their time and showed a sense of proportion and professional courtesy not always found ... Bloedel, Van Dusen and McPhee fellowships, the C.F.S. Science Subvention. Program ...... As trees tend to regularly space themselves this continuous ...
SYSTEMS FOR THE SELECTION OF TRULY RANDOM SAMPLES FROM TREE POPULATIONS AND THE EXTENSION OF VARIABLE PLOT SAMPLING TO THE THIRD DIMENSION
by
KIMBERLEY ILES B.S. Forest Management, Oregon State University, 1969 M.Sc. Forest Biometrics, Oregon State University, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
in THE FACULTY OF GRADUATE STUDIES Department of Forestry
We accept this thesis as conforming to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA June, 1979 ©
Kimberley l i e s , 1979
In presenting
t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements
f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.
I f u r t h e r agree that permission f o r extensive
copying
of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives.
I t i s understood that
copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.
Kim
Department of
Forestry
The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place, Vancouver, B.C., Canada V6T 1W5 Date:
lies
- i i Supervisor:
Donald D. Munro ABSTRACT
Means of drawing t r u l y random samples from populations of trees d i s t r i b u t e d non-randomly i n a plane are p r a c t i c a l l y unknown. Only the technique of numbering a l l items and drawing from a l i s t i s commonly suggested.
Two other techniques are developed, reducing p l o t
s i z e and s e l e c t i n g from a c l u s t e r w i t h p r o b a b i l i t y (1/M) where M i s larger than the c l u s t e r s i z e .
The exact bias from some other s e l e c t i o n
schemes i s shown by the construction of "preference maps".
Methods
of weighting the s e l e c t i o n by tree height, diameter, basal area, gross volume, v e r t i c a l c r o s s - s e c t i o n a l area and combinations of diameter and basal area are described. None of them require a c t u a l measurement of the tree parameters.
Mechanical devices and f i e l d techniques are
described which s i m p l i f y f i e l d a p p l i c a t i o n .
The use of projected angles,
such as are used i n V a r i a b l e P l o t Sampling i s c e n t r a l to most of these methods. C r i t i c a l Height Sampling Theory i s developed as a g e n e r a l i z a t i o n of V a r i a b l e P l o t Sampling.
The f i e l d problem i s simply to measure the
height to where a sighted tree i s "borderline" with a relaskop.
The
average sum of these " c r i t i c a l heights" at a point m u l t i p l i e d by the Basal Area Factor of a prism gives a d i r e c t estimate of stand volume without the a i d of volume tables or tree measurements.
Approximation
techniques which have the geometrical e f f e c t of changing the expanded tree shape are described. The s t a t i s t i c a l advantages of using the
system were not found to be l a r g e , and the problems of measuring the c r i t i c a l height on nearby trees was severe.
In general use
there appears to be no advantage over standard techniques of V a r i a b l e P l o t Sampling, however i n s i t u a t i o n s where no volume tables e x i s t i t may have a p p l i c a t i o n , and the problem of steep measurements angles to nearby trees can be overcome by using an o p t i c a l c a l i p e r .
The
system can a l s o overcome the problem of "ongrowth" f o r permanent sample p l o t s .
Donald D. Munro
- iv -
TABLE OF CONTENTS
Page
ABSTRACT.
i i
TABLE OF CONTENTS
iv
LIST OF TABLES
v i i
LIST OF FIGURES
.viii
ACKNOWLEDGEMENTS
xi
INTRODUCTION
1
Reasons t o Sample I n d i v i d u a l Trees
2
Reasons to Sample Randomly
6
Sampling Without Replacement
10
Weighting S e l e c t i o n P r o b a b i l i t i e s
11
SAMPLING SELECTION SYSTEMS PROPORTIONAL TO VARIOUS PROBABILITY WEIGHTINGS
.12
Frequency Weighting
12
"Nearest Tree" Methods
12
Bias i n Subsampling From a Cluster
17
S e l e c t i o n of the Closest Tree i n Each Cluster
21
The "Azimuth Method"
23
Methods f o r the E l i m i n a t i o n of Bias
29
P l o t Reduction Method
29
The E l i m i n a t i o n Technique
31
Increasing E f f i c i e n c y
32
Diameter or Circumference Weighting
35
S e l e c t i o n P r o p o r t i o n a l to Diameter Alone
35
S e l e c t i o n P r o p o r t i o n a l t o Diameter and a Constant ( C ) . . . 41 g
Problems of Scale and F i e l d Use
42
- v -
Tree Height Weighting
49
An Adaptation of L i n e - I n t e r s e c t Sampling to Standing Trees. . 51 Basal Area Weighting
54
Basic Ideas of Point Sampling
55
Bias From S e l e c t i o n of a Single I n d i v i d u a l From Every Cluster.57 Bias From Random S e l e c t i o n From Each Group
57
Nearest Tree Method
60
The Azimuth Method With V a r i a b l e P l o t s
71
Non-Random But Unbiased Methods For Subsampling
71
Height Squared Weighting
73
Combining Diameter Squared and Diameter Weightings
73
Mechanical Devices to A i d S e l e c t i o n
83 2
Adding a Constant, S e l e c t i o n P r o p o r t i o n a l to aD. + bD^ + c. . 8 6 A Generalized Instrument Gross Volume Weighting
86 90
Basics of the C r i t i c a l Height Method
90
Random S e l e c t i o n From a Cluster P r o p o r t i o n a l to C r i t i c a l Height of the Tree, a Biased Method S e l e c t i o n of the Tree With Largest C r i t i c a l Height,
96
a Second Biased Method.
98
V e r t i c a l Cross-Sectional Weighting
101
Cylinder Volume Weighting
101
CRITICAL HEIGHT SAMPLING,HISTORICAL DEVELOPMENT AND LITERATURE REVIEW
104
ADVANTAGES AND APPLICATIONS
107
APPROXIMATION METHODS
110
The Rim Method
118
Approximating C r i t i c a l Height With the Rim Method
120
- v i-
FIELD APPLICATION
128
Relaskop Use
131
Log Grading
135
VARIABILITY OF THE SYSTEM CONCLUSIONS
136 . 139
LITERATURE CITED
140
APPENDIX I .
144
L i s t of Symbols and Terms
- vii -
LIST OF TABLES
Table
Page
1
Computations involved f o r the example i l l u s t r a t e d i n Figure 2
20
2
C a l c u l a t i o n s f o r the example shown i n Figure 17
59
- viii -
LIST OF FIGURES Figure 1
Page Construction of D i r i c h l e t c e l l s around stem mapped trees on a tract to be sampled
15
I l l u s t r a t i o n of the computation of sampling probabilities when a single tree i s chosen randomly from a l l those on a fixed plot
19
Preference map when selection i s based on the closest tree within a fixed plot
22
4
Construction of preference maps based on the azimuth method
24
5
Azimuth method applied with fixed plots
27
6
Horizontal l i n e sampling, basic idea of the selection rule
36
7
I l l u s t r a t i o n of selection probability with l i n e sampling
37
8
Construction of l i n e sample preference map
39
9
Band width for the preference map when sampling i s proportional to + C 43
2
3
g
10
Band width for the preference map when sampling i s proportional to D, - C 44 g
11
Using the angle gauge to establish the reduced diameter of a tree
12 & 13
Field systems to select trees proportional to D. - C l s Field system to select trees proportional to D. + C l s Use of an angle gauge on f l a t ground for v e r t i c a l l i n e sampling Use of two measurements on sloping ground
14 15 16 17
Plots of variable size i n horizontal point sampling
45 46 47 50 52 56
- ix -
18 19 20 21 22
Constructions with "nearest tree" s e l e c t i o n and v a r i a b l e p l o t s
61
I l l u s t r a t i o n of terms used i n proof of bias toward smaller trees
62
I l l u s t r a t i o n of construction r u l e s f o r the nearest tree preference map
67
Completed preference map, with v a r i a b l e p l o t s
69
nearest tree method
Completed preference map f o r the azimuth system and v a r i a b l e p l o t s
72
23
P l o t area composed of 3 simple f i g u r e s
74
24
2 P l o t of area p r o p o r t i o n a l to aD^ + bD^
76
25
Geometry used to determine the angle 9
79
26
2 S e l e c t i o n p r o p o r t i o n a l to aD . + bD^
27 28 29 30 & 31 32 33 34
81 2 P r i n c i p l e s of s e l e c t i o n p r o p o r t i o n a l to aD^ - bD^ • . • 82 Device p r i n c i p l e s for sampling p r o p o r t i o n a l to aD + bD i i Device p r i n c i p l e s f o r sampling p r o p o r t i o n a l to aD -bD i l P r i n c i p l e s of s e l e c t i o n p r o p o r t i o n a l to aD + bD + c i l Prism device f o r s e l e c t i o n of trees by p l o t s indexed by diameter alone 2
.84
2
85
2
87
The "expanded t r e e " , c r i t i c a l height c r i t i c a l point
89
and 91
I l l u s t r a t i o n of some of the basic concepts of c r i t i c a l height determination
93
S e l e c t i o n p r o b a b i l i t i e s p r o p o r t i o n a l to c r i t i c a l height
97
36
S e l e c t i o n of tree with l a r g e s t c r i t i c a l height
99
37
S e l e c t i o n p r o p o r t i o n a l to v e r t i c a l crosss e c t i o n a l area of the stem
35
102
- x -
38
39
The step f u n c t i o n formed by the sum of VBARS of overlapping expanded trees i n standard v a r i a b l e p l o t sampling Side view showing sum of c r i t i c a l heights as a smooth continuous f u n c t i o n
40
41 42
43
109 m
The e f f e c t of "ongrowth" i n permanent sample points with v a r i a b l e p l o t v s . c r i t i c a l height sampling systems
112
The proportion of occasions the c r i t i c a l point w i l l be i n the crown.
114
C a l c u l a t i o n of the p r o b a b i l i t y that the c r i t i c a l point w i l l be at too steep an angle for accurate measurement
116
Example of c a l c u l a t i o n s when only the "rim" of the expanded stem i s sampled
119
44
I l l u s t r a t i o n of the terms used to develop an estimating system f o r c r i t i c a l height . . . . . . . 121
45
C r i t i c a l height as measured by hand held relaskop
133
C r i t i c a l height as measured by t r i p o d mounted relaskop
134
C o e f f i c i e n t of v a r i a t i o n f o r 5 sampling methods
137
46 47
- xi -
ACKNOWLEDGEMENTS
I would f i r s t l i k e to thank Dr. Donald D. Munro, whose a t t e n t i o n to d e t a i l , coupled w i t h h i s tolerance f o r the many e x t r a projects i n which I was engaged, puts me s o l i d l y i n h i s debt.
I
could not have had b e t t e r guidance, advice or breadth of opportunity, and I am g r a t e f u l . I would also l i k e to thank the r e s t of my committee, Drs. Kozak, Demaerschalk, Eaton and W i l l i a m s , who were generous with t h e i r time and showed a sense of proportion and p r o f e s s i o n a l courtesy not always found i n graduate committees, and which made my time here more p r o f i t a b l e . My appreciation to the Biometrics Group, both s t a f f and students, f o r the s t i m u l a t i o n , information and help they provided as w e l l as the sheer pleasure of t h e i r company. I would l i k e to acknowledge Dr. Frank Heygi and Mr. Rob Agnew of the B r i t i s h Columbia Forest Service who k i n d l y provided data f o r the simulation s t u d i e s , and Mr. Davis Cope and Ms. Karen Watson who provided timely mathematical i n s i g h t and an enjoyable exchange of ideas. C r u c i a l f i n a n c i a l assistance was provided by the MacMillan Bloedel, Van Dusen and McPhee f e l l o w s h i p s , the C.F.S. Science Subvention Program and teaching a s s i s t a n t s h i p s through the Faculty of Forestry.
- xii -
I am indebted to the Faculty of Forestry i n general, who always seemed to r e a l i z e that i t was the student himself, rather than c r e d i t s , papers and p r o j e c t s , that was the end product of a graduate program. This a t t i t u d e has made my time at U.B.C. not only a f i r s t c l a s s educational adventure but three years of hard work that were tremendous fun.
- 1 -
INTRODUCTION
This t h e s i s w i l l develop two basic areas of i n t e r e s t . The f i r s t part w i l l examine the problem of drawing a t r u l y random sample of trees d i s t r i b u t e d i n a non-random pattern i n a plane. I t i s d i f f i c u l t to b e l i e v e that such systems have not been developed i n the past.
Other than the c l a s s i c a l method r e q u i r i n g l a b e l l i n g
of a l l trees i n the population the author has found l i t t l e mention of methods to accomplish t h i s .
Two major methods of assuring a random
sample w i l l be given, and one of them (the e l i m i n a t i o n method) w i l l be used to s e l e c t trees w i t h p r o b a b i l i t y p r o p o r t i o n a l to a number of d i f f e r e n t parameters f o r that tree.
S e l e c t i o n p r o p o r t i o n a l to diameter,
height, basal area, gross volume and weighted combinations of diameter and basal area w i l l be of p a r t i c u l a r i n t e r e s t . The biases
inherent
i n i n c o r r e c t attempts to gather random samples w i l l be examined and methods of s p e c i f y i n g the magnitude of such biases w i l l be developed i n the form of "preference maps" constructed from standard stem l o c a t i o n maps.
In several cases mechanical aides w i l l be devised f o r f i e l d work
i n s e l e c t i n g samples with p a r t i c u l a r weightings. The second part of the t h e s i s w i l l develop the theory of C r i t i c a l Height Sampling f o r the inventory of tree stands without the use of volume t a b l e s .
The basic concept i s to extend the theory of
B i t t e r l i c h s ' V a r i a b l e P l o t Sampling to the t h i r d dimension, so that not only the tree basal area, but the e n t i r e tree volume i s expanded by a constant and then d i r e c t l y sampled.
A large s c a l e f i e l d t r i a l w i l l not
- 2 -
be attempted, but the f i e l d work w i l l be described and some f i e l d experience w i l l be gained to a n t i c i p a t e problems i n a p p l i c a t i o n and to develop a l t e r n a t i v e measurement schemes to solve them.
Approximation
techniques w i l l be discussed and some computer simulation w i l l be used to determine the r e l a t i v e v a r i a b i l i t y of the system i n comparison to standard Fixed P l o t or V a r i a b l e P l o t techniques.
A p p l i c a t i o n s and
timber types f o r which the system i s p a r t i c u l a r l y suited w i l l be discussed. Emphasis throughout the t h e s i s w i l l be on the geometrical reasoning involved, since an understanding of t h i s w i l l g r e a t l y a i d attempts to adapt these methods to l o c a l c o n d i t i o n s .
Reasons to Sample I n d i v i d u a l Trees
S e l e c t i o n of trees f o r sampling, as opposed to complete enumeration, i s now standard p r a c t i c e i n f o r e s t inventory.
In most
cases, the trees are selected as a c l u s t e r , and the value of i n t e r e s t i s a sum of i n d i v i d u a l values i n that c l u s t e r .
S t a t i s t i c s are then
applied to that sum as a s i n g l e observation and u s u a l l y expanded on an area b a s i s .
I n t h e i r c l a s s i c a p p l i c a t i o n both f i x e d p l o t sampling and
v a r i a b l e p l o t sampling are examples of t h i s process of s e l e c t i n g a c l u s t e r of t r e e s .
The c l u s t e r has four major advantages, one i s
s t a t i s t i c a l and the other three procedural. The s t a t i s t i c a l advantage i s that the sum of a c l u s t e r i s often l e s s v a r i a b l e than i n d i v i d u a l tree values, both because i t incorporates several observations which may tend to "average out" under even random
- 3 -
spacing arrangements, and because there i s a tendency f o r trees to react to each others presence.
This "competition e f f e c t " tends to
cause higher v a r i a t i o n between i n d i v i d u a l s , but lower v a r i a t i o n among the groups, since as one tree increases i t s growth i t i s often at the expense of i t s neighbors. The process of competing f o r a shared amount of a v a i l a b l e l i g h t , water and n u t r i e n t s serves to r e t a i n a constant e f f e c t on a group of trees even while increasing i n d i v i d u a l d i f f e r e n c e s . This reasoning c e r t a i n l y a p p l i e s to growth, i f not to t o t a l volume. We are thus l e d to the c l a s s i c s i t u a t i o n of v a r i a t i o n w i t h i n (rather than between) c l u s t e r s
which gives c l u s t e r sampling i t s s t a t i s t i c a l
advantage. The f i r s t procedural advantage i s that once a p l o t center i s established i t i s u s u a l l y small a d d i t i o n a l e f f o r t to determine several trees f o r sampling at the same time.
With the expense involved i n
t r a v e l time, p a r t i c u l a r l y i n random sampling, i t often decreases t o t a l project cost to measure a c l u s t e r even when v a r i a b i l i t y between and w i t h i n c l u s t e r s would not i n d i c a t e a s t a t i s t i c a l advantage. The second procedural advantage i s that most sampling systems are based on the concept of volume measured on a land area b a s i s .
I t often
i s easy to determine the t o t a l area involved i n an inventory, but rather hard to determine the t o t a l number of t r e e s , hence a system based on volume i n an i n d i v i d u a l tree requires a more ingenious approach to sampling.
Such approaches as Triangle Sampling by Fraser (1977), and
the same concept applied e a r l i e r by Jack (1967) are examples of using a land area base f o r i n d i v i d u a l t r e e s , but they l a c k widespread acceptance.
- 4-
)
The t h i r d advantage to s e l e c t i o n of a c l u s t e r , i s that simple r u l e s f o r the unbiased s e l e c t i o n of a c l u s t e r of trees are easy to develop and r e a d i l y a v a i l a b l e , while unbiased s e l e c t i o n r u l e s f o r i n d i v i d u a l trees are d i f f i c u l t to f i n d or simply u n a v a i l a b l e . S e l e c t i o n of i n d i v i d u a l s by even so simple a c r i t e r i o n as frequency i s d i f f i c u l t indeed.
The f o l l o w i n g quote i s from P i e l o u (1977).
In order to choose a random i n d i v i d u a l from which to measure the distance to i t s nearest neighbor, the only s a t i s f a c t o r y method i s to put numbered tags on a l l the plants i n the population and then to consult a random numbers table to decide which of the tagged plants are to be included i n the sample. In doing t h i s , we acquire w i l l y - n i l l y a complete count of the population from which i t s density automatically f o l l o w s . There i s another method of p i c k i n g random p l a n t s , but i t , too, requires that the s i z e of the t o t a l population be known. I f a sample of s i z e n, say, i s wanted from a population of s i z e N, the p r o b a b i l i t y that any given plant i n the population w i l l belong to the sample i s p=n/N. We must then take each population member i n turn and decide by some random process whose p r o b a b i l i t y of "success" i s p whether that member i s to be admitted to the sample. Even i f we are w i l l i n g to guess the magnitude of N i n t u i t i v e l y and assign to p a value that w i l l give a sample of approximately the desired s i z e , i t i s s t i l l necessary to subject every population member to a " t r i a l " i n order to decide whether i t should be included i n the sample; as the successive t r i a l s are performed, a complete census of the population i s automatically obtained.
The sampling of tree c l u s t e r s based on a f i x e d area has three main disadvantages. advantage i n doing so.
F i r s t , there may be l i t t l e
statistical
I n an area where the growth of a c l u s t e r may
be very consistent the t o t a l volume may be q u i t e d i f f e r e n t .
Dollar
- 5 -
values, p a r t i c u l a r l y where d i f f e r e n t species are involved, are l i k e l y to be even more v a r i a b l e . S t r a t i f i c a t i o n , e i t h e r before or a f t e r data c o l l e c t i o n , may be only p a r t l y h e l p f u l i n reducing t h i s variance. Secondly, as trees are measured with greater accuracy
and
for m u l t i p l e c h a r a c t e r i s t i c s , the cost d i f f e r e n c e between e s t a b l i s h i n g the sample point and measuring the tree diminishes, and i t i s l e s s reasonable to measure many trees "as long as you are there anyway". The concern for accurate net volumes i n lump sum sales has led the United States Bureau of Land Management (BLM) to " F a l l , Buck and Scale" c r u i s i n g (Johnson, 1972).
This kind of d e s t r u c t i v e , i n t e n s i v e
measurement can only be j u s t i f i e d when sample s i z e s are as small as s t a t i s t i c a l l y feasible.
"Extra" trees can no longer be measured j u s t
because i t w i l l s i m p l i f y the s e l e c t i o n process to measure a c l u s t e r rather than an i n d i v i d u a l . Third, the t o t a l land area involved may be more d i f f i c u l t to determine than the number of trees.
I t may be complexly
defined,
i n t r i c a t e i n pattern or there may be d i f f i c u l t y p h y s i c a l l y measuring the border.
In a d d i t i o n there may be "boundary effects" f o r trees
p h y s i c a l l y near the border.
This has r e s u l t e d i n a great number of
papers, f o r instance (Martin et a l . , 1977; Beers, 1966; Beers, B a r r e t t , 1964).
1976;
I t might be an advantage, where p o s s i b l e , to avoid
rather than solve these problems.
Reasons to Sample Randomly
Random sampling does not always mean s e l e c t i o n of i n d i v i d u a l s with equal p r o b a b i l i t y , although i t i s often discussed i n t h i s manner. I t does mean that an i n d i v i d u a l item has a p a r t i c u l a r p r o b a b i l i t y of being included i n a sample, and u s u a l l y that the s e l e c t i o n of the i n d i v i d u a l does not a f f e c t the further p r o b a b i l i t y of s e l e c t i n g any a d d i t i o n a l member of the population f o r the sample.
A more rigorous
d e f i n i t i o n of random sampling can be found i n Brunk (1965).
We w i l l
consider, f o r the f o l l o w i n g discussion, only equal p r o b a b i l i t y s e l e c t i o n w i t h replacement because i t i s simple and i l l u s t r a t e s the main points of the f o l l o w i n g
topics.
In equal p r o b a b i l i t y s e l e c t i o n with replacement each permutation of observations has an equal p r o b a b i l i t y of occurrence.
This usually
means that during the sampling process each observation has an equal chance of s e l e c t i o n at any time.
This i s d e s i r a b l e , since i t permits unexpected
termination of the sampling process without a f f e c t i n g the randomness of the smaller sample. This type of random s e l e c t i o n i s presumed f o r most estimates of population variance. Many sampling schemes can be considered of t h i s type by s u i t a b l e d e f i n i t i o n s of what s h a l l be considered "an observation".
Two main points are of i n t e r e s t .
F i r s t , such a sample y i e l d s an unbiased estimate of the mean 2 ( M ) and variance ( C T ) of the population. Unbiasedness
i s s t i l l considered
by many to be a very d e s i r a b l e feature f o r an estimator. There are other sampling schemes which also y i e l d unbiased estimates of the mean.
- 7-
One way such estimates are e a s i l y produced i s by simply assuring that each observation has an equal p r o b a b i l i t y of being sampled. Systematic sampling can often have t h i s e f f e c t , and i s u s u a l l y meant to. Unbiasedness i s not hard to produce i n an estimator, nor i s i t u n i v e r s a l l y accepted as a d e s i r a b l e feature. expected mean square error
E £ (x -
The smaller
which can be produced by
Baysian estimation, many robust methods and other techniques, i s o f t e n gained by accepting very small biases.
The problem i s to assure the
researcher that these biases can indeed be expected to be small. The major arguments f o r the concept of unbiasedness are given by Brunk (1965).
1. To s t a t e that an estimator i s unbiased i s to s t a t e that there i s a measure of c e n t r a l tendency, the mean, of the d i s t r i b u t i o n of the estimator, which i s equal to the population parameter. This i s simply the d e f i n i t i o n of unbiasedness. An equally appealing property, however, from t h i s point of view, might, f o r example, be that the median (page 348) of the estimator be equal to the population parameter. 2. For many unbiased estimators one can conclude, by applying the law of large numbers, that when the sample s i z e i s large the estimator i s l i k e l y to be near the population parameter. However, t h i s i s the property of consistency, discussed below; and the argument here i s not p r i m a r i l y i n favor of unbiasedness, but i n favor of consistency. For
J
°^ t*ie
example, the unbiased estimator, |^ ^ " 1 population variance has thi,s property; but so also does the sample variance s i t s e l f . 3. An important advantage from the point of view of the development of the theory of s t a t i s t i c a l inference i s that i n many respects unbiased estimators are simpler to deal with. The l i n e a r properties of the expectation are p a r t i c u l a r l y convenient i n dealing with unbiased estimators. I f , f o r example
- 8-
6 i s a parameter having 6^ and Q^ as unbiased estimators i n two d i f f e r e n t experiments, every weighted mean ad^ + with a + /3 = 1 i s also an unbiased estimator of 6 . We note, however, that non-linear transformations do not i n general preserve unbiasedness. For example, 6 i s an unbiased estimator of2 6, then $ is not an unbiased estimator of 6 . One point of view i s that the class of a l l possible estimators of a p a r t i c u l a r parameter i s unmanageably large. A way of approaching the problem i s to r e s t r i c t attention to an important subclass, such as the class of unbiased estimators.
From the point of view of most p r a c t i c a l research we can dismiss the random sample from further consideration i f unbiasedness i s the only c r i t e r i a of interest. much of sampling
The use of the random sample i n
theory i s not necessary, but rather a device f o r
simplifying the mathematics. The second main feature of a random sample i s the known standard deviation of the mean
( 6
between 1 and the maximum expected value f o r
~y ' DBH_^" .
Grosenbaugh's
system f o r 3P sampling i s s i m i l a r , except that tie ^ deals with the e n t i r e population rather than a subset. The discussions i n t h i s t h e s i s w i l l generally regard the occasion when a sample i s not drawn at a random point as a waste of time and e f f o r t .
This i s not s t r i c t l y true, since the occurrence of vacant
p l o t s might be used f o r p o s t - s t r a t i f i c a t i o n to increase the p r e c i s i o n of the estimator or f o r other uses. 2 6 p r o p o r t i o n a l to DBH
"
In the example of choosing a tree
the l i s t s of DBH
on p l o t s where no sample was
chosen might be used as a d d i t i o n a l information f o r estimating the parameter of i n t e r e s t , much l i k e Grosenbaugh's use of the "adjusted" rather than the "unadjusted"
estimator (Grosenbaugh, 1976).
e f f i c i e n c y under these considerations hecomesheavily and
Since
complexly
involved with the choice of estimator, a simpler c r i t e r i o n w i l l be applied.
The c r i t e r i a
of e f f i c i e n c y i s the p r o b a b i l i t y of s e l e c t i n g a
- 34 -
random sample w i t h minimum f i e l d work, which g e n e r a l l y means s e l e c t i n g a tree at each random sample point. Systems which screen the population through two or more stages can be constructed i n a v a r i e t y of ways. p r o b a b i l i t y s e l e c t i o n scheme.
Consider the equal
At the f i r s t stage we could s e l e c t a
c l u s t e r of t r e e s , each s e l e c t i o n being p r o p o r t i o n a l to tree b a s a l area (or any other c r i t e r i a which would s i m p l i f y f i e l d s e l e c t i o n ) .
At the
second stage t h i s manageable subset could s e l e c t an i n d i v i d u a l p r o p o r t i o n a l to (1/basal area) by the e l i m i n a t i o n system.
The product
of the two p r o b a b i l i t i e s would be:
M4 • s
where:
*(-=7)
-(-c^f)
< 6 4
>
BA. = the b a s a l area of tree i 1 C^,C2= constants depending on the exact s e l e c t i o n system used. C^ depends on the 1st stage s e l e c t i o n p r o b a b i l i t y (probably on the c r i t i c a l angle discussed l a t e r ) . C2 depends on the l a r g e s t p o s s i b l e value f o r i=l The product i s a constant, regardless of tree b a s a l area. The
r e l a t i v e p r o b a b i l i t y of s e l e c t i o n i s therefore 1/N. Such a m u l t i p l e stage s e l e c t i o n system may be of advantage where simple mechanical means can be contrived to s e l e c t trees w i t h compensating or more stages.
p r o b a b i l i t i e s at one
The v a r i a b i l i t y of these schemes would be of primary
i n t e r e s t , and would have to be robust under a v a r i e t y of s p a t i a l
- 35 -
distributions.
Simulation of stem maps would probably be the best
device f o r assuring t h i s property. Because of the f i e l d s i m p l i c i t y of s e l e c t i o n s based on f i x e d distances or angles, and l i s t s e l e c t i o n from short l i s t s , the e l i m i n a t i o n method w i l l be of primary i n t e r e s t .
Diameter or Circumference Weighting
S e l e c t i o n P r o p o r t i o n a l to Diameter Alone
This problem was e s s e n t i a l l y solved by Strand (1958) with the i n t r o d u c t i o n of h o r i z o n t a l l i n e sampling.
Trees are sighted a t r i g h t
angles along transects through the sample area.
A tree can be selected
when the tree stem subtends the angle projected perpendicular line.
Figure 6 shows an example.
crosses an unseen p l o t surrounding basal area.
to the
A tree w i l l be picked i f the l i n e the tree and p r o p o r t i o n a l to i t s
The use of the angle gauge determines when you are i n the
p l o t , the magnitude of the angle determines the absolute s i z e of the unseen p l o t .
The dashed l i n e s i n Figure 6 i n d i c a t e the unseen borders
of the p l o t s around each t r e e .
To s i m p l i f y f i e l d work the trees are
sometimes sighted to only one side of the transect. equations used are simple.
Changes to the
The assumption i n the f o l l o w i n g discussions
i s that trees are sighted to both sides of the transect. The p r o b a b i l i t y of s e l e c t i o n obviously i s p r o p o r t i o n a l to the diameter of the unseen p l o t around each t r e e . principle.
Figure 7 i l l u s t r a t e s the
The angle, when small, acts i n the same way as c a l i p e r i n g
- 36 -
Figure 6.
H o r i z o n t a l l i n e sampling, basic idea of the s e l e c t i o n r u l e .
transect
- 37 -
Figure 7.
I l l u s t r a t i o n of s e l e c t i o n p r o b a b i l i t y w i t h l i n e sampling.
-©-
distance = D.*PDF x
e e— d i r e c t i o n of transect
i
— &
e
- 38 -
a tree f o r diameter ( D j and m u l t i p l y i n g by a constant which we s h a l l c a l l the P l o t Diameter Factor (PDF).
Under the assumption that trees
are "convex outward" i n c r o s s - s e c t i o n ( i . e . f l a t or curved outward so that a s t r i n g wrapped around i t would always touch the surface) the average of a l l p o s s i b l e c a l i p e r measurements i s equal to the perimeter divided by ir .
The h o r i z o n t a l l i n e sample therefore s e l e c t s trees
p r o p o r t i o n a l to the perimeter or average diameter.
Grosenbaugh (1958)
has pointed out that large angles can cause c e r t a i n biases, since the angle gauge no longer nearly resembles the c a l i p e r measurement.
This
i s of l i t t l e concern f o r most p r a c t i c a l work which uses angles i n the range of .03 to .08 radians (roughly 1.7 to 4.5 degrees). In the case where angles must be large f o r some reason, or trees cannot be considered
convex outward there i s always the option
of sampling with a p l o t , then choosing from a l i s t of cumulative diameters or perimeters by the e l i m i n a t i o n method.
Two v a r i a n t s of h o r i z o n t a l l i n e
methods are biased, and the bias can be c a l c u l a t e d by the construction of preference maps. With the f i r s t method a transect i s s t a r t e d from a random point and the f i r s t tree " i n " w i t h the angle gauge i s chosen f o r sampling. A preference map can be drawn only where the d i r e c t i o n of the l i n e i s assumed.
A l i n e centered at the t r e e , and of length (PDF * D^) i s
constructed
(see Figure 8 ) . D^ i s the diameter of the tree at the
s i g h t i n g point of the angle gauge.
-
Figure 8.
39
-
Construction of l i n e sample preference
tree number
map.
- 40 -
The PDF i s a constant r e l a t i n g tree diameter to the diameter of an unseen p l o t surrounding the tree.
The value depends upon the
" c r i t i c a l angle" ( 6 ) which i s used.
This tree w i l l be chosen f o r sampling whenever a point i s chosen along i t s band.
The bands run opposite the d i r e c t i o n of the sampling transects
u n t i l intercepted by another tree or the area border.
The p r o p o r t i o n a l
area of these bands (ab^) determine the p r o b a b i l i t y of sampling the tree.
Specifically:
P r o b a b i l i t y of sampling a p a r t i c u l a r tree:
(7.1)
Relative probability i s :
Pr
(7.2)
i=l The p r o b a b i l i t y of sampling a tree along a transect beginning at a random point i s :
N
(7.3)
- 41 -
If the transect l i n e s are of f i x e d length (L^) there i s only a slight modification. at length
A l l bands of the preference map are truncated'
from the tree.
the revised diagram.
A l l equations above w i l l then apply to
There i s an obvious bias toward s o l i t a r y trees
and those on one edge of clumps.
As the diameter of the p l o t s increases
we approach a l i m i t i n g s i t u a t i o n where only the distance to the next tree i n the population ( i n the d i r e c t i o n of the transect) i s the determining factor. These diagrams lend themselves plotting.
e a s i l y to computer
The s t r a i g h t l i n e s , truncated only by i n t e r c e p t i o n of another
tree or the area border, are simply c a l c u l a t e d and drawn, and areas are e a s i l y computed.
When the transect i s run other than North-South,
i t is
easy to t r a n s l a t e a l l the coordinates and act as though the transects were North-South. A second method i s to extend a l i n e from a random point i n the t r a c t , and sample the f i r s t tree intercepted. This i s e s s e n t i a l l y the same as the previous system, but i n t e r c e p t i o n along a simple compass l i n e i s used instead of using an angle gauge.
The bands of the preference
map w i l l be longer and narrower, and a mathematical approach w i l l probably be needed because of the small scale involved. The width of the band i s simply the diameter of the tree.
S e l e c t i o n P r o p o r t i o n a l to Diameter and a Constant
(C ) g
A d i f f e r e n t v a r i a t i o n i s to sample the f i r s t tree encountered by running a s t r i p of s p e c i f i e d width (W ) centered on a transect
- 42 -
I f the tree i s included when any part of i t i s w i t h i n the s t r i p sampling i s p r o p o r t i o n a l to diameter plus a constant.
When the tree
i s sampled only i f the tree center i s included t h i s takes the form of sampling with a rectangular p l o t of varying length. map for the f i r s t method can be e a s i l y constructed. i s simply D. + W , centered at the tree. 1 s same.
Figure 9 shows the basic idea.
The preference
The band width
Further construction i s the
The e l i m i n a t i o n technique can be
used, with a constant length s t r i p , to obtain a random sample. s e l e c t i o n i s to be p r o p o r t i o n a l to required.
- C
g
When
only a small change i s
In t h i s case, the tree i s chosen only when i t i s e n t i r e l y
w i t h i n the s t r i p .
Figure 10 shows the construction of the preference
band of such a t r e e .
Problems of Scale and F i e l d Use
I f the r a t i o of s t r i p width to diameter i s too large or too small the diameter can be "expanded" or "reduced" to make i t easier to do the sampling.
Reduction can be done by c a l i p e r i n g and using
(1/x) * (D > as the diameter. i
A l t e r n a t e l y an angle gauge can be used
to get a p r o p o r t i o n a l reduction on the p r i n c i p l e s of the Biltmore S t i c k . Unfortunately, such a system i s very dependent on a round c r o s s - s e c t i o n . Figure 11 shows the p r i n c i p l e .
\
- 43 -
Figure 9.
Band width f o r the preference map when sampling i s p r o p o r t i o n a l to + C. g
Band width i s W + D. s 1
center l i n e of s t r i p . s t r i p of width W
The tree w i l l be chosen whenever the s t r i p passes between e i t h e r of these l i m i t i n g s i t u a t i o n s . The distance between the center l i n e s then determines the p r o b a b i l i t y of s e l e c t i n g that tree.
- 44 -
Figure 10.
Band width f o r the preference map when sampling i s p r o p o r t i o n a l to D. - C . i s r
Band width i s W - D. s 1
s t r i p of width W
D.
The tree i s chosen when the s t r i p i s between these two extremes. The distance between the c e n t e r l i n e s determines the p r o b a b i l i t y of selection.
- 45 -
Figure 11.
Using an angle gauge to e s t a b l i s h the reduced diameter of a tree.
.The angle of the instrument determines the proportion of D. to D . The distance between the contact p o i n t s 1 r i s then used as the reduced diameter of the tree.
- 46 -
P r o p o r t i o n a l expansion of the tree diameter i s easy to e s t a b l i s h with the angle gauge.
Often the use of the expanded diameter
makes the constant term a more reasonable distance to measure. Figures 12 and 13 show two ways to s e l e c t , i n the f i e l d , sample trees proportional to D. - C . S i m i l a r l y Figure 14 shows a system f o r X s s e l e c t i o n p r o p o r t i o n a l to D. + C . A l a t e r discussion w i l l o u t l i n e i
s
the use of the Wheeler Pentaprism to automatically account f o r the s t r i p width.
- 47 -
Figure 12.
F i e l d systems to select trees p r o p o r t i o n a l to D. minus C . i s
I f the tree i s " i n " along the transect move back a distance W and sight again. Sample the tree only i f i t i s " i n " at both p o i n t s .
- 48 -
Figure 14.
F i e l d system to s e l e c t tree proportional to D. plus C . x s
i
w
s
r
Select the tree i f i t i s " i n " with the prism to one side of the transect or_ w i t h i n the distance W from the transect on the other side.
- 49 -
Tree Height Weighting
This problem i s solved by using v e r t i c a l angles i n much the same way that a h o r i z o n t a l angle was used on tree diameters. A tree i s to be sampled when a transect passes w i t h i n a distance p r o p o r t i o n a l t o t r e e height.
A simple method, i n f l a t t e r r a i n , i s
to sample the tree when i t f a l l s w i t h i n a p a r t i c u l a r v e r t i c a l . a n g l e ( C ) . v
Figure 15 shows the general idea.
The angle i s being used to
e s t a b l i s h the proportion of height to distance. H i r a t a (1962) describes t h i s method to sample f o r e s t areas f o r volume.
I n sloping
t e r r a i n t h i s angle s e l e c t i o n method s t i l l s e l e c t s p r o p o r t i o n a l to height, but the exact proportion i s no longer given by the tangent of the angle. In such cases, the d i f f e r e n c e between two tangents i s used, read from a s u i t a b l y c a l i b r a t e d instrument such as the Sunto Clinometer.
Figure 16
shows the p r i n c i p l e involved. The preference map i s constructed i n exactly the same way as with h o r i z o n t a l l i n e sampling.
The system a p p l i e s equally w e l l to some
segment of the tree height such as distance to the f i r s t limb, merchantable height, crown length, e t c .
The reasoning concerning the e l i m i n a t i o n
method and a p p l i c a t i o n of a constant term are the same as with h o r i z o n t a l l i n e sampling.
The band width i s determined by height rather than
diameter, but otherwise there i s no d i f f e r e n c e i n the p r o b a b i l i t i e s or preference maps.
-
Figure 15.
50
-
Use of the angle gauge on f l a t ground f o r v e r t i c a l sampling.
- 51 -
The e f f e c t of leaning trees i s discussed by Loetsch e_t a l . (1973).
I f the tree i s not e a s i l y seen along a perpendicular
from
the transect i t can be tested from any other point which maintains the same distance as from the line, t o the tree.
An Adaptation
of L i n e - I n t e r s e c t Sampling to Standing Trees
I f a l l the trees on a t r a c t were f e l l e d perpendicular random transect of length L, the t r a c t volume could be simply as a s p e c i a l case of l i n e - i n t e r s e c t sampling.
to a estimated
The estimate, from a
s i n g l e transect, of t o t a l t r a c t volume would be: n CA. V =
where:
CA
i
i=l
i s the c r o s s - s e c t i o n a l area of a stem crossed by the transect,
This i s the average depth of tree cross-sections along the transect m u l t i p l i e d by the t r a c t area.
The trees w i l l not i n fact be
conveniently l a i d across the transect, but could t h i s s i t u a t i o n be simulated i n some way?
The key point i s that the diameter to measure
i s the one which i s the same distance up the standing tree stem as the h o r i z o n t a l distance from the transect to the tree.
That point can
e a s i l y be found on l e v e l ground by looking up the tree at a [""M] radian (45°) v e r t i c a l angle.
On sloped ground a "% s c a l e " as i l l u s t r a t e d
i n Figure 16 can e a s i l y be used.
We thus have a simple way of using
l i n e - i n t e r s e c t sampling theory on standing trees.
-
Figure 16.
52
-
Use of two measurements on sloping ground.
(tan b - tan a) = tan C^, g i v i n g correct h o r i z o n t a l sampling distance i n sloped areas.
- 53 -
If
T / 4 radians i s an inconvenient angle f o r some reason,
the estimating equation can be s l i g h t l y changed. a tree w i l l be sighted with a v e r t i c a l angle
distance = tree height
*
The distance over which
is:
cotan C
Since the tree i s "stretched" out over a longer distance, the estimate must be correspondingly decreased.
The f i n a l formula being:
Z i BA
i=l
V =
*
T
*
tan C v
A recent work by Minowa (1976) i n the Japanese language appears to have been based on the same idea.
Although a complete t r a n s l a t i o n i s not
a v a i l a b l e h i s formula i s c l e a r l y equivalent t o the one j u s t derived. Minowa gives i t as f o l l o w s :
V
J
\ 4 * L * c o t
\ ^ J
y
g y
From the a r t i c l e c i t e d and subsequent work (Minowa, 1978), i t seems obvious that Minowa has not recognized that t h i s part of h i s work i s a s p e c i a l case of l i n e - i n t e r s e c t sampling.
This i s an important step ,
since t h i s form of sampling has accumulated a considerable amount of l i t e r a t u r e , f i e l d t e s t i n g and acceptance by f i e l d f o r e s t e r s .
- 54 -
Basal Area Weighting
Since the development of the sampling method of B i t t e r l i c h (1948), u s u a l l y c a l l e d h o r i z o n t a l point sampling or v a r i a b l e p l o t sampling, a great deal of f o r e s t inventory has been done with s e l e c t i o n of trees p r o p o r t i o n a l to basal area.
Since a simple count
of trees which are " i n " with an angle gauge only provides an
estimate
of basal area, several trees are u s u a l l y subsampled for volume characteristics. This i s done mainly to e s t a b l i s h the "Volume to Basal Area R a t i o " (VBAR).
This r a t i o i s u s u a l l y much l e s s v a r i a b l e than the number
of " i n " trees.
Only a few of these trees should be measured f o r maximum
sampling e f f i c i e n c y .
The cost of measuring t r e e s , compared to simply
counting those which subtend the angle used, i s quite high. cost perhaps only one i n twenty trees should be measured.
To minimize The problem
i s to choose the volume sample trees at l e a s t unbiasedly i f not at random. One attempt to evade the r e l a t i v e cost problem i s to construct "Diameter-Height Curves" showing the tree heights f o r each diameter c l a s s . The diameters of a l l " i n " trees are then measured, at l i t t l e expense, and heights are read from the curve.
I t i s open to question whether the
extra e f f o r t might not be invested more p r o f i t a b l y i n taking more p l o t s for tree count only.
Once such a Diameter-Height Curve i s established
i t c o n s t i t u t e s a bias i n subsequent measurements which should be accounted for
statistically.
- 55 -
The process by which subsamples are chosen w i l l be examined i n some d e t a i l i n regards to possible biases and non-randomness.
Basic Ideas of Point Sampling
In point sampling a tree i s counted when the stem subtends an angle established with some form of angle gauge.
The angle radiates
from a s i n g l e point on the land area being sampled.
There i s therefore
a c i r c l e around each tree of diameter (D_^ * PDF) where that tree w i l l be counted " i n " with the angle gauge.
Figure 17 shows an example.
The
i l l u s t r a t i o n i s very much l i k e Figure 2, except that the c i r c l e s have an area p r o p o r t i o n a l to the basal area of the stem.
The p r o b a b i l i t y
of p i c k i n g a tree with an angle gauge from a random point i s obviously p r o p o r t i o n a l to the basal area of the stem.
More formally the
p r o b a b i l i t i e s are:
For p i c k i n g an i n d i v i d u a l t r e e :
(8.1)
P
R e l a t i v e p r o b a b i l i t y of p i c k i n g a tree:
(8.2)
i=l
i=l
- 56 -
Figure 17.
P l o t s of v a r i a b l e s i z e i n h o r i z o n t a l point sampling.
tree 1
- 57 -
S l i g h t l y more d i f f i c u l t are the p r o b a b i l i t i e s of sampling when one of the trees i s randomly chosen from a c l u s t e r which are a l l " i n " from a p a r t i c u l a r point.
Choosing c l u s t e r s of trees makes a sample
non-random, but s e l e c t i n g one tree from each c l u s t e r can seldom f a i l to be biased as w e l l .
Bias From S e l e c t i o n of a Single I n d i v i d u a l From Every Cluster
S e l e c t i o n of one i n d i v i d u a l has a l l the biases discussed e a r l i e r with f i x e d p l o t s .
I t favors c l u s t e r s with small numbers of trees, but
t h i s c l u s t e r i n g e f f e c t i s changed because smaller trees have smaller plots.
In a d d i t i o n , when the s e l e c t i o n i s not random, further biases
can e x i s t .
Each of the simple systems can be explored by preference
maps.
Bias From Random S e l e c t i o n From Each Group
A f t e r the mapping of p l o t s around each of the trees a group of compartments i s formed. trees involved
For each of these t compartments the number of
(n^) and i t s area (a^) are determined. The expectations
f o r sampling are
then calculated as follows:
- 58 -
P r o b a b i l i t y of sampling a p a r t i c u l a r tree:
t. 1 1.0 T
(2.1)
R e l a t i v e p r o b a b i l i t y of sampling a tree:
Pr
(2.2)
N
i=l These are simply the equations used e a r l i e r with f i x e d p l o t s . 2 The d i f f e r e n c e i s that the area of the p l o t s i s now (BA_^ * PDF ) rather than f i x e d .
To complete the example, the c a l c u l a t i o n s are given f o r
Figure 17 i n Table 2. In general, there i s a decrease i n i n d i v i d u a l p r o b a b i l i t y as more p l o t s overlap, i n d i c a t i n g a bias again toward sparsely d i s t r i b u t e d trees.
The average number of trees counted from a random point i s a
f u n c t i o n of the basal area of the t r e e s , not of t h e i r numbers. basal area i s evenly d i s t r i b u t e d bias i s reduced.
Where
Stratification w i l l
help to reduce the bias when i t i s done to equalize basal area.
The
overlap of p l o t s w i l l have a greater e f f e c t on smaller t r e e s , where the overlap i s a greater proportion of t o t a l area, therefore the mixture of s i z e classes favors s e l e c t i o n of l a r g e r t r e e s .
- 59 -
Table 2.
C a l c u l a t i o n s f o r the example shown i n Figure 17 " kl a
Compartment
tree 2 3 4
1
a
*
b
*
c
*
*
d
*
5
3
k
k
n
15.5
1
15. 5
A
4.83
2
2.42
*
1.33
3
0. 44
*
3.33
2
1. 67
e
*
3.00
1
3. 00
f
A
*
1.83
2
0. 92
g
* *
*
* A
0.5
3
0. 17
0.17
2
0 08
i
A A
0.5
2
0 25
3 k
A
2.17
1
2 17
A
4.83
1
4 83
1
A A
1.83
2
0 .92
m
A
43.17
1
43 17
h
83.00
Relative Probability actual
Tree
theoretical
-4.5%
1
20.02
,2412
25/99*
2
6.28
,0757
10/99
-25%
3
3.51
.0423
5/99
-16%
4
9.11
.1098
14/99
-22.5%
5
44.08
.5311
45/99
+17%
83.00 nr *
Selection Bias
sum of p l o t areas = 99 m , tree 1 has a p l o t of 25 m .
- 60 -
b
)
Nearest Tree Method
When the nearest tree which i s " i n " with an angle gauge i s chosen f o r sampling the preference
map i s b a s i c a l l y the same as
i t was w i t h f i x e d p l o t s , but the s i z e of the p l o t around the t r e e v a r i e s with tree s i z e .
The area formed by the overlap of two p l o t s
i s a l l o c a t e d to the smaller one unless the b i s e c t o r f a l l s w i t h i n the overlap.
See Figures 18c to 18e. As shown i n Figure 18, the i n i t i a l increase i n s e l e c t i o n
p r o b a b i l i t y f o r two p l o t s i s i n favor of the smaller (See 18b). The bias i n r e l a t i v e s e l e c t i o n - p r o b a b i l i t y eventually equalizes as p l o t centers approach each other, and favor the l a r g e r p l o t only w i t h very close spacing as i n Figure 18d and 18e. As w i l l now be shown, a random spacing favors the smaller t r e e . Consider the p l o t s of two trees (a) i s the smaller and (b) i s the l a r g e r .
See Figure 19. We w i l l choose one p l o t , and i t s
associated t r e e , f o r sampling when we are w i t h i n that p l o t and that p l o t center i s nearest the random point P. when a and b overlay the random p o i n t ,
Consider what w i l l happen
b w i l l never be chosen when
a and the outer r i n g of b ( b ) occur at the p o i n t , since the center of Q
b
would always be f u r t h e r away.
Since b^ i s the same s i z e as a i t has
an equal chance of being c l o s e s t to P when a and b^ occur together. The c o n d i t i o n a l p r o b a b i l i t y of choosing b when a and b overlay, P i s therefore reduced,
b has a 50% chance of s e l e c t i o n only i f b.
and a overlay P, and no chance otherwise.
l
- 61 -
-
Figure 19.
62
-
I l l u s t r a t i o n of terms used i n proof of bias toward smaller trees.
area b. 1
= area a
area b
= area b. + area b 1 o
- 63 -
Ignoring edge e f f e c t s , the exact p r o b a b i l i t i e s are noted below.
A = area of p l o t a B = area of p l o t b B = area of outer r i n g of p l o t b as i n Figure 19, o i t i s the same as B-A. B_^= area of the inner c i r c l e of p l o t b, i t i s the same s i z e as p l o t a.
P r o b a b i l i t y that a occurs alone, hence a w i l l always be chosen.
P r o b a b i l i t y that b occurs alone, hence b w i l l always be chosen.
P r o b a b i l i t y that a and b occur together.
since B. = A = B * 1
C o n d i t i o n a l p r o b a b i l i t y of choosing a and b given that a and b^ occur together:
- 64 -
p \ a| > }
~ h
(9.4)
P |b | a , b |
= h
(9.5)
a
h
±
i
P r o b a b i l i t y that a and b
Q
occur together, hence a i s always chosen.
(9.6)
The e f f e c t of the r e l a t i v e p r o b a b i l i t i e s i s not d i f f i c u l t to derive formally using these equations i n combination.
The p r o b a b i l i t y of choosing a i s :
P { S
a
} =
p
|a,b| +
4~[ P
_A_ T
+
_A_ T
| ' i } ] a
b
+
P
I > of a
b
. T
J
T
A
+
B-A
_A_ T
[
_A_ T
[ r ] - + [-f
( 9
VT
- > 7
/
(T-B) + (hk) + (B-A) T
J
The p r o b a b i l i t y of sampling a i s
t
(9.8)
£A/T (1 - R )J , where the term R FL
a
stands f o r the proportional reduction due to both p l o t overlap and the s e l e c t i o n system.
We can do the same thing f o r tree b.
- 65 -
p
{ S (
=
b
p
{ b,a
\
(9.9)
{*S>±\\
+4"[P
•fM-M+W] T
T J
\
T
K)
T
4-]
[f
+
B
(9.10)
I^B/T B/T
We now have the form
I f R, i s l a r g e r than R D
((1 l - R^
there i s a greater p r o p o r t i o n a l reduction i n the cL
large p l o t than i n the small one.
p
{^}
i s
p r o p o r t i o n a l l y reduced i f
and only i f :
R
h
>
\
iff
(f) - |>
> *
+]
iff I
B y
T
T
iff
_]
* PDF * D
,
* TT J
±
|
[ ,
.
PDF
+
*
2
j
I
-2
*
(10.4)
±
* J
(10.5)
T o t a l area = i n s i d e p l o t + s t r i p area + small c i r c l e area PDF
2
* TT * D.
2
l
+
£w
g
*
D
*
±
TT
*
PDFJ +
Jw
g
2
7rj
(10.6)
Here we have solved the problem except f o r the f i n a l complicating term
W
s
2
w
, the area of a c i r c l e of radius W . The easiest way out ' s
of t h i s i s to remove i t from the center of the p l o t .
We then have a
p l o t which has a void i n the center (of radius W ) as shown i n Figure 24. s 2 r
The area of t h i s p l o t i s aD^
+ bD^
The angle gauge used
to pick the p l o t determines a, while b i s determined by the s t r i p width. An example may help to c l a r i f y the procedure. We wish to sample trees p r o p o r t i o n a l to the equation:
P |si|
CX
7.8 D
+ 3 D.
2 ±
We also wish to use a s t r i p width of 2 meters f o r convenience i n the field.
As already shown:
^ PDF
*
j
D
^2
+
jp
D F
* ^ * w
„. *
D
J=
plot
area
- 76
Figure 24.
-
P l o t of an area p r o p o r t i o n a l to aD. plus bD
- 77 -
The problem i s to choose PDF. By d e f i n i t i o n :
PDF
* *
D. i
2
=aD. i
2
=
7.8 D. l
2
and |PDF * W
[
s
*T*D.l=bD.=3D
ij
i
i
by c a n c e l l i n g the D^ term from both sides we obtain:
PDF
*
w
= a
and
j^PDF * W * TTJ = b g
and m u l t i p l y i n g by a/b:
(a/b) * PDF *
therefore (a/b) * PDF *
W
s * ' ] ' [PDF
c a n c e l l i n g terms we get the general equation:
(a/b) * W
s
* 4 = PDF J
i n s e r t i n g the example values we have:
PDF =
(7.8/3) * 2 * 4|= 20.8
* TT *
1/4
=a
- 78 -
The angle needed to produce t h i s r e l a t i o n s h i p i s
0 = 2 *
arcsin
I PDF I
Figure 25 shows the basic geometry needed to derive t h i s formula. To check the r e s u l t s we compute the following examples. Tree 1,
diameter = 20 cm, PDF = 20.8, s t r i p width = 2 meters.
7.8 (.2) + 3 (.2) = .912 2
p l o t area =
20.8
T
(.2)
z
+ (20.8 * TT * 2)
(.2)
39.73
39.737.912=
43.563
Tree 2, diameter = 1 meter
7.8 (1) + 3 (1) = 10.8
p l o t area
20.8
-]
470.485
470.485 / 10.8 = 43.563
(1)
+
(20.8 * TT * 2) (1)
- 79 -
Figure 25.
Geometry used to determine the angle 9
hence:
—r-
.
=
.0480 radians (2.755°)
e =
.0962 radians (5.511)
- 80 -
The a c t u a l p l o t i s 43.563 times as large as the number 2 given by the formula (7.8D_^ + 3D^) due to the s p e c i f i e d s t r i p width of 2 meters.
I f i t were d e s i r a b l e we could reduce both the l i n e a r
dimensions of the P l o t Diameter Factor and the s t r i p width by a factor of
43.563 to make the p l o t area i n square meters have the
same magnitude. Once the s t r i p width and angle gauge have been established the f i e l d work i s s t r a i g h t forward.
F i r s t , a random point P i s selected.
Second, a l l trees around P are sighted with the angle gauge at a distance W
g
from P.
Figure 26 shows the f i e l d s e l e c t i o n scheme.
A random
s e l e c t i o n can then be made from trees by the e l i m i n a t i o n method. A recent a r t i c l e by Schreuder (1978) uses j u s t such a s e l e c t i o n scheme i n a method c a l l e d "Count Sampling".
Although Schreuder does not
consider i t "a p r a c t i c a l system" i t can be considerably
s i m p l i f i e d and
improved f o r f i e l d a p p l i c a t i o n by modifying the Wheeler Pentaprism as described i n a f o l l o w i n g
section.
S e l e c t i o n of trees with a p r o b a b i l i t y proportional to J a D ^ 2 requires a s l i g h t m o d i f i c a t i o n .
The area W
g
2
- bD^j
TT must be added to the
p l o t , and t h i s can be done by adding 'it to the center of the p l o t and g i v i n g the t r e e two chances of s e l e c t i o n when a random point f a l l s i n t h i s area.
A s t r i p i s "removed" from the basic p l o t by s i g h t i n g across the
point from a distance W^.
Figure 27 shows the s e l e c t i o n r u l e .
- 81 -
Figure 26.
S e l e c t i o n proportional to aD. plus bD..
To s e l e c t trees from point P, view a l l trees with an angle gauge at distance W from P.
The p l o t around an i n d i v i d u a l tree shows the locus of a l l points where the tree i s " i n " with the s e l e c t i o n r u l e described above.
- 82 -
Figure 27.
P r i n c i p l e s of s e l e c t i o n proportional to aD. minus bD..
To s e l e c t trees from point P view tree across the point with a prism from distance W .
This diagram shows the p l o t around each tree. The tree i s counted once at a l l points w i t h i n the s o l i d c i r c l e or twice when w i t h i n the dotted c i r c l e .
- 83 -
Mechanical Devices to A i d S e l e c t i o n
The s e l e c t i o n method shown i n Figure 26 could be simulated without moving from point P by mechanical device which functions much l i k e a split-image rangefinder. Figure 28 shows the geometry of the device and the views to be expected through the eyepiece.
Sampling
2
i s p r o p o r t i o n a l to aD_^ + bD^.
a)
tree i s w i t h i n the distance W
b)
of point P. s tree i s outside W , but " i n " with the angle gauge.
c)
tree i s "out" with the angle gauge.
r
g
Figure 29 shows the same basic idea used f o r sampling p r o p o r t i o n a l to aD.
2
- bD..
l
l
Such a device could be b u i l t by simply attaching prisms i n f r o n t of the two lenses of a Wheeler Pentaprism which i s a v a i l a b l e commercially. This allows easy determination without leaving the sample point as long as the tree can be seen.
F a i r l y wide border s t r i p s could be accommodated
i n t h i s way. Using an angle of .0262 radians (1.5°) a 100 cm base would allow a 38 meter width f o r W . g
Alignment, measurement, and slope problems
would make such a s t r i p i n f e a s i b l e under f i e l d conditions by other methods.
Line of s i g h t w i l l remain a major problem.
Figure 28.
Device p r i n c i p l e s f o r sampling proportional to aD^ plus
bD^
Figure 29.
Device p r i n c i p l e s f o r sampling proportional to aD. minus bD..
" i n " , count twice
" i n " , count once
- 86 -
Adding a Constant, S e l e c t i o n P r o p o r t i o n a l to aD.
+ bD. + c
Adding t h i s f u r t h e r r e s t r i c t i o n to the previous methods forms a cumbersome system.
The only reasonable f i e l d method would be to apply 2
the methods o u t l i n e d f o r aD
+ bD
to a
ir
radian (18CT) sweep,
while using a f i x e d p l o t to select trees through the opposite ir radians. Figures 30 and 31 show the basic idea.
I t would appear d e s i r a b l e to
generalize the rather awkward method and allow s e l e c t i o n with any p l o t s i z e computed by an equation using only diameter as a v a r i a b l e .
One
method, using only standard c r u i s i n g prisms, requires only that the diameter be measured, or adequately estimated.
A prism i s then rotated
to form an angle which w i l l e s t a b l i s h the p l o t s i z e around the tree. For accurate work, p a r t i c u l a r l y with wider angles, two prisms, each of which e s t a b l i s h h the i n i t i a l c r i t i c a l angle ( 9^)
should be counter-
rotated.
The geometrical theory f o r t h i s adaptation i s given by Beers
(1964).
For s i m p l i c i t y the case of a s i n g l e prism w i l l be used f o r the
following discussion. A Generalized
Instrument
For each diameter tree the p l o t s i z e i s l i s t e d . formula below
9, i s e s t a b l i s h e d f o r each diameter
Using the
tree. 2
- 87 -
P r i n c i p l e s of s e l e c t i o n p r o p o r t i o n a l to aD
Figure 30.
+ bD + c.
- 88 -
A prism of d e f l e c t i v e angle
0
i s used where
6
max 0^.
larger than
i s always max
J
A round prism i s best, preferably with a hole
i n the middle f o r easy mounting.
0^ can be formed by r o t a t i o n of
the prism around i t s center by an angle ( A ) . r
e
d
= max * G
C O S
ure 41.
The proportion of occasions the c r i t i c a l point w i l l be i n the crown.
crown base
expanded stem
distance f o r which c r i t i c a l point w i l l be i n the tree crown
- 115 -
A second approach would be to use a taper equation with distance t o the t r e e , tree height and DBH t o c a l c u l a t e where the c r i t i c a l height should f a l l .
I t might be wise to check nearby
v i s i b l e sections to place l i m i t s on the range of p o s s i b l e values, p a r t i c u l a r l y i n the case where frequent broken tops are encountered i n the stand.
Such an " i n d i r e c t " method was suggested by B i t t e r l i c h
(1976) and i n f o r m a l l y by Beers (1974) and Bonnor (1975).
Any b i a s i n
such a procedure w i l l only a f f e c t those measurements where the c r i t i c a l point i s obscured. The second f i e l d problem occurs even when the c r i t i c a l point i s below the crown.
The angle of measurement to the c r i t i c a l point w i l l
probably be considered too steep when the tangent of the angle exceeds 1.5 (about .98 radians or 56 degrees).
This happens i n an area around
the tree where the expanded t r e e radius i s no l e s s than 2/3 of the height to that p o i n t .
This w i l l depend on the taper of the tree and the angle
gauge used. As an example consider a cone where the height i s 50 times the base radius and the p l o t diameter f a c t o r (PDF) i s 100. This leads t o the geometry shown i n Figure 42. The distance we wish t o f i n d i s B, from the tree t o a point where the angle to the c r i t i c a l point has tangent 1.5. The proportion of c r i t i c a l height to t o t a l height i s the same as the proportion of distance from the p l o t boundary to the t r e e . This r e l a t i o n s h i p i s simple because of the c o n i c a l shape.
Since we
wish the r a t i o of c r i t i c a l height t o distance from B to the tree to be 3/2 i t i s p o s s i b l e t o solve f o r p.
- 116 -
Figure 42.
C a l c u l a t i o n of the p r o b a b i l i t y that the c r i t i c a l point w i l l be at too steep an angle f o r accurate measurement.
Side View
p*100
Top View
(l-p)*100
- 117 -
[
3 "I
I" c r i t i c a l height 1
=
I
2 J P
distance B
=
J
["
p * 50
|_ ^"P) *
_ 1 0 0
J
p L ( P) 1 _
2
J
3 (1-p)
p = 3 - 3p 4p = 3
p = (3/4)
hence B = 1/4 * 100 = 25
The general form of the equation f o r p, assuming a c o n i c a l tree form, i s : p
[
where:
"[
ii*
J
(15.1)
(maximum acceptable tangent) * (plot diameter f a c t o r ) (tree height to radius r a t i o ) J
The proportion of times a tree crown w i l l not be measurable w i l l then be ( 1 - p ) . 2
The r e s u l t i s unaffected by the height of the t r e e , although i t i s d i r e c t l y changed by the p l o t diameter f a c t o r and the r e l a t i o n s h i p between the tree height and tree radius.
This example c a l c u l a t i o n t e l l s
us that one sixteenth o f the time the c r i t i c a l point could not be observed on a tree.
This proportion
stems grow t a l l e r and narrower. 30:1
can r i s e sharply as the expanded
At a r a t i o of 50:1 f o r tree height and
f o r p l o t diameter f a c t o r (a more reasonable r a t i o i n p r a c t i c e ) p
becomes .474, g i v i n g an intercept i n the crown about 28% of the time.
- 118 -
One method, when the angle to the c r i t i c a l point i s too large, i s to move away from the tree X times the distance to the tree and use an angle which has (1/X) the tangent of the o r i g i n a l .
For
example, w i t h the wide scale relaskop normally using 2 bars we could double the distance and use one bar width to f i n d the c r i t i c a l point. We could also move 4 times as f a r away and use 1/2 bar to f i n d the same point.
This would probably not be too d i f f i c u l t since the
distances would be small, but i t might be best to avoid the whole issue by not sampling trees i n these cases.
The Rim Method
To avoid these cases we must assume some kind of tree form. We could simply ignore the center parts of the expanded stem, sampling only the "rim" which remained and where the angle to the c r i t i c a l point was not too steep.
Let us again assume a cone shape w i t h a
50:1 height to radius r a t i o (HRR) and a 100:1 p l o t diameter f a c t o r (PDF). Figure 43 i l l u s t r a t e s t h i s example. In the f i e l d a l l trees which covered more than 4 times the usual angle would be ignored. i n s i d e the c e n t r a l area.
These are the cases when the sampler i s
The problem here i s to s p e c i f y the change i n
expected c r i t i c a l height caused by not measuring c r i t i c a l height w i t h i n that section of the expanded stem.
In p r a c t i c e t h i s means s p e c i f y i n g
tree form and s o l v i n g mathematically,. or sampling f o r the proportion. The c o r r e c t i o n term (C
) which would afterwards be applied to c a l c u l a t e rm the f u l l tree average c r i t i c a l height would be:
- 119 -
Figure 43.
Example of calculations when only the "rim" of the expanded stem i s sampled.
(1) cylinder volume
25 *
(2) small cone volume (top)
25 * T * 12.5
(3) entire expanded stem
100 * T * 50
(4) "rim" volume
(3)-(2)-(l) =441,786.5 or 84.4% of f u l l cone
• ^ 37.5 = 73,631.1 unit
= 8,181.2 =523,598.8
3
- 120 -
C rm
cone volume rim volume
Approximating C r i t i c a l Height With the Rim Method
From the viewpoint of a f i e l d worker the ideal sampling system would require no measurements at a l l . l i k e to make very simple counts. an angle gauge.
At most perhaps one would
An example would be tree counts with
I t i s also very l i t t l e trouble simply to read a
v e r t i c a l angle to a point on a tree, since instruments which do this quickly and easily are commercially available. Providing that the v a r i a b i l i t y of any sampling method using v i r t u a l l y no measurements were low enough that method would be very desirable. The problem usually reduces to one of instrumentation, and often to the specific problems of correction for slope or distance.
I f a system requires measurement
of some sort one seeks to base i t on the simplest measurements for f i e l d work. Kitamura has attempted to solve this problem (Kitamura, 1968) but the translation i s very d i f f i c u l t to follow. The following l i n e of reasoning was developed by the author and i s somewhat different, but w i l l be much easier to understand. Let us assume a cone shaped tree of radius B=l and some height K. We note that only the height and form affect the Volume to Basal Area Ratio, so we can work with any diameter we desire. See Figure 44. We w i l l measure the c r i t i c a l height of the tree whenever
- 121 -
Figure 44.
I l l u s t r a t i o n of the terms used to develop an estimating system f o r c r i t i c a l height.
- 122 -
we f a l l w i t h i n the range A to B, so we w i l l be measuring
critical
height i n the "rim" area j u s t discussed. We wish to develop a method to estimate the average c r i t i c a l height of the tree using only the tangent of the angle (0 ) and tree diameter. CH
approach can be used, r e f e r r i n g to Figure 44.
The f o l l o w i n g
We would l i k e to have
a system f o r estimating c r i t i c a l height i n the form:
and
where:
CH
= B*Q*(tan
CH
= B *Q *T
0_„)
Q
a constant, as yet unknown
tan 0, CH
the
(16.01)
(16.02)
tangent of the angle from the base of the
tree to the c r i t i c a l height when viewed from
T
the
sample point.
the
average tangent f o r a t r e e .
We begin by f i n d i n g the average tangent (T):
B
(16.03) A
B
(16.04) A
- 123 -
The cumulative density f u n c t i o n f o r x, given that x i s w i t h i n distance B, i s : 2 CDFN
1
=
2 .5B
=
(16.05)
hence the p r o b a b i l i t y density f u n c t i o n i s :
PDFN
" 2x
=
(16.06)
_B _ 2
Using T as the tangent from a random point x we solve the f o l l o w i n g formula:
B
T
/
=
A
2x
(16.07)
dx
B
K T
dx
=
(16.08)
X
c a n c e l l i n g x and removing 2K/B
yields:
dx
(16.09)
- 124 -
T
=
2K
%(x-B)'
(16.10)
_B _ 3
•B/2 T
=
T
=
0
(16.11)
4- B
2K
B
3
2
j
(16.12)
(16.13)
Knowing that the volume i n the modified cone i s h a l f that of the o r i g i n a l cone we can now solve f o r Q i n the form we wish to have our f i n a l estimator:
CH
E
Volume "I Basal area J
1/2 * 1/3 K B
T
B* 2
(16.14)
Combining t h i s r e s u l t w i t h equation 16.02 we have:
[-H=
CH
=
B*Q*T = B*Q
Therefore the value of Q i s :
Q =
(16.15)
- 125 -
The estimator f o r the c r i t i c a l height w i l l then be:
CH
e
=
B) tan 0
(16.16)
C R
tan 0
C H
(16.17)
A second way to prove t h i s would be as follows: —
=
r Volume I L Basal area J
expressing the volume of a s o l i d of r e v o l u t i o n by the Theorem of Pappus we get:
CH
where:
C
g
2
TT
B
C f
S
(16.18)
TT
= the center of g r a v i t y of the cross-section of the expanded tree stem curve.
S
= the c r o s s - s e c t i o n a l area of the stem from A to B 2
CH
=
(-y- B)
TT
J
B
(x) tan 0
^
B
C R
dx (16.19)
TT
- 126 -
s h i f t i n g terms gives:
CH
=
(+•)
/"
) tan 0
C R
tan 0
CH
dx
dx
(16.20)
(16.21)
At t h i s p o i n t , we can recognize that the term i n square brackets i s the p r o b a b i l i t y of a p a r t i c u l a r value of the tangent occurring (the p r o b a b i l i t y density f u n c t i o n of x ) .
When sampling randomly i n the-
plane we would be choosing the tangent w i t h that p r o b a b i l i t y and under a random sampling process we can drop that term.
tan 0
CH
which gives the same r e s u l t as before.
dx
This leaves:
(16.22)
I t i s of i n t e r e s t because the
method i s very general, and a p p l i e s to any c r o s s - s e c t i o n (stem curve) that i s of i n t e r e s t .
Thus f o r any stem curve we may use the approximating
formula f o r c r i t i c a l height as f o l l o w s :
CH
= C e
g
* tan 0 1
(16.23)
- 127 -
Where C i s the center of g r a v i t y of the expanded stem curve on g one side of the v e r t i c a l a x i s . Kitamura (1968) develops a method s i m i l a r t o the rim method and a l s o uses a s i m i l a r estimator, but there are d i f f e r e n c e s i n application.
Instead of a l l o w i n g the hollow center h i s system
requires the sampler t o back up from the tree u n t i l he i s a c e r t a i n p r o p o r t i o n a l distance from the stem, and then measure the angle to the c r i t i c a l point.
I n e f f e c t , he would be " f i l l i n g " the otherwise hollow
section w i t h a constant.
This i s a great deal of t r o u b l e i n the f i e l d .
A l l t h i s appears to be much ado about very l i t t l e indeed. In adopting the estimation scheme we lose one of the major advantages of c r i t i c a l height sampling - the unbiased sampling procedure to tree form.
sensitive
I f we are w i l l i n g to assume some t r e e form why not j u s t
measure (or estimate) t o t a l height and get VBAR d i r e c t l y ? ?
Certainly
i f one goes to a l l the t r o u b l e of making Kitamura's scheme work i t i s more e f f o r t than simply measuring the distance t o the tree.
Indeed the
whole business seems t o be an awkward contrivance simply to avoid the one h o r i z o n t a l measurement.
S t i l l , there may be an advantage which
Kitamura has overlooked. Consider the two procedures f o r estimating c r i t i c a l height from a random p o i n t . (a)
c r i t i c a l height = distance t o t r e e * tan 0
(b)
c r i t i c a l height = a constant * tan
0^
CH
- 128 -
Method (a) i s the d i r e c t method and measures the height of an imaginary s h e l l around the t r e e .
The estimator therefore has the
same d i s t r i b u t i o n and s t a t i s t i c a l c h a r a c t e r i s t i c s as the tree bole itself.
The mean, variance and density functions are p r o p o r t i o n a l
to the stem form. Method (b) on the other hand has a d i s t r i b u t i o n which i s not the same as the tree bole.
In e f f e c t we have created an "expanded
t r e e " w i t h the same volume (or known proportion thereof) but w i t h an e n t i r e l y d i f f e r e n t "shape".
The d i s t r i b u t i o n of the estimator of tree
volume p h y s i c a l l y surrounding the tree w i l l overlap d i f f e r e n t l y with the estimator of nearby t r e e s .
By manipulating the form of the estimator
we can then change the variance of the sum of c r i t i c a l heights which depends on tree spacing.
I t may be that i n f o r e s t stands, or perhaps
i n the measurement of objects i n other f i e l d s of study, such manipulation of the overlapping shapes could s i g n i f i c a n t l y reduce the sampling variance.
FIELD APPLICATION
The f i e l d work f o r the c r i t i c a l height system has been described from a t h e o r e t i c a l point of view.
As w i t h any sampling system there
w i l l be adjustments necessary f o r p r a c t i c a l f i e l d a p p l i c a t i o n .
Several
p l o t s were e s t a b l i s h e d i n a D o u g l a s - f i r stand near the U n i v e r s i t y of B r i t i s h Columbia t o i d e n t i f y problems i n a p p l i c a t i o n and p o s s i b l e solutions.
- 129 -
The most s t r i k i n g problem i n a p p l i c a t i o n i s with trees which are close to the sample point.
With nearby trees a number
of measurement problems become serious. source of e r r o r .
Tree lean can be a large
Although the c r i t i c a l point w i l l s t i l l be accurately
located the c r i t i c a l height measurement w i l l often be u n r e l i a b l e . The maximum i n t e r c e p t bias i s p o s s i b l e , as discussed by Grosenbaugh (1963).
Correction f o r t h i s type of b i a s can be made f o l l o w i n g h i s
suggestions.
The c r i t i c a l point of nearby trees tends to be i n the
crown, and obscured by f o l i a g e or branches. On the other hand, taper i s r a p i d i n the top s e c t i o n of the t r e e , and there i s a great advantage to the depth of f i e l d f o r d i s t i n g u i s h i n g between the subject tree and the background.
The depth of
f i e l d advantage seems noticeable up to about 11 meters distance from the tree.
While i t i s p o s s i b l e to move away from the tree i n m u l t i p l e s
of the distance between the tree and the point center t h i s was found to be awkward f o r very short distances.
I t i s c l e a r that e i t h e r some
technique to bypass the nearby trees or some other method of c r i t i c a l height measurement i s needed. One a l t e r n a t e method i s to c a l c u l a t e the diameter at the c r i t i c a l point (from tree diameter and distance) then l o c a t e that point and i t s c r i t i c a l height using an o p t i c a l f o r k l i k e the Wheeler Pentaprism.
This o p t i c a l f o r k can be used from any point where the
stem i s v i s i b l e .
The method bypasses problems of steep measurement
angles and considerably reduces the d i f f i c u l t y of seeing the tree stem.
- 130
-
Taper equations can be used, but doing so waives the main advantage of a system designed to be s e n s i t i v e to a c t u a l tree form. Taper f u n c t i o n use would only be advisable i f i t helped to maintain another p o s s i b l e advantage of the system - lower v a r i a b i l i t y due to the d i s t r i b u t i o n c h a r a c t e r i s t i c s of the tree stems.
This would c e r t a i n l y
be i n d i c a t e d on permanent growth p l o t s f o r instance.
Lacking any
proof
that the use of c r i t i c a l height sampling w i l l reduce sampling variance, i t would seem advisable to use the same taper functions i n a normal v a r i a b l e p l o t sampling procedure. The "rim method" discussed previously i s one way to ignore the nearby trees altogether,
but the sampler must c a r e f u l l y keep track of
" i n " t r e e s , e s p e c i a l l y i f the Basal Area Factor i s reduced i n order to increase the tree count at each p o i n t .
This method seems to be the most
promising f i e l d adaptation even though i t too requires an assumption about tree shape. Not a l l problems are removed by s i g h t i n g trees i n the lower bole.
The lower s e c t i o n of the stem has l e s s taper, i s more often obscured
by brush and often has a bad background f o r s i g h t i n g w i t h the relaskop. In a d d i t i o n the stem i n t h i s area i s more l i k e l y to be e l l i p t i c a l rough on the surface.
and
The base of trees i s often impossible to see,
although a f l a s h l i g h t held at stump height w i l l help a great deal i n brush. A more p r a c t i c a l method might be to s i g h t the lower reading on a c o l l a p s a b l e f i b e r g l a s s pole and d i r e c t l y add the distance l a t e r i n the computations.
- 131 -
Relaskop Use
Several h i n t s about relaskops may be u s e f u l . eyes open when using the relaskop.
Keep both
The use of binocular v i s i o n
decreases the d i f f i c u l t y of a poor background on the tree stem. Moving very s l i g h t l y from side to side w i l l often r e v e a l an adequate o u t l i n e of an upper stem even when the crown i s rather dense. often l e s s trouble to read the degree scale i n the relaskop convert afterwards to percent.
It is
and
The percent scale i s frequently hard
to read and abruptly changes scale without adequate l a b e l l i n g .
It is
h e l p f u l i f the BAF i s chosen such that an odd number of bars i s used. This way
they can both be white or black against the stem p r o f i l e as
may best s u i t the background to the t r e e . Adjustment to the sunshade can r e s u l t i n an almost
transparent
image of the relaskop s c a l e , which helps i n l o c a t i n g the c r i t i c a l point. The i n t e n s i t y of the scale can be v a r i e d by moving a thumb i n front of the forward c i r c u l a r window of the relaskop. the scale nearly disappears.
When t h i s window i s covered
Causing the scale to " b l i n k " by moving a
f o r e f i n g e r on and o f f the window i s sometimes h e l p f u l i n f i n d i n g the c r i t i c a l p o i n t , p a r t i c u l a r l y i n low l i g h t conditions. Two m o d i f i c a t i o n s to the relaskop were u s e f u l .
A threaded
i n s e r t a v a i l a b l e at most camera stores w i l l change the metric European t r i p o d thread at the base to a standard E n g l i s h system thread f o r easy t r i p o d mounting.
The second m o d i f i c a t i o n involved removing the side
panel of the relaskop to expose the wheel bearing the measuring scale.
- 132 -
The negative side of i t was marked with a red transparency i n overhead p r o j e c t o r s .
pen used
Shallow negative angles were then very
apparent when the scale turned b r i g h t red.
I t i s necessary to insure
that the marking pen i s not the permanent type, so that e r r o r s can be corrected. To explore the p r e c i s i o n of measuring a tree under f i e l d conditions c r i t i c a l height was determined repeatedly on a 60 cm D o u g l a s - f i r from the distances 12, 16, 20 and 24 meters.
The background
and crown condition of the tree was t y p i c a l f o r a D o u g l a s - f i r
stand,
but the tree was chosen so that no brush would i n t e r f e r e with s i g h t i n g s on the lower bole.
The points on the tree where i t was obviously " i n "
and "out" were a l s o recorded at the same time the c r i t i c a l point estimated.
The r e s u l t s are shown i n Figure 45 f o r hand held
was
relaskop
readings and Figure 46 f o r readings with a t r i p o d mounted relaskop. The increased p r e c i s i o n i s obvious with the t r i p o d .
Every e r r o r of 1
meter i n c r i t i c a l height implies an e r r o r of (1 m cubic meter * i n volume per hectare.
In t h i s case, the BAF was
1.0.
The greater p r e c i s i o n of the t r i p o d mounting was during the f i e l d work.
BAF)
impressive
Differences i n c r i t i c a l height s t i l l appeared,
and tended to occur i n clumps j u s t as they d i d w i t h the hand held instrument, however the cause was e a s i l y determined.
The
relaskop
scale was so s e n s i t i v e that i t was p i c k i n g up the bumps and overgrown knots on the tree stem.
I f anything the relaskop was too s e n s i t i v e ,
even without magnification.
Often the c r i t i c a l point occurred at two
or three places, and whether you moved up or down the tree determined
- 133 -
Figure 45.
24
C r i t i c a l height as measured by hand held relaskop.
20
16
12
distance from tree i n meters
- 134
-
distance from tree i n meters
- 135 -
which one was f i r s t noticed.
This problem was not as serious w i t h
species such as hemlock and cedar where taper was e i t h e r smoother or more r a p i d . In general, there i s no problem determining c r i t i c a l height to acceptable accuracy providing that the tangent of the angle i s not beyond about 1.50 and the l i n e of view i s c l e a r .
I f no assumptions
can be made about tree form i t i s recommended that the diameter of the c r i t i c a l point be c a l c u l a t e d and then located using an o p t i c a l c a l i p e r .
Log Grading
The grading of logs w i t h c r i t i c a l height sampling i s the same as i n standard v a r i a b l e p l o t c r u i s i n g .
In place of the use of a VBAR
for each grade s t a t i n g the cubic volume of wood per u n i t area i n a p a r t i c u l a r grade we have a c r i t i c a l height f o r each grade.
The p o r t i o n
of the stem between the c r i t i c a l point and tree base i s divided i n a s e r i e s of c r i t i c a l heights a t t r i b u t e d to the grades of those sections. The volume i n a grade at each sample point i s then computed by:
Volume
, per ha = grade
V^CH . * BAF X—< grade
These estimates are averaged over a l l points i n the c r u i s e .
With the
c r i t i c a l height method the amount of sampling i n a grade i s p r o p o r t i o n a l to the volume i n the grade.
In standard v a r i a b l e p l o t c r u i s i n g the sampling
i n a grade i s p r o p o r t i o n a l to the basal area of the trees containing that grade.
- 136 -
VARIABILITY OF THE SYSTEM
The v a r i a b i l i t y of c r i t i c a l height sampling was b r i e f l y explored using a s i m u l a t i o n study on an a c t u a l stand of D o u g l a s - f i r trees.
The stand used was established i n approximately 1860, and
contained 192 trees ranging i n diameter from 14 to 160 cm and i n height from 15 to 47 meters.
The median tree was approximately ,4
cubic meters. A c o n i c a l form was assumed f o r a l l trees.
The stand
was clumped and more v a r i a b l e than u s u a l . Repeated simulations were done over 200 random points throughout the area.
BAF and p l o t s i z e were v a r i e d on each run and
variance of the t o t a l volume estimate was c a l c u l a t e d .
The r e s u l t s
are shown i n Figure 47, recorded by the average number of trees measured i n each t e s t .
A few a d d i t i o n a l runs with d i f f e r e n t random
points and l a r g e r sample s i z e s were made to v e r i f y that these r e s u l t s were r e p r e s e n t a t i v e . There appears to be no s t a t i s t i c a l advantage i n the c r i t i c a l height method.
The v a r i a b i l i t y of both c r i t i c a l height sampling or
standard v a r i a b l e p l o t sampling are the same f o r p r a c t i c a l purposes. The advantage of CH sampling i s that i t i s an unbiased estimate of stand volume.
The disadvantage i s that the trees near the sampling
point are d i f f i c u l t to measure.
The approximation to the rim method
using the percent angle and tree diameter proved to have a c o e f f i c i e n t of v a r i a t i o n about 10% higher than the f i r s t two meth'ods.
The l o s s
- 137 -
gure 47.
C o e f f i c i e n t of V a r i a t i o n f o r 5 sampling methods.
140-
F Fixed
plot
A Approximated Rim Method R Rim Method * C r i t i c a l Height
120H
+ Variable Plot
100H
80-
6 0 -
40-
20-
i
4
i
6
•
8
10
12
average number of trees per p l o t
14
- 138 -
i n e f f i c i e n c y does not seem warranted
simply to eliminate the
measurement of the distance to the t r e e .
The standard rim method,
simply i g n o r i n g trees which were more than twice the c r i t i c a l
angle
at the base, was more competitive w i t h the standard c r i t i c a l height technique and also eliminated the problem of measuring nearby t r e e s . Fixed p l o t sampling was competitive as long as the average number of trees measured was kept above 6 trees per p l o t .
The range of 6-10
trees per point seems to be most e f f i c i e n t f o r sampling purposes i n stands of t h i s type.
- 139 -
CONCLUSIONS
In the f i n a l a n a l y s i s the use of the c r i t i c a l height method w i l l depend on the importance of the bias i n volume tables and the d i f f i c u l t y
of measuring the nearby trees f o r c r i t i c a l height.
The most promising method f o r measuring these d i f f i c u l t trees seems to be the use of the Wheeler Pentaprism. A p p l i c a t i o n of the system w i l l probably be l i m i t e d t o cases where volume tables are very u n r e l i a b l e due t o v a r i a b i l i t y of the merchantable top, continuous f o r e s t inventory where "ongrowth" i s a problem, and use of the method to s e l e c t trees randomly w i t h p r o b a b i l i t y p r o p o r t i o n a l to gross volume.
- 140 -
LITERATURE CITED
Barrett, J.P. Correction for edge effect bias i n point sampling. Forest Science, Volume 10, pages 52-55. Beers, T.W.
and C.I. M i l l e r . 1964. The Purdue point-sampling block. Journal of Forestry, Volume 54, pages 267-272.
Beers, T.W.
1966. The Direct Correction, for Boundary-line Slopover i n Horizontal Point Sampling. Research Progress Report 224, February 1966, Purdue University Agricultural Station, Lafayette, Indiana.
Beers, T.W. Beers, T.W.
1974.
Private Communication.
1976. Practical Boundary Overlap Correction. Journal Paper 6175, Purdue University Agricultural Station, Lafayette, Indiana.
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International
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Feinmechanische-
B i t t e r l i c h , W. 1976. Volume sampling using indirectly estimated c r i t i c a l heights. Commonwealth Forest Review, Volume 55, Number 4, pages 319-330. Bonnor, M.
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Private communication.
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- 141 -
Finlayson, W. 1969. The Relascope. Feinmechanische-Optische Betriebsgesellschaft, Salzburg. Fisher, R.A. 1936a. The Design of Experiments. Boyd Ltd.
Oliver and
Fisher, R.A. 1936b. Has Mendels work been rediscovered? Annals of Science, Volume 1, pages 115-137. Fraser, A.R. 1977. Triangle Based Probability Polygons for Forest Sampling. Forest Science, Volume 23, Number 1. Grosenbaugh, L.R. 1958. Point-Sampling and Line-Sampling: Probability Theory, Geometric Implication, Synthesis. Southern Forest Experiment Station, Occasional Paper 160. Grosenbaugh, L.R. 1963. Optical Dendrometers for out-of-reach Diameters: A conspectus and some new theory. Forest Science Monograph, Number 4", 47 pages. Grosenbaugh, L.R. 1974. STX 3-3-73: Tree Content and Value Estimation Using Various Sample Designs, Dendrometry Methods, and V-S-L Conversion Coefficients. USDA Forest Service Research Paper SE-117. Grosenbaugh, L.R. 1976. Approximate Sampling Variance of Adjusted 3P Estimates. Forest Science, Volume 22, Number 2, pages 173-176. Hirata, T. 1955. Height Estimation Through B i t t e r l i c h s Method, Vertical Angle Count Sampling. Japanese Journal of Forestry, Volume 37, pages 479-480. Hirata, T. 1962. Studies on methods for the estimation of the volume and increment of a forest by angle count sampling. B u l l . Tokyo Univ. For., Volume 56, pages 1-76. Husch, B., C.I. M i l l e r and T.W. Beers. 1972. Forest Mensuration, 2nd Edition, Ronald Press Company, New York, 410 pages. l i e s , K.
1974. Penetration Sampling - An Extension of the B i t t e r l i c h System to the Third Dimension. Oregon State University. Unpublished manuscript.
Jack, W.H.
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APPENDIX I LIST OF SYMBOLS AND TERMS
area of a compartment formed on a preference map by overlapping of p l o t s , as w e l l as c e l l , i n d i c a t i n g the f i r s t tree from a given azimuth. area of a subcompartment formed by the overlapping of plots. The angle of r o t a t i o n used with a prism. area of a band associated with a p a r t i c u l a r t r e e . area of a f i x e d p l o t used i n the sampling process. The basal area of tree i . Basal Area Factor. Area of a c e l l around a tree i n which that tree's c r i t i c a l height i s greater than any other tree. Center of g r a v i t y f o r a p o r t i o n of a stem curve. C o r r e c t i o n term to c a l c u l a t e f u l l average c r i t i c a l height from.the c r i t i c a l height estimated by the "rim method". a general constant, the exact value of which depends on the d e t a i l s of the sampling scheme. a v e r t i c a l angle used to s e l e c t a t r e e f o r p o s s i b l e sampling. c r o s s - s e c t i o n a l area of a part of a tree stem crossed by a transect. average c r i t i c a l height. Estimated c r i t i c a l height. C r i t i c a l height of a p a r t i c u l a r t r e e . Cumulative density f u n c t i o n . Diameter at the base of the t r e e crown. Diameter a t some point on tree i .
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Diameter at the lowest s i g h t i n g point on the t r e e , presumed to be the l a r g e s t diameter as w e l l . Diameter at breast height (1.3 meters) on tree i . Area of a D i r i c h l e t c e l l around tree i . Expansion f a c t o r of an angle gauge. (plot area/tree basal area).
Equal to
A f a c t o r used i n c a l c u l a t i n g p. The number of " i n " trees at a point A r b i t r a r y height of a cone. length of l i n e s used i n l i n e sampling or one of i t s variations. area of a modified D i r i c h l e t c e l l , l a r g e s t number of trees selected- as a c l u s t e r . The maximum expected sum of c r i t i c a l heights, sample s i z e Number of p o s s i b l e observations i n the population. Usually the number of trees i n an area. number of trees involved i n a compartment. The l a r g e s t number of trees selected i n any c l u s t e r throughout the sample area. number of trees present i n a c l u s t e r chosen by f i x e d or v a r i a b l e p l o t s . A proportion of the distance to the edge of the p l o t from the tree located at the center. p r o b a b i l i t y of sampling tree i . p r o b a b i l i t y of sampling a c l u s t e r of np t r e e s . p r o b a b i l i t y of sampling a tree a f t e r e s t a b l i s h i n g a random point on the t r a c t . a random point on the area to be sampled.
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r e l a t i v e p r o b a b i l i t y of sampling tree i compared to any other tree on the t r a c t . A constant r e l a t i n g tree diameter to the diameter of an unseen p l o t surrounding the t r e e . P r o b a b i l i t y density f u n c t i o n . Proportion of p l o t radius,- to be m u l t i p l i e d by tan i n approximating "rim method". a uniform random number between 1 and some s p e c i f i e d upper l i m i t . c r o s s - s e c t i o n a l area of part of the stem p r o f i l e . sum of c r i t i c a l heights at a sample point. t o t a l number of subcompartments formed by p l o t s or s t r i p s .
overlapping
T o t a l area of the t r a c t of land on which sampling i s conducted. Average tangent throughout the p l o t area. tangent of the angle to the c r i t i c a l point from a random point w i t h i n the p l o t radius. T o t a l number of compartments formed by overlap of p l o t of tree i with other p l o t s . the tangent of the angle to the c r i t i c a l point. (CH/distance to t r e e ) . a p a r t i c u l a r tree from the
Equal to
population.
The estimated volume w i t h l i n e - i n t e r s e c t sampling. The volume of a p a r t i c u l a r tree i . Volume to Basal Area r a t i o . width of s t r i p used i n s e l e c t i n g a tree with a p a r t i c u l a r sampling system. Sometimes the distance between two transects. The number of points on a g r i d placed on the t r a c t to be sampled.
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number of compartments on preference map favoring s e l e c t i o n of tree i . t o t a l number of compartments on a preference map. The angle used to s e l e c t a tree with v a r i a b l e p l o t sampling. V e r t i c a l angle to the c r i t i c a l point i n c r i t i c a l height sampling.
