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A . :trv... .. we?!“ v {{1}}; K SA 4.9” L I B R A R Y Mange-.11 Stem Uinversity will/gluyll/l WI lull/11211118111111: TH 59:0 This is to certify that the thesis entitled A Preliminary Investigation Of Narrow—Strip Sampling As Applied To Forestry presented by James Edward Kearis has been accepted towards fulfillment of the requirements for Ph.D. degree in FOIEStI‘y [Aka/7W Major pressor Date April 3, 1979 0-7639 OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. © Copyright by James Edward Kearis 1979 ‘3" I " A PRELIMINARY INVESTIGATION OF NARROW—STRIP SAMPLING AS APPLIED TO FORESTRY By James Edward Kearis A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Forestry 1979 LAVA... M‘ .i ’- ~ kjyl' ABSTRACT A PRELIMINARY INVESTIGATION OF NARROW—STRIP SAMPLING AS APPLIED TO FORESTRY By James Edward Kearis Narrow—strip sampling consists of sampling a forest stand by means of strips so narrow that the trees may be linearly ordered along the strips. The term 'narrow-strip' has been used in two basically different senses in previous research. On the one hand the 'narrowness' meant that a strip of a width much narrower than usually used was run through the stand. On the other hand, as used in this paper, 'narrowness' means that the trees which are sampled by a narrow—strip are ordered, one after the other, along the narow-strip. Pielou (1962) used narrow—strip sampling in the second sense but her theoretical analyses are totally different from those developed in this paper. In this study, a theoretical basis is given for narrow—strip sampling, a major componet of which is the derivation of the expected distance between adjacent stems along a narrow-strip given the density of stems per unit area, the average diameter breast high of the stems, and the width of the narrow—strip. Monte—Carlo techniques are used to verify theoretical statements and to examine biases and precisions of James Edward Kearis narrow-strip estimators. A range of real and computer- generated forest stands is used in the study and consists of combinations of random, regular, and clustered stem pat— terns with a realistic range of densities and diameter distributions generated from an empirical distribution function. Density and diameter considerations follow from field data gathered in stands located in the New Jersey Pine Barrens. Distance sampling is anr unpractical method of sampling for density but has been studied because of its theoretical appeal. Three supposedly robust methods of estimating density using distance sampling are compared with a density estimator using narrow-strip sampling in a Monte-Carlo study of 34 real and computer-generated stem maps. It is found that narrow-strip sampling gives highly robust estimates of density outperforming the best of the distance estimators. Narrow-strip sampling, circular-plot sampling, and point sampling are used to sample the 34 stem maps for basal area, species proportion, diameter, and density using both random and systematic location of the three types of sampling elements which define clusters of stems. Strip width and length, plot radius, and basal area factor are chosen such that all three methods have an equal expected sample size. Methods are compared on the bases of bias and precision. On these bases, narrow-strip sampling generally performs as well as the other two methods. When data gathered in the field in one oak—dominated James Edward Kearis and in one pine-dominated stand are analyzed, it is found that narrow-strip sampling performs very well. Narrow- strip sampling takes about one—half the time of plot sampling to estimate basal area, density, diameter, species proportions, and stand table entries. These estimates are close to the plot sample estimates. One low-intensity narrow-strip sample takes half the time that a point sample does to estimate basal area but, in addition, yields good density and diameter estimates. Diameters were not taken in the point sample in order to avoid tree-to-tree travel time, but, since there is no travel time involved in measur- ing diameters of trees which are narrow—strip sampled, diameters were taken in the low—intensity narrow-strip sample. It seems likely that many areas of application lie ahead for narrow-strip sampling, and a few are described briefly. However, additional research will be required to verify results of this study under different conditions. Narrow—strip sampling is an attractive and practical alter- native to circular-plot sampling, point sampling, and dis— tance sampling for estimation of basal area, species pro— portions, diameter, density, and stand table entries in forest stand situations included in this study. Theoretically there i£§ no reason why narrow-strip sampling for these characteristics cannot be applied to any forest situation. ACKNOWLEDGMENTS Appreciation is given to the Forestry-Wildlife Section of the Horticulture-Forestry Department at Cook College, Rutgers University. The Section made available funds and facilities for student assistance, office work, and computer- time. The students who helped gather the field data are Ms. Margaret Clark, Mr. Joseph Iozzi, Ms. Donna Orsini, and Ms. Keelin Reardon. In addition Ms. Reardon performed data analysis and clerical work. Access to the areas used in the field study was graciously permitted by Mr. Rodgers Todd, Supervisor of Wildlife Management Areas, New Jersey Division of Fish, Game, and Shellfisheries. Thanks is given to the Doctoral Committee members who gave useful comments and criticisms as the study progressed. The committee members are: Dr. Daniel Chappelle, Department of Resource Development Dr. Dennis Gilliland, Department of Statistics and Probability Dr. James Kielbaso, Department of Forestry Dr. Wayne Myers, Department of Forestry (chairman until 8/78) Dr. Victor Rudolph, Department of Forestry (chairman as of 8/78) TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS Chapter I. II. III. IV. VI. INTRODUCTION LITERATURE REVIEW . DESCRIPTION OF DATA AND METHODS OF ANALYSIS . A. Description of stands used in field studies B. Description of stem maps sampled by computer . . . C. Methods of analysis THEORETICAL BASIS OF NARROW-STRIP SAMPLING A. Estimation of density . B. Estimation of other characteristics C. General considerations . . . . . COMPARISON OF NARROW—STRIP SAMPLING WITH DISTANCE SAMPLING FOR ESTIMATING DENSITY A. Definition of distance estimators B. Monte— Carlo narrow- strip density estimator . C. Comparison of the estimators COMPARISON OF NARROW—STRIP SAMPLING WITH CIRCULAR-PLOT SAMPLING AND POINT SAMPLING . Sampling methods and formulas Random sampling . . . Systematic sampling Conclusions . . DOCUB> iv Page Vi ix ll 11 15 19 23 25 37 49 49 52 53 Chapter Page VII. PRACTICAL APPLICATION OF NARROW- STRIP SAMPLING . . . . . . . . 69 A. Design of a narrow— strip sample . . . . 69 B. Results of narrow- strip cruises in two stands . . . . . . . . . . . . . . . . . 75 VIII. RECOMMENDATIONS...............84 IX. SUMMARY AND CONCLUSIONS . . . . . . . . . . . 92 LITERATURE CITED . . . . . . . . . . . . . . . . . . . 96 APPENDICES Appendix A. Generating a density equation . . . . . . . . 100 Table p KOOO\IO\U1 Al. A2. A3. A4. A5. LIST OF TABLES Stand tables Characteristics of the stem maps Example of stem being sampled by the three techniques . . Bias of density estimators Random sampling comparisons Systematic sampling comparisons Choice of width Basal area Species proportions, dbh and density Expected inter—tree distance Exponential regression coefficients. NRSTRP sample input NRSTRP output Program NRSTRP vi Page l3 16 40 55 61 66 72 78 79 101 105 106 110 113 Figure l. N O‘U‘l-DUJ 10. ll. 12. 13. 14. 15. 16. 17. LIST OF FIGURES Trees ordered along a narrow—strip The three basic stem patterns. Point sampling Sample plot layout Patterns for computer-generated stem maps. Given one stem is sampled, the other stem either is or is not sampled . . . Derivation of the probability that a stem is narrow- strip sampled given that a neighboring stem has been narrow— strip sampled . . . Assume the population stem radii to be constant and the center of the already sampled stem falls on the narrow—strip centerline Expected distance between two adjacent stems along a narrow—strip . . . . . . . . Form for a density estimation table using estimates from a narrow—strip sample Geometry of stem sampled by the three techniques Comparisons of stem being sampled by the methods Narrow- strip sampling of nearest neighbor distances as a PPS method . . . . . Continuous sampling of variation Distances used to define 'narrowness' Geometry of Cox's estimator Systematic layout of sampling elements Page l4 18 26 28 29 31 34 38 41 44 45 48 50 57 Figure Page 18. Bias in estimating inter-tree distance using the quick method of measurement . . . . . . . . 64 19. Locations of sample plot centers . . . . . . . . 70 20. Spacing between narrow-strip centerlines . . . . 73 21. Narrow-strip data sheet . . . . . . . . . . . . . 76 22. Stand 1 stand tables. . . . . . . . . . . . . . . 80 23. Stand 2 stand tables. . . . . . . . . . . . . . . 81 24. Two possible applications of narrow—strip sampling. . . . . . . . . . . . . . . . . . . . 87 25. Another density estimator using narrow-strips . . 90 Al. Density related to inter-tree distance. . . . . .103 A2. Regression constants related to diameter. . . . .107 viii p(t) E(D) DN e(DN) e*(DN) Pr(E) LIST OF SYMBOLS conditional probability of a stem being narrow- strip sampled given that it is a distance t from a stem which has been narrow-strip sampled random variable of the distance between two successive stems along a narrow—strip expected value of D density of a stand in stems per unit area the narrow—strip estimator of density derived from the infinite series expansion of E(D) the number of stems in a stand of species 1 number of stems in a stand random variable of stem radius breast—high expected value of diameter breast-high basal area of a stand in square units per unit area number of stems narrow-strip sampled h nearest neighbor distance random variable of the jt Probability Proportional to Size Cox's distance sampling estimator of density Diggle's distance sampling estimator of density Monte—Carlo version of e(DN) derived by sampling truncated tj distributions from stem maps narrow-strip width probability of the event E ix ArwB A={-} Pr(AIB) XmU(a,b) dbh AB BUF BAF PRF intersection of the two sets A and B the empty set notation for the set A='the set of all x such that (z) x has property P' (A={xz x has Pl) conditional probability of the event A occurring given that the event B has occurred the random variable X has the uniform distribution on the interval (a,b) diameter breast high arclength from point A to point B blow—up factor is the reciprocal of the sampling intensity Basal area factor used in point sampling plot radius factor used in point sampling 43560 _1 1/2 __ BAF PRF —{ 576 } CHAPTER I INTRODUCTION To describe a forest stand, a decision must be made to employzisampling technique which will provide desired information on selected stand characteristics. After the stand characteristics have been selected, the choice of a sampling technique is affected by available resources (funds, manpower, time, and equipment) and by the desired accuracy and precision of the required estimates. All else equal, the most cost-efficient sampling technique is preferred. A new method of sampling called narrow—strip sampling is investigated in this paper. Narrow—strip sampling con— sists of sampling a forest stand by means of strips so narrow that the trees may be linearly ordered along the strips. No matter how a narrow—strip is placed in the forest, no two trees can lie side-by-side along the narrow— strip (see trees A and B in Figure l); i.e., the trees are linearly ordered along the narrow—strip. This means that for any pair of trees sampled by the narrow—strip, one tree must precede the other. Because of the narrowness of the strip, inter-tree distances between neighboring stems along the narrow-strip can be taken and this allows an perpendicular to centerline tree A and tree B illustrate what is meant by two stems lying ‘\ . side-by-side along a narrow—strip tree A / / >/ / °\ 5 \ \ tree 3 , 0 tree centerline of narrow-strip defined by lOO-foot metal tape //’ (direction of travel ////// is along centerline as indicated by r the arrows) first tree edge of 3-foot wide narrow-strip defined by placing Biltmore stick over centerline, tree is sampled if any part of stem at breast height meets any part of the narrow-strip Figure 1. Trees ordered along a narrow-strip. estimate to be made of density (the number of stems per unit area). Though trees can grow with stems in contact at breast height, this is rare and can be adjusted for in practice. Besides density, volume and biomass are also important characteristics of a forest stand. Estimation of these characteristics using narrow-strip sampling could not be covered in this paper because of the preliminary scope of the study. However, diameters are measured as a part of the narrow-strip samples, and since heights can be measured as in any other sampling technique, volumes can also be estimated. In fact, narrow strip sampling provides a convenient frame from which to sample for any characteristic desired. This paper demonstrates that narrow-strip sampling can provide cost—efficient estimates of density, diameter, basal area, and species proportions while achieving an accuracy and precision which compare quite favorably to accuracies and precisions experienced with circular—plot and point sampling. A major conclusion is that narrow—strip sampling may be an attractive and practical alternative to circular-plot and point sampling. CHAPTER II LITERATURE REVIEW Narrow-strip sampling has not been used before as used in this study. Warren (1971, 1972) mentioned the possibility of some fruitful research in narrow-strip sampling of forest stands. He pointed out that Pielou (1962, 1963) used narrow-belt transects (another name for narrow—strips) to study runs of one species with respect to another and to study runs of healthy with respect to diseased trees along the strips but he adds: . we have only one type of individual, and we record the distance between succes— sive individuals as projected on the long axis of the plot.’ Warren also points out some practical advantages of narrow- strip sampling: 1. Searching for boundaries, a difficult task in heavy understory, is eliminated. 2. It is a simple matter to tell whether or not a tree is in the sample. 3. The crew is not as prone to feel it is wasting time in travelling from sampling point to sampling point since the crew is involved in cruising continuously along the strip. Narrow-strips can be used to estimate density. Because of its effect on density estimation, spatial pattern is an important characteristic of a plant community. Pattern is also difficult to assess. Three basic patterns are generally recognized (Figure 2). The three areas have the same density but possess widely divergent spatial patterns. (The use of the word 'pattern' instead of the word 'distri- bution' will be consistent throughout this paper to avoid the connotation of an underlying statistical distribution which the word 'distribution' carries with it.) The fact that most plant populations have a clustered pattern is generally accepted throughout plant ecology. Most of the studies of pattern occur in forestry because trees are generally very easily distinguishable as individuals. Two measures of the presence of a species in an area of interest are density, the number of plants per unit area, and ggygr, the proportion of the total area covered by the vertical projection of aerial shoots of the species (Greig- Smith, 1964). Another measure of presence important in forestry is basal arga, the number of square units of stem per unit area occupied by trees using the diameter of the stem at breast height, 4.5 feet above the ground. There are a variety of ways to sample for these indicators of the degree of presence of a species. In quadrat sampling for density (Cowlin, 1932; Greig- Smith, 1952; Thompson, 1958) the number of individuals con— tained in a rectangle of fixed size is examined. Circular- plots are also used for this purpose. The rectangles are located randomly, systematically or contiguously. The size of the rectangle is important and can cause problems with the bias of the density estimate (Pielou, 1977). regular Figure 2. random clustered The three basic stem patterns. Cover can be estimated by using the line-intercept method of sampling. The proportion of the length of lines that the species intercepts is used to estimate cover. The lines are generally placed systematically (Johnston, 1957). The line-intercept method assumes that aerial portions of plants do not overlap or intermingle; i.e., individual plant boundaries are well defined. That this is not the case can cause a serious bias in the estimate. The most popular method of estimating basal area in forestry is the point sampling method (Grosenbaugh, 1952; Dilworth and Bell, 1968). A fixed angle is projected (Figure 3) and depending on the distance of thetnxfiafrom the sampling point and on the tree's diameter breast high, the tree appears larger than, smaller than or the same size as the projected angle. If the tree is a true 'borderline' tree (one which is exactly the same size as the projected angle), it is counted as a sample tree. Each sample tree represents the same amount of basal area. The basal areas sampled are averaged over all sampling points to obtain the estimate. Density can also can be measured by distance sampling. There are two types of distances measured in distance sampling: point-to-plant and plant—to-plant. In point—to-plant sampling, a point is chosen at random in the forest and the distance from the point to the nearest stem is measured. Plant-to- plant sampling assumes a stem is chosen at random and the distance to the nearest neighboring stem is measured. Distances . t can also be measured from a po1nt or a plant to the k h nearest “‘\sample Odo not sample pk. sampling point fixed sampling ’ . borderline angle = or. . Figure 3. Point sampling. plant where k>l. Various estimators of density have been derived using one or the other or some combination of the two methods (for reviews see Mneller-Dombois (1974), Persson (1971), and Pielou (1977)). All derivations assume a random pattern of stems. Distance sampling is laborious, contributes nothing to estimating stand characteristics other than density and pattern, and must be carried out independently of the sampling for these other characteristics. As a result, distance sampling is rarely done except for research purposes. A good study using distance methods to estimate density is Persson (1964). The introduction of bias into density estimates due to non-random patterning of stems is covered by Persson (1971). One way density can be estimated is to estimate the expected value of ti where t1 is the distance between a stem and its nearest neighbor. If e(ti) is such an estimate, then ne(ti) is the estimated average area occupied by a stem and the density estimate (in stems per acre if tl is in feet) is 43560/@e(ti)). A small bias in e(ti) can cause a very large bias in the density estimate. Robust techniques of density estimation are currently being tested (Cox, 1976). Robust techniques are those which are not seriously affected by non-random patterns of stems. The term 'narrow-strip' denotes strips narrower than the usual 1/2-chain or l-chain widths which have been used in the past to cruise a certain proportionrfifthe stand area —-- the same as circular—plot sampling does today. 'Narrow' 10 means 16.5 feet to Meyer (1942) and 3.3 feet to Cottam and Curtis (1949). At the other extreme, line-intersect sampling uses lines to sample logs lying askew on the forest floor (Bailey, 1969, 1970; DeVries, 1973a, 1973b, 1974; Warren and Olsen, 1964). An interesting summary of similar methods used in wildlife studies is given in Eberhardt (1978). Of particular interest in Eberhardt's paper is a denisty estimator using strips which are not necessarily narrow. Eberhardt's density estimator will be discussed in detail in Chapter VIII. Studies using narrow-strip sampling have been conducted by Pielou (1962, 1963, 1965). Pielou places narrow—strips systematically and records the occurrence of one or the other type of tree (diseased/healthy or first—species/second—species) along the strips. When studying the incidence of disease, she assumes that healthy trees can either be placed in disease-free 'gaps' or infected 'patches'. She can conclude from her analysis whether or not the two types of trees are randomly mingled along the strips. She obtains probabili— ties of runs of length k=l,...,5 (see Vitayasai, 1971); i e., she finds, for example, the probability of three healthy trees occurring in succession. Most recently Birth (1977) stated: 'Rock outcrops, minor slope changes, and other site factors often result in micro—stands that are substantially different from the major stand encompassing them. Sampling with continuous, narrow, fixed-area strips that traverse the entire stand and are oriented across terrain features is an appropriate way of including the variation.’ CHAPTER III DESCRIPTION OF DATA AND METHODS OF ANALYSIS A description of two real stands which are used in field applications of narrow-strip sampling along with a description of 34 real and computer—generated stem maps used in Monte-Carlo analyses are given in Sections A and B. General discussions of the methods of analysis which are used in Chapters IV-VII are given in Section C. Chapter IV is the theoretical basis for this study. Chapters V and VI are Monte-Carlo studies of 34 stem maps, and any statement about bias or precision of an estimator in these chapters refers only to these stem maps unless otherwise stated. Chapter VII gives results aftwo field applications in the stands mentioned in Section A. Full details of the methods of analysis precede the disucssions of results in the appropriate chapters. A. Description of stands used in field studies. The two stands which form the empirical basis for this paper are located on lands managed by the New Jersey Division of Fish, Game, and Shellfisheries. Stand 1 is 62.5 acres and Stand 2 is 43.0 acres. Stand 1 is dominated by white oak (Quercus alba L.) and red oak (Quercus velutina 11 12 L.). Pitch pine (Pinus rigida Mill.) is also present. Stand 1 is contained in the Peaslee Wildlife Management Area approximately six miles northwest of Tuckahoe. Stand 2 is contained in the Bevans Wildlife Management Area located approximately six miles south of Millville. Stand 2 is dominated by pitch pine but contains some white and red oak. Distribution of stems by species and dbh (diameter bnxmt high) is given in Tablelu These distributions were obtained from stem counts in twenty—five 0.2-acre circular- plots located systematically in each stand. These distri- butions are also used to generate species and diameters for computer-generated stem maps. At each circular—plot center a 2-chain and 4-chain rectangular plot was laid out (Figure 4). A point sample was taken from the plot center and three narrow—strips were run parallel to the 4-chain sides of the rectangular plot and equidistant from each other. The strips were three feet wide. Both stands are in an area of very little topographic relief. Elevation in the area is 80 to 120 feet above sea level. The area is part of a broad sand, silt and gravel plain sloping gently southwestward into Delaware Bay. The area has a mild climate with high humidity. Drought is not usually a problem. Since the area is well drained and the soil is acidic and coarse textured, fires can be a problem. Stand 2 was burned in the past ten years as evidenced by scars on the pitch pine stems. l3 Table 1. Stand tables Stand 1 red white pitch dbh oak oak pine species propor— (inches) (stems per acre) tions 4 9.87 37.62 0.60 red oak 0.49 5 15.90 33.46 1.39 white oak .49 6 13.50 26.28 2.98 pitch pine .ll 7 15.10 10.95 2.98 1.00 8 14.50 6.37 4.17 9 13.90 3.60 4.77 10 7.05 —- 2.58 11 4.23 -- 2 98 l/ 12 2.82 -- 4.18 96.87 118.68 26 23 density = 242.18 Stand 2 pitch dbh oaks pine species proportions (inches) (stems per acre) 4 22.35 19.42 oaks 0.25 5 12.44 33.24 pitch pine .75 g 6 9.00 33.62 1.00 7 7.35 27.22 8 5.69 23.41 9 2.55 19.82 10 4 07 12.01 11 -- 8.60 12 -- 5 20 13 -- 7.82 63.45 190 36 density = 253.18 1/ 12 inches (13 inches) and up counted as one dbh class. l4 circular-plot and point sampling center 4 chains — narrpw=strip centerline 2 chains Figure 4. Sample plot layout. Orientation of rectangular plots to stands shown in Figure 19. 15 B. Description of stem maps sampled by computer. There are 34 stem maps analyzed by Monte-Carlo tech- niques. The maps are numbered F = 1,...,34. Important char- acteristics of the maps are listed in Table 2. F = 1,...,30 are computer-generated and F = 31,...34 are actual stem maps. The patterns used in the generation of F = 1,...,30 are: 1. random (F = 1,...,5). stem center = (X,Y). XmU(0,173') and YmU(0,346') where U denotes the uniform probability density function. ii. regular (Figure 5a.) a. square (F = 6,. .,10) b. rectangular (F = 11,. .,15) c. equilateral—triangular (F = 16,...,20) iii. clustered ( Figures 5b and 5c.) a. square (F = 21, ..,25) b. equilateral-triangular (F = 26,...,30) Clusters consist of stems equally spaced around the circum— ference of a circle of radius 10 feet with a stem at the cen- ter. Cluster centers are then located either on a square or equilateral-triangular pattern. The number of stems per cluster is constant for a given stem map since only a cer- tain number of clusters could fit within a 173 foot by 346 foot map. The number of stems per cluster varies from 7 to 12 depending on the desired density of the map as discussed below. Species and diameter generation are accomplished using probability density functions empirically determined from an eight percent systematic sample by 0 . 2-acre circular—plots located l6 0.0 0.0 0.5 0.0 0.00 00 50H 05 00m 00m 00 0.0 0.0 0.5 0.0 0.05 00 00a HOH 000 05m 00 0.0 0.0 0.5 0.0 0.H0 NN 00H 00 N00 00m 00 0.0 0.0 0.5 0.0 0.00 00 0NH N0 000 000 00 H.0 0.0 0.5 0.0 0.00 00 0HH 00 000 000 mm 0.0 0.0 0.5 5.0 5.H0 0H 00 00 00m 000 am 0.5 0.0 H.5 0.0 H.00 5N 00H NHH H00 00H 00 0.0 0.0 0.5 0.0 0.50 00 00a 00 N00 000 0H H.0 0.0 0.5 0.0 H.00 00 NNH 00 500 000 0H 0.0 0.0 0.5 0.0 0.00 00 NHH 00 0H0 0mm 5H 0.0 H.0 0.5 0.0 0.00 5H 00 00 N00 00H 0H 5.0 0.0 H.5 0.0 H.05 00 00H 5HH 00¢ H00 0H 5.0 0.0 0.5 5.0 0.05 00 00a 0HH 050 000 0H 0.0 0.0 H.5 0.0 0.00 00 «NH N0 000 000 0H 0.0 0.0 0.5 0.0 0.00 00 0HH 55 000 000 NH 0.0 0.0 0.5 0.0 0.00 0H 50 05 000 00H Ha 5.0 0.0 0.5 0.0 0.00 00 00H 00H N00 000 0H 0.0 0.0 0.5 0.0 0.00 00 00H HHH 000 000 0 0.0 0.0 H.5 0.0 0.00 00 HHH 00H 000 000 0 0.0 0.0 0.5 0.0 0.50 00 HOH 00 000 0am 5 0.0 0.0 0.5 0.0 0.00 am 00 55 000 00H 0 5.0 0.0 H.5 0.0 0.05 00 00H HHH 00¢ H00 0 H.0 0.0 0.5 0.0 0.00 00 00a 00H 000 500 q H.0 0.0 H.5 0.0 0.00 mm 00H H0 000 000 0 0.0 0.0 H.5 0.0 0.00 50 HHH N0 H00 0am N 0.0 0.0 0.5 0.0 0.H0 0m 00 05 500 00H a Amuom saw :00 000 Ammaocav \um.amv spamcme suamame Amuom mafia xmo xmo £20 mohm mafia xmo xuflmcmw awe Hmm \mamumv awe souwm ouwsz 0mm \M o0mum>¢ Hammm soufim wuflsz xmo 0mm \Hmaoum zuflmaom Eoum .mmmfi Eoum osu mo mowumflhouomhmso .N manwe 17 .0maHHEoo moHoomm HH< \N .Ammuom 0.0V ummm 00m 00 “wow 00H mum 500.....H0v mama Emum Hmsuom 0am Ammpom 50.Hv umom 000 00 Doom 05H mum 500.....H0 mama wouwnocmwnHmDDQEoo \w 0.0 u: I: 0.0 0.HHH 000 u: I: 05H 0mm 00 0.0 0.0 0.5 0.5 0.00 0HH 55 00 00H 000 00 0.0 H.0 H.5 0.0 0.00 50 00H 00H H00 000 N0 0.0H 0.0 0.0 0.5 0.50 H0 00H H 05H 00m H0 5.0 0.0 0.5 0.0 0.05 00 00H 5HH N00 000 00 0.0 0.0 0.5 5.0 0.H5 0m 00H HHH 000 00m 00 0.0 0.0 0.5 0.0 0.00 00 NNH N0 000 000 00 0.0 0.0 0.5 0.0 0.00 50 HOH 00 000 000 50 Ammom 000 000 000 AmmsocH \.um.amv suflmcme suamcme Amuom xmo xwo xmo £00 mohw ocHa xmo kunaow ama Mom \maoumv awe gouHm mUHSB 0mm \Mm0MHm>< Hmwmm souHm wuH£3 xmo 0mm \MmEoum kuHmCmm Emum woDGHucoo .N mHHmH o 0 ‘ . .i—. ' O Cir-'1 o 18 O I O 1""; ' 1 ix " .-~’-. . 0 x Square Rectangular Equilateral Figure 5a. Figure 5b. Figure 5c. Figure 5. Patterns Regular patterns. 0 O O O c 0 l o 0 O O O o O O O ' . 0 c o ’ 0 o o 0 0 o 0 " o , o o , ‘ 0 o 0 o ' ' . o ' ' O O c O 0 ° ' o o ' o o ' ' Equilateral clustered. for computer-generated stem maps. 19 in Stand 1. The method of generating a specific stem map is as follows: choose a pattern. choose a density. generate (X,Y). DWNH for each (X,Y): a. generate a species. b. generate a diameter. F = 31,...,34 are from field maps using a Suunto compass and a 100 foot metal tape. F = 31 and F = 32 were made in Stand 1 and F = 33 and F = 34 were made in Stand 2. All values in Table 2 are population values for the 34 stem maps. Density in Stand 1, averaged over twenty—five O 2-acre circular—plots, is 242.2 stems per acre. This average density is denoted by DN. Densities in the stem maps cycle from '1ow' (0.8DN) through ‘medium-low' (0.9DN) through 'average' (DN) through 'medium-high' (1.1DN) through 'high' (1.2DN) as the F cycle through 5j+1, 5j+2, 5j+3, 5j+4, 5j+5 for j = 0,1,....,5. Densities, diameters, and species proportions for F = 31, ..,34 are exactly as mapped. For F = 1,...,30 the area is 173 by 346 feet or 1.37 acres. For F = 31,. .,34 the area is 132 by 264 feet or 0.8 acres. Patterns for F = 1,...,30 are as previously discussed. Patterns for F = 31 and F = 33 were found to be clustered and patterns for F = 32 and F = 34 were found to be random by use of Pielou's index of non-randomness (1959). C. Methods of analysis 1. Theoretical basis. Narrow—strip sampling is a cluster sampling technique (Cochran, 1963) and a PPS (Probability Proportional to Size) technique. Chapter IV includes the theoretical frame— work of narrow-strip sampling upon which Monte-Carlo studies of Chapters V and VI and field applications of Chapter VII are based. Narrow—strip estimators of density, diameter, basal area, species proportions, and stand table entries are defined. Since the density estimator for narrow—strip samp- ling establishes the narrow—strip technique as a viable and practical sampling method, this estimator is the most impor- tant theoretical concept of this paper. The effect upon the density estimator of narrow—strip width, stem radius, h nearest neighbor and probability density functions of jt distances is contained in the derivation of the expected distance between two successive stems along a narrow-strip. General considerations about narrow-strip sample size, the ability of narrow strips to sample variation continuously, effects of wrongly including or excluding trees from the sample, and a mathematical definition of 'narrowness' are also given in Chapter IV. 2. Comparison with distance sampling. Monte—Carlo studies are discussed in detail in Chapters V and VI. Such studies are designed to model particular situations. A general reference dealing with the components and design of such studies is Schmidt and Taylor (1970). Some papers dealing with simulation studies in forestry are Mawson (1968), Newnham (1966, 1968), Newnham and Maloley (1970), O'Regan and Palley (1965), and Payandeh (1970a). 21 A paper by Mohn and Stavem (1974) disucsses randomly located, non-intersecting discs in the x—y plane. An article on simulation of distance sampling is Diggle, Besag, and Cleaves (1976). Basic to the study of all systems containing stochastic components is the generation of random numbers. Generators are available which generate psuedo—random numbers. The generator chosen for this paper is a modified version of IBM's RANDU and it was found to be sufficiently random over the range of values generated and over subsets of that range according to the purposes for which the numbers were used. No degenerate tendencies were exhibited and the cycle length was at least ten times that of the number of psuedo-random numbers generated. In Chapter V distance sampling estimators of density are compared with the narrow—strip density estimator. Three supposedly robust estimators of density (Diggle, 1975; Lewis, 1976; Cox, 1976) are chosen to compare with the narrow-strip density estimator. Dependence of the accuracy of density estimation on spatial pattern prompted selection of all robust estimators. Thelfluxuaestimators chosen appeared to be successful. 3. Comparison with plot and point sampling. Use of 'small' stem maps was necessitated by the enormous number of calculations required for each of the 100 Monte-Carlo realizations run for each sampling method over every stem map. Since narrow-strip, plot, and point sampling were being studied as statistical methods, sizes of 22 stem maps were large enough so that results could be extended to larger stands. Similarly, the rectangular shape of stem maps offers no real restriction to generaliztion of results. Systematic location of sampling units was also employed to study the three sampling methods. Behaviour of the three methods was analyzed with respect to estimation of the following characteristics: density, average diameter (total and by species), basal area, and species proportions. Be- haviour was analyzed for random and systematic location of sampling units over all 34 stem maps. Methods are compared on the basis of absolute value of bias (accuracy), sign of the bias, and the coefficient of variation of estimates averaged over 100 trials (precision). The range of densities and patterns, combined with empirical distribution functions for species and diameter—given-species, allowed for a comprehensive study of how well the three sampling methods estimated the nine characteristics. 4. Field applications. In Chapter VII the design of a narrow—strip sample is discussed by means of two field applications. One application is in Stand 1 and the other is in Stand 2. The design includes specification of narrow—strip width to be used in the cruise, choice of sampling intensity, location of narrow- strips, equipment, methods, and cost-efficiency. Results of the narrow—strip technique are compared with results of circular-plot and point samples taken in both stands. CHAPTER IV THEORETICAL BASIS OF NARROW-STRIP SAMPLING This chapter discusses narrow—strip sampling theoreti— cally to explain how the method works and to provide a basis for the Monte-Carlo studies in Chapters V and VI. Accuracy and precision of the estimators will be considered in Chapters V and VI because they provide the basis upon which the judgements about the efficiency of narrow—strip sampling will be made. Section A is the derivation of E(D) = the expected distance between two successive stems along a narrow—strip. Derivation of E(D) results in a relationship which is used to define the narrow-strip density estimator, e(DN). Narrow- strip density estimation is accomplished in three parts: (1) derivation of p(t) = the probability that a stem of radius r which is a distance t from its neighbor (a neigh- boring stem along the narrow-strip) is sampled by a narrow— strip of width w given that its neighbor has already been sampled; (2) derivation of E(D) resulting in an infinite series whose terms involve integrals of p(t) and their products; (3) definition of e(DN). Because of the apparently highly robust nature of e(DN), the expression for E(D) eliminates the necessity of either field sampling or Monte— 23 24 Carlo sampling stem maps to find numerical values of e(DN) corresponding to different values of w, r, DN and E(D). Numerical integration programs (see IBM, 1974) can be used to generate tables needed to look up e(DN) once the parameters are specified and an estimate of E(D) is made. The Appendix includes a description of a procedure used to obtain numerical values which define e(DN) and a listing of the Fortran IV program NRSTRP along with sample input and output for that program. NRSTRP generates the table of values needed to define e(DN) numerically for a given situation. In Section B estimators used in Chapter VI are discussed. Narrow-strip sampling is more of a PPS technique than plot sampling, but less of a PPS technique than point sampling; however, simple counts, instead of weighted counts, of stems are used to define estimators for species proportions. Later on (Chapter VIII) use of weighted counts is suggested for this purpose. Diameter and density estimates are used to estimate basal area. The estimated stand table entries, like the basal area estimate, depend on diameter and density estimates but the entries also depend on the accuracy with which species proportions, as functions of species and diameter class, can be estimated. Section C includes a discussion of narrow—strip sample size. Narrow-strip sampling nearest neighbor distances is considered as a PPS technique. This section also includes remarks about the way in which narrow-strips sample variation continuously as they are run through a stand and about the 25 effect that wrongly including or excluding stems in a narrow- strip sample has on estimates. A mathematical definition of 'narrowness' is also given. A. Estimation of density. 1. Derivation of theAprobability that a stem is narrow-strip sampled given that a neighboring stem has been narrow-strip sampled. Let CO be a stem of radius r0 and center x0. Let Cj be a jth nearest neighbor to C where jgs-l and S = the number of O stems in a population. Cj has radius rj and center x,. Let .J tj = the distance between x0 and Xj so that tj is the random variable denoting the jth nearest neighbor distance in the population. Define the jth nearest neighbor circle to be that circle centered at xO of radius tj (Figure 6). Select one edge, e, of the narrow-strip and fix this edge for the following derivation. Consider a line, e', parallel to e and a distance rO from e. For notational purposes let 'NS' denote 'narrow-strip'. Then e' lies outside the narrow—strip. CorlNS # ¢ if, and only if, x0 w+2rO with centerline CL. Let w = width of NS and CL be lies within a strip of width located randomly. Let Q = the distance from xO to e' measured perpendicular to CL. Q is a random variable. Figures 6a and 6b show how sampling of CO and Cj is done andluan is measured. Cj is sampled only if it touches the NS and this implies that xj falls within a strip of width w+2rj centered on CL. Let D = the random variable of the distance between jth nearest neighbor circle Figure 6a. Both stems sampled. Figure 6b. One stem sampled. Figure 6, Given one stem is sampled, the other stem is or is not sampled. Narrow— strip lies between lines e and e' touches the narrow- strip only if x0 falgs in the strip between e' and e" 27 successive stems along a narrow-strip. For the time being, the direction of travel along the narrow—strip is not spec— ified so that D is measuredilleither direction. Since CL is randomly located, the stem pattern does not affect p(tj) = Pr(erWNS # ¢ICOFINS # ¢). Also, the random location of CL implies that Q%U(0,w+2R), independently of stem pattern, where Q = the random variable of the distance between x0 and e' and R = the random variable of stem radius in the population. Refer to Figure 7: p(tj> = 2<£fi+fiE>/<2wtj> 2(0 1+02)tj/(2n tj) = %[Sin_1((q-r0+rj)/tj) + Sin—1( (w+2rO-(q+ro) + _ $9731] . -1 . — ;[S1n ((q-r0+rj)/tj) + S1n ((w rO+rj-q)/tj)] See Figure 8. Assuming r0 = rj = E(R) and denoting E(R) by r: p(tj> = $[Sin'l + Sin‘1<(w+2r—q>/tj>1 The assumption on the radii can be justifed if a uniform diameter distribution is assumed and this assumption simplifies later integrations and so it is made here. Further studies could investigate the effect which non—uniform diameter distri- butions might have on density estimation. Assuming q = E(Q) = W/ 2+1”: =I|N=1|H [Sin-1((w/2+r)/tj) + Sin—1((w+2r-w/2-r)/tj)] sn{l((w+2r)/(2tj)) p(tj) - Finally: if tjil/2(w+2r) p(tj) (l) m": Shf1((w+2r)/(2tj)) otherwise 28 w/2 3' Figure 7. th nearest neigbhor circle (x. can fall anywhere on its circumference) Derivation of the probability that a stem is narrow-strip sampled given that a neighboring stem has been narrow-strip sampled. x = center of stem C of radius r (k = 0,j) and t. = Eistance betw en x and x.. w = w dth of narrow—strip an e isJ fixed edge of that strip. e' = line parallel to and a distance r away from e. Q (= q in the Figure = distance from xO to e'. 29 w/2+r E' CL __ 1 w C' C X. J m 0, I as soon as t. Y>W/2+r, Cj misses the NS -/ — V Figure 8. Assume the population stem radii to be constant and the center of the already sampled stem falls on the narrow-strip centerline. 30 2. Derivation of the expected distance between two adjacent stems along a narrow-strip, Refer to Figure 9. The situation described in Section A1 now exists. The following derivation assumes a random pattern of stems. For D = tj to occur (i e., the narrow- strip distance between two successive stems is in fact the jth nearest neighbor distance in the population), it must be true that erlNS # ¢. D = tj does not occur just because erlNS ¢¢. It must also be true that Ckn NS =¢>for k = 1,. .,j-1 such that no Ck fall between (this is where 'narrowness' is essential) CO and Cj' These j-l events still allow Ckn NS % ¢ to the left of C (since Cj falls to the right of CO in Figure 0 9). These j-l events occur independently of one another and independently of ijlNS #q)with probabilities: 1 - (1/2)p(tk) for k = 1, ..,j—l This implies: j-l Pr(D=t ) = E[(1/2)p(t ) H (1-(1/2)p(tk))] J J k=1 j-l = (1/2)E(p(t )) H [l-(1/2)E(p(tk))] (2) J k=l The (l/2)p(tj) is used insteadcifp(tj) since now the travel is in a fixed direction (it does not matter which direction is chosen) along the narrow—strip so that each inter-tree distance is counted only once. Let Ij be the indicator random variable for the event D = tj: 1 if D = t. I. =“ J J 0 if D ¢ tj Finally: 31 it is assumed that x0 is on the CL P m U xk can fall anywhere on ‘ this arc given that x. falls on Xk X . 1, Vans arc 0 __ a 5, _ /// \\\\\xk cannot fall on this arc Figure 9. Expected distance between two adjacent stems along anarrow—strip. x. falls on either one (right or left) othhe two small arcs. Given this event occurs, the x must fall on the larger arcs like the ohes indicated for k = 1,...,j-1. 32 and: E(D) = ZEEk>) = 1 (letting E> = Ek> k = 1 for notational purposes) 00 [OXKi(t)dt E.(X) 1 . . 2 Ki(t) 2(nm)lt21'1e'“mt /(i-1)! (i = 1,2,...) The probability density functions, Ki’ are for the ti in a random population with density m (Thompson, 1956). 3. Definition of the narrow-strip density estimator. Equation (3) gives E(d) in terms of w, E(R), and m for a random pattern. For English units m = DN/4356O stems per square foot and for Metric units m = DN/lOOO stems per square meter. In practice w is known while E(D) and E(R) are estimated by e(D) and e(R) from a narrow-strip sample. What is then required is an estimate, e(DN), of density. A brief description of how to obtain e(DN) from a table of values generated by numerical integration will now be given. A full description is contained in the Appendix. Suppose that in an area of interest it is known: 33 7 inches :E(R) i 10 inches 200 i DN 1 280 Suppose also that it has been decided to sample with strips of width w = 3 feet (see Chapter VII for a determination of w). By numerically integrating (3) a table of the following form may be generated (refer to Figure 10). The dbh range from 5.0 to 12.0 inches and the densities range from 180 stems per acre to 300 stems per acre. These ranges have been extended from the previously mentioned 7.0 to 10.0 inches and 200 stems per acre to 280 stems per acre to faciliatate interpolation. To generate the table in Figure 10, w is fixed at 3 feet while the dbh varies from 5.0 to 12.0 inches in steps of 1.0 inches. For each dbh the densities are allowed to vary from 180 stems per acre to 300 stems per acre by steps of 20. Each of the 56 tabular values is the result of one numerical intergration. Different forest situations will require different ranges and increments for the dbh and densities as well as adifferent choice of w. The integra— tions can be formed for any situation. After a narrow-strip sample of width w = 3 feet is com- pleted, we have estimates of the average dbh and E(D) of a a forest stand. All that is required to find the narrow- strip estimate of density, e(DN), is to search down the column corresponding to the dbh estimate until the estimate of E(D) is reached and then proceed to the left to read e(DN). Some interpolation will usually be required (see the Appendix). Chapter V uses this same method of interpolation DN Figure 10. 34 w = 3 feet dbh = 5.0 6.0 7.0 8.0 9.0 10.0 11.0 180 2:; 200 220 a 240*****x*********xxE(D) 260 280 300 Form for a density estimation table using estimates from a narrow-strip sample. w = width of narrow strip (feet); dbh = diameter breast high (inches); DN = density (stems/acre); E(D) = narrow-strip nearest neighbor distance expected for the given w, dbh, and DN (feet). 12. 0 35 applied to the 34 stem maps. Monte-Carlo values of E(D) are generated and differ significantly from the numerically integrated values in the Appendix. The reason for these differences is that the distributions of the tj are truncated by the sizes of the stem maps. B. Estimation of other characteristics 1. Estimation of species proportions. Simple (unweighted) counts are used to estimate species proportions in this paper. Weighted counts will be mentioned in Chapter VIII. If: Si = number of stems in a stand of species 1 (i = 1,...,s) s S = 2 Si = number of stems in stand 1 = 1 si = number of stemscflfspecies i which are narrow-strip sampled s 3* = 231 i = 1 then the narrow-strip estimator of the proportion of the ith species in a stand is: e(Si/S) = Si/Slk 2. Estimation of diameter. If: . .th . . R. . = rad1us of the j stem of spec1es 1 1,] (j = 1,...,si) 36 ri j = radius of the jth stem of species 1 which is narrow-strip sampled (j = l""’Si) then: s E(2Ri) = % ZlRi j = average diameter of 1th species 1 j = l 2 3 S1 E(2R) = g') X Ri j = average diameter of all stems_ 1.: lj = 1 and the narrow-strip estimators of these quantities are: _ g i e(E(2Ri)) — s. ri,j 1 . J==1 2 3 Si 8(E(2R)) = S71: 2 XrLj 1.: 1 j = 1 3. Estimation of basal area. If: n Si 2 BA. = (— E R. .) DN = basal area in a stand 1 S 1,j J = l for species 1 n S Si 2 BA = (§ 2 E Ri j) DN = basal area of stand i.= lj = 1 S. _ 1 1 2 e(BAi) — (8* Z ri J.)e(DN) J = 1 _ n S Si 2 e(BA) — E. Z Z ri .)e(DN) 1==1.j = l 4. Estimation of stand table entries. If: 37 Si j = number of stems in a stand of species 1 ) which are also in diameter class j = 1,...,vi si j = number of stems of species 1 in class j S which are narrow-strip sampled then the (i,j)th entry in the stand table is: (Si,j/S)DN and the narrow-strip estimator of this quantity is: e((Si,j/S)DN) = (si j/s*)e(DN) C. General considerations. 1. Narrow-strip sample size. Suppose that F0 is a forest stand of area A. Let C be a stem of radius r such that C is located randomly in F0. Let NS denote a narrow-strip of width w and length 1 such that the centerline is located randomly over F0. Let PT denote a circle of radius (2r)(PRF) located randomly in F0 where PRF = some plot radius factor used in point sampling. Finally PL denotes a circular—plot of radius Z located randomly in F0. See Figure 11 from which the derivations of the following formulae follow: + Pr(cn NS 90>) = (2(r/12A+ w)1 = (w :/6)1 2 Pr(CrNPT #¢) = "(ZriRF) 2 Pr(Cn PL #1) = 1(2 Z r/12) It is instructive to examine ratios of these three probabilities because this gives a comparison of how much more (or how much less) likely C is to be sampled according to one (or to another) of the three sampling techniques. To study this behavior, r 38 (2r)(PRF) Figure 11. Geometry of stem sampled by the three techniques. F = forest stand of area A; C = stem of radius r (inches); CL = centerline of narrow—strip of width w (feet) and length 1 (feet); PRF = plot radius factor of point sampling circle PR; PL = circular-plot of radius Z (feet). ”J5“ 39 is allowed to vary while w, 1, Z, and PRF are given the values in Table 3. These values are representative of real situations. To be able to compare the techniques we fix wl = "22:: 0.2 acre. The three ratios of interest are: _ Pr(CerT #¢) 1‘1“) ‘ Pr(Cn PL #4.) _ Pr(Cfl- PT 70>) 1‘2“) ‘ Pr(Cn NS #4)) _ Pr(Cn NS #e) 1‘3“) "' Pr(Cn PL 76¢) These ratios are graphed in Figure 12. The k3 curves in Figure 12 illustrate the phenomenon that plot shape does make a difference when sampling stems as circles as opposed to sampling stems as points. In each situation the narrow-strip has a higher probability of sampling stems of a given radius than does the circular- 2 plot even though wl = Hz This occurs partly because: 2/w = 21/wl = perimeter-to-area ratio for the narrow-strip 2/Z = ZwZ/(nZZ) = perimeter—to-area ratio for the circular-plot so that: Z/w = (2/w)/(2/Z) = relative amount of perimeter in a narrow-strip as compared to a circular-plot In the three situations given in Table 3 we have, respectively, Z/w = 17.55, 8.78, and 6.58. Over the seven random stem maps (F = l,...,5,32,34): 40 Table 3. Example of stem being sampled by the three techniques. DN E(tl) w 1 Z BAF PRF E(R) range of r 250 6.60 3 2904 52.66 10 2.750 3.50 2-10 75 12.05 6 1452 52.66 30 1.588 14.00 10-20 35 17.64 8 1089 52.66 60 1.123 25.00 20—30 DN = density (stems/acre); E(tl) = expected value of lSt nearest neighbor distance in random pattern with density = DN; w = width of narrow-strip; 1 = length of narrow- strop; Z = circular-plot radius; BAF = basal area factor; PRF = plot radius factor; E(R) = average stem radius in population; r = stem radius (inches). ratios of probabilities l l t 41 Note: random stem pattern ‘ l ratios are for a \ 4 6 8 Figure 12. l \ l l ‘ \ L 10 12 14 16 18 20 22 24 26 radius (inches) Comparisons of stem being sampled by three methods. 28 30 42 3* = (l+t)(a/A)S where a/A = sampling intensity a = area covered by narrow-strips A = area of F (1.37 or 0.80 acres) S = number of stems in F l+t proportion of stems narrow-strip sampled in 100 Monte-Carlo realizations as an excess of (a/A)S = number of stems that the narrow-strips should have sampled on a strictly areal basis For these seven stem maps E = .1218 and .0919:t:.l770. Over all 34 stem maps E = .1068 and .0584:t:.l770. This says that narrow-strips sampled, on the average, 10.68% more stems than would be implied purely by their area. Since the k2 curves fall below the line y = 1, the narrow—strip also has a higher probability of sampling a stem of a given radius than does the point sample. The k1 curves show the point to circular-plot sampling relationship and are included for the sake of completeness. 2. Narrow-strip sampling of nearest neighbor distances as a PPS method. In PPS sampling, the size of a population element is a measure of that element's importance in estimating some characteristic of a population. For example, in point sampling for basal area, if two stems, C1 and C2, are such that r2 = krl, then Pr(Czn PT #¢) = kzPr(Cln PT #¢). When narrow-strip sampling D to estimate E(D), the smaller distances 43 are the more likely distances to be sampled. Consider Figure 13 in which a first and second nearest neighbor circles are pictured along with a narrow-strip. In the sense that trees which are closer together exert more of an influence on each other than trees which are further away, tj is, in general, 'more important' than tk when jp(t2) so that k2X2.- i = 1,...,m} l ’ Figure 16a is an example of an 'A-situation' and Figure 16b is an example of a 'B-situation'. A random variable W1i is now defined: 49 50 Figure 16a. 'A-situation'. Figure 16b. 'B-situation'. Figure 16. Geometry of Cox's estimator. For both figures the random point is 0, the nearest stem to O is P and G is P's nearest neighbor. 51 wli = [2H+SinBi-(n+Bi)COSBi]_l where sin((1/2)Bi) = Yli/(lei) Finally: r alifrn:(1/4)N é =j. 62if1n<(1/4)N where 01::(w/2)(1.17-.68m/N)H/N 2: (w/Z)(.20+3.20m/N)H/N _n2 m2 H “ §Z1i+ {Yzi 2 _ 2 211 ‘ (1/“)X1iw1i 2. Diggle's estimator. Diggle (1975) claims his y* is 'moderately' robust. By robust he means that the mean standardized bias is 'small' in absolute value for a wide variety of patterns. Mean standardized bias is defined by: E((Y*-Y)/Y) where Y is the density. y* is defined by: . n 2 YX (TI/I1)§Xi A n 2 YY 07/ani where Xi = random point-to-plant distance n-< ll random plant to nearest neighbor distance i “ _ “ “ 1/2 Y - (YXYY) 1 = 435609'1 'Yl\ 3. Lewis' estimator. Lewis (1975) defines R' as follows: 52 t _ 2 R — 43560/(NR ) where R = —(1/4)R1+(3/2)R2 n R1 = (l/n)§Rli n R2 = (l/n)§R2i Rli = random point to nearest plant distance R2i = random point to second nearest plant distance B. Monte—Carlo narrow-strip density estimator. The interpolation procedure in the Appendix which defines e(DN) is now used to define e*(DN). (See also IVA3). There is one important difference in the two parameters, a and b, which define e*(DN) and e(DN). That difference derives from the fact that in regressing to obtain a and b for e(DN), the 'actual' (or 'numerically integrated') values for E(D) were used. In the case of e*(DN) the values for E(D) which are used to determine a and b are derived from 1500 Monte-Carlo realizations over 34, l 37-acre or 0.80—acre stem maps and so the resulting values for E(D) are substantially less than the numerically integrated values for E(D) used to define e(DN). In effect, the sizes of the stem maps truncate the distributions of the tj' Nonetheless the regressions which yield the values of a and b for e*(DN) exhibit very good fits to the E(D) values. Over populations with random, regular, and clustered patterns and 6.5 inches:E(2R):7.5 inches and 1903DN1290, narrow-strips of width w, where 2 feetgwi9 feet, were randomly 53 located. Fifteen hundred narrow-strips were used per stem map per width. For a given stem.map and a given width the 1500 narrow—strips yielded approximately 10,500 values of D and these values were summed and averaged to obtain one Monte-Carlo value for E(D). The table of E(D) values in the Appendix is for a width of 3 feet. E(D) values used to define e*(DN) were obtained for widths of 2, 3,...,9 feet. Values of a and b were obtained for each of these widths. The lowest r2 for any w was .9596 and the largest average absolute bias was .0637. Absolute bias is the absolute value of the mean standardized bias as defined by Diggle using e*(DN) in place of 7*. The absolute biases are then averaged over the 34 stem.maps for a given width. Eight pairs of a and b values (onepmfi1:for each w) were regressed to obtain over-all values for a and b. The fit for a was linear and the fit for b was quadratic: a = .004633665(w+E(2R))+.10961952 with r2 = .9999 b = -5.9988485((w+E(2R))-4.8391573)2+725.84898 with r2 = .9831 Finally: e*(DN) = be“aEn1+n2) 3. E1(p(t)) is defined on page 32: {random if K = E1(p(t)) pattern = non-random if K ¢ El(p(t)) 4. If pattern is non-random: {clustered if 32(e(t1)) % 0 pattern = . 2 regular 1f 3 (e(t1)) = 0 where sz(e(t1)) = narrow-strip sample variance of the estimate of E(tl) Pattern estimates based on narrow-strip sampling of the 34 stem maps were not always accurate and no theoretical basis (only the intuitive basis of 1-4) could be found; however, it seems that, if pattern is a population charac— teristic important enough to study in some situation, there very well could be some quick method of estimating pattern as a part of a narrow—strip cruise for other characteristics. Tree growth in an even-aged stand can be sampled from stems which are narrow-strip sampled and which are also first nearest neighbors. Nearest neighbors in such a 86 stand should exhibit more significant interaction in terms of competition over their life-span than stems which are further apart. Thinning causes changes in nearest neigh- bors and growth responses could also be studied with respect to this aspect. Considerations similar to those on growth studies also apply to yield studies in conjunctionvfiiflrnarrow- strip sampling. For both growth and yield the narrow- strip method's ability to sample variation continuously might give new insight into the variability of these characteristics. In these cases, narrow-strip sampling offers a simple way (ocular estimation of successive stems along a narrow-strip)in which to choose nearest neighbors for measurement. Narrow—strip sampling could also be combined with other methods of sampling. For example suppose a narrow- strip of length 1 is expected to encounter 100 stems. Also suppose that along a transect of length 1 some points are to be chosen for point sampling. At the tenth stem along a narrow-strip a point sample would be taken. If five points are to be chosen, the other four points would occur at every twentieth stem after the tenth one. Due to an effect called 'waves of density' (Zeide, 1972, 1975) a narrow-strip would determine sampling points which would allow point samplestx>occur in higher density regions thus reducing the variance of a point sampling estimate (Figure 24a). 87 r_narrow-strip nga \7 90th stem narrow-strip_+ th '#/JO stem log length "—density waves l KN l 1 50th stem <§< // th . 0 stem , 10th stem = 1St point sample center Figure 24a. Point sampling. Figure 24b. Fuel samp- ling Figure 24. Two possible applications of narrow—strip sampling. 88 Similar considerations apply to 3P sampling but in this case every kth stem (k = l,...,50) might be tested to see whether or not it fell into a 3P sample. Information on species, diameter, etc. need not be taken at all as a part of a narrow-strip sample in the 3P sampling and point samp- ling cases just mentioned, rather the narrow-strips would simply provide a sampling frame from which to sample stems using one of the other two methods. Stem maps made from aerial photographs (Payandeh, 1970) could be used to gather pre-sampling information for a nar- row strip cruise. The intersection of the major and minor diameters of a tree crown would be used as the coordinates of a stem center. A regression of crown area on the square of the dbh would be used to determine stem diameter. Forest fuel sampling and sampling for logging residue have been done using the line-intersect method of sampling (Bailey, 1969, 1970; DeVries, 1973a, 1973b; VanWagner, 1968); Warren and Olsen, 1964). Narrow—strip sampling could be used in these types of situations by counting a piece of debris in the sample only if the shaded circles (Figure 24b) of diameter equal to the mid-diameter of the piece are met by a narrow strip. The number of pieces sampled should be much lower than in line-intersect sampling so that more intensive measurements of each piece could be accomp— lished in what should prove to be a shorter period of time. Also, because circles in a plane parallel to the forest floor are being sampled, orientation of the piece centerline with respect to the forest floor should not be a problem. 89 In a paper by Eberhardt (1978) discussion is given of strip sampling in which the strips are not assumed to be narrow. Density estimates using the method mentioned in that paper might be studied by running narrow-strips through aiforest stand and comparing the density estimate given by Eberhardt, e'(DN), with the density estimate derived in this paper, e(DN). The derivation in Eberhardt's paper relies on the strip centerline being randomly located (see Figure 25) parallel to one edge of the area of interest. Assume a narrow-strip has the x-coordinate of its centerline located such that X%U(0,W). Let stems Ci of radius ri be located independently of one another over a forest stand for i = l,...,S. Suppose the forest stand is a W by L rectangle and the narrow-strip centerline is located parallel to the side of length L. Let k = the number of Ci sampled by this narrow-strip. Define: r 1 if Cir'NS # ¢ x =4 0 if C.(\NS = 9 q 1 Now: Pr(Cin NS #¢) = (w+2wri)L/(WL) = (w+2ri)/W where w = narrow-strip width and so: E(k) [0.Pr(X(Ci) =(D+l.Pr(X(Ci) = 1)] .J l—‘MCD l—‘Mm (w+2ri)/W 90 1 . ‘ t¢-"-narrow—strip I of width w . at .uarrow—strip L —-—c1rcle Ci centerline l H————forest stand. L Kr——w fl Figure 25. Another density estimator using narrow-strips. 90 4—"—narrow-strip of width w narrow—strip L ——-c1rcle Ci centerline ‘ (———fbrest stand L R w it Figure 25. Another density estimator using narrow-strips. 91 = S(w+2r)/W where E = E(r) = % Z r. This gives a density estimate via: S e(S) = e(k)W/(w+2e(r)) e'(DN) = e(S)/(WL) = e(k)/((w+2e(r))L) where e(S) = estimate stems in e(k) = estimate e(r) = estimate 1 of total number of the stand of E(k) of E(r) As a closing remark it must be emphasized that more research is needed to compare narrow-strip sampling of vary- ing intensities with currently used methods of sampling such as plot, point and 3P sampling. Weighted estimators, especially for dbh and species proportions, should be studied. The study of estimation by means of narrow-strips of impor- tant characteristics other than the ones studied in this paper is also needed to explore this promising sampling technique. Other characteristics which should be studied are volume growth, and biomass. It is hoped that the few possible areas of study listed here will generate enough interest in narrow-strip sampling so that this method may begin to become established as a basic sampling technique in forestry. CHAPTER IX SUMMARY AND CONCLUSIONS Narrow—strip sampling consists of sampling a forest stand by means of strips so narrow that the trees may be linearly ordered along the strips. Narrow—strip sampling is a form of cluster sampling in which the number of elements sampled is a random variable. Narrow—strip sampling is also a PPS technique. A theoretical basis for narrow-strip sampling is given. Monte-Carlo techniques are used to verify theoretical statements and to examine the biases and precisions of narrow—strip estimators. A range of real and computer- generatedforeststands are used in the study and they consist of combinations of random, regular, and clustered patterns with a realistic range of densities and diameter distributions which are generated from an empirical distri— bution function. Density anddiameterconsiderations follow from field data gathered in two stands located in the New Jersey Pine Barrens. Distance sampling is an unpractical method of sampling for density but hasbeen.studied because of its theoretical appeal. Three supposedly robust methods of density esti— mation using distance sampling are compared with a density estimator using narrow—strip sampling in a Monte—Carlo 92 is 93 study of 34 real and computer—generated stem maps. It is found that narrow—strip sampling gave highly robust estimates of density, outperforming the best of the distance estimators. Narrow—strip sampling, circular-plot sampling and point sampling are used to sample the 34 stem maps for basal area, species proportions, diameters, and density using both random and systematic location of the three types of sampling elements which define clusters of stems. Strip length, plot radius, and basal area factor are chosen such that all three methods have an equal expected sample size. Methods are compared on the bases of bias and precision. On these bases, narrow-strip sampling generally performs as well as the other two methods. When data gathered in the field (one oak—dominated 62.5-acre stand and one pine—dominated 43 O—acre stand) are analyzed, it is found that narrow—strip sampling per- forms very well. A high—intensity narrow—strip sample takes about one-half the time of a plot sample to give good estimates of the same characteristics which the plot sample estimates and, in about twice the time it takes to point sample only for basal area, the narrow—strip technique samples for basal area, density, diameters, species pro— portions, and stand table entries. A low—intensity narrow— strip sample is compared to a point sample in the pine— dominated stand. The low—intensity narrow-strip sample performs remarkably well since it takes half the time of 94 the point sample but also gives very good estimates of average diameter (all species combined) and density. It seems likely that many areas of application lie ahead for narrow—strip sampling. A few possible such areas are suggested in Chapter VIII. In the meantime many empirical studies will be required to further verify what this study has demonstrated: narrow-strip sampling is an attractive and practical alternative to circular—plot samp— ling, point sampling, and distance sampling for estimation of basal area, species proportions, diameters, stand table entries, and density in the types of situations included in this study. Theoretically there is no reason why narrow-strip sampling for these characteristics and other characteristics like growth, volume, and biomass cannot be applied to any forest situation. LITERATURE CITED LITERATURE CITED Bailey, G.R. 1969. An evaluation of the line-intersect method of assessing logging residue. Inform. Rep. VP- X-23, Can. Dep. Fish. Forest., Forest Prod. Lab., Van- couver, B.C. . 1970. A simplified method of sampling log- ging residue. For. Chron. 46: 288-94. Birth, E.E. 1977. Horizontal line sampling in upland hardwoods. J. For. 75: 590-91. Cochran, W.G. 1963. Sampling Techniques. 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APPENDIX APPENDIX A Generating a density equation Table A1 is generated by means of a Fortran IV program called NRSTRP. NRSTRP approximates the sum of the infinite series for E(D) over a specified range of diameters and densities and for a given width. Table A2 gives the results of least-squares exponential curve fits for each graph in Figure Al. Figures Al and A2 are graphs of Table A1 and A2. Table A3 gives input card layouts and input values for five test cases using NRSTRP. Table A4 gives the output of the test cases listed in Table A3. Table A5 is a listing of NRSTRP. Since estimated diameters and inter-tree distances encountered in a narrow—strip sample will not in general match those values given in Table A1, interpolation will usually be required to estimate density. One method of interpolation is now discussed. This method results in: e(DN) = beaD where a = a(d) b = b(d) d = diameter estimate D = inter-tree distance estimate e = 2.71828... = the base of the natural logarithms By inspecting Figures A2 it appears that: a(d) = rd+s 100 101 omoomm.mm ooommq.wm moommm.mm mmqmam.o¢ com wHNmNN.oq qoanma.aq Hmmqma.mq quoom.mq owm oammmm.mq qmmmmm.