a i n - s i z e frequency d is trib u tio n s . The main c lu s t e r s g e n e r a lly correspond to 1) sizes coarser than 10; 2) sizes from 10 to 3 .5 0 to 40; and 3) sizes 40 and fin e r. These groups are thought to represen t re s p e c tiv e ly surface creep bed load, mixed suspension bed-load ( s a l t a t i o n and non-uniform suspension), and uniform suspension. D is c rim in a n t-fu n c tio n analysis o f sediment samples from the Sleeping Bear Point/M anitou Passage area demonstrates t h a t g r a i n - s i z e d is t r i b u t i o n s can be used f o r d is c rim in a tio n providing t h a t a v a ila b le g r a in - s iz e s do not l i m i t the r e s u ltin g d is t r ib u t io n s and t h a t the environments are c h a ra c te riz e d by d i f f e r e n t energy c o n d itio n s . D is c rim in a n t-fu n c tio n a n a ly s is and Q-mode f a c t o r a n a ly s is o f the L i t t l e Sable Point sediment samples show t h a t when a v a ila b le g r a in sizes are lim ite d in s iz e range, s i m il a r g r a in - s iz e d is t r i b u t i o n can occur in environments t h a t are c h a ra c te riz e d by supposedly d i f f e r e n t energy co n d itio n s. I t i s apparent from t h i s study t h a t source areas which provide a v a r i e t y o f g r a i n - s i z e s , the presence o f sediment in the s iz e range 40 and f i n e r and d i f f e r i n g energy conditions are p re ­ r e q u i s i t e f o r e f f e c t i v e environmental d is c r im in a tio n . R ich a rd Lee Chambers I t has been shown th a t sediment te x tu r e { L i t t l e Sable Point sediments) i s not always e f f e c t i v e f o r d is c rim in a tin g rece n t en viron ­ ments. The problem o f id e n t if y in g a n c ie n t environments is f u r t h e r complicated because o f 1 i t h i f i c a t i o n and diagenesis o f the sediment. Even i f sedimentary environments, be they recent or a n c ie n t, cannot be uniquely i d e n t i f i e d by g r a in - s iz e s tu d ie s , the knowledge o f energy tra n s p o rt le v e ls obtain ed by principal-com ponents an a ly s is o f pre­ s e n tly o p e ra tiv e systems w i l l aid in the i n t e r p r e t a t i o n o f p a le o -flo w regim es. For my p aren ts, A ilee n and Bob ii ACKNOWLEDGMENTS I would l i k e to take t h is o pp o rtu n ity to thank Dr. Sam B. Upchurch, d is s e r ta tio n ad visor and good f r i e n d , f o r his continued encouragement throughout t h is p r o je c t . Sam o r i g i n a l l y suggested t h is study and co n trib u te d much to i t s completion. We had some good discussions on the g o lf course, sorry about a l l the times you l o s t Sam. I would also l i k e to thank the o th e r members o f my committee f o r t h e i r comments on the manuscript, they a re: Drs. J. Alan Holman, James F is h e r, Maynard M. M i l l e r and A rthur P. Pinsak. Many thanks to Dr. "Chip" Prouty f o r jo in in g the committee only one week before the o ra l defense. Thanks are extended to Dr. Edward Rothman, U n iv e rs ity o f Michigan, Department o f S t a t i s t i c s , f o r his time discussing various aspects o f the m u l t iv a r ia t e techniques used in th is study. Special thanks to the Great Lakes Environmental Research Laboratory, NOAA,ERL. Dr. Pinsak made i t possible f o r me to be permanently employed w ith the Limnology D iv is io n o f the Lake Survey Center (now p a rt o f the NOAA,ERL,GLERL research group in Ann Arbor, Michigan) and work f u l l time on my d is s e r t a t i o n . The Lake Survey's research vessel SHENEHON was made a v a ila b le to me f o r sample c o l l e c ­ t io n in the Sleeping Bear Point/M anitou Passage area in northern Lake Michigan. The tim e , funds, and m a te ria ls made a v a ila b le to me by these o rg an iza tio n s are g r a t e f u l l y acknowledged. iii L a s t, but not l e a s t , thanks to my w ife Linda who put up w ith a l l the chaos t h a t goes w ith a d is s e r t a t i o n . iv TABLE OF CONTENTS Page LIST OF TABLES........................................................................................................vi 1 LIST OF FIGURES................................................................................................. v i i 1 NOMENCLATURE......................................................................................................... INTRODUCTION ......................................................................................................... 1 PREVIOUS WORK..................................................................................................... 6 LOCATION AND PROCEDURES ............................................................................... 10 Location and C o lle c tio n o f Samples.............................................10 Sample Preparation and Size A n a ly s is ........................................ 17 V a r i a b l e s .............................. 23 DATA ANALYSIS.......................................................................................................... 24 INFORMATION CONTENT IN GRAIN-SIZE FREQUENCY DISTRIBUTIONS ..................................................................................................... 29 I n t r o d u c t i o n ............................................* ........................................29 L i t t l e Sable P oint Depositional Area ............................... 35 * ...................... 39 .................................... 67 ..................................................... 71 Principal-components an a ly s is D is c rim in a n t-fu n c tio n an a ly s is Q-mode f a c t o r an a ly s is . . . Sleeping Bear Point/M anitou Passage Depositional A r e a ................................................................................ 78 D is c rim in a n t-fu n c tio n a n a ly s i s ............................................ 79 Principal-components an a ly s is ........................................ B5 COMPARISON OF THE LITTLE SABLE POINT AND THE SLEEPING BEAR POINT/MANITOU PASSAGE ENVIRONMENTS ............................................ 105 v T able o f C ontents ( c o n t 'd ) Page CONCLUSIONS............................................................................................................. H ° APPENDICES............................................................................................................. I 15 APPENDIX A .................................................................................................... 116 APPENDIX B .....................................................................................................119 APPENDIX C .....................................................................................................122 LIST OF REFERENCES............................................................................................126 vi LIST OF TABLES age Principal-components r e s u lts o f te s t cases . . - 34 Principal-components r e s u lt s o f a l l L i t t l e Sable Poin t samples.............................................................. 41 Principal-components r e s u lts o f L i t t l e Sable Poin t beach samples . * ................................... 46 Principal-components r e s u lts o f L i t t l e Sable Poin t bar and trough samples ........................... 50 Principal-components r e s u lts o f L i t t l e Sable P oint S i lv e r Creek bed samples ...................... 54 Principal-components r e s u lts o f L i t t l e Sable P o in t a e o lian samples ................................... 58 D is c rim in a n t-fu n c tio n c l a s s i f i c a t i o n o f the fu n c tio n a l groups te s te d a t L i t t l e Sable P o i n t ....................................................................... 68 D is c rim in a n t-fu n c tio n c l a s s i f i c a t i o n o f the fu n c tio n a l groups te s te d a t the Sleeping Bear P oint/M anitou Passage area ........................... 82 Principal-components r e s u lts o f the Sleeping Bear Point/M anitou Passage shoal and nearshore samples ......................................................... 86 Principal-components r e s u lt s o f the Sleeping Bear P o int/M anitou Passage beach samples . . . 90 Principal-components r e s u lts o f the Sleeping Bear Point/M anitou Passage profundal samples . 94 Principal-components r e s u lts o f the Sleeping Bear Point/M anitou Passage t r a n s it io n a l samples 98 vi i LIST OF FIGURES F ig u re 1. Page Index map o f study areas located on the eastern shore o f Lake Michigan. L i t t o r a l currents and wind vectors (W) are in d ic a te d bya r r o w s ............................. 5 2. Sample lo c a tio n s a t L i t t l e Sable P o in t, Lake Michigan. S it e 1 locates a e o lia n samples west o f S i l v e r Lake. S ite 2 lo c a te s beach, S i l v e r Creek and o ffsh ore bar and trough samples. S ite 3 lo c ate s the beach samples c o lle c te d a t the morainal source a r e a ..................................................................12 3. Bathymetric c h a rt o f the Sleeping Bear P o in t/ Manitou Passage a re a , LakeMichigan ...................................... 16 4. Sample lo c a tio n s a t the Sleeping Bear P o in t/ Manitou Passage a rea , Lake Michigan. Piston cores and surface samples were taken a t lo c a tio n s in d ic a te d by a tria n g le . Only surface samples were c o lle c te d a t s ite s in d ic a te d by a dot ...............................................................19 5. Shipek surface sampler (a ) and an Alpine p is to n co rer (b ) used f o r sample c o l le c t i o n a t the Sleeping Bear P o in t/M an ito u Passage a r e a , Lake Michigan . . . . 21 P h i- p r o b a b il i t y graph o f 4 o f the 9 t e s t cases used to determine the e f f e c t o f clo su re on c o r r e la tio n c o e f f i c ie n t s ........................................................................................ 32 Rankit graph o f the October 1972 sample set c o lle c te d a t L i t t l e Sable P o in t. S o lid l i n e is the R a n k it normal cu rv e , dashed l i n e is the observed curve .................................................................................... 37 6. 7. 8. Rankit graph o f the dune 1973 sample set c o lle c te d a t L i t t l e Sable P o in t. S o lid l i n e represents the Rankit normal cu rve, dashed l i n e is the observed c u r v e ....................................................................... 38 9. P h i- p r o b a b il i t y graph o f the grand frequency d i s t r i b u t i o n o f a l l L i t t l e Sable Poin t samples . . . . vi i i 43 Page F ig u re s ( c o n t 'd ) 10. P h i- p r o b a b il i t y graph o f the grand frequency d i s t r i b u t i o n o f the L i t t l e Sable Point beach samples ............................................................. 11. P h i-p ro b a b i1i t y graph o f th e grand frequency d i s t r i b u t i o n o f the L i t t l e Sable Point bar and trough s a m p l e s ...........................................................................52 12. P h i- p r o b a b ilit y graph o f the grand frequency d i s t r i b u t i o n o f the L i t t l e Sable Point S ilv e r Creek bed samples ......................................................... 56 P h i-p ro b a b i1i t y graph o f the grand frequency d i s t r i b u t i o n o f the L i t t l e Sable Point a e o lia n samples ............................................................................... 60 13. 14. Loading p r o f i l e s of each L i t t l e Sable Point e n v ir o n m e n t ............................................................................................ 63 15. D is c rim in a n t-fu n c tio n a n a ly sis c l a s s i f i c a t i o n o f the L i t t l e Sable Poin t environments ........................... 70 Ternary diagram o f normalized Q-mode f a c t o r loadings o f the L i t t l e Sable Point samples .................... 74 Areal d i s t r i b u t i o n o f two Q-mode fa c to rs f o r the L i t t l e Sable Point Samples ............................................ 77 Index map o f the environments a t the Sleeping Bear P o in t/M an ito u Passage area used i n the d is c r im in a n t-fu n c tio n a n a ly s is . The environments were defined a f t e r French (1964) 81 16. 17. 18. 19. 20. 21. . . . . D is c rim in a n t-fu n c tio n c l a s s i f i c a t i o n o f the Sleeping Bear Point/M anitou Passage environments ...................................................................................... 84 P h i- p r o b a b il i t y graph o f th e grand frequency d i s t r i b u t i o n o f the Sleeping Bear P o in t/ Manitou Passage shoal and nearshore samples .................. 88 Phi-probabi 1 i t y graph o f the grand frequency d i s t r i b u t i o n of the Sleeping Bear P o in t/ Manitou Passage beach samples ................................................. 92 ix Page F ig u re s ( c o n t 'd ) 22. 23. 24. 25. P h i- p r o b a b ilit y graph o f the grand frequency d i s t r i b u t i o n o f the Sleeping Bear P o in t/ Manitou Passage profundal samples .................................. 96 P h i-p ro b a b i1i t y graph o f the grand frequency d i s t r i b u t i o n o f the Sleeping Bear P o in t/ Manitou Passage t r a n s i t i o n a l samples ......................... 100 Loading p r o f i l e s o f each Sleeping Bear Point/M anitou Passage environment ................................... 103 Diagrammatic g r a in - s iz e frequency d is t r i b u t i o n s . Dashed l i n e represents a narrow g r a i n - s i z e d i s t r i b u t i o n which r e f l e c t s only one energy l e v e l , w h ile the s o lid l i n e represents a w id er g r a i n - s i z e d i s t r i b u t i o n r e f l e c t i n g th re e energy le v e ls ............................................................................... Ill x NOMENCLATURE Canonical v a r ia b le these a re weights r e f e r to the standard scores o f the responses, - X. sj where X -- r e f e r s to the $ . i t h in d iv id u a l *J J on the j t h measurements, X. equals the J mean and $ . equals the standard d e v ia tio n s , J computed from the c o r r e la t io n m a trix . th is value in d ic a te s the degree to which the e x tra c te d f a c t o r s , w ith eigenvalues >1 represent v a r ia tio n s in a c o r r e l a t i o n m a trix . Communality Eigenvalue and eig en ­ vector a value X such th a t Ax - Xx = 0 , where A is some symmetrical m a tr ix , has a non-zero s o lu tio n i s c a lle d an eigenvalue ( c h a r a c t e r i s t i c ro o t) o f the m a trix A, and the non-zero so lutio ns o f t h is equation are c a lle d eigenvectors ( c h a r a c t e r i s t i c v e c to rs ) corresponding to X. Factor analysis m u l t i v a r i a t e technique used to discern s im p lify in g r e la tio n s h ip s in complex data. I t may be used to study re la tio n s h ip s between v a ria b le s (R-mode) o r between cases (Q-mode). Factor axes these re p re s e n t best f i t t e d lin e s through data c lu s te rs in a h y p e r -e llip s o id a l space. H y p e r-e llip s o id a l space- an e l l i p s e described m a th e m a tic ally as one w ith more than 3 orthogonal axes. Loading p r o f i l e a g rap h ic al re p re s e n ta tio n o f loading values versus the v a r ia b le s . Loading value t h is valu e a v a ria b le the c lo s e r b e tt e r the represents the degree to which c o rr e la te s w ith a f a c t o r a x is ; th e value is to u n it y the c o rre la tio n . 2 Mahalanobis D th is is a measure o f the d is tan ce between two sample c l u s t e r m u l t iv a r ia t e means. the j t h principal-com ponent o f the sample o f P - v a r i a t e observations i s the l in e a r compound P rin c i pal-components Yj = Al j x l + “ ,+APjx P whose c o e f f i c i e n t s are the elements o f the c h a r a c t e r i s t i c vector o f the sample m atrix corresponding to the j t h la r g e s t root , the c o e ff ic ie n t s o f the i t h and j t h components are orth og on al. The sample v a ria n c e o f the j t h component is xy Profundal in lim nology, th is term means deep w ater, with q u ie t c i r c u l a t i o n . Ranki ts a non-param etric s t a t i s t i c used to t e s t n orm ality o f values f a l l i n g i n t o some ranked o rd e r. S a lt a t io n the mode o f movement o f p a r t i c l e s in a series o f jumps upwards from the bed; i t is in te rm e d ia te in ch aracter between surface creep and suspension. Surface creep in sedim entological terms, t h i s re fe rs to l a r g e r p a r t i c le s transported by r o l l i n g o r s l i d i n g , i n t e r m i t t e n t l y along the bed. Suspensi on - the mode o f movement o f p a r t i c l e s which are supported by the f l u i d ; two types o f suspension are recognized: in nonuniform suspension there is an increased g ra d ie n t o f p a r t ic le s nearer the bed, w hile uniform suspension denotes wholly f l u i d supported p a r t i c l e s . Varimax r o t a tio n - one o f several types o f ax is ro ta tio n s used in f a c t o r a n a ly s is ; f o r t h i s type o f r o t a t io n the axes remain orthogonal to each o th e r . INTRODUCTION Sedimentological models o fte n use concepts based on funda­ mental r e la tio n s h ip s between g r a in - s iz e frequency d is t r ib u t io n s and dep ositio nal environments. Considerable controversey has developed o ve r the usefulness o f sediment te x tu r e and summary s t a t i s t i c s (e .g ., mean, standard d e v ia t io n , skewness, and k u rto s is ) as environmental in d ic a to r s . This dispute a ris e s in p a rt from the m u ltitudinous g r a i n - s i z e measurements and computational methods developed since the turn o f the cen tu ry. Many authors ( e . g . , Friedman, 1967; V is h e r, 1969; Upchurch, 1970a) have demonstrated the usefulness o f d i f f e r e n t combinations o f summary s t a t i s t i c s f o r environmental d is ­ c r im in a t io n , w h ile others ( e . g . , Shepard and Young, 1961; Schlee, e t a l , , 1964; Solohub and Klovan, 1970) using the same or d i f f e r e n t s t a t i s t i c s claim the environments cannot be d is crim in ate d on te x tu re alo n e . F urther com plications in the use o f g r a in - s iz e s t a t i s t i c s to make environmental in te r p r e t a t io n s are d erived from the i n s e n s i t i ­ v i t y o f the s t a t i s t i c s to su b tle d iffe re n c e s in sample te x tu re s and to polymodal mixtures o f sediment from d i f f e r e n t sources. In order to e f f e c t i v e l y d is c r im in a te between d e p o s itio n a l environments i t is im portant to understand the sources o f en viron m entally s e n s it iv e t e x t u r a l in fo rm a tio n w ith in sediment samples. This re q u ire s i d e n t i ­ f i c a t i o n o f the most useful s t a t i s t i c s f o r c h a r a c te r iz a t io n o f se d i­ ments. These s t a t i s t i c s are those which are l e a s t v a r ia b le w ith in 1 2 sediment types and most v a r ia b le across sediment types (D a v is , 1973). The use o f m u l t iv a r ia t e s t a t i s t i c s in sedimentology ( e . g . , Im b rie, 1963; Im brie and Van A ndel, 1964; Klovan, 1966; Davis, 1970; Upchurch, 1970a, 1972; A l l e n , e t a l . , 1971; Mather, 1972; Davies and E th rid g e , 1975) has shown a considerable increase in the l i t e r a t u r e during the past twelve years stemming from the a v a i l a b i l i t y o f high speed computers. With the computational ca p acity o f the computer i t has become possible to use a la rg e number v a ria b le s in order to c h a ra c te r iz e s iz e d i s t r i b u t i o n s , r a th e r than l i m i t i n g comparisons to summary s t a t i s t i c s . For example, the percent o f sediment in 1/2 phi in t e r v a l s can be used as v a r ia b le s . Use o f s iz e frequency data as v a ria b le s allows principal-com ponents and Q-mode f a c t o r analyses to i d e n t i f y those size frequency v a ria b le s which c lu s t e r in t o indepen­ dent groups and are en viro n m en tally s e n s itiv e v a r ia b le s . These v a ria b le s may then be r e la te d to s p e c if ic sediment d is t r ib u t io n s which r e f l e c t d e p o s itio n a l environments and to the energy le v e ls in these environments th a t co n tro l sediment te x tu r e s . This study te s ts the e ffe c tiv e n e s s o f sediment s iz e d is t r ib u t io n s as environmental d is c rim in a to rs by means o f m u l t i v a r i a t e techniques. This study also shows why f o r c e r t a in sediment types d is c rim in a tio n is not always p o s s ib le . Two c o n tra s tin g Recent d ep ositio nal systems were chosen f o r t h is study to provide an o p p o rtu n ity to ev alu ate the e f f e c t s o f provenance and hydrodynamics on sediment te x tu r e in adjacent en v iro n ­ ments. The f i r s t system studied is located a t L i t t l e Sable P o in t, Michigan ( F ig . 1 ). This area was chosen because of the lim it e d g r a in -s iz e range a v a ila b le to the system (Upchurch, 1970b). Most o f the sediments in the beach, o ffs h o re bar and trough, a e o lia n , and f l u v i a l environments o f the main study area are lim ite d such th a t d iffe re n c e s in te x tu r e between the re s p e c tiv e environments should be a function o f hydrodynamics, r a th e r than o f provenance and hydro­ dynamics. The second study area is located in the Manitou Passage area o f Sleeping Bear P o in t, Michigan (F ig . 1 ) . This s i t e was se lec ted because o f the presence o f a v a r i e t y o f sediment provenances and hydrodynamic environments (French, 1964). The sediment in t h is area ranges from lag gravels and coarse sands on shoals to f in e fr a c tio n s in clu d in g deep water s i l t and c la y . 4 Figure 1: Index map o f study areas located on the eastern shore o f Lake Michigan. L i t t o r a l currents and wind vectors (W) are in d ic a te d by arrows 5 88 87 86 85' 46 46' W'nd/ E.c.n.6. A4»ckm«W'-^ C ity .Chi ^ /S L E E P IN G BEAR P O IN T / M ANITOU P A S S A G E / 45' 45 T r*««r*a C ity , Michigan 44 ° 44 Lake Michigan W in d / Milwauki L IT T L E SA B LE P O IN T 'G r i n d H i v i n 43' 43 42 42 k il o m e t e r s mry 20 20 40 so SO STATUTE M ILES 20 86 40 100 PREVIOUS WORK Every d ep ositio nal environment is assumed to have charac­ t e r i s t i c energy co n dition s and flu c tu a tio n s through time and space (Sahu, 1964). Recent workers (Folk and Ward, 1975; Sahu, 1964; V is h e r, 1965, 1969; Klovan, 1966; Friedman, 1967; Upchurch, 1970a, b; A lle n e t a l . , 1971; and B u lle r and McManus, 1972) have suggested th a t sediment te x tu re s seem to r e f l e c t d i f f e r e n t energy co n d itio n s. However, Visher (1969) states th a t s i m i l a r tra n s p o rt mechanisms operate w it h in a number o f environments and the consequent sediment te x t u r a l responses may be s i m il a r . Klovan (1 9 6 6 ), Bagnold (1 9 6 7 ), and Davis (1970) concluded th a t the e n t i r e g r a in - s iz e spectrum r e f l e c t s tra n s p o rt energy and should be analyzed in o rder to charac­ t e r i z e environments o f d ep o s itio n . Many authors have shown th a t most s ize d is t r i b u t i o n s are composed o f two o r more log-normal sub-populations r a t h e r than a s in g le , l o g - l i n e a r curve (Krumbein, 1937, 1938; O tto , 1939; Doeglas, 1946; Inman, 1949; Bagnold, 1956; F u l l e r , 1961; Spencer, 1963; V is h e r, 1969; and Upchurch, 1970b). Moss (1962, 1963) suggests th a t th ree sub-populations could e x i s t w it h in the same sample and th a t they are produced by d i f f e r e n t tra n s p o rt mechanisms: creep; 2) s a l t a t i o n ; and 3) suspension. 1) surface Moss r e la te d these tra n s p o rt mechanisms in terms o f f l u i d mechanics concepts developed e a r l i e r by Bagnold (1 9 5 6 ). 6 7 In 1969, V ish er discussed a number o f processes which he f e l t were uniquely depicted in lo g -p ro b a b i1i t y p lo ts o f g r a i n - s t z e d is t r i b u t i o n s . These processes include: 1) c u rre n ts ; 2) swash and backwash; 3) waves; 4) t i d a l channel c u rre n ts ; 5) f a l l suspension; 6) t u r b i d i t y c u rre n ts ; and 7) wind. out from Visher suggested th a t log-normal sub-populations are re la te d to the mechanisms des­ crib e d by Moss (1962, 1963); and th a t the number o f sub-populations present in the t o t a l d i s t r i b u t i o n , the percent o f sediment in each su b-population, the s iz e range o f each sub-population, the s o rtin g o f each sub-population, and mixing between the sub-populations vary s y s te m a tic a lly in r e l a t i o n t c provenance, tra n s p o rt mechanisms p re­ sent, and r e l a t i v e energy i n t e n s it y . Bagnold (1 9 6 6 ), A lle n , e t a l . ( 1 9 7 1 ) , and B l a t t , e t a l . (1972) have attempted to r e l a t e s p e c ific g r a in - s iz e i n t e r v a l s to tra n s p o rt processes in terms o f f l u i d mechanics, mainly on an experimental and th e o r e tic a l basis. Bagnold (1966) noted t h a t grains co a rse r than 20 tend to be moved as bedload a t the onset o f g ra in motion, w hile those grains from 20 to approxim ately 3.5 0 to 40 are transported by nonuniform suspension ( A l l e n , e t a l . , 1 9 7 1 ), which denotes an increase in p a r t i c l e d e n s ity nearer the bed. Bagnold (1966) states th a t uniform suspension would be f u l l y developed in g rain s f i n e r than 3 .3 0 . B la tt, et a l. (1972) noted t h a t sand in the s iz e range o f -1 0 to 30 tends to move m ainly by t r a c t i o n and i n t e r ­ m i t t e n t suspension w h ile f i n e r sizes are probably tran sp o rted by suspension. T h e re fo re , these d i f f e r e n t s iz e modes represen t d i f f e r e n t sediment tra n s p o rt mechanisms, and the r e l a t i v e portions and s iz e ranges o f the modes should r e f l e c t d i f f e r e n t energy l e v e l s . 8 Davis (1 9 7 0 ), A lle n , e t a l . (1971) and Mather (1972) were the f i r s t to use a form o f R-mode f a c t o r an a ly s is in an attem pt to show s t a t i s t i c a l r e la tio n s h ip s between g r a in - s iz e i n t e r v a l s . Davis (1970) showed t h a t f o r samples from B a r a ta r ia Bay, L ouisiana, s ix t y percent o f the sample v a r i a b i l i t y could be explained by simply com­ paring the r a t i o o f sediment f i n e r than 4 0 to the m a te ria l coarser than 4 0. In a study o f the sediment o f the Gironde Estuary, France, A lle n e t a l . (1971) showed t h a t se ven ty-th ree percent o f the t o t a l g r a i n - s i z e v a r i a b i l i t y was accounted f o r by th ree f a c t o r axes. sediments in the estuary range in s ize from -3 0 to 6 0. The Factor I is associated w ith sizes coarser than 0 .6 0 , representing the surface creep process. Factor I I is associated w ith s ize s f i n e r than 3 0 and is in te r p r e te d to represent a pure suspension load. The sediment sizes between 0 .6 0 and 3 0 are associated w ith f a c t o r I I I . e t a l . (1971) suggested th a t f a c t o r I I I two tra n s p o rt mechanisms. A lle n r e f le c t s the in t e r a c t io n o f The grains between 0 .6 0 and 1.6 0 are in te r p r e te d to represent the s a l t a t i o n mode, w h ile those between 1.6 0 and 3 .0 0 represent a non-uniform suspension mode. Mather (1972) noted t h a t six fa c to r s were needed to e x p la in the t o t a l variance in g l a c i o - f l u v i a l sediments and each f a c t o r was r e la t e d to a small p a rt o f the t o t a l d i s t r i b u t i o n . He did not r e l a t e these fa c to r s to any process, but noted th a t the sand fr a c t io n s accounted f o r s i x t y - e i g h t percent o f the v a ria n c e. The terminology and f l u i d mechanics concepts used in t h is study are taken from these previous workers. V ish er (1969) has developed a very useful approach f o r an alyzin g g r a in - s iz e 9 d is t r i b u t i o n curves, and comparisons between the m u l t iv a r ia t e approach used in th is study and V is h e r's approach w i l l be made when­ ever possible. LOCATION AND PROCEDURE Location and C o lle c tio n o f Samples Two independent Recent d ep o s itio n a l systems were chosen f o r th is study; the L i t t l e Sable Point area (F ig . Bear Point/M anitou Passage area ( F ig . 1 ) . 1 ) , and the Sleeping Both are located in Michigan on the eastern shore o f Lake Michigan. L i t t l e Sable Point - The s u r f i c i a l deposits o f t h is area are pre­ dominately o f g la c ia l o r ig in (S aylor and Hands, 1970). At the lakeshore, Recent, beach, a e o lia n , and f l u v i a l sediments are accumu­ la t in g ( F ig . 2 ) . The source m a te ria l f o r sediments in the S i lv e r Lake sand dunes and ad jacen t Lake Michigan beach system comes from Pleistocene morainic sediment approximately 2.5 km south o f S i lv e r Lake (Upchurch, 1970b). Upchurch (1970b) found the fo llo w in g s o rtin g conditions a t the source area: 1) the pebble-cobble f r a c t io n forms a lag beach d eposit; 2) the s i l t - c l a y population is winnowed and c a rr ie d lakeward; and 3) the sand population is transpo rted by litto ra l c u rre n ts . The dominant l i t t o r a l c u rre n t f o r th is area is northward (Ayers, e t a l . 1958) (F ig . 1 and 2 ) . Thus, beach sediment transported northward from the source area has a r e s t r i c t e d g r a in s iz e d i s t r i b u t i o n , lim it e d by the sand s iz e population a t the source and competence o f the tra n s p o rtin g mechanism, namely l i t t o r a l d r i f t . There is much feedback between the beach and adjacent dunes (Upchurch, 1970b) and presumably between the beach, o ffs h o re bars, 11 Figure 2: Sample lo c a tio n s a t L i t t l e Sable P o in t, Lake Michigan. S it e 1 locates ae o lia n samples west o f S i lv e r Lake. S ite 2 locates beach, S i lv e r Creek and o ffsh o re bar and trough samples. S it e 3 locates the beach samples c o lle c te d a t the morainal source area littoral drift 3t • ' First Bar 42» • now f* n c « ' litto ra l d fi,t '" : Lake -0 • “ - 27• 29* 19* 15* 18* 25* 14* ' - '1 0 * - 21* 13ft '2 4 * 12* 11* Michigan BEACH 5 km o <0 r* fo e^ S-Oft o t ft p ° o - . cT «' o * ° ■% ■* 0 ( a o o 0 a™ — =0 0113 3426 * ■ » * ■ 3 a 2 i\ C 1" • 2 >r "J 86*** -"J r> s 18 ! o» « ! • a> m s o: («» "• 2»-= ®* 21 f r tt o r a i d r if t 40# 17* 22* I “ L 13 dunes, and S i lv e r Creek. The L i t t l e Sable Point d ep ositio nal system provides an e x c e lle n t opportunity to study, under the c o n stra in ts o f a s in g le source and a l im it e d g r a in - s iz e range, the g r a in - s iz e d i s t r i b u t i o n produced by d i f f e r e n t , adjacent hydrodynamic environments. Two sample sets were c o lle c te d : one set during e a r l y w in te r conditions in October 1972 (n = 5 2 ) * , the o th e r during the spring conditions in l a t e June 1973 (n = 4 2 ). Sampling was randomized by c o lle c t in g sediment from the environments l i s t e d below during a random walk through an environment then noting t h e i r approximate p o s itio n on a sketch map. The environments sampled inclu d e: 1) beach sediment a t the m orainic source area (n = 9 } ; 2) beach s e d i­ ment on Lake Michigan west o f S i lv e r Lake {n = 2 0 ); 3) o ffs h o re bars and troughs in th e area o f the beach j u s t described (n = 1 8 ); 4) bed sediment from S i l v e r Creek (n = 2 2 ) ; and 5) a e o lian sediment between S i lv e r Lake and Lake Michigan (n = 2 5 ). As suggested by Visher (1 9 6 9 ), the samples were c o lle c te d from the upper c e n tim eter to r e f l e c t the physical conditions j u s t p r i o r to sampling. The data are l i s t e d in Appendix A. Sleeping Bear P o in t/M an ito u Passage area - This area (F ig s . 1 and 4) is a region o f both nearshore and open w ater environments, w ith d i f f e r e n t sedimentary processes o peratin g in an area o f mixed s e d i­ ment provenances. •k The s u r f i c i a l sediments o f Sleeping Bear P o in t N is the number o f random samples c o lle c te d . 14 and North and South Manitou Islands a re moralnic and g la c ia l o u twash, w ith larg e a c t iv e sand dunes on Sleeping Bear P o in t (French, 1964). Figure 3 shows the bathymetry f o r t h is a rea . The lake bottom physiography is complex and has been c a lle d the rid g e -an d v a lle y province o f Lake Michigan (Hough, 1958). lo g ic a l zones in the study area a re : The major morpho­ 1) the sh a llo w -w a te r, Sleeping Bear Shoal, extending approxim ately 8 km northwest o f Sleeping Bear P o in t, w ith water depths from 10-20 m; 2) deep, q u ie t water basins north and south o f the shoal w ith w ater depths from 75-100 m; and 3) an in te rm e d ia te depth area north to n ortheast o f Sleeping Bear Point w ith depths from 20-75 m (French, 1964). The most prominent fe a tu re o f Sleeping Bear P o in t, the west fa c e , has undergone extensive erosion in the past f i f t y years and has re t r e a t e d more than 15 m during t h a t time ( G i l l i s and Bakeman, 1963). According to French (1 9 6 4 ), the shoal areas w ith depths less than 20 m are flo o re d w ith clean sands or w ith a pebblecobble-boulder pavement overlaying c la y t i l l . The rocky-bottomed c e n tra l p ortions o f the shoal appear to be covered w ith lag concen­ t r a t e s eroded from the underlying t i l l . The northeast t i p o f Sleeping Bear Point has undergone a t le a s t two major episodes o f mass wastage since the tu rn o f the c cen tu ry. In 1906, approximately 8x10 square meters o f unconsoli­ dated g la c ia l outwash and sand dune m a te ria l slumped in to the ad jac en t lake w ater (Upchurch, 1973). Eighty thousand square meters slumped in to the la k e again in March 1970 (U.S. Department o f Com­ merce News, 1971) due to heavy ra in s and high w ate r. 15 Figure 3: Bathymetric c h a rt o f the Sleeping Bear P oint/M anitou Passage a rea , Lake Michigan 61 0 m Sleeping B e a \ ir$. Point \ 45.8 m C O N TO U R INTERVAL = 15 2m (5 0 ft) KILOM ETERS 17 Forty-tw o surface samples and eighteen piston cores were 2 c o lle c te d from a 284 km area encompassing the above environments (F ig . 4) during a one-week cru is e in Ju ly 1973. A ll samples were c o lle c te d from the Lake Survey research vessel SHENEHON. Bottom 3 samples were c o lle c te d a t each s t a tio n w ith a 400 cm Shipek s e d i­ ment sampler ( F ig . 5 a ). The sampler recovers the upper 10 cm in an e s s e n t i a ll y undisturbed co n d itio n . Cores were c o lle c te d w ith a 160 kg Alpine piston c o re r (F ig . 5b) t h a t is capable o f p en etra tin g up to 3.5 m; w ith p e n e tra tio n dependent upon sediment type. F u ll p en etra tio n was possible in the s i l t - c l a y r ic h a re a s , w h ile l i t t l e p e n e tra tio n was possible in the coarse sand areas. Sample P re p ara tio n and S ize Analyses Sample P rep aratio n - Samples from the Sleeping Bear Point/M anitou Passage area were f i r s t tre a te d w ith 10% HC1 a c id to remove sh e ll m a t e r ia l, and w ith 30% to remove organic d e t r i t u s . The sam­ ples were d rie d and weighed, then s p l i t a t the 4 .0 0 in t e r v a l by w e t-s ie v in g methods. The coarse f r a c t i o n was r e - d r ie d and weighed and the percent o f t o t a l weight c a lc u la te d . ready f o r s e t t l i n g - t u b e a n a ly s is . This p o rtio n was then The sediment f i n e r than 4 .0 0 was omitted from the m u l t iv a r ia t e te s ts to avoid the problems generated during m u lt ip le c o r r e la t io n an alysis o f closed number systems (page 2 9 ). The above procedures were not necessary f o r the L i t t l e Sable P o in t samples as they contained no sh e ll fragments o r organic d e t r it u s and were tru n cated a t approxim ately 30. 18 Figure 4: Sample lo c atio n s a t the Sleeping Bear Point/M anitou Passage a re a , Lake Michigan. Piston cores and surface samples were taken a t lo c a tio n s in d ic a te d by a tria n g le . Only surface samples were c o lle c te d a t s it e s in d ic a te d by a dot 86°10' 86°05' 86°oo' ▼7 ▼8 ▼9 ▼6 ▼5 ▼17 ▼16 •M 47 ▼18 •14 *26 •10 ▼4 •12 * 3 •25 's le e p in g B e a \ •16 •2 4 •23 • •17 Point \ * 6 ▼Core and Surface Sample • Surface Sample •18 22 •19 •21 ▼15 KILOMETERS •20 ▼11 20 Figure 5: Shipek surface sampler ( a ) and an A lp in e p is to n c o re r (b ) used f o r sample c o l l e c t i o n a t th e S leeping Bear P o in t / Manitou Passage a r e a . Lake Michigan 22 Size Analysis - A 12.5 cm in s id e -d ia m e te r s e t t l i n g tube u t i l i z i n g a d i f f e r e n t i a l pressure transdu cer, c a r r i e r demodulator, and s t r i p c h a rt recorder were used to determine f a l l - v e l o c i t y frequency d is t r i b u t i o n s . The tube has a 140 cm e f f e c t i v e f a l l a g r a in - s iz e measuring range o f -1 0 to 40. length w ith F a l l - v e l o c i t y frequen­ cie s were converted to percent frequencies based on f a l l v e lo c itie s o f known quartz sand d is t r i b u t i o n s . The s e t t l i n g tube used in t h is study was b u i l t and c a lib r a t e d by t h is author fo llo w in g the recommendations o f Gibbs (1 9 7 2 ). c a l i b r a t i o n procedures were as fo llo w s : The 1) pure, w ell sorted q u a rtz sands ranging in s iz e from - 0 . 5 0 to 4 .0 0 were c a r e f u l l y sieved a t 1 /4 0 i n t e r v a l s ; 2) ten r e p e t i t i v e 1 gram samples f o r each 1/4 0 in t e r v a l were introduced in to the tube and allowed to s e t t l e ; graphs o f the f a l l times were recorded d i r e c t l y on a P e rk in - Elmer re co rd er; 3) a tem plate was prepared from the mean f a l l tim es; 4) a t e s t sample was prepared by recombining weighed q u a n t it ie s of sediment from each o f the 1/4 0 s iz e i n t e r v a l s ; 5) ten r e p e t i t i v e 1 gram samples were introduced in to the s e t t l i n g tube; 6) f i n a l l y the g r a in - s iz e d is t r i b u t i o n s as determined by the s e t t l i n g tube were compared to the prepared t e s t sample f o r p re c is io n . The mean s iz e o f the sediment as determined by moments c a l ­ c u la tio n s f o r 10 s e t t l i n g tube frequency curves d if f e r e d o nly by an average o f +_ 0.81% from the mean size o f the prepared sample. The range o f d iffe re n c e s was found to be 0.43% to 1.07%. The r e p r o d u c i b i l it y between the 10 runs was found to be b e t t e r than +_ 1%, ranging from 0.65% to 1.34%. As was found by Gibbs (197?) 23 a s e t t l i n g tube is a f a s t (up to 100 samples a day can be a n a ly ze d ), ac c u ra te , and p recise method f o r s iz in g sands. For th is study, a sample was s p l i t in to two sub-samples and the average frequency d is t r i b u t i o n was used. I t should be empha­ sized here t h a t d iffe re n c e s in m in e ra lo g ic a l d e n s itie s are taken in to account because the s e t t l i n g tube measures h y d ra u lic equiva­ lences. V a riab le s - For th is study, the v a ria b le s are the r e l a t i v e frequencies o f sediment from each sample which f a l l w ith in s p e c ifie d s iz e groups, which are a t 0 .5 0 s iz e i n t e r v a l s . These are the same v a ria b le s used to compute the moment s t a t i s t i c s , such as mean and standard d e v ia tio n . DATA ANALYSIS Several m u l t iv a r ia t e methods were used in th is study and include principal-components a n a ly s is , Q-mode f a c to r a n a ly s is , and d is c r im in a n t-fu n c tio n a n a ly s is , and are conmonly c ite d in the geologic l i t e r a t u r e . These methods have been used in a wide v a r ie ty o f f i e l d s such as geochemistry, p e tro lo g y , paleontology, and se d imentology. For e x c e lle n t reviews o f these methods see Im brie (1 9 6 3 ), Harbaugh and Merriam (1 9 6 8 ), and Davis (1 9 7 3 ). S u ffic e to mention the principal-com ponents and Q-mode f a c t o r analyses are mathe­ m atical and s t a t i s t i c a l techniques th a t i d e n t i f y patterns o f c o rr e la tio n s in a data m a trix . Using these techniques, the data are reduced to a small set o f independent fa c to rs which account fo r a c e r t a in portion o f the t o t a l variance in the o rig in a l data m a trix . D is c rim in a n t-fu n c tio n an a ly s is m athem atically determines the d is ­ tance and percent o verlap between m u l t i v a r i a t e normal populations as an in d ic a to r o f separation o f supposedly d i f f e r e n t p opulations. By p rincipal-com ponents, which is an R-mode s o lu t io n , i t is possible to examine in t e r - r e l a t i o n s h i p s between the v a r i a b l e s , in th is case the g r a i n - s i z e f r a c t i o n s , and f i n d the most e f f i c i e n t l i n e a r combination o f the v a r ia b le s . Principal-components are the eigenvalues o f the o r i g i n a l data m a trix . There are as many eigenvalues as th ere are v a r ia b le s , w ith each eigenvalue accounting f o r a c e r ta in p ro po rtion o f the t o t a l v a ria n c e . 24 Normally the f i r s t 25 two o r th re e components account f o r the g re a te s t varian ce (Davis, 1973). Principal-components an a ly s is was used in an e f f o r t to group g r a in - s iz e classes in to c lu s te rs th a t are c o r r e la t e d , which may aid in i d e n t if y in g major hydrodynamic processes operating w ith in an environment. Principal-components an a ly s is fo llo w s three basic steps: 1) p re p a ra tio n o f a product-moment c o r r e la tio n m a tr ix ; 2) e x tra c ­ tio n o f i n i t i a l f a c t o r s ; and 3) a varimax (orthogonal) ro ta tio n to a term inal s o lu tio n ( N ie , e t a l . , 1970 ). The fa c to rs represent best f i t t e d axes through c lu s te rs o f data vectors in a hypere l l i p s o i d a l space. a b ility The f i r s t f a c t o r accounts f o r the most v a r i ­ (th e most prominent c l u s t e r ) , w h ile fa c to rs 2 , 3 n represent the best f i t t e d axes through the resid ual c lu s te rs . Although the axes are orthogonal, the v a ria b le s they represent are not n e c e s s a rily independent in nature making subsequent re s u lts sometimes d i f f i c u l t to i n t e r p r e t . In o ther words, a v a ria b le can be r e la te d to more than one f a c t o r a x is . However, p rin c i pal - components an a ly s is is a useful technique in the absence o f a hypothesis to account f o r the observed v a ria tio n s in the measured parameters. Factor analysis is somewhat d i f f e r e n t from p r in c i pal components a n a ly s is . Although f a c t o r analysis can be c a rrie d out in the R-mode i t is g e n e ra lly a Q-mode s o lu tio n , which shows re la tio n s h ip s between samples. F acto r analysis is commonly regarded as a s t a t i s t i c a l technique r a th e r than a mathematical p r i n c ip a lcomponents a n a ly sis (D avis, 1973). Q-mode f a c t o r a n a ly s is was used 26 in t h is study to group the samples in to s i m il a r te x tu r a l s u ite s t h a t should represent d i f f e r e n t sedimentary environments. Q-mode f a c t o r a n a ly s is follow s the same steps as principal-components except t h a t in step one a cosine theta m a trix is prepared, which measures the angles o f the vectors p ro jected in t o the data p o in t c lu s te r s . This m atrix shows b e t t e r r e la tio n s between samples as the variance i s c a lc u la te d across the v a ria b le s (D a vis, 1973). Unlike principal-com ponents, Q-mode f a c t o r an a ly s is requ ires some a p r i o r i knowledge about the s tru c tu re o f the data on the p a r t of the i n v e s t ig a t o r . This means t h a t in the f a c t o r model, the s i g n i ­ f i c a n t number o f fac to rs p , must be known p r i o r to the a n a ly s is . Other d iffe re n c e s between principal-components and Q-mode f a c t o r an a ly s is can be found in Davis (19 73 ). Like principal-components a n a ly s is , the number o f fa c to r s retained in Q-mode f a c t o r a n a ly sis i s s m a ll, since only a few fa c to r s contain most o f the varia n c e. Some analysts only r e ta in the fa c to rs which have eigenvalues >1, t h a t i s , only those fa c to r s which have variance g re a te r than o r equal to the variance o f the o r ig in a l standardized v a ria b le s are re ta in e d (D a v is , 1973). The i n i t i a l and ro ta te d f a c t o r axes in Q-mode f a c t o r an alysis are orthogonal to each o ther and a sample can be r e la t e d to more than one f a c t o r . This is to be expected as sharp boundaries between ad jacen t environments are very ra re indeed. D is c rim in a n t-fu n c tio n a n a ly sis is a technique which fin d s the best l i n e a r combination o f the v a ria b le s which provides the maximum d iffe r e n c e between the p re v io u s ly defined groups. To avoid c y c lic arguments, the v a ria b le s used in the an a ly s is cannot in clu d e those 27 c r i t e r i a used f o r the i n i t i a l assumption o f d iffe re n c e s between the groups to be d is c rim in a te d (Krumbein and G r a y b il l , 1965; M orrison, 1967; and D avis, 1973). D is c rim in a n t-fu n c tio n a n a ly sis consists o f fin d in g a transform which gives the minimum r a t i o o f d iffe re n c e s between the m u l t iv a r ia t e means to the m u lt iv a r ia t e variance w ith in the groups in question, which transforms the o r ig in a l data in t o a s in g le d is c rim in a te score. This score re p re ­ sents the samples lo c a tio n along a l in e d efined by the d is c rim in a n tfu n c tio n eq u ation . This a n a ly s is follow s s ix basic steps: 1) i t fin d s row v e cto r means o f the groups; 2) i t fin d s the d iffe re n c e s in the row v e cto r means; 3) i t c a lc u la te s the corrected sums o f square f o r each group m a t r ix ; 4 ) the pooled variance-co variance m a trix o f the two groups i s c a lc u la te d , 5) the variance-covariance m a trix is in v e rte d ; f i n a l l y 6) step 2 is m u l t ip l ie d by step 5 to fin d the d is c rim in a n t c o e f f i c i e n t s . The d is tan ce between the 2 groups is the Mahalanobis D , which is expressed in terms o f the pooled va ria n c e. This value can be tested f o r s ig n ific a n c e by s u b s titu tin g i t f o r [ S ] ~ \ H o te llin g T^ s t a t i s t i c the variance-co variance m a tr ix , in the (M o rriso n , 1967; H avis, 1973); which has the fo llo w in g form: where N-| is the number o f samples in group 1 and Ng is the number o f samples in group 2. This converts a m u l t iv a r ia t e problem in t o u n iv a r ia te terms which can be tested using the F - s t a t i s t i c by 28 s u b s titu tin g T 2 in to the fo llo w in g equation: c h N1 + N2 - M - 1 t 2 (N1 + N2 - 2)M 1 where the N's have the same meaning as above and M equals number o f v a r ia b le s . The value o f F is tested f o r the s ig n ific a n c e as an F w ith the fo llo w in g degrees o f freedom: F (1 - a ); M, N1 + N2 - 2 - 1 Tests o f s ig n ific a n c e f o r principal-components and f a c t o r analysis are a v a i l a b l e , however they are g e n e ra lly not used because o f the d i f f i c u l t y in checking assumptions o f m u l t i v a r i a t e n orm ality and e q u a lity o f the m a trix s tru c tu re (M orrison, 1967). Non- n o rm ality and in e q u a lit y o f the m atrices does not seem to g r e a tly a f f e c t the r e s u lts o f d is c r im in a n t-fu n c tio n an a ly s is as t h is pro­ cedure is robust to the above d e v ia t io n s , th e re fo re the F - t e s t can be used w ith c e r t a in confidence (Rothman, personal communication, 1975). The computer programs used in th is study were obtained from several sources. et a l. The principal-com ponent program is from N ie, (1 9 7 0 ), the Q-mode fa c to r program is from Davis (1973) and the d is c r im in a n t-fu n c tio n program i s from the UCLA, Biomedical Package, 1972. INFORMATION CONTAINED IN GRAIN-SIZE FREQUENCY DISTRIBUTIONS In tro d u c tio n The nature o f the data used in the m u lt iv a r ia t e te s ts requ ires some exp lanatio n th a t w i l l help place the re s u lts in p ers p ec tiv e. The data are expressed as frequency percentages, and as percentages, the data form a closed a rra y (th e sum o f the rows equal 100%) (Chayes, 1960, 1962; Krumbein and G r a y b i l l , 1965; D avis, 1973). This constant sum ra ise s troublesome t h e o r e tic a l questions concerned w ith induced negative c o r r e la tio n s in the principal-components a n a ly s is . The c o r r e la t io n m a trix c a lc u la te d from a closed data m a trix is overdeterm ined; t h a t i s , i t has more rows and columns than necessary, and some o f these c o r r e la tio n s are formed by the closed ta b le and not true r e la tio n s h ip s in the data. For example, i f A, B and C are known and i f the t o ta l is f ix e d , then th e re is more i n f o r ­ mation than is needed, because A and B determine the value o f C. In c o r r e la t io n re la te d a n a ly sis each v a r ia b le can be represented by an eigenvalue and eig en vecto r w ith magnitude e q u iv a le n t to the v a r i a b i l i t y in the o r i g i n a l data a r r a y . T h e re fo re , i f one o f the v a ria b le s is determined by constant summation, then one o f the eigenvalues and corresponding eig en vecto r o f the m a trix w i l l be zero and spurious c o r r e la tio n s are formed (Davis 1973). Even though the v a ria b le s c o n s t it u t e a closed array in p r i n c ip a l- 30 components a n a ly s is , the a r ra y is not closed in Q-mode f a c t o r a n a ly s is . Real negative c o r r e la tio n s can occur in an open o r closed a r ra y and show re al inverse r e la tio n s h ip s between the v a ria b le s (D a v is , 1973). In o rder to avoid an overdetermined c o r r e la tio n m a trix , clo su re can be prevented by excluding e i t h e r A, B, o r C from the o r ig in a l data m a trix before the a n a ly sis is performed. For th is study, one v a r ia b le (th e percentage o f sediment w ith in a size class i n t e r v a l ) was e lim in a te d to open the data a rra y . For the L i t t l e Sable Poin t a e o lia n , bar and trough, and f l u v i a l samples the 0 0 v a ria b le was e lim in a te d . This v a r ia b le never accounted f o r more than 2% o f the t o t a l sample and m a te ria l was present in only 12 o f 65 samples. in t h is size class The 3(9 v a r ia b le was e l i m i ­ nated from the beach samples because more sediment was present in the 0 0 s iz e i n t e r v a l . Only 7 o f the 29 beach samples had sediment in the 30 s iz e class and represented less than 3% o f the t o t a l sample. Even though many o f the samples s t i l l sum to 100£, 19 samples do not and t h e o r e t i c a l l y th is should c o n s titu te an open data m a trix (Huang and Rothman, personal communication, 1975). A lso , i f approxim ately 6 or more v a r ia b le s are used in a closed a r ra y , the e f f e c t o f closure on induced negative c o r r e la t io n is reduced (Chayes, 1960, 1962; Krumbein and G r a y b i l l , 1965). Nine t e s t cases were prepared to e v a lu a te the e f f e c t o f c lo su re on a closed data a r ra y using principal-components a n a ly s is ( F ig . 6 ) . These t e s t cases represent a t r u e , log-norm al, sediment 31 F igure 6: P h i - p r o b a b i l i t y graph o f 4 o f the 9 t e s t cases used to determ ine the e f f e c t o f c lo s u re on c o r r e l a t i o n c o e f f i c i e n t s CUMULATIVE PERCENTAGE (A> ro PHI SCALE 33 d i s t r i b u t i o n s im ila r to several a e o lia n samples c o lle c te d a t the S ilv e r Lake S tate Park sand dunes. The v a ria b le s used are per­ centages o f the t h e o r e tic a l d is t r i b u t i o n s in each 0 .5 0 s iz e i n t e r ­ val from 0 .5 0 to 2 .5 0 . Examination o f the c o r r e la tio n m atrix (Table 1) shows th a t fo u r s i g n i f i c a n t negative c o r r e la tio n s were formed; in a closed (3x3) m a trix a t le a s t two n egative c o rr e la tio n s are induced (Chayes, 1960, 1962). S ig n ific a n c e cannot be attached to these negative c o r r e la tio n s because in a closed m a trix the n ull hypothesis, p = 0 , may be misleading (Chayes, 1960, 19 62 ). The f a c t o r m a trix (Table 1) shows th a t o n ly one f a c t o r was e x tra c te d and accounts f o r 76.4% o f the t o ta l v a ria n c e . A ll the loading values (the degree to which a v a r ia b le is associated w ith a f a c t o r ) are very h ig h , w ith v a r ia b le s 2.00 and 2 .5 0 showing h ig h -n e g a tiv e loading values. An in t e r e s tin g comparison can be made between these t e s t sam­ ples and f a c t o r I I I o f A lle n , e t a l . (1 9 7 1 ). T h e ir f a c t o r I I I represents the in te rm e d ia te sand sizes from 0.6 0 to 3 . 0 0 , which is a s i m i l a r s iz e range o f the t e s t samples in th is study. They d iv id e d th is f a c t o r into two sub-groups on the basis o f the signs o f the f a c t o r lo a d in g s , hence one group is associated w ith s iz e 3.00 to 2.00 (n e g a tiv e loading v a lu e s ), w h ile the other group i s represented by sizes 2 .0 0 to 0 .6 0 ( p o s itiv e loading v a lu e s ). These groups were in t e r p r e te d to represen t non-uniform suspension and s a l t a t i o n pro­ cesses, r e s p e c tiv e ly . Returning to Table 1, i t is e v id e n t th a t these groups are present in the t e s t samples and the loading value signs a r e also the same f o r each group. A lle n , e t a l . (1971) did 34 TABLE 1 PRINCIPAL COMPONENTS RESULTS OF TEST CASES Sumroary S t a t i s t i c s 0 .5 1 .0 1.5 2 .0 2 .5 Grand Means 0.01 2 .2 43.5 50.4 3 .6 Grand Standards D eviations 0.00 0 .6 2 .4 3 .0 0 .5 Phi Size Product-Moment C o rre la tio n M a trix _____________ Phi Size 0 .5 0 .5 1 .0 1.00 1.0 1.5 .7 8 * 1.0 0 1 .5 2 .0 .7 8 * -.6 3 * .8 3 ** 0 .8 4 ** 1.00 2.0 -.9 3 ** 1.00 2 .5 I - . 68 * 1.000 s ig n ific a n t a t a 5 0 .0 5 *, a = 0 .0 1 * *, n = 9 , df = 7 Factor 2 .5 Eigenvalue Cumulative Percent o f Eigenvalue 3 .8 2 76.4 Rotated Factor M a trix Phi S ize 0 .5 1. 0 1.5 2. 0 2 .5 I 0 ,8 10 0.931 0.957 -0 ,9 3 2 - 0 .7 1 2 35 not take in to account the e f f e c t o f c lo s u re , th e re fo re t h e i r in t e r p r e t a t i o n may be erroneous. Bagnold (1966) noted t h a t a change in sediment process occurs a t approxim ately 20. With the exception o f the negative f a c t o r loadings i t appears t h a t , even when clo su re is p resen t, the fa c to rs i d e n t i f i e d in principal-components a n a ly ­ sis are in s e n s it iv e to closure and are v a l id i f caution is used in th e ir in te rp re ta tio n . With the problem o f closure and induced negative c o r r e la tio n s i d e n t i f i e d the re s u lts o f th is study w i l l be presented. L i t t l e Sable Poin t Depositional Area S t a t i s t i c a l Analysis - The sediment c o lle c te d f o r th is study comes from an area w ith e s s e n t i a ll y one source o r parent m a te ria l (Upchurch, 1970b) and any d iffe re n c e s in the s iz e frequency d i s t r i ­ butions from o ffs h o re bars to beach, to a e o lia n , to f l u v i a l en viron ­ ments should r e f l e c t changes in the processes operating in those environments. I f the sediment from these environments are derived from one major source, the s iz e d is t r ib u t io n s should be r e l a t i v e l y constant through time and space due to the l im it a t io n s imposed by the parent d i s t r i b u t i o n . S t u d e n t 's - t and Chi-square (x ) te s ts were used to t e s t f o r d iffe re n c e s between the June and October sample s e ts. S tu d e n ts -t t e s t was used because the tru e population mean and variance are not known but are estim ated from the c o lle c te d samples. An amount o f u n c e rta in ty i s associated w ith these e s tim a te s , so decisions based upon them cannot be as p re c is e as those based on 36 the Z t e s t . When the number o f samples are I n f i n i t e , the t - d i s t r i - bution is equal to the Z o r normal d is t r i b u t i o n (D a vis, 1973). Three assumptions are necessary to perform t h is t e s t (D a v is , 1973). One is th a t the samples were c o lle c te d randomly; which is so f o r th is study. The second is t h a t the samples are normally d is t r i b u t e d . The Rankits method o f te s tin g f o r n orm ality (Sokol and R o h lf, 1969) was used because the sample means have an apparent bimodal o r s l i g h t l y skewed d i s t r i b u t i o n . n orm ality in the d a ta . This method allow s somewhat f o r non­ Figures 7 and 8 show a p lo t o f Rankit values versus the ranked mean phi s ize o f each sample f o r the October and June sampling p erio d s. Although the observed d i s t r i ­ bution deviates from the normal Rankits d i s t r i b u t i o n , the d e v ia tio n is not a s i g n i f i c a n t a t a = 0.001 as c a lc u la te d by Chi-square. The t h i r d and most c r i t i c a l assumption is the e q u a l i t y o f the grand variances o f the June and October sample s e ts . The c r i t i c a l value o f F between the two grand sample variances is 1.16 and is not s i g n i f i c a n t a t a = 0 .0 5 . T h e re fo re , there i s no reason to suggest a d iffe re n c e in the grand sample variances. Having es tab lis h e d the v a l i d i t y o f the th re e basic assumptions, the S tu d e n ts -t t e s t can be used to t e s t f o r d iffe re n c e s between the grand p h i- s iz e means o f the June and October sample se ts. 1 .3 9 , w ith 92 degrees o f freedom. The c a lc u la te d value o f t is This value o f t does not f a l l in to the r e j e c t i o n region ( a = 0 . 0 5 ) , and i t must be concluded t h a t th ere is no reason to suggest t h a t the two sample sets came from populations w ith d i f f e r e n t grand means. VALUE RANKIT OCTOBER SAMPLES X* (calc) = 2.983 * 2 (0.001,6) = 22.458 -2 2.0 RANKED PHI MEAN Figure 7: Rankit graph of the October 1972 sample set collected at L ittle Sable Point. Solid lin e is the Rankit normal curve, dashed lin e is the observed curve 2.2 JUNE SAMPLES X2 (calc.) =0.981 x2 (.001,9) = 27.877 hi D —I < > H * Z < CC -2 1.4 1.5 1.6 1.7 1.9 2.0 2.1 RANKED PHI MEAN Figure 8: Rankit graph of the June 1973 sample set collected at L it t le Sable Point. Solid lin e represents the Rankit normal curve, dashed lin e is the observed curve 39 Chi-square was used to t e s t f o r s i g n i f i c a n t d iffe re n c e s in the grand frequency d is t r ib u t io n s o f the June and October sample s e ts . The c a lc u la te d value is 3 .3 3 , w ith s ix degrees o f f r e e ­ dom is not s i g n i f i c a n t a t a = 0 .0 5 . This suggests th a t there is no reason to assume t h a t the two sample sets came from populations w ith d i f f e r e n t frequency d is t r i b u t i o n s . Results o f these te s ts lend strong support to the previous assumption t h a t one sediment source, o r a t le a s t a very lim ite d range in s iz e s , i t a v a ila b le to the study area. Since these samples come from a lim it e d parent p o p u la tio n , i t is possible to study te x t u r a l responses to d i f f e r i n g energy conditions in the various environments w h ile discounting v a r i a b i l i t y imposed by sediment introduced from m u lt ip le sources. Principal-Components Analysis Principal-components a n a ly sis was used to determine s iz e l i m i t s and r e la tio n s h ip s between the sub-populations ( e . g . , those deposited by surface creep, s a l t a t i o n , and suspension) presumed to c o n s titu te the sediment (V is h e r , 1969). A uniform suspension population is not expected to be present in the L i t t l e Sable Poin t samples, because the sediment is truncated a t approximately 30. l i m i t s represented by each f a c t o r w i l l The s iz e be compared to the same sub­ populations p lo tte d as cumulative p r o b a b i l it y curves a f t e r the method o f Visher (1 9 6 9 ). Although V ish er (1969) studied sediment from a number o f environments, the L i t t l e Point environments 40 do not appear to be d i r e c t l y analogous. This is la r g e ly due to the f a c t th a t Visher c o lle c te d samples over a wide geographical area t h a t represents many source types. A ll L i t t l e Sable P o in t Samples - Table 2 l i s t s the r e s u lts o f the principal-components an a ly s is o f a l l p le s . the L i t t l e Sable P o in t sam­ The product-moment c o r r e la tio n m a trix shows the in t e r c o r r e ­ l a t i o n among the v a r ia b le s th a t are s i g n i f i c a n t a t a = 0 .0 5 and 0 .0 1 . Two fac to rs w ith eigenvalues >1 were e x tra c te d by p r i n c ip a l- components analysis and account f o r 74.1% o f the t o t a l variance. The values l i s t e d in the ro ta te d f a c t o r m a trix are loading values which show how each v a r ia b le is associated w ith the e x tra c te d fa c to rs . The c lo s e r a f a c t o r loading approaches u n i t y , the b e t t e r the v a r ia b le is e x p la in e d . Communalities are the cumulative sums o f squares o f the f a c t o r loading in a row wise manner. The com­ munal i t ie s in d ic a te how w ell the fa c to r s w ith eigenvalues >1, describe the v a r ia b le s ; as communalities approach u n ity the b e t t e r the exp la n a tio n . Figure 9 is a cumulative p r o b a b i l it y p lo t o f the grand means f o r each size class f o r a l l the L i t t l e Sable P oint samples. Based on the segmentation o f the l i n e , th re e sub-populations are in d ica te d on t h is graph: 1) sub-population A between 0.5 0 and 1 .0 5 0 ; 2) sub­ population B between 1.050 and 2 .5 0 ; and 3) sub-population C f i n e r than 2 .5 0 . These sub-populations correspond w ith those i d e n t i f i e d by principal-com ponents a n a ly s is . 41 TABLE 2 PRINCIPAL-COMPONENTS RESULTS OF ALL LITTLE SABLE POINT SAMPLES Summary S t a t i s t i c s Phi S ize 0 .5 1 .0 1.5 2 .0 2.5 3 .0 Grand Means 0 .6 3 .6 23.3 52.8 17.2 2 .2 Grand Standard D e viatio n s 0.7 4 .6 19.2 14.2 13.0 3 .2 Product-Moment C o rr e la tio n M atrix Phi Size 0 .5 0 .5 1.5 1 .0 1.00 1 .0 2 .0 2 .5 3 .0 -.6 4 ** 0 .4 4 ** -.2 8 ** -.6 7 ** -.8 0 ** -.5 5 ** .2 3 * .5 6 ** 1 .0 0 1.00 1 .5 1.00 2 .0 2 .5 1.00 .7 9 ** 3 .0 1.00 s i g n i f i c a n t a t a = 0 . 0 5 * . a. = 0.01 * * , Fac to r Eigenvalues I II 2.995 1.451 Rotated Factor M a trix I n = 94 , d f = 92 Cumulative Percentage o f Eigenvalues 49.9 74.1 Communalities II Phi Size 0 .5 - 0 .9 0 0 0.388 0 .1 5 9 1 .0 0.395 0.758 0.731 1 .5 0 .7 6 5 0.541 0 .8 8 0 2 .0 -0 .0 8 6 -0 .9 3 4 0.881 2 .5 -0 .9 5 6 -0 .1 0 3 0 .9 2 5 3 .0 -0 .9 1 8 0.158 0 .8 6 9 42 Figure 9: P h i- p r o b a b il i t y graph o f the grand frequency d i s t r i b u t i o n o f a l l L i t t l e Sable Point samples 43 99.99 U ttl* S a b i* Point LL ENVIRONMENTS 99 9 99 CUMULATIVE PERCENTAGE 90 50 01 -1 0 1 PHI SCALE 2 3 4 44 Factor I accounts f o r 49.9% o f the t o t a l va ria n c e . This fa c t o r is associated w ith sizes f i n e r than 20, which have high negative lo adin gs; and w ith the 1.50 s iz e i n t e r v a l , which has a f a i r l y high p o s it iv e loading value. The o th e r v a ria b le s have n e g lig ib le loadings values on f a c t o r I . Sub-population C on the p r o b a b ilit y graph accounts f o r 2.5% o f the d i s t r i b u t i o n and corresponds w ell w ith the 2 .5 0 and 3.00 i n t e r v a ls o f f a c t o r I . Factor I I accounts f o r 24.2% o f the t o t a l va ria n c e . f a c t o r is associated w ith sizes 20 and co arser. loads on both fa c to r s I and I I , fa c to r I I . a tiv e . This The 1.5 0 s iz e class but only moderately high on The loading value o f the 2 .0 0 s iz e class is h ig h ly neg­ The 2 .5 0 and 3 .0 0 s iz e class have n e g lig ib le loading values on f a c t o r I I . Except f o r the 0 .5 0 s iz e i n t e r v a l the communalities are high to very high and the two fa c to r s ex p la in the data w e ll. Sub-populations A and B o f the p r o b a b i l it y graph account f o r 97.5% o f the d i s t r i b u t i o n and g e n e ra lly corresponds to f a c t o r I I and the 1.50 in te r v a l o f f a c t o r I . I t is im portant to note in principal-com ponents a n a ly s is th a t a given g r a i n - s i z e in t e r v a l can be a ffe c t e d by more than one f a c t o r , in t h is case the 1.50 i n t e r v a l , and a unique s o lu tio n i s not always possible. Although the f a c t o r axes are orthogonal the v a ria b le s they represent are not e n t i r e l y independent as is shown in the c o r r e la tio n m a trix . Sub-population A on Figure 9 would probably be in te r p r e te d by Visher (1969) as a surface creep tra n s p o rt mechanism, and sub-population B and C as a s a l t a t i o n tra n s p o rt mechanism. The high c o r r e la t io n s 45 show t h a t the tra n s p o rt mechanisms surface creep and s a l t a t i o n may not be e n t i r e l y independent in a h yd raulic sense. T h e re fo re , s iz e in t e r v a ls loading on more than one fa c t o r can be expected. In principal-components a n a ly sis a break in the d i s t r i b u t i o n is sug­ gested as approxim ately 20, a t the change in sign o f the loading values. This break is the s iz e a t which Bagnold (1966) noted a change between the bedload and non-uniform suspension-transport mechanisms. On t h is b as is , the sediment approximately 20 and f i n e r is in te rp re te d as a non-uniform suspension, w h ile the sediment coarser than 20 probably represents the surface creep and s a l t a t i o n tra n s p o rt mechanisms. Beach Samples - The source beach samples and the beach sam­ ples c o lle c te d on Lake Michigan west o f S i l v e r Lake S tate Park ( F ig . 2) can be considered as a s in g le , beach subsample f o r the 2 fo llo w in g reason. The c a lc u la te d x between the grand frequency d is t r ib u t io n s o f the two samples (source beach and Lake Michigan beach) is 10.83 and is d is t r ib u t e d approximately as x g which is not s i g n i f i c a n t a t a = 0 .0 5 . This suggests t h a t the frequency d is t r ib u t io n s o f the two samples are not d i f f e r e n t . Table 3 l i s t s the r e s u lts o f the principal-com ponents a n a ly ­ s is f o r the beach samples. Two fa c to rs w ith eigenvalues >1 were e x tra c te d and account f o r 75.4% o f the t o t a l va ria n c e. n a l i t i e s are moderately high to very high. The commu- V a ria b le s w ith c o rre ­ la tio n s s i g n i f i c a n t a t a = 0.01 are also l i s t e d . Figure 10 is a cumulative p r o b a b ilit y graph o f the grand means o f each s iz e class f o r the beach subsamples. Two 46 TABLE 3 PRINCIPAL-COMPONENTS RESULTS OF LITTLE SABLE POINT BEACH SAMPLES Summary S t a t i s t i c s Phi Size 0 .0 0 .5 1 .0 1.5 2 .0 2 .5 Grand Means 0 .6 0 .6 6 .4 35.3 49.5 7 .3 Grand Standard Deviations 1.0 0 .8 7.2 16.4 17.7 6 .6 Product-Moment C o rre la tio n M a trix Phi Size 0 .0 0 .0 1 .0 0 .5 1 .5 2.0 2.5 -.8 7 ** -.4 8 ** -.9 1 ** -.7 7 ** 1.00 0 .5 .4 8 ** 1 .0 0 1.00 1.0 1 .5 .6 8 ** 1.00 2 .0 1.00 2.5 .5 4 * * 1 .0 0 s i g n i f i c a n t a t a = 0 . 0 1 * * , n = 29 , d f = 27 Eigenvalues Cumulative Percentage o f Eigenvalues I 3.308 55.1 II 1.212 75.4 Factor Rotated Factor M a trix I II Communalities Phi Size 0 .0 -0 .0 4 2 0.837 0.702 0.5 0.274 0.745 0.631 1.0 0.846 0.237 0.772 1.5 0.956 -0 .0 4 2 0.917 2 .0 -0 .9 5 0 -0 .0 5 9 0.907 2 .5 -0 .7 4 0 -0.201 0.5 89 47 F ig u re 10 P h i-p ro b a b i 1 i t y g ra ph o f th e g ra n d fre q u e n c y d i s t r i b u t i o n o f th e L i t t l e S a b le P o in t beach samples 48 9 9 99 Little Sable Point BEACH 99 9 99 CUMULATIVE PERCENTAGE 90 50 10 1 1 01 -1 0 1 PHI SCALE 2 3 4 49 sub-populations are in d ic a te d in th is p lo t : 1) sub-population A coarser than 1.3(9; and 2) sub-population B f i n e r than 1 .3 0 . Factor I accounts f o r 55.1% o f the t o t a l variance and is associated w ith sizes f i n e r than 1 .0 0 . Size 1.50 and 2.0 0 have high p o s itiv e loading va lu e s , w hile the 2.5 0 and 3.00 have high negative loading values. Sub-population B which is in d ic a te d on the p r o b a b i l it y graph accounts f o r 98% o f the d i s t r i b u t i o n and corresponds very w ell w ith f a c t o r I . Factor I I has high p o s itiv e loading values associated w ith 0 .5 0 and 1 .0 0 . This f a c t o r accounts f o r 19.3% of the t o ta l v a ria n c e. The o th e r s iz e classes have very low loading values and are not associated w ith th is f a c t o r . Factor I I corresponds w ell w ith sub­ population A on the p r o b a b i l it y graph which accounts f o r 2% o f the d is trib u tio n . Offshore bar and trough samples - Table 4 l i s t s the r e s u lts of the principal-components an a ly s is fo r the o ffs h o re bar and trough sam­ ples. Two fa c to rs w ith eigenvalues >1 were e x tra c te d and account fo r 71.9% o f the t o t a l va ria n c e . f i c a n t a t a = 0.01 are l i s t e d . V a ria b le s w ith c o r r e la tio n s s i g n i ­ The communalities are very hig h, except f o r the 0 .5 0 and 1.0 0 s iz e classes. Figure 11 is a cumulative p r o b a b i l it y graph o f the grand means o f each s iz e class f o r the bar and trough samples. The d i s t r i b u t i o n is almost log-normal^ however two sub-populations are in d ic a te d : 1) sub-population A co arser than 1 .1 0 ; and 2) sub-population B f i n e r than 1 .1 0 . 50 TABLE 4 PRINCIPAL-COMPONENTS RESULTS OF LITTLE SABLE POINT BAR AND TROUGH SAMPLES Summary S t a t i s t i c s Phi Size 0 .5 1 .0 1.5 2 .0 • CM 3.0 Grand Means 0 .4 3 .3 25 .8 34.4 15.2 1.2 Grand Standard D eviations 0 .5 2.7 20.1 16.6 10.2 1 .4 Product -Moment; C o rr e la tio n M a trix Phi Size 0 .5 0 .5 1 .0 1.5 2 .0 2 .5 1.00 -.8 2 ** -.6 0 ** 3 .0 1.00 1 .0 1.00 1.5 2. 0 1.00 2 .5 1.00 3 .0 . 88 * * 1. 0 0 s i g n i f i c a n t a t a - 0 . 0 1 * * , n = 18, d f = 16 Factor Eigenvalues 2.706 1.608 I II Rotated Factor M a trix I Cumulative Percentage o f Eigenvalues 45.1 71.9 Communalities II Phi S ize 0 .5 -0 .4 8 9 -0 .0 0 7 0.240 1 .0 0.367 0.500 0.386 1 .5 0.308 0.907 0.9 18 2 .0 0.252 - 0 .9 5 8 0,981 2 .5 - 0 .9 2 0 -0 .3 1 2 0.944 3 .0 - 0 .9 1 8 0.033 0.844 51 F ig u re 11 P h i-p ro b a b i1i t y graph o f th e grand fre q u e n c y d i s t r i b u t i o n o f th e L i t t l e Sable P o in t b a r and tro u g h samples 52 9 9 99 L ittle S able Point BARS AND TROUGH 9 9 .9 99 CUMULATIVE PERCENTAGE 90 50 01 1 0 1 PHI SCALE 2 3 4 53 Factor I is associated w ith sizes f i n e r than 2.0 0 (high negative loading values) and is moderately associated w ith 0 .5 0 . This f a c t o r accounts f o r 45.1% o f the sample varia n c e. The o th e r size i n t e r v a l s have small loading values on t h is f a c t o r . Factor I I , which accounts f o r 26.8% o f the t o ta l va ria n c e , is associated w ith sizes f i n e r than 0 .5 0 and coarser than 2 .5 0 . The 1.00 and 1.50 size class have moderate to very high p o s it iv e loading va lu e s, w h ile the 2 .0 0 s ize class has a high negative value. The o th e r s iz e classes have small loading values on th is f a c t o r . The f a c t o r axes do not c o r r e la t e very w ell w ith the sub-populations on the p r o b a b i l it y graph. Sub-population A accounts f o r 4% o f the d i s t r i b u t i o n , w h ile sub-population B accounts f o r 96% o f the d is trib u tio n . S ilv e r Creek bed samples - Table 5 l i s t s the re s u lts o f the principal-components a n a ly s is o f the S i l v e r Creek samples. V a riab le s w ith c o r r e la tio n s s i g n i f i c a n t a t a - 0.01 are l i s t e d in th is ta b le . Two fa c to r s w ith eigenvalues >1 were e x tra c te d and account f o r 81.5% o f the t o t a l variance. Communalities are high to very high. Three sub-populations are in d ic a te d on the cumulative prob­ a b i l i t y graph (F ig . 12): 1) sub-population A w ith sizes f i n e r than 1.2 50 ; 2) sub-population B w ith sizes between 1.250 and 2 .5 0 ; and 3) sub-population C w ith sizes f i n e r than 2 .5 0 . Factor I accounts f o r 59.8% o f the t o t a l variance and is re la te d to sizes f i n e r than 0 .5 0 . The s iz e s 1 .0 0 and 1.50 have high p o s it iv e loading va lu e s , w hile those f i n e r than 1.50 have high 54 TABLE 5 PRINCIPAL-COMPONENTS RESULTS OF LITTLE SABLE POINT SILVER CREEK BED SAMPLES Summary S t a t i s t i c s Phi Size 0 .5 1 .0 1.5 2 .0 2 .5 3 .0 Grand Means 0 .7 2 .5 19.1 57.3 18.6 1 .4 Grand Standard D eviations 0 .7 1 .3 17.9 9 .7 12.2 1 .4 Product-Moment C o rre la tio n M a trix _______________ Phi Size 0 .5 0 .5 1.00 1 .0 1 .0 1.5 1.0 0 1.5 .7 5 ** 1.00 2. 0 2 .0 2 .5 3 .0 -.6 8 ** -.6 1 ** -.5 3 ** -.7 5 ** -.8 8 ** -.7 4 ** 1. 00 2 .5 1.00 3 .0 .7 7 ** 1.00 s i g n i f i c a n t a t a = 0 . 0 1 * * , n + 22, d f = 20 Cumulative Percentage OF Eigenvalues_____ Factor Eigenvalues I 3.589 59.9 ** 1.300 81 .5 Rotated Factor M atrix I II Comnunalities Phi Size 0 .5 -0 .0 8 9 0.875 0.775 1.0 0 .8 35 0.201 0.737 1 .5 0.975 0.069 0.956 2 .0 - 0 .7 0 5 - 0 .5 4 8 0.798 2 .5 -0 .8 7 5 0.236 0.822 3 .0 - 0 .8 1 6 0.362 0.799 55 F ig u re 12: P h i-p ro b a b i 1 i t y g ra p h o f th e g ra n d fre q u e n c y d i s t r i b u t i o n o f th e L i t t l e P o in t S ilv e r C reek bed samples S able 56 9 9 99 L ittle S ab le Point SILVER CREEK 99 9 99 CUMULATIVE PERCENTAGE 90 50 01 -1 0 1 PHI SCALE 2 3 4 57 negative values. This fa c t o r c o r r e la te s w ell w ith the sub-populations B and C defined on the p r o b a b ilit y graph, w ith sub-population B corresponding to the sizes w ith p o s it iv e loading v a lu e s, and C corresponding to the s iz e classes w ith negative loading values. Sub-populations B and C on the p r o b a b i l it y graph account f o r 95% o f the t o t a l d i s t r i b u t i o n . Factor I I is re la te d to the 0 .5 0 s ize class and accounts f o r 20.7% o f the t o t a l varia n c e. r e la te d to t h is f a c t o r . The 2.00 s iz e class is moderately This fa c t o r c o rr e la te s to sub-population A on the p r o b a b i l it y graph, which accounts f o r o nly 5% o f the t o t a l d is trib u tio n . A eolian samples - Table 6 l i s t s the principal-com ponents re s u lts f o r the aeo lian samples. a - V a ria b le s s i g n i f i c a n t l y c o rr e la te d a t 0.0 5 and 0.01 are l i s t e d . Three fa c to r s w ith eigenvalues >1 were e x tra c te d and account f o r 83.1% o f the t o t a l varia n c e. Com­ munal i t i e s are moderately high to very high. Figure 13 i s a cumulative p r o b a b i l it y graph o f the grand frequency d is t r i b u t i o n o f the a e o lia n samples. the presence o f two sub-populations: The graph shows 1) sub-population A w ith sizes coarser than 1 .2 5 0 ; and 2) sub-population B w ith sizes f i n e r than 1.2 50 . Factor I accounts f o r 42% o f the t o t a l variance and is r e la t e d to the 1 . 5 , 2 . 5 , and 3.00 s iz e classes. The sizes 2 .5 0 and f i n e r have high n egative loading va lu e s, w h ile 1.5 0 has a high p o s it iv e loading v a lu e . 58 TABLE 6 PRINCIPAL-COMPONENTS RESULTS OF LITTLE SABLE POINT AEOLIAN SAMPLES Summary S t a t i s t i c s Phi Size 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 Grand Means 0 .6 1 .5 11.4 51.4 28.8 5 .7 Grand Standard Deviations 0 .8 1 .2 14.3 10.1 11.6 4 .0 Product-Moment C o r r e la tio n M a trix Phi Size 0 .5 0 .5 1 .0 1.5 2 .0 2.5 3 .0 -.7 3 ** -.5 5 ** 1.00 1.00 1.0 1.00 1.5 2.0 -.4 8 ** 1.00 2 .5 .8 0 ** 1.00 3 .0 1 .0 0 s i g n i f i c a n t a t a = 0 . 0 5 * , a = 0.01 * * , n * 26, d f = 24 Eigenvalues Factor Cumulative Percentage o f Eigenvalues I 2.519 4 2 .0 II 1.450 6 6 .2 III 1.013 83.1 Rotated Factor M a trix I II III Communalities Phi Size 0 .5 0.106 0 .2 9 8 0.736 0 .6 42 1.0 -0 .0 1 3 -0 .4 6 2 0 .6 8 8 0.6 87 1.5 0.885 - 0 .4 2 3 -0 .0 5 6 0 .9 67 2.0 0.127 0.937 0.0 53 0 .8 97 2.5 -0 .9 3 7 -0.211 -0.071 0.9 27 3 .0 -0 .8 4 3 -0 .3 6 0 -0 .1 4 3 0.861 59 F ig u re 13 P h i- p r o b a b il it y g ra ph o f th e g ra n d fre q u e n c y d i s t r i b u t i o n o f th e L i t t l e S a b le P o in t a e o lia n samples 60 99 9 9 L ittle S ab le Point AEOLIAN 99 9 99 CUMULATIVE PERCENTAGE 90 50 10 1 1 01 -1 0 1 PHI SCALE 2 3 4 61 Factor I I i s r e la t e d to the 2.00 s iz e class and accounts f o r 24.4% o f the t o t a l v a ria n c e , w ith a high p o s itiv e loading va lu e . Sub-population B o f the p r o b a b ilit y graph accounts f o r 96% o f the t o ta l d is t r i b u t i o n and c o rr e la te s very w ell w ith fa c to rs I and I I . Factor I I I accounts f o r 16.9% o f the t o t a l variance and is associated w ith sizes coarser than 1 .5 0 . This f a c t o r c o rr e la te s well w ith sub-population A on the p r o b a b ilit y graph, which accounts f o r only 5% o f the t o t a l d i s t r i b u t i o n . Discussion o f the Principal-Components Analysis and P r o b a b ilit y Graphs o f the L i t t l e Sable Point Samples Assignment o f the sediment sub-populations i d e n t i f i e d in t h i s study, on the basis o f principal-components an a ly s is and cum ulative p r o b a b i l it y graphs, to d i s t i n c t hydrodynamic sedimentary processes follow s the r e s u lts o f e a r l i e r workers ( e . g . , Bagnold, 1966; V ish er, 1969; and A lle n e t a l . 1971). An im portant fe a tu r e to n o tic e on the p r o b a b ilit y graph is th a t the size frequency d is t r i b u t i o n can be d ivid ed in to two o r more l o g - l i n e a r sub-populations (V is h e r , 1969) and th a t each sub­ population can be c o rr e la te d in most cases (b a r and trough samples are an exception) to one or two p rin c ip a l axes. These axes are orthogonal, hence s t a t i s t i c a l l y independent, th e re fo re each sub­ population may r e f l e c t a d i f f e r e n t tr a n s p o r t a tio n a l/d e p o s itio n a l h is to r y (A lle n e t a l . , 1 9 7 1 ) . Figure 14 i l l u s t r a t e s fa c t o r loading p r o f i l e s f o r each environment. These graphs show the r e la t io n s h ip o f each s iz e cla ss 62 F ig u re 14 Loading p r o f i l e s o f each L i t t l e Sable Point environment BARS AND TROUGHS LU 1 D SILVER CREEK NS SC FACTOR II < 0.5 > O z 0 D 0.5 < -0 .5 FACTOR I 1 BEACH AEOLIAN LU O1 1 < > 0.5 o 0 z FACTOR < Q 0.5 FACTOR II FACTOR I < -0 .5 < - 0.5 O -i -t 1 PHI SIZE CLASS PHI SIZE CLASS SC-Surface Creep 2 S Saltation NS=Non-Uniform Suspension 64 to the e x tra c te d fa c to rs by p lo t t in g the loading values on each f a c t o r by each s iz e class i n t e r v a l . The r e la t io n s h ip o f each s iz e class in t e r v a l cannot be in t e r p r e te d r i g i d l y , but these p lo ts do show th a t breaks occur in the loading p r o f il e s when a l l fa c to rs are p lo tte d to g e th e r. On the basis o f th is study and A lle n , e t a l . (1 9 7 3 ), the sediment ranging from approxim ately 10 to 30 can be assigned to the s a lt a t io n process o f Visher (1 9 6 9 ). However, A lle n , e t a l . (1971) suggested t h a t t h is sand-size group might represent a t r a n s i t i o n between two end-member tra n s p o rt processes (su rfa ce creep and sus­ pension). I t was found in t h is study and in A lle n , e t a l . (1971) th a t a d e f i n i t e break occurs a t 20 corresponding to a change in sign o f the f a c t o r loading v a lu e s , also noted by Bagnold (19 66 ). Bagnold (1966) suggested t h a t sediment coarser than 20 would be transported as bedload and those sizes f i n e r than 20 would be tra n s ­ ported as a non-uniform suspension ( c f . F ig . 1 4 ). The sediment 10 and co arser is in te rp re te d to represent the surface creep mode and corresponds w ell to the i n t e r p r e t a t i o n of Visher (1969) and A lle n , e t a l . (1 9 7 1 ). W ithin the l im it a t io n s o f the s iz e range o f the sediment c o lle c te d in t h is study, the percent o f sediment in each sub­ population and the number o f sub-populations and the tru n c a tio n points on the p r o b a b ilit y graphs are comparable to the same en viron ­ ments studied by Visher (1 9 6 9 ). I t has also been shown in t h is study th a t the sub-populations d e lin e a te d on p r o b a b i l it y graphs c o r r e la te w ell w ith a principal-com ponents a n a ly s is o f the same 65 samples. There are some major d iffe re n c e s in the inform ation ex tra cte d by these methods however. The major d iffe re n c e is in the amount o f variance explained by the two methods. Visher (1969) states t h a t i f i t is assumed th a t each sediment tra n s p o rt process is r e f le c t e d in a separate log-normal p o p u la tio n , the pro po rtion o f each population should be re la te d to the r e l a t i v e importance o f the corresponding process in the in fo rm atio n contained in the whole d is t r i b u t i o n . Visher assumes th a t the t o t a l variance o f the sample is accounted f o r by the sum o f the proportions o f the sub-populations, but does not take in to account variance produced by sampling e r r o r , measurement e r r o r and lo c al flu c t u a t io n s in the p r e v a ilin g energy c o n d itio n s . I f these variances are taken in to account, then the proportions w i l l account f o r something less than 100% o f the to t a l varia n c e. Another type o f e r r o r is in h e re n t in the p lo t t in g o f data on lo g -p ro b a b i1i t y paper (H o el, 1962). In t h is type o f graph the confidence l i m i t s placed about the mean d i s t r i b u t i o n are very wide in both t a i l s o f the graph ( i . e . , in the 95 to 99.99% region and in the 0.001 to 5% region on the graph p a p e r), thus the frequency per­ centages in these regions are overwhelmed by t h e i r own variances. Results of the principal-components an a ly s is show t h a t the t o t a l variance is not explained in each environment, assuming o f course t h a t only those f a c t o r axes w ith eigenvalues >1 are considered as s i g n i f i c a n t , whereas the tru n c a tio n points and number o f popu­ la tio n s are b a s ic a lly id e n tic a l paper. to those d efined on p r o b a b i l it y 66 Another d if fe r e n c e between principal-components an a ly s is and p r o b a b ilit y graphs is more s u b tle than the variance problem and is concerned w ith the number o f populations d e lin e a te d in the size range o f 10 to 30. In most cases only one population (one lo g - l i n e a r segment) is defined on the p r o b a b i l it y paper. I t has been suggested by A llen e t a l . ( 1 9 7 1 ) and th is w r i t e r , on the basis o f principal-components a n a ly s is , t h a t two sub-groups are present in th is size range in every environment as denoted by a change in sign o f the f a c t o r loadings. However, i t is possible in t h is study th a t the m atrix was a c tin g as a closed a rra y and negative c o r r e la tio n s were induced r e s u lt in g in high n egative f a c t o r loadin g. This is c e r t a i n l y the case f o r A lle n , e t a l . ( 1 9 7 1 ) as they made no apparent attem pt to open t h e i r m a trix . V ish er (1969) s ta te s t h a t in the samples he analyzed only the beach samples contained two sub-groups in t h is size range and th a t the break occurs a t 20, which he a t t r i ­ buted to swash-backwash beach processes. On the basis o f principal-components and p r o b a b i l it y graphs the s iz e frequency d is t r ib u t io n s were d ivid ed in to th ree energy/ process r e la t e d sub-populations which are described below in the terms o f Moss (19 62 , 1963), Bagnold (1 9 6 6 ), V ish er (1 9 6 9 ), and A lle n , e t a l . (1 9 7 1 ): Sub-population I - t h is population is g e n e r a lly described by sediments coarser than 10 and r e la te d to a surface creep process. 67 Sub-population I I - t h is population is g e n e r a lly described by two sub-groups in the s i z e range o f 10 to 30; Group A - g r a in -s iz e s from 10 to 20, transported by s a l t a t i o n . Group B - g r a in -s iz e s from 20 to 30, transported as non-uniform suspensi on. Principal-component fa c to rs account f o r 74-84% o f the to ta l g r a in - s iz e variance in the L i t t l e Sable P oint environments, the remainder o f the varian ce not explained can be assigned to random f lu c t u a tio n s in the tra n s p o rt processes, sampling and measurement e rro rs ( Im b r ie , 1963; and A lle n , e t a l . , 1 9 7 1 ) . In sedim entological terms the populations l i s t e d above are the best te x tu r a l d e lin e a to rs o f the hydrodynamic processes operatin g in the L i t t l e Sable Point environments. P rin c ip a l- components a n a ly sis seems to r e f in e and q u a n tify the grap h ical con­ clusions o f Visher (1969) and support the t h e o r e tic a l r e s u lts about ju n c tu re points between processes as described by Bagnold (1966). D isc rim in an t-F u n c tio n Analysis D is c rim in a n t-fu n c tio n a n a ly sis was used on the L i t t l e Sable P oin t samples to t e s t f o r s i g n i f i c a n t d iffe re n c e s between the sam­ pled environments. The a p r io r i c r i t e r i a used to t e s t f o r d i f f e r ­ ences in the L i t t l e Sable Point samples i s based on th e mode o f sam­ p le c o l le c t i o n ; t h a t i s , each sub-set o f samples was c o lle c te d from a d i f f e r e n t environment. 68 2 The Mahalanobis D f o r the beach, a e o lia n , bar and trough, and f l u v i a l m u l t i v a r i a t e means (not th e tru e means, but the D o r Z score means between the four p o p u latio ns) is 109.03. This value 2 is d is tr ib u te d approxim ately as x -jg and a t a >0.001 is h ig h ly s ig n if ic a n t and suggests t h a t a t l e a s t one environment i s s i g n i f i ­ c a n tly d i f f e r e n t from the o th ers. Table 7 l i s t s the d is c rim in a n t c l a s s if i c a t io n o f the fu n c tio n a l groups te s te d . TABLE 7 D is c rim in a n t-fu n c tio n Cl assi f i c a t i o n o f the Functional Groups Tested a t L i t t l e Sable Poi n t Functional Group Tested Bars and Troughs Aeolian Si 1ver Creek Beach 17 5 0 3 25 S ilv e r Creek 1 11 10 0 22 Beach 0 5 23 1 29 Bars and Troughs 2 6 9 1 18 Aeolian Total Comparing Table 7 to Figure 15, a graphical re p re s e n ta tio n o f the f i r s t canonical v a r ia b le versus th e second, suggests th a t the aeolian samples c o n s titu te one t e x t u r a l group (environment) w hile the beach, o ffs h o re bars and trough, and S i l v e r Creek samples c o n s titu te another t e x t u r a l group. The beach samples d is c rim in a n t b e t t e r (les s samples were m i s - c l a s s i f ie d ) than th e o th e r groups, however as can be seen in Table 7 and Figure 15, th e re i s much more t e x t u r a l overlap 69 F ig u re 15 D is c r im in a n t -f u n c tio n a n a ly s is c l a s s i f i c a t i o n o f the L i t t l e Sable P o in t environments 2.4 i ---------------- 1---------------- 1---------------- 1----------------1----------------1---------------- 1— E B 1.6 E B_ B OB B E CANONICAL VARIABLE 2 0.7 0.3 *• - 0E 06 1.2 ‘ V 5$$ B E - B 2.2 GROUP MEANS ^ GROUP OVERLAP E AEOLIAN S SILVER CREEK o BARS AND TROUGHS B BEACH ♦ -3.1 -4.1 DISCRIMINANT-FUNCTION ANALYSIS Little Sable Point Lake Michigan s i_________ i_________ i_________ i_________ i_________ i_________ i_________ i__________ -4 .5 -3 .6 -2 .6 -1.7 -0 .8 0.2 1.1 CANONICAL VARIABLE 1 2.1 3.0 4.0 71 between the beach, S i l v e r Creek and o ffsh ore bars and trough samples than th e re is between the a e o lia n samples and the o th e r groups. The bar and trough samples show the g re a te s t t e x t u r a l overlap o f any group, being c l a s s i f i e d mainly as beach and f l u v i a l sediment. This may r e f l e c t an i n t e r - p l a y between the b e a c h -flu v ia l and l i t t o r a l d r i f t component in the area showing a feedback between these environments. S aylor and Hands (1970) t r i e d to d is tin g u is h te x t u r a l d iffe re n c e s in the bar and trough sediment but w ith l i t t l e Very l i t t l e success. is known about t h is environment ( B l a t t , e t a l . , 1972). Almost h a l f o f the S i lv e r Creek samples were c l a s s i f i e d as beach sediment. As w i l l be more c l e a r l y seen in th e next section (Q-mode f a c t o r a n a l y s i s ) , the beach processes seem to o ve rrid e the f l u v i a l processes a t the mouth o f S i l v e r Creek. I t is apparent from the d is c rim in a n t a n a ly sis t h a t th ere is a g re a t deal o f t e x t u r a l overlap between adjacent environments a t L i t t l e Sable P o in t, much more than would normally be expected on the basis o f tra n s p o rt competency o f the processes in these environments. The s in g le sediment source in t h is area seems to be a dominant f a c t o r f o r environmental d is c r im in a tio n . Unless a l l sizes are a v a i l ­ ab le to the system, the d iffe re n c e s in energy le v e ls o f the d i f f e r e n t environments w i l l not be r e f le c t e d in the sediment. Q-Mode Factor Analysis Q-Mode fa c t o r a n a ly s is was used to c l a s s if y th e sediments in to s i m il a r groups by c a lc u la tin g the varian ce across th e sample v a r ia b le s . Four fa c to r s were re ta in e d f o r varimax r o t a t io n , one f a c t o r f o r each environment. Under id e a l conditions each environment would be 72 associated w ith one and o nly one f a c t o r . be expected f o r two reasons: However, t h is should not 1) on the basis o f d is c rim in a n t- fu n c tio n a n a ly s is i t has a lre a d y been shown t h a t th e re is a g rea t amount o f t e x t u r a l overlap in samples from these environments; and 2) one would not expect p e r fe c t separation in nature because o f t r a n s it io n a l zones between adjacent environments. The r e s u lts o f the Q-Mode f a c t o r a n a ly s is f o r a l l L i t t l e Sable Point samples are l i s t e d in Appendix C. Only th ree indepen­ dent fa c to rs w ith eigenvalues >1 were e x tra c te d and these e x p la in 99.5% o f the t o t a l sample va ria n c e . fa c to r s are a l l f o r fa c t o r I I I Communalities f o r the th ree very high, exceeding 0 .9 9 2 . Although the eigenvalue i s >1, i t only accounts f o r 2.7% o f the t o t a l varian ce and has i n s i g n i f i c a n t f a c t o r loadings. Figure 16 i s a te rn a ry diagram of normalized f a c t o r loadin gs. Each fa c t o r loading was normalized by d iv id in g the square o f the loading value by i t s corresponding communality (Klovan, 1966). This procedure provides re s u lts s i m il a r to the o b liq u e -p r o je c tio n method o f Imbrie and Van Andel (1964) in th a t more emphasis is placed on the la r g e r f a c t o r loadings; however, the axes are s t i l l orthogonal. The samples loading hig hest on f a c t o r I are the a e o lian samples and those c o lle c te d from the main p a rt o f S i l v e r Creek, which flows across the dune f i e l d . Beach, source beach and the f l u v i a l samples from the mouth o f S i l v e r Creek load on f a c t o r I I . Offshore bar samples are d is t r ib u t e d between both fa c to r s I and I I . I t is evident in Figure 16 th a t th e re 1s g re a t t e x t u r a l o verlap in the samples. 73 F ig u re 16 Tern ary diagram o f norm alized Q-mode f a c t o r loadings o f the L i t t l e Sable P o in t samples FACTOR I F SILVER CREEK • BARS AND TROUGHS E AEOLIAN 1 BEACH FACTOR III *F FACTOR II 75 As shown in Figure 17 f o r the most p a r t , the beach and nearshore trough samples load on f a c t o r I I . Five beach samples load f a c t o r I , fo u r o f these are south o f a snow fence which may s l i g h t l y impede l i t t o r a l d r i f t to the north and cause a change in t e x tu r e . N o tice th a t the samples c o lle c te d in and near the mouth o f S i l v e r Creek load on f a c t o r I I , w hile the o th e r f l u v i a l samples load on f a c t o r I . Those f l u v i a l samples loading on f a c t o r I are from the stream w ith in the dune t r a i n , which serves as a sediment source f o r S i lv e r Creek, w h ile those loading on f a c t o r I I are dominated by in tro d u c tio n of m a te ria l by l i t t o r a l d r i f t and beach processes. B eall (1970) noted the dominance o f beach pro­ cesses over f l u v i a l processes in a b e a c h -d e lta ic system. The bar and trough samples load on f a c t o r I o r I I , o r load moderately on both I and I I . The a e o lian dune samples load almost e n t i r e l y on f a c t o r I , w h ile the source beach samples load on f a c t o r I I . Q-Mode f a c t o r an alysis emphasizes a lack o f environmental d i f f e r e n t i a t i o n as shown in the previous section w ith d is c r im in a n tfu n c tio n a n a ly s is . The beach processes, due to wave surge, seem to o v e rrid e th e f l u v i a l processes a f f e c t i n g S i l v e r Creek as much as 6-10 meters upstream of the mouth. The o ffs h o re bar and trough samples show g r e a t te x tu r a l v a ria n c e , which probably r e f l e c t s fee d ­ back between the beach and S i lv e r Creek sediment. Samples from the second trough load on f a c to r I and may represent m a te ria l deposited by backwash a c tio n from the f i r s t b a r. Although t h is study deals w ith major environmental groupings ( i . e . , f l u v i a l , s t r a n d lin e , and a e o lia n ) th e re appears, as should be expected, to be d i f f e r e n t sub-environments w ith in each major group. 76 F ig u re 17 A re a l d i s t r i b u t i o n o f two Q-mode f a c t o r s f o r th e L i t t l e Sable P o in t Samples LAKE MICHIGAN i ltt o r a i drift -2 .5 w" * t ; x x, f -1 0 0 m I* • n f f / j / slream mouth ~ 5 m wide snow fence M O R A IN A l SOURCE AREA — crest - ^ W ind IPWM trough r_~_| Factor I Factor II • 'U U I - • - -crest v - A — Factor I, II HUES* ‘ crest Q i- t t t * i t t l l I T T I Little Sable Point Michigan .c r e s t !_ „ Factor I £ 3 Factor II m Factor I, II Sand Dunaa Location 1 78 Figure 17 i l l u s t r a t e s the s p a tia l d i s t r i b u t i o n o f the Q-Mode fa c to rs f o r the L i t t l e Sable Point samples. D is c rim in a n t- fu nctio n a n a ly s is has shown t h a t environmental d i f f e r e n t i a t i o n o f these sediments is somewhat tenuous when d ea lin g w ith sands from one source w ith a truncated or winnowed d i s t r i b u t i o n . It is important to note however in Figure 17 t h a t there is some s p a tia l d is t r i b u t i o n o f sediment p a tte rn s . Although samples were c o lle c te d from supposedly d i f f e r e n t energy environments, the energy r e la te d processes in t h is study area are g e n e ra lly not s u f f i c i e n t l y d is ­ t i n c t enough to cause s i g n i f i c a n t changes in the s iz e d is t r i b u t i o n s in the sands w ith the above c h a r a c t e r i s t i c s . Sleeping Bear Point/M anitou Passage Depositional Area The sediment in the Sleeping Bear Point/Manitou Passage area have m u ltip le source areas w ith a range in g r a in -s iz e s from lag gravel deposits and coarse sand on Sleeping Bear Shoal to f i n e s i l t and c la y in the profundal basin north o f the shoal (F ig . 1 8 ). The f a c t th a t the sediments are not tru n cated a t the f i n e end o f the d i s t r i b u t i o n may provide f o r a b e t t e r means o f environmental d is c r im in a tio n . Many authors have shown t h a t the f i n e r sediments are deposited under lower hydrodynamic co n dition s which can be depth dependent ( e . g . , Passega, 1957, 1972; V is h e r, 1969 ). The fin e s winnowed from the higher energy shoal environment should be deposited in the lower energy profundal environment. Before any o f the Sleeping Bear Point/M anitou Passage samples were analyzed the study area was d iv id e d in to fo u r areas 79 th a t appear to be d i f f e r e n t d e p o s itio n a l/e ro s io n a l environments. The a p r i o r i c r i t e r i o n used f o r t h is c l a s s i f i c a t i o n is based on the morphological open water zones noted by French (1 9 6 4 ), which are based on the bathymetry o f the area. The f i r s t environment is the shoal and nearshore zone where twenty-one samples were c o lle c t e d . Five o f the samples were not analyzed because o f t h e i r coarse c l a s t s iz e . These samples appear to be p a r t o f the lag deposits des­ cribed by French (1 9 6 4 ). The second environment includes the beach area on the n orth e as te rn t i p o f Sleeping Bear P oint where twentynine samples were c o lle c te d . The west side o f the p o in t was not sampled because the base o f the west face consists o f lag beach gravels o f m orainic sediments (French, 1964). The nine samples from the profundal area north o f Sleeping Bear Point c o n s titu te the t h ir d environment, w ith the fo u rth environment making up the t r a n s ­ i t i o n a l area between the profundal basin and the shoal and near­ shore. F ifte e n samples were c o lle c te d in the t r a n s it io n a l area. The data are l i s t e d in Appendix B. D isc rim in an t-F u n c tio n Analysis Figure 18 is an index map o f the Sleeping Bear P o in t/ Manitou Passage area and locates the environments d efined on the 2 basis o f bathymetry. The Mahalanobis D of the fo u r m u lt iv a r ia t e means is 1634.35. This value is d is t r ib u t e d approxim ately as p X 27 arK* a t a>0 . 00 1, is h ig h ly s i g n i f i c a n t and suggests th a t a t le a s t one environment is s i g n i f i c a n t l y d i f f e r e n t from the o th e rs. Table 8 l i s t s the d is c rim in a n t c l a s s i f i c a t i o n o f the fu n c tio n a l 80 Figure 18: Index map o f the environments a t the Sleeping Bear Point/Manitou Passage area used in the d is c rim in a n t-fu n c tio n a n a ly s is . The environments were defined a f t e r French (1964) PROFUNDAL T RANSI TI ONAL 20 m — SHOAL AND NEARSHORE BeacW' i . ''S le e p in g Bear Point Lag Gravel KILOMETERS TRANSI TI ONAL 82 groups te s te d . Comparing Table 8 to Figure 19 , a graphical p lo t o f the f i r s t canonical v a r ia b le ag ain st the second, shows t h a t a l l the environments are s i g n i f i c a n t l y d i f f e r e n t . Three beach samples were c l a s s i f i e d w ith the shoal and nearshore samples, w h ile three shoal samples were c l a s s i f i e d in the t r a n s i t i o n a l group. Some te x tu r a l o verlap between the beach and shoal and nearshore en v iro n ­ ments should be expected as they are both areas o f high energy wind induced wave processes. TABLE 8 D isc rim in an t-F u n c tio n C l a s s i f i c a t io n o f the Functional Groups Tested a t the Sleeping Bear P o in t/ Manitou Passage Area Profundal Nearshore and Shoal Profundal 9 0 Nearshore & Shoal 0 T ra n s itio n a l Beach T ra n s i­ tio n a l Beach Total 0 0 9 13 3 0 16 0 0 15 0 15 0 3 0 26 29 The d is c rim in a n t c l a s s i f i c a t i o n provides a basis f o r e v a lu a tin g , by means o f principal-com ponents a n a ly s is , d i f f e r e n t tra n s p o rt processes in these environments. D is c rim in a n t-fu n c tio n analysis shows s t a t i s t i c a l l y th a t the s iz e d is t r ib u t io n s represent d is tin c e environments on the basis o f t h e i r m u l t iv a r ia t e means. 83 F ig u re 19 D is c r im in a n t - fu n c t io n c l a s s i f i c a t i o n o f the Sleeping Bear P o in t/M a n ito u Passage environments i 4.9 - DISCRIMINANT-FUNCTION ANALYSIS Sleeping Bear Point/ Manitou Passage Lake Michigan CANONICAL VARIABLE 2 3.5 1.6 - 0.2 - 1-------- 1---------1-------- 1— \ -1 .3 - * 8 8 T P -3.1 - -4 .9 I - 6.0 -3 .8 - 1.6 GROUP MEANS BEACH SHOAL AND NEARSHORE TRANSITIONAL PROFUNDAL I 1 1 1 0.6 2.7 4.9 7.1 CANONICAL VARIABLE 1 9.3 11.5 13.7 85 Principal-Components Analysis The shoal and nearshore, profunda1 , t r a n s i t i o n a l , and beach samples were subjected to principal-components an a ly s is to d e te r ­ mine th e r e la tio n s h ip s between the sub-populations o f t h e i r size frequency d is t r ib u t io n s in an e f f o r t to d e lin e a te tra n s p o rt pro­ cesses o p e ra tin g in the fo u r environments. The sediment f i n e r than 40 was not included in t h i s an a ly s is f o r two reasons; 1) excluding th is f r a c t i o n w i l l open the data a rra y and avoid the closure pro­ blem discussed e a r l i e r ; and 2 ) i t has been assumed by several workers t h a t t h is s iz e group represents the suspension tra n s p o rt mechanism and should th e r e fo r e load on an independent f a c to r { e . g . , Bagnold, 1956; Passega, 1957, 1972; V is h e r, 1969; Davis, 1970; A l l e n , e t a l . , 1971). Shoal and nearshore area - Table 9 l i s t s the principal-components re s u lts f o r the shoal and nearshore samples. V a ria b le s c o rr e la te d a t a = .05 and .01 a re l i s t e d in t h i s ta b le . Two fa c to rs w ith eigenvalues >1 were e x tra c te d and account f o r 80.4% o f the t o t a l v a ria n c e . The communalities are high to very high. Figure 20 is a c u m u la tiv e -p r o b a b ility graph o f the grand frequency d i s t r i b u t i o n o f the shoal and nearshore samples. Four sub-populations are in d ic a te d in t h i s d is t r i b u t i o n w ith tru n c a tio n points a t : 1) 2.0 0 between A and B; 2) 3.3 0 between B and C; and 3) 4 .0 0 between C and D. Factor I is c lo s e ly r e la te d to the s ize classes 0 .5 0 to 1.50 (h ig h p o s it iv e loadin g values) and the s iz e class 2 .5 0 36 TABLE 9 PRINCIPAL-COMPONENTS RESULTS OF SLEEPING BEAR POINT/ MANITOU PASSAGE SHOAL AND NEARSHORE SAMPLES Summary S t a t i s t i c s Phi Size 0 .5 1 .0 1.5 2.0 2 .5 3 .0 3 .5 4 .0 Grand Means 4 .7 12.1 25.6 30.6 13.8 6 .3 3 .4 0 .8 Grand Standard Deviations 3 .9 11.7 13.4 8 .7 8 .3 6.6 1.2 Product-Moment C o rr e la tio n M atrix 0 .5 1 .0 2.0 3 .0 1 .5 3 .5 4 .0 Lf> * CM Phi Size 1.0 0 0*5 8.8 .8 5 * * 1.00 1.0 .58 * -.5 3 * .6 4 * * 1.00 1.5 2.0 -.5 5 * -.7 2 ** -.5 5 * -.5 1 * -.5 0 * * -.6 5 ** -.7 1 **-.5 7 * -.6 0 * 1 .00 2.5 .6 5 ** 1.00 . 88* * 1 .00 3.0 3.5 1.00 4 .0 .7 2 * * .6 3 * * 1.00 s i g n i f i c a n t a t a = 0 . 0 5 * . a = 0 . 01* * , n = 16, d f = 14 Factor Eigenvalues Cumulative Percentage o f Eigenvalues I 4.501 56.3 II 1.930 80.4 Rotated Factor M a trix Communalities Phi Size 0 .5 0 .8 38 0.226 0.753 1 .0 0 .9 6 0 - 0.001 0.922 1.5 0.781 0.365 0.743 2.0 -0 .4 5 9 0.835 0.892 2 .5 - 0 .7 6 4 -0 .2 0 5 0.626 3 .0 - 0 .5 2 4 -0 .8 0 9 0.929 3.5 -0 .5 3 1 -0 .8 5 4 0.853 4 .0 - 0 .3 4 5 -0 .7 6 7 0.708 87 Figure 2 0 : P h i-p ro b a b il f t y graph o f the grand frequency d i s t r i b u t i o n o f the Sleeping Bear P oint/M anitou Passage shoal and nearshore samples 88 99 99 99 9 S leep in g B ear Point / M anitou Passage SHO AL AND NEARSHORE 99 CUMULATIVE PERCENTAGE 90 50 10 1 1 .01 -1 0 1 PHI SCALE 2 3 4 89 (moderately high n eg ative loading v a lu e ) . 56.3% o f the t o ta l v a ria n c e . This f a c t o r accounts fo r Sub-population A on the p r o b a b ilit y graph accounts f o r 70% o f the t o t a l d i s t r i b u t i o n and c o rr e la te s w ell w ith th is f a c t o r . Factor I I accounts f o r 24.1% o f the to ta l variance and is r e la te d to the s iz e classes 3.00 and f i n e r (high neg ative loadings) and w ith the 2.00 s iz e class (high p o s it iv e lo a d in g ). Sub­ population C on the p r o b a b i l it y graph accounts f o r 4% o f the t o t a l d i s t r i b u t i o n and seems to c o rr e la te w ith th is fa c t o r . Sub-population B on the p r o b a b i l it y graph accounts fo r 25% o f the t o ta l d i s t r i b u t i o n and c o r r e la te s w ith both f a c t o r I (high negative loading values) and f a c t o r I I (high p o s it iv e loading v a lu e s ). 8each samples - Table 10 l i s t s the p r in c ip a l components re s u lts f o r the beach samples. Two fa c to rs w ith eigenvalues >1 were ex tra c te d and account f o r 85.2% o f the t o t a l v a ria n c e . high to very high. The communalities are V a ria b le s s i g n i f i c a n t l y c o rr e la te d a t a = .05 and .01 are also l i s t e d . Figure 21 i s a cu m u lative -p ro b ab i1i t y graph o f the grand frequency d is t r i b u t i o n f o r these samples. Three sub-populations are shown on th is graph w ith tru n c a tio n points a t : 1) 0 .5 0 between A and B; and 2) a t 1 .7 0 between B and C. Factor I i s r e la t e d to the 1 .0 0 s ize class (high p o s itiv e lo a d in g ) and those 2 . 0 0 and f i n e r (v e ry high negative lo a d in g s ). This fa c t o r accounts f o r 30.3% o f th e t o t a l varia n c e. This f a c t o r 90 TABLE 10 PRINCIPAL-COMPONENTS RESULTS OF SLEEPING BEAR POINT/ MANITOU PASSAGE BEACH SAMPLES Summary S t a t i s t i c s Phi Size 0 .0 0 .5 1.0 1.5 2 .0 2 .5 Grand Means 1.7 4 .6 4 4 .6 32.1 12.0 3.1 Grand Standard Deviations 2 .5 3 .9 18.3 11.6 11.5 6 .3 Product-Moment C o rre la tio n M a trix Phi Size 0.0 0 .0 1.00 0 .5 0 .5 1.0 .7 5 ** 1.0 0 1.5 2 .0 2 .5 -.6 7 ** -.5 4 ** -.3 7 * -.8 7 ** -.7 4 ** -.5 3 ** .4 2 * * 1.00 1.0 1.00 1.5 2.0 1.00 . 88* 2 .5 1 .00 s i g n i f i c a n t a t a = 0 . 0 5 * , a = 0.01 * * , n = 29, df = 27 Eigenvalues Cumulative Percentage o f Eigenvalues I 3.291 54.9 II 1.821 85 .2 Factor Phi Size Rotated F ac to r M a trix I II Communalities 0.0 0.165 -0 .8 3 6 0 .7 2 7 0 .5 0.382 -0 .8 6 3 0 .8 9 2 1.0 0.873 -0 .1 9 3 0 .8 0 0 1.5 0.114 0.890 0.805 2.0 -0 .9 6 0 0.207 0.9 65 2.5 - 0 .9 5 8 -0.041 0.920 91 F ig u re 21 P h i- p r o b a b ilit y graph o f th e grand fre q u e n c y d i s t r i b u t i o n o f th e S le e p in g Bear P o in t/M a n ito u Passage beach samples 92 9 9 99 99 9 Sleeping Bear Point / M anitou Passage BEACH 99 CUMULATIVE PERCENTAGE 90 50 10 1 1 01 1 0 1 PHI SCALE 2 3 4 93 re la te s w ell w ith sub-population A on the p r o b a b ilit y graph which accounts f o r 50% o f the t o t a l p o p ulatio n. Factor I I accounts f o r 54.9% o f the t o ta l varia n c e. The s iz e classes r e la te d to t h is f a c t o r are the 0.0 0 to 0 .5 0 (high negative loadings) and the 1.50 i n t e r v a l (high p o s itiv e lo a d in g ). This f a c t o r corresponds w e ll w ith sub-populations B and C on the p r o b a b ilit y graph which account f o r 50% o f the t o ta l d i s t r i b u t i o n . Profunda! sediments - Table 11 l i s t s the principal-components re s u lts f o r the profundal sediments. Two fa c to rs w ith eigenvalues >1 were e x tra c te d and account f o r 89.9% o f the t o t a l v a ria n c e. C o rre la tio n s s i g n i f i c a n t a t a = .05 and .01 are l i s t e d . The com­ munal i t i e s are high to very high. Figure 22 shows the grand frequency d is t r i b u t i o n o f the profundal sediments. Three sub-populations are present w ith tru n c a tio n points a t : 1) 2.0 0 between A and B; and 2) a t 4.00 between B and C. Factor I is r e la te d to the s iz e classes 2 .0 0 and coarser (high to very high loading v a lu e s ). This fa c to r accounts fo r 67.7% o f the t o t a l variance and c o r r e la te s w ell w ith sub-population A on the p r o b a b i l it y p lo t which accounts f o r 10% o f the t o t a l s iz e d is trib u tio n . Factor I I accounts f o r 22.2% o f the variance and is r e la te d to sizes f i n e r than 2.0 0 (high p o s itiv e lo a d in g s ). This f a c t o r c o rr e la te s w ell w ith sub-population B on the p r o b a b ilit y graph which 94 TABLE 11 PRINCIPAL-COMPONENTS RESULTS OF SLEEPING BEAR POINT/ MANITOU PASSAGE PROFUNDAL SAMPLES Phi Size Summary S t a t i s t i c s 0 .5 1.0 1 .5 2 .0 2 .5 3.0 3.5 4 .0 Grand Means 0.6 1.7 2.7 5 .0 1 .9 4 .0 3 .5 1 .5 Grand Standard Deviastions 0 .4 1.4 2.2 2 .9 1 .0 5.5 3.2 1.8 3.0 3 .5 4 .0 Product-Moment C o rre la tio n M a trix Phi Size 0 .5 0 .5 1.0 0 1.0 1.0 1 .5 .9 5 ** 1.00 .9 2 ** .7 4 * .9 9 ** .8 5 * 1.00 1.5 2 .0 2.0 2 .5 . 88* * 1.00 2 .5 1.0 0 3 .0 .72 * 1.00 . 86* * .7 8 * .9 1 ** .7 0 * 4 .0 1.00 s ig n ific a n t a t a = 0 .0 5 *, a = 0 .0 1 **, n = 9, d f - 7 Factor I II Eigenvalues 5.415 1.778 Rotated Factor M a trix 1 tl Cumulative Percentage o f Eigenvalues 67.7 89 .9 Communalities Phi Size 0 .5 1.0 1.5 2.0 2 .5 3 .0 3 .5 4 .0 0.9 40 0 .9 6 9 0 .9 52 0.844 0.295 0.186 0.110 0.565 0 .1 3 8 0.2 1 5 0 .2 8 8 0.2 73 0.8 63 0 .9 0 8 0.981 0.731 0.903 0.987 0.991 0.788 0.832 0.860 0.975 0.855 95 Figure 2 2 : P h i-p ro b a b i1i t y graph o f the grand frequency d is t r i b u t i o n o f the Sleeping Bear Point/M anitou Passage profundal samples 96 99 99 S leeping B ear Point / M anitou Pasaaga PROFUNDAL 99 9 99 CUMULATIVE PERCENTAGE 90 50 10 1 1 01 0 1 PHI SCALE 2 3 4 97 accounts f o r 10% o f the d i s t r i b u t i o n . Sub-population C on the p o r b a b ilit y graph is re la te d to s i 2es 4.00 and f i n e r and accounts f o r 80% o f the t o t a l d is t r i b u t i o n . T ra n s itio n a l sediments - Table 12 l i s t s the principal-components re s u lts f o r the tr a n s it io n a l sediments. Two fa c to r s were e x tra c te d w ith eigenvalues >1 and account f o r 79.4% o f the t o t a l v a ria n c e. The communalities are moderately high to high in va lu e . Figure 23 shows the grand frequency d i s t r i b u t i o n o f the t r a n s i t i o n a l samples. Three sub-populations are present w ith tru n c a tio n points a t : 1) 3.250 between A and B; and 2) a t 4 .0 0 between B and C. F acto r I is r e la te d to s ize s 1.50 and coarser {high p o s it i v e ) loadings) and sizes 2.00 and 2.5 0 (moderately high negative lo a d in g s ). This f a c t o r accounts f o r 47.3% o f the t o t a l v a ria n c e. F ac to r I I accounts f o r 32.1% o f the t o t a l variance. The sizes r e la t e d to th is f a c t o r a re the i n t e r v a ls 3.00 (high n egative loading) and 3.5 0 to 4.00 (high p o s itiv e lo a d in g s ). Sub-populations A and B on the p r o b a b i l it y graph are r e la te d to fa c to r s I and I I ; t o t a l p o p u la tio n . these sub-populations account f o r 86% o f the Population C r e la t e s to sizes f i n e r than 40 and accounts f o r 13% o f the p o p u la tio n . 98 TABLE 12 PRINCIPAL-COMPONENTS RESULTS OF THE SLEEPING BEAR POINT/ MANITOU PASSAGE TRANSITIONAL SAMPLES Suntnary S t a t i s t i c s Phi Size 0 .5 1 .0 1 .5 2.0 2 .5 3 .0 3 .5 4 .0 Grand Means 1.6 5.2 8.1 15.4 17.0 25 .6 10.5 3 .8 Grand Standard Deviations 1.6 4 .5 6.6 8.8 12.9 12.3 7 .4 2.7 3 .0 3 .5 4 .0 Product-Moment C o rr e la tio n M a trix Phi Size 0.5 0 .5 1 .0 1.5 1.00 -.6 4 ** -.7 2 ** 1.00 1 .0 2 .0 2 .5 -.6 1 * . 86* * -.6 7 ** 1.00 1 .5 2 .0 -.7 4 ** 1.00 2.5 -.6 4 ** -.7 7 * * -.6 1 * 1.00 3 .0 -.6 9 1.00 3 .5 1.00 4 .0 1 .00 s i g n i f i c a n t a t a = 0 . 0 5 * , a = 0. 01* * » n = 15 , d f = 13 Factor Eigenvalues Cumulative Percentage o f Eigenvalues I 3.780 4 7 .3 II 2.575 79.4 Rotated Factor M a trix I Communalities II Phi Size 0 .5 1 .0 1 .5 2.0 2 .5 3 .0 3 .5 4 .0 0.812 0 .9 0 9 0 .9 5 7 0.4 23 -0 .6 6 2 -0 .7 6 9 -0 .1 5 2 0.107 -0 .1 4 9 0.135 -0 .0 9 8 -0.761 -0 .7 0 8 0.520 0.7 75 0.837 0.681 0.846 0.927 0.7 58 0.9 40 0.863 0.624 0.7 13 99 Figure 23: P h i-p r o b a b i1i t y graph o f the grand frequency d i s t r i b u t i o n o f the Sleeping Bear P o in t/M a n ito u Passage t r a n s i t i o n a l samples 100 9 9 99 99 9 Slaeplng B ear Point / M anitou Pasaaga TRANSITIO NAL 99 CUMULATIVE PERCENTAGE 90 50 10 1 1 01 -1 0 1 PHI SCALE 2 3 4 101 t Discussion of the Principal-Components and P ro b a b ility Graph Results f o r the Sleeping Bear Point/Manitou Passage Environments Figure 24 is a p lo t o f the loading values versus phi s iz e f o r each o f the four environments. The reader i s reminded here th a t the pure suspension mode (>40 size s ) is not represented in the principal-components a n a ly s is , omission o f t h i s s ize f r a c t i o n should not a f f e c t the loading values of the o th e r sizes on the fa c to r s . The sediment f i n e r than 40 in the profundal sediments accounts f o r 50-95% o f the t o t a l sample by w e ig h t, 15-35% in the t r a n s it io n a l sediments, less than 5% in the shoal and nearshore samples, and none in the beach sediment. As t h i s m a te ria l accounts f o r a g re a t proportion o f the t o t a l weight in th e profundal s e d i­ ment, the amount o f variance accounted f o r by the two fa c to rs is grossly over estim ated. The variance accounted f o r by the two facto rs in the t r a n s i t i o n a l and shoal and nearshore sediments should be only moderately over estim ated. T h e re fo re , only trends in the loading p r o f ile s can be used s i g n i f i c a n t l y f o r the profundal sediments. The c u m u lative -p ro b ab i1i t y graphs ( c f . , Figs. 20, 21 , 22, and 23) o f the Sleeping Sear Point/Manitou Passage sediments c o r r e ­ l a t e well w ith the fa c to rs e x tra c te d by the principal-com ponents analyses ( F ig . 2 4 ). The p r o b a b i l i t y graphs suggest the presence o f 3 to 4 su b -po pu lation s, w h ile the principal-components r e s u lt s suggest t h a t the s ize d i s t r i b u t i o n s can be d iv id e d in to 3 sub­ groups, which probably correspond to the fo llo w in g tra n s p o rt mechani sms: 102 -F ig u re 24: Loading p r o f i l e s o f each Sleeping Bear P o in t/M a n ito u Passage environment PROFUNDAL v ' VALUE 0,5 y ns 0 ■ 0.5 sc + s I -1 1 ^ XT* 1 2 3 o ft 1 Fa c to r *?/ " fa c to r ■ LOADING LOADING VALUE SHOAL AND NEARSHORE sc + s i TRANSITIONAL VALUE LOADING LOADING VALUE NS sc + s 0 2 3 4 PHI SIZE CLASS SC=Surface Creep . i NS i BEACH 1 1 / / v PHI SIZE CLASS PHI SIZE CLASS 0.5 I SC 0.5 0 - 0.5 1 0 1 2 PHI SIZE CLASS S=Saltation NS-Non’ Uniform Suspension 3 i 104 Sub-population I Sub-population I I - coarse sand ( 1 . 0 0 and coars er) a f f e c t e d by surface creep. interm ed ia te sand sizes ( 1 . 0 0 to 4 . 0 0 ) a f f e c t e d by two mechanisms: 1) 2) Sub-population I I I - s a l t a t i o n ( 1.00 to 2 . 0 0 ) non-uniform suspension (2 .0 0 to 4 . 0 0 ) s i l t and c la y (40 and f i n e r ) a f f e c t e d by a pure suspension mechanism. These sub-populations described here are very s i m i l a r to the sub-populations described f o r the L i t t l e Sable Point samples. COMPARISON OF THE LITTLE SABLE POINT AND THE SLEEPING BEAR POINT/MANITOU PASSAGE ENVIRONMENTS Even a cursory examination o f the m u l t i v a r i a t e r e s u l t s o f the environments studied i n d i c a t e d if f e re n c e s i n the e f f e c t iv e n e s s o f environmental d i s c r im in a t i o n on the basis o f g r a i n - s i z e s alone (cf. Figs. 15 and 19). D isc rim in atio n o f environments using t e x t u r a l data was most e f f e c t i v e a t the Sleeping Bear Point/Manitou Passage ar ea, w hile l i t t l e d is c r im in a t i o n was possible f o r the L i t t l e Sable Point environments. The g r a i n - s i z e d i s t r i b u t i o n s o f the two study areas d i f f e r s i g n i f i c a n t l y in terms o f the range o f sizes a v a i l a b l e to the systems. The samples at L i t t l e Sable Point have a very r e s t r i c t e d s i z e range, which is l i m i t e d to the sand-size range (00 to 3 0 ) , w hile the samples from the Sleeping Bear Point/Manitou Passage area range from lag gravels to very f i n e deep water s i l t and c la y . Both study areas have a wide range i n energy conditions from the lower energy profundal environment a t the Sleeping Bear Point/Manitou Passage area and the a e o lian environment a t L i t t l e Sable Point to the higher energy beach environments in both areas. I f i t is assumed t h a t t e x t u r a l m o d ific a tio n s are energy-dependent, passage i n t o or through any environment should r e s u l t in m o d if i ­ cati on o f the o r i g i n a l g r a i n - s i z e d i s t r i b u t i o n , provided t h a t the new environment has s u f f i c i e n t energy to modify the d i s t r i b u t i o n , 105 106 or t h a t the sediment remains in any environment long enough to acquire the t e x t u r a l c h a r a c t e r i s t i c s o f t h a t environment. The loading p r o f i l e s ( c f . Figs. 14 and 24) o f the en viron­ ments from each study area shows t h a t t ru n c a t io n points o f g r a i n s i z e frequency d i s t r i b u t i o n s occur a t r e l a t i v e l y the same phi size in te rv a ls . This suggests t h a t regardless o f sediment ty p e , the same basic t ra n s p o rt mechanisms are r e f l e c t e d in samples from d i f f e r e n t environments. The proportions o f the sub-populations may vary between environments which is e v id e n t in the Sleeping Bear Point/Manitou Passage environments. The problem o f induced negative c o r r e l a t i o n s formed by closed m a trice s, p a r t i c u l a r l y concerning the change in sign o f f a c t o r loading values a t 2 0 , can now be f u r t h e r evaluated by com­ paring the f a c t o r loading p r o f i l e s ( c f . two study areas. Figs. 14 and 24) from the I t was stated t h a t the break a t 20 f o r the t e s t cases was due to c l o s u re , but t h is same break occurs in the sam­ ples from L i t t l e Sable P o i n t , which t h e o r e t i c a l l y form an open matrix. The principal-components r e s u l t s o f the t r a n s i t i o n a l and profundal environments form an open m a trix when the sediment f i n e r than 40 is excluded from the an a ly sis. The t r a n s i t i o n a l sediments have a percentage range of sediment f i n e r than 40 from 15 to 35%, w hile those from the profundal environment range from 50-95% f i n e r than 40. The 20 break in the loading p r o f i l e f o r the t r a n s i t i o n a l sediments (Fig . 24) occurs with a change in sign o f the loading value (high negative va lue ) as b efo re , however, f o r the profundal 107 samples, the 20 break occurs where f a c t o r I changes from high to low p o s i t i v e va lu e s , and f a c t o r I I changes from low to high posi­ t i v e values ( F ig . 24). These r e s u l t s tend to support Bagnold's (1966) assumption t h a t a change in hydrodynamic process occurs a t approximately 2 0 . I t is not possible to compare the amount o f explained v a r i ­ ance between the two study areas because the amount o f t o t a l v a r i ­ ance co n tribu ted by the s i l t and c l a y by the Sleeping Bear Point/ Manitou Passage samples is not known. Environmental I n t e r p r e t a t i o n o f Sedimentary Environments The profundal sediments in the Sleeping Bear Point/Manitou Passage area are ch a rac teriz e d by a predominance o f s i l t and clay (50-95%) which i s c a r r i e d as a suspended load and deposited by gravitational s e ttlin g . Coarser g ra ined material may be i n t r o ­ duced by s a l t a t i o n and/or surface creep during storms, o r by slumping from the adjacent Sleeping Bear Point. Airborne materia l may als o be derived from the a d jac en t Sleeping Bear sand dunes. Decreasing water depths w it h a consequent increase in wind generated wave energy in to the shoal and nearshore area re su lts in the winnowing o f the s i l t / c l a y f r a c t i o n . This area i s ch aracteriz ed by lag gravels and coarse to medium sand sizes. The sand is pre­ dominantly c a r r i e d as s a l t a t i o n , non-uniform suspension and/or surface creep. The maximum depth o f erosion in t h i s area is approximately 20m (French 1964). French (1964) noted t h a t much o f the m a teria l forming the extensive sediment plume on th e northeast 108 t i p o f Sleeping Bear Point is f i n e sand derived from erosion on the s h o al, the s i l t / c l a y f r a c t i o n is c a r r i e d lakeward by strong cu rr ents in t h is area. Most o f the f i n e sand and s i l t in the t r a n s i t i o n a l se di­ ments, Sleeping Bear Point/Ma nitou Passage, is derived from the shoal and nearshore and is probably tra ns ported as a s a l t a t i o n and/or non-uniform suspension load. F i n a l l y , in the beach environment o f both areas two types o f s a l t a t i o n modes were d i s t i n c t l y recognized, and probably r e s u l t from the swash and backwash in the foreshore zone as described by Visher ( 1 9 6 9 ) . The swash, being the stronges t process, probably tra n s p o rts material as p r i m a r i l y non-uniform suspension and/or suspension, while the backwash processes probably t ra n s p o rt sand mainly as s a l t a t i o n and/or non-uniform suspension. The bar and trough samples in the L i t t l e Sable Point area show much t e x t u r a l overlap w it h the beach samples, presumably the same processes and in r e l a t i v e l y the same proportions are a f f e c t i n g these samples. The s i g n i f i c a n c e o f s a l t a t i o n in wind t ra n s p o rt o f sediments in the a e o li a n environment has been emphasized by Williams ( 1 9 6 3 ) , Hoyt ( 1 9 6 5 ) , and Visher ( 1 9 6 9 ) . In a d d it io n to the dominant s a l t ­ a t i o n mode, these authors have also shown t h a t small amounts o f suspended m a teria l may be p re sent (not present in the L i t t l e Sable Point a e o li a n samples), but very l i t t l e ( l e s s than 1-25&, V isher, 1 9 6 9 ) . surfa ce creep m a t e r ia l Despite the d if f e r e n c e s in beach 109 and dune transport mechanisms, these two environments a t L i t t l e Sable Point cannot be p e r f e c t l y separated w ithout some overlap on the basis o f t e x t u r e alone. Others have also noted t h a t in many cases beach and dune sands cannot be dis tin g uish ed t e x t u r a l l y (Shepard and Young, 1961, Schlee, e t a l . , E t h r i d g e , 1975). 1964, Davies and CONCLUSIONS Several m u l t i v a r i a t e methods, principal-components, Q-mode f a c t o r , and d is crim in an t f u n c t i o n , analyses were applied to an e n t i r e range o f g r a i n - s i z e s to t e s t the e f f e c t i v e n e s s o f sediment s i z e d i s t r i b u t i o n s as environmental d is c r im in a t o rs f o r several d i f f e r e n t environments. In the case o f the L i t t l e Sable Point samples, the r e s u l t s showed l i t t l e separation o f the environments s tu d ie d, however, the environments in the Sleeping Bear P o i n t / Manitou Passage area discrim in ated very w e l l . These r e s u l t s can be explained in terms o f a very simple causal r e l a t i o n s h i p : g r a i n - s i z e d i s t r i b u t i o n o f the parent m a t e r ia l (i.e ., the those grains a v a i l a b l e to the system) seems to contro l the e f f i c i e n c y o f environ­ mental d is c r i m i n a t i o n . I f i t i s assumed t h a t t e x t u r a l m o d ifica tio n s are energy-dependent then systems w ith a l i m i t e d g r a i n - s i z e range, such as the L i t t l e Sable Point system, may not r e f l e c t the e n t i r e energy spectrum o f the system, whereas systems with u n lim it e d g ra in s i z e ranges, such as the Sleeping Bear Point/Manitou Passage system, can r e f l e c t d i f f e r e n t energy le v e ls in t h e i r sediment d i s t r i b u t i o n s ( F i g . 25 ). The r e s u l t s o f t h i s study tend to confirm, on the basis of principal-components a n a l y s i s , t h a t g r a i n - s i z e d i s t r i b u t i o n s are composed o f two or more sub-populations, which may be r e l a t e d to 110 RELATIVE FREQUENCY PERCENT 111 Level 1 Level 2 Level 3 G R A IN -S IZ E Figure 25: Diagrammatic g r a i n - s i z e frequency d i s t r i b u t i o n s . Dashed l i n e represents a narrow g r a i n - s i z e d i s t r i b u t i o n which r e f l e c t s only one energy l e v e l , w h i l e the s o l i d l i n e represents a wider g r a i n - s i z e d i s t r i b u t i o n r e f l e c t i n g t h r e e energy level s 112 d i f f e r e n t t ra n s p o rt mechanisms. Although t h i s technique does not determine which mechanisms control g r a i n - s i z e d i s t r i b u t i o n s i t does show t h a t two, and in one case t h r e e , f a c t o r axes are needed to ex pla in a major portio n o f the variance in a se t o f g r a i n - s i z e data. I t has also been shown t h a t c e r t a i n g r a i n - s i z e e n t i t i e s c l u s t e r around the same f a c t o r s and are comparable in most cases to s t r a i g h t l i n e segments on lo g - p r o b a b i 1i t y g r a i n - s i z e graphs. Although Q-mode f a c t o r ana ly sis was used on a l i m i t e d basis in th is study, because the broad categories o f dep ositional environ­ ments were known before hand, i t i s f e l t t h a t t h i s technique has much p o t e n t ia l in the f i e l d o f sedimentology. Samples can be c l a s s i f i e d i n t o groups which show both geographical and geological meaningful trends w it h ou t any a p r i o r i knowledge o f the s p a t i a l positio n o f the samples (Klovan, 1966). point. This i s a very important For y e a r s , many i n v e s t i g a t o r s have used a r b i t r a r y indices in an e f f o r t to c h a r a c t e r iz e sediment from d i f f e r e n t environments. A r b i t r a r y in the sense t h a t these indices were defined by the author in a special way. For example, Mason and Folk (1958) suggested t h a t a graph o f skewness versus k u rto s is was ap p ro pria te f o r d i s ­ c r im in a t in g between d i f f e r e n t environments. However, even w ith d i f f e r e n t symbols, i t was d i f f i c u l t to see any trends in the samples. Klovan (1966) s ta tes 11 I t is p r e c i s e l y from such u n q ua ntified data t h a t we t r y to reco nstru ct environments o f d e p o s i t i o n . . . " If we can not d is t i n g u i s h d if f e r e n c e s between recent dep ositio nal environments our task is confounded iiranensely when we turn to the a n c ie n t dep osits. 113 Passega (1957, 19 72 ), Bagnold ( 1 9 6 6 ) , Klovan ( 1 9 6 6 ) , K o ld ijk ( 1 9 6 8 ) , and others suggest t h a t the presence o f the f i n e ta il ( s i l t / c l a y f r a c t i o n ) is a s e n s i t i v e environmental i n d i c a t o r . This f r a c t i o n was shown, i n d i r e c t l y , to be a s i g n i f i c a n t f a c t o r f o r environmental d is c r im in a t i o n o f the Sleeping Bear Point/Manitou Passage samples. I n d i r e c t l y , in the sense t h a t the m a t r i x was normalized p r i o r to analysis which took i n t o account the absence o f t h i s f r a c t i o n from the a n a l y s i s . The presence o f the f i n e t a i l in the Sleeping Bear Point/Manitou Passage area samples allowed f o r a more e f f e c t i v e r e f l e c t i o n o f the energy spectrum in t h i s deposi­ t i o n a l system. The present study shows only the r e s u lt s o f applying several m u l t i v a r i a t e techniques to two areas o f recent sedimentation. The causal process o f sediment t r a n s p o r t postualted here are not all inclusive. Many other processes or mechanisms not studied here are su b je ct to review ( i . e . , t u r b i d i t y cu rr e n ts ) as well as o ther aspects such as r a t e o f sediment supply, r a t e o f subsidence, storm frequencies and i n t e n s i t y changes i n source areas, which would complicate the patter ns found. As shown in t h i s study, i t is d i f f i c u l t to s t a t e w it h c e r t a i n t y whether dep ositional environments can be uniquely i d e n t i ­ f i e d in recent deposits using only g r a i n - s i z e a n a ly s is . The problem o f environmental d i s c r i m i n a t i o n is f u r t h e r complicated in ancient deposits p a r t i c u l a r l y w it h respec t to l i t h i f i c a t i o n and diagenesis o f the sediment. There i s a g r e a t need a t t h i s time f o r more experimental flume and wave tank studies with much more so p histic ated in s i t u measurements and c o l l e c t i o n techniques in ord er to e s t a b l i s h the hydrodynamic aspects o f sediment t ra n s p o rt mechanisms which up to now have been mainly studied by some type o f g r a i n - s i z e analys Once these techniques have been es tab lis he d and a reasonable data base e s t a b l is h e d , the m u l t i v a r i a t e techniques used in t h i s study could be used to e s t a b l is h a more o b j e c t iv e c l a s s i f i c a t i o n o f sediment types and processes. Even i f environments can not be d i s t i n c t l y i d e n t i f i e d , in the f o s s i l d e p o s its , knowledge o f e n e rg y -tra n s p o rt r e l a t i o n s h i p s of recent or experimental deposits w i l l help to b e t t e r i n t e r p r e t p aleo -flo w regimes. APPENDICES 115 APPENDIX A 116 APPENDIX A L i s t of Phi Size Classes and Frequency Percentages o f Each L i t t l e Sable Point Environment Source Beach 0.50 1.00 1.50 2.00 2.50 3.0j 1 0.00 0.00 16.0 54.0 28.0 1.9 0.0 2 0 .0 0 1.00 2 .5 31.7 5 8 .3 6 .4 0.0 3 0.00 0.00 2.0 27 .5 67.5 2.9 0.0 4 2.00 3.00 30.5 49 .4 1 3 .6 1.4 0.0 5 1.00 0.50 16 .5 56.5 24.0 1.7 0.0 6 1.80 0 .3 0 2 .4 30.5 VO 4 .9 0.0 7 3.00 0.00 2 .0 48.0 42.3 4 .6 0.0 8 0.00 0 .0 0 1 .8 42.2 52 .5 3.4 0.0 9 0.00 0.00 2.0 42.0 5 3 .0 2.9 0.0 7.8 4.5 7.5 8.2 3 .5 13.5 2.8 6.0 3,5 13.3 4.8 2.5 30.0 2.7 9.5 4.8 18.5 19.5 2.8 15.5 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.5 0.0 1.0 0.0 0.0 0.0 0.0 0.5 0.7 2.0 0.0 0.0 2.5 o* 0.00 Beach Samples West o f S i l v e r Lake 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 0 .0 0.0 2.3 3.0 0.0 0.0 1.5 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.0 0.4 0.0 0.0 0.0 1.7 0.5 0.0 1 .0 1.0 0.8 2.5 1.5 1.0 1.0 0.0 0.0 0 .5 1.0 0.0 0.0 1.1 0.0 2 .0 2 .5 1.0 0.30 13 .0 0 .8 0 7.80 10.4 10.0 1.0 4.5 19.0 2.0 9.5 2.0 12.0 1.5 2.0 8.3 1.5 23.5 32.5 19.0 15.9 47.5 14.2 39,0 52.3 46.5 10.5 50.5 50.5 8.0 58.7 14.5 54.0 15.0 13.5 50.2 25.0 66.7 60.5 68.5 71.1 36.0 7 0 .5 45.9 30.0 37.0 7 2 .7 3 9 .2 2 7 .0 60.0 29.1 72.5 26 .5 63.0 65.0 37 .2 55 .5 117 Offshore Bars and Trough Sample Number 00 .50 10 1.50 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 0.0 0.0 0.6 0.0 0.0 0.0 0.0 0.0 1.5 1.5 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.0 0 .5 0.0 0.0 0.0 1.0 0.1 0.9 1.5 0.5 0.0 o.o 1.0 0.0 1.0 2.5 2.0 1 .3 6.5 3.5 2.0 3.0 2.0 T.7 2.9 3.4 3.0 2 .0 3.0 13.0 3.0 2 .2 3.0 17.5 10.8 77.5 40.5 16.5 12.0 28.0 27.0 22.0 56.5 36.7 16.5 7.5 16.5 54.5 9.0 8.8 6.0 20 70.0 76,2 3,5 44 .8 64 .5 63.0 6 1 .5 65.8 65 .3 44.7 4 9 .8 55.0 61 .0 57.5 32.3 64.5 50.5 50 .0 2.50 30 10.0 11.0 14.5 7.40 15.0 20.5 7.5 5.2 8.5 4.3 9 .2 22.5 2 6 .5 2 0 .8 0.2 20.0 34.0 36 .5 0.0 0.0 2.0 0.6 0.0 2.5 0.0 0.0 0.0 0.0 0.0 1.5 1.5 2.2 0,0 2.5 4.5 3 .5 Si 1ver Creek Bed 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 00 .50 10 1.50 20 2.50 30 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.5 0.0 0.0 0.0 0 .0 1.5 0.9 1.0 1.5 0.0 1.5 0.5 0.0 0.0 1.0 1.5 1.0 2 .0 1.5 0.0 0.5 0.5 2.0 1.0 2.0 2.0 3.0 3.0 1.1 3.0 1.5 3.0 2.0 3.5 3.8 1.0 1.0 1.5 6.0 1.3 5.5 2.5 2.8 2 .7 26.5 5.5 5.0 7.0 9.0 7.0 3 .8 35.5 5.0 4.0 3.0 25.5 42 .7 6.5 9.0 4.5 4 3 .0 2.7 56.5 30.7 41 .2 4 6 .8 6 1 .5 65 .5 59.0 6 6 .0 67 .5 63.0 69.0 59.5 51.5 56.8 66.5 60.0 50.7 6 6 .0 67.0 55.0 42.7 53.0 31.0 56.3 50.5 41 .8 8.5 26.4 32.0 24 .0 15.2 23 .4 24.2 0.0 37.5 34.7 24.5 9.0 2.8 24.7 20.1 34.5 7.3 3 6 .0 5.5 8.7 4.0 7.2 0.0 1 .6 2 .0 1.0 3.3 2.1 1.0 0.0 3.0 1.5 2 .5 0.0 0.0 1.8 1.9 3.0 0.0 5 .0 0.0 l.L 0.0 0.0 118 Aeolian 00 .50 10 1.50 20 2.50 0.0 1.0 1.0 2 .5 65.5 14.5 71 0.0 0.0 1.8 0.2 6 7 .0 26.5 72 0.0 0.0 1.0 56.5 37.0 5.5 73 1.8 1.2 2 .0 7.0 6 3 .5 21 .0 74 0.0 0.0 1.0 3 .0 47 .8 39 .8 75 0.0 2.0 1.0 2.0 52.0 34 .5 76 2 .0 1.0 3.6 28.6 4 6 .0 16.0 77 0.0 0.5 0.5 4.4 46.1 37.5 78 0.0 0.0 0.0 2.5 4 0 .0 44 .0 79 0.0 0.0 0 .0 6 .5 55.0 31.5 80 0.0 0.0 0 .5 10.5 66.0 20.5 81 0 .0 2 .2 2 .0 2.2 39.7 44.1 82 0.0 1.5 0 .7 4.0 59.3 30.7 83 0.0 2 .5 1.5 31.0 49.2 14.0 84 0.0 0.0 1 .0 4.0 50.0 37.5 85 0.0 1.0 3.4 19.1 46.3 24.7 86 0.0 0.0 5.0 3.2 47.7 37.1 87 0 .0 1.5 1.5 1 .5 63.5 29.5 88 0.7 0.8 1.0 2 .0 72.5 2 2 .0 89 0.0 0.0 1.5 14.0 4 4 .5 32.5 90 0 .0 0.9 0.9 39.2 48 .5 10.5 91 0.0 0.0 0.5 5.0 48.8 41.7 92 0.0 0.0 1.5 26.5 53.0 17.0 93 0.0 0.0 1 .8 4.0 36.2 44.3 94 0.0 0.0 4.0 6.5 41 .0 42.3 APPENDIX B 119 APPENDIX B L i s t o f Phi Size Classes and Frequency Percentages o f Each Sleeping Bear Point/Manitou Passage Environment Shoal and Nearshore Sample Number S6 Sll S14 SI 6 SI 7 S19 S20 S21 S24 S25 S26 M47 Cl 2 CIS C16 0.5 3 .7 0.0 6 .5 1.0 2 .0 4.0 8.0 4.5 6.5 9.0 7 .0 5.0 0.5 1.7 15.0 1.0 0 1 4 .8 1 .96 15 .0 10 .0 3.50 15 .9 18 .5 6.70 11.3 17.7 10.3 6 .0 0 1.45 13.3 38.4 1.50 20 2.50 30 2 3 .0 8 .8 0 32.4 30 .0 2 1 .0 4 8 .7 2 8 .4 19.4 20.2 38 .6 3 7 .6 2 1 .0 2.1 22.5 3 6 .5 17 .8 10 .8 37.5 4 5 .9 52 .4 8 .4 0 36 .5 43.2 4 1 .8 28 . 8 37.5 31.8 26.0 27.9 9.00 14.7 21.5 6.5 11.0 17.5 15.4 7 .5 20 .9 16.5 5.6 6.9 30.2 30.5 7.0 0.9 14.6 27.5 1.9 1.9 3.4 5.1 1 .0 4.9 3.4 0 .2 0.6 5.7 2 4 .0 4.7 0 .2 3.50 8.0 25 .4 0.1 0.0 0.0 1.8 0.0 0.1 0.2 0.0 0.0 0.2 8.7 5.5 0.0 40 3.0 2.0 0.0 0 .0 0 .0 0.1 0.0 0 .0 0 .0 0 .0 0.0 0 .0 2.9 2.6 0.0 >40 0.4 2.0 0.1 0.1 0.1 0 .6 0.1 0.2 0.1 0.1 0.1 0.0 3.3 14.8 0.1 Profundal Sample Number 0.5 1.00 SI S2 C5 C6 C7 C8 C9 C13 C14 0.1 1.6 0.9 0.5 0.5 0 .4 0.3 0.6 0.5 0.2 5.1 1.5 1 .3 1.3 1.0 0.8 2.2 1.6 1 .50 20 2.50 0.2 7.7 2.0 2.5 2.4 1 .4 1.1 3 .6 3.0 0.5 9.7 3.9 4.3 6.0 2.7 3 .5 8.9 5.6 0.2 2 .6 1.6 2.0 1.9 2.2 1.4 1.5 3 .7 30 0.2 6.7 2.0 0.6 0.3 0.9 0.9 7.8 16.7 3.50 0.3 4.9 3.4 2.9 0.8 2.6 2.4 2 .8 11.1 40 >40 0.1 4 .8 0.4 1.9 0.4 0.7 0.7 0.6 4.2 98 .2 56 .8 8 4 .3 84.0 86.4 88.1 8 8 .9 7 2 .0 5 3 .6 120 Beach 1.50 2.00 101 2.6 6.6 55.3 32.0 3 .5 0.0 102 1.8 5.7 52.0 33.3 7.2 0.0 103 0.0 0.6 23.2 56.5 15.7 2.0 104 0.0 1.3 24.0 50.2 23 .0 1.5 105 0.0 3 .3 35.9 47.6 106 0.0 0.0 4.5 10.8 51 .2 2.0 26.0 108 1.3 3.2 6.5 23.6 35 .4 20.0 111 0.0 1.8 15.8 42.7 2 9 .0 6.6 112 1.2 4.8 59.0 28.5 3.9 0.6 113 0.5 1.0 14.0 36.0 32 .2 11.0 114 2 .5 1 .0 4 7 .0 40 .9 7.8 7.8 115 2 .2 4.6 54.2 30.8 6.5 1.7 116 1 .0 2.5 46.5 38.0 1 0 .5 1.5 117 0.0 4.0 39.8 41.2 7.0 4.0 118 1.8 4.2 54 .0 33.7 6.3 0 .0 119 1.5 6.8 68 .0 19.2 3.0 1.5 120 0.8 4.7 57.8 28.7 6.5 1.5 121 11.2 17.8 28.7 11.0 2.8 0.0 122 3.2 2 .5 51 .4 32.8 1 0 .0 0.0 123 8.0 10.5 60 .5 16.2 4.0 0.8 124 0.5 11.8 67 .2 16.0 4,5 0.0 125 0.0 2.0 40.5 45.0 12.5 0.0 126 0.0 5.2 51.5 33.0 8.0 0.0 127 1 .2 3.3 53.2 32.3 10 .0 0.0 128 0.0 2.7 4 6 .8 39.5 1 0 .3 0.7 129 0.0 4 .0 54 .5 32.5 9.0 0 .0 130 0.0 4.0 6 2 .0 28.0 6.0 0 .0 131 3 .0 3 .0 48 .5 37.3 8.2 0.0 132 4.2 10.6 69.9 13.3 2.0 0 .0 ro 1.00 I'd 0.5 0 • 0 .0 0 2.50 121 Transi t i o n a l Sample Number .50 10 S3 3 .4 10.5 8.2 10.1 S4 2 ,9 14.1 23.4 2 1 .8 S5 0.6 1.9 3.3 S7 0.0 0.9 1.0 S8 1.2 1.6 3.2 S9 4.0 12.2 17.6 S10 2.4 6.3 5.9 Cl 0.8 6.5 12.2 C2 0.0 0.0 1.9 C3 0.0 0.0 0 .9 C4A 0,0 6.3 8.8 C4B 1.8 2.3 CIO 4.8 Cl 7 1.50 30 3.50 40 >40 4.9 25.3 17.3 4.6 15.7 6.2 4.2 2.0 2.3 22.7 5.30 19.7 36.5 12.6 3.9 16.1 4.20 35 .3 34.8 12.5 3 .3 7.1 32.1 35.1 16.9 1 .9 0.9 7.1 17.8 6.6 13.0 5.3 4.4 19.1 13.7 24.1 23.1 2 .6 13.0 25.6 1 0 .9 13.5 13.7 5.7 11.0 19.0 45.3 24.6 2.2 1.1 5 .9 4.4 56.6 15.7 8.9 11.3 13.3 6.9 18.9 24 .0 5 .2 16.7 5.5 21.5 26.6 26.6 7.1 0,9 7.6 4 .3 15.5 25.0 12.7 10.7 1 .5 4.8 20.7 1.6 8.1 10.0 14.2 9.1 22.7 10.1 4.8 19.4 Cl 8 0.0 3.1 3.7 18.0 40.0 7.8 3.4 14.3 Cll 0.8 8.4 19.3 2.3 4.8 1 .6 19.7 20 8 .9 0 2.20 9.70 34.9 2.5 0 8.0 APPENDIX C 122 APPENDIX C L i s t o f Q-Mode Factor Results f o r L i t t l e Sable Point Samples Cumulative Percentage of Eigenvalues Factor Eigenvalue 1 78.65 83.67 2 12.29 96.75 3 2.62 99.53 4 0.34 99.90 5 0 .0 7 99.97 6 0.02 100.00 7 0 .0 0 100.00 * • a * 0 .0 0 94 100.00 Rotated Factor M a trix Sample Number 1 2 3 4 5 6 7 8 9 I 0 .1 8 0 .6 0 0.61 0.03 0.13 0.60 0 .3 8 0.47 0.47 Factor II 0 .9 7 0 .6 8 0 .6 0 0 .9 2 0.98 0 .6 6 0.8 8 0.8 0 0.7 9 III Communali ty 0.08 0.41 0 .5 0 -0.08 0 .0 3 0 .4 4 0 .2 4 0.35 0.36 0.998 0.999 0.999 0.998 0.999 0 .999 0 .9 9 8 0.999 0.999 123 R otated F a c to r M a tr ix Factor II III Communality 10 0.67 0.56 0.47 0.999 11 0 .5 8 0 .6 8 0.43 0.999 12 0 .7 0 0.50 0.5 0 0.997 13 0.72 0.45 0.51 0.998 14 0.32 0.92 0.1 7 0.999 15 0.77 0.43 0.46 0.999 16 0.45 0 .8 2 0.32 0.999 17 0.26 0.96 0.07 0.999 18 0 .3 3 0.91 0.20 0.999 19 0 .7 8 0.38 0 .4 8 0.999 20 0.34 0.91 0.19 0.999 21 0.19 0.96 0.07 0.995 22 0.18 0.97 0.08 0.999 23 0.90 0.33 0.24 0.999 24 0 .7 4 0.44 0.5 0 0.999 25 0 .2 0 0.97 0.04 0 .998 26 0.81 0.45 0.37 0.999 27 0.81 0.42 0.37 0.999 28 0 .3 0 0.93 0.1 9 0.999 29 0.72 0 .6 0 0 .3 2 0 .9 9 9 30 0.72 0 .4 8 0.48 0.999 31 0.76 0 .3 8 0.51 0.999 32 - 0.01 0.94 -0.28 0.999 33 0 .4 8 0.83 0 .2 5 0 .9 9 9 34 0.77 0.47 0.41 0.999 35 0 .8 3 0.41 0 .3 5 0.999 36 0.64 0.63 0 .4 3 0.999 37 0.63 0 .6 0 0.4 8 0.999 38 0.69 0 .5 4 0.46 0.999 124 R otated F a c to r M a tr ix I Factor II III Communality 0 .3 4 0.91 0.21 0.999 0 .5 5 0.77 0.30 0.999 41 0.83 0 .4 8 0 .2 6 0.999 42 0 .8 9 0.3 3 0.2 9 0.998 43 0.82 0.4 8 0.30 0.999 -44 0.20 0.9 6 0.1 4 0.999 -45 0.84 0.37 0 .3 7 0.999 46 0 .9 3 0 .3 3 0.10 0.999 47 0.95 0 .2 9 0 .0 8 0.997 48 0.66 0.61 0.43 0.999 49 0.89 0.30 0 .3 2 0.9 98 50 0 .9 3 0 .2 9 0.21 0.997 51 0.87 0.33 0.3 4 0.998 52 0.81 0.37 0 .4 4 0.998 53 0.87 0 .3 3 0.3 3 0.999 54 0 .8 8 0 .2 8 0.37 0.999 55 0.51 0.71 0.46 0.998 56 0.6 5 0 .7 4 - 0.0 2 0.997 57 0 .9 4 0.27 0.17 0.995 58 0 .8 9 0 .2 8 0.35 0.999 59 0.66 0.61 0.42 0.999 60 0.45 0.81 0 .3 4 0.999 61 C.88 0.32 0.34 0.999 62 0.84 0.36 0.3 9 0.999 63 0.94 0.27 0.15 0.997 64 0.4 5 0.86 0.23 0.999 65 0.96 0 .2 4 0 .1 2 0 .9 9 8 66 0 .2 4 0 .9 6 0.08 0.999 67 0.62 0 .6 8 0 .3 7 0.998 125 R o ta ted F a c to r M a trix Facto r II III Communality 0.34 0.999 69 0.41 0.88 0.21 0.999 70 0.83 0.28 0.46 0.995 71 0.91 0 .2 3 0.33 0.999 72 0.29 0 .9 3 0.14 0.998 73 0.86 0.34 0.36 0.998 74 0.97 0 .2 2 0.01 0.998 75 0.33 0.92 0.11 0.998 76 0.68 0.69 0.22 0.995 77 0.96 0.25 0.01 0.993 78 0.96 - 0.12 0.999 79 0.84 0.38 0.37 0.999 80 0.93 0.30 0.17 0.998 81 0.97 0.19 - 0.12 0.998 82 0.93 0.2 7 0.23 0.999 83 0.65 0.70 0.27 0.998 84 0.96 0 .2 5 0.06 0.999 85 0.83 0.53 0.14 0.997 86 0.96 0.24 0.04 0.999 87 0.92 0.24 0.28 0.998 88 0.86 0.26 0.41 0.999 89 0.90 0.42 0.03 0.998 90 0.54 0.78 0.27 0.999 91 0.96 0.25 0.01 0.996 92 0.72 0 .6 2 0.28 0.999 93 0.95 0.20 -0 .1 8 0.986 94 0.95 0.27 - 0 .0 9 0.998 CD 0.80 • 0.47 o I LIST OF REFERENCES 126 LIST OF REFERENCES A l l e n , G. 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