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Systems for the selection of truly random samples from tree populations and extension of variable plot.. 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 thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t fr e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of th i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Forestry The University of B r i t i s h Columbia 2075 Wesbrook Place, Vancouver, B.C., Canada V6T 1W5 Kim l i e s Date: - i i - Supervisor: Donald D. Munro ABSTRACT Means of drawing t r u l y random samples from populations of trees distributed 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 plot size and selecting from a cluster with probability (1/M) where M i s larger than the cluster s i z e . The exact bias from some other selection schemes i s shown by the construction of "preference maps". Methods of weighting the selection by tree height, diameter, basal area, gross volume, v e r t i c a l cross-sectional area and combinations of diameter and basal area are described. None of them require actual measurement of the tree parameters. Mechanical devices and f i e l d techniques are described which simplify f i e l d application. The use of projected angles, such as are used i n Variable Plot Sampling i s central to most of these methods. C r i t i c a l Height Sampling Theory i s developed as a generalization of Variable Plot 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 multiplied by the Basal Area Factor of a prism gives a direct estimate of stand volume without the aid of volume tables or tree measurements. Approximation techniques which have the geometrical effect 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 large, 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 Variable Plot Sampling, however i n situations where no volume tables exist i t may have application, 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 also overcome the problem of "ongrowth" for permanent sample plots. Donald D. Munro - i v - TABLE OF CONTENTS Page ABSTRACT. i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES . v i i i ACKNOWLEDGEMENTS x i INTRODUCTION 1 Reasons to Sample Individual Trees 2 Reasons to Sample Randomly 6 Sampling Without Replacement 10 Weighting Selection 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 Selection of the Closest Tree i n Each Cluster 21 The "Azimuth Method" 23 Methods for the Elimination of Bias 29 Plot Reduction Method 29 The Elimination Technique 31 Increasing E f f i c i e n c y 32 Diameter or Circumference Weighting 35 Selection Proportional to Diameter Alone 35 Selection Proportional to Diameter and a Constant ( C g ) . . . 41 Problems of Scale and F i e l d Use 42 - v - Tree Height Weighting 49 An Adaptation of Line-Intersect Sampling to Standing Trees. . 51 Basal Area Weighting 54 Basic Ideas of Point Sampling 55 Bias From Selection of a Single Individual From Every Cluster.57 Bias From Random Selection From Each Group 57 Nearest Tree Method 60 The Azimuth Method With Variable Plots 71 Non-Random But Unbiased Methods For Subsampling 71 Height Squared Weighting 73 Combining Diameter Squared and Diameter Weightings 73 Mechanical Devices to Aid Selection 83 2 Adding a Constant, Selection Proportional to aD. + bD^ + c. .86 A Generalized Instrument 86 Gross Volume Weighting 90 Basics of the C r i t i c a l Height Method 90 Random Selection From a Cluster Proportional to C r i t i c a l Height of the Tree, a Biased Method 96 Selection of the Tree With Largest C r i t i c a l Height, 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 136 CONCLUSIONS . 139 LITERATURE CITED 140 APPENDIX I. L i s t of Symbols and Terms 144 - v i i - LIST OF TABLES Table Page 1 Computations involved for the example i l l u s t r a t e d i n Figure 2 20 2 Calculations for the example shown i n Figure 17 59 - v i i i - LIST OF FIGURES Figure Page 1 Construction of Dirichlet cells around stem mapped trees on a tract to be sampled 15 2 Illustration of the computation of sampling probabilities when a single tree is chosen randomly from a l l those on a fixed plot 19 3 Preference map when selection is 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 line sampling, basic idea of the selection rule 36 7 Illustration of selection probability with line sampling 37 8 Construction of line sample preference map 39 9 Band width for the preference map when sampling is proportional to + C g 43 10 Band width for the preference map when sampling i s proportional to D, - C g 44 11 Using the angle gauge to establish the reduced diameter of a tree 45 12 & 13 Field systems to select trees proportional to D. - C 46 l s 14 Field system to select trees proportional to D. + C 47 l s 15 Use of an angle gauge on f l a t ground for vertical line sampling 50 16 Use of two measurements on sloping ground 52 17 Plots of variable size in horizontal point sampling 56 - i x - 18 Constructions with "nearest tree" selection and variable plots 61 19 I l l u s t r a t i o n of terms used i n proof of bias toward smaller trees 62 20 I l l u s t r a t i o n of construction rules for the nearest tree preference map 67 21 Completed preference map, nearest tree method with variable plots 69 22 Completed preference map for the azimuth system and variable plots 72 23 Plot area composed of 3 simple figures 74 2 24 Plot of area proportional to aD^ + bD^ 76 25 Geometry used to determine the angle 9 79 2 26 Selection proportional to aD . + bD^ 81 2 27 P r i n c i p l e s of selection proportional to aD^ - bD^ • . • 82 28 Device p r i n c i p l e s for sampling proportional to aD 2 + bD .84 i i 29 Device p r i n c i p l e s for sampling proportional to aD 2 -bD 85 i l 30 & 31 P r i n c i p l e s of selection proportional to aD 2 + bD + c 87 i - l 32 Prism device for selection of trees by plots indexed by diameter alone 89 33 The "expanded tree", c r i t i c a l height and c r i t i c a l point 91 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 93 35 Selection p r o b a b i l i t i e s proportional to c r i t i c a l height 97 36 Selection of tree with largest c r i t i c a l height 99 37 Selection proportional to v e r t i c a l cross- sectional area of the stem 102 - x - 38 The step function formed by the sum of VBARS of overlapping expanded trees i n standard variable plot sampling 109 39 Side view showing sum of c r i t i c a l heights as a smooth continuous function m 40 The effect of "ongrowth" in permanent sample points with variable plot vs. c r i t i c a l height sampling systems 112 41 The proportion of occasions the c r i t i c a l point w i l l be i n the crown. 114 42 Calculation of the probability 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 43 Example of calculations 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 for 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 46 C r i t i c a l height as measured by tripod mounted relaskop 134 47 Coefficient of v a r i a t i o n for 5 sampling methods 137 - x i - ACKNOWLEDGEMENTS I would f i r s t l i k e to thank Dr. Donald D. Munro, whose attention to d e t a i l , coupled with his tolerance for the many extra projects i n which I was engaged, puts me s o l i d l y i n his debt. I could not have had better guidance, advice or breadth of opportunity, and I am grateful. I would also l i k e to thank the rest of my committee, Drs. Kozak, Demaerschalk, Eaton and Williams, who were generous with thei r time and showed a sense of proportion and professional courtesy not always found i n graduate committees, and which made my time here more prof i t a b l e . My appreciation to the Biometrics Group, both s t a f f and students, for the stimulation, information and help they provided as well as the sheer pleasure of their 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 kindly provided data for the simulation studies, and Mr. Davis Cope and Ms. Karen Watson who provided timely mathematical insight and an enjoyable exchange of ideas. Crucial f i n a n c i a l assistance was provided by the MacMillan Bloedel, Van Dusen and McPhee fellowships, the C.F.S. Science Subvention Program and teaching assistantships through the Faculty of Forestry. - x i i - 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 projects, that was the end product of a graduate program. This attitude has made my time at U.B.C. not only a f i r s t class educational adventure but three years of hard work that were tremendous fun. - 1 - INTRODUCTION This thesis w i l l develop two basic areas of interest. 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 distributed i n a non-random pattern i n a plane. It i s d i f f i c u l t to believe that such systems have not been developed i n the past. Other than the c l a s s i c a l method requiring 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 elimination method) w i l l be used to select trees with probability proportional to a number of different parameters for that tree. Selection proportional 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 interest. The biases inherent i n incorrect attempts to gather random samples w i l l be examined and methods of specifying the magnitude of such biases w i l l be developed i n the form of "preference maps" constructed from standard stem location maps. In several cases mechanical aides w i l l be devised for f i e l d work in selecting samples with p a r t i c u l a r weightings. The second part of the thesis w i l l develop the theory of C r i t i c a l Height Sampling for the inventory of tree stands without the use of volume tables. The basic concept i s to extend the theory of B i t t e r l i c h s ' Variable Plot Sampling to the t h i r d dimension, so that not only the tree basal area, but the entire tree volume i s expanded by a constant and then d i r e c t l y sampled. A large scale 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 anticipate problems i n application and to develop alternative 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 Plot or Variable Plot techniques. Applications and timber types for 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 thesis w i l l be on the geometrical reasoning involved, since an understanding of th i s w i l l greatly aid attempts to adapt these methods to l o c a l conditions. Reasons to Sample Individual Trees Selection of trees for sampling, as opposed to complete enumeration, i s now standard practice i n forest inventory. In most cases, the trees are selected as a cluster, and the value of interest i s a sum of individual values i n that cluster. S t a t i s t i c s are then applied to that sum as a single observation and usually expanded on an area basis. In their c l a s s i c application both fixed plot sampling and variable plot sampling are examples of th i s process of selecting a cluster of trees. The cluster 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 cluster i s often less variable than individual 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 for trees to react to each others presence. This "competition effect" tends to cause higher v a r i a t i o n between individuals, 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 for a shared amount of available l i g h t , water and nutrients serves to ret a i n a constant effect on a group of trees even while increasing individual differences. This reasoning certainly applies to growth, i f not to t o t a l volume. We are thus led to the c l a s s i c situation of v a r i a t i o n within (rather than between) clusters which gives cluster 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 plot center i s established i t i s usually small additional e f f o r t to determine several trees for sampling at the same time. With the expense involved i n travel 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 cluster even when v a r i a b i l i t y between and within clusters would not indicate 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 basis. 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 trees, hence a system based on volume i n an indi 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 for indi v i d u a l trees, but they lack widespread acceptance. - 4 - ) The th i r d advantage to selection of a c l u s t e r , i s that simple rules for the unbiased selection of a cluster of trees are easy to develop and readily available, while unbiased selection rules for individual trees are d i f f i c u l t to find or simply unavailable. Selection of individuals 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 following quote i s from Pielou (1977). In order to choose a random individual from which to measure the distance to i t s nearest neighbor, the only satisfactory 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 follows. There i s another method of picking random plants, but i t , too, requires that the size of the t o t a l population be known. If a sample of size n, say, i s wanted from a population of size N, the probability 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 probability 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 clusters based on a fixed area has three main disadvantages. F i r s t , there may be l i t t l e s t a t i s t i c a l advantage i n doing so. In an area where the growth of a cluster may be very consistent the t o t a l volume may be quite 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 different species are involved, are l i k e l y to be even more variable. S t r a t i f i c a t i o n , either before or after data c o l l e c t i o n , may be only partly helpful i n reducing this variance. Secondly, as trees are measured with greater accuracy and for multiple c h a r a c t e r i s t i c s , the cost difference between establishing the sample point and measuring the tree diminishes, and i t i s less 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" cruising (Johnson, 1972). This kind of destructive, intensive measurement can only be j u s t i f i e d when sample sizes 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 just because i t w i l l simplify the selection process to measure a cluster 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 physically measuring the border. In addition there may be "boundary effects" for trees physically near the border. This has resulted i n a great number of papers, for instance (Martin et a l . , 1977; Beers, 1966; Beers, 1976; Barrett, 1964). I t might be an advantage, where possible, to avoid rather than solve these problems. Reasons to Sample Randomly Random sampling does not always mean selection of individuals with equal pr o b a b i l i t y , although i t i s often discussed i n this manner. I t does mean that an indiv i d u a l item has a part i c u l a r probability of being included i n a sample, and usually that the selection of the individual does not affect the further probability of selecting any additional member of the population for 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, for the following discussion, only equal probability selection with replacement because i t i s simple and i l l u s t r a t e s the main points of the following topics. In equal probability selection with replacement each permutation of observations has an equal probability of occurrence. This usually means that during the sampling process each observation has an equal chance of selection at any time. This i s desirable, since i t permits unexpected termination of the sampling process without affecting the randomness of the smaller sample. This type of random selection i s presumed for most estimates of population variance. Many sampling schemes can be considered of th i s type by suitable 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 interest. F i r s t , such a sample yields 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 desirable feature for 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 easily produced i s by simply assuring that each observation has an equal probability of being sampled. Systematic sampling can often have this effect, and i s usually meant to. Unbiasedness i s not hard to produce i n an estimator, nor i s i t universally accepted as a desirable feature. The smaller expected mean square error E £ (x - which can be produced by Baysian estimation, many robust methods and other techniques, i s often 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 for the concept of unbiasedness are given by Brunk (1965). 1. To state that an estimator i s unbiased i s to state that there i s a measure of central 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 this point of view, might, for 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 size i s large the estimator i s l i k e l y to be near the population parameter. However, this i s the property of consistency, discussed below; and the argument here i s not primarily i n favor of unbiasedness, but i n favor of consistency. For example, the unbiased estimator, |̂  ̂  " 1 J °̂  t*ie 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 linear 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 , for example - 8 - 6 is a parameter having 6^ and Q^ as unbiased estimators in two different experiments, every weighted mean ad^ + with a + /3 = 1 is also an unbiased estimator of 6 . We note, however, that non-linear transformations do not in general preserve unbiasedness. For example, 6 is 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 particular parameter is unmanageably large. A way of approaching the problem is to restrict attention to an important subclass, such as the class of unbiased estimators. From the point of view of most practical research we can dismiss the random sample from further consideration i f unbiasedness is the only c r i t e r i a of interest. The use of the random sample in much of sampling theory is not necessary, but rather a device for simplifying the mathematics. The second main feature of a random sample is the known standard deviation of the mean (<r—)or "sampling error" as i t is often called in the biological literature. Random sampling does not by any means minimize this sampling error. It is 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 in precision 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 right angles to the drainage pattern" (Husch, et a l . , 1972) and similar advice in many texts. - 9 - Fisher (1936a) covers the systematic sample thoroughly. His criticism i s mainly in 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 in phase with a periodically arranged population, and w i l l then also have an apparent decrease in 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 nul 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 solidly 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 sizes give d i s t r i b u t i o n s of means which are roughly normal. If the population size frequency d i s t r i b u t i o n 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 usually assumed before theoretical calculations can be made about the behavior of the sample mean. I t i s this 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 error 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 additional time and ef f o r t to do so. I t therefore seems desirable to produce methods of obtaining such random samples. We s h a l l primarily be interested i n the case where a population of trees i s dispersed on a tr 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 individuals. This i s , i n many respects, no different than selecting 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 cross-section of the tree stem or other special features, these methods w i l l have application 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 correction factor". In many cases, sampling without - 11 - replacement i s desirable, but when the f i n i t e population correction cannot be computed accurately, and because of the reasons mentioned i n the l a s t section, i t may be desirable to sample with replacement. The methods capable of sampling with replacement are easily modified for 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 results i n the d e f i n i t i o n of sampling without replacement giving equal probability 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 thesis. The methods developed i n th i s thesis w i l l be applicable to sampling with or without replacement. Weighting Selection P r o b a b i l i t i e s Individuals i n a population are often selected for sampling with a probability proportional 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 selection without actually measuring that c h a r a c t e r i s t i c . There are several reasons to make weighted selections. F i r s t , i t may be desirable to have more precise answers regarding some size classes of the items i n the population. Often larger trees are more valuable than smaller trees. Second, larger sample sizes are often required f o r some classes of items because they are more variable. Third, i t i s sometimes mathematically easier to sample by weighting the selection - 11a - probability (and give the measurements equal treatment thereafter) than to sample with equal probability 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 result can be less variable when selection p r o b a b i l i t i e s are varied. A very general equation for the estimation of a population t o t a l can be written: . where: V = the value measured on item i from the population T = the estimated population t o t a l for the type of population value measured p = t* l e Probability of sampling item i from the population. The variance of the t o t a l i s proportional 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? Clearly t h i s can be done by sampling with a probability proportional to the measured value of each item. One or more of these reasons often applies i n sampling forest stands, therefore considerable e f f o r t w i l l be made to derive means of selection for random samples proportional to several tree c h a r a c t e r i s t i c s , and to devise ways of selecting trees without the actual measurement of those c h a r a c t e r i s t i c s . Sampling methods w i l l f i r s t be derived for sampling with equal pr o b a b i l i t y for each i n d i v i d u a l and some of these techniques w i l l then be adapted for sampling with other p r o b a b i l i t i e s . - l i b - ) Much of the f i r s t part of the thesis i s concerned with "bias" and how to avoid i t . More s p e c i f i c a l l y the concern i s with "selection bias" where objects may i n fact be selected with a probability much different than the one intended by the sampler. This i n turn w i l l generally result 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 for changes to estimating equations which w i l l compensate for selection bias, but the emphasis w i l l be on selecting 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 for selecting members of a population i s one based on equal frequency, where each of the N members of a population i s measured with the same probability. In addition to t h i s requirement a random sample would also give equal selection probability to each permutation of observations. Surprisingly l i t t l e advice can be found on the problem of selecting a random sample of trees from a forest area. The only common system suggested for drawing such a sample i s f i r s t to lab e l a l l of the N individuals, to draw a random number from 1 to N, and to fi n d and measure that i n d i v i d u a l . The process i s repeated to select a sample of desired size. The ef f o r t involved i n t h i s process usually eliminates i t from serious p r a c t i c a l consideration. "Nearest Tree" Methods A common method i n practice i s to f i r s t f i n d a random point on the tract to be sampled. This i s simply accomplished by the intersection of two random coordinates. A tree "near" t h i s point i s then chosen subjectively, or the tree nearest to the point i s chosen by measurement. - 13 - This "nearest tree" idea w i l l be examined i n some d e t a i l throughout the thesis. The bias i n the f i r s t system cannot be calculated, but that of the second system i s easy to examine. I t can most easily be studied by the use of polygons constructed around each tree. These polygons are called 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 Available" (Brown, 1965) or "Occupancy Polygons (Overton et_ al_., 1973) . Their use i n forestry i s i l l u s t r a t e d by such publications 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 incloses a l l points to which the tree i s closer than any other tree i n the population. The size 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 follows: " the smallest polygon that can be obtained by erecting perpendicular bisectors to the horizontal l i n e s j oining 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 entire tract without overlap or gap, that they are easy to construct; and that they are not affected by the shape or size of the c e l l constructed around other trees, which insures the same c e l l regardless of the order of construction. Jack (1967) gives a number of useful equations for determining which trees are c r i t i c a l to the construction of the c e l l - 14 - and for calculating the c e l l area. He also mentions a computer program for automatically doing this from f i e l d data giving the bearing and distances to surrounding trees. Newnham and Maloley (1970) provide a computer routine for 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 bisection. Figure 1 shows the general process involved, and examples of the c e l l s around 3 trees. for sampling w i l l be called "preference maps" for that selection system. I t i s obvious that the probability of choosing the tree nearest to a random point i s d i r e c t l y proportional to the area of the D i r i c h l e t c e l l around the tree. The exact probability of choosing the tree i would be: Such diagrams, noting the area i n which a tree may be chosen (1.1) where: = The probability of selecting tree i for sampling under the selection system. DC. = The area of the D i r i c h l e t c e l l around tree i I T = The t o t a l area of the tract where sampling i s conducted. A l l symbols i n th i s work w i l l be defined when f i r s t used, and also l i s t e d i n Appendix 1 for easy reference.  - 16 - There i s every reason to believe, on the basis of experience, that large trees are spaced at wider intervals than small trees i n the same conditions. At the very least we know that different forest areas of the same size have different numbers of trees. I t i s therefore apparent that trees from both areas cannot have the same probability of selection by the nearest tree method, and that the bias i s very probably i n favor of larger trees, 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 select trees with equal frequency, i s there a simple system which does?? The use of any size fixed plot w i l l assure that every tree has exactly the same probability of selection. This i s probably the easiest way to select a sample where each tree has an equal chance of selection. This, unfortunately, does not allow us to pick a random indi v i d u a l as eas i l y . Often such a cluster of trees picked by a plot w i l l serve the sampling purpose but occasionally a subsample i s required or a t r u l y random selection of individuals 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 selection of more than one tree from a plot would be non-random even though i t might be unbiased. If clusters with positive covariance among trees are selected, but treated as a random sample of individuals, the sampling error w i l l be underestimated. The reverse s i t u a t i o n can also occur, as stated e a r l i e r . - 17 - t Bias i n Subsampling From a Cluster 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 plot. The tree corresponding to the random number i s chosen for sampling. If only one tree i n each fixed plot i s chosen i n this way (or any other), 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 . The probability of sampling an in d i v i d u a l on a particular plot i s (1/np). For an indiv i d u a l tree i n a given plot this can r i s e to a maximum of 1 or decrease to a minimum of 1/n , max where n i s the greatest number of trees found on any plot i n the study max J v j area. The expectation of an indiv i d u a l for sampling can be calculated from a preference map constructed i n the following way: 1. Construct a plot around each of the N trees on the area with the s i z e , shape and orientation desired. 2. Determine the number of trees which share each compartment formed by overlaps of the plots, and the area of each. In complex situations t h i s w i l l best be done by d i g i t i z e r or planim ter, i n simpler cases perhaps by equation. 3. Calculate the expectation of sampling tree i from a randomly located single point by the formula: - 18 - N - 1.0 k=l (2.1) where: t ^ = t o t a l number of compartments formed by overlap of tree i with other plots a^ = area of compartment k n^ = number of trees sharing compartment k T = t o t a l area of tract sampled. The probability that tree i has been selected given that some tree has been selected (hereafter called r e l a t i v e probability) i s : Pr { s . j . T * p { s . } N (2.2) L-W EH i = l k=l where: t = 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 calculations using c i r c u l a r fixed 2 area plots of 25 m . The probability 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 probability when a single tree i s chosen randomly from a l l those on a fixed plot. - 20 - Table 1. Computations involved for the example i l l u s t r a t e d i n Figure 2. Compartment from diagram tree 1 2 3 4 5 a * 12.0 1 12.0 b * * 4.0 2 2.0 c * * 2.6 3 • 0.87 d * * 2.7 2 1.35 e * * * * 3.1 4 0.78 f * * * 0.6 3 0.2 g * 2.9 1 2.9 h * * 8.0 2 2.0 i * * * 5.1 3 1.7 j * 6.2 1 6.2 k * * 2.0 2 1.0 1 * 5.3 1 5.3 m * * 2.9 2 1.45 n 22.1 1 22.1 79.5 Tree E L V \ ] Relative Probability Selection Bias 1 2 3 4 5 17.2 11.8 13.88 13.1 23.52 .22 .15 .17 .16 .30 +10% -25% -15% -20% +50% Totals: 79.50 1.0 - 21 - This i s simply the r e l a t i v e area covered by the plot 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 cluster then that observation should be weighted proportional to the size of the cluster. This w i l l remove any bias from estimation based on the sample mean. Selection of the Closest Tree i n Each Cluster The equations are the same once the preference map has been drawn. There are fewer compartments but construction i s more tedious. Figure 3 shows res u l t s . When plots overlap the bisector i s drawn. This i s p a r t i c u l a r l y easy, since most of the bisector construction i s provided already by the overlap of the c i r c l e s . The nearest bisectors reduce the size of the plot as i n a D i r i c h l e t c e l l . Relative reduction i s most severe i n clumps of trees, decreasing the selection probability of trees i n dense spacings. As the size of plots increases the bisectors are increasingly 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 d i s t r i b u t i o n . 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 selecting an indi v i d u a l tree: r e l a t i v e probability of selecting a tree: (3.2) i = l probability of selecting some tree at a random point: N (3.3) The "Azimuth Method" The system, sometimes used i n forest inventory, i s to establish a sample point, then choose for subsampling the f i r s t tree whose center i s encountered i n a clockwise (or counter-clockwise) dir e c t i o n from the plot center. The starting azimuth can be fixed (such as always starting from North) or randomly chosen. A preference map can be constructed for any given st a r t i n g d i r e c t i o n . Figure 4 shows the construction of such a map. Examples of each of the following steps are noted. A North starting azimuth and clockwise rotation are assumed for this example. - 24 - Figure 4. Construction of preference map based on the azimuth method. - 25 - Construction steps are: 1. Construct, a l i n e (called the "stem l i n e " ) from each tree South to the area border. 2. Starting at the West border lab e l the stem li n e s 1-N from West to East. 3. Starting with tree 1 (tree^) and continuing sequentially, 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 after tree N-l. 4. Consider only the f i n a l tree N. From tree^ 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 tree^ to the border i n a dire 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 strikes the area border west of the tree^ stem l i n e . Polygons i n which a part 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 tree^, 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 is 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 is selected as a subsample, the construction is slightly more d i f f i c u l t . A map is drawn as in Figure 2, indicating the polygons in which a particular subset of trees may be chosen. For each of these subsets a further construction is made as described in Figure 4 but only considering the trees of that subset, and drawn across the polygon being considered. This is 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 is done by hand. As a f i r s t step construct only the stem lines which are in polygons occupied by two or more trees. Second, label the obvious situations f i r s t , particularly cases where only one tree is eligible. 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 in 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 fixed plots. - 28 - z . 1 y a • where: 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 allocated to a particular tree. Relative probability of sampling a tree i s : Pr {s.| = „ { Z ) (4.2) i = l Probability of choosing at least one tree from a random point i s : z P a c I M " T where: z P = the t o t a l number of compartments i n the area being sampled. - 29 - Methods for the Elimination of Bias Much attention has been given to the d i s t r i b u t i o n of individuals i n plots of various size under random spacing. Matern (1971) discusses such a d i s t r i b u t i o n . In f a c t , we know that tree d i s t r i b u t i o n s i n the plane are not usually random. The situation was nicely described by Warren (1972) as follows: Trees are, of course, not points. The diameter of a tree, conventionally measured at breast height (4 f t . 6 i n . or 1.3 m) i s generally not ne g l i g i b l e with respect to the distance between trees. Further, competition between trees ultimately produces an area about each tree within which no other tree can exist. These r e s t r i c t i o n s are often conveniently neglected i n theoretical studies (notably the w r i t e r ' s ) ; exceptions are the theoretical derivation of Matern (1960, p.47) and the simulation studies of Newnham (1968) and Newnham and Maloley (1970). The idea that there i s a truncation of actual d i s t r i b u t i o n s at some minimum plot s i z e , as w e l l as an actual upper l i m i t for the number of trees i n a given plot si z e , gives r i s e to two systems for selecting random samples. These w i l l be called the "plot reduction method" and the "elimination method". Plot Reduction Method Since the problem causing bias i s the overlap of p l o t s , they can be reduced i n size to a diameter equal to the smallest distance - 30 - between members of the population. This means maximum diameter i n the case of non-circular plots. The probability of being chosen i s now equal for a l l members of the population and the choice of not more than one member i s assured with each plot. In addition, observations selected are independent, insuring a truly random sample where any permutation of objects i s equally l i k e l y . The probability of finding a p a r t i c u l a r sample tree i s given by: where: ap = the area of the fixed plot used. The probability 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 equalizing 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 plot size and minimizing the probability of a vacant plot. I t i s not the tree i t s e l f which must be selected, but something which can be associated with the tree, and detectable on the plot. Even though tree stems may be quite close together the upper parts of the tree w i l l usually 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. This more even d i s t r i b u t i o n w i l l lead to more e f f i c i e n t sampling. In addition, such methods are useful 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 their high potential for automation. The Elimination Technique Reduction of plot size can become quite severe under conditions of high clumping, this leads to i n e f f i c i e n c y i n sampling due to unoccupied plots. The elimination technique may be more effective i n this case. A plot of size ap i s chosen, and the maximum number of trees which could f a l l into such a plot i s symbolized by Mp. For any plot 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 plot that tree i s chosen, otherwise a new plot i s established. This system again gives a random sample of individuals. The probability of selecting a single tree i s calculated by: which i s a constant. The r e l a t i v e probability i s simply 1/N. If 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 actual probability for a particular tree can then be computed by the standard formula except i n those compartments where np > Mp. Where the plots 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 N (6.2) i = l where the maximum of the values (1/np, 1/Mp) has been used i n equation 6.1 to calculate p • Increasing E f f i c i e n c y The way to minimize the occurrence of unoccupied plots 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 plot j *p |selecting one of those trees | more formally: Mp np=l (6.3) where: p jnp > Mpj =0 p | | i s the probability of selecting a plot with np trees present for possible selection. - 33 - The p r o b a b i l i t i e s of selecting particular numbers of trees might be calculated from theoretical considerations or by f i e l d tests. The elimination procedure i s a very general one, and can be used to choose a f i n a l sample from the equal probability f i r s t stage selection with plots. Suppose one i s interested i n choosing a random sample of trees from a population with probability proportional to 2 ^ DBH_/ . One way to make the selection i s to choose a subset with equal probability using a p l o t , then select from the manageably short l i s t of 2 6 DBH.' by means of a random number. The random number would be chosen l np 2 > 6 between 1 and the maximum expected value for ~y ' DBH_̂ " . Grosenbaugh's system for 3P sampling i s s i m i l a r , except that tie ^ deals with the entire population rather than a subset. The discussions i n this thesis 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 plots might be used for p o s t - s t r a t i f i c a t i o n to increase the precision of the estimator or for other uses. In the example of choosing a tree 2 6 proportional to DBH " the l i s t s of DBH on plots where no sample was chosen might be used as additional information for estimating the parameter of interest, much l i k e Grosenbaugh's use of the "adjusted" rather than the "unadjusted" estimator (Grosenbaugh, 1976). Since effi c i e n c y under these considerations hecomesheavily and 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 eff i c i e n c y i s the probability of selecting a - 34 - random sample with minimum f i e l d work, which generally means selecting a tree at each random sample point. Systems which screen the population through two or more stages can be constructed i n a variety of ways. Consider the equal probability selection scheme. At the f i r s t stage we could select a cluster of trees, each selection being proportional to tree basal area (or any other c r i t e r i a which would simplify f i e l d selection). At the second stage t h i s manageable subset could select an ind i v i d u a l proportional to (1/basal area) by the elimination system. The product of the two p r o b a b i l i t i e s would be: Ms4 • *(-=7) - ( - c ^ f ) < 6 4 > where: BA. = the basal area of tree i 1 C^,C2= constants depending on the exact selection system used. C^ depends on the 1st stage selection probability (probably on the c r i t i c a l angle discussed l a t e r ) . C2 depends on the largest possible value for i = l The product i s a constant, regardless of tree basal area. The re l a t i v e probability of selection i s therefore 1/N. Such a multiple stage selection system may be of advantage where simple mechanical means can be contrived to select trees with compensating p r o b a b i l i t i e s at one or more stages. The v a r i a b i l i t y of these schemes would be of primary interest, and would have to be robust under a variety of s p a t i a l - 35 - di s t r i b u t i o n s . Simulation of stem maps would probably be the best device for 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 selections based on fixed distances or angles, and l i s t selection from short l i s t s , the elimination method w i l l be of primary interest. Diameter or Circumference Weighting Selection Proportional to Diameter Alone This problem was essentially solved by Strand (1958) with the introduction of horizontal l i n e sampling. Trees are sighted at 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 to the l i n e . Figure 6 shows an example. A tree w i l l be picked i f the l i n e crosses an unseen plot surrounding the tree and proportional to i t s basal area. The use of the angle gauge determines when you are i n the plo t , the magnitude of the angle determines the absolute size of the unseen plot. The dashed l i n e s i n Figure 6 indicate the unseen borders of the plots around each tree. To simplify f i e l d work the trees are sometimes sighted to only one side of the transect. Changes to the equations used are simple. The assumption i n the following discussions i s that trees are sighted to both sides of the transect. The probability of selection obviously i s proportional to the diameter of the unseen plot around each tree. Figure 7 i l l u s t r a t e s the pr i n c i p l e . The angle, when small, acts i n the same way as calipering - 36 - Figure 6. Horizontal l i n e sampling, basic idea of the selection rule. transect - 37 - Figure 7. I l l u s t r a t i o n of selection probability with l i n e sampling. -©- distance = D.*PDF x e — & e— i d i r e c t i o n of transect e - 38 - a tree for diameter ( D j and multiplying by a constant which we s h a l l c a l l the Plot Diameter Factor (PDF). Under the assumption that trees are "convex outward" i n cross-section ( i . e . f l a t or curved outward so that a st r i n g wrapped around i t would always touch the surface) the average of a l l possible caliper measurements i s equal to the perimeter divided by ir . The horizontal l i n e sample therefore selects trees proportional to the perimeter or average diameter. Grosenbaugh (1958) has pointed out that large angles can cause certain biases, since the angle gauge no longer nearly resembles the caliper measurement. This i s of l i t t l e concern for 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 for some reason, or trees cannot be considered convex outward there i s always the option of sampling with a plot, then choosing from a l i s t of cumulative diameters or perimeters by the elimination method. Two variants of horizontal l i n e methods are biased, and the bias can be calculated by the construction of preference maps. With the f i r s t method a transect i s started from a random point and the f i r s t tree " i n " with the angle gauge i s chosen for 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 tree, and of length (PDF * D̂ ) i s constructed (see Figure 8). D̂  i s the diameter of the tree at the sighting point of the angle gauge. - 39 - Figure 8. Construction of l i n e sample preference map. tree number - 40 - The PDF i s a constant r e l a t i n g tree diameter to the diameter of an unseen plot 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 for sampling whenever a point i s chosen along i t s band. The bands run opposite the dire c t i o n of the sampling transects u n t i l intercepted by another tree or the area border. The proportional area of these bands (ab^) determine the probability of sampling the tree. S p e c i f i c a l l y : P robability of sampling a p a r t i c u l a r tree: i = l The probability of sampling a tree along a transect beginning at a (7.1) Relative pr o b a b i l i t y i s : Pr (7.2) random point i s : N (7.3) - 41 - If the transect lines are of fixed length (L^) there i s only a s l i g h t modification. A l l bands of the preference map are truncated' at length from the tree. A l l equations above w i l l then apply to the revised diagram. 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 plots 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 in the population (in the dir e c t i o n of the transect) i s the determining factor. These diagrams lend themselves easily to computer pl o t t i n g . The straight l i n e s , truncated only by interception of another tree or the area border, are simply calculated and drawn, and areas are easily computed. When the transect i s run other than North-South, i t i s easy to translate 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 essentially the same as the previous system, but interception 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. Selection Proportional to Diameter and a Constant (C g) A different 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 specified width (W ) centered on a transect - 42 - If the tree i s included when any part of i t i s within the s t r i p sampling i s proportional to diameter plus a constant. When the tree i s sampled only i f the tree center i s included this takes the form of sampling with a rectangular plot of varying length. The preference map for the f i r s t method can be easily constructed. The band width i s simply D. + W , centered at the tree. Further construction i s the 1 s same. Figure 9 shows the basic idea. The elimination technique can be used, with a constant length s t r i p , to obtain a random sample. When selection i s to be proportional to - C g only a small change i s required. In th i s case, the tree i s chosen only when i t i s ent i r e l y within the s t r i p . Figure 10 shows the construction of the preference band of such a tree. Problems of Scale and F i e l d Use If 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 calipering and using (1/x) * (Di> as the diameter. Alternately an angle gauge can be used to get a proportional reduction on the princ i p l e s of the Biltmore Stick. Unfortunately, such a system i s very dependent on a round cross-section. Figure 11 shows the p r i n c i p l e . \ - 43 - Figure 9. Band width for the preference map when sampling i s proportional to + Cg. 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 either of these l i m i t i n g situations. The distance between the center l i n e s then determines the probability of selecting that tree. - 44 - Figure 10. Band width for the preference map when sampling i s proportional to D. - C . r i s 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 centerlines determines the probability of selection. - 45 - Figure 11. Using an angle gauge to establish the reduced diameter of a tree. .The angle of the instrument determines the proportion of D. to D . The distance between the contact points 1 r i s then used as the reduced diameter of the tree. - 46 - Proportional expansion of the tree diameter i s easy to establish 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 select, i n the f i e l d , sample trees proportional to D. - C . Sim i l a r l y Figure 14 shows a system for X s selection proportional to D. + C . A late r discussion w i l l outline i s the use of the Wheeler Pentaprism to automatically account for the s t r i p width. - 47 - Figure 12. F i e l d systems to select trees proportional to D. minus C . i s If 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 points. - 48 - Figure 14. F i e l d system to select 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_ within 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 horizontal angle was used on tree diameters. A tree i s to be sampled when a transect passes within a distance proportional to tree 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 within a particular vertical.angle (C v). Figure 15 shows the general idea. The angle i s being used to establish the proportion of height to distance. Hirata (1962) describes this method to sample forest areas for volume. In sloping t e r r a i n t h i s angle selection method s t i l l selects proportional to height, but the exact proportion i s no longer given by the tangent of the angle. In such cases, the difference between two tangents i s used, read from a suitably calibrated 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 horizontal l i n e sampling. The system applies 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, etc. The reasoning concerning the elimination method and application of a constant term are the same as with horizontal l i n e sampling. The band width i s determined by height rather than diameter, but otherwise there i s no difference i n the p r o b a b i l i t i e s or preference maps. - 5 0 - Figure 15. Use of the angle gauge on f l a t ground for v e r t i c a l sampling. - 51 - The effect of leaning trees i s discussed by Loetsch e_t a l . (1973). If the tree i s not easily 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, to the tree. An Adaptation of Line-Intersect Sampling to Standing Trees If a l l the trees on a tract were f e l l e d perpendicular to a random transect of length L, the tract volume could be simply estimated as a special case of li n e - i n t e r s e c t sampling. The estimate, from a single transect, of t o t a l tract volume would be: n V = i = l CA. where: CA i s the cross-sectional area of a stem crossed by the transect, i This i s the average depth of tree cross-sections along the transect multiplied by the tract area. The trees w i l l not i n fact be conveniently l a i d across the transect, but could this 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 horizontal distance from the transect to the tree. That point can easily 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 "% scale" as i l l u s t r a t e d i n Figure 16 can easily 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. - 52 - Figure 16. Use of two measurements on sloping ground. (tan b - tan a) = tan C^, giving correct horizontal sampling distance i n sloped areas. - 53 - If T / 4 radians i s an inconvenient angle for some reason, the estimating equation can be s l i g h t l y changed. The distance over which a tree w i l l be sighted with a v e r t i c a l angle i s : distance = tree height * 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: V = ZBAi i = l * 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 translation i s not available his formula i s c l e a r l y equivalent to the one just derived. Minowa gives i t as follows: V J \ ^ y \ 4 * L * c o t J g y From the a r t i c l e cited and subsequent work (Minowa, 1978), i t seems obvious that Minowa has not recognized that this part of his work i s a special 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 testing and acceptance by f i e l d foresters. - 54 - Basal Area Weighting Since the development of the sampling method of B i t t e r l i c h (1948), usually called horizontal point sampling or variable plot sampling, a great deal of forest inventory has been done with selection of trees proportional 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 usually subsampled for volume chara c t e r i s t i c s . This i s done mainly to establish the "Volume to Basal Area Ratio" (VBAR). This r a t i o i s usually much less variable than the number of " i n " trees. Only a few of these trees should be measured for maximum sampling efficiency. The cost of measuring trees, compared to simply counting those which subtend the angle used, i s quite high. To minimize cost perhaps only one i n twenty trees should be measured. The problem i s to choose the volume sample trees at least 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 for each diameter class. 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 pro f i t a b l y i n taking more plots for tree count only. Once such a Diameter-Height Curve i s established i t constitutes a bias i n subsequent measurements which should be accounted for s t a t i s t i c a l l y . - 55 - The process by which subsamples are chosen w i l l be examined in some d e t a i l i n regards to possible biases and non-randomness. Basic Ideas of Point Sampling angle established with some form of angle gauge. The angle radiates from a single 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 proportional to the basal area of the stem. The probability of picking a tree with an angle gauge from a random point i s obviously proportional to the basal area of the stem. More formally the pr o b a b i l i t i e s are: For picking an ind i v i d u a l tree: In point sampling a tree i s counted when the stem subtends an P (8.1) Relative probability of picking a tree: (8.2) i = l i = l - 56 - Figure 17. Plots of variable size i n horizontal 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 cluster which are a l l " i n " from a par t i c u l a r point. Choosing clusters of trees makes a sample non-random, but selecting one tree from each cluster can seldom f a i l to be biased as w e l l . Bias From Selection of a Single Individual From Every Cluster Selection of one indi v i d u a l has a l l the biases discussed e a r l i e r with fixed plots. I t favors clusters with small numbers of trees, but t h i s clustering effect i s changed because smaller trees have smaller plots. In addition, when the selection i s not random, further biases can exist. Each of the simple systems can be explored by preference maps. Bias From Random Selection From Each Group After the mapping of plots around each of the trees a group of compartments i s formed. For each of these t compartments the number of trees involved (n^) and i t s area (a^) are determined. The expectations for sampling are then calculated as follows: - 58 - Probability of sampling a part i c u l a r tree: t. 1 1.0 T (2.1) Relative probability of sampling a tree: Pr (2.2) N i = l These are simply the equations used e a r l i e r with fixed plots. 2 The difference i s that the area of the plots i s now (BA_̂  * PDF ) rather than fixed. To complete the example, the calculations are given for Figure 17 i n Table 2. In general, there i s a decrease i n individual probability as more plots overlap, indicating a bias again toward sparsely distributed trees. The average number of trees counted from a random point i s a function of the basal area of the trees, not of their numbers. Where basal area i s evenly distributed bias i s reduced. S t r a t i f i c a t i o n w i l l help to reduce the bias when i t i s done to equalize basal area. The overlap of plots w i l l have a greater effect on smaller trees, where the overlap i s a greater proportion of t o t a l area, therefore the mixture of size classes favors selection of larger trees. - 59 - Table 2. Calculations for the example shown i n Figure 17 tree n k " a k l Compartment 1 2 3 4 5 3 k a * 15.5 1 15. 5 b * A 4.83 2 2. 42 c * * * 1.33 3 0. 44 d * * 3.33 2 1. 67 e * 3.00 1 3. 00 f A * 1.83 2 0. 92 g * * * 0.5 3 0. 17 h * A 0.17 2 0 08 i A A 0.5 2 0 25 3 A 2.17 1 2 17 k A 4.83 1 4 83 1 A A 1.83 2 0 .92 m A 43.17 1 43 17 83.00 Tree Relative Probability actual theoretical Selection Bias 1 2 3 4 5 20.02 6.28 3.51 9.11 44.08 ,2412 ,0757 .0423 .1098 .5311 25/99* 10/99 5/99 14/99 45/99 -4.5% -25% -16% -22.5% +17% 83.00 nr * sum of plot areas = 99 m , tree 1 has a plot of 25 m . - 60 - b ) Nearest Tree Method When the nearest tree which i s " i n " with an angle gauge i s chosen for sampling the preference map i s b a s i c a l l y the same as i t was with fixed plots, but the size of the plot around the tree varies with tree si z e . The area formed by the overlap of two plots i s allocated to the smaller one unless the bisector f a l l s within the overlap. See Figures 18c to 18e. As shown i n Figure 18, the i n i t i a l increase i n selection probability for two plots i s i n favor of the smaller (See 18b). The bias i n r e l a t i v e selection-probability eventually equalizes as plot centers approach each other, and favor the larger plot only with very close spacing as i n Figure 18d and 18e. As w i l l now be shown, a random spacing favors the smaller tree. Consider the plots of two trees (a) i s the smaller and (b) i s the larger. See Figure 19. We w i l l choose one plo t , and i t s associated tree, for sampling when we are within that plot and that plot center i s nearest the random point P. Consider what w i l l happen when a and b overlay the random point, b w i l l never be chosen when a and the outer ring of b (b Q) occur at the point, since the center of b would always be further away. Since b^ i s the same size as a i t has an equal chance of being closest to P when a and b^ occur together. The conditional probability of choosing b when a and b overlay, P i s therefore reduced, b has a 50% chance of selection only i f b. l and a overlay P, and no chance otherwise. - 61 - - 62 - Figure 19. 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. = area a 1 area b = area b. + area b 1 o - 63 - Ignoring edge effects, the exact p r o b a b i l i t i e s are noted below. A = area of plot a B = area of plot b B = area of outer ring of plot 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 plot b, i t i s the same size as plot a. Probability that a occurs alone, hence a w i l l always be chosen. Probability that b occurs alone, hence b w i l l always be chosen. Probability that a and b occur together. since B. = A = B * 1 Conditional probability of choosing a and b given that a and b^ occur together: - 64 - p \ a | a>h±} ~ h P |b | a , b i | = h (9.4) (9.5) Probability that a and b Q occur together, hence a i s always chosen. (9.6) The effect 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 probability of choosing a i s : P { S a } = p | a , b | + 4 ~ [ P | a ' b i } ] + P I a > b o f ( 9- 7> _A_ T _A_ T _A_ T _A_ T + [ . T J T V T / A + B-A (T-B) + (hk) + (B-A) T [ Jr t]- + [-f (9.8) The probability of sampling a i s £A/T (1 - R F L)J , where the term R a stands for the proportional reduction due to both plot overlap and the selection system. We can do the same thing for tree b. - 65 - p { S b( = p { b,a \ + 4 " [P {*S>±\\ •fM-M+W] T \ T J T T B K) + [f 4-] We now have the form I B/T ( l ^B/T (1 - R^ (9.9) (9.10) If R, i s larger than R there i s a greater proportional reduction i n the D cL large plot than i n the small one. p{^} i s proportionally reduced i f and only i f : R h > \ i f f i f f i f f ( f ) - |> +] > * I B y T T < i which i s always the case when A i s smaller than B. - 66 - I t i s admittedly of minor interest what w i l l happen under random spacing, since we know that trees are not distributed i n this manner. As the angle gauge changes to produce larger plots the preference map uses more bisectors 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 simp l i f i e d by some of the following rules about l i n e s . The c i r c l e d numbers on Figure 20 refer to use of the rules noted below. 1) There i s a distance, called the "plot radius" for each tree i , computed by PIL = (D * PDF/2) 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 within the plot radius of the smaller tree or_ closer to another tree than a bisector with that tree. 3) Bisectors between trees are not drawn when the distance from the point on the bisector to the smaller tree i s less than the distance PR. of the smaller tree. l 4) Bisectors are not drawn at points which are within the plot radius of a t h i r d tree and are closer to that t h i r d tree than to either of the trees used for the bisector. Using these rules, parts of the preference map can be drawn as each tree i s considered. The c e l l around an indi v i d u a l tree w i l l not usually be completed u n t i l several other trees are also plotted. This - 67 - Figure 20. I l l u s t r a t i o n of construction rules for the nearest tree preference map. - 68 - i s p a r t i c u l a r l y true of larger 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 for convenience. Probability of selecting an individual tree: N -( .) Relative probability of sampling: P r { S i } = ~ —^-^ ( 3 - 2 ) i = l P r o bability of selecting a tree at some random point: N I8.} *—T— In summary, the system w i l l have the following general properties. The actual biases toward small trees w i l l probably be even larger 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 selection of one tree per group. An increased mixture - 69 - Figure 21. Completed preference map, nearest tree method with variable plots. - 70 - of size classes w i l l increase the bias toward smaller trees, as w i l l greater variation i n size classes. The plot reduction method would be most d i f f i c u l t to apply with variable plots, and the elimination method would be the best way to obtain a random sample. In this case, the " i n " trees are numbered sequentially, 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 this case, the probability of sampling a tree i s : (10.1) i=l Relative probability i s : (10.2) i= l The probability of choosing some sample at a random point i s : Mp (10.3) The la 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 Variable Plots This i s just a modification for the system based on fixed plots. After the d i f f e r e n t sized plots have been constructed the compartments are further 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 well spaced or on the border of clumps. Figure 22 shows the preference map with extra lines removed. I t i s based on the same spacing and sizes as Figure 21. Non-Random But Unbiased Methods For Subsampling Selection of clusters of a l l " i n " trees at a point assures a probability based s t r i c t l y on basal area. Individual trees can also be picked with a fixed probability i n 2 ways. F i r s t , each time a tree i s " i n " i t i s picked with a probability (say 1/10) with a randomizing device. This allows more permutations of samples than systematically choosing every tenth tree, although either 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 this sum can be chosen to select 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 operational convenience and the fact that the sampler does not know when the next sample w i l l occur. - 72 - Figure 22. Completed preference map for the azimuth system and variable plots. - 73 - Choice of one sample tree at 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 selection within clusters must be done for some reason, weighting the sample proportional to tree count can remove the bias. Height Squared Weighting This i s an extension of horizontal point sampling, but uses tree height rather than tree diameter. Developed by Hirata (1955) for sampling the height of forest 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 size of the c i r c u l a r plot 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 horizontal 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 proportional to 2 aD^ + bD^. The perimeter of a plot i s proportional to diameter, and so, very nearly, i s a s t r i p around the plot. Figure 23 i l l u s t r a t e s the following l i n e of reasoning. - 74 - Figure 23. Plot area composed of 3 simple figures. c i r c l e of radius W inner c i r c l e (PDF*D.) TT - 75 - Area of inside c i r c l e = [ PDF * D. 1 | PDF2 * I -2 _ > _ ] , . [ , j * ± Outside ring area = ĵ Wg * PDF * D ± * TT J + * J (10.4) (10.5) Total area = inside plot + s t r i p area + small c i r c l e area 2 2 PDF * TT * D. l + £ w g * D ± * TT * PDFJ + J w g 2 7 r j (10.6) Here we have solved the problem except for the f i n a l complicating 2 term W w , the area of a c i r c l e of radius W . The easiest way out s ' s of this i s to remove i t from the center of the plot. We then have a plot which has a void i n the center (of radius W ) as shown i n Figure 24. r s 2 The area of t h i s plot i s aD^ + bD^ The angle gauge used to pick the plot 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 proportional to the equation: P | s i | CX 7.8 D ± 2 + 3 D. We also wish to use a s t r i p width of 2 meters for convenience i n the f i e l d . As already shown: ^ PDF * j D^2 + j p D F * w ^ * „. * D J = plot area - 76 - Figure 24. Plot of an area proportional to aD. plus bD - 77 - The problem i s to choose PDF. By d e f i n i t i o n : PDF * * 2 2 2 D. =aD. = 7.8 D. i i l and |PDF * W * T * D . l = b D . = 3 D [ s i j i i by cancelling the D̂  term from both sides we obtain: PDF * w = a and ĵ PDF * Wg * TTJ = b and multiplying by a/b: (a/b) * PDF * therefore (a/b) * PDF * W s * '] ' [ PDF * TT * 1/4 = a cancelling terms we get the general equation: (a/b) * W * 4 = PDF s J inserting the example values we have: PDF = (7.8/3) * 2 * 4|= 20.8 - 78 - The angle needed to produce this relationship i s 0 = 2 * arcsin I PDF I Figure 25 shows the basic geometry needed to derive this formula. To check the results we compute the following examples. Tree 1, diameter = 20 cm, PDF = 20.8, s t r i p width = 2 meters. 7.8 (.2) 2 + 3 (.2) = .912 plot 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 plot area 20.8 - ] (1) + (20.8 * TT * 2) (1) 470.485 470.485 / 10.8 = 43.563 - 79 - Figure 25. Geometry used to determine the angle 9 . hence: —r- = .0480 radians (2.755°) e = .0962 radians (5.511) - 80 - The actual plot i s 43.563 times as large as the number 2 given by the formula (7.8D_̂  + 3D^) due to the specified s t r i p width of 2 meters. I f i t were desirable we could reduce both the linea r dimensions of the Plot Diameter Factor and the s t r i p width by a factor of 43.563 to make the plot 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 straight 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 Wg from P. Figure 26 shows the f i e l d selection scheme. A random selection can then be made from trees by the elimination method. A recent a r t i c l e by Schreuder (1978) uses just such a selection scheme i n a method called "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 for f i e l d application by modifying the Wheeler Pentaprism as described i n a following section. Selection of trees with a probability proportional to JaD^ 2 - bD^j 2 requires a s l i g h t modification. The area Wg TT must be added to the plot, and th i s can be done by adding 'it to the center of the plot and giving the tree two chances of selection when a random point f a l l s i n thi s area. A s t r i p i s "removed" from the basic plot by sighting across the point from a distance Ŵ . Figure 27 shows the selection r u l e . - 81 - Figure 26. Selection proportional to aD. plus bD.. To select trees from point P, view a l l trees with an angle gauge at distance W from P. The plot around an indi v i d u a l tree shows the locus of a l l points where the tree i s " i n " with the selection rule described above. - 82 - Figure 27. P r i n c i p l e s of selection proportional to aD. minus bD.. To select trees from point P view tree across the point with a prism from distance W . This diagram shows the plot around each tree. The tree i s counted once at a l l points within the s o l i d c i r c l e or twice when within the dotted c i r c l e . - 83 - Mechanical Devices to Aid Selection The selection 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 proportional to aD_̂  + bD^. a) tree i s within the distance W of point P. s r b) tree i s outside Wg, but " i n " with the angle gauge. c) tree i s "out" with the angle gauge. Figure 29 shows the same basic idea used for sampling proportional to aD.2 - bD.. l l Such a device could be b u i l t by simply attaching prisms i n front of the two lenses of a Wheeler Pentaprism which i s available 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 in this way. Using an angle of .0262 radians (1.5°) a 100 cm base would allow a 38 meter width for Wg. 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 sight w i l l remain a major problem. Figure 28. Device prin c i p l e s for sampling proportional to aD^ plus bD^ Figure 29. Device p r i n c i p l e s for sampling proportional to aD. minus bD.. " i n " , count twice " i n " , count once - 86 - Adding a Constant, Selection Proportional to aD. + bD. + c Adding t h i s further 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 outlined for aD + bD to a ir radian (18CT) sweep, while using a fixed plot to select trees through the opposite ir radians. Figures 30 and 31 show the basic idea. I t would appear desirable to generalize the rather awkward method and allow selection with any plot size computed by an equation using only diameter as a variable. One method, using only standard cruising 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 establish the plot size 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 establish 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 for this adaptation i s given by Beers (1964). For s i m p l i c i t y the case of a single prism w i l l be used for the following discussion. A Generalized Instrument For each diameter tree the plot size i s l i s t e d . Using the formula below 9, i s established for each diameter tree. 2 - 87 - P r i n c i p l e s of selection proportional to aD + bD + c. Figure 30. - 88 - A prism of deflective angle 0 i s used where 6 i s always max max J larger than 0^. A round prism i s best, preferably with a hole in the middle for easy mounting. 0^ can be formed by rotation of the prism around i t s center by an angle ( A r ) . e d = Gmax * C O S <V <U-1> The prism i s mounted and the angle of rotation i s marked with the tree diameter. Figure 32 shows such a mechanism set for checking a 50 cm tree. The decision 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 selection system has two advantages. F i r s t , size of plot can be derived by any means, as long as i t can be indexed only by diameter. Second, the plot i s a s o l i d c i r c l e centered at the tree. Such areas around trees are often described when dealing with problems such as competition, rooting zones, moisture depletion etc. When interpolation cannot be done a programmable calculator can easily be used to establish the rotation angle. If the differences i n the values for 0^ are small the adjustment may be d i f f i c u l t . When the angles of rotation are nearly a l i k e the bias of setting the instrument may be large. In such a case two prisms should be used. One fixed prism establishes the basic angle (A . ) , while the other rotatable thinner main one ( Ap"L u s) corrects i t by larger angles of rotation. The f i n a l angle 0, i s then given by: - 89 - Figure 32. Prism device for selection of trees by plots indexed by diameter alone. TOP VIEW pxvo c lear round prism view through prism i s offset - 90 - 6, = A . + (cos A ) * (A . ) (11.2) d mam r plus This method w i l l increase the range of rotation of the smaller prism and simplify the problem of setting 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 effect i s to "expand" the diameter by a constant a l l along that stem. This gives a simple method to expand the stem of a tree. 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 tree, and proportional 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 distance, called the " c r i t i c a l height" w i l l allow us to select trees proportional to t o t a l volume. The tree need only be selected on the basis of i t s c r i t i c a l height. This basic idea was discovered by M. Kitamura i n 1962, but no English translations of the work were available 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 called i t "Penetration Sampling" ( l i e s , 1974). A more detailed history w i l l be given i n the second part of the thesis. - 91 - Figure 33. The "expanded tree", 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 within 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 horizontal plane. A prism, relaskop or any other angle gauge would serve the purpose. If the tree image i s " i n " when sighting horizontally towards the tree for 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 for a distance of 12 meters. For the system to be p r a c t i c a l the physical problem of moving up and down the v e r t i c a l l i n e with the angle gauge must be circumvented. The solution to this problem i s fortunately quite simple. I t i s commonly known that the relaskop (or newer telerelaskop) has an "automatic slope correction". The geometric form of that correction does not seem to be as well known. When an observer views a tree diameter at any v e r t i c a l angle the automatic correction 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 horizontally 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 correction then makes i t possible to th e o r e t i c a l l y " l e v i t a t e " above or below the sample point along a v e r t i c a l axis and "observe the tree horizontally". These are the exact requirements of the system. We now have a p r a c t i c a l method for 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). 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 this " c r i t i c a l point" actual tree bole tree, base 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" 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 "borderline"). 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 tree, or the point below which cubic volume i s not to be considered. I t i s computationally convenient i f these angles are measured in A . 3) The slope distance to the base of the tree. 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 calculate the horizontal distance to the tree. Both v e r t i c a l angles along with t h i s horizontal 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 finding system i s used for 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 proportional to tree volume the elimination method may be used to choose a single tree. This allows selection of a random individual or a random sample proportional to t o t a l volume and without prior assumptions about tree shape or any use of volume tables. - 95 - When E i s the expansion factor of the angle gauge used and calculated by: E = ( s i n plot area tree basal area (12.1) • [ - expanded tree volume actual tree volume - ] • E(V.) V. and using M as the maximum expected sum of c r i t i c a l heights at a sum point, the p r o b a b i l i t i e s with the elimination method are: I'.) • [ • [ • probability of sighting the tree with the prism BA. * E 1 V. BA. M sum expected c r i t i c a l height of tree E * V. l M L sum J ,M L sum -1 (12.2) The r e l a t i v e probability i s : Pr N- N E M M i = l N E T i i = l (12.3) The probability of choosing some tree at a random point i s N given by: Z ( E * v i ) M sum average sum of c r i t i c a l heights M sum (12.4) - 96 - Random Selection From a Cluster Proportional to C r i t i c a l Height of the Tree, a Biased Method If one tree i s chosen using the sum of c r i t i c a l heights at a point (S c) and drawing a random number uniformly from the i n t e r v a l 1-Sc, the problem becomes very d i f f i c u l t . The overlap of trees i s now i n 3 dimensions. Figure 35 shows a side view for 2 trees only. In the area indicated by "0" there i s a probability of selection 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 intersect i n order to compute the selection 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 solution i s too d i f f i c u l t . The only p r a c t i c a l solution i s by numerical approximation methods. Using a grid of xp points across the land area, the estimated probability of selection i s then given by: r XP where S > 0 c / (13.1) where: CH. i s the c r i t i c a l height of tree i , (13.2) i= l - 97 - - 98 - The probability of selection of a tree at a random point i s l i k e that of horizontal point sampling. The maximum diameter i s used to construct a plot around each tree. The proportion of land area covered by at least one plot gives the desired probability. 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 their basal area. Selection 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 selection 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 selection 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 larger. Unfortunately, t h i s l i n e cannot be located as easily as with other systems we have examined. One way to establish the l i n e by computer i s to plot perimeter points on c i r c l e s of size (stump * PDF) except where two or more such c i r c l e s overlap. When two c i r c l e s overlap only the intersection points w i l l be drawn, for diameters at intervals up the stem, u n t i l they no longer overlap and providing that no other c i r c l e overlaps this intersection. 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 alternative, where software plotter f a c i l i t i e s are available, i s to plot the surface of each of the trees from a top view with hidden lines removed. This w i l l produce the same kind of mapping. 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 are: - 99 - gure 36. Selection of tree with largest c r i t i c a l height. tree c tree d - 100 - (14.1) (14.2) E^ . i = l The probability of choosing a tree at a random point i s : N i= l T (14.3) Ta l l e r trees have increased probability of selection as do trees of better form and those which are we l l distributed r e l a t i v e to their basal area. As the expansion factor increases the bias towards t a l l e r trees increases because more smaller trees are completely enclosed i n the larger expanded tree. same cross-sectional shape but different magnitudes. To calculate t o t a l volume of the objects the relationship of cross-sectional area to plot area w i l l have to be known, and this 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 free sections, etc.) i t measures outside the bark and cannot deduct for breakage or r o t , except perhaps by the samplers estimation of % reduction as i n most The system i s applicable to any set of objects which have the - 101 - systems. The use of the system to sample for t o t a l volume w i l l be discussed shortly. V e r t i c a l Cross-Sectional Weighting We have established a method of sampling proportional to horizontal cross-sectional area with horizontal point sampling. C r i t i c a l height l i n e sampling can be used to sample proportional to v e r t i c a l cross-sectional area of a tree. Consider the v e r t i c a l cross-section 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 proportional to the cross-sectional 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 elimination method. Cylinder Volume Weighting 2 Tree volume can be calculated by V. = ( TT /4)D. H.F.. F. relates the volume of the tree to the volume of a cylinder of the 2 same diameter and height ( ir /4 * D̂, * R\) . When F^ i s the same for a l l trees, sampling can be done by a two stage selection system. F i r s t , the trees are picked proportional to basal area by the use of an angle gauge, then they are measured for 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 selection proportional to cylinder - 102 - Figure 37. Selection proportional to v e r t i c a l cross-sectional area of the stem. The c r i t i c a l height calculated 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 right angles to the transect and centered at the tree. - 103 - volume. Grosenbaugh (1974) uses a system l i k e this one for what he c a l l s the "Point-3P" sampling method. Unfortunately, he s t i l l has the problem of actually measuring tree volume on a l l the trees at some of the points. The c r i t i c a l height method has other advantages besides freedom from volume tables, and they w i l l be discussed shortly. - 104 - CRITICAL HEIGHT SAMPLING , HISTORICAL DEVELOPMENT AND LITERATURE REVIEW In one of B i t t e r l i c h ' s early a r t i c l e s concerning variable plot sampling ( B i t t e r l i c h , 1956), he "indicated that the volume contributed by each tree i n an angle count sample i s related to i t s ' c r i t i c a l height' " ( B i t t e r l i c h , 1976). It i s strange that B i t t e r l i c h himself did not find the exact form of that relationship. 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 his use of the angle gauge. At any rate his system developed into a two phase sample. 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 for the "Volume to Basal Area Ratio" (VBAR) which established the cubic volume r e l a t i n g to each square unit of basal area. This i s usually done by selecting sample trees and dividing 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 in d i v i d u a l ratios i f necessary. The tract volume i s then estimated by In 1962, Masami Kitamura delivered a paper laying out the basic technique of c r i t i c a l height sampling (Kitamura, 1962). The basic - 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 forest could be used to sample d i r e c t l y for 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 indirect 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 directions for the wide scale relaskop (Finlayson, 1969). In 1973, the author independently derived the system as a special case of a more general system of random l i n e s penetrating a volume of space i n the forest. The method was called "Penetration Sampling" and was l a t e r developed as a class 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 translated 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 establish 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 tests of the c r i t i c a l height system as one of the components of forest 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 for the f i r s t meeting of the International Association of Survey S t a t i s t i c i a n s at the 39th Session of the International S t a t i s t i c a l I n s t i t u t e i n Vienna. I t described the 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 forest inventory (Loetsch, 1973). These l a s t two works contained the f i r s t discussions of the method i n the English language. Due to the lack of other English translations 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 authority on the variable plot technique, had s t i l l not heard of the method (Beers, 1974). As late 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 English journal 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 material published i n Allgemeine Forstzeitung ( B i t t e r l i c h , 1975) and printed for 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 with 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 descriptions of s i m i l a r i t i e s of Kitamurafe and Minowate systems were outlined 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 publications and knowledge of the existence of the system i s s t i l l unusual. - 107 - ADVANTAGES AND APPLICATIONS The basic 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 insight i s that the s o l i d content of the forest can be sampled by passing random v e r t i c a l l i n e s through the tract area and sampling the "depth" of wood encountered. 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 e a r l i e r . There i s no reason, other than operational convenience, for the sampling l i n e s to be oriented v e r t i c a l l y . A 45 degree angle to the ground would probably be less variable. 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 direct sample for 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 highly variable, and cannot be predicted from taper equations. This i s often the case i n hardwood species, especially where log grade i s an important factor. The length to which in d i v i d u a l standing trees w i l l be cut simply cannot be predicted, but i s e a s i l y judged while looking at the p a r t i c u l a r tree. The system i s sensitive to actual tree form, and does not require any assumptions. This makes i t useful for situations where standards of u t i l i z a t i o n change frequently or where no research has been available on a species. When the c r i t i c a l height i s divided 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 proportional to basal area i s automatically implemented since trees are selected with 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 variable plot system, with a l l trees measured for VBAR, a tree i s either 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 result i s a "step function" over the area. Figure 38 i l l u s t r a t e s the sum of VBARs for the three trees on a sample area. The preference map for horizontal point sampling establishes the size 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 for the t r a c t . This i s shown by the form of the volume estimate from a single variable plot. I estimated volume per hectare BAF i = l VBAR. where: BAF = the Basal Area Factor of the angle gauge used. I = the number of " i n " trees at a point. This E VBAR term changes dis c r e t e l y . The same reasoning applies to fixed plots 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 function formed by the sum of VBARs of overlapping expanded trees i n standard variable plot 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 in 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 in the use of the variable plot technique for continuous forest inventory i s "ongrowth", where a tree which was previously "out" grows sufficiently to be included at a subsequent measurement. If the tree i s quite large the increase in 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. Figure 40 demonstrates the effect. 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 in 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 visible 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 . - I l l - 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 effect of "ongrowth" i n permanent sample points with variable plot vs. c r i t i c a l height sampling systems. cn Pi < PQ > to •H X. O U o o e CD second measurement change at second measurement due to "ongrowth" tree location" second measurement 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 locating 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 probability of f a l l i n g into 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 within 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 within the crown i s given by Jbc where: D, = the diameter at the base of the crown, be 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 for calculating 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 solution i s an approximate interpolation scheme. Using nearby v i s i b l e points and the number of bar widths on the relaskop one can interpolate to find the c r i t i c a l point. This i s of course a possibly biased approach, but the bias 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 for 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. expanded crown base stem distance for 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 to the tree, tree height and DBH to calculate 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 possible 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 informally by Beers (1974) and Bonnor (1975). Any bias i n such a procedure w i l l only affect 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 tree radius i s no less than 2/3 of the height to that point. 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 plot diameter factor (PDF) i s 100. This leads to the geometry shown i n Figure 42. The distance we wish to fi n d i s B, from the tree to 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 plot boundary to the tree. This relationship i s simple because of the conical shape. Since we wish the r a t i o of c r i t i c a l height to distance from B to the tree to be 3/2 i t i s possible to solve for p. - 116 - Figure 42. Calculation of the probability that the c r i t i c a l point w i l l be at too steep an angle for accurate measurement. Side View p*100 (l-p)*100 Top View - 117 - [ 3 "I = I" c r i t i c a l height 1 = [" p * 50 _ p 2 J I distance B J |_ ̂ "P) * 1 0 0 J L ( 1 _ P ) 2 J P 3 (1-p) p = 3 - 3p 4p = 3 p = (3/4) hence B = 1/4 * 100 = 25 The general form of the equation for p, assuming a conical tree form, i s : (15.1) where: p " [ i i * J [ (maximum acceptable tangent) * (plot diameter factor) (tree height to radius ratio) 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 result i s unaffected by the height of the tree, although i t i s d i r e c t l y changed by the plot diameter factor and the relationship between the tree height and tree radius. This example calculation 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 can r i s e sharply as the expanded stems grow t a l l e r and narrower. At a r a t i o of 50:1 for tree height and 30:1 for plot diameter factor (a more reasonable r a t i o i n practice) p becomes .474, giving 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, with the wide scale relaskop normally using 2 bars we could double the distance and use one bar width to find the c r i t i c a l point. We could also move 4 times as far away and use 1/2 bar to find 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 with a 50:1 height to radius r a t i o (HRR) and a 100:1 plot diameter factor (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. These are the cases when the sampler i s inside the central area. The problem here i s to specify 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 within that section of the expanded stem. In practice t h i s means specifying tree form and solving mathematically,. or sampling for the proportion. The correction term (C ) which would afterwards be applied to calculate 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 is sampled. (1) cylinder volume 25 * 37.5 = 73,631.1 unit • ^ 3 (2) small cone volume (top) (3) entire expanded stem 25 * T * 12.5 100 * T * 50 = 8,181.2 =523,598.8 (4) "rim" volume (3)-(2)-(l) =441,786.5 or 84.4% of f u l l cone - 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 . At most perhaps one would like to make very simple counts. An example would be tree counts with an angle gauge. It i s also very l i t t l e trouble simply to read a vertical angle to a point on a tree, since instruments which do this quickly and easily are commercially available. Providing that the var 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. If 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 line 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 for c r i t i c a l height. - 122 - we f a l l within the range A to B, so we w i l l be measuring c r i t i c a l height i n the "rim" area just 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 C H) and tree diameter. The following approach can be used, re f e r r i n g to Figure 44. We would l i k e to have a system for estimating c r i t i c a l height i n the form: CH = B*Q*(tan 0_„) (16.01) and CH = B *Q *T (16.02) where: Q a constant, as yet unknown tan 0, CH the tangent of the angle from the base of the tree to the c r i t i c a l height when viewed from the sample point. T the average tangent for a tree. We begin by finding the average tangent (T): B (16.03) A B (16.04) A - 123 - The cumulative density function for x, given that x i s within distance B, i s : 2 2 CDFN = 1 = .5- (16.05) B hence the probability density function i s : " 2x _ B 2 _ PDFN = (16.06) Using T as the tangent from a random point x we solve the following formula: T = B / A 2x dx (16.07) T = B K X dx (16.08) cancelling x and removing 2K/B yields: dx (16.09) - 124 - T = 2K _ B 3 _ %(x-B)' •B/2 (16.10) T = 0 (16.11) T = 2K B3 j 4- B 2 (16.12) (16.13) Knowing that the volume i n the modified cone i s half that of the o r i g i n a l cone we can now solve for 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 B2* (16.14) Combining t h i s result with equation 16.02 we have: [-H= CH = B*Q*T = B*Q (16.15) Therefore the value of Q i s : Q = - 125 - The estimator for the c r i t i c a l height w i l l then be: CHe = B) tan 0 C R (16.16) 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 revolution by the Theorem of Pappus we get: 2 TT C S CH f (16.18) B TT where: C = the center of gravity of the cross-section of the g expanded tree stem curve. S = the cross-sectional area of the stem from A to B B 2 (-y- B) TT J (x) tan 0 C R dx CH = ^ (16.19) B TT - 126 - s h i f t i n g terms gives: CH = (+•) / " ) tan 0 C R dx (16.20) tan 0 CH dx (16.21) At t h i s point, we can recognize that the term i n square brackets i s the probability of a pa r t i c u l a r value of the tangent occurring (the pr o b a b i l i t y density function of x ) . When sampling randomly i n the- plane we would be choosing the tangent with that probability and under a random sampling process we can drop that term. This leaves: tan 0 CH dx (16.22) which gives the same result as before. I t i s of interest because the method i s very general, and applies to any cross-section (stem curve) that i s of interest. Thus for any stem curve we may use the approximating formula for c r i t i c a l height as follows: CH = C * tan 0 e g 1 (16.23) - 127 - Where C i s the center of gravity of the expanded stem curve on g one side of the v e r t i c a l axis. Kitamura (1968) develops a method similar to the rim method and also uses a similar estimator, but there are differences i n application. Instead of allowing the hollow center h i s system requires the sampler to back up from the tree u n t i l he i s a certain proportional distance from the stem, and then measure the angle to the c r i t i c a l point. In e f f e c t , he would be " f i l l i n g " the otherwise hollow section with a constant. This i s a great deal of trouble 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 sensitive to tree form. I f we are w i l l i n g to assume some tree form why not just measure (or estimate) t o t a l height and get VBAR directly?? Certainly i f one goes to a l l the trouble of making Kitamura's scheme work i t i s more effo r t than simply measuring the distance to the tree. Indeed the whole business seems to be an awkward contrivance simply to avoid the one horizontal measurement. S t i l l , there may be an advantage which Kitamura has overlooked. Consider the two procedures for estimating c r i t i c a l height from a random point. (a) c r i t i c a l height = distance to tree * tan 0 C H (b) c r i t i c a l height = a constant * tan 0 ^ - 128 - Method (a) i s the direct method and measures the height of an imaginary s h e l l around the tree. 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 characteristics as the tree bole i t s e l f . The mean, variance and density functions are proportional 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 effect we have created an "expanded tree" with the same volume (or known proportion thereof) but with an en t i r e l y different "shape". The d i s t r i b u t i o n of the estimator of tree volume physically surrounding the tree w i l l overlap d i f f e r e n t l y with the estimator of nearby trees. 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 forest 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 for the c r i t i c a l height system has been described from a theoretical point of view. As with any sampling system there w i l l be adjustments necessary for p r a c t i c a l f i e l d application. Several plots were established i n a Douglas-fir stand near the University of B r i t i s h Columbia to i d e n t i f y problems i n application and possible solutions. - 129 - The most s t r i k i n g problem i n application i s with trees which are close to the sample point. With nearby trees a number of measurement problems become serious. Tree lean can be a large source of error. 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 unreliable. The maximum intercept bias i s possible, as discussed by Grosenbaugh (1963). Correction for t h i s type of bias can be made following his suggestions. The c r i t i c a l point of nearby trees tends to be i n the crown, and obscured by foliage or branches. On the other hand, taper i s rapid i n the top section of the tree, and there i s a great advantage to the depth of f i e l d for d i s - tinguishing 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 possible to move away from the tree i n multiples of the distance between the tree and the point center t h i s was found to be awkward for very short distances. I t i s clear that either 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 alternate method i s to calculate the diameter at the c r i t i c a l point (from tree diameter and distance) then locate that point and i t s c r i t i c a l height using an o p t i c a l fork l i k e the Wheeler Pentaprism. This o p t i c a l fork 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 sensitive to actual tree form. Taper function use would only be advisable i f i t helped to maintain another possible 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 characteristics of the tree stems. This would cert a i n l y be indicated on permanent growth plots for 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 variable plot 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 " trees, especially i f the Basal Area Factor i s reduced i n order to increase the tree count at each point. 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 sighting trees i n the lower bole. The lower section of the stem has less taper, i s more often obscured by brush and often has a bad background for sighting with the relaskop. In addition 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 and rough on the surface. 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 sight the lower reading on a collapsable fiberglass 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 hints about relaskops may be useful. Keep both eyes open when using the relaskop. 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 reveal an adequate outline of an upper stem even when the crown i s rather dense. I t i s often less trouble to read the degree scale i n the relaskop and convert afterwards to percent. The percent scale i s frequently hard to read and abruptly changes scale without adequate l a b e l l i n g . I t i s helpful 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 suit the background to the tree. Adjustment to the sunshade can result i n an almost transparent image of the relaskop scale, which helps i n locating the c r i t i c a l point. The intensity of the scale can be varied by moving a thumb i n front of the forward c i r c u l a r window of the relaskop. When th i s window i s covered the scale nearly disappears. Causing the scale to " b l i n k " by moving a forefinger on and off the window i s sometimes help f u l i n finding the c r i t i c a l point, p a r t i c u l a r l y i n low l i g h t conditions. Two modifications to the relaskop were useful. A threaded insert available at most camera stores w i l l change the metric European tripod thread at the base to a standard English system thread for easy tripod mounting. The second modification 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 pen used i n overhead projectors. Shallow negative angles were then very apparent when the scale turned bright red. I t i s necessary to insure that the marking pen i s not the permanent type, so that errors can be corrected. To explore the precision 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 Douglas-fir 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 for a Douglas-fir stand, but the tree was chosen so that no brush would interfere with sightings on the lower bole. The points on the tree where i t was obviously " i n " and "out" were also recorded at the same time the c r i t i c a l point was estimated. The results are shown i n Figure 45 f o r hand held relaskop readings and Figure 46 for readings with a tripod mounted relaskop. The increased precision i s obvious with the tripod. Every error of 1 meter i n c r i t i c a l height implies an error of (1 m cubic meter * BAF) i n volume per hectare. In t h i s case, the BAF was 1.0. The greater precision of the tripod mounting was impressive during the f i e l d work. 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 just as they did with the hand held instrument, however the cause was e a s i l y determined. The relaskop scale was so sensitive that i t was picking up the bumps and overgrown knots on the tree stem. I f anything the relaskop was too sensitive, 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. C r i t i c a l height as measured by hand held relaskop. 2 4 2 0 1 6 1 2 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 with species such as hemlock and cedar where taper was either smoother or more rapid. 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 clear. 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 calculated and then located using an o p t i c a l caliper. Log Grading The grading of logs with c r i t i c a l height sampling i s the same as i n standard variable plot cruising. In place of the use of a VBAR for each grade stating the cubic volume of wood per unit area i n a pa r t i c u l a r grade we have a c r i t i c a l height for each grade. The portion of the stem between the c r i t i c a l point and tree base i s divided i n a series of c r i t i c a l heights attributed to the grades of those sections. The volume i n a grade at each sample point i s then computed by: Volume , per ha = V^CH . * BAF grade X—< grade These estimates are averaged over a l l points i n the cruise. With the c r i t i c a l height method the amount of sampling i n a grade i s proportional to the volume i n the grade. In standard variable plot cruising the sampling i n a grade i s proportional 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 simulation study on an actual stand of Douglas-fir 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 conical form was assumed for a l l trees. The stand was clumped and more variable than usual. Repeated simulations were done over 200 random points throughout the area. BAF and plot size were varied on each run and variance of the t o t a l volume estimate was calculated. The results are shown i n Figure 47, recorded by the average number of trees measured i n each test. A few additional runs with different random points and larger sample sizes were made to v e r i f y that these results were representative. 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 variable plot sampling are the same for 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 coe 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 loss - 137 - gure 47. Coefficient of Variation for 5 sampling methods. 1 4 0 - 1 2 0 H F Fixed plot A Approximated Rim Method R Rim Method * C r i t i c a l Height + Variable Plot 100H 8 0 - 6 0 - 4 0 - 2 0 - i 4 i 6 • 8 1 0 1 2 1 4 average number of trees per plot - 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 tree. The standard rim method, simply ignoring trees which were more than twice the c r i t i c a l angle at the base, was more competitive with the standard c r i t i c a l height technique and also eliminated the problem of measuring nearby trees. Fixed plot sampling was competitive as long as the average number of trees measured was kept above 6 trees per plot. The range of 6-10 trees per point seems to be most e f f i c i e n t for sampling purposes i n stands of t h i s type. - 139 - CONCLUSIONS In the f i n a l analysis 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 for c r i t i c a l height. The most promising method for measuring these d i f f i c u l t trees seems to be the use of the Wheeler Pentaprism. Application of the system w i l l probably be limited to cases where volume tables are very unreliable due to v a r i a b i l i t y of the merchantable top, continuous forest inventory where "ongrowth" i s a problem, and use of the method to select trees randomly with p r o b a b i l i t y proportional to gross volume. - 140 - LITERATURE CITED Barrett, J.P. 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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 pl o t s , as well as c e l l , indicating the f i r s t tree from a given azimuth. area of a subcompartment formed by the overlapping of plots. The angle of rotation used with a prism. area of a band associated with a particular tree. area of a fixed plot 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 gravity for a portion of a stem curve. Correction term to calculate 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 de t a i l s of the sampling scheme. a v e r t i c a l angle used to select a tree for possible sampling. cross-sectional 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 part i c u l a r tree. Cumulative density function. Diameter at the base of the tree crown. Diameter at some point on tree i . - 145 - Diameter at the lowest sighting point on the tree, presumed to be the largest 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 factor of an angle gauge. Equal to (plot area/tree basal area). A factor used in calculating p. The number of " i n " trees at a point Arbitrary 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 , largest number of trees selected- as a cluster. The maximum expected sum of c r i t i c a l heights, sample size Number of possible observations i n the population. Usually the number of trees i n an area. number of trees involved i n a compartment. The largest number of trees selected i n any cluster throughout the sample area. number of trees present i n a cluster chosen by fixed or variable plots. A proportion of the distance to the edge of the plot from the tree located at the center. probability of sampling tree i . probability of sampling a cluster of np trees. probability of sampling a tree after establishing 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 probability 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 plot surrounding the tree. Probability density function. Proportion of plot radius,- to be multiplied by tan i n approximating "rim method". a uniform random number between 1 and some specified upper l i m i t . cross-sectional 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 overlapping plots or s t r i p s . Total area of the tract of land on which sampling i s conducted. Average tangent throughout the plot area. tangent of the angle to the c r i t i c a l point from a random point within the plot radius. Total number of compartments formed by overlap of plot of tree i with other plots. the tangent of the angle to the c r i t i c a l point. Equal to (CH/distance to tree). a particular tree from the population. The estimated volume with l i n e - i n t e r s e c t sampling. The volume of a particular tree i . Volume to Basal Area r a t i o . width of s t r i p used i n selecting 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 grid placed on the tract to be sampled. - 147 - number of compartments on preference map favoring selection of tree i . t o t a l number of compartments on a preference map. The angle used to select a tree with variable plot 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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