INFORMATION TO USERS
The most advanced technology has been used to photo­
graph and reproduce th is manuscript from the microfilm
master. UMI film s th e text directly from the original or
copy submitted. Thus, some thesis and dissertation copies
are in typewriter face, while others may be from any type
of computer printer.
The quality of th is reproduction is dependent upon the
quality of the copy submitted. Broken or indistinct print,
colored or poor quality illustrations and photographs,
print bleedthrough, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a
complete manuscript and there are m issing pages, these
w ill be noted. A lso, i f unauthorized copyright m aterial
had to be removed, a note will indicate the deletion.
Oversize materials (e.g., maps, drawings, charts) are re­
produced by sectionin g the original, beginning at the
upper left-hand corner and continuing from left to right in
equal sections with sm all overlaps. Each original is also
photographed in one exposure and is included in reduced
form at the back of the book. These are also available as
one exposure on a standard 35mm slide or as a 17" x 23"
black and w h ite photographic print for an additional
charge.
Photographs included in the original m anuscript have
been reproduced xerographically in th is copy. H igher
quality 6" x 9" black and w hite photographic prints are
available for any photographs or illustrations appearing
in this copy for an additional charge. Contact UMI directly
to order.

UMI
U n iversity M icro film s Intern ation al
A B e ll & H ow ell Inform a tion C o m p a n y
3 00 N o rth Z e e b R oad, Ann Arbor, Ml 4810 6-13 46 USA
3 1 3 /7 6 1 -4 7 0 0
8 0 0 /5 2 1 -0 6 0 0

O rder N u m b er 8923 8 5 4

E v a lu a tio n o f m u ltilev el sa m p lin g tech n iq u es for fo rest in ven tory
in n o rth ern M ich ig a n
Humphreys, Rubens Dias, Ph.D .
Michigan State University, 1989

UMI

300 N. Zeeb Rd.
Ann Arbor, MI 48106

EVALUATION OF MULTILEVEL SAMPLING TECHNIQUES FOR
FOREST INVENTORY IN NORTHERN MICHIGAN

By
Rubens Dias Humphreys

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Forestry

1989

ABSTRACT

EVALUATION OF MUL TILEVEL SAMPLING TECHNIQUES FOR
FOREST INVENTORY IN NORTHERN MICHIGAN

By

Rubens Dias Humphreys

There are
to

forest

several

inventory.

sampling

The

selection of

one depends on several aspects
the

area;

funds
etc.

size

of

available;

the

multistage

or

area;

remotely sensed

objectives
of

multilevel

multiphase
imagery

the most

appropriate

: number of forest type(s)

availability

In large areas,

methods that can be applied

remotely

sampling

can

of

be

inventory;

sensed

imagery,

techniques

used.

is essential for

the

in

such as

Incorporation

of

the application of

such methods.
Two

multilevel

multistage

method

sampling

whic h

has

techniques were

been

used

quite

evaluated:
frequently

a
in

inventories in the USA and a multiphase technique which has
been

recently

stages.

The

developed.

first

1:107,000;

the

photographs

and

measurements.

The

technique

where

stage
second
the

The

former

was

composed

of

three

used LANDSAT TM imagery enlarged to
stage
third

utilized

stage

later method was a
stratification

of

1:24,000

incorporated
stratified
the

area

CIR
ground

two phase
was done by

Rubens Dias Humphreys
computer classification of the digital representation of the
imagery.
random

The

first

sample

phase

within

each

sub-sample of the first,
The

study

was

of

approach

stratum;

the

was

second

a

simple

phase

was

a

where field measurements were done.

conducted

Wexford County,

this

Michigan.

on

an

area

of

4 7,850

acres

in

Field measurements were done using

point sampling and the volumes of trees were estimated using
specific

volume

size,

the

lower

than

sampling
that

variables used
the

second

forest

equations.

for

for

the

sampling

contained

on

The

size of

for

the

of

the

multiphase

sample

technique

technique.

The

units

(estimated

secondary

unit)

was
size

selection probabilities

the primary

for

area

of

the

and

for

the

were not

sampling units

chosen

a) only two P S U ’s were contained

red oak stratum;

subdivision

reduced

(percent crown closure),

was not convenient because:
within

the

the

third stage sampling units
appropriate.

the

using a

multistage

to calculate

stage

type

error

Although

b)

it was

conifer

too large

stratum

and

to be used
c)

produced

variations on the selection probabilities calculated for the
second

stage,

which

resulted

in

negative

between its size variable and predicted volume.

correlations

This work is specially dedicated to my
parents,

Brigida D. Humphreys and

Dr. Jorge Humphreys

(in memoriam)

ACKNOWLEDGMENTS

I

would

Brazilian
Cultura

like

to

Government

express

through

my

the

appreciation

Ministerio

da

to

the

Educacao

e

for the financial support of my studies at Michigan

State

University.

Also

to

the

Instituto

de

Pesquisas

Tecnologicas do Estado de Sao Paulo SA - IPT for allowing me
to leave my duties in order to pursue my degree.
My appreciation
Dr.

Carl

during

W.

all

Ramm
the

and

for

his

steps

graduate committee

respect

of

Dr.

goes to my major professor

support,
my

patience

graduate

Robert Marty,

Dr. Karen Potter Witter

and

program

Dr.

guidance

and

to

my

David P. Lusch and

for their guidance

and constructive

critiques of my work.
My

most

sincere

appreciation

to

Geraldo

Jose

Zenid,

Marcelo de Andrade and Joyce de Andrade for their invaluable
support in taking care of our things;

to Mr.

Sergey G. Guins

and his wife M r s . Katherine Guins for their love and help in
hours

we

needed

most.

To

other

friends

from

the

Wood

Division that directly or indirectly gave their contribution
and

support,

s ig ni fic ant .

even

if

small,

but

that

sure

ends

up

being

To the Department
to Dr.

Dennis W.

of Natural Resources

Hu dson from the Center

of Michigan and

for Remote

Sensing,

for allowing me to use the imagery of the study area.
Finally,
gratitude

to

but
my

not

wife

the

least

Berenice

for

important,
her

love,

my

deepest

encouragement

and support during this laborious journey and to my daughter
Sheila,

although

of understanding.

still

a

child but with

a great

capability

TABLE OF CONTENTS

Page

LIST OF T A B L E S ................................................. ix
LIST OF F I G U R E S ............................................... xi
CHAPTER 1 ...................................................
Introduction .
Statement of the Problem and Assumptions .........

1
1
3

CHAPTER 2 ...................................................
7
Literature R e v i e w .....................................
7
Background ..........................................
7
Simple Random S a m p l i n g ...........
10
Stratified Random Sampling .....................
12
Cluster Sampling ................................
15
Multilevel Sampling D e s i g n s ........................ 15
Double S am p l i n g ...................................... 17
Systematic S a m p l i n g ................................. 17
Multistage Sampling T e c h n i q u e .......................... 19
Multiphase Sampling T e c h n i q u e .......................... 26
CHAPTER 3 ...................................................... 30
M e t h o d s ................................................... 30
Multistage Sampling T e c h n i q u e ........................ 31
Sensitivity Analysis ............................
35
Multiphase Sampling T e c h n i q u e ........................37
Study A r e a ............................................ 40
L o c a t i o n ............................................ 40
General Characteristics. . . . . .
40
Data S e l e c t i o n .............
42
Remotely Sensed I m a g e r y ............................ 42
Field M e a s u r e m e n t s ................................. 43
Data Pro c es si ng ........................................ 45
Ph ot oin te rpr eta ti on ................................. 45
Multistage Sampling T e c h n i q u e ..................... 46
First stage Sample Selection ................
46
Second stage Sample S e l e c t i o n ................... 48
Third stage Sample Selection ................
49
Multiphase Sampling T e c h n i q u e ..................... 50
Digital Image Processing .....................
50

Page

Phase one Sample S e l e c t i o n ..................... 57
Phase two Sample S e l e c t i o n ..................... 59
Location and Transfer of the Samples from
the Imagery to the Base Map. .
60
Equations Used for Volume Estimation ........
61
Area Es t i m a t i o n ......................................64
Time Measurements and C o s t s ........................64
CHAPTER 4 ...................................................... 66
R e s u l t s ................................................... 66
Multistage Sampling T e c h n i q u e ........................67
Proportionality Between Measure of Size
and Predicted Volume .........................
72
Sensitivity A n a l y s i s ................
77
Multiphase Sampling Technique. ..................
88
Digital Image Processing .......................
88
Estimated Volume, Variance and Time
Measurements ...................................
88
96
CHAPTER 5 ................................
D i s c u s s i o n ............................................... 96
Red Oak S t r a t u m ......................................97
Conifer Stratum ................................... 105
Hardwood Stratum. .
114
General C o ns id er a ti on s ..............................120
Population Estimates. . . .
..................... 125
Digital Image P r o c e s s i n g ........................... 127
Time Measurements and C o s t s .......................140
Number of S a m p l e s ................
144
CHAPTER 6 .....................................................147
Summary and C o n c l u s i o n s ................................ 147
APP ENDIX A ....................................................156
AP PENDIX
B ................................................... 159
APPENDIX.. C ....................................................165
APPENDIX.. D ....................................................166
AP PE NDI X ..E ....................................................167
LITERATURE C I T E D .............................................. 169

LIST OF TABLES
Page

Table 1

Total Number of Sampling Units on Each,
S t r a t u m ............................................ 57

Table 2

Values of the Sampling F r a c t i o n ................. 60

Table 3

Coefficients for the Volume Equation by
S p e c i e s ............................................ 62

Table 4

Estimated Total Volume, Standard Deviation
Coefficient of Variation and Confidence
Intervals for Each Stratum and the
P o p u l a t i o n .........................

68

Table 5

Total and Merchantable Basal Area for Each
S t r a t u m ............................................ 70

Table 6

Estimated Areas for Each Stratum,
Government and Private Land and
T o t a l ............................................... 71

Table 7

Estimated Volume per Acre and Standard
D e v i a t i o n .......................................... 71

Table 8

Time Spent on the Activities for the
Multistage Technique .........................

73

Table 9

Changes in the Estimated Total Volume
and its Variance with Changes in the
Selecton Probabilities for C o n i f e r s ............ 78

Table 10

Effects of Changing the Selection
Probabilities on the Estimated Total
Volume and its Variance for C o n i f e r ............ 79

Table 11

Changes in the Estimated Total Volume
and its Variance with Changes in the
Selection Probabilities for Hardwood . . . .

80

Page

Table 12

Effects of Changing the Selection
Probabilities on the Estimated Total
Volume and its Variance for Hardwood

. . . .

81

Table 13

Changes in the Estimated Total Volume
and its Variance Associated with Changes
in the Selection Probabilities
for Red O a k ........................................ 83

Table 14

Effects of Changing the Selection
Probabilities on the Estimated Total
Volume and its Variance for Red O a k ............ 84

Table 15

Table 16

Minimum and Maximum Values for the
Total Volume and the V a r i a n c e ................... 89
Univariate Statistics for the Study
Area .

90

Table 17

Results of the Supervised Classification
of the Study A r e a ................................. 90

Table 18

Estimated Volume per Acre, Standard
Deviation and Coefficient of Variation
for the ith Population S t r a t a .................. 91

Table 19

Estimated Volume per Acre, Standard
Deviation and Coefficient of Variation
for the Four Phase one Strata and the
P o p u l a t i o n ........................................ 91

Table 20

Total and Merchantable Basal Area for
the Phase one S t r a t a ............................ 93

Table 21

Time Spent on the Activities for the
Multiphase Technique .........................

94

LIST OF FIGURES

Page

Figure 1

Location of the Study A r e a ...................... 41

Figure 2

Prediction Variable versus Predicted
Volume for the Second stage for
a) Hardwood, b) Conifer and c) Red Oak

. ..

75

Prediction Variable versus Predicted
Volume for the Third stage for
a) Hardwood, b) Conifer and c) Red Oak

. . .

76

Figure 3

Figure 4

Effect of Changing the Selection
Probabilities on a) Volume and
b)
Variance
for
H a r d w o o d ................... 85

Figure 5

Effect of Changing the Selection
Probabilities on a) Volume and
b)
Variance
for
C o n i f e r . ................... 86

Figure 6

Effect of Changing the Selection
Probabilities on a) Volume and
b)
Variance
for
Red O a k ......................87

Figure 7

Feature Space For Bands

4 and 5 ............... 129

Figure 8

Feature Space For Bands

4 and 6 ............... 130

Figure 9

Feature Space For Bands

4 and 7 ............... 131

Figure 10

Feature Space For Bands

5 and 6 ............... 132

Figure 11

Feature Space for Bands

5 and 7 ............... 133

Figure 12

Feature Space for Bands

6 and 7 ............... 134

Figure 13

Feature Space for Principal Components
1 and 2 ........................................... 135

CHAPTER 1
INTRODUCTION

The

term

International
itemized

inventory

is

Dictionary

list

of

of

current

applied to commercial

defined
the

by

the

English

assets".

The

enterprises where

W e b s t e r ’s

Language
term

was

it was

Third

as:

"an

initially

necessary to

control the stock of raw materials and the items produced.
The
bearing
size,

term

exploited or not,

water

defined

capable

by

FAO

as

dominated

by

of producing

:

"all

regime,

or providing

shelter

for

lands

trees of
wood or

of exerting an influence on the climate

any

other

or on the

lifestock and

wild­

(Loetsch g_t _al• 1964 )".
By

combining

inventory.
species

The

that

"itemized

as

both

occur

precision.

we

form

the

term

forest

generally refers to the tree

a

specific

to

the

area

tabulated

of

interest.

information

The
about

This information can be presented in several ways

by

of

in

refers

tree

species,

information normally
number

words

"current assets"

list"

the trees.
such

is

a vegetation association

products,

life

forest

refers

trees

occurring

The

information

diameter
to
in

classes,

quality.

estimated volume
the

may

be

area,

with

expressed

an
on

and to

The
the

associated
a

per

unit

are basis (hectare or acre),

and

the precision

is a measure

of its reliability.
The

information

depends
1964;

to

be

on the objectives

Nyyssonen,

i s , for

1976).

instance,

obtained
of the

If

future

the

future

instance,

land

implementation

inventory

is

with

purpose

the

trees,
tree

specifying

the available
gathered

on

of

aim

on

appropriate.

quality.

making

the

use

area,
type

1976,

inventory.

The

an answer

(Nyyssonen,

much

species,

1978):

"How

dimensions and grade,
time

alternative

periods
cost

this question
use

of

for

reconnaissance

inventory
volume

of

diameter

1978).

inventories

inventory is to provide

to

1976).

is

done

standing

classes

and

the uses of

The information

is of

little use

in

investment decisions.

pre-investment

given

inventory

recreational

is given to

Information for this objective
a

a

total

emphasis

reconnaissance

the

al .

identify projects

species,

stock (Nyyssonen,

of

planning,

is to

estimating

Normally no

(Loetsch

(Nyyssonen,

the

by

inventory

there is no need to gather

This

volume

forest

objective

information related to tree volume
for

a

inventory

studies or waters he d management,

If,

by

the

at

limits

depends

available

should be obtained

purpose

industrial

stock.

wood,

can be made

volume

on the type
The

this

type

to the following

tentative
per

of

from
of

question

specified

by

available within

mill
unit?".

sites
The

within

answer

to

of industry that is going
requirement

for

a

paper

mill,

plywood mill

or

sawmill

of acceptable species,

are quite

different

in

terms

dimensions and grades.

If the owner of a forest tract wants information aimed
at

management

planned

to

quantity

decisions,

obtain

of

then

estimates

timber

cut.

the

of

the

Other

inventory
volume

types

of

should

growth
forest

and

be
the

inventory

exist and some are discussed by Husch (1971).
It must be emphasized that the information generated by
alternative

types

interchangeable.

of

Although

information,

including

type,

advised

it

is

for each situation
1.1

specific

methods

on

that

may

maker and taking
accessibility

on

basic

step
of

agencies,
be

the

existing
of their

inventory

designed

Problem and
in

There

Assumptions A
order

timber

in

are

for

to

planning

of

the

a
or

and

sampling

conducting

objectives

in

companies

several

forest

obtain

products

private

involved

available

i nv ent or ies . Depending

of

not

1976).

i n v e nt or ie s .
are

use

is

regardless

specific

availability

foresters
forest

a

of the

Government

make

inventories

make

the

inventories

can

(Nyyssonen,

the

area.

consulting
executing

to

constitutes

information

one

past

Statement

inventory

forest

forest
decision

into consideration factors such as size and

of

the

area,

forest

types

present

and

funds

a v a i l a b l e , an appropriate selection of a sampling method can
be done.
The
problem:

decision
selecting

maker
a

may

be

multilevel

faced

with

sampling

the

following

technique

for

reconnaissance inventory when a large area to be inventoried
is

involved

case,

and

limited

funding

either a multistage

is

available.

or multiphase

In

sampling

such

a

techniques

can be considered.
In

order

selecting

to

give guidance

the

technique,

for the

most appropriate

a multistage

decision

maker

multilevel

in

sampling

and a multiphase method were

chosen

from the several multilevel sampling methods available to be
evaluated on the present
for the

selection of

research project.

the methods were:

for the use of remotely sensed
and
when

satellite
large

satellite
digital

imagery

areas to
imagery

format

inventory

may

or

imagery,

methods

should
as

is

well

inventoried

the

may
the

be
as

not

the

have

of

and

applicable in situations where

c)

convenient
b)

to

printed
the

since

transparent

responsable

imagery

allow

airphotos

involved;

facilities

form

digital,

are

in printed,

satellite

in

including

are

organization

requirements

they should

such images

available

and

digital

images

be

a)

imagery,

because

The

for
be

for

and
the

processing
used

or

on

the

transparent

methods

stratification of

should
the

be

forest

is possible.
Based
selected

on

was

these
the

r eq ui r em en ts ,

one

multiphase method was
The

first

inventories

method

developed

multistage

Langley

been

in the west

coast

used

more

region of

the

method

( 1975 ) and

the one developed by Johnston

has

The second method was

by

the

(1982).

frequently
United

only recently developed.

the

for

States.

No record of

its

(practical)

data

could

approach

selecting
to

between

taken
the

found

independently

except

Forest

size

the

is

plots

were

within

with

simulated

an

area

northern

equivalent

to

selected

maker

in

techniques,

located

lower

on

Michigan.

the
The

two

townships.

for

each

method

and,

point

samples

were

equations were

used

randomly

chosen

sample volume.

Regional

volume

per acre volume.

decision
sampling

two

to estimate

plot,

the

them on
in

independently

each

to

area

help

multilevel

apply

National

of

to

two

Manistee

used

be

(Johnston ,e_t. .al. , 1983).
The

was

use

Stratification

Sample

of the area was

done by photointerpretation of 1:24,000 CIR

(color infrared)

t r a n s p a r e n c i e s . As the multiphase method requires the use of
digital

imagry,

on

ERDAS-400

the

Sensing

at

the cost
several
The

the processing
microcomputer

Michigan

for each

State

activities

on each

total

field and

office

for

method,

each

consult ing

of

time
the

foresters

Center
order

required

cost as

average
working

per

Michigan

average price was estimated by a telephone
among registered consulting foresters

to

was

in

Remote

estimate

measured.

terms

of the

for each

by measuring

to estimate

in

for

calculated

estimated

price

performed

to perform the

procedures

total volume

t i m e . In order
an

the

procedures was done

precision of the estimated
and by the

at

image was

U n i v e r s i t y . In

m e t h o d , the

evaluation of both

method

of this

the total

hour
was

charged
used.

the
cost
by
This

survey conducted

in Michigan.

For

the

assumptions

purpose
were

objective was

of

made.

this
First,

to perform

an

estimation

desired.

Second,

pulpwood

values

order

of

the
given

to estimate

total

costs,

was

assumed

this type of
volume

estimated
in

it

project,

a reconnaissance

area. As previously defined,
when

research

cubic

of

volume
feet

that

inventory of

the
the

inventory is done
standing

would

per

several

acre.

the work would be

trees

refer

to

is
the

Finally,

in

performed as

if

it were done by an individual consulting forester.
The
value

to

results

of

future users

this

research

are

of the methods

northern region of Michigan.

in

expected

to

inventories

be

of

in the

CHAPTER 2
LITERATURE REVIEW

2.1

Ba ckground

In

the

planning

inventory it is necessary to define,
unit of assessment.
executed,
In

If a national

phase

of

a

among other things,

forest inventory

forest
the

is to

be

the unit of assessment could be a state or region.

smaller

properties,

it

could

be

the

whole

property,

a

stand or areas with similar forest types,

age or site index.

These

or

units

may

be

remotely sensed

delineated

imagery

(e.g.

on

a

map

airphotos,

on

any

satellite

form
or

of

radar

i m a g e r y ).
The use

of

remotely

provides an

immense

of a forest

inventory.

area can be

sensed

amount

imagery,

mainly

of information

By means

for the

may

also

be

areas

may

not

crown

diameters,

be

identified
of

and

immediate

crown

execution

of photointerpretation,

stratified into several forest types.

areas

airphotos,

closure

segregated

interest.
and,

in

Nonforest

since

these

Measurements
some

the

cases,

of
tree

height can be made precisely.
Although
airphotos,

it

all

these

is only

data

in rare

can

instances

be
that

obtained
timber

from
volume

can be directly estimated

(Husch,

1971).

As

a result,

field

measurements are always part of any forest inventory.
If the forest area is small or the trees very valuable,
all

the

trees

on

case

we

would

obtain a census

Care

must

tree

be

twice

the

taken

or

to

property could be

in

such

omit

or

cases

any

of

a complete
not

to

the

these

of

acres,

cases,

method

has

are

a census
to

be

the

object

would not

chosen

in

be

In

this

enumeration.

measure

trees

Normally larger areas of forest lands,
thousands

measured.

the

(Husch,

same

1971).

involving hundreds or
of

the

inventory.

feasible

order

to

method,

it

and

In

a sampling

estimate

the

parameters of interest.

the

In applying

any

sampling

population

be

clearly

considered

as the

is chosen

(Freese,

are

the

minimum

aggregate
1962;
size

defined.
of units

Cochran,

unit

(Chapelle,

interest

(population) is divided

specified size.
it

is

not

sampling
Once
sampling

This

to

that

population

from which
The

which

the

the

is

sample

sampling

units

population

is

In a forest inventory the area of
into

sampling

units with a

size has to be kept constant

possible

apply

the

laws

of

otherwise

probabilities

of

(Loetsch ei a l . 1964).
the

population

units

subdivision
forest

1985).

The

1977).

into

divided

is essential

types

may

of

a

be

defined,

specified

based

or upon

similar conditions.

is

on

results

past
of

Independent

it

size

is subdivided
and

experience

previous

shape.
with

inventories

of the size or

into
This

common
under

shape of the

9
sampling

units,

unbiased

interest can be obtained.
and

size

of

the

estimates

a

sampling units

available,

map
it

of

can

sampling units.
the whole
The map
the

are

frame

that

the cost

of

used

to

of the

survey

(Husch g_t ,al• 1982).

property

population and

of

parameter

to

be

subdivide

inventoried

the

area

is

into

the

One restriction is that the units must cover

showing

list

the

be

the

What will be affected by the shape

and the precision of the estimate
If

of

the

all
the

not

overlap

sampling units

units
sample

measurements. A

must

on

the

constitutes

p o p u l a t i o n . It

of units

sampling

(Cochran,

will

frame

can

be

the frame
is

selected

also

be

1977).

from
for

or

this
field

established

by

using airphotos or other forms of remotely sensed data

such

as

when

satellite

large

areas

or

radar

i m a g e r y . The

covering

thousands

latter
of

are

acres

used

are

to

be

i n v e n to ri ed .
There are several sampling methods that can be
to

forest

in v e n t o r y . The

method depends

on the area of

composed of only one
of

the

selection of

a r e a ; the

objectives

The

sampling

selected

randomly

the

of probability

s e l e c t i o n , once

interest:

of

the

each

systematically.
are

first

t y p e s ; the

inventory; the

is

extent
costs

of remotely sensed i m a g e r y .

units wi thin
or

the

appropriate

whether the area

or of several forest

i n v o l v e d ; the availability

laws

the most

applied

involved

sampling
In

method may

random

whereas

in

selection,
systematic

unit is selected the positions

the remaining units are automatically d e f i n e d .

be

of

10
2.1.1
consists

Simple

of

Random

selecting

Sampling

sampling

units

(SRS)

from

-

the

This

method

population

such that every possible combination of n units has an equal
chance of being
advantage

of

estimate

of

selected

the

SRS

the

(Freese,

method

is

parameter

estimation of the

sampling

1962;

that

of

Cochran,

it

yields

interest

error,

which

1977).
an

and

The

unbiased

allows

the

is a measure of the

precision of the estimate.
The

sampling

replacement.

In

units

the

can

first

be

selected

case,

a

unit

can

sample more than once and in the second,
appear

in

the

sample

only

once.

The

with

or

without

appear

in

the

a selected unit may

formula

for

computing

standard error is different depending upon which replacement
system

is

used;

replacement
The

formula

(Freese,

use

traveling

the

of

time

possibility of

is

simpler

1962; Cochran,

SRS

has

some

dispersed

selecting

sampling

atypical

estimates

of

sampling

units

concentrated

volume

applying

inventory,

some

This

irregular
the

with

1977).

sampling

such

units

units which may

parameter
in

of

of

the

and

the

result

interest

areas

as

when

high

in
the

or

low

(Husch e.Jt a l . 1982 ).

In

sizes.

are

sampling

disadvantages

between

the

for

length does

of

the

happens

shapes

border

SRS

of

and

the

not

for

selecting

sampling
when

some

the

units

units

may

areas

of

of the measurement

area or,

in

case of

in
have

a

forest

different

interest
units

strips,

fall

their

fall within the limits of the area.

have
on

total

In this

11
case,

the

laws

since they

of probability can not

require

that

size (Loetsch ei. &1.
To

solve

may be used.

this

all

sampling

units

problem,

ratio or

regression

In ratio estimation,

and the corresponding

sampling

unit.

of

timber,

the

known.

The

formula

for

on

the

the

area

total

subject
ft±. al.

an

to sampling
(1964),

ratio

same

of

to

estimate

area

of

the

the

estimate
error.

area

the

from

a

Husch

the

forest

of the

be

total
If

the

is simpler than

sample
&X.

total

must

measurement.

formula

in

to the mean

order

error,

estimators

estimator

the estimated variance

precision

is

The

of the mean volume

In

area is measured without
if

the

such as area of the unit

timber volume.

volume

depends

of

two related variables have

unit,

this case would be the ratio
per

be

1964).

to be enumerated on the same

area

be rigorously applied

and

(1982)

therefore

and

Loetsch

present such equations.

The ratio estimator is biased although the bias becomes
negligible as
order to use
that a
and

the

linear

that

(Freese,
aJL

the

size increases

1962;

(Cochran,

ratio estimator effectively

correlation

the

1982).

sample

regression

exists
line

Loetsch .&£ a l .

between
goes

1964;

If this is not the case,

the

In

it is required
two

through

Cochran,

1977).

variables

the

origin

1977; Husch ei

a regression rather than

a ratio estimator should be used.
As
used

to

with

a

ratio

increase

variable xi

that

estimate,

precision

by

the

regression

the

is correlated with yi

use

of

estimate
an

(Cochran,

is

auxiliary
1977).

If,

12
for instance,
the

xi

is the area of the sampling unit

corresponding

coefficient would

volume,

express

the

the

estimated

average change

and yi

is

regression

in volume

per

unit change in area between the sampling units in the sample
and the population

(Husch e_l „al. 1982).

The re are certain conditions
to

use

the

regression

that must

estimator

population mean for the independent
must
the

be

known;

the

independent

relationship

variables

has

be met in order

(Freese,

1962).

The

(supplementary) variable

between

to be

the

dependent

reasonably

and

linear;

and

the variance of the dependent variable about its mean should
be constant for all values of the independent variable.
2.1.2

Stratified

applied when the
into

smaller

forest

areas

called strata.

Random

to be

of

independent

to

Freese

-

inventoried can be

more

An

Sampling

homogeneous

This

method

subdivided

characteristics

random sample

is drawn within

each stratum.
According

presents several
allows separate
stratum and,

(1962),

advantages
estimates

for a

given

over

stratified
simple

random

sampling

random sampling.

It

of the mean and variance for each
sampling

intensity,

it gives

more

precise estimates of the parameter of interest.
Some

requirements

more

efficient

are

(Bickford,

independently

have

sampling
1961):
from

the

to

be met

by

means

1)

the

sampling;

of

in order

stratification.

strata
2)

to obtain a

must

there

be

must

These
define

be

real

is

13
differences among

strata means and variances;

and 3)

there

must be a proper distribution of units per stratum.
Several criteria may be used to
homogeneous strata,
types,

density

geography,

and

such as

topographical

classes,
stand

separate

height,

conditions,

the area

features,

age,

(Bickford,

into

forest

site

classes,

1961;

Husch

e..fc

jal. 1982).
Stratification is normally done by photointerpretation.
In fact,
for

this

forest

stratify

is one

inventory.

should

interest,

since

(Bickford,

of the most valuable

be

When

possible,

closely

this

would

1961; Cochran,

related
give

uses of airphotos

the
to

large

criterion used

to

the

of

parameter

gains

in

precision

1977; Husch ,e.±. .&!. 1982).

When a heterogeneous population is stratified the total
variation

is

strata and
1964).
than

into

two

parts:

the variation between the

the

variation

strata

within

(Loetsch e_t a l .

If the variation among units within a stratum is less

the

variation

population
random

divided

estimate

sampling

within

a

is

stratum

among
will
used
is

units
be

more

(Freese,
reduced,

h o m o g e n e o u s , a precise

estimate

obtained

sample

from

a

small

from

different

precise
1962).
since

of the

(Cochran,

strata,

than
As

it

if

the
is

stratum
1977).

the

simple

variation
internally

mean can
The

be

greater

the difference among stratum m e a n s , the more advantage there
is to use stratified sampling in comparison to simple random
sampling

(Bickford,

1961).

14
The

third

requirement

relates

sample units within each stratum.
used to allocate
stratum

such

allocations
In

the

as:

advanced

allocation,

procedures may be

optimum,

and

balanced

number

This

area of the

allocation

procedure

stratum
requires

this

be

as

the

if

random

a

simple

total

used for the inventory (Husch
Optimum allocation

sample

sampling

is used when costs

to obtain a precise estimate

mean

fixed

a

allocation,
depends

the

cost

(Husch

distribution

on whether the

per

stratum

sampling

unit.

proportional
standard

units

to

In

the

the

(Cochran,

balanced

is taken on

This

going

to

be

are

involved,

or

et

al.

of the

second
of

last

stratified

1982).

In

optimum

sampling units

per

stratum

sampling unit

proportional

product

can

was

In the first case,

inversely

deviation.

allocation
In

is

of

cost per

the same in each stratum.

size

a l . 1982).

when we want
for

each

a

If

available,

within

in

order to estimate the total number of samples to measure.

calculated

variance

units

in

not

the

of

stratum

is

of

the

to the relative

1982 ).

knowledge

of

1962; Husch ei jai.. 1977).

proportional

e_t. al.

distribution

total number of sampling units to each

stratum is proportional
(Husch

the

Several

proportional,

(Freese,

to

case,
the
case

the
sample

stratum
is

or

is

the sample size

to

the

varies

also

cost
size

area

and

called

per
is
its

Neyman

1977).

allocation

an

each stratum.

eaual
This

number

type

of

sampling

of allocation

is

15
less

efficient

than

proportional

or

optimum

(Bickford,

1961).
2.1.3

Cluster

Sampling

-

In

some

cases

of

forest

inventory it may be costly or impractical to select a sample
by the SRS method. This may happen when the distance between
sampled units is so large that the time and cost of covering
the entire

area

is prohibitive,

identify each unit
situations,
number

the

of

mutually

Each cluster

simple

random

units

wi thi n
is

the universe

sampling

units.

process

in

called

units

exclusive

is impossible

(Frayer,

can

be

groups

1979 ).

In

aggregated

or

clusters

to

such

into
of

of

each group

selected
one

is

cluster

stage

or

drawn

are

and

all

measured.

simple

cluster

a

such

should have an equal number of units.

sampling
each

or when it

A

the
This

design

(F r a y e r , 1979).
When

the

units

of

regional or national

assessment

forest

are

large,

inventories,

such

as

in

the measurement

of

all units within a cluster is costly or impracticable.
convenient
units

of

in

such

cases

hierarchical

to

order.

subdivide

the

This process

It is

clusters

into

constitutes

what

may be called multilevel sampling designs.
2.1.4

Multilevel

level implies
be used
1979).

that more

in the
These

Sampling

Designs

than one

source

-

The

term

of information will

estimation of population parameters

sources generally,

multi­

(F r a y e r ,

but not necessarily,

one or more types of remotely sensed imagery (e.g.

involve

satellite

16
images,

high,

medium

and

low

altitude

aerial

photographs)

and ground measurements.
Multilevel

sampling

sampling

and

designs,

a population

sampling

units,

secondary

multiphase

1982).

random

or

which

an

a

number

subdivided
units

primary

into

smaller

may

(Loetsch

in

selection probabilities.

last

called

multiphase

sampling,

variables

is

used

from

is also

population

parameters

(Frayer,

"sampling

remains

the

unit".

same,

information used.
second phase

For

1979,

1981).

multiphase

independent

of

the

A first phase unit

unit,

and

units are partitioned

so on.
into

varying
sampling

on

phases

between multistage and multiphase sampling
the

or

The

(P P S ).

information

various

(Husch s„t

e.t a.l • 1964).
equal

In

be

is taken by

be done with

with probability proportional to size

turn

units

random selection process can
The

sampling

of

independent sample

selection

multistage

multistage

into

is

into

tertiary sampling

stage,

systematic

In

secondary

smaller,

At each

classified

divided

of

The

be

sampling.

is

each

units.

subdivided into
al.

may

auxiliary

in

estimating

The

difference

is in the size of

sampling
number

is

of

size

phases

the same

In multistage

smaller units

unit

size as a

sampling,

at each

of

the

succeeding

stage.
There are two types of multiphase sampling,
regression

estimators

Regression estimators
is,

for

instance,

a

and
are

those

using

those using

stratification.

used

when the

auxiliary

photo plot

estimate

of

variable

the variable

of

17
interest

(i.e.,

volume

or

density).

Stratified

are used when the auxiliary variable
variable,

specifying

a given phase
2.1.5

into which

(Johnston,

Double

- This

phases.

the

use

estimators

and

total

of

the

random

falls

at

1982).

Sampling

regression

is an indicator

stratum a sampling

sampling with only two
of

estimators

auxiliary

is

a

form

This method
whan

variable

multiphase

is suitable

the
is

of

population
unknown

for
mean

(Freese,

1962).
In the first phase,
the

auxiliary

have a precise
In the
first

second
sample

variable)

variable

phase,
and

A regression
on

(independent

variable)

in

order

to

estimation of its population mean and total.
a random
the

is measured

measurement

a large random sample is taken from

variable

of

is

taken

interest

from

the

(dependent

(Husch e.±. a l . 1982 ).

equation can

both

subsample

variables.

then be
The

developed using

form

of

the

the

regression

equation does not necessarily have to be linear.
The use of remotely sensed imagery such as airphotos is
possible with

this

could be,

instance,

for

sampling

obtained from photo
al.

The

auxiliary

the estimated volume

variable

per unit

interpretation of photo plots

area

(Husch ei

1982).
2.1.6

form

method.

of

systematic

Systematic
nonrandom
sample

to a pre-specified.

Sampling
sampling.

are

not

pattern

- Systematic
The

units

selected at
(Freese,

sampling
included

random but

1962 ).

The

is
in

a
a

according

population,

18
numbered
where

1

to

K=N/n.

N,

A

is

split

random

into

number

n

is

groups

of

selected

K

units

between

each

1 and

K,

say r * , and the r*h element of each of the K groups comprise
the sample

(Cochran,

Systematic
unbiased
totals,
other

1977).

sampling

estimates
and being

has

of

advantages

homogeneous

faster

probability

the

sampling

methods

between

(Husch

faster

their distribution pattern

Lastly,

the

means

and

and cheaper to execute compared

travel

locaiton.

providing

population

Additionally,
because

of

size

sampling

of

the

e.±. .aJL•

units

1982 ).

is

usually

facilitates

population

to

need

their
not

be

known since they are selected at fixed intervals.
The
there

great

is

no

(Husch at. al.
for this
minimum

disadvantage
valid

that

of

two

selected,

method

the

systematic

for

1982; Cochran,

is

systematic

of

estimating

1977;

Frayer,

calculation

randomly

sampling,

the

1979).

Another

disadvantage

is

that

systematic

error

The reason

units.

unit

the others being taken at a constant

that

requires

sampling

first

is

sampling

of variance

selected

only

sampling

is

a
In

randomly

interval.
sampling

may

give poor precision when unsuspected periodicity

is present

in

case,

the

on

the

the

population

variance

of

the

(Cochran,
estimators

1977).

In

change

this

depending

correspondence between the periodic nature of the population
and the sampling interval
If

the

systematic

population
sample

is

(Frayer,

of

more

1979).

interest
effective

has

a

than

linear
a

trend,

simple

a

random

19
sample
in

(Cochran,

"random"

1977;

order,

equivalent to

simple

Frayer,
then

1979).

If

systematic

random sampling.

the population

sampling
This

is

would

be

is one condition

where an unbiased estimate for the variance of the estimator
exists and

can be calculated by using

the equations

derived

for simple random sampling.
2.2
may

be

Multistage

used

for

which little

sample

smaller

and

stage

surveying

is known.

primary

Sampling

units

smaller

(Anderson,

Technique

vegetation

The population

which

are

units.

1979;

on

large

Meyers

areas

sampling

about

is divided into large

selectively

Each

Multistage

subdivided

subdivision

s_t al.

1980;

into

constitutes
Husch

e_t

a

al.

1982 ) .
The
forestry
(1964),
sampling

multistage
for

many

discussed

sampling

years.

technique

Freese

( 1962 ) and

the applications

techniques

to

forest

has

of

two

inventory.

been

used

Loetsch

si

and three
The

in
al.

stages

primary

units

were composed of blocks of equal or unequal sizes, which can
be delineated on maps of the area or on airphotos.
The use
surveys

may

development

of airphotos

in multistage

or

be

of

may
the

have been developed
application.
Johnston
techniques,
Formulas

not

method.

sampling

necessary,
Multistage

that do not

depending
sampling

require

They

up

four

were

to

derived

for

described
stages,
the

for

on

the

techniques

airphotos for their

Such development was done by Frayer

(1980).

for forest

(1979)

multistage
resource

estimation

of

and

sampling
inventory.

means,

totals

20
and

proportions,

application

of

and

the

the

methods

respective

developed

was

variances.

The

illustrated

with

simulated numerical examples.
Yandle

e.±. a l .

sampling procedure

( 1977)
for

developed

forest

use of aerial photography.
using a
with

relascope,

the

replacement

and

proportional
updating

a

to

size.

survey

a

simpler

The first stage is point sampling
stage

involves

with selection
This

estimated by measuring

in
a

stage

inventory which does not make

second

or

two

method
surveys

sampling

probabilities

is

more

in

which

subsample

list

appropriate

of trees

the

volume

from the

for
is

first

stage.
If aerial photography of the area to be inventoried can
be obtained

at

stage.

number

The

several

proportional to

scales,

of

each

stages

the size

scale may

one

can

be used as a

include

is not

of the area to be inventoried.

Two

and three stages sampling designs using airphotos and ground
measurements
areas,

have

varying

been

between

used

in

100,000

forest
to

1.6

surveys
million

of

large

acres.

The

objectives of the surveys varied but satisfactory results in
terms

of

precision

and cost were
1969; Hall

of

obtained

the

estimated

(Wert,

1968

parameter

and

1969;

of

interest

Heller ei flJL.

&JL. 1979; Harris e± a l . 1983 ).

The launch of spaceships such as in the APOLLO program,
the

SKYLAB

satellites,
development

and

more

recently

the

LANDSAT

series

of

opened a new perspective for the application and
of

multistage

sampling

techniques.

The

images

21
obtained

from

space

are

multistage designs.

normally used

Although

the

as

a

first

resolution of

stage

such

images

may not be appropriate for detailed photointerpretation,
advantage

that

covered.

One

inventoried.
increased

they

offer

image
The

is

may

in

contain

introduction

image

resolution,

photointerpretation,

terms

may

the

of

of

entire

the

for

large
area

subsequent

allowing

increase

the

the
area

to

stages

more

in

be
with

detailed

efficiency

of

sample

selection at each stage.
Langley
variable

(1969,

1971

and

probability

1975)

sampling

developed

a

technique

multistage
which

may

incorporate satellite images or high-altitude photography as
a

first

stage.

inventorying
several
(Wert,

His

large

method

areas

occasions

to

proved

and

has

estimate

1968 and 1969; Anderson,

Satellite images
as a digital

image.

first

in

stage

to

been

the

sampling

techniques.

classified

subdivided

ground

an

area

or

was

Nichols

e.t

classification of LANDSAT digital

in a three stage

the

of

Lee s_l a.l • 1984).

data as a first stage

of

volume

in

Both forms of the image can be used as a

&JL. ( 1973 ) used automatic

classes.

for

successfully used

total

1979;

appropriate

can be obtained as a printed image

multistage

image

be

into

sampling design.
four

timber

The

volume

The second stage was composed of aerial photography
selected

primary

measurements.

The

effective when compared

to

units.
method
the

The

third

was

10 points

stage

shown

to

was
be

the
cost

system applied

to

22
the

same

area.

The

sampling error

was

8.2%,

which

was

far

below the acceptable value of 20% for the area inventoried.
Multistage sampling

designs

have

been applied not

only

to estimate total volume but other parameters of interest as
well.

Hall ei. al.

for collecting
and width,
species
grazing

( 1979) used a two stage sampling procedure

data on vegetation parameters

percent

for

cover,

general

and wildlife

such as

relative density and

land

use

planning

management.

Color

height

frequency by

with

emphasis

on

infrared photographs

(CIR) at a scale of 1:12,000 and line transects were used as
the first and second stages,
per unit

respectively.

Although the cost

area of the inventory was considered high,

the CIR

photographs proved useful for inventory stratification.
Gialdini
followed
based

in

e.t. „&! •
the

design

information

studies

in

(1973),
and

the

order

to

the

where

the

^he

demons!

inventory
sampling

imagery

described

the

implementation

anagement

wac

LANDSAT

and

system,

system for timber
discussed

(1975)

author
ate

the

The

first

by

used

useful

as

was
a

this

timber

particular
management

parameters
area,

basal

addition

to

study,

purposes.

such as number
area growth
volume.

might

In

have

not

the

second

On

of acres,

and

surface

this

case,

area
a

three

five

case

of

case

the

study

e.t.

a.l •

appropriate
first

stage.

the objective

been

number

be

imagery-

Nichols

Estimating only the total merchantable volume,
of

an

to

applicability

described

proved

<

escribed

planning.

procedure

steps

enough

study,

of trees,

other
basal

were estimated
stage

for

in

stratified

23
sampling procedure was used.

LANDSAT data were

first

used

stage

but

probabilities.

were

not

Instead,

to

LANDSAT

used on

calculate

data

selection

were

used

for

stratification and for area measurement within strata,
on

computer

the

use

classification.

of

the

LANDSAT

surveys,

inflated

parameter

if

used and

data

variances

probability

reason
was

can

with

were

each

be

considered

The

this

in

to

in

multiparameter

size

for

results

s a t i s f a c t o r y , although

some

sampling

positively and

overall

based

change

obtained

proportional

other.

for

that

if the variables are not all

correlated
survey

The

the

highly
of

the

is

the

relative

standard error associated with some parameters was above the
5% specified

for the

v o l u m e . Langley

(1978),

discussed

possibilities of using remote sensing in multistage
methods

for

difficulties
The

multiresource

al.

case

stage

used

in

a

objectives evaluated
of manual

simple

each
size)

For

this

three

stage

sampling.
the

the

sampling

The

ground

digital

sampling

to

of

S190

Other

S190

CIR

2 ) the

imagery

application
units

PPS (probabi lit y proportional

condition

showed
and

that
the

CIR

1) the efficiency

for selecting sampling

results

by

data as the

method.

enlarged

classes and

detail

SKYLAB

in the analysis were:

timber volume

random

analysis,

instead of LANDSAT

s t a g e , as opposed

between

presented

was discussed in more

photointerpretation

in identifying
of

study

(1973) .

photography was
first

sampling

involved in such i n v e n t o r i e s .

third

Titus

i n v e n t o r i e s , and

the

the

at
to

correlation

photointerpreter *s

24
timber
into

volume
two

estimations

classes

promising
higher

class

for

a

(timber
first

correlations

and

stage

with

were

very

low.

non-timber)

conditions.

division

seemed

stratification,

ground

A

due

This

more

to

the

suggested

that PPS sampling would be more efficient than simple random
sampling.

The probability of selection would be proportional

to the percent of the primary unit covered by t i m b e r .
The

feasibility

sampling design

of using

for regional

LANDSAT data
forest

in a

multistage

inventory was

by Nichols el .al. (1976) and Harding at al.

discussed

( 1978) . The area

involved was 10 million acres located in western Washington.
The

objectives

volume

data

of

base

c l a s s , within

the

inventory were

for

a large

a limited

accuracy.

Basal

ownership

categories

h a r d w o o d . The

to

region,

by

time and to an

area

per
for

acre

both

was

second

specified precision

of

provide
broad

a

forest

ownership

acceptable

level of

estimated

by

growth

10%

f ive

conifer

for the

and

estimated

mean was achieved for four of the five ownership ca t e g o r i e s .
The

study

using

did

LANDSAT

not
data

positively
for

showed the promise of
carried

out

to

forest

confirm

the

management

feasibility

inventory

of

but

it

such d a t a . Further research was to be

compare

a

detailed

LANDSAT

classification

with the current forest management in v e n t o r y .
Stratif

ication

i nve nto ry . The
strata

of

criterion.

area

is
to

be

homogeneous
This

allows

a

common

inventoried

composition,

for

practice
is

forest

subdivided

based

a reduction on

in

on

into

specific

the total

number

25
of sampling units,
strata,

of

more

since

they will

homogeneous

be located

variance.

The

on areas,

end

result

or

is

a

reduction on the cost of the survey. Multistage sampling has
been used with

satisfactory

results in situations where

area to be inventoried was stratified (Wert,
al.

1969;

Titus e ±

a l . 1973;

1975; Harris ei al.
sampling

techniques

techniques with
.at al.

1983).

the

1968; Heller et

Gialdini ei al.

1975;

Langley,

On other a p p l i c a t i o n s , multistage

have

been

satisfactory

combined

results

with

multiphase

(H e g y i , 1980;

Peterson

1983 ) .

Although multistage sampling has been successfully used
for

forest

has

some

surveys

of

in

technique

the

error

that

number

for

of

for
a

may not

completely
plots

of

application

when

the

is

done

as

over

multistage

disadvantage

stratification

disadvantage

population

ground

o bj e c t i v e s , the

p l o t s . Because

randomly distributed

sampling
than

ground

multistage

variability
plots

different

d i s a d v a n t a g e s . One

distribution
use of

for

of

would

sampling

random

sampling

is
is

vegetation

inventory
the

is

of

much

the

to
with

can

be

of

larger

the

1980) .

same

Another

related

to

its

stratified.

If

type,

it

tends

to

(A l d r e d , 1980) .

economic o n e . In some

interest

of

number

be

apply a multistage procedure

basically an

parameter

the

a r e s u l t , the

sampling
.al •

localized

aspect,

equal

tends

interfere with the efficiency of the method
The decision to

as

the a r e a . As

population
by

an

the

this

detect

(Meyers

multistage

is

technique

estimated

to

forest

situations
within

the

26
established
However,

precision

as

the

area

introduction of more
1978).

and

cost

to

be

stages

On the other hand,

by

using

only

inventoried

one

gets

becomes more

stage.

larger,

feasible

the

(Langley,

since the ratio of travel costs

to

the costs for installing a field plot is high for multistage
design,
used

the

method,

(Meyers g.i al.
2.3

despite

its disadvantages,

tends to

be

1980).

Multiphase

Sampling

Technique

Multiphase

sampling

makes use of information in auxiliary variables from various
phases,

or

levels,

(Frayer,

1979,

in

estimating

population

parameters

1981).

If an n-phase

sampling

design

is used,

at

each of the

first n-1 phases selected,

sampling units are located on the

corresponding

imagery.

correlated

levels

with

the

the final phase,

of

variable

of

An

auxiliary

interest,

variable,

is measured.

ground measurements may be taken

At

(Johnston,

1982 ) .
The difference

between multistage

the size of the "sampling units".
unit

size

phases

of

size

as

a

the

units

is

constant,

those

is

information used.
second
are

are

using

phase

and

independent

so

(F r a y e r , 1979,
two

types

regression

of

on.

into

is

in

In a multiphase design the

A first

partitioned

succeeding stage
There

it

and multiphase

of

phase
In

the

unit

is the

multistage

smaller

number

units

of

same

sampling,
at

each

1981 ) .
multiphase

estimators,

sampling
also

dependent phases and those using stratification,

designs:

called

with

also called

27
with

independent

type

is

used

phases

when

the

auxiliary

estimate

of

density),

which

is measured

is

when

the

used

the

(Loetsch

variable

indicating

falls

a

at

given
on

the

multiphase

sampling,

(Johnston,
Double

limited
can also

two

used

is

photographs

of

the

as the first phase.

photo

variable

For

plots
of

and

or ratio
(Hush

useful

area

are

of photo plots

For

plot

the

selected

interest.

the

quite

example,

is

and second phases.
mean

first

unit

of

the

type

the

multiphase

sampling

are

A

developed between data

the

type

indicator

Most

discuss

or

of

second

sampling
estimators

si .al.

1982).

when

aerial

available.

are

randomly

In

such

selected

A supplementary or auxiliary variable is

photointerpretation.
the

of

Regression

inventory

a large number

measured.

form

with double

estimators

then

a

phases.

Regression

cases,

second

sampling

the

articles

volume

an

a

plot

1982).

only

be

is

1982).

with

first

a photo

The

stratum

deals

The

(i.e.

variable

few

is

ground.

(Johnston,

while

sampling

to

on the

subject

1964).

interest

which

phase

literature

type

into

&JL •

variable

of

auxiliary

variable,

si

volume

second
for

linear

is

phase,

field

a

estimated

by

subsample

of

measurement

regression

equation

of
is

from the plots measured on the

the
then

first

The equation can be utilized to estimate
total

the

of

interest

is

costly

or

not be

The

sampling design is useful when the measurement

variable

need

interest.

This type
the

variables

of

between

of

two

parameter

relationship
of

the

of

for

linear.

updating

a

28
forest

inventory

(Freeze,

1962;

Hush

location of the photo

plots

on the

systematically

than

randomly,

starts

to

rather

overcome

the

problem

s±

&1»

1982).

first phase

of

with

can be done

multiple

estimating

The

the

random

sampling

error (Shine <e± .al. 1962).
Double
been

sampling

shown

to

precision of

be

the

used

when

common

sampling

efficient

estimate

1978;
the

a

very

or tropical forests
Hutchinson,

is

in

terms

when applied

inventory

1972; C o c h r a n , 1977;

1984).

unit

has

been

It

has

cost

and

temperate

Temu e..fc. .al• 1971;

Double

J o h n s t o n , 1982 ).

of

in either

(Bickford e_t a l • 1963;
Mattila,

method.

sampling

stratified

In some

can

be

(M c L e a n ,

applications,

double sampling with regression estimators has been shown to
be

more

ef f ic ient

in

terms

of

cost

error than simple random sampling

and

estimated

(Wear

al.

sampling

1964 ) .

Multiphase sampling is not 1 imited to two p h a s e s . Three
and four phase methods have also been developed and applied
in practice
1982;

(Loetsch s± a l • 1964;

La Bau _e:t al«

1983;

F r a y e r , 1979;

Johnston,

Jeyaratnam e..t a l . 1984;

Li s..fc al.

1984).
As
level

seen

f rom

sampling

inventory

of

the

literature

techniques
natural

have

review
many

resources.

p r e s e n t e d , mul ti­

applications
The

in

the

introduction

of

satellite imagery has greatly contributed to the development
of new multilevel

d e s i g n s . These images

of covering

areas

large

assisted classification,

have

the advantages

and of allowing the use of computer
which

is

used as a

first

level

on

29
several

of

the

satellite

methods

sensors

(Systeme

Pour

of

meters

10x10

application
more

discussed.

with

higher

of

one of

efficient

due

la Terre)

its

multilevel

sensors

sampling

to

the

development

resolution,

1 ’Observation de
in

With

the

such

with a

better

SPOT

resolution

(Jensen,

techniques

as

of

1986),
may

the

become

capability

for

stratification either visually or automatically.
It

is

methods

possible

with

both

creating

more

sampling

procedure

then

it

multistage
previous

usefulness

will
used

the

in place

a

multiphase
experience

conditions

cannot

be

Some

of a more

with

properly
may

regions.

their

a result,

convenient to evaluate.

as

is

simpler

objectives,
design.
be it a

facilitated

method

have

such,

been

a

complex

on

their

applicability on

used

practical

Other

applied

if

similar

only been

evaluated.

not have

thereby

technique,

the

methods

and,

If

inventory

design,

sampling

designs,

techniques.

achieve

although widely used,
As

available

multilevel

sampling

is available.

simulated

of

the

of an appropriate sampling

successful

conditions
on

or

be

combine

types

complex

should

The selection

to

on

methods,
certain

such areas

is

CHAPTER 3
METHODS

This

chapter

sampling

methods

section,

the

This

presents
selected

multistage

technique

was

units

stages were

for

by

method,

i.e.

being

selected

considered

analysis was performed

general

aspects

evaluation.

In

the

is

presented.

sampling

developed

variable probability
sampling

the

in

technique
Langley

at

(1975).

It

the probabilities
each

the present

on each

of

stage
study.

sampling

first

is
of

varies.
A

the

a

the

Three

sensitivity

technique

evaluated

and is also described in the first section.
The

second

section

technique selected.
(1982).

It

describes

the

multiphase

sampling

This technique was developed by Johnston

is a double

sampling

for

stratification

and

may

be considered as an extension of stratified random sampling,
wherein

a

two

phase

sample

rather

than

a

simple

random

sample is taken in each stratum.
The third section gives a description of the area where
the selected sampling techniques were independently applied.
This

area

is

located

in

northern

Michigan.

The

fourth

section describes the remote sensing imagery (air photos and

31
satellite

imagery)

used

for

the

project.

It

also

presents

the method used for field measurements.
The

fifth

section

describes

photointerpretation and digital
the

study

parts:

area.

This

de scription

section
of

the

processing
also

the

methods
of the

includes

process

used

used

for

imagry of

the

following

for

sampling

selection at each stage of the multistage sampling technique
and at each phase
equations

used

of the multiphase

for volume

sampling technique,

estimation,

the

process

for

the
area

estimation and time and costs considerations.
3.1

Mu ltistage

sampling,
units

a sample

(P S U ) is

a sample

of

first stage

randomly

of secondary

selected.

The

been selected

Sampling

selected.

Technique

continues

from each

stage.

Within

each PSU

(SSU)

until

is also

sampling

The variable

of

then measured in the last stage units (Johnston,
The
refers

above

to

process of
selecting
from

the

description of multistage

randomly
random
every

selected

selection
possible

popu la tio n

selection is made with
selected

more

than

combination

in

the

1962;

replacement,
once,

selection is given by P=l/N,
units

samples

popu la tio n

at

each

the

of

Hush

(Anderson,

randomly

interest

is

1982).
technique

stage.

The

probability of
sampling

sX

have

units

a l . 1982 ).

If

or the same unit can be

constant

where N

n

selected,

units

sampling

implies an equal

(Freese,

multistage

units or primary sampling

sampling units

process

In

probability

of

is the total number of
1979).

The

estimate

of

the total timber volume when the probabi lit y of selection is

32
constant

can

be

expressed

as

(Langley,

1975;

Anderson,

1979):

y = (1 / n ) Z (yi/P)

(1)

i = 1

where

yi

is

the

estimated

total

timber

volume

of

the

ith unit;
n is the number of sample units measured;
P is the constant probability of selection.
The term yi/P
volume Y,

based

sampling units
Another

on

is

an estimate of

the measurements

(Anderson,
alternative

variable probability.

this case,
at

the

is

to

draw

predicted

timber

(Langley,

1975).

is

to size

of

the

n

sample

is called
- PPS

from

obtained

units

selection with

(Cochran,
the

a

ith

1977).

aerial

In

sample unit

criterion

from

by

such

as

photographs

This probability is given by:

Pi
xi

derived

volume

on each

select

the probability of selecting

jth

where

made

timber

1979).

This process

probability proportional

the population

N
= X i / S Xi

is

the

(2 )

predicted

characteristic of interest and ZPi

volume

or

(i = 1,...,N)

some

other

is equal to

unity.
By

substituting

equation

2

into

equation

1

the

estimator of the total timber volume will be given by:
n
y = (1/n) Z yi/Pi
i=1

(3)

new

33
The

estimator

y

is unbiased

with probabilities

Pi

if

the

and with

sample

replacement

units

are

drawn

(Cochran,

1977).

Equation 3 reflects the simplest case of multistage sampling
where only one stage is considered.
Any

estimate

of

associated measure
PPS

the

variance

(Langley,

1975;

a

population

of variability.
of

the

In the case

estimate

Cochran,

parameter

given

by

has

of one

an

stage

equation

3

is

1977):
N

var(y)

where N

is

the

= (1/n) 2Pi[(yi/Pi)
i=1
number

of

units

in

y]2

the

(4)

pop ulation

and

the other terms are as already defined.
As
the

seen from equation

squared

deviations

determination of the

of

4,

the variance

the

ratios

yi/Pi

probabilities of

of

y depends

from

y.

on

If

the

selection of the

ith

unit is based on a variable with a high correlation with the
variable
the

of

interest,

ratios

yi/Pi .

the

As

less

this

variability

ratio

is

an

there
estimate

is

among
of

parameter y, the squared deviations would be small and,
result,

the

(Langley,
If

variance

1975;

the

of

Anderson,

area

to

be

the

order

to

inventoried

In the present
estimate

the

would

be

as a

reduced

1979).

large primary sampling units,
necessary.

estimator

the

large,

resulting

in

a second or third stage may be

study,

total

is

three

volume.

respective variance are given as:

stages where used

The

estimator

and

in
its

34
ni

d

t ij

y = 1/m (2 1/Pi*ru)(2 l/Pij*tij)(Z yijk/Pijk)
i =1
j =1
k=1

(5)

M

var(y)

= 1/m 2 Pi[(yi/Pi)
i=1
M

1/m

(6 )

Ni

2 1/Piru 2 Pi j [ (yi j/Pi j )- yi ]2
i=1
j= 1
M

1/m

where:

- y]2 +

Ni

Tij

21/Pi ru 21/Pijtij 2Pi j k [ (yi jk
i =1
j =1
k=1

m = number of primary units
m

+

/Pi jk )-yi j ]2

on the sample

= number of secondary units

drawn within the

ith primary unit
tij

= number
jth

of tertiary units drawn within

secondary unit within

the

ith

the

primary

unit
yi j

= volume measured

on the

jth

secondary unit

of the ith primary unit
Pi j =

selection
secondary

probability of drawing
unit

given

the

by

the

ith

the

kth

third

jth

primary

unit
yi j k

= volume

measured

unit in the

jth

in

second stage unit

stage
in the

ith primary unit
Pi j k

= probability

of

selection associated with

yi jk .
The

other

probabi li ty Pij

terms
and Pijk

are

as

previously

defined.

are given respectively by:

The

35
Ni

Pij

= xij/S xij

(7 )

j=l
Tij

Pi J k = X i j k / Z X i j k

(8)

k=1

where:

xij

= the characteristic of interest for the jth

second stage unit within the ith
xijk

first stage unit

= the characteristic

k* h third stage unit within the jth
the

ith

first

calculate

the

stage

unit.

The

of

interest

second stage unit within
formulas

optimum allocation

for the

of

to

sampling

be

used

units

to

to
each

stage are presented on Appendix A.

3.1.1
performed

Sensiti vit y Analysis
in order

to

A sensitivity analysis was

evaluate what

effect

selection probabilities for the first
third stages
and

its

values

change

in the

second (P ij ) and

(Pi j k ) would have on the estimated total volume

variance.
of

(Pi),

a

the

This

size

was

done

variables

by

for

randomly

each

changing

stage.

A

table

random numbers was used to derive the following rule:
random

number

is

variable by

10%;

The

for

reason

even,

increase

the

value

of

if the random number is odd,
using

10%

was

to

be

of

if the

the

subtract

compatible

the

size
10%.

with

the

intervals on the crown density scale used.
After
stratum,

applying

this

rule

the probabilities were

for

each

stage

within

each

recalculated and entered

in

equations 5 and 6 for the estimation of the total volume and
its

variance,

respectively.

performed on each

A

total

stage within each

of

five

stratum,

trials

were

except for the

36
first

stage

stage

units

stratum,
repeat

of

red

(primary

only

the

the

three

stratum.

units)

were

trials

same value

the selection

oak

could

of the

probabilities

changed to calculate
probabilities

for

the

the

Since

only

contained

be

obtained

two

first

within
that

this

did

selection probabilities.
for

a given

total volume

other

stage

and

stages

were

were

not
When

being

its variance,
kept

with

the

their

original values.
In order to further evaluate the effect of changing the
probabilities

on

the

estimated

total

its

those

themselves.

A total of 125 combinations of the probabilities

Pi j and

conifer
from

and

the

Pi j k

resulted from

hardwood
red

oak

strata.

125

strata formed
selection

combinations
five

groups,

Pi,

probabilities Pi

75

stratum, since

Pi j

were

combined

five

trials

combinations

only

three

among

for

the

resulted

trials

were

as explained above.
for

the

each with

probabilities.

probabilities

the

Only

possible for the first stage,
The

were

and

variance,

Pi,

probabilities

volume

In
and

the

conifer

and

25 combinations of the
combinations

Pi j k

hardwood

for

group

kept constant and equal

of

the

A,

the

to the values

for trial# 1 shown in Appendix B.
The

following

example will

combinations were

performed.

combination among

Pi,

The number

Pi j

illustrate

Within

and Pi j k

group A,

the

obtained:

way

the

following
3,

5.

1 refers to the two values of the probability

Pi

for trial #1 shown on Appendix B,

was

the

1,

for a given stratum.

These

37
values

were

kept

constant

for

within

group

A.

The

number

the probability

Pi j

for

3

the

other

refers

trial

#3.

25

to the

The

combinations

four values

number

the twelve values of the probability Pijk,

5,

of

refers

to

for trial #5.

The set of 25 combinations for each of the other trials
B, C, D and E were obtained
probability

Pi

would

in the

refer

calculated for trial #2,

#3,

same way except that

respectively

to

the

the

values

#4 and #5.

The three trials for the red oak stratum were called A,
B and C and were
the

also formed by a set of 25 combinations

probabilities

explained above.
the

three

Pi ,

Pi j

and

For each of the

strata,

the

total

Pijk

in

325

the

same

way

total combinations

volume

and

its

of

variance

as
for

were

calculated.
3.2

Multiphase

sampling

technique

follows:

consider

stratified into

Sampling

selected
a

universe

L population

known number of units Ni

can
of

Technique

be
N

generally
sampling

strata.

the

first phase,

(i = 1,...,L)

a simple

where

ui

is a

units

as

which

is
a

(9)

random sample

= Ui #Ni

sampling

described

such that:

is drawn within the ith population stratum,

ni

multiphase

Each strata contains

i
N = S Ni
i =1

At

The

size m

given by:

(10)

fraction chosen

the ith phase one stratum and 0 < ui < 1 .

of

in advance

for

38

being

Within

the

ith

the

number

population

of

phase

population stratum the m
strata such that the
contains m j

stratum

one

define

strata.

Then

in

units are partitioned

(ij)th

(j

=

the

Ii

the

into the

phase one

as
ith
Ii

stratum

sampling units and:

Ii

ni

The

unknown

= S m j
j= l

number

of

(11)

sampling

units

in

the

contained in the (ij)th phase one stratum is Nij

universe

and:

Ii

Ni

= S Nij

(12)

i= 1

At phase
the

two,

a subsample of

size mij

is chosen within

(ij)th phase one stratum:

mi j = vi j * m j

where vij

(13)

is the phase two sampling fraction and

0 < vi j < 1.
The

characteristic

of

interest

yi j k for

k =

l,...,mij

is then measured.
The estimators

of the

population

strata parameters

given as:
Ii

yi . . = 2 (nij/ni)yij.

(14)

j= i

d

and

yi j . = 1/mij

ij

(2 yi j k )
k=1

(15)

are

39

where:

yi . . = estimated population mean

for the ith stratum

yij . = estimated population mean

for the (ij)th phase

one stratum
nij

= number of sample units on the

(ij)th phase

one

within

ith

stratum
ni

= number

of

samples

drawn

the

population stratum.
The variance of y i ..

is estimated by:

Ii
var(yi . . ) = (Ni-l ) { S [ ( m j - 1 ) - (mi j - 1 ) ]
_______
j = l _______
Ni
m
Ni -1

Ni -ni
__________
Ni(m-l)

where: wi j

si j =

( w i j si j 2 ) }
__________

mi j

ii
[2 wi j (yi j . - yi . . )a
l=1

= nij/m

+

(16)

and

[ 1/ (mi j -1 )]2 (yijk
k= 1

- yij. )2

(17)

and the other terms are as previously defined.
The unbiased

estimate of

the population

mean

is

given

by:
L

ys t = 2 wi *yi . .
i= 1

where

wi

(18 )

= Ni/N.

The estimated variance of the estimated population mean

is:

40
L

var (ya t ) = S wi 2 var (y i . . )

(19)

i=1

where

v a r ( y i . . ) is given by equation 16.

3.3 Study Area
3.3.1
within

the

Location The

study

Manistee

National

Michigan.

It

(Henderson

Twp, ) and

area
are

of
in

land.

includes

these

two

The

T.22N.

present

ownership
study

(Figure

Forest

two

townships

government

area

1)

in

is

(Boon

47,852

and

County,

T.21N.

R.11W.

T w p ).

acres;

15,650

was conducted

located

Wexford

townships,
R.11W.

is

The

total

32,202

acres

acres

only

is

on

private

government

land.
This
access,

area

the

was

selected because

availability

of

of

its

information

related

per unit area for some of the forest types,
of

imagery,

and

the

fact

that

the

location

area

to

and

volume

the availability
also

has

a

well

developed network of roads.
3.3.2 General Characteristics The environs of the study
area

consists

topography

is

of

typical

Elevations

range

level.

of

Most

underlain
drained.

by

coarse

the
thick

of

between
area,

textured
this
840

end

moraines.

subdistrict
and

of

1,700

feet

both

hills

and

deposits of

sandy

drift

Areas of gently

plains are also present

state.

above

depressions,

sloping ground moraine
(Albert g_t &!•

the

Hilly

1976).

and

is

sea
is
well

and outwash

41

m
Wexford

Figure 1 Location of the Study Area

42
The

total

annual precipitation

in

the

region

is

30

inches. The mean annual temperature in this part of Michigan
is

44

degrees

Fahrenheit,

with

an

annual

extreme

minimum

temperature of -18 degrees Fahrenheit.
In

presettlement time,

covered

the

scientific
maple

outwash plains

names)

forests.

intense,
species

such

and

red

maple

has

been

red

and

the

as

have

and

to

bigtooth

communities

of

has

and

are a result

1940s.

Some

been

of plantations

of

these

not

successional

trembling

of

the

beech-sugar

in abundance.

white pine

also

for

aspen

Hemlock

logging,

post­

The occurrence of white and

stands that are present in the area,
pine,

and

forests

although

Early

combination

and deer browsing.
and

agriculture,

increased

a

B

supported

composition.

birch,

due

forests,

hardwoods

fire

greatly

pine-oak

(see Appendix

moraines

forest

paper

reduced

pine

the

Logging

changed

logging fire,

pine and

within

reduced.

The

conifer

most being red and jack

established

stands,

northern

mainly

red

in the
pine,

1930s

are

now

being managed by the US Forest Service.

3.4 Data Selection
3.4.1
evaluated
imagery,

Remotely Sensed Imagery
in this

as

following

study

already
imagry

require

stated.
was

the

For

used:

use

the
1)

Both sampling procedures
of remotely

multistage
color

transparencies at a scale of 1:24,000

taken

and

true

2)

LANDSAT

composites

TM

transparencies

of

sensed

method

the

infrared

(CIR)

in April,

1977;

and

(scene # E-4Q094 - 15554; November,

false
4 1982),

color
at a

43
scale of
air

1:750,000.

photos

LANDSAT

used,

in

digital

30,

The

imagery,

1981)

CIR

Michigan

air

both

area

from

purpose of
their

Sensing

data

821973

were

from

1544

3X0;

-

was

at Michigan

longer

(S ya n- Wi t t g e n s t e i n ,

than

a

the

LANDSAT
at

the

University.

The

in

order

is located on the
hardwood

The oaks have

1961),

from

The

used

northern

other

loan

available

State

were

red oak stand that
surrounding

on

Resources.

digital,

images

stratification.

leaves

#

obtained

Natural

and

satellite

the

satellite

(scene

were

of

analog

discriminate the

the

form

photos

Remote

fall

but

the same CIR

in its four bands.

Department

Center for
late

were

MSS

October,

For the multiphase method,

to

study

stands

for

a tendency to keep

northern

hardwood

characteristic

that

species
improves

discrimination during image processing.
3.4.2
volume

Field

per

was used

unit

Measurements

area

for

each

In

forest

within the selected plots

the

field

measurements

person

and

because

of

time

type,

to

estimate

point

sampling

instead of measuring the

whole plot for both sampling methods.
because

order

This method was chosen

would
and

be

done

financial

by

only

one

constraints.

Although point sampling might be considered as another stage
in the multistage method,
data obtained

from the

the same way as

it

is

sampling

important
points

sample plot data.

The

unit area obtained by point sampling
of

the

population

parameter

of

to

can

note that the

be processed

in

estimated volume per

is an unbiased estimate

interest

(De

Vries,

1986).

the

44
The

plot

dimensions

were

of

73x73

meters

square

and

56x56

meters square for the multistage and the multiphase sampling
methods,

respectively.

Within
located.

each

After

plot,

two

locating

sample

one

of the

the field,

two random numbers were

generated

list.

These

numbers

coordinates

(on a theoretical

directions)

of the

For

the

was

followed,

Once

a

pair

coordinates

of

the

returning

of

random

of

tree,

measured.

depending

the

variables
equation
(red pine;

The

the cardinal

procedure

marked
to

corner.

locate

the

incorporating

reason

level,
for

the height,

was

this

also

a

available.

For

several

important

jack pine;

type

available.
with

paper birch and beech),
For

the

height
These

other

and

equations

as

are

the

volume
species

aspen;

sugar

local volume

species

DBH

with

on

of

timber

black, white and red oak;

equations

used.

the

up to 4

measured

distinction

to

section 3.5.4.

(X ,Y )

it was discarded.

related

was

the

same

used

on the species,

cherry;

variables,

to

the

was

equation,

in

diameter at breast high (D B H ) was always

and red maples;

volume

plot

reached by pacing.

measured

were

as

Bitterlich design,

outs ide-bark-diameter

relascope.

the

from a computer

initial,

were

randomly

10, was used for measuring all points.

For each "in"
But

point,

the

numbers

the

basal area factor of

inches

to

of

used

grid oriented

of a point sample,

A relascope

selected

were

second

were

corners

first point which was

selection
by

points

a

general

independent
presented

in

45
For
trees

the

estimation

with

DBH

merchantable.

of

the

above

Appendix

volume

4.0

C

per

inches

presents

unit

area,

were

the

only

considered

tally

sheet

used

during the field work.

3.5 Data Processing
3.5.1
scale

Photointerpretation

photographs

covered

A

the

total

of

whole

thirty

study

1:24,000

area.

Interpretation was done using a mirror stereoscope.

The area

was stratified

forested

areas

covered

into
with

four

strata:

mixed

three

hardwoods,

strata were

conifers

and

red

oak;

one stratum was considered as non-forest and was composed of
agricultural areas and water.
sampling procedures.
not

considered

for

The

These strata were used in both

towns

located within

stratification

purposes

the

since

area were
they

were

of no interest for the project.
As only government
the

limits

of

property was included on the

private

properties

from

U.S.

maps were transferred to the photo-overlays.
for

each

township,

although

validated

by

comparing

published

in

1983.

properties

to

the

reflecting projector
scale

of

the maps

the

with

Transfering
overlays
at the

airphotos

(1:31,680).

them

published
a

the
was

Center

( 1:24,000)

plat
limits

Forest

was

Service

These maps,
in

1977,

book
of

Remote

one
were

which
the

accomplished
for

study,

private
using

Sensing.

adjusted

to

was

that

a
The
of

46
3.5.2 Multistage Sampling Technique

3.5.2.1
used

the

First

stage

LANDSAT

transparencies.

TM

The

Sample

imagery

images

Selection

in

were

both

The first

stage

true and false

color

enlarged

from

1:750,000

to

1:107,000 by means of an optical projector at the Center for
Remote Sensing.
screen

of

the

An overlay was placed on the rear-projection
equipment

were delineated.
its
its

northern

difference

is

border.

corners

imagery,

the

limits

approximately
At

corresponds

Since the

the

of

the

study

area

The study area is not a perfect rectangle -

border

southern

and

the

to

484

scale

feet

of

was

approximately

considered

as

than

1:107,000,
1.4

of the area were not well

area

shorter

a

this

millimeters.

defined in the
rectangle

with

dimensions of 9.6 by 19.2 centimeters.
The area was
2,530.3

acres

subdivided

each

(corresponding

side) which established
for

the

primary

into eighteen primary units

a sample frame.

units,

evaluated but rejected.

to a

1,384

acres

square

3.2

cm

on

of
a

An alternative

size

in

also

area,

was

Several of these smaller units would

have fallen completely within private property.
Within

each

primary units
not permit
have

stratum

was done.

the

increased

an
Time

independent
and

selection

financial

of

two

constraints

did

selection of more primary units,
the number

of tertiary units

which would

to be

measured

in the field.
For
for the

the

calculations

primary units,

of

the

selection

the size variable used

probabilities
for the

three

47
strata

was

the

estimated

total

contained wi thin each unit.
dot

grid overlay.

The

area

of

the

forest

type

This measurement was done with a

assumption

in this case

was that

the

more area occupied by a certain forest type within a primary
unit,

the

(1969),
same

greater

Titus

size

the

volume

et .&! • ( 1975 ) and

variable

probabilities.
reforested

wood

for

Lee

area

per

probabilities.

This

difference being

that

the

a.t

on

that

Anderson

(1979)

determination

al.

primary

(1984)
unit

unit.

of

used

to

used

the

this

selection

percentage

calculate

the

Langley

is

basically

same

the

total area of the

of

selection
thing,

the

forest type was

divided by the area of the prima ry unit.
The

process

followed

the

of

procedure

(1979) . A cumulative
certain

forest

Selection
estimated
sum of
the

primary

total

these

of

list

area

of

a

by

probability

to

study

Anderson
area for a

units was

calculated

is equal

this

and

estimated total

each prima ry unit by

probabilities

requirements

for

(1975)

all primary

were

for

selection

Langley

of the

type within

probabilit ies

unit

compiled.

dividing
its

one which

distribution

sum.

the
The

satisfies
(L a n g l e y ,

1975) .
From a table of random n u m b e r s , two numbers between one
and

the

sum

of

the

total

estimated

area

per

primary

unit

were selected with r ep l a c e m e n t . A sampling unit was selected
if the random number fell within its range on the cumulative
1ist.

48
The false color composite
the

area

occupied

by

units. Conifers were
to

distinctive

red

each

imagery was used to estimate

forest

type

wi thin

the

primary

easier to identify on this imagery due
and

pine,

respectively.

Red

color

composite

to

due

dark

tones

oak
its

was

for

red

pine

distinctive

brighter

red

and

jack

on

the

false

color.

The

mixed

hardwoods could also be easily identified on the false color
imagery.

No

distinction

between

private

and

government

property was done on the 1:100,000 imagery.
3.5.2.2

Second

primary units
to
the

the

on each

1:24,000

second

to

the

place.

As

Selection

strata were

photographs,

dimensions

CIR

a

Sampling

of the three
aerial

s t a g e . The

transferred
decimal

CIR

Stage

of the

photographs

result,

the

which

were

primary

13.3

cm

on

a

side.

The

constituted

rounded
units

difference

selected

transferred

primary units

photographs had a smaller area (2,518 acres)
drawn

The

on

when

to

one

the

CIR

since they were

in

area

is

small

(0.5%) and can be considered negligible.
The

primary

secondary units,
these

secondary

selected

were

then

divided

each

with

an area

units

were

selected wi thin

primary

secondary units,
selection

units

unit.

The

of

157.4

selection

into

sixteen

acres.
each

Two

of

previously

process

for

the

and the size variable used to calculate the

probabilities,

were

the

same

as

for

the

primary

units.
At this
properties

stage,

that

it was possible

were

located

to consider the private

within

the

secondary

units.

49
Whenever

a

forest

private property,

type was

located

Third

Stage

secondary unit was divided
units

the

limits

of

no area measurement was done.

3.5.2.3

tertiary

within

of

1.3

Sampling

into one

acres.

Selection

hundred

Three

Each

selected

and twenty

tertiary

units

one
were

selected within each previously selected secondary unit. The
selection process was
size

calculation
closure

of

made

their

estimates

use

scale which was

to 95%.

The assumption,

crown closure within

of

this

selection

were

density

calculation

of

the

( 1979 ) and

(1984 )

the

but

probabilities was percent crown c l o s u r e . An derson
a.i.

for

above,

selection

ei

used

same as described

the

Lee

variable

the

size variable

probabilities.

The

photointerpreted using a
graduated

in 10%

of course,

for

the

crown

crown-

intervals

from

5%

is that higher percentage

a tertiary unit

reflects

higher

timber

because

a high

volume.
This assumption

is not necessarily true

value of crown closure

is not

timber volume

area.

a

dense

population

overstocked
percentage
have

per unit

young
crown

another

of

always associated with a high
The observer may

thin

stands.
closure

independent

Aerial

as

forest

because

of

types

the

(Avery

an

such

at

impossibility

as

independent

is available
.&!•
of

is

found

volume tables

v a r i a b l e , such

height of t r e e s . Such a table
mixed

trees,

be looking

1959) .

as

at
in

that use

variable
average

must
total

for the northern
It

estimating

was

not

total

used

height

50
from the airphotos

used,

since the ground could not be seen

in the points where the tertiary units were located.
The

ideal

photographs

situation

showing

selected into

the

the

field.

would
location

be

to

of

the

take

the

CIR

tertiary

units

This was not possible because

they

were available for office work only. The tertiary units were
therefore transferred

from the CIR photographs

of each township using a reflecting projector.

to base maps
The maps with

the location of the tertiary units were taken to the field.

3.5.3 Multiphase Sampling Technique

3.5.3.1
multiphase
research
area.
the

Digital Image Processing As already stated,
sampling

required

In this

steps

study

processing,
on the

the use

to

be

of digital

evaluated

imagery

section only an overview will

taken

area.

technique

to

For

digitally

more

detailed

process

the

treatments

by

this

for the

study

be presented of
subscene

of

digital

subject

such

The digital

as

Lillesand

and

Kiefer

(1987),

image

processing

analysis

software

modular interactive
a color

by 256 pixels

Hudson

(1986).
of the image was performed on a

micro-computer from the Center for Remote Sensing.

display

the

the reader should consult specialized literature

(1986) and Jensen

digital

of

software

image

package

was

used.

This

system that allows

composed of three bands

(picture element)

subscene

A Landsat
is

a

the user to
within a 240

(Hudson,

1986).

the

51
In

order

subscene

(240

created.
the

to
x

digitally
256

process

pixels)

of

the

the

Landsat

study

area

With the computer compatible tape

tape

drive

unit

of

the

ERDAS-400,

located using the READBIL program.

image,
had

to

a
be

(C C T ) loaded into

the

study

area

was

This program searches the

CCT for a specific area of interest based on the coordinates
of the

center

aligning

of

the area.

a transparent

These

grid

the row and column numbers

over

coordinates
the

Landsat

for the area of

are

found

image

to

by

show

interest and

the

center coordinates.
Once

the

located,

subscene

a file

containing

(with extention

the

.LAN)

was

program LOADBIL.

These

written

band-interleaved-by-line

in

the

this format,

1 band

3,

tape i.e.,

line

1 band

the

.LAN

line 1 band 1,

4,

etc.

file was

calculate

some

subsequent

processing

Jensen,

1986).

This

program

which

information
columns

and

corner;

etc)

standard

has

bands;
or

the

deviation;

of

two
on

the

the

(X ,Y )

minimum

it

In

a)

(e.g.

(e.g.

for

necessary

by

1986;

the

BUILDH

or

modify

add
number
the

to
for

(Hudson,

histogram;

maximum

line

if all

statistics

to

by

1986).

was

were created

header

and

is useful

file

coordinates

trailer

format.

line 1 band 2,

(Jensen,

data

functions:
the

(B I L )

univariate

statistics

contained

created using

format

formed,

fundamental

These

was

bands are written line

bands are to be used in the analysis
Once

area

two programs are utilized with C C T ’s

the data for all the

line onto the

study

of

upper
mean;

brightness

rows,
left
mode;

values;

52
user

assigned

band

number)

and

b)

to

compute

statistics

describing the image.
To

display

program allows

the

image

the

READ

prog ram

the display of up to three

was

used.

selected bands of

a .LAN file as an intensity-modulated

image of red,

blue,

color composite

superimposed

(Hudson,

1986).

to create

The

image

a false

can

be

This

displayed,

green or
image

magnified

or

reduced.
The image displayed on the screen was not geometrically
rectified

to

correct

distortion).
necessary

The

for

process

some

measurements

of

not

with

do

the
the

case

area,

rectification of
the

associated
obtained

and

of

an

in

of

the

brightness

As

process

of

geometric
(row,
same

values

this

was
to

geometric

relationship

column)
point

are

is

accurate

it was decided not

the

location

coordinate

require

distances.

the exact

pixel

the

or

earth-rotation
rectification

that

study,

Besides,

image,

(i.e.

geometric

direction

the present

input
map

skew

applications

rectification.

between

for

and
is

also

the

rarely
modified

(J e n s e n , 1986).
The subscene
larger

than the

obtained

study

from the CCT encompassed an area

area.

The

CURBOX program

was used to

define the study area within the subscene extracted from the
CCT. A rectangular
left and

lower

Locating

the

image

was

box is drawn on the screen and the upper

right
corners

very

corners of
of

difficult

the

the study area are defined.

study

because

of

area
the

on

the

displayed

resolution

of

the

53
image.

The CIR photographs

were used

to

help

identify

them

as precisely as possible.
After

the

(X,Y)

opposite corners

coordinates

of the

area

fills

with

the

zeros,

part

area

were

of

diagonally-

defined,

the SUBSET

image surrounding

eliminating

new file.

obtained

two

study area to a new file.

the

thereby

unwanted data on the

the

study area were

program was used to copy the
program

of

after

the

study

possibility

The statistics

it

the

This

for the

was isolated

of

study

within

the

subscene.
Once the new file was formed,
again

to

display

HISTOEQ,

which

the

image.

performs

a

An

the READ program was used

enhancement

histogram

program

equalization

called
of

the

displayed image was used. This program enhances the image by
maximizing the color contrast.
Before
program,
data,

the

which

classification
calculates

was used.

was performed,

the

principal

The number of principal

to the

number

of bands

case,

there

were

component

principal

PRINCE
of

the

components is equal

on the original data.

four

the

In the present

components

since

the

program

was

original data had four bands,
To
used.

do

the

classification,

the MAXCLASS

This program performs a supervised classification of a

.LAN

data

the

maximum

supervised
Hudson

file

by

either

likelihood

classification

(1986)

which

the

minimum

classifiers.
was

done

distance-to-means
The

based

on

found that unsupervised

or

selection

of

the

by

study

classification,

54
on a site with similar
low

classification

classification.

accuracies

The

distance-to-means
likelihood

forest cover as the

same

was

algorithm

as

author

more
for

study area,

compared

to

gave

supervised

also

found

that

minimum

accurate

than

the

maximum

the

same

test

area

referred

to

above.
Although
scene,
the

as opposed

minimum

classify
the

these

results

to

were

a fall

scene as

distance-to-means

the

study area.

selection of this

obtained

using

a

winter

in the present

study,

classif ier

Another point

classifier

that

is the

was

used

contributed

fact

that

to
to

it takes

less computer time to do the classification than the maximum
likelihood algorithm.
The minimum distance-to-means classifier calculates the
spectral distance
each training

between each pixel and the mean value

site.

which the distance
A

The

file

be

of

the

created.

the program called FIELD.
save,

is assigned to

is smallest

requirement

signature

pixel

in a signature

(Hudson,

file

class

for

1986).

MAXCLASS
This

the

for

program
was

is

created

that
by

a

using

This program was used to build and

file,

the

statistics

of training

sites

within the study area.
By using

a joystick,

a polygon was drawn on the

which encompassed an area of known cover type.

screen

Based on the

histogram and on the summary statistics shown on the screen,
a

total

training

of

thirty

stage

for

three

training

supervised

sites

were

classification

obtained.
is

The

critical

55
since the success of the classification relies on it, and is
more of an art
The CIR
the

than a science

photographs

training

sites.

independently
area,

were

on

used to

The

both

(Lillesand and Kiefer,
help

training

townships

in

the

areas

that

selection

were

Henderson
types:

training

Township

red

and

sites

(T.21N.

pine;

agriculture

selected

constitute

the

jack

water.

were

initially

study

R.11W.)

pine;
From

on

red

the

oak;

obtained

following
mixed

Boon Township

(T.22N.

except

since it does not occur in the area.

The

program

signature
This

is

to

the

earth

utilized
spectral

targets.

sensing

literature,

unique.

The

may

be

used

to

create

stored on a

by

MAXCLAS.

file of

The

term

response measured

The

term

implies

spectral

"signature"
a pattern

patterns

quantitative,

may be distinctive,

obtained for all the same

was

names which were

file

refers

SIGNAM

but

cover

R.11W.),

sites were

red oak,

but

that

observed

in

they

not

are

they are not

a

unique

types

list

of

signatures.
"signature"

by remote
as

from

hardwood;

sixteen training

data

of

but the classification was done on the whole area.
Seventeen

over

1987).

used

sensors

in

remote

is absolute
remotely

and

sensed

absolute;

they

(Lellesand

and

K i e f e r , 1987).
Since
conifer,
was
ones

there

were

mixed hardwood,

decided

to

initially

several

sites

red oak and agriculture

synthesize
obtained

training

a new

for

class

those

program was used to accomplish this.

from

cover

for
strata,

several

types.

The

of

the
it
the

ADDSIG

This program allows the

56
combination
added to
were as

of

the

several

end

follows:

five;

one

mixed

hardwoods

jack

two

red pine

five;

is

that

The

as

a

new

given

file

synthesized

to

two

red oak

two

agriculture

new synthesized
former

result

classification

have

of

a

the

was

from

from four;
from

classes

be

classes

classes were synthesized

and

the
a

from

The

from four;

from

values

classes.

file.

between the

classes,

brigtness

the

pine

The difference
old

of

signatures

two

five.

and

the

broader

range

combination

of

the

using

the

new

the

screen

done

by

of

synthesized classes.
The
using

classified

the

DISPLAY

image

was

displayed

program.

The

on

difference

between

by
the

DISPLAY and the READ program is that the former is used when
a file

with

a

.GIS

extent ion

is

to be

the case for the classified image.

displayed,

which

is

The later program is used

when a file with a .LAN extention is involved.
In order
map

of

the

eliminate

to obtain a hard copy in the form of a number
study

the

area,

zeros

that

the

PUPDATE

surrounded

program
the

was

area.

used

After

to

this,

the NPRINT program was used to print the number map from the
GIS

file.

in

the

The hard
data

coordinates.

copy or number map shows how each pixel

file

was

a

result,

As

within the number map

classified
it

was

the clusters

and

its

possible
of

pixels

respective

to

delineate

that

belonged

to the cover types of interest.
The number
was

then

used

map with
in

the

the delineated

selection

process

clusters
of

the

of

pixels

sample.

To

57
locate

the

selected

the CURSES
on

the

scene.

pixels

(sampling

program was used.

screen
The

and

allows

units)

on

the

image,

This program displays a cursor

the

user

screen and data file

to

move

it

around

coordinates of the

sampling units were shown on the screen,

the

selected

allowing them to be

located within the study area.
3.5.3.2
was

Phase one Sample Selection The whole population

divided

agriculture
sampling

into
and

water.

units

classified

four

strata:
Each

(pixels),

image.

The

conifer,

stratum
which

number

had

were

of

units

a

mixed

hardwood,

known

number

obtained
for

each

from
stratum

of
the
is

presented in Table 1.

Table 1
Total Number of Sampling Units
on Each Stratum

Stratum

Sampling Units
(# of pixels)

Conifer

8,538

Mixed Hardwood

35,929

Agriculture

12,775

Water

Only

the

interest

in

estimate

wood

conifer

the

213

and

present

volume.

As

mixed

study
a

hardwood

since

result,

the
the

agriculture and water were not considered.

strata
objective

non-forest

were

of

was

to

strata

The towns located

58
within the area were included on the non-forest stratum,
classified

as

such,

objectives

of

the

since

they were

study.

For

the

of no
two

interest

strata

of

and

for the

interest,

sampling fractions were arbitrarily chosen so as to allocate
a reasonably
(Johnston,

large number of sampling
1987).

This

information on the use

was

units

done

to each

because

of the method was

no

stratum
previous

available

for the

study area.
Sampling fractions of ui
chosen

for

the

respectively.
1,

2)

conifer

phase

one

(ni )=

123;

sampling

red

units

pine

mixed

values
10,

were

of

hardwood
the ui

the

strata,

and Ni

following

obtained:

mixed hardwood stratum

(Ii ) phase

into

the

the

1 in equation

At this point,
two

and

By entering

from Table

= 0.0144 and U 2 = 0.00315 were

(i =

number

conifer

of

stratum

(n 2 ) = 113.

each stratum was further stratified into

one

strata.

and

jack

The

pine

conifer
phase

stratum was

one

strata.

divided

The

mixed

hardwood stratum was divided into hardwood and red oak phase
one strata.
The

samples

randomly

selected

one stratum.
the

number

cover
the

types
number

identify
and

and

column

(ni and

n2 )

within

each

and classified into one

of

stratum
the

were

two phase

The selection process was done with the help of
map.
of

Previous
interest

map.

The CIR

locate

the

numbers) of

to

were

this

selection

identified

photographs
cover
these

types.

and

were
The

cover-type

process,

the

delineated
used

on

to

help

coordinates

(row

clusters

were

59
obtained

from

the

number

map

in

order

to

the

sampling

define

their

positions wi th in the study area.
The

selection

process

The first number of the selected pair was assigned

column.
262

on

the

number

Since

the

study

columns,

defined the
within

the

parcels

area was

selection

limits.

within

discarded.

and

of

(X ,Y ) coordinates

those

located

map

The

of

the

Selected

composed
pair

of

property

and

was

to

rows

units

a

and

which

unit was done

used

to

386

numbers,

private

were

numbers

number

of

sampling
of

of

second

a sampling

photographs

private

the

the

boundaries

CIR

pair

a

selected.
row

a

used

random

a

from which

units

table of

to

numbers,

of

only

that

were

property

were

to

identify

determine

the

whether

a

selected sampling unit was within their limits.
The

selection

distribution of

process

resulted

sampling units

per phase

one

following

stratum:

from

units,

= 100 were red pine;

from the n 2 = 113 mixed hardwood units,

=

10

were

red

oak

and

m 2

=23

the

the ni = 123 conifer

n 21

mi

in

=

were jack pine and ni 2

103

were

hardwood.

These

results are in agreement with equation 11.

3.5.3.3
subsample of

Phase
size mij

sampling fractions
chosen for each
units

per

two

Sample

Se lection

given by equation 13

vij

for

At

phase

is selected.

two,
The

i = j = 1,2 were again arbitrarily

stratum in order to obtain a sample of five

stratum,

except

for

the

red

oak

samples were measured due to its small size.

in

which

six

a

60
Table 2 presents the sampling fractions vij

for each of

the stratum considered.
Table 2
Values of the Sampling Fraction

The

second

random-number

Stratum

vi j

Jack pine

0,22

Red pine

0.05

Hardwoods

0.05

Red Oak

0.60

phase

sampling

table

by

units were

selecting

selected using

five

numbers

in

a

the

intervals between 1 and each n i j .

3.5.3.4 Location and Transfer of the Samples from the
Imagery to the Base Map

The

location

representation
using

the

number

of

CURSES

map

shows

of

the

the

selected

imagery

program,
the

was

as

samples
done

already

coordinates

of

on

on

the

the

computer

stated.
each

digital

Since

sampling

by
the

unit,

their location on the imagery was relatively easy.
Once
imagery,
better

a
the

selected
CIR

identify

sampling

photographs
its

location

unit
were

in

the

was

located

consulted
study

in

area.

in

the

order
This

to
was

done by calculating the distance of the sampling unit to two
distinct

features

on

the

imagery,

such

as

road

61
intersections.

The

distance

was

determined by

number of pixels to the reference points.
not

geometrically

corrected,

the

counting

the

As the imagery was

pixels were

considered

as

representing an area of 56 x 56 meters square

(Lusch,

After

sampling

unit

values

were

finding

relative to

the

some

"coordinates"

feature

in

the

of

image,

transferred to the CIR photographs to
of

the

cover

type

and

to

the

the

base

these
identify

map

by

the

1988).

location

converting

the

calculated distances to the appropriate scale.

3.5.4
estimate
during
for

the

volume

the

the

Service

Equations used

field

of

individual

measurements,

Huron-Manistee
were

for Volume Estimation In order

used

volume

National

(Hesse,

trees

1987).

considered
equations

Forest
The

bythe

general

as

"in"

developed

US

Forest

form

of

the

equation is:
Vi
where:

= A( 1 - e< c * DBH > )B

Vi= volume of the ith

(20)

tree in cubic feet

A, B and C are coefficients
DBH = diameter at breast height in inches.
This

equation

gives

net

volume without

bark

to 4 inch

minimum top diameter.
Table 3 shows the regression coefficients for the above
equation,

for

the species

Appendix C presents

found during

the scientific name of

the

field

the trees.

work.

to

62

Table 3
Coefficients for the Volume Equation by Species

Species

A

B

C

Red pine

119.83517

4.10017

-.08516

Jack pine

100.91007

4.15180

-.0878

White oak

120.63656

3.99791

-.07755

Red oak

185.60448

3.78645

-.06456

Aspen

81.04818

4.90735

-.1164

Sugar maple

49. 16485

6.25352

-.1631

Red maple

105. 78777

4.19832

-.09

Cherry

160. 14361

3.71334

-.0668

Beech

161.83706

3.82825

-.0675

63

For

the

species

general volume

not

listed

on

Table

2,

equation presented by Beer

the

following

g.t al.

(1966)

was

used:
vi = (D 2 (D+190)/100,000>((H(168-H)/6400)+(0.3 2 / H ) )(79)
(2 1 )

This equation

gives

without

from

bark,

the
a

where merchantability
minimum

diameter

1

volume
foot

is

of

stump

limited

(Beers,

the

1964).

stem

to

a

in

point

by branches,
In

the

cubic
on

feet,

the

bole

deformity

present

study,

or
the

minimum diameter used was 4 inches.
This

general

volume

equation

was

originally

for estimating volume in standard cords.
greater

flexibility

substituted

for

in

the

conversion factor 79

usage,

number

merchantable

of

8

introduced

foot

(Beers,

height

bolts

was

and

the

1964).

calculation of the volume per unit area from each

sample point within a sampling unit,
was used

In order to achieve

(to convert from standard cord to cubic

feet without bark) was
For the

its

developed

(De Vries,

the following equation

1986):
N

Vj

where:

Vj

= k S Vi/gi
i= 1

= volume per acre for the jth point sampling

k = basal area factor
vi

(22)

(BAF = 10)

= volume of the ith "in" tree given by one

of the above equations,

in cubic feet

gi - basal area of the ith

"in" tree,

in sq

ft

N = total number of "in" trees with DBH > 4in.

64

The

final

estimate

of

the volume

per

sampling unit was obtained by averaging

unit

area

for

a

the volume per unit

area for the two point sampling.

3.5.5

Area

estimation
done

of

using

with the

an

parts

the

cover

or

study

planimeter.

The

CIR

the

forest

for

area

area

were

photographs

types

attached

to

The specific planimeter used allowed for the

svibtraction of
CIR

readjustments.

This

this

measurements
the

delineating

the

in

of

The

types within

electronic

overlays

them were used.
addition

Es timation

phase

the

photograph

readings

done

without

the

characteristic

of

the

work

by

of

the

in

different

necessity

equipment

accelerating

the

of

helped
process

without loosing the precision of the readings.
Some
not very

sections
obvious

of

on

the

boundary

the CIR

of

the

photographs.

study
As

a

area

were

result,

the

total area of the study area was estimated by using the base
maps.

As

property,

these
they

maps
were

also

show

used

to

the

limits

of

the

private

estimate

the

areas

of

those

3.5.6 Time Me asu rements and Costs

The

number of hours

properties.

required for each ac tivity was recorded in order to estimate
the

total

individual

cost

each

activities

classification
selection,

for

during

sampling

were:

evaluated.

photointerpretation,

digital

area measurements,

technique

processing,

and field work.

The

computer
sampling

65
Since the time was
each activity,

an

measured in terms of hours spent

estimation

of the

price

per

hour

on

that

a

consulting forester would charge to do the same type of work
would

be

survey

necessary.

was

consulting
assumed

done

To
on

a

foresters

that

the

estimate

this

random

in

the

price

price,

sampling

State

per hour

a

of

telephone
registered

of Michigan.

included

It

was

mileage.

The

average price obtained from this survey was used to estimate
the costs involved for each sampling procedure.
The

cost

for

purchasing

satellite

images)

was

available

for use

from

Remote

Sensing.

microcomputer

The

was,

rate established

that

office hours,

included

the

price

by the

per

the computer

since

the

hour

photographs

these

for

included.

Center

For

(CIR

Michigan DNR and

however,

University clientele.
assumed

not

imagery

It

for Remote

the Center

the
is

images

be used

were
for

use

of

the

based

on

the

Sensing

for

purpose of this study,

would

and

only during

and that no operator would be required.

it

nonwas

normal

CHAPTER 4
RESULTS

This

chapter

presents

the

results

obtained from

the

application of the sampling techniques selected to the study
area.

No discussion

of

the

results

are made

in the present

chapter as these follow in Chapter 5.
First the results obtained from the multistage sampling
technique
volume

are

and

variation

its

also

in

variable

and

analysis

graphical

include

estimated

deviation,

the

coefficient

intervals

for

for the estimated

separately.

and

the

confidence

total

sensitivity

These

standard

and

population
stratum

presented.

The results
are

form.

presented
The

the estimated

in

total

several

for the

for

each

from

the

tables

and

between

the

second and

size
third

stages are presented in graphical form for each stratum.
estimated basal

area and

acreage

for each

of

estimated

obtained

relationship

volume

the

total

stratum are

The
also

shown.
For the

multiphase

technique,

the

results

include

the

numbers obtained from the digital processing of the imagery,
the

estimated

the

population

area

for

each

total
and

volume
for

stratum

and

each

its

standard

stratum.

The

are also presented.

deviation

estimated
Time

spent

for

basal
for

67
each activity and

the

respectives

costs

are

shown

for

both

sampling techniques.
4.1
on the

Multistage Sampling Technique During the field work

conifer

stands

had

regional

red oak strata,

been

thinned.

office

Cadillac,
thinned

and

of

Michigan,

during

1986

This

-

which

probabilities
had

to

be

that

1987.

of

the

The

Service,

red oak

conifer

course,

were

for

Forest

obtained

were thinned in 1986

thinning,

photographs

Information

U.S.

showed

red and jack pines,

units

the

it was noticed that

the

did

taken

in

selection

recalculated

of

since

stand had

show
As

the
they

been

1987).
the

CIR

result,

the

tertiary
were

in

including

on

a

the

located

(Norton,

1977.

from

stands,

not

the

sampling

based

on

the

percent crown closure.
In

order

to

acquired

1:15,840

taken

1987

the

in

1:24,000

and

its

black

were

used.

and the

percent

crown

variance

for

the

probabilities,

and

white

The

overlays

CIR photographs

the new photos
estimate

recalculate

were

infra-red

photographs

originally

same crown density scale was used

to

the

red

oak

the

for
of

The

to

used
scale

closure.

enlarged

newly-

estimated

and

total

conifer

volume

strata

were

recalculated based on the new probabilities.
The estimated total volume,
coefficient
the three

of

variation

and

its standard deviation,

the

confidence

strata and the population,

4. For the multistage technique,
obtained by

the

summation of the

intervals

the
for

are presented in Table

the population estimate was
estimated

total volume per

68

Table 4
Estimated Total Volume, Standard Deviation,
Coefficient of Variation and Confidence
Intervals for Each Stratum and the
Population

MULTISTAGE
Stratum

Conifer
Hardwood
Red Oak
Population

Total Volume
(cuft)

131,112,680
234,377,358
14,082,693
379,572,731

Standard
Deviation

4. 13866E+7
6.92547E+7
1 .29223E+7
8.17072E+7

Confidence Intervals

Conifer
Hardwood
Red Oak
Population

C . V . (%)

31.6
29. 6
91.8
21. 5

(cuft)

[48,339,405 , 213,885, 955]
[95,867,927 , 372,886, 789]
[0 , 39,927, 371]
[219,158,447 , 542,987, 014]

MULTIPHASE

Stratum

Conifer
Hardwood
Red Oak
Population

Total Volume
(cuft)

115,621,019
93,652,660
104,316,573
76,336,702

Standard
Deviation

1.36771E+7
4.02768E+7
2.54317E+7
1.04770E+7

Confidence Intervals

Conifer
Hardwood
Red Oak
Population

[88,266,838
[13,099,005
[53,453,232
[55,382,703

C. V . (%)

11.8
43.0
24.4
13.7

(cuft)

, 142,975, 200]
, 174,206, 314]
, 155,179, 913]
, 97,290, 699]

69
stratum

(Langley,

calculated
was done
of

K

by

1975).

summing

The

the

variance

variables

is

variances of the K variables
Table 5 presents
for the

three

stratum

refers

trees.
(red

the

strata.
to

pine

strata,

and

the

considered
diameter

as

at

on

breast

area

both

hight

of the
sum

as

among

values

refer

with

DBH greater

values

of basal

area

shown represent

4.0

the

the

pine
trees
oak

all

trees

of

their

basal

inches.

average

two points per unit.

both

red

independent

than

on

and
to

area

conifer

conifer

Merchantable

trees

the

basal

"in"

hardwood

(DBH).

sum

of

for the

the

to

with

area

points,

This

1986).

scattered

refers

sampling units,

the

considered

For

was

stratum.

area refers to only

basal

"in"

basal

species

pine).

total

to

(De Vries,

trees

basal

jack

each

total

the variance

equal

total

including hardwood
Merchantable

the

total and merchantable

The

all

of

for

based on the property that

independent

points,

variance

area

All

of

the

twelve

The number within

the parentheses express the range of the basal area found on
the plots measured on each stratum.
Approximate 95% confidence
for
This

each

stratum

requires

the

can

be

limits

determined

following

for the

by

assumptions

total

volume

y±t( o .95)/ v a r (y ).
(De

Vries,

1986):

ignoring the number of degrees of freedom on the variance of
the

total;

distributed
squaring

assuming
about
the

that
its

standard

the

estimated

parameter
deviation

with
and

total

is

variance
setting

normally
given

t~2.

by

These

70
Table 5
Total and Merchantable Basal Area
for Each Stratum

Stratum

Total

Conifer

120

(85 to 180)

117

(70 to 180)

Hardwood

122

(75 to 150)

112

(75 to 145)

Red Oak

116

(85 to 160 )

111

(85 to 160)

are

also shown

confidence limits

Merchantable

(s q f t / a c r e )

in Table

4 for each of the

stratum and for the population.
Table

6

shows

the government
study area.
sum

to

covered

estimated

and private

government

the
with

net
a

area

the areas
total

areas.

forest

in

area

It

type

land

figures

add

to

since

means
were

the

the

of each

that

total

stratum

for

total for the
stratum do not
those

the

subtracted

process of measurement on the CIR photos.
private

each

properties and

The values for

the

represent

the

values

areas

not

during

the

The government and

since

these

readings

its

standard

were obtained from the base maps.
The

estimated

deviation

for

shown in Table
variance

per

divided,
squared
per

each

volume
stratum,

acre

for

and

7. To calculate
acre,

total

respectively,
(shown

per

each

6).

plot

for

and
the

population,

are

the estimated volume and the

volume

by the

in Table

acre

and

its

variance

were

total area and the total

Appendix

measured

E presents
on

the

the

three

area

volume
strata.

71

Table 6
Estimated Areas for Each Stratum,
Government and Private Land and
Total

Area (acres)

Stratum

7 ,783

Conifer

16,349

Hardwood
Red oak

1 ,529

Government

32,203

Private

15,649

Total

47,852

Table 7
Estimated Volume per Acre and Standard
Deviation

Conifer

Volume
(c u f t / a c r e )
Std.

Dev.

2,740

864.888

Hardwood

4,898

1,716.333

Red Oak

Population

294

7,932

270.048

1,707.495

72

Table

8

presents

activities related
the
from

respective
the

time,

in

to the multistage

costs.

survey

the

The

of

average

selected

hours,

spent

sampling

price

per

on

the

technique

and

hour

consulting

obtained

foresters

in

Michigan was $29.00.
Within
represents

the
the

parentheses,
total

time

the

required

plots selected on each stratum.
the

parentheses

between plots
reach the
for

the

refer to

within

first
first,
the

refers

to

each

plot

number

to measure

the

total

stratum and

and

measurement

the

left

the twelve

The right-hand number within

third

time
the

measured on that

second

on

time

day.

stages

spent

required

The
for

and selection of

the

moving

times

each

to

shown

stratum

sample

units

in those stages.
4.1.1 P r o p ortion Between Measure of Size and Predicted
Volume
The propo rt ion ali ty
variable

of

proportional
best

measure

probabilities

between

interest
to

size
of

would

variable of interest

in

is

an

size
be

the

for

the

sampling
important

variable
with

1977;

and

the

The

selection

is proportional to
De Vries,

the

probability

characteristic.

calculating

one that

(Cochran,

size

1986):

the

73

Table 8
Time Spent on the Activities for the
Multistage Technique

Activity

Photointerpretation
Sampling Measurement
Conifer
First stage
Second stage
Third stage
Field

Time

Cost

(hours)

($)

18.3

530.70

15
10.6
13
16.7

435.00
307.40
377.00
481.40

(9.5 + 7.2)

Hardwood
First stage
Second stage
Third stage
Field

304.50
10. 5
43.50
1.5
646.70
22. 3
27.4 (14.3+13.1 ) 794.60

Red oak
First stage
Second stage
Third stage
Field

5.8
12
17.8
23.7

Area Measurements

Total

(9.1+14.6)

168.20
348.00
516.20
687.30

41.3

1,197.70

235.9

$6,841.10

74
Pi
where

yi

is the

= yi/Y

size of

the variable

of

interest

and Y

is

the total for the variable of interest.
This

implies

being sought,
to

work

that

the

population

is known in advance.

with

probabilities

well

correlated

with

that

probabilities calculated
optimum

probabilities

the estimator.

the

are

However,

it is common

proportional

variable

would

the

estimate

to

an

(X i ), one that should also

this way are

that

the

In practice,

easily assessable measure of size
be

total,

of

approximations

minimize

higher

interest.

the

the

The

of the

variance

correlation

of

between

the chosen measure of size and the variable of interest,

the

more

the

the

optimum

of

the

volume

for

already

the

stated,
stages

three

between
second
the

1975;

were

closure,

are

the

two variables.
since

stratum.

the

approximate

will

be

a

Anderson,

variables

simple

type

Figures

size

third

estimated

percent

crown

strata,

and

size

forest

stage,

result

(Langley,

corresponding

graphs

end

will

smaller
1979;

De

1986).

relationship

third

The

total

For each of the
the

probabilities

probabilities.

variance
Vries,

calculated

variable
stages
used

total

within

2

the

respectively.

3 present

and

predicted

respectively.

for

area

and

the

second

occupied

sampling
Also

and

by

the

unit,

and

shown

correlation coefficients

As

on

these

between

the

No relationships were developed for the first
only

two

primary

units

were

selected

on

each

75

4.0E+04

2.8E+04 —

Yij (cuft)

2.4E +04 —
3.0E+04

r = —.508

r= -.8 0 1

2, 2.0E +04

—

2.0E+04
1.6E+04 —
□

□

1.0E+04

1.2E+04
1 1

90

I

105

1

1

I'

1

120

I

1

135

1

1

I

150

50

AREA (acres)

70

90

110

AREA (acres)

(a)

(b)

.8 E + 0 4 -!

^

1.4E+04 —

o
.0E+04

r = -.9 4 9

6.0E+03

100

120

140

160

AREA (acres)

(c)

Figure 2

Prediction Variable versus Predicted

Volume fo r the Second Stage fo r a) Hardwood
b) Conifer and c) Red Oak

130

76

Yijk (cu fi/acre)

4000

4 0 0 0 —,
□

r = .1 11

r=,540

a

3000

o 3000 —

2000

2000 -

>1000

1000
50

95

20

95

PERCENT CROWN
CLOSURE

PERCENT CROWN
CLOSURE

(b)

(<0

3400

^ 2500 —

1600
95
PERCENT CROWN
CLOSURE

(c)

Figure 3

Prediction Variable versus Predicted

Volume fo r the Third Stage for a) Hardwood
b) Conifer and c) Red Oak

77
4.1.2 Sensitivity Analysis
Table

9

presents

the estimated

total

variance for each of

the five trials of

with changes

selection

in

probabilities
B) are:

Pi

the

(the

actual

volume

for the first stage;

are

its

the conifer stratum

probabilities.

values

and

The

presented

selection

in

Appendix

Pi j for the second stage and

Pi jk for the third stage.
Table

10 shows

specific

selection

originally

summary of the effects of changing one

a

probability,

calculated.

keeping the

The percentage

shown

others

on

the

as

upper

portion of the letters D (for decrease) or I (for increase),
refers

to

volume.
the

the

change

that

occurred

The percentage shown on the

change

that

occurred

in the

in

the

estimated

lower portion,

total

refers to

estimated variance

of

the

and

its

total.
Table
variance

11 presents
for

each

stratum with

the estimated

of

changes

the five

in

the

total

trials

selection

volume

for

the

hardwood

probabilities.

These

probabilities are presented in Appendix B.
Table
specific

12 shows a

selection

summary of

probability

as originally calculated.

the

effects

of changing

while

keeping

the

The percentage

other

a

two

shown on the upper

portion of the letters D (for decrease) or I (for increase),
refers
volume.
the

to

the

that

occurred

The percentage shown on the

change

total.

change

that

occurred

in the

in

the

estimated

lower portion,

total

refers to

estimated variance

of

the

78

Table 9
Changes in the Estimated Total Volume and its
Variance with Changes in the Selection
Probabilities for Conifers

Probability

Trial#

Volume

(cuft)

Variance

1
2
3
4
5

114,368,973.46
142,900,588.66
120,958,301.38
141,564,788.60
119,531,602.72

1.50015E+15
1 .86155E+15
1.58 407 E+ 15
1 .84474E+15
1.56592E+15

1
2
3
Pi j
(second s t a g e ) 4
5

133,326,981.90
132,174,896.31
135,191,295.14
127,151,392.80
118,945,960.84

1.77107E+15
1.74086E+15
1 .82094E+15
1 .61084E+15
1.40974E+15

1
2
3
4
5

138,831,787.85
135,303,420.66
133,761,233.23
145,720,643.25
128,288,304.01

1 .92385E+15
1 .82587E+15
1.78387E+15
2 . 12247E+15
1.63872E+15

Pi
( first s t a g e )

Pi j k
(third s t a g e )

79

Table 10
Effects of Changing the Selection Probabilities
on the Estimated Total Volume and its Variance
for Conifer

Trial#

Probability

1

2

14.6%

9.0%

D

Pi

1. 7%
I
3.4%

5 .9%

3 .2%

I

Pi j k

I
12.3%

6 .6%

D = Decrease

3 .1%
D
6.3%

2.0%

11.1%
I
23.9%

4 .1%

10.2%
D

6.3%

I

9.4%

7.7%

3 .1%
I

1 .6%

9.7%
D

I
8.1%

0.8%

I

Pi j

D
8.7%

5

8.0%

8.4%

I
14.2%

4

3

21.5%

2 .2%
D
4.5%

I = Increase

x% (percent change in estimated total volume)
D
y% (percent change in estimated variance of
the t o t a l )

80

Table 11
Changes in the Estimated Total Volume and its
Variance with Changes in the Selection
Probabilities for Hardwood

Probability

Trial#

Volume

(cuft )

Variance

1
2
3
4
5

222,257,862.32
251,759,795.15
222,399,384.29
213,690,995.25
254,642,391.44

4 .55412E+15
5 .32574E+15
4 . 55690E+15
4 . 38260E+15
5 . 19950E+15

1
2
3
Pij
4
(second s t a g e ) 5

218,435,668.84
218,623,101.65
240,506,803.61
236,512,959.40
216,074,784.30

4 . 16542E+15
4 . 1 7066E+15
5.05065E+15
4 . 88136E+15
4 . 07536E+ 15

1
2
3
4
5

245,628,425.85
238,970,339.44
245,690,373.36
236,115,681.76
264,367,392.86

5.27383E+15
4 . 988 46E+15
5. 27652E+15
4 . 868 53E+15
6.119 62E+15

Pi
(first stage)

Pi jk
(third stage)

81

Table 12
Effects of Changing the Selection Probabilities
on the Estimated Total Volume and its Variance
for Hardwood

Trial#

Probabil ity

1

2

5.5%
Pi

D

7.4%
I

5 .3%

Pi j

15.0%

9.7%

5.4%

15.0%

8.4%

9.4%

0.9%

2.3%
5.3%

8.5%
D

I

I

8.6%
I

D
5.3%

7.2%
D

5

4

D
11.0%

7 .3%
D

3

17.7%

1.8%

«»

4.8%
Pi j k

I

1 .9%
I

9.9%

4.8%

4.0%

10.0%

D = Decrease

12.8%

0.7%
I

I

I
1.5%

27.6%

I = Increase

x% (percent changes in estimated total volume)
D
y% (percent changes in estimated variance of
the t o t a l )

82
Table
variance
third

13

for

each

stages

selection

presents

of

of

the

the
the

red

estimated
five

oak

probabilities.

trials

stratum

The

total
for

volume
the

achieved

selection

and

its

second

and

with

varying

probabilities

are

shown in Appendix B. The reduced number of first stage units
contained

in

that

not

did

probability

this

stratum

repeat

as

resulted

the same

calculated

for

inonly

value
other

three

for

trials,

the

trials

selection

(see

section

3.1.1).
Table
specific

14 shows

a

summary of

the effects

calculated.

The

percentage

portion of the letters D (for decrease)

volume.
the

Pi

to

the

are

equal

change

that occurred

shown

on

the

a

that

occurred

The effects
not
to

shown

the

for
since

in

the

in

the estimated

estimated

trials #2 and #3
values

probabilities

of

Pi

upper

or I (for increase),

The percentage shown on the lower portion,

change

total.

changing

selection probability and keeping the other two as

originally

refers

of

these

total

refers to

variance

of

the

for the probability
probabilities

originally

were

calculated.

For

trial #5, the probabilities were equal to those of trial #4.
The results

of changing

the estimated total
combinations,

volume

for each

and

4,

5 and 6.

on Figures 4 and

5 and A,

selection probabilities

its variance

stratum,

shown on Figures

groups of combinations.

the

The

can be

seen

letters A,

for

all

on

of the

from the

plots

B,

and E

B and C on Figure 6,

C,

D,

refer to the

83

Table 13
Changes in the Estimated Total Volume and its
Variance associated with Changes in the
Selection Probabilities for Red oak

Probability

Trial#

Volume

(cuft)

Variance

1
2
4

13,683,197.60
14,082,692.81
14,571,130.86

1 .62261E+14
1 .66987E+14
1 .72763E+14

1
2
3
Pij
4
(second stage) 5

13,234,186.48
14,487,685.39
13,385,862.14
13,792,302.19
15,652,432.94

1 .47472E+14
1 .76724E+14
1 .50872E+14
1 .60168E+14
2.06284E+14

1
2
3
4
5

14,510,836.58
13,936,557.96
14,456,470.97
13,889,661.25
14,679,645.13

1 .77307E+14
1.63536E+14
1 .75979E+14
1 .62435E+14
1 .81461E+14

Pi
(first stage)

Pi jk
(third stage)

84

Table 14
Effects of Changing the Selection Probabilities
on the Estimated Total Volume and its Variance
for Red Oak

Trial #

Probability

1

2

3

NC

NC

3.5%

2.9%
D

Pi

I

2.9%

Pi j

2.9%
I

13.2%

3.0%
Pi jk

I

10. 7%

1.0%

2.6%

NC = no change

5.4%

D = Decrease

23.5%

4.2%

4.2%

1.4%
D

I
2.1%

I

D

5.8%

11.1%

2.1%

5.2%
D

D
6.2%

same as
trial 4

3.5%

6.4%
D

5

4

I
2.8%

8.7%

I = Increase

x% (percent change in estimated total volume)
D
y% (percent change in estimated variance of the
total)

85

3.0E+08 —

2.7E+08 ■

JUl

N

2.4E+08 ■

1

P i

Til

P i

n

1

i

1

p i
p i

2.1 E+08 ■

l i

i i

i

i

in

1.8E+08 ■
125
COMBINATIONS OF PROBABILITIES

(a)
7.5E +15 —

6.5E +15 ■

M
H

5.5E+15 ■

p i
4 .5 E + 1 5 —

11

P I

11

1

P

P iU

1
1 1 1

1

in
1

i

3.5E +15 ■

2.5E+15 •

125
COMBINATIONS OF PROBABILITIES
(b)

Figure 4

E ffe c t o f Changing the Selection

P ro b a b ilitie s on a) V olum e and b) V a ria n ce
fo r H ardw ood

86

1.8E+08 —i

1.6E+08 —

VOLUME (cuft)

pJ

PJ

P-J pj N

1.4E+08 —

1.2E+08 —

pj

IN

IN
N

N

1.0E+08 —

8.0E+07 ■
125
COMBINATIONS OF PROBABILITIES

(a)
2.5E+15 —

VARIANCE

2.1E+15 —

1.3E+15 —

9.0E+14 —

5.0E+14
125
COMBINATIONS OF PROBABILITIES
(b)

Figure 5

E ffe c t of Changing the Selection

P ro b a b ilitie s on a) Volum e and b) Variance
fo r C onifer

87

1.7E+07 —

VOLUME (cuft)

1.6E+07 —

1.4E+07 —

1.3E+07 —

1.1E+07

1.0E+07

0

75
COMBINATIONS OF PROBABILITIES

(a)
2 .4 E + 1 4 —i

VARIANCE

2.1E+14 —

1.8E+14

1.5E+14 —

1.2E+14 —

9.0E+13
75
COMBINATIONS OF PROBABILITIES
(b)

Figure 6

E ffe c t of Changing the Selection

P robabilities on a) V olum e and b) Variance
fo r Red Oak

88

the

Table

15

total

volume

obtained

presents

as

a

the

and

minimum

and

its

variance

of

all

result

maximum
for

the

values

each

for

stratum,

combinations

of

the

selection probabilities.
4.2 Multiphase Sampling Technique
4.2.1

Digital

Image

Processing

The

univariate

statistics for the digital representation of the study area,
composed of 386 rows and 262 columns,

are presented in Table

16.
Table 17 presents
the

digital

shown

are

results of the classification

representation
from

generated

the

the

after

percentages

on

non-zero points,

18

the standard

study

listing

classification

third

column

area.
for

was

were

The

the

numbers

GIS

file

completed.

The

calculated

based

on

which totaled 57,455.

4.2.2 Estimated Volume,
Table

the

histogram

the
the

of

of

presents

the

Variance and Time Measurements

estimated volume

deviation and

the coefficient

per unit

area,

of variation

for

the i* h population strata (conifer and hardwood).
The

estimated

deviation

and

phase

strata and

one

the

volume

per

coefficient

unit
of

area,

the

variation

for

for the population

are

standard
the

four

shown on Table

19. Appendix E presents the volume per plot for each of the
four strata.
The

total

coefficient

of

volume
variation

and
and

its

standard

confidence

deviation,

intervals

for

the
the

89

Table 15
Minimum and Maximum Values for the
Total Volume and the Variance

Volume

Min.

Max

C

101,520,931.79

163,762,490.93

H

198,464,956.22

294,736,993.87

R

12,682,506.40

16,881,819.60

Variance

Min.

Max.

C

1.18135E+15

2.4 5 3 00 0E +1 5

H

3 . 78008E+15

7.154320E+15

R

1.39393E+14

2.319200E+14

C = Conifer

H = Hardwood

R = Red Oak

90

Table 16
Univariate Statistics for the Study Area

Bands

4

5

7

6

18.654

17.377

65.036

70.918

2.913

5.387

15.425

19.891

Median

18

15

66

71

Mode

18

14

84

92

Minimum

10

7

6

0

Maximum

47

70

101

114

Mean
Std.

Dev.

Table

17

Results of the Supervised
Classification of the
Study Area

Classes

Zero Points

# of Points

43,824

%

0

Red pine

7,229

12.58

Jack pine

1,309

2.28

28,230

49.14

Red oak

7,699

13.40

Others*

12,988

22.60

Hardwood

♦Includes pixels classified as
non-forest areas

91

Table 18
Estimated Volume per Acre, Standard Deviation
and
Coefficient of Variation for the i*h
Population Strata

Stratum

Volume
(c u f t / a c r e )

Standard
Deviation

C . V . (%)

Conifer

2,416

285.820

11.8

Hardwood

1,977

343.471

17.4

Table

19

Estimated Volume per Acre, Standard Deviation
and Coefficient of Variation for the Four
Phase one Strata and the Population

Strata

Volume
(cuft/acre)

Standard
Deviation

C . V . (%)

Red pine

2,625

777.164

29.6

Jack pine

1, 507

278.006

18.4

Hardwood

1,957

841.695

43.0

Red oak

2, 180

531.465

24.4

Population

1,595

218.946

13.7

92
conifer,

hardwood and red oak strata and

for the population

are presen ted on Table 4.
The
refer

values

to

the

initially

second

stratum
other

in Table
stratum

divided.

hardwood
strata:

shown

The

was

18

into

first

and

the hardwood

which

stratum

further

hardwoods

for

divided

red

oak.

stratum

the

population

was

was

conifers.

The

into
The

two

phase

numbers

one

shown

in

Table 4 refer to these phase one strata.
In order to obtain the total volume and the variance of
the

total

for

each

volumes per unit
respectively,
The

total

stratum

and

for

the

population,

area and their variances

by the

area

total

was

area and the

used

so

that

were

the

multiplied,

total area squared.

the

results

could

be

comparable with the multistage technique.
The approximate
volume
were

of

the

95% confidence

strata

calculated

and

by using

the

intervals

population

the

equation

for the

shown

presented

total

in Table
in

4

section

4.1 and taking into consideration the same assumptions.
Table

20

presents

merchantable basal
averages
stratum

(2

refer to
areas.

are

For

areas

based

points

the

on
per

ranges

the

the

red

averages

for the four phase
a

total

plot).

of the

of

10

The

total and

and jack

pine

or

species.

Merchantable

conifer trees with DBH above 4 inches.

total

and

one strata.

in

points

total

The
per

parentheses

the merchantable

strata,

basal

the

sampling

numbers

refers to all trees considered as "in",
DBH

for

basal

basal
area

independent of their
area

refers

to

only

As in the case of the

93
Table 20
Total and Merchantable Basal Area
for the Phase one Strata

Basal Area
(ft2 / a c r e )
Strata

plots

Merchantable

Total

Red pine

134
(75-185)

132
(75-175)

Jack pine

110
(90-130)

97
(80-130)

Hardwood

101
(70-155)

94
(65-150)

Red Oak

108
(65-140)

92
(65-120)

measured

on

the

multistage

sampling

technique, few

hardwood species were found among the conifers.
For

the

hardwood

all "in" trees,

stratum,

total

basal

independent of their DBH.

area

refers

to

Merchantable basal

area refers to all "in" trees with DBH above 4 inches.
For the red oak stratum,
all

"in"

trees

the total basal area refers to

regardless

merchantable basal area,

of

their

DBH.

For the

only red oak trees with DBH above 4

inches were considered.
Table
different

21

presents

activities

the

and

time,

their

multiphase sampling technique.

in

hours,

respective

spent

costs

on the

for

the

94

Table 21
Time Spent on the Activities for the
Multiphase Technique

Activity

C o s t ($ )

Time(hours)

Computer Class.

336.50

67.30

Phase one Strata
Conifer Stratum
P. I .*
Sampling selec.
Sampling trans.
Hardwood Stratum
Sampling selec.
Sampling trans.

18.30
36.80
2.00

530.70
1 ,067.20
58.00

27.80
2.00

806.20
58.00

Phase two Strata
Conifer Stratum
Red pine
Field
Area measur.
Jack pine
Field
Area measur.
Hardwood Stratum
Hardwood
Field
Area measur.
Red Oak
Field
Area measur.

Total
*P.I.

8.60
8.00

(3.8+4.8)

249.40
232.00

7.50
8.00

(4.0 + 3.5 )

217.50
232.00

15.40
16.00

(5.0+10.4)

446.60
464.00

(3.9+3.4)

211.70
269.70

7.30
9.30

234.30
= Photointerpretation

$5,179.50

95
Appendix A
the

sampling

allocation,
for

the

presents

fractions
and

the computer

u

for the

multistage

considered was

equations

of

and

v,

for the

calculations

considering

proportional

calculation of sample

method,

$29.00.

the

average

size

price

(n).

per

the

scene was

of $5.00

As

hour

The price per hour for the use

in classifying

of

of

(Lusch,

1988) .
On

Table

21,

parentheses on the
time

the
lines

spent measuring

parentheses
between

refers

plots.

the
to

The

number

to

the

left

within

for field work refers

to the

plots.

the

the

The number

total

activity

time

called

to

required
sampling

the
total

right
to

in

move

selection

consisted of: delineation of forest types on the number map;
establishment
stands;

of

coordinates

for

the

conifer

selection of the phase one samples

the selected
selection
cartographic

samples
(mi j ’s ) .
transfer

on the LANDSAT
Sampling
of the

selected

CIR airphotos to the base maps.

hardwood

(m ); location of

imagery;

transfer

and

final

refers

field

plots

sampling
to

the

from

the

CHAPTER 5
DISCUSSION

The

discussion

of

separated by

stratum.

oak stratum,

the

second

and

the

third

finally

the

The

results

first

presented

section deals

section covers
section

deals

the

herein

with the

conifer

with

is

the

red

stratum
hardwood

stratum.
Section

5.4

sensitivity
obtained

discusses

analysis.

for

the

the

Section

population

implications

5.5

discusses

estimates,

the

of

the

results

from

both

sampling

performed

just

for

the

discussion

of

the

techniques evaluated.
Digital
multiphase
aspects

image

processing

sampling

related

the different

to

technique
the

strata

is

and

a

classification
presented

5.7 and 5.8 discuss repectively,
measurement and the

was

costs

in

of

the

imagery

section 5.6.

into

Sections

the aspects related to time

involved in each technique and

the number of samples to be measured.

to

97
5.1
for the
total

Red O a k Stratum
multistage

volume

for

other strata.
area,

as

sampling
the

Being

red

the

The

results presented

technique
oak

show

stratum

a low

as

it

estimated

compared

smallest stratum that

seen from Table 6,

on Table 4

contained

to

the

occurs in the

only two primary

sampling units.
The par ticular multistage
has

the

stage

tendency

on

would
1971).

areas

have

a

For

the

contained

of

being

unit was

This

units had

of

stratum

the

technique
that

had

(Langley,
to

be

the

each

since

they

(Langley,

primary

and

fact,

had
this

implies

each time

at

selection
of

In

evaluated

units

volume,

area

a new independent

to be done

secondary units

more

one

selected.

means

selected,

stratum,

the

as

or

technique

sampling

probability

oak

of

twice,

the

higher value

red

of

replacement.

concentrate

higher

most

probability
selected

to

sampling

the

units

a

high

unit

was

sampling

with

primary sampling

selection of secondary

1975).

As a result,

selected within

the

the

same

four

primary

unit.
Due

to

the

characteristic

of

the

method

sampling units

on areas

of higher volume,

units

for the

first draw of the

selected

the

same

as

for

two

secondary

the

units

to

concentrate

the two

secondary

primary unit were

second draw of the primary unit.
had

highest

probability

of

These

selection

(see Appendix B).
The
expansion

selection
factors

probabilities
to

estimate

are
the

used to
total

calculate

the

volume.

The

98
expansion factors

for the

respectively by the
as

shown

in

first and second

sum of the ratios

equation

5.

Due

to

the

stages are given

1/Pi*m
high

and

1/Pij*tij

probabilities

selection of the primary and secondary u n i t s , the
factors

for

these

these

factors

stages

were

population total

had

a

relatively

multiplied

given by

the

by

the

sum of

low

of

expansion

value.

estimate

When

of

Yi j k /P i j k , the

the

result

was a low estimated total v o l u m e .
Under

the

variation was
its low

multistage

extremely

estimated

t e c hn iq ue ,

high for the

total

volume,

the

coeff icient

red oak

as

of

stratum due to

seen in Table 4.

Also,

the volume per unit area was low

as shown in Table 7. On the

other

the

hand,

the

resulted

in

a

stratum,

with

application

high
an

of

estimated total
estimated

multiphase

volume

standard

for

technique

the

deviation

red

oak

almost

two

times as large as that obtained with the multistage sampling
method.
Table

4

also

technique.

The

stratum

very

is

multistage

total
the

of
in

volume
same

technique,

low

the

of

of

by

is

the

the

one

multiphase
the red

obtained

This

difference

due,

of

magnitude

as

technique

that

have given

for

by
in

c o u r s e ,to

total v o l u m e . If the

the multistage

former might

for

variation for

te c h n i q u e .

variation

given

results

compared with

the estimated

order

of variation.

the

coefficient

sampling

coefficient
difference

shows

the

a lower

oak
the
the
the

estimated

had

been

of

multiphase
coefficient

99
As

seen

in

Appendix

E,

the

plots

measured

for

the

multistage technique show slightly higher volumes than those
measured on the multiphase technique.
in the

same

stand,

with

In some areas where
80

(Buchnan,

the

volumes

per

in

from Tables

5 and

red

oak

20,

for

technique

are

technique.

These

the

factor

the

red oak.

site

index of

d e n s i t y . As

seen

total and merchantable basal

area

plot

plots

higher

than

that

the

the

the

was

the

measured

on

those

differences

multistage

for

determined

may

management done in the stand.
of

appropriate

plots were measured with

1985),

differences

for

soils

The plots were located

the

for

be

a

multistage

the

multiphase

reflection

of

the

It also indicates the tendency

technique

to

concentrate

the

sampling

units in areas of higher volume.
The
estimated
based on

approximate

95%

total

of

volume

the estimations

confidence
the

given by

(Table 4) is extremely large.
information
volume)

about

is vague

total volume

the

red

for

stratum,

the

calculated

the multistage

technique

This is an indication that the

parameter

(De Vries,

oak

interval

being

estimated

(total

1986). The low estimation of the

associated with a high variance

determined the

large confidence interval.
The confidence
technique

(Table

multistage
basis,

method.

however.

interval

4)

If

calculated

interval
is

much

This

calculated

smaller than that

can

only

considered
from

from the multiphase

the

be

said

on

separately,

multiphase

given by the
a

the

comparative
confidence

technique

is

also

100

quite

large.

Al though

information

about

large,

the

it

total

nevertheless

volume

than

conveys

the

more

confidence

interval calculated from the multistage technique.
In

order

samples

to

should

sampling

reduce

the

have been

techniques

confidence

measured

evaluated.

in

In

interval,

each

the

of

case

the

two
multi­

more primary,

would have to be

selected.

Since only two primary units

were

located

the

oak

them

encompassed most
of more units
in

this

of

the

would

case,

red

area of the

not have

would

stratum

and

the

stage technique,

within

secondary

of

more

and

been

one

stratum,

been possible.

have

tertiary units

to

use

the

The

of

selection

alternative

smaller

primary

sampling units.
The

relationship

predicted volume
for the third

between

for the

stage

the

second

for the

size

stage

variable

and

and estimated

red oak stratum,

were

the

volume

shown on

Figures 2c and 3c. None of the correlations were significant
at the

0.05

level.

Contrary

correlation between
negative.
from the
the

This

result

is

due

to:

a)

the

in Figure 2c

six plots

average

volume

and

b)

located on an area
for

the

these

selection

secondary unit in question was high
value

for

the

plots

expansion

that was
was

(0.23511).

factor

for

the

is

for

thinned

2,113

probability

the

selected

secondary sampling unit with the highest value

feet/acre)

a low

be expected,

the two variables shown

size variable were

(the

to what would

for

cubic
the

This produced
second

stage

101

(1/Pij*tij).

As

a

result,

the

predicted

volume

for

the

secondary unit in question was low.
According
office

of

original

to

information

the

U.S.

basal

area

feet/acre.

It

average

total

between

95

Forest
of

was

red

thinned
area

120

square

and

Service

the

basal

obtained

to

the

(Norton,

oak
90

from

stand

measured

in

feet/acre

1987),

was

square

regional

110

the

square

feet/acre.

this

and

study

the

The

varied

merchantable

basal area between 85 and 115 square feet/acre.
The other

six plots

measured on the red oak stand were

located in areas that had not been thinned.
selected

from

the

value

for

the

plots

was

2,645

merchantable

secondary

size

variable.

cubic

basal

sampling
The

varied

unit

average

feet/acre.

area

These plots were

Both

with
volume

the

between

a

smaller

for

total

85

and

these

and

160

the

square

fe e t / a c r e .
The selection probability
consideration
secondary
0.16319.
factor

unit
This

for

plot,

also

high,

previously
created

the

secondary unit.
per

was

a

second

secondary unit
lower

considered.

stage

a

but

larger

This fact,

determined

for the

value

when

than
Its

for

for

the

value

was

the

compared

under

to

expansion
the

other

associated with the higher volume
larger

predicted

volume

for

the

and

the

secondary unit.
The

correlation

predicted
relatively

volume
low

between

for

(Figure

percent

the

third

3c).

This

crown closure

stage

is

positive,

low correlation

is due

but
to

102

the

high

plots.
0.05

variability

The

There

are

This

volumes

shown

plots

but with quite

crown closure
volume.

the

regression line

level.

closure

of

within

is not

with

the

for instance,

measured

significant
same

different volumes.

of 85%,

the

at

percent

The

the

crown

plots with

show a great variety

a
of

is an indication of the inadequacy of measuring

percent crown closure as a size variable.
Table
volume

13

and

shows

its

the

changes

variance

for

in

the

the

estimated

red

oak

total

stratum

as

calculated by changing one of the selection probabilities at
a time,

while

As

in

seen

keeping

Table

13,

probability

Pi:

same values

for the

as the values
technique.

the
only

trials

three

#1,

as originally
trials

2 and 4.

estimated total

shown

This

others

in Table

is

because

4,

are

Trial
volume

calculated.

shown

for

the

#2 presented

the

and its variance

for the multistage

the

values

of

the

sampling
selection

probability for trial #2 were equal to the values originally
calculated.

The

probability Pi
Table

14

shown

in

shows

that

volume

and

Table

4,

probability

P i j . The

an

on

increase

happened

to

the

value

of

the

for trial #3.

estimated total
values

same

the

the

largest

its variance,
occurred

values

of this

estimated

total

increase

in

as compared to

with

trial

#5

probability
volume

of

for

the
the
the

determined

11.1%

and

on

the variance of 23.5%.
The larger

reduction on the estimated total

volume

and

its variance occurred with trial #1 for the probability P i j .

103
The

estimated

total

volume

was

reduced

by

6.4%

and

its

variance by 13.2%.
Trial
changes

#2

for

the

probability

on the estimated

compared

with

volume was

the

total

values

increased

by

on

just

Pi jk

volume

Table

gave

and

4.

smallest

its variance,

The

1.0% and

the

estimated

its

variance

as

total

by only

2 .1%.
Figure
selection

6a

and

6b

shows

probabilities

its variance.

on

The pattern

Group A refers

the

the

is

effect of

combining

estimated total

identical

to the combinations

volume

for both

1 to 25;

the
and

variables.

group B to the

combinations 26 to 50 and group C to the combinations 51

to

75 .
The
on both

pattern of the
graphs.

graphs)

1-1-1

of

values
were

left,
the

of

a

the

first

to the Y-axis,

selection

to

combination

For example,

close

repeats

line parallel

represents

probabilities.
to the

Each

curve

#1

of

the

introduced

in

equations

the

in each

X-axis

of

the

small

refers

probabilities.

trial

itself

(on

and

horizontal

line

to the combination

This

6.

both

selection

means

that

probabilities Pi ,Pi j and
5

group

the

Pi j k

Combination

1-1-2

means that the values for trial #1 of probability Pi and Pij
and trial
All

the

stratum,
its

#2
75

for Pijk

were introduced

combinations

that

in equations

resulted

for

the

5 and
red

6.
oak

and their effects on the estimated total volume and

variance

respectively,

can

be

followed

and Table 14.

by

Figure

6a

and

6b,

104
When the first five combinations were completed and the
values for trial

#2

in equations

5 and

total

and

volume

This

increase

for the probability Pi j were
6,

its

a

sudden

variance

occurred

at

increase

occurred

in

estimated

6a and

1-2-1.

Table 14, the values of the probability Pij
of Pijk

the

(Figure

combination

introduced

As

6b).

shown

in

for trial #2 and

for trial #1 determined the observed increase in the

estimated total volume.
The
groups

last

have

five
the

combinations

highest

volume and the variance.

values

trial

increase

#5
in

of
the

probability

for

the

Pij

the probability
estimated

total

Trial #5 of the probability Pijk

the

total

seen in Table

determined

volume

all

of the combined

and Pi j k . As
Pi j

on

estimated

This is the result

effects of the probabilities
13,

of

and

the

its

highest

variance.

also determined an increase

on the estimated total volume and its variance.
The
obtained
1,
for

minimum

by the values

1 and 4,
each

produced

value

of
a

for Pi,
the

for

estimated

total

of the probabilities

in

the

probabilities
estimated

for

total

volume

for the

Pi j and Pi j k , respectively.

selection

decrease

the

15,

represents

the value in Table 4,
The

maximum

a reduction

trials

The values

those

trials

volume

(Table

14). The value for the minimum estimated total volume,
on Table

shown

of 11.0% as compared to

for the multistage sampling technique.

value

for

the

estimated

total

volume

obtained with the values of the probabilities for trials
5 and

5,

for Pi,

Pij

was

and P i j k , respectively.

The values

was
4,
for

105
each of the
13)

selection probabilities for those

determined

an

The maximum value

increase
for

the

in

the

trials

estimated

estimated total

total

volume,

(Table
volume.

shown

on

Table 15, represents an increase of 20.0% in relation to the
value in Table 4.
The

minimum

variance

of

the

and

maximum

total

values

were

also

for

the

obtained

by

combinations of probabilities as for the volume.
value of the estimated variance of the total,
15,

represents

in Table

4.

a reduction

The maxi mum value

in relation to the value
5.2

Conifer

evaluated,
pine

and

jack pine

pine

For

The minimum

to the value

increase of 39%

was

the

case

multistage

subdivided
for

the

technique

into jack
multiphase

The basic reason for not subdividing was that the

pine

technique,

the

stratum was not

as

plantations

jack

same

in Table 4.

in the study area

of the area of the conifer stratum.
the

the

shown in Table

in relation

represents an

Strat um

the conifer
red

technique.

of 20%

estimated

as

a

separate

a different

represent only

As a result,

stratum

by

structure would be

7.3%

to evaluate

the

multistage

required

in terms

of the number of stages and the sizes for the sampling units
at

each

stage.

For

the

multiphase

technique,

however,

the

subdivision of a stratum is a requirement of the procedure.
Comparing
the multiphase
the

conifer

technique

the

results

sampling

stratum,

gave

an

it

obtained

techniques
can

estimated

be

by

the

multistage

presented in Table
seen

total

that

volume

the
13%

and

4 for

multistage
higher

than

106
that

estimated

standard
larger
the

by

the

deviation

than that

technique was

from

for

coefficient

multiphase

of

the

the

technique.

multistage

multiphase

variation

larger than

for

the

The

method

was

approach.
the

estimated
3

As a

multistage

coefficient

times

result,
sampling

of variation

for

the multiphase technique.
The
the

approximate

multistage

calculated
larger

95% confidence

technique

for

the

standard

is

much

multiphase

deviation

interval
larger

method.

This

the

former

of

calculated

for

than

one

is

the

due

to

the

technique

as

compared to the later.
The

results

obtained

reflect mostly the volume
measured,
plots

only

one

contained

had

only a

from
of

the

multistage

red pine.

Of

a predominance

of

few jack pine

trees

the

technique

twelve

jack
and

plots

pine.

Four

seven plots

were composed of only red pine.
The results

obtained

from the multiphase

be considered as a more realistic

appraisal

on

the

the

conifer

subdivided into

stratum.
red pine

Since

technique

of total

stratum

and jack pine,

had

plots were

can

volume
to

be

measured

on both substrata.
Table
substratum.

19

presents

As

shown,

higher volume per unit

the

the
red

the

pine

pine

stands.

substratum

for

each

presents

a

The higher standard deviation

associated with the management

red

obtained

area and a higher standard deviation

than the jack pine substratum.
may be

results

The

jack

pine

that is being done on
stands,

on

the

other

107
hand,

are not being managed.

They are relatively old and may

have reached a stage of nearly homogeneous volume.
The

above

technique
volume.
pine

discussion

presented

explains

a higher

value

why

for

the

multistage

the estimated

total

Most of the plots me asured had a predominance of red

trees

or were

pure

red pine

stands.

These

stands

have

higher values for the volume per unit area and for the basal
area (see Table 20) than the jack pine stands.
In the multiphase technique,

the estimated total volume

for the conifer stratum had to be averaged between the plots
measured on the red pine and on the jack pine substrata.
The

pine

plantations

are

concentrated

southeast portion of the study area,

selected withi n

the

conifer

multistage technique were the same.
primary
with

unit

was

selected

replacement,

selected within the
selected,

a new

has to be done
Figure
variable

and

sampling

of

twice.

As

the

the

four

stratum

same

primary unit.

for

units

is

were

Each time

the

only one

sampling

secondary

done
also

a unit

on the subsequent

is

stages

(sampling with replacement).

2b

shows

the
the

the

predicted
conifer

relationship
volume

stratum.

for

Each

represents a secondary sampling unit.
on

the

The two primary

In other words,

the

selection of units

in

although there are some

small-area plantations in the northern part.
sampling units

mostly

X-coordinate

at

secondary unit which was

70

acres

between
the
point

the

size

second

stage

on

graph

the

The two points aligned

correspond

selected twice.

As was

to

the

same

the case of

108
the

red

oak

stratum,

the

conifer

stratum

exhibits

a

high

negative correlation between the two variables.
The

secondary

estimated

area

unit

of

the

predicted volume.

Among

particular

had

(0.18470)

unit
which

for

the

though the

the

stratum

highest
a

low

for the unit.

plots

had

unit

value

measured

for

in pure

red pine

this

probability

the

question

in this

lowest

selected,

As a result,
in

greatest

the

selection

expansion

the predicted
was

low

even

secondary unit

an average volume of 2,544 cubic feet/acre.
all located

the

secondary units

the

secondary

three

contained

conifer

determined

factor (1/Pi j *ti j )
volume

that

had

These plots were

stands wi th merchantable

basal

areas between 90 and 130 square feet/acre.
The

secondary

quite different
Figure

2b.

2,966

varying

between

in

a

hand,

cubic

selected

plots

and

chosen

the

red

pine

and

The
jack

had

as seen on
time

the

estimated volumes

feet/acre.

stands.

also

first

with merchantable

180 square

red pine
of

twice

its estimated volume,

feet/acre

110

stand

was

selected had higher

in pure

mixed

that

for

three

was

average

plots were

values

The

secondary unit

unit

basal

Two

of

on

area
these

third was measured
pine.

On

the

other

the three plots that were selected the second time the

secondary
cubic
between

unit

was

feet/acre,
70

factor value

and

chosen
with

130

for this

had

an

average

merchantable

square

feet/acre.

volume

basal

area

Since

the

of

1,918

varying
expansion

twice-chosen secondary unit was high,

109
the reason for the difference in predicted volume was simply
the differences in the measured volumes of the plots.
The

secondary

second

highest

within

this

stands.

unit

predicted

secondary

Their average

merchantable
feet/acre.
secondary

with

basal
The

volume.

unit

area

was

were

varying
of

the

the

with

the field

of

area

three

located

cubic
90

among

red

pine

feet/acre with
and

130

in

square

for

four,

lowest. This,

the

measured

pure

factor

the

has

plots

in

between

highest
was the

location

All

expansion

selection probability
the

smallest

volume was 2,342

value

unit

the

this

since

its

conjunction

plots, determined

its

high

predicted volume.
Figure

3b

shows

the

variable

and

the

samples.

The

correlation

(r = 0 .54).
achieved
3c.

As

This

was

estimated

is

for the
the

relationship
volume

is

slightly

red

oak

case

for

positive
greater

stratum

between
the

and

than

for the red

oak

size

third

stage

relatively
the

(r = 0 .512)

the

low

correlation

shown on

stratum,

Figure

there

are

several plots in the conifer stratum with the same value for
the percent
low

crown closure,

correlation

is,

of

but with
course,

different

volumes.

associated

variability of the estimated plot volumes.

with

The
the

The inadequacy of

the percent crown closure measurement as a size variable for
calculating

the

demonstrated.

Both

selection
regression

probabilities
lines

for the

were not significant at the 0.05 level.

is

again

conifer

stratum

110

Table 9 presents
variances

that

probabilities
Each

for

the

estimated total

resulted

selection

keeping

the

each

from

stage

others

as

the

highest

increase

volume

and

its

variance

was

Pi jk . The
increase

shown in Table 4,
The

largest

of

trial

this

decrease

#1

in

multistage

technique.

in

in variance

turn,

in

As

estimated

with

trial

volume

was

while

shown

both

observed

increase

in

calculated.

total

#4

11%

in

of

the

coupled

as compared with the values

decrease

observed

wi th

values

for

trial,

selection

for the multistage sampling technique.

v o l u m e , compared
with

the

changed

originally

10,

with a 24%

the

was

Table

probability

changing

of

probability

volumes and their

in

the probability

the

the

the

volume

estimated

other h a n d , occurred with

in

was

estimated

Table

4,

Pi . Under
reduced

variance
trial

the

of

#5 f or

was

the

15%.
the

total

observed

conditions

The

largest

total,

on

the

the probabil ity

Pi j

where a reduction of 22% was o b s e r v e d .
The

least

its

variance,

were

observed

volume was
1.6%.
Pi j

As
for

variation
in

in

the

com parison

with

trial

increased by

#2

only

shown

in Ap pe n di x B,

trial

#2

are

very

estimated

with
for

the
the

total

results

volume
in Table

probability

0.8% and the

and
4,

Pi j . The

variance

by only

the values of the probability
close

to

the

ones

originally

calculated.
Figure

5

probabilities
variance.

shows
on

Group

the

the
A

effect

of

estimated

refers

to

changing
total

the

the

volume

combinations

selection
and

its

of

the

Ill
selection probabilities 1 to 25; group B to the combinations
26 to

50;

group C to

the combinations

the combinations

76 to

51

100 and group E

to

75;

group D to

to the combinations

101 to 125.
The

pattern

for

the

volume

curve

(Figure

5a),

was

basically determined by the changes in the probability Pi j k .
The

125

combinations

and

their

effects

on

the

estimated

total volume can be followed by Figure 5a and Table 10. Once
again,

each

of

the

small

horizontal

lines

on

the

graph

represent a combination of probabilities.
The

first

combination

of

(1-1-1)

determined

a certain

volume.

The

combination

value

second

for

the

estimated

the

the

effect

decreasing

of

for

(1-1-2)

probabilities

the estimated
determined

volume

than

the

Pi

was

first.

for trial

the estimated total

probability

total

a smaller

the

for trial #1 and Pijk

of increasing

effect

selection

value

total

Although the probability Pij
#2 had

the

volume,

dominant.

The same happened for the third combination (1-1-3).
When

the

values

for

trial

were introduced in equation 5,
estimated
because
effect

total

trial
on

#1

#4

for

increasing

combination
trial

volume

with

for the

the

#4

the

the

probability

Pi j k

a noticiable increase in the

occured.
the

of

This

probability
estimated

increasing

was
Pi j k

total

effect

due,
had

volume.
of

probability Pi j , surpassed

effect of trial #1 for the probability P i .

in

part,

a

strong

This,

in

the

values

of

the

decreasing

112

The

introduction

probability Pijk

of

the

values

for

trial

#5

of

the

in equation 5 resulted in a marked decrease

on the estimated total volume observed on combination 1-1-5.
As

seen

on

Table

10,

the

values

for

trial

probability Pijk

had the effect of decreasing

total

This

volume.

decreasing

effect

effect,

of trial

in

#1

#5

the

the estimated

combination

for the

of

with

the

probability Pi,

were

responsible for the decreased estimated total volume.
The
due to

the outcomes

P i j . As
on

decreasing

seen

those

trend

the

of trials

in Table

trials

of

curve

#4 and

within

5 for

group

A

is

the probability

10, the values of the probability Pij

resulted

in

a

continual

decline

of

the

estimated total volume.
The sudden
occurred at the

increase

in the estimated

beginning

of group B was

total

volume

that

due to probability

P i . At that point the values of trial #2 for the probability
Pi were

introduced

shown

in

Table

total

volume

into

10,

for

the

all

equation
effect

three

5

of

(combination 2-1-1).
increasing

probabilities

the

As

estimated

resulted

in

the

sharp increase.
The values of the first three trials of the probability
Pijk

had

volume,
noted,
trial

the

as

trial.

This aspect
three

of

seen on the

though,
to

effect

that

the

last

combinations

line

the

which

it

the

of Table

percentage

Actually

determined

increasing

of

repeats

10.

from

pattern
itself

total

It should be

increase varies

decreases

step-like

estimated

5.9%

from

to

2.0%.

for the

first

at

constant

113
intervals,
the

as can be seen on Figure 5a. As a result,

combination

increase

the

of

the

estimated

combinations

2-1-1

probabilities
total

to

volume,

2-1-3,

has
as

the

a

tendency

occurred

pattern

even if

of

with
the

to
the

curve

remains because of the different percentages of increase.
The pronounced
of group B was
the

beginning

increase

that

occurred

not repeated at the
of

probability Pi

group

were

C,

the

beginning of group C.

values

introduced

at the beginning

into

for

trial

equation

5.

#3
As

of
seen

At
the
in

Table 10, those values decreased the estimated total volume.
As

a

result,

the

pattern

decreasing tendency.
beginning

of

of

the

Another abrupt

group

D,

because

of

curve

remained

in

a

increase occurred at the
the

introduction

of

the

values of trial #4 for the probability P i .
The discussion above is also valid for the curve of the
variance
this

of

the

curve

is

total,

shown on

similar

to

the

Figure

pattern

5b.
of

The

the

pattern

volume

of

curve

(Figure 5 a ).
The minimum

values

its variance

were

Pi,

Pi j k ,

Pi j and

percentage
and

of

obtained

the estimated
with the

respectively.

decrease

its variance

for

are

on

the

both

trials

As

seen

the

largest

total
#1,

on

estimated

for

corresponds

value

on

Table

to
4.

a

reduction
The

of

variance

5 and

Table

10,

total

29%

in

the

volume
The

shown on Table

relation

decreased

and

5 for

each probability.

minimum value for the estimated total volume,
15,

volume

by

to

45.0%

the
in

114
relation

to

the

value

on

Table

4,

for

the

multistage

sampling technique.
The

maximum

its variance
Pi j and

value

were

The

and

its

of

These

increasing

variance,
for

estimated

with trial

P i j k , respectively.

maximum value

Table

the

obtained

highest percentages
volume

for

for

each

total

#2,

trials

volume

and

4 for

Pi ,

3 and

correspond

both the

estimated

probability

the estimated total

to

total

(Table

volume,

the

10).

shown

on

15, corresponds to an increase of 25.0% in relation to

the value

on Table

4. The

increase

in variance,

corresponds

to 43% in relation to the value on Table 4.
5.3

Hardwood Stratum Comparing the results obtained for

the multistage
in

Table

4

multistage

and multiphase

for

the

sampling

hardwood

technique

gave

stratum,

a

much

techniques

presented

it

seen

can

higher

be

estimated

that
total

volume and standard deviation than the multiphase technique.
The
2.5

estimated
times

estimated

total

higher

volume
than

standard

for the

the

multistage

multiphase

deviation

is

1.7

technique

technique
times

is

and

the

larger.

The

coefficient of variation for the multiphase technique on the
other hand,
to

the

is much

lower

higher than that for the multistage

estimated

total

volume.

This

total volume produced in turn an extremely
interval.

The

multistage

technique

the other method.

confidence
is also

interval
large

In both cases,

lower

estimated

large confidence

estimated

but not

due

as

by

large

as

the
for

the information provided by

115
the confidence intervals about the parameter being estimated
is vague.
The multistage
areas

of

higher

technique
The

had

technique
volumes.

an

average

average volume

technique,

of

The

the

multiphase

technique

20 and

average volume per plot

measured

2,202

measured

sample

for

this

cubic

feet/acre.

in the

multiphase

was 1,957 cubic feet/acre.

basal

technique

(Tables

of

the plots

on the other hand,

its tendency to

plots

volume

lower average merchantable
in

showed

area of the plots

as

compared

5 respectively)
and

also

the

to

the

The

measured

multistage

reflects

the

lower

lower estimated

total

volume.
Figure
variable

2a

and

shows

the

the hardwood

the

relationship

predicted

stratum.

Each

secondary unit selected.

volume
point

for
on the

between

the

the

second

graph

size

stage

of

represents

a

As shown there is no secondary unit

that was selected twice as occurred on the other two strata.
The
much

correlation
lower

between

than

the

two

variables

is

correlation

for

the

two

secondary unit

with

the

estimated

smallest

secondary
the

unit

largest

correlation

the

with

the

estimated
between

the

highest
area

predicted
of

lowest
area
two

other

of

is

strata.

The

contained

whereas

volume

hardwood.

variables

and

volume

hardwood,

predicted

negative

the

contained

This

contrary

negative
to

what

would be expected.
The

secondary

the hardwood

unit

that

stratum was the

contained the

largest

area

one that had a large value

of
for

116
its

selection

probability (0.14945).

This

high selection

probability determined a low value for the expansion
for the second stage
volume

for

though

the

the

(1/Pij*tij ) . As a result,

secondary

three plots

unit

in

measured

question

on this

area of

these

three

plots

the predicted
was

low,

even

secondary unit

an average volume of 2,071 cubic feet/acre.
basal

factor

had

The merchantable

varied between

100 and

115

square feet/acre.
At the other extreme,
the

smallest

largest

area

of

predicted

probability

the

the secondary unit that contained
hardwood

volumes

(0.06253).

stratum

because

This

low

had

of its

value

of

one

of

the

low selection
the selection

probability determined a high value for the expansion factor
(1/Pi j *ti j ) which
the

in term

secondary unit

in

determined the

question.

three plots

measured on

2,107 cubic

feet/acre.

the

The

higher volume

average

secondary unit

volume

for

for

the

in question

was

The merchantable basal

area of these

three plots varied between 100 and 115 square feet/acre.
Figure
variable

and

3a

shows

the

the estimated

relationship between the
volume

for the

third

size

stage.

The

correlation is positive but extremely l o w . As in the case of
the other

two

strata,

percent crown closure
low

correlation

is

there are

but with quite different volumes.
associated

among the plot volumes.
used

is evident.

The

several plots with the

with

the

high

same
The

variability

The inadequacy of the size variable

high variability of the

volume

in the

three strata is clear from Figure 3. As with the conifer and

117
red oak

strata,

both

correlations

for

the hardwood

stratum

were not significant at the 0.05 level.
Table 11 presents the effects of changing the selection
probabilities

on the

estimated

variance,

for

the hardwood

performed

as

for the

probability at a time

total

stratum.

other

two

volume

The

its

calculations were

strata,

and keeping the

and

by

others

changing

as

one

originally

cal c ul at ed .
The highest increase in both estimated total volume and
its variance was

obtained with trial

Pi jk

The

(Table

12).

the variance
the

by

multistage

decrease

in

total volume was

occurred

decrease

with

technique

estimated

#4 for the probability
The largest

for the probability
increased

28% as compared to actual
sampling

the

#5

4).

The

for

largest

occurred with

trial

Pi when the volume was reduced

9.7%.

in the estimated variance

trial

13% and

sample values

(Table

total volume

by

#5

for

the

of the total

probability

P i j . The

reduction in the variance was of 17.7%.
The least variations
its

variance

occurred

in the estimated

with

trial

#4

total

for

the

volume and
probability

P i j k . The volume was increased by only 0.7% and the variance
by

1.5%.

observed
estimated
variance
selection

Another
with

trial

total
by

small

1.8%.

variation on

#4

for

volume was
As

probabilities

original probabilities.

the

in

for

estimates

probability

increased

seen

both

by

Appendix
trial #4

0.9%
B,

are

the

was

P i j • The
and

its

values

close

to

of
the

118
Figure
selection
its

4a

and

b

shows

probabilities

variance,

on

the

the

effect

of

estimated

respectively.

Group

changing

total

A

volume

refers

to

the
and
the

combinations of the selection probabilities 1 to 25; group B
to the combinations 26 to 50; group C from 51 to 75; group D
from 76 to 100 and group E from 101 to 125.
The pattern of
the

first

the

five

curve

shown on Figure

combinations

was

probability Pi j ic . As shown in Table
had,

in all

estimated

of the

total

trials,

volume

that the percentage
There

is

sudden

an

for

in

occurred

trial

#5

respectively.

when

As

both

pattern

the

estimated

total

on

trial

estimated
by

noticiable

variation.
volume

the

#5 gave highest
and

probability

and

probability

equations

12,

volume

is

the

the

into

Table

total
the

of

It

from trial to trial.

to

introduced

for

generated

variance.

values

the

increasing both the

character

the

seen

probability Pijk
on

the

were

its

by

12, the probability Pi j k

increase varies

oscillating

increase

variance

of

determined

an effect of

and

4a and b for

5

values
rate of

its

Pijk

and
of

its
Pi jk
6,
the

increase

variance.
repeats

The

The

itself

through the c u r v e s .
The other marke d increases and decreases are due to the
introduction

of

the

into the equations
The start
the

other

to

two

probabilities

estimate the volume

(Pi j

and

Pi)

and its variance.

of each group of combinations is done by changing

probability

Pi.

As

a

result,

just

and Pijk were changed within each group.

the

probability

Pi j

119
As

seen

on

total volume
greater

both

and

than

curves,

the

its variance

those

for

values

for

the

estimated

for group B and E are overall

the

other

three

groups.

These

differences are due to the probability Pi. As shown in Table
12,

the

trials
and

effect

#2

and

its

5

the

is to

values

within these

of

increase

variance. These

the beginning

were

of

trials

the

determined,

groups
as

estimated

Pi

total

for

volume

respectively

to

The pattern of both curves

are similar

stated,

probability

correspond

of groups B and E.
two

the

by

to the other groups

the

probabilities

and

Pi j

and

volume

and

Pi jk .
The

maximum

value

for

its variance did not occur,

the

estimated

as happened on the other strata,

at the same combinations of probabilities.
for the
5 for

estimated

Pi,

Pij

the values

rate

maximum

respectively.

As

three probabilities

of

The maximum value

volume occurred at trials

and Pijk,

for the

the highest
The

total

total

seen

#5,

in Table

at those trials

increase on the estimated total

value of

the

estimated

3 and

total

12,
gave

volume.

volume shown

on

Table 15, corresponds to an increase of 25.8% in relation to
the value on Table 4,
The
total,
Pi,

maximum

for the multistage sampling technique.

value

for

on the other hand,

Pi j

and

at

the

corresponding

those

trials

estimated

maximum

estimated

variance

occurred at trials #2,

Pijk, respectively.

probabilities
increase on

the

value

The

for

the

highest

variance
for

the

(Table
estimated

the

3 and 5 for

values

gave

of

the

three

rate
12).

of
The

variance,

120

shown

in

Table

15,

represent

an

increase

of

49.2%

in

relation to the value on Table 4.
The
and

minimum values

its variance,

for both the

occurred at

probabilities

trials

#4,

respectively.

As

in

trial

#4

gave

total volume
same

the

and

happened

the trials
volume

seen

5

and

Table

highest

its variance.

for

trial

its

#5.

variance,

same

4

12,

volume

Pi,

for

Pi j

of

and

the

Pijk,

the probability

of

both

the

Pi,

estimated

For the probability P i j , the

For

the

probability

increasing

but

total

combination

for

reduction

had the effect of

and

the

estimated

the

Pijk,

the estimated

rate

of

increase

all

total

was

the

lowest.
The value
in Table

15

the value

of the

minimum estimated

represents

in

Table

a reduction of
For

4.

represents a reduction of

the

5.4

General Considerations From

negative
always

2a,

b

and

correlation

occurs

with

general explanation
correlation
volume
caused,

a

the

cases,

may

in relation

the

minimum

value

the value

the
and

impression

the

field

in

predicted

te c h n i q u e .

There

that

volume
is

a

common behavior observed on the
variable
The

and

the

predicted

size of

the primary units

a high variability

for the estimated

area of the stratum contained on the 16 secondary units.
places where

to

the discussions related

variable

size

stage.

shown

technique.

get

multistage

for the

second

in most

one
size

the

between

for the

c,

volume

26.9% in relation to

for the multistage sampling

Figure

18.1%

variance,

Table 4,

to

total

plots were

measured also

The

influenced

121
the

behavior

of

the

curves.

These

effects

were

discussed

separately for each stratum.
For
within
value

the

the

red

primary

for the

possible

oak

stratum,

unit

of

the

selected

estimated

twice,

area of the

within

outside

the

the

limits

four

units

had

a

zero

There are two

the secondary unit was

of private

boundaries of

secondary

stratum.

reasons for the zero value:

located

16

land,

the stratum.

Two

or

was

located

secondary

units

had very low values for the estimated are a of the stratum.
For

the

conifer

stratum,

of

the

within the primary unit selected twice,
zero and
the

one had a very

stratum.

The

low value

reasons

16

secondary

units

seven had an area of

for the estimated area of

for the

zero

area

estimates

were

the same as a b o v e .
As

a result

estimated

area

above strata,
units

was

of

on

the

several

of

the

secondary

units

on

low
the

the total sum of the areas of the 16 secondary

relatively

selection

probabilities

estimated

area

probabilities
stratum.

zero area estimates and of the

of
for

low.

the
units

This

for

secondary

stratum,
with

resulted

low

and

in

units

ve ry

estimated

low

very

high

with

high

selection

area

of

the

These selection probabilities are used to calculate

the expansion factor for the second stage (1/Pij*tij).
If, within a primary unit,

a secondary unit

which has a high selection probability
of the stratum)

is selected

(large estimated area

and another secondary unit is selected which

has a low selection probability

(small

estimated area of the

122

stratum),
quite

the expansion factor

different.

The

first

have a low expansion
probability.

The

for these

secondary

unit

factor because of

second

secondary

two units would be
selected

its higher

unit

would

would

selection

have

a

high

expansion factor because of its lower selection probability.
The

end

largest

result

estimated

predicted
area of
not

would

volume

the

be

area
than

that
of

the

the

stratum.

the

stratum

unit

with

This would

large differences

on

secondary
would

the

happen

the volumes

unit
have

smaller
only

of

the

selected

in

with

if

a

the

lower

estimated
there

were

plots

measured

the

hardwood

within each secondary unit.
Two

primary

stratum.

The

units

f irst

government

property.

units

a zero

had

stratum.
the

were
was

As

completely

a

value

result,
for

the

contained

none

of

the

estimated

within
secondary

area

of

the

Only one secondary unit had a low estimated area of

stratum. Accordingly,

all

the

selection

probabilities

calculated for the 16 secondary units were smaller than 0.1,
and they did not vary as much as in the other strata.
other two

strata,

some

much higher than 0.1
For the

two

hardwood-stratum
the
was

(e.g.,

primary

observed (Figure
correspond

units

selected

u n i t , the

the stratum

2a).
to

selection probabilities

were

in the red oak stratum).

secondary

estimated area of

volumes

of the

On the

The

the

two
two

within the

proportionality
and

between

the predicted

points
secondary

with

the

units

first

volume
highest

selected

within the first primary u n i t . The secondary unit with small

123
estimated
volume,

area

and

of

stratum

vice-versa.

because of the
selection

the

This

probabilities.

has

to

occur

primary unit.

It

among

a

small

proportionality

should

units

may

be

secondary

was

and

units

that

withi n

of

the

the predicted

between the

independently

observed

the calculated

a

single

size variable

among secondary units

occur

predicted

noted

size variable

The proportionality

and the predicted volume
primary

had

lower variability found among

proportionality between the
volume

also

of different

the

variation

existing on the selection probabilities.
For

the

two

primary unit,
of the

secondary

can also be observed.

selected

secondary
had
of

zero
the

units

within

stratum.

the

second

in

This

probabilities.

those

the

selected

the

a

Several

the

primary
second

a relatively

produced

very

The two points located on the

second

four had

selection

0.1.

the

although not

2a correspond to

contained

area and

for

withi n

the predicted volume,

lower portion of Figure
units

chosen

the proportionality between the estimated area

stratum and

evident,

units

large

two

secondary

u n i t . Of
primary

one

low estimated

area

variation

units,

16

unit,

among

probabilities,

secondary

the

had

the

including

values

above

The consequence was that the expansion factor for these

secondary units were the lowest among the four.
these

secondary

units

had

the

lowest

As a result,

predicted

volumes

although they had large estimated areas.
One
has to be

conclusion

from

the

above

discussion

is

that

one

alert when choosing the size of the primary units

124
in situations where stratification is necessary.

The stratum

with the smallest area should be used as a guide in order to
define

the

most

this

is

size

variables

volume

done

appropriate

carefully,

for

not

a

the

being

stage

size

for

the

problems

primary

that

proportional

may

be

result

to

avoided

the

and

unit.
from

If
the

predicted

more

precise

estimations may be obtained.
From the discussions
the

selection

and

its variance,

probabilities
there

be noted,

however,

the

stage

first

related
on

is no

to the effect

the

estimated

consistent

of changing
total

effect.

It

volume
should

that if the change in the probability for

( P i ) determined an

increase

or decrease

on

the estimated total volume of x % , the variance was increased
or decreased
hand,
third

if

by approximately the same amount.

the

change

stages,

Pi j

decrease

in

would

increased

be

indication
selection

the

that

in the
and

estimate
by

the

probability

Pijk, determined
total

volume

approximately
size

probabilities

for

variables
for

the

2x%.

the
an

of

x%,
This

used

to

second

On the other

and

second
increase

the

and
or

variance

aspect

is

calculate
third

an
the

stages

should be measured as accurately as possible.
Of course
or the variance
parameters

are

in practice,
is
not

no

one would know if the volume

being over or under
known.

But one

estimated

should

be

since the

aware

of the

effect that an error on the measurement of the size variable
might cause on the estimated total volume and its variance.

125
5.5

Population Estimates Comparing the results in Table

4 it can be
a

much

seen that

higher

multiphase

the multistage

estimate

method.

technique has produced

of

the

total

volume

Actually,

the

difference

than

is

the

almost

a

factor of five times.
According
can

be

to

used

population

Langley

in

is

(1975),

situations

necessary.

independently

volumes

for the population

of the estimated
volume

on each

volume

stratum

estimated

population

estimated

variance

multiphase

on

technique,

rather than a

stratum (Johnston,
The
volume
for

on

the

major

be

for each

given

stratum.

is

each
on

the

technique

by

other

the

method

is

total

by the

summation

As

estimated

the

variance
the

(De

of

estimated

the

given

stratum

the

The

is independent,

of

sum

Vries,

hand,

the

of

the

1986).

The

is

based

on

the difference being that a two-

simple

random sample

is

taken

in each

1982).

contributor

the multistage
hardwood

estimated total

cases,

stratum.

would

total

for

multistage

stratification

such

each

stratified random sampling,
phase,

where

In

applied

the

to

technique

stratum.

volume

for

by the multistage method

the

As

higher
is

total

the estimated

seen

from

the hardwood

is 2.5

estimated

Table

stratum

volume
4,

the

calculated

times larger than the value

calculated by the multiphase technique.
The

estimated

reflects

the

average

volume

total

availability
per

acre

volume

for

of

pine,

red
than

the

jack

conifer

which
pine.

has

stratum
a

This

higher
also

126
contributed

to

the

by the multistage
total volume

larger

estimated population

technique.

for the

The multiphase

conifer

stratum was

measured on both conifer types.
average

volume

average

per

acre

estimated

consequently,
Had the

volume

estimated

from

order

magnitude

technique,

red

for

given

estimate of the

composed of

plots

jack pine has a lower

pine,

the

this

conifer

lowered
stratum

the
and,

its estimated total volume.

calculated
of

than

Since

total

the

the

total

volume for the

multistage
as

the

difference

technique
value

between

red oak

been

for

the

of

the

stratum

the

same

multiphase

estimated

population

total for the methods would have been larger.
Another

aspect

to

estimated population
simply

the

stratum.

consider

total

summation

of

for

is

the

fact

the multistage

the

estimated

that

the

technique

total

for

each

In the multiphase t e ch ni qu e, on the other h a n d , the

estimate of

the population total is obtained

the results

from equation 18 by the estimated total area.

is not

just

stratum.
weight

is

a

simple

addition

of

the

Equation 18 has the term wi

- which

is

a reduction

by multiplying

estimated

volume

It
per

= Ni/N - called stratum

f a c t o r , to

compensate

by the

size of the stratum.
In r e a l i t y , as
is

not

known,

population

it

total

long as the parameter of the population
is

difficult

obtained

by

the

to

say

if

multistage

the

estimated

technique

is

overestimating the population volume or if the estimation of
the multiphase

is u n d er e st im at in g. But what can

be said

is

127
that,
more

although
precise

multistage

for

the

the

The

positive

certainly had

an effect

This

correlation
on the

should

similar

be

forest types,

given a

for the second
in

the

in forest

the

in

stage

population
lower

given by the

inventories

multiphase

first consideration

the

stage

the

total volume

is
the

third

determined

As a result,

for

between

higher estimated

of the

total

than

correlation

higher variance

multistage technique.
of

population

technique

negative

precision for the estimate

areas

estimated

and the predicted volume

low

variance.

the

multiphase

technique.

size variable
and

smaller,

in

technique

the

process

of

selecting a multilevel sampling technique.
The
evaluated
Forest

results obtained
can

not

Service.

county level.
as a result,

be

sites.
cover

digital
These

types

training

sites

interest

(red

were
and

areas

composed of a sample
the numerical
each

surveys

Digital

agricultural

feature

with

those

statistics

techniques

from
are

the

U.S.

accurate

study area was composed of two

classification of

various

sampling

at

townships,

this comparison could not be recommended.

5.6

training

both

compared

Forest

The

by

Image

Processing

imagery

implies

are
of

obtained
jack
and

water.

of pixels

of

In

from

pines,

representation
type

the

"representative"

interest.

the

the

whose

of the

interest.

areas

of

present

and

training
brightness

the

study,

types
red

of

of

oak),

sites

were

values

are

attributes

for

classification

is

spectral
The

supervised

selection

forest

hardwood
These

The

128
ultimately based on the statistics obtained from the various
training sites selected.
It is of interest for the image analyst to evaluate the
spectral
using

"separability" of the various cover types selected,

different

evaluation,
discriminate
graphical

combinations

of

which

helps

select

a

the

classes,

can

means

(Jensen,

be

1986).

restricted

to

two-dimensional

because

its

simplicity

of

spectral
set

of

done

plots,

statistical

procedure,

that

graphical

will

best

be

or

the

by

method,

presented

of understanding.

graphs are normally called feature space.
use of the

bands

This

statistically

Only the

and ease

bands.

These

For details on the

reader

is referred to

the above citation and to Goodenough gi „al, 1974.
Figures
possible

for

brightness

plots

were

within

through

two-band

b a n d s , and
The

7

each

show

combinations

the

first

values

obtained
of

13

the

and

used
by

a

the

feature

of

the

second

to

construct

sites

four

the

all

LANDSAT

components.

feature

sampling

selected

for

MSS

principal

systematic

training

space

on

of

space
pixels

the

study

a r e a . The number of pixels selected was proportional to the
size of each training s i t e .
The
obtain

feature
valuable

space

plot

insights

may

allow

into

the

the

interpreter

structure

of

to
the

multispectral data s e t . It may also provide an indication of
incorrectly chosen pixels

(Donker ei a l . 1977 ) .

129

120— |

105-

Band

60-

Brightness

75-

Values

5

90-

□

AGRICULTURE

a

RED PINE

o

JACK PINE

*
v

HARDWOOD
WATER

+

RED OAK

□

45-

0
□

□

30-

15-

i

0—

1

0

r

□

I**

T---- !---- 1---- 1
---- 1
---- 1
---- 1

16

32

Brightness Values Band 4
Figure 7

Feature Space f o r Bands 4 and 5

48

130

120— |

105-

A

RED PINE

O

JACK PINE

*

HARDWOOD

V

WATER

+

RED OAK
AGRICULTURE

Band

6

90-

□
+++**

75-

Brightness

Values

□
□

50-

+*

* *

□
□

B

□

45A2
a2

\A“ “0 □

30-

$
o

15V
V

0—

T

0

I

|

16

I

I

I

j

I

I

32

Brightness Values Band 4
Figure 8

Feature Space f o r B ands 4 and 6

I

]

48

131

1 20—j

□
**

a

*1611 *

0
*
v

105-

Brightness

Values

Band

7

90-

AGRICULTURE
RED PINE
JACK PINE
HARDWOOD
WATER

+
++

RED OAK

iX

75-

**

□
□

60-

A

a Aa
A

d d o o B lj
o

a!

d d EE

6

45-

a a ^I

□
□

□□
d

n
D

□□

§□□□□
D

VVW O

30-

15-

0

—

—

0

i—

16

i—

i—

i—

|—

i—

i—

32

Brightness Values Band 4
Figure 9

F eature Space fo r Bands 4 and 7

i—

i

48

132

1 2 0 —i

105* *
*

Brightness

Values

Band

6

90-

A

RED PINE

0

JACK PINE

*

HARDWOOD

V

WATER

+
□

RED OAK
AGRICULTURE

+

75'

+
* +
*

*

* *
+*
*

60'

□

□
□

□ □
□

m

□

□

a*
o

45

'A

A

ffS

□

0

A

30-

15
v
0

T"

0

14

28

42

Brightness Values Band 5
Figure 10

Feature Space f o r Bands 5 and 6

56

133

12 0 — |
*
*

105-

HARDWOOD

A

RED PINE

O

JACK PINE

□

AGRICULTURE

V

WATER
RED OAK

*
* *
+
* * *

907

J

i I +

+

+++ *

Band

*

+ * 1 *

75-

* *

Brightness

Values

+*
□

60-

□

cn

&

□
□

□

□

□

□ □ cPS

AM

45-

□

A

|

A ^ A E f P
W
^A^>
a
W

30-

V 0

&
V

V

15-

0—
0

14

28

42

Brightness Values Band 5
Figure 11

Feature Space f o r Bands 5 and 7

56

134

120i

105

Band

7

90

A

RED PINE

o

JACK PINE

*

HARDWOOD

V

WATER

+

RED OAK
AGRICULTURE

□

*
*
*
* *4*1

75

/T
V
T
s/T
V
T
S /IVT>

HAKf
HXOKft
*
sv
T7KT 1
1
**
*
*
*
*
** □

*

Values

+

Brightness

*

vxKiXlKP ^1/
?1
X

*

1

>?

60'

45-

30<$>

15'

1 1 1 1 1 1 1

0-

0

15

30

|

1 1 1 1 1 1

45

60

75

, ,
1 1

90

1 1 1 1 1

105

Brightness Values Band 6

Figure 12

F e a tu re Space fo r Bands 6 and 7

120

135

140
□□□ □□□□□
□
□ □□
□
□□
□□□□
□□□□

□ □
□
□

□

□□ □
□ □□
□

130

+
**

*

2

w
w ^ v y
w
/ I \ /TV /TV /TV /T \ /TV /TV 7N

*+

Principal Component

**
*

o
O

120

****

A
AA
A AAA

O
AAA
*(7 A A A

S7 A A A
V

A

110

+

RED OAK

v

WATER

□

AGRICULTURE

o

JACK PINE

a

RED PINE

*

100

“i

i

i

i

50

|

75

i—

i—

i—

i—

|—

i—

i—

100

i—

i—

|—

HARDWOOD
i—

i—

125

i—

i—

j—

150

i—

i—

i—

i—

175

Principal Component 1
Figure 13

Feature Space f o r Principal C o m p o n e n ts

1 and 2

|

136
Figures

7

and

12

illustrate

the

marked

correlation

between the two visible-light bands 4 and 5 and the two near
infrared bands
much

6 and 7.

data

is

higher

the

agricultural

In Figure

than

in

areas

12, the dispersion of

Figure

have

7.

an

Figure

7 shows

elongated

the
that

green-red

reflectance cluster which overlays somewhat the hardwood 4 5 feature

space.

areas

can

also

bands

6

and

The

be
7

elongated cluster

observed

(Figure

in

12)

difficult to discriminate

Figures

of the
8,

9,

11.

On

areas would

be

c o n i f e r s , especially

red

agricultural

from the

agricultural

10

and

pine.
On

the

(Figure 13),
easily

principal

component

agricultural

because

of their

areas
high

analysis

of

the

imagery

would be discriminated very

brightness

values.

Note

the

total lack of correlation between principal components 1 and
2.

This

is

a

characteristic

of

this

type

of

statistical

transformation.
In
present

terms

of the

forest

p r o j e c t , forest and

differentiated

in

general clusters

the

types

ofinterest

non-forest

s c e n e . Within

the

for

the

areas could
forest

could be s e p a r a t e d , hardwood

group,

be
two

and c o n i f e r s .

This would warrant the division of the area into two s t r a t a .
In bands

4 and

5 (Figure

d i f f i c u l t , if not
the

infrared

portion

bands,

this

separation would be very

i m p os si b le . Notice that when the bands

i n c l u d e d , the clusters
infrared

7),

band

of

7)

are

became more e v i d e n t . Between the

two

7

the

gives

spectrum

a

(bands

better

6

and

in

separation

of

the

137
hardwood and
by

conifer

comparing

clusters

Figures

8

and

than band

6. This

9,

band

where

can be seen

4

is

plotted

against bands 6 and 7, respectively and in Figures 10 and 11
where band 5 is plotted against bands 6 and 7, respectively.
In terms of subdividing the hardwood and conifer strata
into hardwood verses
some problems

red oak and red pine verses jack pine,

may occur.

Obviously,

always intermingled within the
pixels.
the

the

red

same cluster

oak pixels

are

as the hardwood

Although this spectral overlap is to be expected,

printed

pixels that
spatial

number

map

composed

it

was

possible

the red oak stand.

homogeneity,

the

samples

to

identify

on
the

As a result of this

could

be

selected

and

measured.
For

the

conifers,

pixels that are mixed,
a

separate,

albeit

red

small,
where

present

9,

8,

and

jack

pine

have

some

but some of the jack pine pixels form

mainly on the plots
(Figures

pine

cluster.
the

10,

This

infrared

11 and

12).

can

bands
The

be

detected

(6 and

best

7) are

separation

between these two conifer species would be obtained by using
the principal
run

the

species

component

imagery,

classification.
form

distinct

As

mainly PC

seen

clusters

on

with

1 and

Figure
the

PC

13,

red

2,

the

pine

to
two

having

brighter values than the jack pine.
One
the

aspect

cluster

reflected
Kiefer,

of

to be

noticed

on all

water

pixels.

Water

infrared

portion

1987 ). Bands

6

and

of

the

7 are

of the plots
absorbs

spectrum

located

shown

energy

is

in

the

(Lillesand

and

in this

region of

138
the

spectrum

and,

as

a

result,

very low brightness values.
there

are

some

c l u s t e r . In
is

an

water

Figure

example

sites

which

water

bodies

small.

As

of

were

this

not

that

in

the

some

selected

pixels

might

have

the

giving

much

higher

pond

were
water

within

result,

show

zero

or

with

the

conifer

particularly n o t i c i a b l e . This

"pure"

located

a

intermingled

is

pixels

should

As seen on Figures 8 through 12,

pixels

7

water

selected
brightness

study

of

been

for

area

them,

a

located

brightness

training

values.

are

relatively

portion

on

The

the

values.

of

the

margins

of

As

in

seen

Figure 13, jack pine and water could not be separated if the
principal

components

classification.
between

jack

occurred
printed

Even
pine

with

using
and

the

number

1

and

2

the

four

bands,

would

also

water

hardwood

map,

it

was

and

were

red

possible

used

to

some
be

oak

for

confusion

present.

strata,

identify

the

in

stands

As
the
of

red pine and jack pine and to select specific samples.
The

problem

classes exists,

that

occurs

as in the present case,

omission and commission errors.
pixel

is

errors

not assigned

occur

when

does not belong
Based

only

when

a

to the
pixel

(Jensen,
on

the

overlap

between

two

is the occurrence of

Omission errors occur when a
appropriate

is

class.

assigned

to

space

plots

a

Commission

class

that

it

1986).
feature

shown,

one

can

expect that the percentage of omission and commission errors
might have been large on the classification performed on the
study area.

At

this

point

a question arises:

What would be

139
the effect of these errors on the estimated total volume and
its variance for the multiphase sampling technique?
The
units
are

numeric

results,

(pixels),

used

on

obtained

equations

used respectively
estimated

the

its variance
Just

terms

from

the

16,

18

of

number

computer

and

19.

of

sampling

classification

These

equations

are

for the estimation of the variance of the

population

(v(yi.. ));

in

mean

estimation

for
of

the

ith

population

the population

mean

stratum

(yst)

and

(v(ys t)).

for the

sake

of a quick reference,

the

number of

sampling units shown in Table 1 for the conifer and hardwood
strata

were

changed

original values
16 were
mean

increasing

by 5% and 10%.

its
In

variance

other

were

words,

increased

or

decreased

equations

18

and

19

were

but

the
or

also

decreasing

the

given by equation

the estimated population

changed

if
5%

and

The values

only slightly changed,

and

amount.

by

by

roughly

numbers
10%,

the

increased

on

the

Table

values
or

same

1

were

given

by

decreased

by

approximately the same amount.
This was
are

used

stratum.

to
The

N constant,

to

be

expected

since

numbers

fraction

on

Table

the

result

increasing or decreasing N i , keeping

can be vis ualized by examining

(wi ) for

1

calculate
of

sampling

the

each

equations 18 and

19.
It can
depending

on

be noticed
the

different results

from the

algorithm

discussion presented

used

for

the

that,

classification,

could be obtained for the estimated

total

140
volume

and

its

variance.

calculated total
since

the

choice
order

volume and

population

of

the

to

as

errors.

one

cannot

its variance

parameter

classifier

reduce

commission

Of course,

has

much

values

to

as

Previous

be

if

the

is accurate or not
are

done

possible

experience

say

not

known.

The

appropriately
the

in

omission

with

the

and

classifier

on the area of interest is important to consider when making
the selection.
5.7
total

Time

time

in

Measurements

hours

necessary

sampling procedures,
only

1.6 hours.

This

relation to the
difference,
method.

Tables
can

The

reason

for

this

by

hours

lower

price

was
per

the price

for

hour.

computer

w a s n ’t

only

imagery.

It

also

used

within

the

units

was

(pixels)

factor

which

greatly

must

used

for
to

study

inflated

computing

hours

required

to

on

human hours

and

of

the

$29.00.

28.7% of

which

has

the

selected

For

this

computer

contained

tape.
classify

The

total

the

the
much

that

the

of

the

sampling

study,

time
the

On

a

classification

area.

scene

the

emphasized

locate

from

full

in

in

hour

time

the

window that

the

of

for each

because

technique,

be

selection of the data
within

insignificant

is

per

computer

It

as

hours were

the multiphase

used

each

shows a difference

difference

multiplied

total

of

cost

these were

for

execution

of

between the total

the total

hand,

comparison

A significant

technique,

other

A

for both methods.

multistage

the

the

be considered

exists
for

Costs

8 and 21,

total hours

however,

and

one

was

study

the
area

number

scene

can

of
be

the

141
significantly

reduced

if

the

operator

has

experience

in

working with the equipment.
By

analyzing

stratum,

the

expected.

Table

8

work

had

field

For the conifer

the selection
hours used

of

for

first

field

conifer

stratum was

As more

experience

it

is

the

seen
highest

stratum,

the

stage sampling

work.
the

was

The

that,
cost

total
units

reason

the

as

each

would

hours used

be
for

is close to the

for this

first one measured
gained,

within

is that the

in the

total hours

process.

used for the

selection of first stage sampling units decreased.
The

time

used

for

sampling units

also

varied among the

is

with

associated

selected.

The

the

located

the

As

a

selection

place

primary

stratum were
stratum.

the

units

on

the

strata.

were

the

the

second

for

spent

unit

the

homogeneous

time

stage

This variation

primary

selected

relatively

result,

of

on

was

hardwood

portions

measuring

of
the

area of the stratum contained within each secondary unit was
low.

This

strata.
the

was

the

case

The measurement

secondary

parts

not

of

units

other

of

had

strata

for

the

to

the

conifer

area of

be

done

(hardwood,

each

more

and

red

stratum

within

carefully

agricultural

oak

since

areas)

were

also contained within the secondary units.
The

selection

relatively large
in the

unit.

the

third

number of hours.

evaluation

121 tertiary

of

of

the

percent

sampling units,

stage
Most

units
of the

crown

within

also

used

a

time was used

closure

each selected

within

the

secondary

142
With

the

field

measuring plots
for travel

measurements,

the

time

the

between plots.

This is because most

time

and

stratum,

travel

hardwood

was

the

occurred with the
was

a

The

located

in one area. As

on an

stratum.

difference

in only

in

the

time

the

difficult
as

stratum,
used

for

time. The red oak stratum

one portion

time

of more

the red oak

of the

the selected plots were relatively close.
larger difference

within

in only one place,

For

For

measurement

plots

area

between

measuring the plots and for travel
is concentrated

between

selected

not concentrated

conifer

large

difference

small.

stratum were

access and also were

there

of the plots

the time for moving between plots was reduced.

hardwood

for

in the conifer stratum was greater than that

within the conifer stratum were concentrated
a result,

used

was the

fact

study area,

and

The reason for the
that

the

stand

had

been thinned recently and it was extremely difficult to walk
due to the slash left on the ground.
As related

to the

multiphase

technique

(Table

21),

it

can be seen that the time used for sampling selection on the
hardwood
stated,
map.

and
the

The

conifer

selection of

first

step

clusters of pixels
of interest.
and the
process,

strata

in

is relatively

the samples
this

high.

was done

process

was

to

As

already

on the number
identify

that were classified w it hin each

the

stratum

The coordinates of the clusters were identified

selection process was started.

During

the

selection

a large number of selected sampling units had to be

143
discarded

because

they did

not

fall within

the

forest

type

of interest.
The

hours

required

multiphase method
measured.

Table

total time
and jack
same

required

for

move

twelve

the

ten for the

were

moving
time,

that,

plots.

between

fewer

in

the

samples

were
the

the

fewer

(16.7

for

the

within the

the

jack

measuring

for the

+he

multistage
conifer

pine

plots

technique

stratum,
the

and

red

pine
This

time needed to

technique.

multiphase

The

stratum).

larger

the

hours).

(five within

multistage
the

was practically

method

For

in part,

plots,

even though

hours)

technique

within

for

since

time used

measured

five

plots

measurements

for the conifer stratum,

(16.1

in the

difference accounts,
measure

less,

multistage

multiphase

and

field

for measuring the plots on the red pine

between

plots

stratum

shows

pine plantations

as

to

were much

21

difference occurs
and

for

In

terms

technique

plots were measured.

used

The

of

more

reason

is

that the multistage technique tends to concentrate the plots
in one

region.

randomly
tendency
used

In

selected
for

them

the

multiphase

within
to

to move between

be

each

spread

plots

technique,
stratum

out.

tends

the

and

As a

plots

there

result,

is

the

to be larger than

are
a

time

for the

multistage technique.
The difference
plots

and

hardwood

to

move

stratum.

between the time used
between
Besides

plots
the

is

for measuring

quite

distance

large

between

the

for

the
the

plots,

144
some of them were located in areas of difficult access which
increased the time to reach the plot.
For the red oak stratum,

the time used in measuring the

plots and in moving between plots were practically the same.
Of the
phase

six plots measured within this stratum by the mu lt i­
technique,

thinned.
access

The
and

reduction

two were

other
were

of

both

located

plots

not

were

in

an

located

thinned. This

the

time

area
in

that

areas of

contributed

for measuring

and

was
easy

to

the

the time

for

moving between plots.
The average time required for measuring the 36 plots in
the

multistage

minutes.

For

measuring
minutes.

considered

as

was

the multiphase

the
The

technique

21

plots

of the present work,

of

of

significant,

one

hour

technique,

was

difference

of

one

only

as

and

fifty

the average

hour
six

time

and

fourty

minutes

can

a result,

for the

both methods were equivalent

four
for

eight
not be

conditions
in

terms

of the time spent for field work.
5.8

Number

of

Samples

The equations

presented

in

Appendix A are used to determine the number of samples to be
measured for each method.
determination
measured
(Langley,
minimizing

at

of

the

each

1975).

For the multistage

required

stage

is

Optimum

variance

for

number

based

on

allocation
a

specific

cost for a specific variance.

of

technique,
samples

optimum
can

cost

be
or

to

the
be

allocation
achieved

by

by

minimizing

145

For

the

multiphase technique

calculate the
the

sampling

second phases,

sampling
one

st ra ta

and

proportional
stratum.

fractions

fractions

the

for

assumes

The calculation of only two

the

for
of

the

second

phase,

sampling

units

discussed

sample
first

criterion
the

as

three

size
and

n

second

given

above.

sampling

are

and

unequal

that

the

This

last

sampling

techniques
In

specif ied,

case

fractions

requires

calculations

to
of

21
to

classification.

third

unequal

use

of

calcula tio n of

the

costs

to

on

both

the

related

sampling
three

This

eq uation

can be

first

on

the

approach

equal

approach
and

and

fractions
assumes

unspecified.

non-linear

programing

costs

fractions

the

u

sections.

and

The

associated

the

refer to the
u

associated

with

the

one

can

v,

first
with

section
computer

considered as the cost Cl

calculate

and v.

equations
to

the

into

second section would

costs

sampling

sampling

are

the

estimate
the

Table

refer

C2

the

of

to solve the problem.

order

subdivide

and

ways

are

the

each

, 1983.

based

The

fractions

The second assumes that

in

explicit

phases,

phase

implies

possible

and

unspecified.

costs

first and

four

the

that

first

one for each of the ith

(1979)

optimality

the

the

This treatment is given by Johnston g £

determining

on

necessary to

instead of

allocation

Frayer

would

is

u and v for

respectively.

fractions

it

on

sampling
costs

These

conifer

and

u.

The

associated with the

may be

Appendix

fraction

shown

A,

the

referred
and

as

the

include

the

hardwood

strata,

146
presented in Table
costs

associated

Appendix A,
and

would

21.

The third

with

the

section would

calculation

of

refer to the

v.

As

seen

in

these costs would be associated with the term C3
refer

to

the

field

and

area

measurements

costs,

for each of the four strata.
The

calculations

of

the

optimum

number

each stage

of the multistage technique,

fractions

for

possible

the

on the

multiphase

present

performed

by one person

variances

are

large

case.

on a small

and

the

costs

samples

and of the

technique,

This

of

scale.
are

sampling

would

is because
As

on

not

be

the work was
the estimated

rather

small,

the

number of samples to be measured for each method,

calculated

by applying

realistic.

the

The number of
arbitrarily

equations

sample

on Appendix A,

plots measured

established,

based

available for the field work.

on

for
the

are not

each technique
time

and

was

funds

CHAPTER 6
SUMMARY AND CONCLUSIONS

This

s t u d y ’s

multilevel
northern

main

sampling

Michigan.

A

objective

was

procedures

for

multistage

and

to

evaluate

forest

two

inventory

a multiphase

in

sampling

technique were evaluated.
Multistage
also

referred

to size
area

sampling

to

as

sampling with

(P P S ). This

of

uses unequal

method

interest.

sampling

and

uses

analog

sampling

imagery for the area of interest;
random

probability

requires

Multiphase

pr obability

two

sampling,

proportional

imagery
requires

for

the

digital

it is based on stratified
phase

sample

rather

simple random sampling in each stratum (Johnston,

than

1982).

The following conclusions should be emphasized:
a)

The

population
multiphase

multistage

total

that

technique.

the high variability
which
some

resulted
of

the

in

were

This

5

(e.g.

produced estimates

times

discrepancy

in the volume

different

strata

technique

larger

is basically

found

estimates
hardwood).

than

in the

of

total

The

due

of
the
to

population,
volume

differences

for
in

structure of both methods also contributed to differences in
estimated population totals.

the

148
b) Although the results must be viewed with caution due
to

the

large

sizes,

variances

the

multiphase

estimating

the

multistage

technique.

precise

for

of

all

and

technique

population

estimates

technique

encountered

total

The

was

and

total

costly

than

stratum,

sample

precise

technique

volume

individual

small

more

less

multiphase

the

the

than

gave

the

in
the
more

multistage

except

for

the

hardwood stratum.
c)
the

Contrary

the

size

to

expectations,

variable

and

the

the

correlation

predicted

volume

between
for

the

second stage of the multistage technique was negative.
is

an

indication

method,
the

that,

estimated total

secondary

sampling

in

the

present

application

area occupied by forest
unit

is

not

an

This

of

the

type within

appropriate

size

variable to calculate the selection probabilities.
d)

The

size

variable

selection probabilities
closure)

was

not

between percent

used

for

for

the

appropriate

crown closure

the

third

calculation

stage

either.

(percent

The

significant.

of such correlations

the

crown

correlations

and estimated volumes

third stage were positive for all three strata,
But the values

of

for the

as expected.

were low and not

highly

A high percent crown closure is not necessarily

associated with high plot volume.
e) Although the correlations
and the predicted volume
for

all

strata,

the

for the
second

between the
second
stage

size variable

stage were negative
gave

the

smallest

149
contribution to the variance

of

the estimated

total

at

all

strata.
f) The changing of the selection probabilities for each
stage

of

the

total

volume

multistage
and

its

probabilities are
estimate
the

those

effect

estimated

of

an

technique

variance.
integral

parameters.
changing

total

volume

This

occurred

rule

can

selection

and

the

estimated

because

its

be

established

probabilities

variance,

the

second

and

third

stages,

probabilities at these

stages

variance

than

of

the

total

since

to
for

on

however.

should be taken when measuring the size variables,
for

the

part of the equations used

No

the

influenced

the
Care

specially

changes

in

the

reflect more on the estimated

the

probabilities

of

the

first

the primary

sampling unit

(P S U ) used

stage.

the

g) The

size of

present

study

reasons:

was

in secondary

for the size variable;
P S U ’s

several
size

contained

secondary

variables.

variability
variation,
the

size

stage.
the

convenient

for

the

following

it produced only two P S U ’s for the red oak stratum;

it resulted

all

not

in

in

variable

There

primary

is

no

units.

conifer

sampling

the

units

having

zero

and for the conifer stratum,

some

These

in turn,

sampling

units

zero

plantations,
with

readings

s t r a t a ’s

zero
in

selection

it

although
produced

values

turn

values

for

caused

the

great

probabilities.

This

caused the negative correlations between
and

the

predicted

volume

rule of

thumb

In

application

each

for

to determine
of

the

the
the

second
size

method,

of
one

150
has to evaluate

the

the

stages

number

of

most

being

imagery at each stage,
k)

The

use

appropriate

utilized,

of

the

principal

study area

scene would

both species

of conifer

could be

agricultural

clusters.

Although

areas

the

imagery

combinations,
number

map

between

the
of

jack

and

inventory

depends,

among

use

of

large

remotely

photographs
necessary.
advantage

The
the

the

been

imagery
better

formed

between

to

since
and

separate

red

oak

and

not only on the principal

on

all

the

be

scene.

would

other

identified
The

not

sampling
other

be

band
on

the

discrimination
possible

Multilevel

areas are to be
imagery

more

recently,

methods

evaluated

use

procedure

things,

sensed

and,

of

a

the method.

when

of

on

the

imagery.
of

very useful

have

stand could

water

selection

experience with

scale

component

hardwoods

classified

and

principal component
The

also

red oak

pine

the

discrimination

but

the

a function of

better discriminated,

mixed hardwood was very difficult,
component

as

and the size of the field plots.

classify the

because

size

of

remotely

in

on

for

familiarity

sampling

methods

inventoried,
the

form

satellite
by

this

sensed

forest

and

of

research
in

are
the

aerial

imagery

imagery

and

is
take

either

its analog or digital form.
The
available
taking

imagery used
at

the

large-scale

for

start

of

aerial

both methods was
the

project.

photographs

No

were

photographs are very helpful for multilevel

simply what

was

provisions

for

available.
sampling.

Such

151
The

scales

influence

the

stage

units has
stage).

an

size

is

scale of the

for

of the

the

size

the

size

is

of the

of

the

also

increase

convenient

to

level of

and

measure

imagery,

variable may be difficult.

had

sampling

such as

primary

the

A

depending

This evaluation

(last

size of

vice-versa.

measurement

in

sampling

field plots

but,

an

In situations

of the P S U ’s is increased,

will

last

study

fixed,

influence on the size of the

If the

plot

used

levels of imagery

application,

field plots

field

imagery

of the multistage method.

the number of

present

the

the

on the determination

units at each
where

of

small
on

of the

the
size

- size of PSU vs

size of field plots - has to be done carefully in situations
were

stratification

necessary.

The

considered

first

of

the

smallest
and

area

stratum

the

best

to
in

each

stage

defining
is

the

the

variation

information is available,
enough primary
stratum

to

sampling

ensure

size

the

area

of
in

should

selection

should

relative

be

to

the

another variable to be
the

sampling

the

volume.

or can be anticipated
units

is

should be taken.

Besides the size of the strata,
in

inventoried

the

judgement,

size of the primary sampling units,

considered

be

of

be
the

units
If

at

this

in some way,

placed within
required

each

number

in

order to attain the specified precision of the estimate.

If,

on

the

the

size
for

other

hand,

of the primary
more

than

two

such

information

sampling
P S U ’s

to

the strata of smaller areas.

units
be

is

not

should be

completely

available,

such to allow

contained

within

152
The

correlation

estimated

volume

analyzed
those
all

in

for

some

happen.

strata,
It

the

detail.

two variables

three

between

is

for

size

second

Although
the

this

the

probable

the

third

stage

are

in

and

the

stages

was

correlations
were

not mean that
that

secondary sampling units

and

second

does

variable

negative

this will

situations

larger,

between
for

always

where

a positive

the

correlation

will be established.
Another
variable

aspect

"area"

would be the

is

to

inferior

ideal

superior

to

assessors

(Loetsch

consider

is

volume

e..t

fact

to an estimate

size variable

ocular

the

to use.

An

the

of volume,
"Area"

estimates

, 1963).

that

by

size
which

however

is

inexperienced

alternative

to

percent

crown closure for use

if multistage sampling is chosen could

be

is

crown

percent
1963).

area.
crown

When the

emergent
to be

This
cover

by

scale

canopies

is

the

could

available

by

stratum

of the

reinventoried and

area

obtained

also

multiplying
area

the

(Loetsch

mean

g.£.

imagery permits,

the number

be tried.

area

If an

a.l,
of

is going

informations on the volume per unit
from

previous

inventories,

informations can be used as size variables.

these

The advantage of

using past informations is their highly correlation with the
variable of interest
The
rather

multistage

complex

comments,
important

(new estimation of volume).

the

sampling

structure
definition

effect

on

technique

and,
of

the

as
the

seen
size

results

evaluated
from

of

the

obtained

the
PSU

has

a

previous
has

mainly

an
in

153
situations

where

aspects for
the

stratification

future

multistage

where

recommended

to

P S U ’s,

the

when

the

area

is

done.

Some

research could be pointed out related to

technique:

conditions

of

perform

simulated

stratification

define

the

number

most

by

of

in

types

is

forest

appropriate

of levels

studies

size

imagery

is

for

the

fixed.

The

same study could be done but,

without the limitations on the

number

in

of

levels

of

imagery

order

to

define

both,

the

size of the PSU and the most economic number of stages to be
considered.
could

be

For a not
evaluated

so complex
in

a

study,

situation

the

when

last
the

homogeneous or no stratification is required.
of

alternative

size

variables

(not

condition
forest

is

The evaluation

predicted

volume)

may

also be considered as another line of study.
The application of the multistage
where
some

stratification
limitations

instance

the

of the

to

the

conifer

area

use

stratum

of
on

is

method

in situations

recommended may

the

method.

the

impose

Consider

present

study.

for
This

stratum was formed of small plantations of red and jack pine
spread

all

over the

study area and of

large

plantations

of

red pine concentrated in the south east portion of the area.
Due

to

this

characteristic

of

this

stratum,

it

was

considered as one and not subdivided into red and jack pine.
The field measurements were made on the large plantations of
red

pine

since

was selected
As

a

result,

the

PSU

contained

twice,

due

to its

the

estimated

within

large
total

these

plantations

selection probability.
volume

of

the

conifer

154
stratum

is

reflecting mostly

The multiphase technique,
more

realistic

stratum

measurements

the

to

of

total

structure

be

of

red

pine.

on the other hand, probably gave a

appraisal

because

the availability

made

volume

of

on

the

on

the

method

both

red

conifer

allowed

and

for

jack

pine

pl a n t a t i o n s .
The multiphase
simpler

structure

sampling.

As

this

method

since

the

representation

sampling
it

method

of

the

is,

of

technique
is

based

requires

image,

a

course,

the

image.

one may ask the question:
the

use

lack

available,

for

stratified
of

limitation

process

method

on

the

digitally

appropriate

the

evaluated has a much

If

the

for

of

such

random
digital

the

use

facilities
facilities

of
to
are

"What would be the most

classification of

the

imagery?"

The sensitivity analysis performed on the data obtained from
the digital
showed

processing

that

the

of

effect

the

of

imagery on

changing

the

classified in each stratum of interest,
on

the

changing
method.

estimate
of

the

This

Of

percentages

the

population

selection

means

classification
image.

of

that

algorithms

course,

the

of

omission

particular case

should be

this

information

number

parameter

be

algorithm

available

and

used
that

before

or

pixels

to

the

multistage

classify
the

errors
In

did

unsupervised

gives

chosen.
the

than
the

comission

the one

multiphase sampling technique,

of

supervised

could

of

study

has much less effect

probabilities

either

the present

the
least

for

a

order to have

application

of

the

it is necessary to know which

155
classification algorithm would give a better accuracy for a
certain situation.
Taking

the

This could be an area of future research.

several

cover

types

that

occur

in

Michigan

and

performing both supervised and unsupervised classifications,
using

imagery

algorithms
available

from

would
in

different

be

more

advance,

seasons,

to

accurate.

the

future

If

users

evaluate
these

of

which

data

the

are

multiphase

sampling technique could benefit from them.
Based

solely

application

of

on

both

the

conditions

sampling

established

techniques,

it

is

for

clear

the
from

the results that the multiphase method was more adequate for
the inventory of the particular area selected.
mean

that

the

inventories
or

an

in

other

method

northern

organization

should

Michigan.

wants

to

use

not

If a
a

be

This does not

considered

consulting
multilevel

for

forester
sampling

technique for an inventory it is not advisable to jump right
into the multiphase technique evaluated,
more

precise

analysis,

as

and

less

previously

costly.

A

discussed,

before making the final selection.

just because it was

series

of

variables

have

to

be

and

considered

APPENDIX A
EQUATIONS FOR THE OPTIMUM ALLOCATION OF SAMPLES AT EACH
STAGE AND FOR CALCULATION OF THE SAMPLING FRACTIONS

A. Optimum Allocation at Each Stage
A .1 First Stage
m = (D* )/(Ei/Di )/(/EiDi

+ / E 2D 2 + / E 3 D 3 )

A . 2 Second stage
n = / E 2 D 1 //E 1 D 2
A . 3 Third stage
t = -/E 3 D 2 / V E 2 D 3

M

Ei = 2 Pi (Vi /Pi
i= 1

- V)2

Ni

E2 = (2 Pij(Vij/Pij

- Vi)2 )/Pi

j = 1

M

E3 =

Ni

Ni

(2 2
03 i j//Ni ) 2 03 i j-/Ni
i= 1 j = 1
j= 1

Di = average cost of measuring a first-stage unit,
includes cost of ennumerating the predictions.

157
D 2 = average cost of measuring a second stage unit,
includes cost of ennumerating the predictions.
D 3 = average cost of measuring a third stage unit,
includes travel costs.
D*

= expected cost of the survey

B. Calculation of Sampling Fractions
B.l Phase one Sampling Fraction
u = /C iS * 2 //C 2 S*i

B.2 Phase two Sampling Fraction
v = 4 C 2 S* 3/4C 3 S* 2
B.3 Number of Samples
n = C* / (Ci + C 2 u + C3uv)

S* 1 = s2 - s ’

L

S* 3 = 2
i= 1

S *2

= s' - S*3

Ii

2

(mi j m / m i n )/si j 2

j= 1

L

s ’ = 2 (n i / n )si 2
i= 1

s2 =

Nn (v(y)
N-l

L
- 2 (ni/n)2vi)
i= 1

158
L

v(y) = n(N-l)

[ N-n

2 (m/n)(yi

_______

- y)2 +

i=1

N(n-l)

n(N-l)

N - n

l

--------- ^

T-»

l
/

.

i

V

•>

\i ii- x/a i

,

/

r

n2 (N - l ) i =i

.

v ni

/

.

\ /

/ ... .

/ ii / \ vm

/....

S

/ *T

..

V

/ n /- \ w - n /

i-i

... . 1

vi j

----n(N-l)

xi
vi

= m - 1

2

(mij-1

j= i

bij-1)

si j 2

+

_____

ni -1

Ii
2
(mi j /mi ) (yi j
j= l

-mi

mi j

_____

mi -1

ni

m

-

_____

mi bi j

-

yi )2

ni (mi - 1)

Ii

si2

= mini

[Vi

- 1 2

(mij / m i ) s i j 2 ((mij /bij ) -

1)]

___ j = i

(ni -mi )

mi

b ij
si j 2 = 1/ (bi j -1) 2
(yijk
k =1

- yij)2

C*

= expected cost of the survey

Cx

= fixed cost per phase one unit

C 2 = cost per second phase unit
C3 = cost per third phase unit
s2 = estimated population variance
si2 = estimated population variance of the ith phase
one stratum
si j2 = estimated population variance of the (ij)th
phase two stratum
The other terms are as defined on the text.

APPENDIX B
Selection Probabilities for the Hardwood,
and Red Oak Strata

A.

Hardwood Stratum

A.l Original Probabilities:

Pi

Pi j

Pi J k

.09987
.07918

.06253
.08348
.11689
.14945

.00809
.01059
.01059
.00873
.00989
.00989
.01061
.01061
.00687
.00898
.00804
.00898

Conifer

160
A . 2 Changed Probabilities:

Pi
Trial#l

Trial#2

Trial#3

Trial#4

Trial#5

.10531
.0835

.10449
.06779

.10524
.08345

.10953
.08685

.09192
.07288

Pi j
Trial#l

.06867
.09168
.13817
.13209

Trial#2

Trial#3

Trial#4

Trial#5

.07354
.08033
.15971
.12493

.05863
.09568
.1353
.10583

.07051
.07702
.13855
.10837

.06898
.0921
.15727
.12301

Pi j k
Trial#l

.00925
.00925
.01171
.00763
.00881
.01116
.01169
.00923
.00554
.00831
.00733
.00928

Trial#2

Trial#3

Trial#4

Trial#5

.00945
.00945
.00945
.0099
.00874
.00874
.01196
.01196
.00819
.00828
.00731
.00828

.00954
.01209
.00954
.00995
.00878
.00878
.00948
.00948
.00569
.00822
.00726
.00919

.00928
.01176
.01176
.00757
.00874
.01107
.01179
.01179
.00558
.00826
.00923
.00826

.00706
.00963
.00963
.0074
.00854
.00854
.00939
.00939
.00563
.00831
.00733
.00929

161
B. Conifer Stratum

B.l

Original Probabilities:

Pi

Pi i

Pi j k

.10280
.10280

.10640
.18470
.08790
.10640

.01027
.01941
.01941
.01142
.01007
.01007
.01931
.01158
.01931
.01941
.01256
.00571

B.2 Changed Probabilities:

Pi
Trialfl

Trial#2

Trial#3

Traial#4

Trial#5

.11785
.11785

.09432
.09432

.11143
.11143

.09521
.09521

.11276
.11276

162
Pi i
Tr i al #1

Trial#2

Trial#3

.09854
.17108
.09953
.09854

.11303
.19623
.07642
.11303

.09718
.16872
.09816
.09718

Trial#4

Trial#5

.11341
.16110
.09372
.11341

.11727
.20359
.09691
.11727

Trial#3

Trial#4

Trial#5

.00787
.02135
.01685
.01010
.01145
.00875
.01722
.01457
.02252
.02135
.01011
.00787

.00816
.01748
.02214
.01276
.01142
.01142
.01643
.00885
.02149
.01748
.01049
.00350

.01209
.01648
.02088
.01273
.00871
.00871
.01704
.01442
.02228
.02088
.00989
.00769

Pi jk
Tr i a l # 1

Trial#2

.00806
.01728
.02189
.00999
.00866
.01133
.02149
.00885
.02149
.01728
.01037
.00806

.01259
.02174
.02174
.01249
.01118
.00855
.01722
.01457
.01722
.01716
.01487
.00343

163
C. Red Oak Stratum

C.l Original Probabilities:

Pi
.84395
.84395

Pi jk

Pij
.16319
.23511
.23511
.16319

.00793
.00887
.00887
.00741
.00741
.00840
.00840
.00840
.00840
.00887
.00793
.00887

C.2 Changed Probabilities

Pi
Trialll

.86859
.86859

Trial#2

Trial#3

Trial#4

.84395
.84395

.84395
.84395

.81566
.81566

• •

• •

o
o
CO
o
o

• • •
o o ©

• • • •
o o o o
o o o o o
03 CO 00 CO -3
4k 4k h-4 I-4 to
00 00 -3 co t-4

• *
O
O
-3
4k
-3

•
O

• «

• •

• • •

o o o o o o o
o
o o o o o o o o o o o
CO 00 CO co co -3 03 00 00 co CO
o o co co co 4k 4k 4k o o o
o U1 CO CO CO H 4 CO o cn o o

Pi jk

!-» CO cn CO CO CO CO 00 00 cn cn h-4

• • • •
I-4 to CO I-4
-3 4k 4k ■3
h-4 ■3 ■3 t-4
03 CO CO 03
oo 03 03 00

• • • •
I-4 to CO I-4
cn •3 -0 cn
4k h-4M 4k
to cn cn CO
h-403 03 t—4

Trial#5

o o o o o o o o o o o o
o o o o o o o o o o o o
CO -3 00 ~3 -3 -3 *3 00 00 00 00 CO
t-4 h-4 h-4 CO CO CO co co co I—4 h*4 h-4

I-4 CO CO I-4

Trial#4

o o o o o o o o o o o o
o o o o o o o o o o o o
00 CO 00 CO -3 CO CO 00 00 00 00 -3
CO I-4 CO co 4k CO CO co co CO to to
H 4 -3 I-4-3 o -3 ~3 CO CO I-4Hk 4k

• • • •
I-4 CO CO I-4
4k cn cn 4k
03 00 00 03
00 cn cn 00

Trial#3

o o o o o o o O o o o o
o o o o o o o O o o o o
00 -3 co CO ~3 -3 co 00 03 CO 00 CO
o f—4o CO CO CO CO CO co o o o
-3 CO CO oo CO CO 00 o cn to -3 CO

■ . . .
t-4 CO CO t-4
-3 cn cn -3
co o © CO
03 I-4 I-4 03
cn CO CO cn

Trial#2

«

o o o o o o
o o o o o o
00 to to CO CO -3
I—4 I-4 h-4 4k 4k
-J CO co 3J 3 -3

Trial#l

• «

• • • •
H-4 to to t-4
4k t-4 M 4k
03 H-4 I—4 03

00 cn cn 00
CO 4k 4k CO

4
HSB
t-1
=**=

t-4
H

tj
H-

13)
M
=#=
CO

i-3
CJ
H-

SB
I-4
=tt=
CO

H-

SB
H4
=*t=
4k

•-3
H-

SB
I-4
=*♦=
cn

&

APPENDIX C
List of Scientific Names of Trees

Common Name

Scientific Name

American Beech

Fagus grandiflora

Basswood

Tilia americana

Bigtooth Aspen

Po p u l u s g r a n d i d e n t a t a Michs.

Blach Cherry

Prunus

Elm

Ulmus

Iron wood

Ostrya

Jack Pine

Pinus

Oak

Quercus

Red Maple

A c e r r u b r u m L.

Red Oak

Quercus

Red Pine

Pinus

Sugar Maple

A c e r s a c c h a r u m Marsh.

Trembling Aspen

Populus

White Ash

Fraxinus

Ehrh.

L.

s e r o t i n a E h rh.
spp.
virginiana
banksiana

(Mill.)

L amb.

spp.

r u b r a L.

r e s i n o s a Ait.

tremuloides Michx.
americana

L.

K.Koch

APPENDIX D
TALLY SHEET
FIELD COLLECTION DATA SHEET
Summer 1987
Date:

/ /87

Plot#:

BAF:

L o c a t i o n :__
Time start:

Finish:

Point# 1
Tree#

Spp

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Comments:

DBH

Point# 2
!MerchHT Tree#
1
a
8
1

1
1
B
a
8
a

8
a
il

i
ii
1
1

*
il
ii

I
I
1
1

I

*
l
ii
1
i
l
i
1
i
I

»

1
i
1
i

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Spp

DBH

MerchHt|
t
*
1
1
1
a

1

a
1
a

I

a
1
a
1
a
1
a
1
a

I
a
1
a

I
i
1
a
l
a
1
I
a
1
a
1
a
1

a
1
•

APPENDIX E
A. Volume per Plot in cubic feet/acre
for the Multistage Sampling Technique

A.l Hardwood Stratum

A . 2 Conifer Stratum

Plot#

Volume

Plot#

Volume

1 . 1.1

2,047.05
2,206.81
2,066.79
2,589.14
3,455.34
3,100.82
1,537.66
1,431.21
1,772.23
2,216.26
2,176.66
1,821.06

1 . 1.1
1 .1.2
1.1.3
1 .2.1
1 .2.2
1.2.3
2 .1.1
2 .1.2
2.1.3
2 .2.1
2 .2.2
2.2.3

3,042.66
3,668.98
2,185.42
2,963.12
2,137.78
2,532.41
1,976.81
1,924.06
3,123.94
2,568.85
1,769.06
1,414.91

1 .1.2
1.1.3
1 .2.1
1 .2.2
1.2.3
2 .1.1
2 .1.2
2.1.3
2 .2.1
2 .2.2
2.2.3

168
A . 3 Red Oak Stratum

Plot#

Volume

1 .1.1
1 .1.2
1.1.3
1 .2.1
1 .2.2
1.2.3
2 .1.1
2 .1.2
2.1.3
2 .2.1
2 .2.2
2.2.3

1,833.90
2,654.62
2,659.69
2,171.66
2,419.89
2,123.22
2,033.44
1,764.94
2,166.95
3,340.35
2 ,850.24
2,533.65

B. Volume Per Plot in cubic feet/acre
for the Multiphase Sampling Technique

B.l Hardwood Stratum

B.2 Conifer Stratum

Plot#

Hardwood

Red Oak

Jack Pine

1
2
3
4
5
6

2,423.86
1,615.05
1,243.82
3,204.58
1,298.34

1,385.75
2,506.85
2,301.51
1,958.07
1 ,989.80
2,937.91

1,945.29
1,245.04
1,598.59
1,421.78
1,325.95

Red Pine

3,517.55
3,109.69
1,995.50
1,656.17
2,847.39

LITERATURE CITED

Albert, D . A . ; Denton,
S.R.;
Barnes,
B.V.
1986.
Regional
landscape ecosystem of Michigan.
School
of Natural
Resources, The University of Michigan, 32 p.
Aldred, A.H. 1980.
Forest inventory by multistage remote
sensing.
In:
Remote
Sensing Symposium,
Proceedings,
Great Lake Forest Research Center, Sault Ste. Marie,
Ontario, Canada, pp. 7 7 - 84.
Anderson, J. E. 1979. Multistage variable pr ob ab il ity forest
volume inventory. NASA/ERL Report #179, 32 p.
Avery, G . ; Meyer, M.T. 1959. Volume tables for aerial timber
estimating in nor thern Minnesota. Lake States Forest
Experimental Station, station paper #78, 21 p.
Beers, T.W. 1964. Composite hardwood volume tables. Purdue
University, Agricultural Experiment S t a t i o n , Research
Bulletin 787, 12 p.
Beers, T . W . ; Miller, C.I. 1966. Horizontal point sampling
tables. Purdue University, Agricultural
Experimental
Station, Research Bulletin 808, 80 p.
Bichford, C.A. 1961. Stratification for timber cruising.
Journal of Forestry, 59 (10), 761 - 763.
Bichford,
C.A.;
Mayer,
effective sampling
northeastern forest
61 (11) :826 - 833.

In:

C.E.
and Ware,
K.D.
1963.
An
design for forest inventory: The
resurvey. In: Journal of Forestry,

Buchanan,
D.E.
1985.
Soil
survey
of
Lake
Counties Michigan. USDA Soil Conservation
P»

and
Wexford
S e r v i c e , 135

C h a p e l l e , D.E.
1985.
The
research
process
in
natural
r e so ur ce s. Lecture notes for the course RD/FOR 855,
Michigan State University, Forestry Department, 156 p.

170
Cochran, W.G. 1977.
Third Edition,

Sampling Techniques.
428 p.

John Willey & Sons,

De Vries, P.G. 1986. Sampling theory for forest inventory a teach yourself course. S p r i n g- Ve rl a g, 399 p.
D o n k e r , N . H . W . ; Mulder, N.J. 1977. Analysis of MSS digital
imagery
with
the
aid
of
principal
component
transformation. In: ITC J o u r n a l , Vol. 3, pp. 434 - 467.
F r a y e r , W.E. (E d i t o r ) 1979. Multilevel sampling designs for
resource
i nve nto ri es. Report to US Forest S e r v i c e ,
Rocky
Mountain
and
Range
Experiment
S t a t i o n , Fort
Collings, CO, 113 p.
F r a y e r , W.E. 1981. Multi-level sampling designs for resource
in ve nt o ri es .
In:
XVII
IUFRO
World
Congress,
P r o c e e d i n g s , Division 9, Japan, pp. 169 - 173.
F r e e s e , F . 1962. Elementary forest s am p l i n g . Agriculture
Handbook 232, US Department of A g r i c u l t u r e , 91 p.
G i a l d i n i , M . ; T i t u s , S.; N i c h o l s , J . ; T h o m a s , R. 1 9 7 5 . The
integration of manual
and automatic
image analysis
technique with supporting ground data in a multi-stage
sampling f rame work for timber resource in ve n to ri es :
three e xa m p l e s . I n : Proceedings
of the NASA Earth
Resources Survey Symposium, H o u s t o n , T e x a s , Lyndon B .
Johnson Space C e n t e r , pp. 1377 - 1387.
G o o d e n o u g h , D. ; S h l i e n , S. 1974. Automatic classification
methodology.
Canada
Center
for
Remote
Sensing,
Department of E n e r g y , Mines and R e s o u r c e s , Research
Report 74-1, 8 p.
H a l l , J.A.; H a l e s , M.T. 1979. Multistage inventory technique
case
study
- McCoin
V a l l e y . I n : Seventh
Biennial
Workshop on Color Aerial
Photography in the Plant
Sciences
and
Related
Fields,
P r o c ee di ng s,
Davis,
C a l i f o r n i a , pp. 119 - 124.
H a r d i n g , R . A . ; S c o t t , R.B.
1978.
Forest
inventory with
LANDSAT phase
II
- Washington
forest
productivity
study.
State
of
Washington
Department
of
Natural
R e s o u r c e s , Olympia, Washington, 221 p.
Harris,
J.W.E.;
Dawson,
A .F .
and
Brown,
R.G.
1983.
Evaluation of mountain pine beetle damage using aerial
photography taken with a hand-held 70 mm camera, Gold
Bridge - C l i n t o n , B.C., 1981. Canadian Forest S e r v i c e ,
Pacific Forest Research C e n t e r , B C - X - 2 4 5 , 15 p.

171
H e g y i , F. 1980. A new approach to forest inventory using
remote
sensing.
In:
Remote
Sensing
Symposium,
Proceedings, Great Lake Forest Research Center Sault
Ste. Marie, Ontario, Canada, pp. 9 4 - 9 9 .
Heller,
R.C.;
Wear,
J.F.
1969.
Sampling
forest
insect
epidemics with color films.
In: Sixth International
Symposium
on
Remote
Sensing
of
Environment,
Proceedings, Vol. II, pp. 1157 - 1167.
Hesse,

R.A.

1987.

Private communication.

Hudson, D.W. 1986. Classification of coniferous forest cover
types
using
LANDSAT
MSS
digital
data.
Ph.D.
Dissertation, Department of Forestry, Michigan State
University, East Lansing, 251 p.
Hush,

B. 1971. Planning a forest inventory.
Forest Products Studies, #17, 120 p.

FAO Forestry and

Hush,

B. ; Miller,
C.I.
and
Beers,
T.W.
1982.
mensuration. John Willey & Sons, Third Edition,

Forest
402 p.

Hutchinson, I.D. 1978. An example of the use of two phase
sampling
design
for
reconnaissance
inventory
in
tropical forest. In: New Zealand Journal of Forestry,
23( 1 ) :95 - 106.
Jensen, R.J, 1986. Introductory digital image processing - a
remote sensing perspective. P r en ti ce -H a ll , 379 p.
Jeyaratnam, S.; Bowden, D.C.; Graybill,
F.A. and Frayer,
W.E.
1984.
Estimation
in
multiphase
designs
for
stratification. Forest Science, 30(2):484 - 491.
Johnston,
D.C.
1982. Theory and application of selected
multilevel
sampling
designs.
Ph.D. Dissertation,
Department
of Forest and Wood Sciences, Colorado State
University, Fort Collins, 197 p.
Johnston,
D.C.;
Frayer, W.E.
1983.
Stratified two phase
sampling as a base for monitoring multiresources. In:
Renewable Resource Inventories for Monitoring Changes
and Trends, Proceedings of an International Conference.
Edited by John F. Bell and Toby Atterbury. Corvallis,
Oregon, August, 15-19, pp. 448 - 451.
Johnston,

D.C.

1987.

Private conversation.

LaBau,
V.J.;
Schreuder,
H.T.
1983.
A
multiphase,
multiresource
inventory
procedure
for
assessing
renewable natural resources and monitoring change. In:
Renewable Resource Inventories for Monitoring Changes
and Trends, Proceedings of an International Conference.

172
Edited by John F. Bell and Toby Atterbury.
Oregon, August 15-19, pp. 456 - 459.

Corvallis,

Langley, P.G. 1969. New multistage technique using space and
aircraft imagery for forest inventory.
Symposium on
Remote Sensing of Environment, Proceedings, 6(2):11791183.
Langley,
P.G.
1971. The benefits of multistage variable
probability sampling using space aircraft imagery. In:
Application of Remote Sensing in Forestry, joint report
by IUFRO working group "Application of Remote Sensors
in Forestry", IUFRO Section 25, Germany.
Langley,
P.G.
1975.
Multistage
variable
probability
sampling: theory and use in estimating timber resources
from
space
and
aircraft
photography.
Ph.D.
Dissertation, Wildland Resource Science, University of
California, Berkeley, 101 p.
Langley,
P.G.
1978.
Remote
sensing
in
multistage,
multiresource inventories. In: Integrated Inventories
of Renewable
Natural
Resources,
Proceedings of the
Workshop, Rocky Mountain Forest and Range Experiment
Station, general technical report RM-55, pp. 205 - 208.
Lee, D.C.L.; Hernandes, P.F.; Shimabukuro, Y.E.; Assis, O.R.
d e ; Medeiros,
J.S. de
1984.
Forest inventory using
multistage sampling with probability proportional to
size.
Instituto
de
Pesquisas
Espaciais
INPE,
publicacao # 30 84 -PR E/ 494 , Sao Paulo.
Li,

H.G.; Schreuder, H.T.; Bowden, D.C. 1984. Four phase
sampling
estimation
for
the
Alaska
survey.
In:
Inventorying Forest and other Vegetation of the High
Latitude and High Altitude Regions. Proceedings of an
International Symposium, Society of American Foresters
Regional Technical Conference, July 23-26, Fairbanks,
Alaska, pp. 61 - 67.

Lillesand, T.M.; Kiefer, R.W. 1987. Remote sensing and image
interpretation. John Willey & Sons, 721 p.
Loetsch,
F .;
Haller,
K.E.
1964.
Forest
Verlagsgesellschaft Munchen Basel Wien,
Lusch,

D.P.

1987.

Private conversation.

Lusch,

D.P.

1988.

Private conversation

inventory.
BLV
Vol. 1, 436 p.

MacClean, C.D. 1972. Improving inventory volume estimate by
double sampling on aerial photographs. In: Journal of
Forestry, 70(2):748 - 749.

173
Mattila,
E.
1984.
Survey of forest resources of Finish
Lapland
using
multiphase
systematic
sampling.
In:
Inventorying Forest and other Vegetation of the High
Latitude
High
Altitude
Regions.
Proceedings
of
an
International Symposium. Society of Am erican Foresters,
Regional Technical Conference, 23-26 July, Fairbanks,
Alaska, pp. 55 - 60.
Myers,
W.L.;
Shelton,
R.L.
1980.
Surveys
ecosystem management. John Willey & Sons,

methods
403 p.

for

Nichols, J.D.; Gialdini, M. and Jaakkola, S. 1973. A timber
inventory based upon manual and automated analysis of
ERTS-1 and
supporting aircraft data using multistage
probability sampling.
In:
Third
Earth
Resources
Technology Satellite 1, Symposium, Section A, NASA SP351, Goddard Space Flight Center, Washington DC, pp.
145 - 157.
Nichols, J.D.; Harding,
R.A.; Scott, R.B.; Edwards,
J.R.
1976.
Forest
inventory
of
western
Washington
by
satellite
multistage sampling.
In:
Proceedings,
American Society of P h ot o gr am me tr y, Fall Convention,
pp. 180 - 217.
Norton,

J. 1987.

Private conversation.

Nyyssonen, A. 1976. Objectivos y planeamiento de inventarios
forestales.
Curso FAO/FINLANDIA de entrenamiento en
inventario forestal, Finlandia, pp. 64 - 80.
Peterson, D.L.;
Noren,
D . ;Gnauck, G.
1983.
Methods and
results
of
three
unequal
probability
multistage
sampling designs
for timber
volume
in the pacific
northwest.
In:
Renewable
Resource
Inventories
for
Monitoring
Changes
and
Trends.
An
International
Conference, August 15-19, Corvallis, Oregon pp. 326 329.
Shiue, C.J.; John, H.H. 1962. A proposed sampling design for
extensive forest inventory: double systematic sampling
for regression with multiple s t a r t s . I n : Journal of
Forestry 6 0 ( 9 ) :607 - 610.
Sy a n- Wi ttg ens te in, L. 1961. Phenological aids to species
identification
on
air p ho to gr ap hs .
Forest
Research
Branch,
Technical
Note
#104,
Canada
Department
of
Forestry, 26 p.
Temu,

A.B.;
Philip,
M.S.
1981.
Sampling
woodland
for
fuelwood. In: Proceedings of Forest Resource Inventory,
Growth Models, Management Planning and Remote Sensing.
XVII World Congress,
IUFRO, Ed. Masahisa Nishizawa,
Kyoto, Japan, pp. 272 - 279.

174
Titus, S.; Gialdini, M.; Nichols, J. 1975. A total timber
resource
inventory based upon manual
and automated
analysis of LANDSAT 1 and supporting aircraft data
using stratified multistage sampling techniques.
In:
Tenth International
Symposium on Remote Sensing of
Environment, Proceedings, pp. 1093 - 1099.
Wear,

J.F.; Pope, R.R. and Lauterbach, P.G. 1964. Estimating
beetle-killed Douglas-fir by aerial photo and field
plots. In: Journal of Forestry, 6 2 ( 5 ) :309 - 315.

Wert,

S.L.
photos.
1248.

Wert,

S.L.
1969.
A
system
for
using
remote
sensing
techniques to detect and evaluate air pollution effects
on forest stands. In: Sixth International Symposium on
Remote Sensing of Environment, Vol. II, Proceedings,
pp. 1169 - 1173.

1968.
Douglas-fir
beetle
survey with
color
In: Photogrammetric Engineering, 34(12):1234 -

W i n t e r b e r g e r , K.C. 1984. LANDSAT data and aerial photographs
used in a multiphase sample of vegetation and related
resources in interior Alaska. In: Inventorying Forest
and other Vegetation of the High Latitude High Altitude
Regions. Proceedings of an International
Symposium.
Society
of
American
Foresters,
Regional
Technical
Conference, 23-26, July, Fairbanks, Alaska, pp. 157 163.
Yandle, D.O.; White, F.M. 1977. An application of two stage
forest
sampling.
In:
Southern
Journal
of
Applied
Forestry, 1(3):27 - 32.