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Systems for the selection of truly random samples from tree populations and extension of variable plot… Iles, Kimberley 1979

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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  (<r—)or "sampling  c a l l e d i n the b i o l o g i c a l l i t e r a t u r e . means minimize t h i s sampling  error.  error" as i t i s often  Random sampling does not by any I t i s well known that systematic  sampling often has a smaller actual sampling  error.  By forcing the  observations throughout a non-random population dispersion a systematic sample w i l l often obtain higher variance within the sample and subsequently  a lower variance between samples.  This actual increase i n  p r e c i s i o n i s usually accompanied by an apparent decrease when the sample variance i s computed as i f the sample were random.  Intelligent direction  of the systematic grids can add further precision, leading to suggestions that sampling be done "at r i g h t angles to the drainage pattern" (Husch, et a l . , 1972) and similar advice i n many texts.  - 9 -  Fisher (1936a) covers the systematic sample thoroughly. His c r i t i c i s m i s mainly i n two parts.  F i r s t , such a systematic  allocation can be quite variable (even with a so-called "random start") when the systematic pattern i s i n phase with a periodically arranged population, and w i l l then also have an apparent decrease i n sampling error. The v a r i a b i l i t y would actually be worse than a random sample while i t would seem to be better.  If the sampler also  has control over the placement of the systematic grid i t may actually be quite biased.  Attempts to estimate the actual error by successive  differences (Meyer, 1956), multiple systematic surveys (Shine, 1960) and more advanced methods have not been entirely successful. Second, because of the uncertain sampling error, confidence intervals and tests of hypotheses cannot be made with known probabilities. While i t i s true that most confidence intervals and tests with systematic samples w i l l be conservative - that i s the probability (a) of falsely rejecting a true n u l l hypothesis i s even smaller than stated - this i s not guaranteed.  Fisher's examination of Mendel's work (Fisher, 1936b)  would not be correct when the true sampling error was smaller than assumed. In this case the wrong sampling error would have increased the probability that Fisher would accuse Mendel of tampering with the data or of using a different method than Mendel had stated. For these reasons, Fisher came down s o l i d l y against non-random arrangements for most purposes. The behavior of the sampling error (c —) of a normal population i s well known, readily available and thoroughly documented. Research on the behavior of non-normal parent populations seems to indicate that  - 10  -  even rather small sample s i z e s give d i s t r i b u t i o n s of means which are roughly normal.  I f the population s i z e frequency  distribution  i s known (but not i t s s p a t i a l d i s t r i b u t i o n ) a random sample of observations i s s t i l l u s u a l l y assumed before t h e o r e t i c a l c a l c u l a t i o n s can be made about the behavior of the sample mean.  I t i s t h i s known  behavior of the sample mean which makes the random sample so popular with s t a t i s t i c i a n s .  Other methods, even i f known to have smaller  sampling e r r o r are more open to the kind of c r i t i c i s m that researchers would rather avoid, and i t seems l i k e l y that they w i l l continue to spend the a d d i t i o n a l time and e f f o r t to do so.  I t therefore seems d e s i r a b l e  to produce methods of obtaining such random samples. We s h a l l p r i m a r i l y be i n t e r e s t e d i n the case where a population of trees i s dispersed on a t r a c t of land with unknown s p a t i a l d i s t r i b u t i o n . The problem i s to draw a random sample of i n d i v i d u a l s .  This i s , i n many  respects, no d i f f e r e n t than s e l e c t i n g a plant at random on rangeland, a geologic specimen from an area, or a seaweed on the ocean f l o o r .  When  not taking advantage of the c i r c u l a r c r o s s - s e c t i o n of the tree stem or other s p e c i a l features, these methods w i l l have a p p l i c a t i o n i n a number of d i s c i p l i n e s .  Sampling Without Replacement  When each observation i s allowed to enter the sample only once, there i s a decrease i n the sampling error expressable by means of the " f i n i t e population c o r r e c t i o n f a c t o r " .  In many cases, sampling  without  - 11 -  replacement i s d e s i r a b l e , but  when  the  finite  population  c o r r e c t i o n cannot be computed accurately, and because of the reasons mentioned i n the l a s t s e c t i o n , i t may be d e s i r a b l e to sample w i t h replacement.  The methods capable of sampling with  replacement are e a s i l y modified f o r sampling without replacement, but the reverse i s not always true.  When sampling without replacement  there i s a decrease i n the number of permutations allowed as a sample. The decrease r e s u l t s i n the d e f i n i t i o n of sampling without replacement g i v i n g equal p r o b a b i l i t y to a l l combinations of samples.  I t seems  to the author to be more general to use the phrase "permutations of observations" with the understanding that the number of allowable permutations i s implied when dealing with sampling without replacement. This s h a l l be done throughout the t h e s i s .  The methods developed i n  t h i s t h e s i s w i l l be a p p l i c a b l e to sampling with or without replacement.  Weighting S e l e c t i o n P r o b a b i l i t i e s  I n d i v i d u a l s i n a population are often selected f o r sampling w i t h a p r o b a b i l i t y p r o p o r t i o n a l to one of t h e i r c h a r a c t e r i s t i c s .  I t reduces  f i e l d work to be able to make such a s e l e c t i o n without a c t u a l l y measuring that c h a r a c t e r i s t i c . First,  There are several reasons to make weighted s e l e c t i o n s .  i t may be d e s i r a b l e to have more precise answers regarding some  s i z e classes of the items i n the population. Often l a r g e r trees are more valuable than smaller trees.  Second, l a r g e r sample s i z e s are often  required f o r some classes of items because they are more v a r i a b l e .  Third,  i t i s sometimes mathematically e a s i e r to sample by weighting the s e l e c t i o n  - 11a -  p r o b a b i l i t y (and give the measurements equal treatment t h e r e a f t e r ) than to sample w i t h equal p r o b a b i l i t y and weight a l l the subsequent calculations.  F i n a l l y , there i s the s t a t i s t i c a l advantage that the  f i n a l r e s u l t can be l e s s v a r i a b l e when s e l e c t i o n p r o b a b i l i t i e s are varied.  A very general equation f o r the estimation of a population  t o t a l can be w r i t t e n : .  where:  V  = the value measured on item i from the population  T  = the estimated population t o t a l f o r the type of population value measured  p  =  t  *  le  P r o b a b i l i t y of sampling item i from the  population. The variance of the t o t a l i s p r o p o r t i o n a l to the v a r i a b i l i t y of t h i s r a t i o .  How can the v a r i a b i l i t y be reduced?  C l e a r l y t h i s can  be done by sampling w i t h a p r o b a b i l i t y p r o p o r t i o n a l to the measured value of each item.  One or more of these reasons often a p p l i e s i n  sampling f o r e s t stands, therefore considerable e f f o r t w i l l be made t o derive means of s e l e c t i o n f o r random samples p r o p o r t i o n a l to s e v e r a l tree c h a r a c t e r i s t i c s , and to devise ways of s e l e c t i n g trees without the a c t u a l measurement of those c h a r a c t e r i s t i c s .  Sampling methods w i l l  first  be derived f o r sampling w i t h equal p r o b a b i l i t y f o r each i n d i v i d u a l and some of these techniques w i l l then be adapted f o r sampling w i t h other probabilities.  - lib -  )  Much of the f i r s t part of the t h e s i s i s concerned with " b i a s " and how to avoid i t .  More s p e c i f i c a l l y the concern i s with  " s e l e c t i o n b i a s " where objects may  i n f a c t be selected w i t h a p r o b a b i l i t y  much d i f f e r e n t than the one intended by the sampler.  This i n turn w i l l  generally r e s u l t i n a bias i n any parameter estimated from the data gathered on those objects.  In a few cases suggestions w i l l be made f o r  changes to estimating equations which w i l l compensate f o r s e l e c t i o n b i a s , but the emphasis w i l l be on s e l e c t i n g trees with the intended p r o b a b i l i t i e s .  - 12 -  SAMPLING SELECTION SYSTEMS PROPORTIONAL TO VARIOUS PROBABILITY WEIGHTINGS  Frequency Weighting  The common system f o r s e l e c t i n g members of a population i s one based on equal frequency, where each of the N members of a population i s measured w i t h the same p r o b a b i l i t y .  In a d d i t i o n t o t h i s  requirement a random sample would also give equal s e l e c t i o n p r o b a b i l i t y t o each permutation of observations. S u r p r i s i n g l y l i t t l e advice can be found on the problem of s e l e c t i n g a random sample of trees from a f o r e s t area.  The only common  system suggested f o r drawing such a sample i s f i r s t t o l a b e l a l l of the N i n d i v i d u a l s , to draw a random number from 1 t o N, and to f i n d and measure that i n d i v i d u a l . of desired s i z e .  The process i s repeated to s e l e c t a sample  The e f f o r t involved i n t h i s process u s u a l l y eliminates  i t from serious p r a c t i c a l consideration.  "Nearest Tree" Methods  A common method i n p r a c t i c e i s to f i r s t f i n d a random point on the t r a c t to be sampled.  This i s simply accomplished by the i n t e r s e c t i o n  of two random coordinates. A tree "near" t h i s point i s then chosen s u b j e c t i v e l y , or the tree nearest t o the point i s chosen by measurement.  - 13 -  This "nearest t r e e " idea w i l l be examined i n some d e t a i l throughout the t h e s i s .  The bias i n the f i r s t system cannot be c a l c u l a t e d , but  that of the second system i s easy to examine.  I t can most e a s i l y  be studied by the use of polygons constructed around each t r e e . These polygons are c a l l e d Thiessen diagrams (Jack, 1967), "Voroni polygons" and " D i r i c h l e t c e l l s " (Fraser, 1977), "Plant Polygons" (Mead, 1966), "Area P o t e n t i a l l y A v a i l a b l e " (Brown, 1965) (Overton et_ al_., 1973) .  or "Occupancy Polygons  Their use i n f o r e s t r y i s i l l u s t r a t e d by such  p u b l i c a t i o n s as Jack (1967), Overton et a l . (1973) and Brown (1965). The construction of a D i r i c h l e t c e l l around a tree i n c l o s e s a l l points to which the tree i s c l o s e r than any other tree i n the population.  The s i z e of the c e l l i s dependant only on the spacing of  the trees i n the population.  Construction of the c e l l i s defined  concisely by Jack (1967) as f o l l o w s :  " the smallest polygon that can be obtained by e r e c t i n g perpendicular b i s e c t o r s to the h o r i z o n t a l l i n e s j o i n i n g the center of the tree to the centers of i t s neighbors at breast height of the tree center "  The main features of the c e l l s are that they cover the e n t i r e t r a c t without overlap or gap, that they are easy to construct; and they are not affected by the shape or s i z e of the c e l l  that  constructed  around other t r e e s , which insures the same c e l l regardless of the order of construction.  Jack (1967) gives a number of u s e f u l equations f o r  determining which trees are c r i t i c a l to the construction of the c e l l  - 14 -  and for c a l c u l a t i n g the c e l l area.  He also mentions a computer  program f o r automatically doing t h i s from f i e l d data g i v i n g the bearing and distances to surrounding provide  trees.  Newnham and Maloley  (1970)  a computer r o u t i n e f o r computing c e l l areas from stem maps.  The c e l l s are not d i f f i c u l t to construct on large scale stem maps, since the main process involved i s l i n e b i s e c t i o n . Figure 1 shows the general process involved, and examples of the c e l l s around 3 t r e e s . Such diagrams, noting the area i n which a tree may be chosen for sampling w i l l be c a l l e d "preference maps" f o r that s e l e c t i o n system. I t i s obvious that the p r o b a b i l i t y of choosing the tree nearest to a random point i s d i r e c t l y p r o p o r t i o n a l to the area of the D i r i c h l e t c e l l around the tree.  The exact p r o b a b i l i t y of choosing the tree i would  be:  (1.1)  where:  = The p r o b a b i l i t y of s e l e c t i n g tree i f o r sampling under the s e l e c t i o n system. DC.  = The area of the D i r i c h l e t c e l l around tree i  T  = The t o t a l area of the t r a c t where sampling i s  I  conducted.  A l l symbols i n t h i s work w i l l be defined when f i r s t used, and a l s o l i s t e d i n Appendix 1 f o r easy reference.  - 16 -  There i s every reason to b e l i e v e , on the basis of experience,  that  large trees are spaced at wider i n t e r v a l s than small trees i n the same conditions.  At the very l e a s t we know that d i f f e r e n t f o r e s t  areas of the same s i z e have d i f f e r e n t numbers of trees.  It is  therefore apparent that trees from both areas cannot have the same p r o b a b i l i t y of s e l e c t i o n by the nearest tree method, and that the bias i s very probably i n favor of l a r g e r t r e e s , since these often have wider spacing.  The exact bias can be computed from a stem map  on which the D i r i c h l e t c e l l s have been drawn. Since the nearest tree system does not s e l e c t trees with equal frequency, i s there a simple system which does?? of any s i z e  The use  f i x e d p l o t w i l l assure that every tree has exactly the  same p r o b a b i l i t y of s e l e c t i o n . This i s probably the easiest way to select a sample where each tree has an equal chance of s e l e c t i o n .  This,  unfortunately, does not allow us to pick a random i n d i v i d u a l as e a s i l y . Often such a c l u s t e r of trees picked by a p l o t w i l l serve the sampling purpose but o c c a s i o n a l l y a subsample i s required or a t r u l y random s e l e c t i o n of i n d i v i d u a l s i s desired.  A t r u l y random sample requires  that every permutation of observations be equally l i k e l y , so s e l e c t i o n of more than one tree from a p l o t would be non-random even though i t might be unbiased. I f c l u s t e r s w i t h p o s i t i v e covariance among trees are selected, but treated as a random sample of i n d i v i d u a l s , the sampling error w i l l be underestimated.  The reverse s i t u a t i o n can a l s o occur, as stated e a r l i e r .  - 17 t  Bias i n Subsampling From a C l u s t e r  One common approach to subsampling i s to choose a tree by a uniform random number R between 1 and np, where np i s the number of trees found i n a p l o t . i s chosen f o r sampling.  The tree corresponding  to the random number  I f only one tree i n each f i x e d p l o t i s chosen  i n t h i s way (or any o t h e r ) , there i s an obvious bias i n favor of trees of sparse d i s t r i b u t i o n . a particular  The p r o b a b i l i t y of sampling an i n d i v i d u a l on  p l o t i s (1/np).  For an i n d i v i d u a l tree i n a given p l o t  t h i s can r i s e to a maximum of 1 or decrease to a minimum of 1/n , max where nmax i s the greatest number of trees found on any p l o t i n the studyj J v area.  The expectation of an i n d i v i d u a l f o r sampling can be c a l c u l a t e d  from a preference map constructed i n the f o l l o w i n g way:  1.  Construct a p l o t around each of the N trees on the area with the s i z e , shape and o r i e n t a t i o n d e s i r e d .  2.  Determine the number of trees which share each compartment formed by overlaps of the p l o t s , and the area of each. In complex s i t u a t i o n s t h i s w i l l best be done by d i g i t i z e r or planim t e r , i n simpler cases perhaps by equation.  3.  C a l c u l a t e the expectation of sampling tree i from a randomly located s i n g l e point by the formula:  - 18 -  Nwhere:  t^  1.0  (2.1) k=l  = t o t a l number of compartments formed by overlap of tree i with other p l o t s  a^  = area of compartment k  n^  = number of trees sharing compartment k  T  = t o t a l area of t r a c t sampled.  The p r o b a b i l i t y that tree i has been selected given that some tree has been selected (hereafter c a l l e d r e l a t i v e p r o b a b i l i t y ) i s : {s.j.  Pr  L-W N  i=l  where:  t  T*p{s.}  EH  (2.2)  k=l  = the t o t a l number of compartments on the preference  map.  Figure 2 i l l u s t r a t e s the method and c a l c u l a t i o n s using c i r c u l a r f i x e d 2 area p l o t s of 25 m .  The p r o b a b i l i t y that some tree w i l l be sampled with a randomly chosen point i s : 79.5 m 200 m  = 39.75%  (2.3)  - 19  -  I l l u s t r a t i o n of the computation of sampling p r o b a b i l i t y when a s i n g l e tree i s chosen randomly from a l l those on a f i x e d p l o t .  - 20 -  Table 1.  Computations involved f o r the example i l l u s t r a t e d i n Figure 2.  Compartment from diagram  1  a  2  tree 3 4  * *  5 12.0  1  12.0  *  4.0  2  2.0  *  2.6  3  • 0.87  2.7  2  1.35  3.1  4  0.78  0.6  3  0.2  2.9  1  2.9  8.0  2  2.0  5.1  3  1.7  6.2  1  6.2  *  2.0  2  1.0  1  *  5.3  1  5.3  m  *  2.9  2  1.45  22.1  1  b  *  c d  *  *  e  *  *  *  f  *  *  *  *  *  g h  *  *  i  *  *  *  *  j k  *  n  *  22.1  79.5  Tree  E  LV\]  Relative Probability  Selection Bias  1  17.2  .22  +10%  2  11.8  .15  -25%  3  13.88  .17  -15%  4  13.1  .16  -20%  5  23.52  .30  +50%  Totals:  79.50  1.0  - 21 -  This i s simply the r e l a t i v e area covered by the p l o t of one or more trees. As a p r a c t i c a l matter, i f one tree must be randomly chosen from each c l u s t e r then that observation should be weighted p r o p o r t i o n a l to the s i z e of the c l u s t e r .  This w i l l remove any bias from estimation  based on the sample mean.  S e l e c t i o n of the Closest Tree i n Each C l u s t e r  The equations are the same once the preference map has been drawn.  There are fewer compartments but c o n s t r u c t i o n i s more tedious.  Figure 3 shows r e s u l t s .  When p l o t s overlap the b i s e c t o r i s drawn.  This  i s p a r t i c u l a r l y easy, since most of the b i s e c t o r c o n s t r u c t i o n i s provided already by the overlap of the c i r c l e s .  The nearest b i s e c t o r s reduce  the s i z e of the p l o t as i n a D i r i c h l e t c e l l .  R e l a t i v e reduction i s most  severe i n clumps of trees, decreasing the s e l e c t i o n p r o b a b i l i t y of trees i n dense spacings.  As the s i z e of p l o t s increases the b i s e c t o r s are  i n c r e a s i n g l y important, with the D i r i c h l e t c e l l s being the l i m i t i n g distribution.  Using the term nuiL to denote the area of these modified D i r i c h l e t c e l l s , the p r o b a b i l i t i e s of sampling are given below.  For s e l e c t i n g an i n d i v i d u a l tree:  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 n g a tree:  (3.2)  i=l  p r o b a b i l i t y of s e l e c t i n g some tree at a random point: N  (3.3)  The "Azimuth Method"  The system, sometimes used i n f o r e s t inventory, i s to e s t a b l i s h a sample point, then choose f o r subsampling the f i r s t tree whose center i s encountered i n a clockwise (or counter-clockwise) d i r e c t i o n from the p l o t center. The s t a r t i n g azimuth can be f i x e d (such as always s t a r t i n g from North) or randomly chosen. any given s t a r t i n g d i r e c t i o n . a map.  A preference map can be constructed f o r Figure 4 shows the construction  Examples of each of the f o l l o w i n g steps are noted.  of such  A North  s t a r t i n g azimuth and clockwise r o t a t i o n are assumed f o r t h i s example.  - 24 -  Figure 4.  Construction of preference map  based on the azimuth method.  - 25 -  Construction steps are:  1.  Construct, a l i n e ( c a l l e d the "stem l i n e " ) from each tree South to the area border.  2.  S t a r t i n g at the West border l a b e l the stem l i n e s 1-N from West to East.  3.  S t a r t i n g w i t h tree 1 (tree^) and continuing s e q u e n t i a l l y , sweep clockwise from North u n t i l another tree i s encountered. Draw a l i n e from tree^ opposite the tree encountered u n t i l the border or another stem l i n e i s reached.  Special steps must be taken a f t e r tree N - l .  4.  Consider only the f i n a l tree N.  From t r e e ^ sweep clockwise  from South u n t i l a tree i s encountered. 5.  Draw a l i n e from the stem l i n e of t r e e ^ to the border i n a d i r e c t i o n opposite the tree encountered.  6.  Consider the tree l a s t encountered.  From i t sweep clockwise  from South u n t i l another tree i s encountered, and repeat step 5.  Continue t h i s u n t i l the l i n e so constructed s t r i k e s  the area border west of the t r e e ^ stem l i n e .  Polygons i n which a p a r t i c u l a r tree w i l l be chosen are noted on the diagram by c i r c l e d tree number.  Note that s l i g h t additions may  occur due to areas near the border east of t r e e ^ , but otherwise they are i n one part.  - 26 -  The straight lines of such a map simplifies i t s programming for computer plotting.  The area of the polygons i s easily computed  from intersection points using standard surveying  computations.  When fixed plots are used, and the f i r s t tree from an azimuth i s selected as a subsample,  the construction i s s l i g h t l y more d i f f i c u l t .  A map i s drawn as i n Figure 2, indicating the polygons in which a particular subset of trees may be chosen. For each of these subsets a further construction i s made as described in Figure 4 but only considering the trees of that subset, and drawn across the polygon being considered. This i s not too d i f f i c u l t , since few of the lines considered w i l l cross each polygon.  The separate areas are then labelled indicating which  tree w i l l be selected in each case.  Figure 5 shows an example of such a  construction. Three points are worth mentioning when construction i s done by hand. As a f i r s t step construct only the stem lines which are i n polygons occupied by two or more trees. Second, label the obvious situations f i r s t , particularly cases where only one tree i s e l i g i b l e .  Third, as  you move from one polygon to the next the number of trees to be considered normally changes by one.  This helps to assure that no trees are ignored.  Preference maps of this sort are very d i f f i c u l t to draw, except i n large scale.  The most common errors w i l l probably be due to steps 4, 5, and 6.  Once the compartments have been designated on the preference map selection probabilities are calculated as follows:  - 27  Figure 5.  -  Azimuth method applied with f i x e d p l o t s .  - 28 z. 1  y • where:  a  a  = the area of a compartment on the preference map i n which tree i w i l l be selected.  z_^  = the number of a l l the compartments a l l o c a t e d to a p a r t i c u l a r tree.  R e l a t i v e p r o b a b i l i t y of sampling a tree i s :  Pr  {s.| =  „  {  Z  )  (4.2)  i=l P r o b a b i l i t y of choosing at l e a s t one tree from a random point i s : z P  IM" where:  z P  a  c  T  = the t o t a l number of compartments i n the area being sampled.  - 29  -  Methods f o r the E l i m i n a t i o n of Bias  Much a t t e n t i o n has been given to the d i s t r i b u t i o n of i n d i v i d u a l s i n p l o t s of various s i z e under random spacing. (1971) discusses such a d i s t r i b u t i o n .  Matern  In f a c t , we know that t r e e  d i s t r i b u t i o n s i n the plane are not u s u a l l y random.  The s i t u a t i o n was  n i c e l y described by Warren (1972) as f o l l o w s :  Trees are, of course, not p o i n t s . The diameter of a t r e e , conventionally measured at breast height (4 f t . 6 i n . or 1.3 m) i s generally not n e g l i g i b l e with respect to the distance between t r e e s . Further, competition between trees u l t i m a t e l y produces an area about each tree w i t h i n which no other tree can e x i s t . These r e s t r i c t i o n s are often conveniently neglected i n t h e o r e t i c a l studies (notably the w r i t e r ' s ) ; exceptions are the t h e o r e t i c a l d e r i v a t i o n of Matern (1960, p.47) and the simulation studies of Newnham (1968) and Newnham and Maloley (1970).  The idea that there i s a t r u n c a t i o n of a c t u a l d i s t r i b u t i o n s at some minimum p l o t s i z e , as w e l l as an a c t u a l upper l i m i t f o r the number of trees i n a given p l o t s i z e , gives r i s e to two systems f o r s e l e c t i n g random samples.  These w i l l be c a l l e d the " p l o t reduction  method" and the " e l i m i n a t i o n method".  P l o t Reduction Method  Since the problem causing bias i s the overlap of p l o t s , they can be reduced i n s i z e to a diameter equal to the smallest distance  - 30 -  between members of the population. the case of n o n - c i r c u l a r p l o t s . now  This means maximum diameter i n  The p r o b a b i l i t y of being chosen i s  equal f o r a l l members of the population and the choice of not more  than one member i s assured with each p l o t . selected  are  independent,  In a d d i t i o n ,  observations  i n s u r i n g a t r u l y random sample where  any permutation of objects i s equally l i k e l y .  The p r o b a b i l i t y of  f i n d i n g a p a r t i c u l a r sample tree i s given by:  where:  ap  = the area of the f i x e d p l o t used.  The p r o b a b i l i t y of sampling a tree with a random point i s : N  (5.2)  There are several p r a c t i c a l ways of e q u a l i z i n g the s p a t i a l d i s t r i b u t i o n of objects to be sampled, thereby increasing p l o t s i z e and minimizing the p r o b a b i l i t y of a vacant p l o t .  I t i s not the tree i t s e l f  which must be s e l e c t e d , but something which can be associated with the tree, and detectable on the p l o t .  Even though tree stems may  be quite  close together the upper parts of the tree w i l l u s u a l l y be more evenly distributed.  I t w i l l probably be of advantage to use the tops of trees  or perhaps the edge of the crown (say the center of the north edge) to  - 31 -  determine which tree to sample. lead to more e f f i c i e n t sampling.  This more even d i s t r i b u t i o n w i l l In a d d i t i o n , such methods are u s e f u l  i on a e r i a l photographs which have many advantages, p a r t i c u l a r l y t h e i r high p o t e n t i a l f o r automation.  The E l i m i n a t i o n Technique  Reduction of p l o t s i z e can become quite severe under  conditions  of high clumping, t h i s leads to i n e f f i c i e n c y i n sampling due to unoccupied plots.  The e l i m i n a t i o n technique may be more e f f e c t i v e i n t h i s case.  A plot of s i z e ap i s chosen, and the maximum number of trees which could f a l l i n t o such a p l o t i s symbolized by Mp.  For any p l o t the trees  included are numbered from 1 to np and a random number R i s chosen between 1 and Mp.  I f R corresponds to the number of one of the trees i n the p l o t  that tree i s chosen, otherwise a new p l o t i s established. again gives a random sample of i n d i v i d u a l s .  This system  The p r o b a b i l i t y of s e l e c t i n g  a s i n g l e tree i s c a l c u l a t e d by:  which i s a constant. The r e l a t i v e p r o b a b i l i t y i s simply 1/N. I f the expected maximum number of trees i s exceeded, the system w i l l be s l i g h t l y biased.  The a c t u a l p r o b a b i l i t y f o r a p a r t i c u l a r tree  can then be computed by the standard formula except i n those compartments where np > Mp.  Where the p l o t s are not large and Mp  conservatively  - 32 -  estimated t h i s should happen infrequently.  The r e l a t i v e frequency  of any tree w i l l not then be 1/N, but:  Pr  (6.2)  N i=l  where the maximum of the values (1/np, 1/Mp) has been used i n equation 6.1 to c a l c u l a t e p  Increasing  •  Efficiency  The way to minimize the occurrence of unoccupied p l o t s i s to maximize the product of two p r o b a b i l i t i e s :  P | sampling a tree j  P | one or more trees i n p l o t j *p | s e l e c t i n g one of those trees |  more formally: Mp (6.3) np=l where:  p jnp > Mpj p |  =0  | 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 p l o t with np trees present f o r possible s e l e c t i o n .  - 33 -  The p r o b a b i l i t i e s of s e l e c t i n g p a r t i c u l a r numbers of trees might be c a l c u l a t e d from t h e o r e t i c a l considerations or by f i e l d t e s t s . The e l i m i n a t i o n procedure i s a very general one, and can be used to choose a f i n a l sample from the equal p r o b a b i l i t y f i r s t stage s e l e c t i o n with p l o t s .  Suppose one i s i n t e r e s t e d i n choosing a random  sample of trees from a population with p r o b a b i l i t y p r o p o r t i o n a l to 2 ^ DBH_/ . One way to make the s e l e c t i o n i s to choose a subset with equal p r o b a b i l i t y using a p l o t , then s e l e c t from the manageably short l i s t of 2 6 DBH.'l by means of a random number.  The random number would be chosen np 2 > 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  <  i  which i s always the case when A i s smaller than B.  - 66 -  I t i s admittedly of minor i n t e r e s t what w i l l happen under random spacing, since we know that trees are not d i s t r i b u t e d i n t h i s manner.  As the angle gauge changes to produce l a r g e r p l o t s the  preference map uses more b i s e c t o r s to describe the areas,  eventually  forming D i r i c h l e t c e l l s as the l i m i t i n g case. The computer p l o t t i n g of the preference maps can be s i m p l i f i e d by some of the f o l l o w i n g r u l e s about l i n e s .  The c i r c l e d numbers on  Figure 20 r e f e r to use of the r u l e s noted below.  1)  There i s a distance, c a l l e d the " p l o t r a d i u s " f o r each tree i , computed by PIL = (D  2)  * PDF/2)  A point on the c i r c l e of radius PR^ surrounding  the tree  i s not drawn when such a point i s w i t h i n the p l o t radius of the smaller tree or_ closer to another tree than a b i s e c t o r with that tree. 3)  B i s e c t o r s between trees are not drawn when the distance from the point on the b i s e c t o r to the smaller tree i s l e s s than the distance PR. of the smaller tree. l  4)  B i s e c t o r s are not drawn at points which are w i t h i n the p l o t radius of a t h i r d tree and are closer to that t h i r d tree than to e i t h e r of the trees used f o r the b i s e c t o r .  Using these r u l e s , parts of the preference map can be drawn as each tree i s considered.  The c e l l around an i n d i v i d u a l tree w i l l not  u s u a l l y be completed u n t i l several other trees are also p l o t t e d .  This  - 67 -  Figure 20.  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.  - 68 -  i s p a r t i c u l a r l y true of l a r g e r trees.  A more complete  preference  map i s shown i n Figure 21. P r o b a b i l i t i e s of sampling are exactly the same as i n Equations 3.1 to 3.3.  They are l i s t e d again here f o r  convenience.  P r o b a b i l i t y of s e l e c t i n g an i n d i v i d u a l tree:  N  -(  .)  R e l a t i v e p r o b a b i l i t y of sampling:  P r  { i} S  =  ~  —^-^  ( 3  -  2 )  i=l P r o b a b i l i t y of s e l e c t i n g a tree at some random point: N  I .} *— — 8  T  In summary, the system w i l l have the f o l l o w i n g general properties.  The a c t u a l biases toward small trees w i l l probably be even  l a r g e r than with random sampling, since trees tend to overlap the borders of other trees rather than to be quite near them as might occur under random spacing.  The biases are at any rate i n favor of small  trees and trees which are sparsely d i s t r i b u t e d , and to a larger degree than with random s e l e c t i o n of one tree per group.  An increased mixture  - 69 -  Figure 21.  Completed preference map, with v a r i a b l e p l o t s .  nearest tree method  - 70 -  of s i z e classes w i l l increase the bias toward smaller trees, as w i l l greater v a r i a t i o n i n s i z e c l a s s e s . The p l o t reduction method would be most d i f f i c u l t to apply with v a r i a b l e p l o t s , and the e l i m i n a t i o n method would be the best way to obtain a random sample.  In t h i s case, the " i n " trees are numbered  s e q u e n t i a l l y , then a random number would be drawn between 1 and Mp where Mp i s the estimated maximum number of " i n " trees at any point.  A tree  i s only chosen i f i t s number corresponds to the random number, otherwise a new point i s selected and the process i s repeated.  In t h i s case,  the p r o b a b i l i t y of sampling a tree i s :  (10.1)  i=l Relative probability i s :  (10.2)  i=l The p r o b a b i l i t y of choosing some sample at a random point i s : Mp (10.3)  The l a s t equation uses the preference map, and i s i d e n t i c a l to Equation 6.3.  - 71 -  The Azimuth Method With V a r i a b l e P l o t s  This i s j u s t a m o d i f i c a t i o n f o r the system based on fixed plots.  A f t e r the d i f f e r e n t sized p l o t s have been constructed the  compartments are f u r t h e r subdivided based only on the trees involved with that subcompartment.  Equations 5.1, 5.2 and 5.3 are then used.  Bias i s increased toward small trees which are w e l l spaced or on the border of clumps. removed.  Figure 22 shows the preference map with extra l i n e s  I t i s based on the same spacing and s i z e s as Figure 21.  Non-Random But Unbiased Methods For Subsampling  S e l e c t i o n of c l u s t e r s of a l l " i n " trees at a point assures a p r o b a b i l i t y based s t r i c t l y on basal area.  I n d i v i d u a l trees can also  be picked with a f i x e d p r o b a b i l i t y i n 2 ways.  First,  i s " i n " i t i s picked with a p r o b a b i l i t y (say 1/10) device.  each time a tree  with a  randomizing  This allows more permutations of samples than s y s t e m a t i c a l l y  choosing every tenth t r e e , although e i t h e r method would be  unbiased.  Second, i f the approximate number of trees which w i l l be counted on a l l points i s known, then random integers between 1 and t h i s sum can be chosen to s e l e c t the subsample.  The only advantage to such a system  i s that i t allows sampling with replacement.  The f i r s t method i s  probably best on the basis of o p e r a t i o n a l convenience and the f a c t that the sampler does not know when the next sample w i l l occur.  - 72 -  Figure 22.  Completed preference map f o r the azimuth system and v a r i a b l e p l o t s .  - 73 -  Choice of one sample tree a t each point, p a r t i c u l a r l y without s t r a t i f i c a t i o n , i s to be avoided.  I f random s e l e c t i o n w i t h i n  c l u s t e r s must be done f o r some reason, weighting the sample proportional to tree count can remove the b i a s .  Height Squared Weighting  This i s an extension of h o r i z o n t a l point sampling, but uses tree height rather than tree diameter.  Developed by H i r a t a (1955) f o r sampling  the height of f o r e s t stands, the technique uses a v e r t i c a l angle and sights a l l trees from a point randomly chosen on the area.  The s i z e of  the c i r c u l a r p l o t on the preference map i s determined by the v e r t i c a l angle and tree height, otherwise the equations are i d e n t i c a l to h o r i z o n t a l point sampling.  The reasoning concerning biases and spacing i s the same  but with diameter replaced by height.  Combining Diameter Squared and Diameter  Weightings  Let us f i r s t consider the problem of sampling p r o p o r t i o n a l to 2 aD^ + bD^.  The perimeter of a p l o t i s p r o p o r t i o n a l to diameter, and so,  very nearly, i s a s t r i p around the p l o t . f o l l o w i n g l i n e of reasoning.  Figure 23 i l l u s t r a t e s the  - 74 -  Figure 23.  P l o t area composed of 3 simple f i g u r e s .  inner c i r c l e  c i r c l e of radius W  (PDF*D.) TT  - 75 -  Area of i n s i d e c i r c l e =  j^W  Outside r i n g area =  g  [  PDF * D. 1  _>_]  * 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  <V  <U-1>  The prism i s mounted and the angle of r o t a t i o n i s marked with the tree diameter.  Figure 32 shows such a mechanism set f o r checking a 50 cm tree.  The d e c i s i o n of whether a tree i s " i n " or "out" i s the same as i n conventional uses of the prism as an angle gauge. Such a s e l e c t i o n system has two advantages.  F i r s t , s i z e of  p l o t can be derived by any means, as long as i t can be indexed only by diameter.  Second, the p l o t i s a s o l i d c i r c l e centered at the tree.  Such  areas around trees are o f t e n described when dealing with problems such as competition, r o o t i n g zones, moisture d e p l e t i o n e t c . When i n t e r p o l a t i o n cannot be done a programmable c a l c u l a t o r can e a s i l y be used to e s t a b l i s h the r o t a t i o n angle. the values f o r  I f the d i f f e r e n c e s i n  0^ are small the adjustment may be d i f f i c u l t .  When the  angles of r o t a t i o n are nearly a l i k e the bias of s e t t i n g the instrument may be large.  I n such a case two prisms should be used. One f i x e d prism  establishes the basic angle (A . ) , while the other r o t a t a b l e thinner main one ( p"L ) c o r r e c t s i t by l a r g e r angles of r o t a t i o n . The f i n a l angle A  us  0,  i s then given by:  - 89 -  Figure 32.  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.  view through prism i s o f f s e t  TOP VIEW  pxvo  c lear round prism  - 90 -  6, = A . + (cos A ) * (A . ) d mam r plus  (11.2)  This method w i l l increase the range of r o t a t i o n of the smaller prism and s i m p l i f y the problem of s e t t i n g the instrument accurately.  Gross Volume Weighting  Basics of the C r i t i c a l Height Method  When an angle gauge using a c r i t i c a l angle 0  i s used to  sight along a tree stem the e f f e c t i s to "expand" the diameter by a constant a l l along that stem. stem of a tree.  This gives a simple method to expand the  The basal area, and hence the volume i s expanded by the  same constant at each point along the stem.  Average height of the  expanded stem i s obviously the same as the o r i g i n a l t r e e , and p r o p o r t i o n a l to tree volume. The length of the v e r t i c a l l i n e from a random sample point l e v e l with the base of the tree to the point where i t leaves the expanded tree i s a sample of the average tree volume  (see Figure 33).  This d i s t a n c e , c a l l e d the " c r i t i c a l height" w i l l allow us to select trees p r o p o r t i o n a l to t o t a l volume. the basis of i t s c r i t i c a l height.  The tree need only be selected on This basic idea was discovered by  M. Kitamura i n 1962, but no E n g l i s h t r a n s l a t i o n s of the work were a v a i l a b l e and i t was v i r t u a l l y unknown i n North America. In 1974 the author independently discovered the p r i n c i p l e , from a s l i g h t l y more general viewpoint, and c a l l e d i t "Penetration Sampling" ( l i e s , 1974). A more d e t a i l e d h i s t o r y w i l l be given i n the second part of the t h e s i s .  -  Figure 33.  91  -  The "expanded t r e e " , c r i t i c a l point and c r i t i c a l height.  Random Point P  - 92 -  One of the simplest ways to test what part of a v e r t i c a l l i n e through point P l i e s w i t h i n the imaginary expanded tree bole i s to move up the v e r t i c a l l i n e (B) with an angle gauge while sighting the tree bole along the h o r i z o n t a l plane. other angle gauge would serve the purpose.  A prism, relaskop or any I f the tree image i s " i n "  when s i g h t i n g h o r i z o n t a l l y towards the tree f o r a t o t a l of 12 meters, then the v e r t i c a l l i n e passes through the imaginary expanded tree stem f o r a distance of 12 meters. For the system to be p r a c t i c a l the p h y s i c a l problem of moving up and down the v e r t i c a l l i n e w i t h the angle gauge must be circumvented. The s o l u t i o n to t h i s problem i s f o r t u n a t e l y quite simple.  I t i s commonly  known that the relaskop (or newer telerelaskop) has an "automatic slope correction".  The geometric form of that c o r r e c t i o n does not seem to be  as w e l l known.  When an observer views a tree diameter at any v e r t i c a l  angle the automatic c o r r e c t i o n i s made by s l i g h t l y decreasing the c r i t i c a l angle  6 , which i s then being projected along a slope distance.  The  view through the instrument i s exactly the same as i f the observer was f l o a t i n g v e r t i c a l l y above or below the sample point and was  sighting  h o r i z o n t a l l y at the tree with the o r i g i n a l c r i t i c a l angle.  This form of  automatic c o r r e c t i o n then makes i t p o s s i b l e to t h e o r e t i c a l l y  "levitate"  above or below the sample point along a v e r t i c a l a x i s and "observe the tree h o r i z o n t a l l y " .  These are the exact requirements of the system.  now have a p r a c t i c a l method f o r determining the point at which the v e r t i c a l l i n e leaves the unseen expanded tree bole (see Figure 34).  We  Figure 34.  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  v e r t i c a l l i n e through the sampling point  edge of imaginary expanded tree bole  tree i s borderline at t h i s " c r i t i c a l point"  actual tree bole  length through which the v e r t i c a l l i n e penetrates the imaginary expanded tree bole - the " c r i t i c a l height"  tree, base Random Point P  - 94  -  To get the distance desired only 3 measurements are needed.  1)  The v e r t i c a l angle to the c r i t i c a l point (the point where i t becomes " b o r d e r l i n e " ) . This i s the point where the v e r t i c a l l i n e leaves the expanded tree bole.  2)  The v e r t i c a l angle to the base of the t r e e , or the point below which cubic volume i s not to be considered.  It  i s computationally convenient i f these angles are measured in 3)  A  .  The slope distance to the base of the t r e e .  This should  be measured at the same angle of slope as the second reading, and can be done with an o p t i c a l system i f desired.  The second v e r t i c a l angle and the slope distance can be used to c a l c u l a t e the h o r i z o n t a l distance to the t r e e . Both v e r t i c a l angles along w i t h t h i s h o r i z o n t a l distance w i l l give the c r i t i c a l height.  These  measurements can be made without leaving the sample point i f a range f i n d i n g system i s used f o r slope distance. At any random point P we can get the c r i t i c a l heights of a l l the trees.  Since the length of these c r i t i c a l heights are p r o p o r t i o n a l  to tree volume the e l i m i n a t i o n method may be used to choose a s i n g l e tree. This allows s e l e c t i o n of a random i n d i v i d u a l or a random sample p r o p o r t i o n a l to t o t a l volume and without p r i o r assumptions about tree shape or any of volume t a b l e s .  use  - 95 -  When E i s the expansion f a c t o r of the angle gauge used and c a l c u l a t e d by:  E =  (  •[and using M  sum  p l o t area tree basal area  sin  expanded tree volume a c t u a l tree volume - ] •  (12.1)  E(V.) V.  as the maximum expected sum of c r i t i c a l heights at a  point, the p r o b a b i l i t i e s with the e l i m i n a t i o n method are:  •[  I'.) • [ •  expected c r i t i c a l height of tree  p r o b a b i l i t y of s i g h t i n g the tree with the prism  L  BA. * E  E * V. l  V.  1  BA.  M  L  1  (12.2)  M  sum  ,M sum -  sum  J  The r e l a t i v e p r o b a b i l i t y i s :  Pr  N-  N  EMM  E  (12.3)  N T  i  i=l  i=l  The p r o b a b i l i t y of choosing some tree at a random point i s N given by: Z  (  M  sum  E  *  v  i ) average sum of c r i t i c a l heights M sum  (12.4)  - 96 -  Random S e l e c t i o n From a C l u s t e r P r o p o r t i o n a l to C r i t i c a l of the Tree,  a  Height  Biased Method  I f one tree i s chosen using the sum of c r i t i c a l heights at a point (S ) and drawing a random number uniformly from the i n t e r v a l c  1-S , c  the problem becomes very d i f f i c u l t .  now i n 3 dimensions.  The overlap of trees i s  Figure 35 shows a side view f o r 2 trees only.  In the area i n d i c a t e d by "0" there i s a p r o b a b i l i t y of s e l e c t i o n proportional to the r a t i o of the c r i t i c a l heights of the expanded trees.  This r a t i o must be integrated over the land surface where the  tree boles i n t e r s e c t i n order to compute the s e l e c t i o n p r o b a b i l i t i e s . When 6-10  trees a l l overlap, even with the same stem form, an exact  s o l u t i o n i s too d i f f i c u l t . approximation methods.  The only p r a c t i c a l s o l u t i o n i s by numerical  Using a g r i d of xp points across the land area,  the estimated p r o b a b i l i t y of s e l e c t i o n i s then given by: r  X  P  where S  where:  c  > /  0  (13.1)  CH. i s the c r i t i c a l height of tree i ,  (13.2)  i=l  - 97 -  - 98 -  The p r o b a b i l i t y of s e l e c t i o n of a tree at a random point i s l i k e that of h o r i z o n t a l point sampling. to construct a p l o t around each tree.  The maximum diameter i s used The proportion of land area  covered by at l e a s t one p l o t gives the desired p r o b a b i l i t y . One bias i n t h i s case i s toward trees which are widely spaced r e l a t i v e to t h e i r basal area.  S e l e c t i o n of the Tree With Largest C r i t i c a l Height, a Second Biased Method  This again i s a mathematical problem best solved by numerical approximation.  Here s e l e c t i o n i s based on the maximum c r i t i c a l height.  See Figure 36. The area C w i l l favor s e l e c t i o n of tree c, since at a l l points to the l e f t of the dotted l i n e the c r i t i c a l height of a i s l a r g e r . Unfortunately, t h i s l i n e cannot be located as e a s i l y as with other systems we have examined. One way to e s t a b l i s h the l i n e by computer i s to p l o t perimeter points on c i r c l e s of s i z e (stump c i r c l e s overlap.  * PDF) except where two or more such  When two c i r c l e s overlap only the i n t e r s e c t i o n points  w i l l be drawn, f o r diameters at i n t e r v a l s up the stem, u n t i l they no longer overlap and providing that no other c i r c l e overlaps t h i s i n t e r s e c t i o n . The c e l l s around trees so produced w i l l be the points at which that expanded tree i s the highest.  A better a l t e r n a t i v e , where software  p l o t t e r f a c i l i t i e s are a v a i l a b l e , i s to p l o t the surface of each of the trees from a top view with hidden l i n e s removed. same kind of mapping. are:  This w i l l produce the  Using the area of these c e l l s ( C ^ ) the p r o b a b i l i t i e s  - 99 -  gure 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.  tree c  tree d  - 100 -  (14.1)  (14.2)  E^ . i=l  The p r o b a b i l i t y of choosing a tree at a random point i s : N  i=l  (14.3) T  T a l l e r trees have increased p r o b a b i l i t y of s e l e c t i o n as do trees of better form and those which are w e l l d i s t r i b u t e d r e l a t i v e to t h e i r basal area.  As the expansion f a c t o r increases the bias towards  taller  trees increases because more smaller trees are completely enclosed i n the l a r g e r expanded tree. The system i s a p p l i c a b l e to any set of objects which have the same c r o s s - s e c t i o n a l shape but d i f f e r e n t magnitudes.  To c a l c u l a t e t o t a l  volume of the objects the r e l a t i o n s h i p of c r o s s - s e c t i o n a l area to p l o t area w i l l have to be known, and t h i s i s p a r t i c u l a r l y simple with objects c i r c u l a r i n cross-section.  While t h i s method can be used on sections  of trees (merchantable height, height below f i r s t limb, knot f r e e sections, etc.) i t measures outside the bark and cannot deduct f o r breakage or r o t , except perhaps by the samplers estimation of % reduction as i n most  - 101 -  systems.  The use of the system to sample f o r t o t a l volume w i l l be  discussed s h o r t l y .  V e r t i c a l Cross-Sectional Weighting  We have established a method of sampling p r o p o r t i o n a l to h o r i z o n t a l c r o s s - s e c t i o n a l area with h o r i z o n t a l point sampling.  Critical  height l i n e sampling can be used to sample p r o p o r t i o n a l to v e r t i c a l c r o s s - s e c t i o n a l area of a t r e e .  Consider the v e r t i c a l c r o s s - s e c t i o n  of a tree p a r a l l e l to a random transect through the woods as i n Figure 37. The average c r i t i c a l height along random transects i s p r o p o r t i o n a l to the c r o s s - s e c t i o n a l area.  To choose a random tree f i r s t sum the  c r i t i c a l heights along a transect then use the e l i m i n a t i o n method.  Cylinder Volume Weighting  2 Tree volume can be c a l c u l a t e d by V. = ( TT /4)D. H.F.. F. r e l a t e s the volume of the tree to the volume of a c y l i n d e r of the 2 same diameter and height ( ir /4 * D^,  * R\) .  When F^ i s the same f o r  a l l trees, sampling can be done by a two stage s e l e c t i o n system. F i r s t , the trees are picked p r o p o r t i o n a l to basal area by the use of an angle gauge, then they are measured f o r t o t a l height  (an easier task  than measuring c r i t i c a l height) and f i n a l l y selected from a short l i s t of t o t a l heights.  This produces a s e l e c t i o n p r o p o r t i o n a l to c y l i n d e r  - 102 -  Figure 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 c r o s s - s e c t i o n a l area of the stem.  The c r i t i c a l height c a l c u l a t e d i s the same as the distance run across the p r o f i l e _if_ i t were l a i d down at r i g h t angles to the transect and centered at the tree.  - 103 -  volume.  Grosenbaugh (1974) uses a system l i k e t h i s one f o r what he  c a l l s the "Point-3P" sampling method.  Unfortunately, he s t i l l has the  problem of a c t u a l l y measuring tree volume on a l l the trees at some of the p o i n t s .  The c r i t i c a l height method has other advantages besides  freedom from volume t a b l e s , and they w i l l be discussed s h o r t l y .  - 104 -  CRITICAL HEIGHT SAMPLING , HISTORICAL DEVELOPMENT AND LITERATURE REVIEW  In one of B i t t e r l i c h ' s e a r l y a r t i c l e s concerning v a r i a b l e p l o t sampling ( B i t t e r l i c h , 1956), he " i n d i c a t e d that the volume contributed by each tree i n an angle count sample i s r e l a t e d to i t s ' c r i t i c a l height' " ( B i t t e r l i c h , 1976).  I t i s strange that B i t t e r l i c h  himself did not f i n d the exact form of that r e l a t i o n s h i p . Perhaps he was hampered by a view which was r e s t r i c t e d to the two-dimensional plane i n which tree cross-sections were being "expanded" by h i s use of the angle gauge. sample.  At any rate h i s system developed i n t o a two phase  The f i r s t phase consisted of estimating stand basal area by  counting trees chosen with an angle gauge.  The second phase was  sampling f o r the "Volume to Basal Area Ratio" (VBAR) which established the cubic volume r e l a t i n g to each square u n i t of basal area.  This i s  u s u a l l y done by s e l e c t i n g sample trees and d i v i d i n g t h e i r volume by t h e i r basal area.  These estimates of the volume to basal area r a t i o  are then averaged, weighting i n d i v i d u a l r a t i o s i f necessary. volume i s then estimated  The t r a c t  by  In 1962, Masami Kitamura d e l i v e r e d a paper l a y i n g out the basic technique of c r i t i c a l height sampling (Kitamura, 1962).  The b a s i c  - 105 -  geometry has been described i n a previous chapter. The c r u c i a l idea was that a v e r t i c a l l i n e passing through the f o r e s t could be used to sample d i r e c t l y f o r VBAR.  Two years l a t e r , he published  a major paper concerning the theory of the system (Kitamura, 1964). Another a r t i c l e (Kitamura, 1968) discussed i n d i r e c t methods of c r i t i c a l height measurement.  B i t t e r l i c h reported the development of Kitamuras  system i n 1971 ( B i t t e r l i c h , 1971) and a b r i e f summary of the system was included i n the d i r e c t i o n s f o r the wide scale relaskop (Finlayson, 1969). In 1973, the author independently derived the system as a s p e c i a l case of a more general system of random l i n e s penetrating a volume of space i n the f o r e s t .  The method was c a l l e d "Penetration  Sampling" and was l a t e r developed as a c l a s s project ( l i e s , 1974). A search of the l i t e r a t u r e at that time revealed only one t r a n s l a t e d paper (Kitamura, 1968), but i t contained a diagram and one formula which were s u f f i c i e n t to e s t a b l i s h the s i m i l a r i t y of my own work to that of Kitamura. An a r t i c l e by Loetsch i n the IUFRO proceedings from Nancy, France (Nash et^ a l . , 1973) l i s t e d f i e l d t e s t s of the c r i t i c a l height system as one of the components of f o r e s t inventory which needed research. Also i n 1973, B i t t e r l i c h wrote an a r t i c l e f o r the f i r s t meeting of the I n t e r n a t i o n a l A s s o c i a t i o n of Survey S t a t i s t i c i a n s at the 39th Session of the I n t e r n a t i o n a l S t a t i s t i c a l I n s t i t u t e i n Vienna. the  I t described  use of the angle gauge i n implementing the c r i t i c a l height sampling  system ( B i t t e r l i c h , .1973).  The a r t i c l e does not seem to have been  - 106 -  printed with the other papers presented at that meeting.  In 1973  the system was also mentioned b r i e f l y i n a book on f o r e s t inventory (Loetsch, 1973).  These l a s t two works contained the f i r s t discussions  of the method i n the E n g l i s h language. Due to the lack of other E n g l i s h t r a n s l a t i o n s the existence of the system was v i r t u a l l y unknown i n North America.  By 1974, Thomas  W. Beers, an a u t h o r i t y on the v a r i a b l e p l o t technique, had s t i l l not heard of the method (Beers, 1974).  As l a t e as September 1976, a plea  for information about the system i n the newsletter INFO 75 by Mike Bonnor (Bonnor, 1975) brought no responses except from t h i s author. In 1976, the f i r s t E n g l i s h j o u r n a l a r t i c l e on the system appeared i n the Commonwealth Forest Review ( B i t t e r l i c h , 1976).  The  a r t i c l e was adapted from m a t e r i a l published i n Allgemeine Forstzeitung ( B i t t e r l i c h , 1975)  and p r i n t e d f o r the information of users of the  relaskop and telerelaskop by the manufacturer of that instrument ( B i t t e r l i c h , W. and W. Finlayson, 1975). In 1977, Kitamura developed a v a r i a t i o n of a process by Minowa (Kitamura, 1977).  Minowas basic system was described e a r l i e r i n  connection w i t h using a v e r t i c a l angle to do l i n e - i n t e r s e c t  sampling.  Kitamurafe new method and further d e s c r i p t i o n s of s i m i l a r i t i e s of Kitamurafe and Minowate systems were o u t l i n e d i n an a r t i c l e at the IUFRO conference i n Freiburg (Kitamura, 1978).  At present, no a r t i c l e s have  appeared i n North American p u b l i c a t i o n s and knowledge of the existence of the system i s s t i l l unusual.  - 107 -  ADVANTAGES AND APPLICATIONS  The b a s i c theory of the c r i t i c a l height system (hereafter often abbreviated as CH system) i s not d i f f i c u l t .  The fundamental  i n s i g h t i s that the s o l i d content of the f o r e s t can be sampled by passing random v e r t i c a l l i n e s through the t r a c t area and sampling "depth" of wood encountered.  the  The variance of the estimate i s obviously  based on the v a r i a t i o n found at each of these points where the v e r t i c a l l i n e penetrates the stand.  To decrease the v a r i a t i o n the stems can be  "expanded" and the distance that a v e r t i c a l l i n e penetrates the expanded stem (the c r i t i c a l height) can be determined i n the f i e l d as described earlier.  There i s no reason, other than o p e r a t i o n a l convenience,  sampling l i n e s to be oriented v e r t i c a l l y .  f o r the  A 45 degree angle to the  ground would probably be l e s s v a r i a b l e . The f i r s t advantage to c r i t i c a l height sampling i s that i t i s a d i r e c t sample f o r stem volume. The bias involved i n the use of volume tables i s not included i n the volume estimate.  This i s p a r t i c u l a r l y  important  where the top diameter i s h i g h l y v a r i a b l e , and cannot be predicted from taper equations.  This i s often the case i n hardwood species, e s p e c i a l l y  where log grade i s an important f a c t o r .  The length to which i n d i v i d u a l  standing trees w i l l be cut simply cannot be p r e d i c t e d , but i s e a s i l y judged while looking at the p a r t i c u l a r t r e e . The system i s s e n s i t i v e to a c t u a l tree form, and does not require any assumptions.  This makes i t u s e f u l f o r s i t u a t i o n s where  standards of u t i l i z a t i o n change frequently or where no research has been a v a i l a b l e on a species.  When the c r i t i c a l height i s d i v i d e d by t o t a l  - 108 -  tree height i t provides a weighted sampling of the c y l i n d r i c a l form factor.  A weighting p r o p o r t i o n a l to basal area i s automatically  implemented since trees are selected w i t h an angle gauge. A second advantage, dependent on the l o c a l u t i l i z a t i o n standards, species and spacing, may be i n the variance of the system. In the standard v a r i a b l e p l o t system, with a l l trees measured f o r VBAR, a tree i s e i t h e r included or excluded from a c l u s t e r , and therefore that tree's VBAR i s added to the t o t a l or completely ignored. The r e s u l t i s a "step f u n c t i o n " over the area.  Figure 38 i l l u s t r a t e s  the sum of VBARs f o r the three trees on a sample area. map  The  preference  f o r h o r i z o n t a l point sampling e s t a b l i s h e s the s i z e and shape of  these c e l l s .  The sum of the VBARs i n each of these c e l l s determines  the variance of the volume estimate f o r the t r a c t .  This i s shown by  the form of the volume estimate from a s i n g l e v a r i a b l e p l o t . I estimated volume per hectare  VBAR.  BAF i=l  where:  BAF = the Basal Area Factor of the angle gauge used. I  This  = the number of " i n " t r e e s at a p o i n t .  E VBAR term changes d i s c r e t e l y .  The same reasoning  applies to f i x e d p l o t s using tree volume rather than VBAR.  In c r i t i c a l  height sampling each observation i s an estimate of the VBAR of the tree.  The overlapping heights accumulate i n a continuous manner.  - 109 -  Figure 38.  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.  - 110 -  See Figure 39.  As trees tend to regularly space themselves this  continuous function may be less variable than other sampling systems, particularly the fixed plot methods. This situation w i l l be explored later i n the thesis by simulation techniques. A more important consequence of the smoother continuous distribution i s the effect on permanent forest inventory plots.  One  of the serious problems i n the use of the variable plot technique for continuous forest inventory i s "ongrowth", where a tree which was previously "out" grows s u f f i c i e n t l y to be included at a subsequent measurement. I f the tree i s quite large the increase i n the sum of VBARs for the plot can be quite important.  This consequence of the  step function, allowing trees to "jump" into a plot, can be avoided with c r i t i c a l height sampling. With the CH technique the tree overlaps the point on the second measurement, but contributes only a small value to the sum of the c r i t i c a l heights. effect.  Figure 40 demonstrates the  When mortality i s considered the CH method may have a greater  or smaller effect, since the c r i t i c a l height can be larger or smaller than the tree VBAR.  APPROXIMATION METHODS  There are three basic problems to consider i n the f i e l d application of CH sampling.  F i r s t , the c r i t i c a l point may not be v i s i b l e  from the sample point, generally because of foliage. Second, the angle of measurement may be so steep that i t makes measurement d i f f i c u l t .  - Ill -  Figure 39.  Side view showing sum of c r i t i c a l heights as a smooth continuous function.  - 112 -  Figure 40.  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 vs. c r i t i c a l height sampling systems.  cn Pi <  second measurement  PQ >  to  change at second measurement due to "ongrowth"  tree location"  second measurement •H  X. O  U  o  o e CD  permanent sample point  area covered by expanded diameter  tree location  - 113 -  Third, the instrument simply may not be s u f f i c i e n t l y accurate i n l o c a t i n g the c r i t i c a l point even when i t i s c l e a r l y v i s i b l e at a reasonable angle. The p r o b a b i l i t y of f a l l i n g i n t o the crown section of the tree i s easy to compute i f the diameter at the base of the crown i s adequately known.  The c r i t i c a l point w i l l occur i n the crown when  the sample point f a l l s w i t h i n the expanded area of the diameter of the crown base.  The proportion of times the c r i t i c a l point f a l l s  w i t h i n the crown i s given by  J  where:  bc  D, be  = the diameter at the base of the crown,  D  = the diameter of the base of the stem.  Figure 41 i l l u s t r a t e s the p r i n c i p l e . Some method must be devised f o r c a l c u l a t i n g CH when the c r i t i c a l point i s not v i s i b l e .  Perhaps the most immediate s o l u t i o n i s an  approximate i n t e r p o l a t i o n scheme.  Using nearby v i s i b l e points and the  number of bar widths on the relaskop one can i n t e r p o l a t e t o f i n d the c r i t i c a l point.  This i s of course a p o s s i b l y biased approach, but the  b i a s should be small i n the upper parts of the crown where taper i s rapid.  An unbiased sample i s s t i l l maintained f o r the lower bole.  - 114 -  >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. 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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.  1967. Single tree sampling i n even-aged plantations for survey and experimentation. Proc. 14th Congress Int. Union For. Res. Organ., Munich, 1967, Pt. VI, Sect. 25, pages 379-403.  - 142 -  Johnson, F.A. 1972. F a l l , Buck and Scale Cruising. Journal of Forestry, Volume 70, Number 9, pages 566-568. Kitamura, M. 1962. Zur Bestandesmassenermittlung durch die Deckpunkthohensumme. Report of the 73th meeting of the Japanese Forestry Society, pp. 64-67. Kitamura, M. 1964. Theoretical Studies on the Estimation of Stand Volume Through the Sum of Deckpunkthohen. Bulletin of the Yamagata University (Agricultural Science), Vol. 4, No. 4, pp. 365-403. Kitamura, M. 1968. Einfaches Verfahren zur Bestandesmassenermittlung durch die Deckpunkthohensumme. Journal of the Japanese Forestry Society, Vol. 50, No. 11, pp. 331-335. Kitamura, M. 1977. Eine Methode zur Bestandesmassenermittlung mit dem vertikalen kritischen Winkel. Journal of the Japanese Forestry Society, Vol. 59, No. 3, pp. 101-103. Kitamura, M. 1978. Moglichkeiten der bestandesmassenermittlung mit dem horizontalen oder verticalen kritischen winkel. I.U.F.R.O. proceedings, Nancy, France. Loetsch, F., F. Zohrer and K.E. Haller. 1973. Forest Inventory, Volume I I . Munchen, Basel, Wien. 469 pages. Martin, G.L., A.R. Ek and R.A. Monserud. 1977. Control of plot edge bias i n forest stand growth simulation models. Can. Jour. For. Res., Volume 7, pages 100-105. Matern, B. 1971. Doubly Stochastic Poisson Processes i n the plane. S t a t i s t i c a l Ecology, Volume 1, P a t i l , Pielou and Waters, eds. University Park: Penn State University Press, pages 195-214. Mead, R. 1966. A relationship between individual plant-spacing and y i e l d . Annals of Botany, N.S. Volume 30, Number 118, pages 301-309. Meyer, H.A. 1956. The calculation of the sampling error of a cruise from the mean square of the successive differences. Journal of Forestry, Volume 54, page 341. Nash, A.J., T. Cunia and K. Kuusela. 1973. Proceedings of I.U.F.R.O. Subject Group S4.02, Nancy, France. University of Missouri, School of Forestry, Fisheries and W i l d l i f e .  - 143  -  Minowa, M.  1976. Stand Volume Estimation Through Upper-Stem Diameter. Journal of the Japanese Forestry Society, Volume 58, Number 3, pages 112-115.  Minowa, M.  1978. P r e c i s i o n of a New Method with V e r t i c a l Angle f o r the Estimation of Stand Volume ( I I ) , Line Sampling. Journal of the Japanese Forestry Society, Volume 60, Number 5, pages 186-190.  Newnham, R.M. and G.T. Maloley. 1970. The generation of h y p o t h e t i c a l f o r e s t stands f o r use i n simulation studies. Forest Management I n s t i t u t e Information Report. FMR-X-26, Canadian Forestry Service. 75 pages. Overton, W.S., D.P. Lavender and R.K. Herman. 1973. Estimation of biomass and n u t r i e n t c a p i t a l i n stands of old-growth D o u g l a s - f i r . I.U.F.R.O. Biomass Studies, pages 89-104, College of L i f e Sciences and A g r i c u l t u r e , U n i v e r s i t y of Maine, Orono, Main. 532 pages. P i e l o u , E.C. 1977. Mathematical Ecology. 385 pages.  John Wiley and Sons.  Schreuder, Hans T. 1979. Count Sampling i n Forestry. Volume 24, pages 267-272.  Forest  Science,  Shiue, C.J. 1960. Systematic sampling with m u l t i p l e random s t a r t s . Forest Science, 1960, Volume 6, pages 42-50. Strand, L.  1958. Sampling f o r volume along a l i n e . Number 51.  Norske Skogfors^ksv.  Warren, W.G. 1972. Stochastic Point Processes i n Forestry. Stochastic Point Processes, John Wiley and Sons, Inc. pages 801-814.  - 144 -  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 .  - 145 -  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.  - 146 -  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.  - 147 -  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.  

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