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Sampling heights of second growth coastal douglas-fir in fixed area plots Riel, William G. 1994

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SAMPLING HEIGHTS OF SECOND GROWTH COASTAL DOUGLAS-FIR INFIXED AREA PLOTSbyWilliam G. RielB.SF. University of British Columbia 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OF MASTER OF FORESTRYinFACULTY OF GRADUATE STUDIESDepartment of Forestry (Forest Management)We accept this thesis as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1994© William Gavin Riel 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)_________________Department of T c) fl4,J46-/t/0TThe University of British ColumbiaVancouver, CanadaDate QABSTRACTKnowledge of tree heights is important for classifying sites, projecting growth and yieldand estimating stand volume. Tree height is expensive and time consuming to measure sosamples should be taken in the most efficient way possible. The impact of different sampledesigns and sizes on the fitting of height-diameter equations and subsequent prediction ofvolume is explored in this thesis. Several different height-diameter equation forms werecompared for estimating height in second growth Douglas-fir. After selecting the bestequation, a variety of simple sampling designs and sizes were compared using thisequation. It was found that a uniform design, which was based on sampling tree heightuniformly from 3 diameter classes, gave good results for height estimation. A “large”design, which concentrated 50% of its samples in the largest diameter class, gave the bestestimates for tree volume. In plots less than 50 years old, it was found that sampling morethan 16 tree heights produced diminishing benefits in height and volume estimates.IITABLE OF CONTENTSABSTRACT iiTABLE OF CONTENTS iiiLIST OF TABLES vLIST OF FIGURES viACKNOWLEDGEMENTS viii1. INTRODUCTION 12.0 LITERATURE REVIEW 32.1 Sampling for Height-diameter curves 32.2 Desirable Characteristics of a Height-Diameter Equation 62.2.1 Problems With Restricted Regressions 72.3 Comparing Regression Curves 82.4 Curve Fitting Techniques 92.5 Common Height Diameter Equation Forms 112.5.1 Bias due to logarithmic transformations 132.5.2 Features of Nonlinear Models 133.0 METHODS AND ANALYSIS 163.1 Data Preparation 163.2 Fitting Height Diameter Curves 183.3 Simulating Sampling Designs 183.4 Height Estimate Comparisons 223.5 Volume Comparisons 224.0 RESULTS 244.1 Height Diameter Curves 244.2 Sampling Designs 304.2.1 Height estimation 304.2.1.1 Mean deviation in height 304.2.1.2 Average maximum and minimum deviation inheight 344.2.1.3 Average standard deviation of height differences 424.2.1.4 Mean absolute deviation in height 474.2.2 Volume estimation 514.2.2.1 Mean deviation in volume 514.2.2.2 Average maximum and minimum deviation involume 554.2.2.3 Average standard deviation of volume differences 634.2.2.4 Mean absolute deviation in volume 684.3 Ranking of sampling designs 725.0 DISCUSSION 745.1 Model Selection 745.2 Sampling designs 765.2.1 Height estimation 765.2.2 Volume estimation 775.2.4 Application in the field 795.2.5 Measurement errors and costs 81ill6.0 CONCLUSIONS AND RECOMMENDATIONS .837.0 LITERATURE CITED 85APPENDIX 1- ANALYSIS OF VARIANCE FOR HEIGHT-DIAMETERMODELS 88APPENDIX 2- RESULTS OF SAMPLING 91ivLIST OF TABLESTable 1. Common height-diameter models 12Table 2. Height diameter models and authors 19Table 4. Height diameter models compared 25Table 5. Summarized rankings of sample designs 72Table 6. Overall rankings of sample designs 73Table 7. Model 1 Coefficients 88Table 8. Model 1 Analysis of Variance 88Table 9. Model 2 Coefficients 88Table 10. Model 2 Analysis of Variance 88Table 11. Model 3 Coefficients 89Table 12. Model 3 Analysis of Variance 89Table 13. Model 4 Coefficients 89Table 14. Model 4 Analysis of Variance 89Table 15. Model 5 Coefficients 89Table 16. Model 5 Analysis of Variance 90Table 17. Model 6 Coefficients 90Table 18. Model 6 Analysis of Variance 90Table 18. Average mean deviation in height 91Table 19. Average maximum deviation in height 92Table 20. Average minimum deviation in height 93Table 21. Average mean absolute deviation in height 93Table 22. Average standard deviation of height differences 94Table 23. Maximum deviation in height 95Table 24. Minimum deviation in height 96Table 25. Average mean deviation in volume 96Table 26. Average maximum deviation in volume 97Table 27. Average minimum deviation in volume 98Table 28. Average mean absolute deviation in volume 99Table 29. Average standard deviation of volume differences 99Table 30. Maximum deviation in volume 100Table 31. Minimum deviation in volume 101Table 32. Sample rankings for sample size 8 102Table 33. Sample rankings for sample size 12 103Table 34. Sample rankings for sample size 16 104Table 35. Sample rankings for sample size 20 105Table 36. Sample rankings for sample size 24 105Table 37. Sample rankings for sample size 28 106vLIST OF FIGURESFigure 1. Initial data set 17Figure 2. Final data set 17Figure 3. Six models compared 24Figure 4. Residual Plot for Model 1 26Figure 5. Residual plot for model 2 26Figure 6. Residual plot for model 3 27Figure 7. Residual plot for model 4 27Figure 8. Residual plot for model 5 28Figure 9. Residual plot for model 6 28Figure 10. Mean deviation in height for AGE1512 30Figure 11. Mean deviation in height for AGE1SI3 31Figure 12. Mean deviation in height for AGE2SI1 31Figure 13. Mean deviation in height for AGE2SI2 32Figure 14. Mean deviation in height for AGE2SI3 32Figure 15. Mean deviation in height for AGE3SI1 33Figure 16. Mean deviation in height for AGE3SI2 33Figure 17. Average maximum deviation in height for AGE1S12 35Figure 18. Average maximum deviation in height for AGE1SI3 35Figure 19. Average maximum deviation in height for AGE2SI1 36Figure 20. Average maximum deviation in height for AGE2SI2 36Figure 21. Average maximum deviation in height for AGE2SI3 37Figure 22. Average maximum deviation in height for AGE3SI1 37Figure 23. Average maximum deviation in height for AGE3SI2 38Figure 24. Average minimum deviation in height for AGE1S12 39Figure 25. Average minimum deviation in height for AGE1SI3 39Figure 26. Average minimum deviation in height for AGE2SI1 40Figure 27. Average minimum deviation in height for AGE2SI2 40Figure 28. Average minimum deviation in height for AGE2SI3 41Figure 29. Average minimum deviation in height for AGE3SI1 41Figure 30. Average minimum deviation in height for AGE3SI2 42Figure 31. Average standard deviation of height differences for AGE1SI2 43Figure 32. Average standard deviation of height differences for AGE1S13 43Figure 33. Average standard deviation of height differences for AGE2SI1 44Figure 34. Average standard deviation of height differences for AGE2SI2 44Figure 35. Average standard deviation of height differences for AGE2SI3 45Figure 36. Average standard deviation of height differences for AGE3SI1 45Figure 37. Average standard deviation of height differences for AGE3SI2 46Figure 38. Mean absolute deviation in height for AGE1SI2 47Figure 39. Mean absolute deviation in height for AGE1S13 48Figure 40. Mean absolute deviation in height for AGE2SI1 48Figure 41. Mean absolute deviation in height for AGE2SI2 49Figure 42. Mean absolute deviation in height for AGE2SI3 49Figure 43. Mean absolute deviation in height for AGE3SI1 50Figure 44. Mean absolute deviation in height for AGE3SI2 50vFigure 45. Mean deviation in volume for AGE1S12.51Figure 46. Mean deviation in volume for AGE1SI3 52Figure 47. Mean deviation in volume for AGE2SI1 52Figure 48. Mean deviation in volume for AGE2SI2 53Figure 49. Mean deviation in volume for AGE2SI3 53Figure 50. Mean deviation in volume for AGE3SI1 54Figure 51. Mean deviation in volume for AGE3SI2 54Figure 52. Average maximum deviation in volume for AGE1S12 55Figure 53. Average maximum deviation in volume for AGE1S13 56Figure 54. Average maximum deviation in volume for AGE2SI1 56Figure 55. Average maximum deviation in volume for AGE2SI2 57Figure 56. Average maximum deviation in volume for AGE2SI3 57Figure 57. Average maximum deviation in volume for AGE3SI1 58Figure 58. Average maximum deviation in volume for AGE3SI2 58Figure 59. Average minimum deviation in volume for AGE1SI2 59Figure 60. Average minimum deviation in volume for AGE1SI3 60Figure 61. Average minimum deviation in volume for AGE2SI1 60Figure 62. Average minimum deviation in volume for AGE2SI2 61Figure 63. Average minimum deviation in volume for AGE2SI3 61Figure 64. Average minimum deviation in volume for AGE3SI1 62Figure 65. Average minimum deviation in volume for AGE3SI2 62Figure 66. Average standard deviation of volume differences for AGE1SI2 63Figure 67. Average standard deviation of volume differences for AGE1SI3 64Figure 68. Average standard deviation of volume differences for AGE2SI1 64Figure 69. Average standard deviation of volume differences for AGE2SI2 65Figure 70. Average standard deviation of volume differences for AGE2SI3 65Figure 71. Average standard deviation of volume differences for AGE3SI1 66Figure 72. Average standard deviation of volume differences for AGE3SI2 66Figure 73. Mean absolute deviation in volume for AGE1SI2 68Figure 74. Mean absolute deviation in volume for AGE1SI3 68Figure 75. Mean absolute deviation in volume for AGE2SI1 69Figure 76. Mean absolute deviation in volume for AGE2SI2 69Figure 77. Mean absolute deviation in volume for AGE2SI3 70Figure 78. Mean absolute deviation in volume for AGE3SI1 70Figure 79. Mean absolute deviation in volume for AGE3SI2 71viiACKNOWLEDGEMENTSI would like to give special thanks to my supervisor, Dr. Peter Marshall for his advice andpatience during the development of this thesis. Also, to be thanked are the members of mycommittee, Drs Tony Kozak and Va! Lemay for their advice and kindness.A special thanks to Dr. Les Safranyik of the Canadian Forest Service, who’s suggestionsand advice were of great value to me, especially when I was away from U.B.C. I also wishto thank Dr. Terry Shore, whose friendship and patience during the last two years of mystudy were greatly appreciated.I also would like to acknowledge my friends and fellow grad students who have were asource of encouragement and support. Special thanks go to Cris Brack for the “Coffeeand Philosophy” sessions in hut 6. Thanks to Craig Roessler for always being encouraging.And of course, thanks to Regan Dickenson for always giving me something to laughabout.Lastly, I would like to acknowledge my family, who’s support has made a world ofdifference. Especially, I would like to thank Kathreen, who’s prayers and patience havecarried me through some incredibly difficult times.Dedicated to God’s glory...vm1. INTRODUCTIONThe measurement of tree heights is an important factor in forest management in BritishColumbia (B.C.) for many reasons. Selected tree heights are often used for siteclassification and growth and yield projections. Despite being important, the measurementof tree heights can be an expensive and time consuming process (although, the use of alaser device could greatly reduce time and cost of obtaining measurements). This isbecause tree height, unlike tree diameter at breast height1 (dbh), must be indirectlymeasured or estimated on trees more than several meters tall (Ker and Smith, 1957). As aresult, it is common to measure all tree diameters in an area (such as a sample plot), butonly a certain proportion of the tree heights. For example, stand volume is sometimescalculated using measurements of all diameters in a sample and estimates of height basedon a sub-sample of those trees. The objective of this sub-sample is to obtain adequateinformation to represent the relationship between tree height and dbh at a reasonable cost(Ker and Smith, 1957). Height is then related to diameter using some form of amathematical equation (a height-diameter curve) which allows prediction of height forevery dbh within the sample. If stand volume is the objective of the sample, volume pertree may be obtained from a volume table or (more commonly today) a mathematicalequation which uses tree diameter and the measured or estimated height given by theheight-diameter curve (Curtis, 1967).1n B.C., breast height is defmed as 1.3 m above ground level, taken from the high side on sloped oruneven terrain.1Introduction 2The B.C. Ministry of Forests has recommended that for permanent sample plots (psp’s),each plot should have the dbh measured on every tree within the plot boundaries, but haveheight measurements made only on a subsample of 15 trees plus top height trees (ForestProductivity Councils of B.C. 1990). Given these guidelines, it is important to know howthe size and distribution of height samples affects both height and volume estimation.Some of the effects and consequences that result from employing different height samplesizes and designs in second growth Douglas-fir (Pseudotsuga Mensiesii (Mirb.) Franco)on fixed area plots were explored in this thesis. More specifically, the effects of alteringthe number of heights sampled in a plot on height diameter curve construction and volumeestimation were examined. Knowing how sensitive volume estimates are to the number ofheights measured can help avoid under- or over-sampling tree heights in cases wherevolume estimation is an important goal of the sample. Also, different sampling designswere tested to determine the effects that sub-sample allocation has on volume estimationand height-diameter curve construction.This thesis has been organized in the following format. First, a literature review of height-diameter equations and tree height sampling is presented. Next the methods used in fittingheight-diameter equations are described along with the process of simulating the varioussampling designs. Results of the equation fitting and sampling are presented in the nextsection. This is followed by a discussion of results and implications. Finally conclusionsand recommendations are presented in the fmal chapter.2.0 LITERATURE REVIEW2.1 Sampling for Height-diameter curvesWhile much attention has been paid to the development of mathematical height-diametermodels, much less attention has been given to determining the number of sample heightsrequired and which trees are most suitable for sampling. The number of height samplesrequired to provide an estimate of stand volume for a given level of precision will begoverned by several factors including the number of species present, the variation in treeheights, and the degree of correlation of tree height with dbh (Ker and Smith 1957). Thevariation in tree heights is not a concern when that variation occurs among, rather thanwithin, dbh classes. If there is little variation within dbh classes, height-diameter curvescan be derived which give very good results. However, some variation often does occurwithin dbh classes. For a single tree species, this is generally due to the position attainedby the tree within the crown canopy (Ker and Smith 1957).It has been noted that it is not necessary (or even desirable) to sample all trees in a standrandomly when taking a height sub-sample, but to limit the sample to specific diameterclasses, as long as the selection of samples within those diameter classes is not biased(Bruce and Schumacher 1950; Ker and Smith 1957). For example, Trorey (1932)recommended sampling from two dbh classes, one near the maximum diameter presentand the other at one-half that diameter. Modifying Trorey’s method, Alexander (1945)used sample heights ranging throughout all diameter classes on permanent sample plots.Ker and Smith (1955) found that good results could be obtained by sampling two large32.0 Literature Review 4diameter trees and two trees approximately 30 percent of the dbh of those trees, whenapplying the parabolic height diameter equation.In a study on height estimation for red pine (Pinus resinosa Ait.) and white spruce (Piceaglauca (Moench) Voss), Stiell (1965) found somewhat different results than Ker andSmith (1955) using the same parabolic equation. In this study, many more sample heightswere required to obtain acceptable results (two heights per diameter class across thediameter range). Even so, this was a small proportion of the total population.The size and distribution of the sample can be important when regression techniques areused to estimate the height-diameter relationship. Commonly, regression estimation isinefficient because samples are concentrated in one area within the range of theindependent variable(s) while other areas are under-represented. This generally occursbecause sufficient effort has not been taken to plan the sample in accordance with thesample objectives. Those objectives should be clearly stated before sampling begins(Demaerschallc and Kozak 1974). When developing a predictive regression equation thereare two basic objectives which the equation must satisfy (Demaerschallc and Kozak 1974):1. The regression must be useful for a given range of the independent variable(s).2. The regression line should satisfy a minimum precision requirement2at thelowest possible cost.2Demaerschalk and Kozak (1974) defined the minimum precision requirement as “the required maximumconfidence interval of the mean of Yi for different Xi values”.2.0 Literature Review 5These two objectives will drive the selection of both sample size and sample distribution.The sample range of the independent variable should equal or exceed the range specifiedin objective one. In general, a uniform distribution should be used if there is any doubtabout the form of the relationship. If the relationship is known to be a simple linear one,sampling only at the upper and lower extremes of the independent variable will provide themost efficient design; however, this design does not permit a test for lack of fit. Therequired sample size will then depend entirely on the minimum precision (objective 2) andthe sample distribution chosen (Demaerschalk and Kozak 1974).The theoretical study of optimizing the quantity and distribution of samples falls into thefield of optimal design, of which much work has been published. Unfortunately, much ofthe work is theoretical, and little has been written to facilitate the application of the theoryto practical problems (Penner 1989; Ziegel 1984). While many criteria can be used todefine optimality, the most common for linear regression involve minimizing thegeneralized variance of the parameter estimates (called D-optimality) or minimizing themaximum variance of the predicted response over the design region (called G-optimality)(Atkinson 1982).In traditional optimal design, it is assumed that the cost of sampling and the precisionrequirements are constant over the range of the independent variable(s), both of whichmay be untrue in a practical application (Penner 1989). For example, in biomass studies,the cost of sampling a small tree may be many times lower than the cost of sampling a2.0 Literature Review 6large tree. Precision requirements may also vary throughout the range of interest as well.It is common in forestry to require estimates to be within a percentage of the true mean(e.g., volume estimated to ± 10 %). To deal with these situations, Penner (1989)developed a procedure to weigh the variance function3of traditional optimal design bycost and precision. This resulted in a weighted optimum design which minimized the costof the sample for a given precision requirement. In her study she found that a uniformdesign, while not as cost effective as the weighted optimum design, gave excellent resultsand could be preferable to the weighted optimum design in some circumstances despitecosting more.2.2 Desirable Characteristics of a Height-Diameter EquationAccording to Curtis (1967), the function which is used to model the height-diameterrelationship should be reasonable, even when data are not adequate to fully define theshape of the curve. Curtis (1967) suggested that height diameter curves should bemoderately flexible and possess the following characteristics:1. a graph of the curve should show a slope that is positive, approaching zero asdiameter (D) becomes large;2. the y intercept of the curve should occur at breast height (1.3 m); and3. the curve should be easily fitted by linear regression methods.In optimal design, the variance function measures the the gain in knowledge provided by an observationtaken at a given x (Penner 1989).2.0 Literature Review 7Today, requirement 3 is unnecessary with advancements in nonlinear least squares solutionpackages. It may be necessary to use a sigmoidal curve to meet requirement 2 withoutdistorting the curve. If small trees are absent (or are not important), then this requirementmay not be necessary (Curtis 1967). In fact, if small trees are absent, it may beinappropriate to force the model through the origin. In any case, if requirement 2 isapplied, it is important that the potential problems of using a fixed-intercept regression arerecognized.2.2.1 Problems With Restricted RegressionsAs previously mentioned, it is very common to restrict height-diameter equations so thatthey pass through a fixed point on the y-axis (in this case, at 1.3 m height). It is lesscommon to test whether this condition is valid.Even when it is logical to do so, imposing restrictions on regression coefficients must beregarded as a very strong assumption and should be justified before accepting theconditioned model (Kozak 1973). More specifically, this condition should only be appliedif three conditions are met (Kozak 1973):1. there must be good reason to impose restrictions on the coefficients;2. the basic assumptions of the regression analysis must be met after the restrictionis imposed;3. the sampling should be organized in such a way that the restriction is justifiedfor the sample, not for the population only.2.0 Literature Review 8It has been recommend that a conditioned model, called the “hypothesis” model (Freese1964) should only be accepted over the unconditioned or “maximum” model after it hasbeen tested in at least one of the following ways (Kozak 1973):a) the hypothesis that the residual sum of squares for the hypothesis model isnot significantly different from the residual sum of squares for themaximum model is tested;b) the hypothesis model is tested for lack of fit; orc) the residuals are plotted over the independent variable or over the predictedy,s.2.3 Comparing Regression CurvesOften, when regression models are being fit to sample data, the question of comparingdifferent regression curves arises. It is not always immediately clear which model bestdescribes the relationship between the dependent and independent variables. In caseswhere the same dependent variable is used, the root mean square residual (or the standarderror of estimate) is usually adequate to compare regression curves based on the samesample (Fumival 1961). However, it is very common in forestry to apply models in whichthe dependent variable has been transformed. It is meaningless to compare the standarderror of estimate for these models with any which do not use the same dependent variable.For example, the standard error of estimate from the model log(H) = a + blog(D) +clog2(D) (where H represents total height in metres; D represents diameter at breastheight outside bark in cm; and a, b and c are regression coefficients) cannot be compared2.0 Literature Review 9with the standard error of estimate from the model H = a + bD + cD2. In order to dealwith this problem, Furnival (1961) developed an index of fit (“I”) based on relativelikelihoods which “has the advantage of reflecting both the size of the residuals andpossible departures from linearity, normality and homoscedasticity”. While this index wasoriginally conceived and developed for comparing different volume equations to be used inthe construction of volume tables, it is also suitable for comparing height-diameter modelsand has been used for that purpose in the literature (e.g., Curtis 1967).2.4 Curve Fitting TechniquesHeight diameter curves were once plotted in a freehand, graphical style but are nowgenerally fitted using mathematical techniques. While freehand curves were considered tobe accurate enough for use in local volume tables, the advantages of a least squaressolution has long been recognized (i.e., statistical comparisons, construction of confidenceintervals and repeatable results (Meyer 1936)).Today, the least squares technique is commonly employed when fitting height diametercurves to experimental data. However, several different procedures have been usedhistorically in British Columbia. Before electronic computers were readily available, suchcurves were often fit by hand (Ker and Smith 1955). Mathematical techniques other thanleast squares have also been employed. Trorey (1932) developed a simple equation thatgave a close representation of the relationship between total height and diameter at breastheight in the following form: H = a + bD + cD2,which he did not derive using leastsquares. Since the outside diameter (D) at 4.5 feet (breast height in imperial units, equals2.0 Literature Review 101.37 m) is equal to zero, a = 4.5. b represents the initial height growth rate (feet incrementin height per inch increment of diameter) and c, a negative value, represents the rate atwhich the initial rate decreases (in feet per inch). The constants b and c can be determinedwhen the heights at any two diameter classes are known (generally one of the classes wastaken from the middle of the diameter class range and one from the upper end of the range(Ker and Smith 1955)). Average values of height (H) and dbh (D) for each diameter classare determined and substituted into the original equation to form two equations with twounknowns (b and c). The equations are then simultaneously solved for the unknowns.While this was a convenient and simple method for expressing the relationship betweenheight and diameter, its accuracy was dependent upon the assumption that the relationshipis actually parabolic and that the averages selected are truly representative of thepopulation averages (Ker and Smith 1955). Because of these assumptions, Alexander(1945) used an approximation similar to Trorey’s (1932), but included sample treesthroughout the range of dbh classes. To solve for the parameters, he used a short-cutmethod of least squares described by Bruce and Schumacher (1950, pp. 199-200).Three methods of approximating the least squares solution for a parabolic height-diameterequation were recommended by Ker and Smith (1955). They found that these methodsyielded results very close to those obtained by least squares and were superior to othermathematical approximations of the least squares solution for different model forms.2.0 Literature Review 11Today, the proliferation of digital computers and the widespread availability of statisticalsoftware packages has rendered graphical techniques, or mathematical approximations ofleast squares regression, obsolete. However, other regression methods do exist. It may,for example, be desirable to perform least absolute values regression to minimize theinfluence of outliers. However, least squares (both linear and nonlinear) is most commonlyemployed to fit height-diameter equations.The ease with which least squares solutions can be obtained is certainly a huge benefit forall types of modelling problems facing forest managers. Generally, it is possible for peoplewith very modest statistical backgrounds to obtain linear least squares solutions and manypackages offer reasonable documentation to assist in the interpretation of results.However, one area which has not been well documented is the tendency for some modernstatistical packages to incorrectly calculate certain tests of significance when theregression model has been restricted4.Specifically, many popular statistical packagesgenerate incorrect values forR2, significance testing, and incorrectly calculate confidenceintervals when performing least squares regression through the origin.2.5 Common Height Diameter Equation FormsMany different equation forms have been applied to model the height-diameter relationshipover the years. Several forces have driven the change in model forms including evolvingtechnology and advances in biological and mathematical theory. Table 1 (from Johnson4Kozak, A. 1991. Personal communication.2.0 Literature Review 12and Romero 1991) is a summary of height-diameter models which reflects the range offorms. Note that any model in Table 1 which restricts the intercept to 4.5 (4.5 feet is equalto breast height in imperial units) could be fit without that restriction, or could be fit usingan intercept of 1.3 meters for metric units.Table 1. Common height-diameter models (after Johnson and Romero 1991).Model Authors1. H = 4.5 + e”° Dimitrou (1978), Murphy and Farrar(1987), Wykoff et a!. (1982)2. H = 4.5 +b0e(M Curtis (1967), Zakrzewski and Bella(1988)3. H=b0+b1ffi Van Deusen andBiging (1985)4. H = 4.5 + e01) Wang and Hann (1988), West 1979(b2DY’3) Johnson and Romero, 19915. H=b0+b1e6. H = b0 + eQ12b4BDoiph (1989). Larsen and Hann (1987),Wang and Hann (1988)Curtis (1967), Ker and Smith (1955),7. H = 45+b0D+b1D2 Snowdon (1981), Trorey (1932)(_i.(D)b2) Arney (1985)8. H=4.5+b0e9. H = 4.5 + D/./+b1DKer and Smith (1955), West (1979)10. H =45+bo(L0_e(_1)2Ek etal. (1984), Farr et. a!. (1989), Meyer(1940)where: H estimated tree heighte = base of natural logarithm ( e 2.718282)D = dbh outside barkb0, b1, !2, b3, b4 = regression coefficientsBA=basal areaIt can be seen that height diameter models may be linear or nonlinear in their parameters.In some cases (e.g., model 2) a nonlinear model can be transformed into an equivalentlinear form (in this case, by taking logarithms of both sides of the equation). In this case,2.0 Literature Review 13the model is described in the literature as being intrinsically linear (Draper and Smith1966).2.5.1 Bias due to logarithmic transformationsIt has been noted in the literature that logarithmic transformations of the independentvariable result in systematic underestimation (Baskervile 1972, Flewelling and Pienaar1981). There have been several proposed methods of dealing with this bias. Baskerville(1971) suggested that a correction taken from Brownlee (1967) would be appropriate fortransforming predicted values from logarithmic to arithmetic (untransformed) units. Thisinvolved adding ½ of the sample variance of the logarithmic equation to the predictedvalues before transforming to arithmetic units. Snowdon (1990) recommended that a ratioof the arithmetic sample mean and the mean of the back-transformed predicted valuesfrom the regression be used to correct for bias.Some models (e.g., model 6) are nonlinear in their parameters and can not be transformedto an equivalent linear model. These models are described as being intrinsically nonlinear(Draper and Smith 1966). For purposes of convenience, the term “nonlinear” will be usedto describe only this class of model.2.5.2 Features of Nonlinear ModelsA nonlinear regression model can be described as possessing the following properties(Weisberg 1985):In some cases, this class of models has been described as nonintrinsically nonlinear (Draper and Smith,1966).2.0 Literature Review 141. the function relating the dependent variable (response) to the independentvariables (predictors) is a nonlinear function of the parameters;2. unlike the linear model, there is no need for a direct correspondence betweenpredictors and parameters;3. parameterization is not unique, so many nonlinear regression models areequivalent; and4. as in linear regression, the errors are assumed to be independent and normallydisthbuted. Constant variance is also assumed, but this assumption can berelaxed using weighted least squares, as with linear regression.Computing least squares estimates for nonlinear models can be a complex process whichusually involves an iterative function minimization routine. There are many such routines,and it is not uncommon to use several different routines in the search for a least squaressolution for a single nonlinear model (Weisberg 1985). Many of these routines require thatthe first (and sometimes second) derivative of the model be computed. All iterativeroutines require starting values (initial estimates of parameters).The evaluation of nonlinear regression is not as well defmed as it is for linear regression.Inferential statements lean very heavily on normality and are only accurate for largesamples (Weisberg 1985). Estimates of standard errors produced by many computerpackages can be seriously in error (Weisberg 1985; Willdnson 1989).2.0 Literature Review 15Despite these difficulties and potential problems, nonlinear regression is desirable in manyinstances. For example, it may be completely inappropriate to model relationships that arenonlinear in nature with a linear function. Even some models which can be transformed toan equivalent linear form may be better handled with nonlinear techniques if thetransformation results in a log-normal error distribution. Also, it may be safer toextrapolate nonlinear rather than linear functions (Payendah 1983). In some cases, it ispossible to theoretically interpret the parameters of nonlinear models.In any case, both linear and nonlinear models are used to describe the height-diameterrelationship. The final choice of the model form used should ultimately depend on thepurpose of the model, the quality and quantity of the available data, the ease with which itcan be fit, and the overall quality of the fit.3.0 METHODS AND ANALYSIS3.1 Data PreparationThe data for this thesis were provided by Macmillan Bloedel, Ltd. and consisted of secondgrowth Douglas-fir permanent sample plots for which all heights and diameters weremeasured. The piots were mostly 0.1 acres (.0405 ha) in area, although a few were slightlysmaller at 0.040 1 ha. In total there were 252 complete plots, many of which had beenmeasured repeatedly. After receiving the data, a series of computer programs werewritten to facilitate analysis. First, header lines were stripped and delimiters added toenable easy importing to microcomputer statistical packages. Plots which contained lessthan 80 percent Douglas-fir (by stems) were removed, as were those which had beensubjected to thinning. For those plots with more than one measurement, only the initialmeasurement was used to ensure that each tree appeared only once in the database.The data were analyzed graphically to identify any obvious data points which would exertundo influence (i.e. outliers) on height diameter equations. Figure 1 shows the height-diameter relationship for all of the trees (5149 trees in total) and indicates that most of thepoints fall close together, with the exception of a few obvious outliers that wereeliminated. Since there were no comments in the data set, it is not known if these treeswere damaged, had broken tops, or were the result of measurement or data entry errors.The fmal data set (based on 5134 trees from 114 plots) is graphically presented in Figure2.163.0 Methods and AnalysisFigure 1. Initial data setFigure 2. Final data set170I 00 20 40 60 80dbh (cm - outside bark)100 120 1400454035151050000 ,O00’000 0000 10 20 30 40 50dbh (cm - outside bark)60 70 803.0 Methods and Analysis 183.2 Fitting Height Diameter CurvesSix different height-diameter models were fit to the fmal data set. These models werecompared to determine the “best” model for height-diameter prediction. Of the sixdifferent models used, three involved transformations of the dependent variable (height)and three did not. While no rigorous criteria were used in selecting these six models fromthe many available in the literature, these were chosen because they would be relativelysimple to use in the sample simulation. Nonlinear models would have presented difficultiesin estimating regression coefficients. Models with a fixed intercept would have likelyintroduced serious bias in some of the samples. Table 2 shows the models used and thepublisher of each model. In the case of models 3 and 6, the author (Curtis, 1967)suggested that the model be developed using either forward selection or backwardselimination.The models were compared using Fumival’s index of fit “I” (Furnival 1961) and byplotting the curves and the residuals. After comparing, a “best” model was selected for usein testing the sampling designs.3.3 Simulating Sampling DesignsA variety of sampling designs were simulated. The plots were stratified into three siteindex classes and three age classes (Table 3). Site index (based on King 1966) was theheight in metres at a reference age of 50 years (breast height age). An attempt was madeto randomly select 5 plots in each class to reduce the quantity of data generated during thesample simulation and to provide representation across all age and site index classes. This3.0 Methods and Analysis 19proved impossible due to limitations in the data. As can be seen, some classes had no piotsand some had less than 5 plots. It would have been preferable to have plots representingall age and site index classes as this would allow a better interpretation of the samplingrequirements for stands of different age and site classes. The number of trees per plotvaried for the selected plots, ranging from 42 to 124 trees.Table 2. Height (H) diameter (D) models* and authorsModel Equation Author1. H =b,+b1D+b2 Curtis (1967), Strand(1959), Prodan (1965)2. Curtis (1967), StrandH = b0 +b1 log(D) (1959), Prodan (1965)Curtis(1967) - Fitted inH =b0+b1D+b2ThE+b3,Y+b4,YDa stepwise fashion with3. non-significant termsdeleted.4. Curtis(1967),lo?I) = b0 + b1 Zakrzewski and Bella(1988)5. log(H) = b0 + b1 log(D) Curtis (1967)(_.L’ Curtis (1967) - Fittedlog(H) = bo +b1D +b215 + b3 D in a stepwise fashion6. with non-significantb4(j_ ) + 2 ) terms deleted.*th all cases, log refers to base 10 logarithm.Table 3: Number of plots by age and site index classesAge ClassSite Index Class 1. (10 - 30) 2. (31 - 50) 3. (> 50)1. (<25m) - 4 52. (25-35m) 4 5 43. (>35m) 5 3 -3.0 Methods and Analysis 20After selecting the plots (randomly where possible) for the sample simulation, 5 differentsampling designs were simulated for 6 different sample sizes. Each sample design and sizewas repeated 5 times for each plot. The sampling was simulated using a program calledSampleSim (a copy can be obtained from the author on request). Individual plots werefirst extracted from the data set. Within each plot, trees were sorted by diameter. Aftersorting, each plot was stratified into three classes based on diameter size. If the plot couldnot be evenly divided into three classes, the extra trees (two at most) were randomlyassigned to classes ensuring that class sizes never differed by more than one within a plot.Each plot was then “sampled”. The program incremented the number of trees sampled ineach plot - ranging from 8 sample trees to 28 in steps of 4 resulting in 6 sample sizes (n =the number of trees in the sample) for five different designs:1. Random - all n trees were selected randomly from the plot.2. Extremes - 1/2 of the n trees were sampled from the largest and 1/2 from the smallestdiameter classes. None were taken from the middle.3. Small - 1/2 of the n trees were taken from the smallest diameter class, 1/4 from themiddle and 1/4 from the large diameter classes.4. Uniform - The n trees were taken uniformly throughout all diameter classes.5. Large - 1/4 of the n trees were taken from the smallest diameter class, 1/4 from themiddle and 1/2 from the large diameter classes.3.0 Methods and Analysis 21The program used a simple routine to randomly select trees without replacement from adiameter class. This ensured that there was no bias in selecting trees from the establisheddiameter classes.When a plot was sampled in any given design, several different arrays were created. Thefirst held all the dbh measurements and three others held height values: one for all theheight measurements; one for the sample heights (all elements in this array were set tozero at the start of each sample); and, one array of size n to hold measured heights forcalculating regression coefficients. Sampling resulted in replacing some of the zeros in thesample height array with measured heights. These same values were used to estimateregression coefficients for the model selected as the best. The coefficients were then usedto estimate height values for those remaining elements of the sample height array whichhad a value of zero.Occasionally, the regression coefficients produced extremely unusual and unrealistic heightestimations. This generally only occurred with small sample sizes and was characterized bya very large intercept and extremely large height estimations on very small trees. Thiscondition is tested for in the program, and, if detected, another sample was taken using thedesign and sample size in question. While this introduced some bias into the results, it isvery likely that operationally, samples such as these would either be rejected or fit with adifferent equation which did not give such poor estimates.3.0 Methods andAnalysis 22After sampling, the standard error of estimate was calculated for each regression equationand stored in an output file.3.4 Height Estimate ComparisonsThe sample heights were compared against the true heights and the following values weregenerated for each plot and sample and stored in the output file:1. Mean deviation (bias). This is the average of the tree by tree differences betweenmeasured height and estimated height (equal to zero for measured trees). If there wereno bias, mean deviation would be equal to zero.2. Minimum deviation. This is the smallest (largest negative) difference betweenmeasured and estimated height.3. Maximum deviation. This is the largest (positive) difference between measured andestimated height.4. Standard deviation of differences. This is the standard deviation of the differencesbetween measured and estimated height.5. Mean absolute deviation (MAD). This is the average of the absolute value of the treeby tree differences between measured and estimated height (equal to zero formeasured trees).3.5 Volume ComparisonsTwo volumes were estimated for each tree. The “true” volume, based on measured heightsand diameters, and estimated volume, based on estimated heights and measured diameters.In the latter case, if a tree was sampled (i.e. height was measured), the measured height3.0 Methods and Analysis 23was used in the volume calculation (meaning that true and estimated volumes were equalfor sample trees). The B.C. Ministry of Forests volume equation (Watts, 1983) was usedto calculate tree volumes:=10319071+1.813(1.042ubog(11)where: V = estimated volume (m3);D = diameter outside bark (cm);H = total tree height (m); andlog = base 10 logarithm.The estimated volumes were then compared against the true volumes using the samestatistics used for height.4.1 Height Diameter Curves4.0 RESULTSAll models tested were significant at an X = 0.05 level, although there were some verylarge differences among the shapes of the models. Figure 3 shows the shapes of themodels across the dbh range used in fitting (from 0.5 cm to 71 cm). Models 1, 3, 5 and 6performed similarly for diameters of up to about 35 cm. Model 2 predicted negativeheights for any trees below about 4 cm in dbh. Model 4 predicted considerably smallerheights than the other models with dbh’s larger than 15 cm. Model 6 predicted decreasingheights with dbh’s larger than about 50 cm.Figure 3: Six models comparedThe fit statistics for the six models are summarized in Table 4. Normally, the square rootof the mean square residual (or the standard error of estimate) can be used to determine6050403020100-10-20-30510152025303540455055606570—0-—Model 1—O---MOdel 2—ô----Model 3—)(—Model 4—3(E-- Model 5—0—Model 6dbh (cm)244.0 Results 25the model which gives the best fit. However, neither this, nor the multiple R2 values can becompared when the dependent variable has been transformed (Fumival, 1961). Therefore,Fumival’s index of fit “r’ was used as the basis for comparison.Table 4: Height diameter models comparedModel Equation Multiple Standard IR2* Error1 H = —1.8057 + l.287D — 0.0105D2 0.899 3.21092 3.21092 = —16.2470 + 28.3340 log(D) 0.822 4.2628 1 4.2628H = —76.3726 — 1.1632D + 22.22954ffi +3 0.901 3.18431 3.184399.62431”_1_ — 40.01961’Iffi) D)—4 log(H) 1.2629—1.694i 0.427 0.2455 7.3832L\D)0.884 0.11031 3.1313log(H) = 0.0262 + 0.9425 log(D)6 11) —0.444 — 0.0402D + 0.5651J + 0.902 0.10146 2.8800.1543LD)* Standard errors for models 4,5 and 6 are in logarithmic (base 10) units. The sum ofsquares values used to calculate R2 for models 4, 5 and 6 are in logarithmic (base 10)units.Using this method to compare height-diameter curves, Curtis (1967) found that there weresurprisingly few differences between curves he tested. He concluded that “almost anyreasonable and moderately flexible curve will give similar values of I”. For the most partthis was the case here, with models 2, and especially 4, being exceptions. As a result,determining the “best” model also involved other criteria such the ease with which therelationship can be fit, and, very importantly, the result of analyzing graphs of the residualplots. After determining the index of fit for each model, residual plots (residual values4.0 Results 26versus predicted values) were generated to see if any of the models displayed obvious lackof fit (Figures 4 to 9). This process of plotting residuals also proved to be a valuable stepin comparing equations.Figure 4: Residual Plot for Model 1Figure 5. Residual Plot for Model 2ECuCuCu-30 -20 -10 400Estimated Height (m)0000ECuCuCu00000-30 -20 -10 40a,Estimated Height (m)4.0 Results 27Figure 6. Residual Plot for Model 3Figure 7. Residual Plot for Model 430E0-30252015105-20 -10 -5-10-15-20cPEstimated Height (m)U 2.52E0,01.5eI6-2.5 -2 -1.5 -1 -0.5-0.51.5 2Estimated Height (Iog(m))4.0 Results 28Figure 8. Residual Plot for Model 52.52’—. 1.5E0D5-2.5 -2 -1.5Estimated Height (log(m))Figure 9. Residual Plot for Model 62.521.5E000.50-2.5 -2 -1.5 -i -0.5Estimated Height (Iog(m))4.0 Results 29It can be seen from the residual plots that some models fit the data much better thanothers. Models 1, 3 and 6 showed a reasonable distribution of residuals while models 2, 4and 5 displayed a lack of fit. Any of models 1, 3 or 6 would probably be satisfactory.When the input data were sorted by dbh, it was apparent that diameters were repeated,some many times. This is an ideal situation to apply a lack of fit test. With repeatedmeasurements, the sum of squares residual can be partitioned into two new sum of squaresvalues: pure error and lack of fit. A simple F test can be applied to test for lack of fit. Inthe absence of repeated measurements, it is possible to apply an approximate lack of fittest based on clustering the data (Daniel and Wood 1981). However, this test is verysensitive to the clustering algorithm used and will give different results with differentclustering methods (Weisberg 1985).A lack of fit test was applied to models 1, 3 and 6 to facilitate the process of choosing thebest model. Appendix 1 shows the coefficients and analysis of variance for all models, withthe lack of fit test applied for models 1, 3 and 6.Both models 1 and 3 tested significantly for lack of fit at an c = 0.01 level. This was notapparent in the residual plots, but the huge number of points may have obscured sometrends. The lack of fit test was not significant for model 6. Because of this, and the factthat model 6 had the lowest I value (i.e., the best fit index) and a reasonable distribution ofresiduals, it was selected as the “best” model.4.0 Results 304.2 Sampling Designs4.2.1 Height estimationTo clarify results, the sampling simulations were summarized graphically, first comparingthe effects on height estimation and then the effects on volume (refer to Appendix 2 forthe tabular results of sampling). Mean deviation in heightFigures 10 through 16 show the mean deviations in height by age-site classes, averagedfor all plots in each class and for the 5 sample repetitions.Figure 10. Mean deviation in height for AGE1SI20. 12D ExtremeLargerJ RandomDSmaUUniformSample Size4.0 Results 31Figure 11. Mean deviation in height for AGE1SI3Figure 12. Mean deviation in height for AGE2SI100. 16 20C ExtremeLargeC Random1] SmallUniform24 28Sample Size0. ExtremeU::: 1LkLnsiI 20 24 28 Uniform8 12 16Sample Size4.0 Results 32Figure 13. Mean deviation in height for AGE2SI2Figure 14. Mean deviation in height for AGE2SI30. DExtreme-o.i •Uniform-0.2-0.3-0.48 12 16 20 24 28Sample Size0,0,0, ExtremeLar9eIJ RandomDSmallUniform8 12 16 20 24 28Sample Size4.0 Results 33Figure 15. Mean deviation in height for AGE3SI1Figure 16. Mean deviation in height for AGE3SI2These charts show that the mean deviation in height varied considerably with the differentage class - site index groupings. Specifically, the older plots tended to show greater mean0. 12 16 20 24 28ExtremeLargeI] RandomDSmallUniformSample Size0. iLI0ExtremeLargeC Random. OSmallUniformLnLP1U8 12 16 20 24 28Sample Size4.0 Results 34deviation than the younger plots, especially with small sample sizes. In age class 1, therewas not much change with different sample sizes. In age class 3 - site class 1, the meandeviation was reduced when sampling 16 or more trees. In most cases, the extreme designproduced quite large mean deviations. The unifonn design usually displayed small meandeviations. In general, mean deviation was usually positive indicating that tree height wasmost often under-estimated. Average maximum and minimum deviation in heightFigures 17 to 23 show the average maximum deviation in height for the age class - siteindex groupings. The values in the charts represent the average of the largest positivedeviations for a given age - site index class. Figures 24 to 30 show the average minimumdeviation in height for the age - site index groupings. These values represent the averageof the largest negative deviations for a given age - site index class.4.0 Results 35Figure 17. Average maximum deviation in height for AGE1S12Figure 18. Average maximum deviation in height for AGE1SI35. ExtremeLargeC RandomC SmallUniform-0.5 8 12 16 20 24 28Sample Size0)4) ExtremeC LargeC RandomC SmallC Uniform8 12 16 20 24 28Sample SizeCo 0 3 0 Co N 0rssssssLiLICO 0 3___________________0 C,) t1r:I:I:I:ILi:LiLiLiIII*l:I:II:I:!*!:I*ILi:!:ILiI*lt7IMetresPr’010101P?91UI0101Metres0p0101UI01“3 0)P UI0101b):z.:z.tZi I I C, t’a C,,+sssnsrWS/S/S4J/S/wsss-a-’-I 11 ISSSAcørm8!.aBrsa’ssa cwrma—a9CD——o33C’4.0 Results 37Figure 21. Average maximum deviation in height for AGE2SI3Figure 22. Average maximum deviation in height for AGE3SI15. ExtremeLar9eDRandomiSmallUniform-0.5Sample Size5.,0,1.50.5D ExtremeLarge0 RandomSmaIIUniform-0.5Sample Size4.0 Results 38Figure 23. Average maximum deviation in height for AGE3SI2The average maximum deviations showed progressive improvements with larger samplesizes for all age-site classes. In age class 1, these improvements tended to be very small,and the gain was not large when sampling more than 16 or 20 trees. In age class 2- siteclasses 1 and 2, improvements were small when sampling more than 16 trees. Theremaining age-site classes displayed improvements with each increase in sample size. Thedifferences among age classes were not large, although age class 1 generally displayedlower average maximum deviations. Trends among sample designs were not obvious,although the random design was sometimes poor while the extreme design performed wellfor many samples5. ExtremeLargeC RandomDSmaIIUniform-0.5 16 20Sample Size4.0 Results 39Figure 24. Average minimum deviation in height for AGE1S12Figure 25. Average minimum deviation in height for AGE 1S13-0.58 12 16 20 24 28-1.5-2.5-3.5-4.5-5.5-6.5o ExtremeLargeC Random0 SmallUniform-7.5Sample Size-0.58 12 16 20 24 28-1.5-2.5Ia4)-3.5-4.5-5.5-6.5o ExtremeLargeEl RandomZSmallUniform-7.5Sample Size4.0 Results 40Figure 26. Average minimum deviation in height for AGE2SI1Figure 27. Average minimum deviation in height for AGE2ST2-0.58 12 16 20 24 28-1.5-2.54)4)4)-3.5-4.5-5.5-6.5Q ExtremeLargeD RandomSmalIUniform-7.5Sample Size-0.58 12 16 20 24 28-1.5-2.5-3.54)4)1.4)-4.5-5.5-6.5ExtremeO LargeC RandomQSmaIIUniform-7.5Sample Size4.0 Results 41Figure 28. Average minimum deviation in height for AGE2SI3Figure 29. Average minimum deviation in height for AGE3SI1-0.58 12 16 20 24 28-1.5-2.5-3.51.-4.5-5.5-6.5O ExtremeC LargeD RandomOSmallC Uniform-7.5Sample Size-0.58 12 16 20 24 28-1.5-2.5-3.5-4.5-5.5-6.5O ExtremeC Largeo RandomOSmallC Uniform-7.5Sample Size4.0 Results 42Figure 30. Average minimum deviation in height for AGE3SI2The average minimum deviations displayed some differences between age - site indexclasses, but not as pronounced as those of the mean deviations. In age class 1,improvements were quite small when sampling more than 12 trees. In other age-siteclasses, improvements occurred with each increase in sample size, but those improvementswere often very small. Age class 3 - site class 2 displayed the poorest results. In mostcases, the difference among sample designs was quite small. Average standard deviation of height differencesFigures 28 to 33 show the average standard deviation of height differences. These valuesare the standard deviations of the differences between measured and estimated tree heights-0.58 12 16 20 24 28-1.5-2.5‘.55-6.5ExtremeLargeci Random0 Small• Unibrm-7.5Sample SIze4.0 Results 43averaged over each plot and repetition in each age - site index class for each sample sizeand design.Figure 31. Average standard deviation of height differences for AGE1S12Figure 32. Average standard deviation of height differences for AGE1S132.521.50.501D ExtremeLargeDRandomDSmaflUniform8 12 16 20 24 28Sample Size2.521.50.5D ExtremeLargeC RandomD SmallUniform8 12 16 20 24 28Sample Size4.0 Results 44Figure 33. Average standard deviation of height differences for AGE2SI1Figure 34. Average standard deviation of height differences for AGE2SI22.521.510.5El ExtremeLargeDRandomDSmaIIUniform08 12 16 20 24 28Sample Size2.521.510.5El Extremeu LargeDRandomEl SmallE Uniform08 12 16 20 24 28Sample Size4.0 Results 45Figure 35. Average standard deviation of height differences for AGE2SI3Figure 36. Average standard deviation of height differences for AGE3SI12.521.50.501D ExtremeLargeC RandomL]SmalIUniform8 12 16 20 24 28Sample Size2.521.510.50Q ExtremeLargeC RandomDSmaIIUniform8 12 16 20 24 28Sample Size4.0 Results 46Figure 37. Average standard deviation of height differences for AGE3SI2The average standard deviation of height differences displayed little variation amongdifferent sampling designs, although the extreme design generally gave good results.However, the variation that occurred among age - site index classes was very pronounced.Age class 1 and age class 2 - site index 1 had lower average standard deviations than theother classes at all sample sizes. In age class 1, the largest improvement occurred whensample size was increased to 12. The remaining age-site classes showed fairly steadyimprovements with each increase in sample size, although in age class 2 - site class 1 thoseimprovements were quite small.2.521.510.5o Extremeo LargeRandomDSmaUUniform08 12 16 20Sample Size24 284.0 Results 474.2.1.4 Mean absolute deviation in heightFigures 38 through 44 show the mean absolute deviation in height, averaged for each age -site class for sample sizes 8 through to 28.Figure 38. Mean absolute deviation in height for AGE1SI22.521.510.5ExtremeLargeD Random0 SmallUniform08 12 16 20 24 28Sample Size4.0 Results 48Figure 39. Mean absolute deviation in height for AGE1SI3Figure 40. Mean absolute deviation in height for AGE2SI12.521.510.50D ExtremeLargeI] Random0 SmallLbiform8 12 16 20 24 28Sample Size(0(02.521.510.50QExemeLargeDRandom0 SmallUniform8 12 16 20 24 28Sample Size4.0 Results 49Figure 41. Mean absolute deviation in height for AGE2SI2Figure 42. Mean absolute deviation in height for AGE2SI32.521.500.500 ExtremeLargeC Random0 SmallUniform8 12 16 20 24 28Sample Size2.521.50)0I0.5Q ExtremeLargeC RandomO SmallE Uniform08 12 16 20 24 28Sample Size4.0 Results 50Figure 43. Mean absolute deviation in height for AGE3SI1Figure 44. Mean absolute deviation in height for AGE3SI2The differences among age-site classes were not very pronounced for mean absolutedeviations. In general, all age classes improved with each increment in sample size,I2.521.510.50I] ExtremeLargeDRandomD SmallUniform8 12 16 20 24 28Sample Size2.521.510.50CwQ ExtremeLargeD RandomEl SmallUniform8 12 16 20 24 28Sample Size4.0 Results 51although in most cases the largest improvements occurred at or before reaching a samplesize of 16. Age class 3 was an exception to this and displayed steady improvements witheach increase in sample size. Differences among sampling designs were generally not veryclear, except at a sample size of 8 where the random design displayed considerably largermean absolute deviations for some age-site classes. Also, the extreme design generallyperfonned well in age class Volume estimation4.2.2.1 Mean deviation in volumeFigures 45 through 51 show the mean deviations in volume by age-site classes, averagedfor all plots in each class and for the 5 sample repetitions.Figure 45. Mean deviation in volume for AGE1SI20.0170.012Q Extreme0.007 LargeC Random0.002 0 Small0— I I -‘— I . —--- I Uniform-0.003-0.0088 12 16 20 24 28Sample Size4.0 Results 52Figure 46. Mean deviation in volume for AGE1S130.0170.012Q Extreme00.007 LargeC)lRandom0° 0002 QSmallI I I I I Uniform-0.003-0.0088 12 16 20 24 28Sample SizeFigure 47. Mean deviation in volume for AGE2SI10.0170.0120 Extreme00.007 Large0 Random0 fl.0 0.002 N m 0 SmallUniform-0.003-0.0088 12 16 20 24 28Sample Size4.0 Results 53Figure 48. Mean deviation in volume for AGE2SI2Figure 49. Mean deviation in volume for AGE2SI30.0170.0120.007C)U.0z00.002-0.003C ExtremeLarge[I Random[j Smallfl rL . LIJiL UniformU - u..-.-0.0088 12 16 20 24 28Sample SizeC)C)0.000.0170.0120.0070.002-0.003-0.008C ExtremeLargeI] RandomI. ØSmalln J1L. Lb,iform8 12 16 20 24 28Sample Size4.0 Results 54Figure 50. Mean deviation in volume for AGE3SI1Figure 51. Mean deviation in volume for AGE3SI2Some trends in the mean deviation in volume are apparent. First, volume wasunderestimated, on average, although some over-estimation did occur. The mean0U.0zC.,0.0170.0120.0070.002-0.003-0.008DExtremeLargeIJRandomD SmallUniform8 12 16 20 24 28Sample Size0.0170.0120.0070.002-0.003D ExtremeLargeD RandomD SmallUniform-0.0088 12 16 20 24 28Sample Size4.0 Results 55deviations were much larger in age class 3 (especially in site index class 1) than in theother age classes. Overall, the large and uniform designs gave good results. Average maximum and minimum deviation in volumeFigures 52 to 58 show the average maximum deviation in volume for the age class - siteindex groupings. The values in the charts represent the average of the largest positivedeviations for a given age - site index class. Figures 59 to 65 show the average minimumdeviation in volume for the age - site index groupings. These values represent the averageof the largest negative deviations for a given age - site index class.Figure 52. Average maximum deviation in volume for AGE1S120.250.20 Extreme0.15 Large0U Random00.1 QSmaIIWiform0.0:Sample Size4.0 Results 56Figure 53. Average maximum deviation in volume for AGE1SI30.250.2El Extreme0.15 LargeU Random0.1 C SmallUniform0Sample SizeFigure 54. Average maximum deviation in volume for AGE2SI10.250.2C Extreme0.15 LargeU Random0.1 El SmallUniform0Sample Size4.0 Results 57Figure 55. Average maximum deviation in volume for AGE2SI2Figure 56. Average maximum deviation in volume for AGE2SI3I0. ExtremeLargeRandomC SmallUniform8 12 16 20 24 28Sample SizeI2.50E-012.OOE-011 .50E-011 .OOE-015.OOE-020.OOE+00C ExtremeLargeDRandom0 SmallUniform8 12 16 20 24 28Sample Size4.0 Results 58Figure 57. Average maximum deviation in volume for AGE3SI1Figure 58. Average maximum deviation in volume for AGE3SI2Age class 1 and class AGE2SI1 displayed lower average maximum deviation in volumethan the other classes. These classes showed no discernible improvement with largerI0.[I RandomC SmallUniform8 12 16 20 24 28Sample Size00U.0z00. ExtremeU LargeC RandomC SmallUniform8 12 16 20 24 28Sample Size4.0 Results 59sample sizes. Age class 3 - site index 2 gave displayed the poorest results, and showedimprovements with larger sample sizes. In age class 3- site index 1 results improved whensample size was increased to 16. In most cases, the large design performed well, while thesmall design usually gave very poor results.Figure 59. Average minimum deviation in volume for AGE 1S12-0.05D Extreme0.1 Largea Random-0.15 SmaII0 Uniform-0.2-0.25Sample Size4.0 Results 60Figure 60. Average minimum deviation in volume for AGE1S13Figure 61. Average minimum deviation in volume for AGE2SI108 12 16 20 24 28-0.05-0.1C)C)0-0.15-0.2ExtremeLargeRandomD SmallUniform-0.25Sample Size08 12 16 20 24 28LIJ9IflJ-0.05C)-0.1C)C)-0.150-0.2-0.25DExtremeLargeQFiandomDSmallUniformSample Size4.0 Results 61Figure 62. Average minimum deviation in volume for AGE2SI2Figure 63. Average minimum deviation in volume for AGE2SI38 12 16 20 24 28I0- RandomI] SmallUniformSample Size08 12 16 20 24 28-0.05-0.100.00-0.15-0.2DExemeLargeRandomD SmallUniform-0.25Sample Size4.0 Results 62Figure 64. Average minimum deviation in volume for AGE3SI1Figure 65. Average minimum deviation in volume for AGE3SJ2As with the average maximum deviation, the average minimum deviation in volumeshowed that the older age classes produced the worst results. In general, the large designperformed well, though trends were not always clear. The small design often gave very08 12 16 20 24 28-0.05-0.100C).00-0.15-0.20 ExtremeLargeEl Random0 Small• Uniform-0.25Sample Size08 12 16 20 24 28-0.05-0.100U.00-0.15-0.2O Extreme•LargeEl RandomO Small• Uniform-0.25Sample Size4.0 Results 63poor results in age class 3. In age class 1 and the AGE2SI1 class there were no obvioustrends among sample designs. In classes AGE2ST2, AGE2SI3 and AGE3SI1 there waslittle improvement in the large design when taking more than 16 samples. In classAGE3SI2 results improved with each increase in sample size. Average standard deviation of volume differencesFigures 66 to 72 present the average standard deviation of volume differences. Thesevalues are the standard deviations of the differences between measured and estimated treevolumes averaged over each plot and repetition in each age - site index class for eachsample size and design.Figure 66. Average standard deviation of volume differences for AGE1SI20. Extreme0.06 Large0.05 0 Random0.04 SrnaHo 03 Uniform0.02Sample Size4.0 Results 64Figure 67. Average standard deviation of volume differences for AGE1S13Figure 68. Average standard deviation of volume differences for AGE2SI10. Extreme0.06 Large0.05 0 Random0.04 C Small0.03 Uniform0.020.0: l1iSample Size0. RandomC SmallUniform8 12 16 20 24 28Sample Size4.0 Results 65Figure 69. Average standard deviation of volume differences for AGE2SI2Figure 70. Average standard deviation of volume differences for AGE2SI30. ExtremeLargeC Random0 SmallUniform8 12 16 20 24 28Sample Size0. ExtremeLargeC RandomU SmallUniform8 12 16 20 24 28Sample Size4.0 Results 66Figure 71. Average standard deviation of volume differences for AGE3SI1Figure 72. Average standard deviation of volume differences for AGE3SI2The average standard deviation of volume differences showed some trends that were notapparent in the average standard deviation of height differences. In age class 3, there were0. ExemeLargeEl RandomQ SmallUniform8 12 16 20 24 28Sample Size0. ExtremeLargeC RandomO SmallUniform8 12 16 20 24 28Sample Size4.0 Results 67noticeable differences among sample designs, with the small design consistently showingthe largest average standard deviation. In general, the large design had the lowest averagestandard deviation in this age class. Age class 3 displayed a constant reduction in averagestandard deviation when sample size was increased. In age class 3 - site class 1 the mostpronounced improvements occurred when sample size was increased to 16. In age class 2,there were no large differences among the different sample designs. In this age class, siteindex 1 had a lower average standard deviation than site classes 2 and 3. Age class 1showed little difference among sampling designs and little reduction with increased samplesize. There was no clear difference between site index classes for this age group. Mean absolute deviation in volumeFigures 64 to 69 show the mean absolute deviation in volume, averaged for each ageclass-site index class grouping for sample sizes 8 through to 284.0 Results 68Figure 73. Mean absolute deviation in volume for AGE1S120.070.060.05C Extremea)0.04 D Largea)JRandom0.03 Q Small0 Uniform0.020.010Sample SizeFigure 74. Mean absolute deviation in volume for AGE1SI37.OOE-026.OOE-025.OOE-02C Extremea)4.00E-02 LargeI] Random3.00E02 C Small0 Uniform2.OOE-02:::Sample Size4.0 Results 69Figure 75. Mean absolute deviation in volume for AGE2SI1Figure 76. Mean absolute deviation in volume for AGE2SI20.070.060.05I] Extreme0.04 LargeQRandomU 0.03 Q Small0 Uniform0.020.0:Sample Size0. ExtremeLargeO RandomC SmallUniform8 12 16 20 24 28Sample Size4.0 Results 70Figure 77. Mean absolute deviation in volume for AGE2SI3Figure 78. Mean absolute deviation in volume for AGE3SI14.OOE-0200.000.OOEi-00Q Extreme0 Largea RandomQ SmallUniform8 12 16 20Sample Size24 28000. LargeDRandom0 SmallUniform8 12 16 20 24 28Sample Size4.0 Results 71Figure 79. Mean absolute deviation in volume for AGE3SI2Some trends were evident in the mean absolute deviations in volume. In younger ageclasses, the difference among sample designs was quite small, even with small samplesizes. In age class 3 site class 2 the small design displayed very poor results for samplesizes 8 and 12. The large design was almost always the best, and the extreme design alsoperformed well. In age class 1, improvements were minimal with sample sizes larger than12, but age classes 2 and 3 improved with each increase in sample size, with age class 3displaying the largest relative improvement.4.3 Ranking of sampling designsOutputs for six criteria were ranked for each age-site class (Tables 32 to 37 in appendix2). The ranicings were then totalled across the site-age classes, then these totals rankedagain to show the relative position of each sample design for each sample size (Table 5). ExtremeLargeI RandomC SmallUniform8 12 16 20 24 28Sample Size4.0 Results 72Table 5. Summarized rankings of sample designs*Size Sample MDH MDHTMAX MDHTMIN MADHT MDVOL MDVOIJMAX MDVOLMIN MADVOL8 Extreme 5 1 3 3 4 3 2 38 Large 4 2 2 4 2 1 3 18 Random 2 3 1 5 1 2 5 58 Small 1 4 4 2 5 5 4 48 Uniform 3 5 5 1 3 4 1 212 Extreme 2 1 2 1 3 1 1 112 Large 3 4 1 5 1 2 2 212 Random 4 3 3 3 2 3 3 412 Small 5 5 4 4 5 4 4 512 Uniform 1 2 5 2 4 5 5 316 Extreme 5 2 1 1 3 2 2 216 Large 1 1 5 4 4 1 1 116 Random 4 5 3 3 5 5 4 416 Small 2 4 2 2 1 4 5 516 Uniform 3 3 4 5 2 3 3 320 Extreme 5 5 1 2 5 2 2 220 Large 4 4 5 5 2 1 1 120 Random 2 1 2 4 3 3 3 420 Small 1 3 4 1 4 4 5 520 Uniform 3 2 3 3 1 5 4 324 Extreme 5 1 1 2 5 2 2 224 Large 2 4 4 3 1 1 1 124 Random 1 5 3 4 3 3 3 324 Small 4 3 2 1 2 5 5 524 Uniform 3 2 5 5 4 4 4 428 Extreme 5 2 2 1 4 2 1 328 Large 2 4 3 2 2 1 2 128 Random 3 1 4 3 5 3 4 428 Small 4 5 1 4 3 5 3 528 Uniform 1 3 5 5 1 4 5 2*Whe:MDHT is the average mean deviation in height;MDHTMAX is the average of the largest positive mean deviations in height;MDHTMIN is the average of the largest negative mean deviations in height;MADHT is the average of the mean absolute deviations in height;MDVOL is the average mean deviation in tree volume;MDVOLMAX is the average of the largest positive mean deviations in volume;MDVOLMIN is the average of the largest negative mean deviations in volume; and,MADVOL is the average of the mean absolute deviations in volume.Average standard deviations in height and volume generally highlighted differences amongage-site classes rather than differences among sample designs and were,4.0 Results 73therefore, not included in this ranking. Likewise, minimum and maximum deviations inheight and volume were generated, but not presented graphically or ranked (values arepresented in tables 20 and 21, and tables 26 and 27 in appendix 2). These valuesrepresented the largest minimum and maximum deviations in height and volume for asingle tree estimate. In general, there was little difference among designs and littleimprovement with larger sample sizes.To further summarize overall design performance, the rankings from table 5 were summedacross sample size to give the relative positions of the designs for each of the criteria(Table 6).Table 6. Overall rankings of sample designsSample MDHT MDHTMAX MDHTMIN NADHT MDVOL MDVOLMAX MDVOILMIN MADVOLExtreme 5 1 1 1 5 2 2 2Large 2 2 4 5 1 1 1 1Random 3 3 3 4 3 3 4 4Small 4 5 2 2 4 5 5 5Uniform 1 4 5 3 2 4 3 35.0 DISCUSSION5.1 Model SelectionIn selecting a model to describe the height-diameter relationship, many different modelforms are available for consideration. The model selected in this thesis used a logarithmictransformation of the dependent variable (height). As described in chapter 2, this willresult in systematic underestimation, and the results appeared to confirm this. Severalmethods to correct for this underestimation are available, but none were used in this study.Since the main objective was to compare sample sizes and designs, and the same modelwas used for the testing of each design, any bias introduced by logarithmic transformationwas deemed unimportant.Another problem that became visible with the selected model was its prediction ofdecreasing heights for dbh’s greater than about 50 cm when it was fit to the entiredatabase. This was likely due to the huge number of small trees in the database, and thevery small number of large trees (greater than 50 cm). This condition was not duplicatedexactly in the sample simulation, but it may explain in part why the small sample designperformed poorly in estimating volume.The model used in this thesis was linear in its parameters, and was, therefore, fit usingstandard linear least squares techniques. There are several nonlinear models available thatcould have been used. There are both advantages and disadvantages to using a nonlinear745.0 Discussion 75model, several of these were outlined in chapter 2. For this study, some disadvantagesmade the use of nonlinear models impractical. In particular, fitting nonlinear least squaresis considerably more difficult than linear. The computer program written in this study fitsregression coefficients for each plot 150 times: 5 sample designs by 6 sample sizes by 5repetitions. Besides the greater difficulty of coding the algorithms for the nonlinearestimates, the time required to carry out the sample simulations would have been verylarge due to the iterative nature of nonlinear least squares. It also would have presentedextraordinary difficulties if some samples wouldn’t converge, and it would have been verydifficult to determine if convergence was at a local minima. In contrast, writing thealgorithms for the estimates of the selected model was relatively simple, and programexecution was relatively quick.As stated in chapter 2, it is extremely common to fit height diameter models restricted toan intercept of 1.3 meters (breast height). This is logical, but presented a potentialdifficulty in fitting the various samples. Because the selected model was used for a largevariety of sample sizes taken from plots of different ages, the introduction of such acondition would very likely introduce serious bias into at least some of the samples. It wasdeemed more appropriate to use a model without any restrictions.5.0 Discussion 765.2 Sampling designsAlthough the ranldngs presented in Tables 5 and 6 are useful to view the relative quality ofthe various sampling designs, it is important to recognize that much useful information isnot presented. For example, the magnitude of the deviations is completely hidden. It is notclear if one design is only marginally better than another, or if the differences are quitelarge. However, when considered along with the deviations presented graphically inchapter 4, some useful observations can be made.5.2.1 Height estimationIn estimating tree height, the uniform design performed well in MDHT and MADHT,especially with small sample sizes (8 - 12). As sample sizes increased to 16 and beyond,the magnitude of differences in MADHT among designs were generally quite small. Theuniform design did not perform as well as the other designs in the MDHTMIN andMDHTMAX categories, but these categories often didn’t display clear trends amongsample designs. The extreme design performed well for most height categories, although itproduced some large deviations in MDHT. This would probably be due to the extremedesign producing biases that are mostly positive or negative for a given sample. TheMADHT results for the extreme design were similar to the uniform design. Despite itsrelatively high ranking in Table 6, the small design offered little improvement over randomsampling, except in MADHT with a sample size of 8. Random sampling gave very poor5.0 Discussion 77results in MADHT with a sample size of 8. The large design performed similarly to thesmall design, but generated better results in MDHT.In general, the deviations increased with age and site quality (although MADHT did notvary much across classes). This implies increased height variation in older plots, andincreased height variation with higher site quality.When evaluating the effects of increasing sample sizes, it becomes clear that different age-site index classes will require different sample sizes to achieve the same results. In ageclass 1, and age class 2 site index 1, there was little to be gained by sampling beyond 16,even though these classes had more trees per plot than the other classes (average trees perplot for AGE1SI2 was 67, 58 for AGE1SI3, and 73 for AGE2SI1). Other age classesshowed improvements with successive increases in sample size. Even in these classes, themost dramatic improvements occurred at or before reaching a sample size of 20. Theaverage number of trees per plot for these classes were quite similar, with values rangingfrom 50 to Volume estimationIn estimating volume, the large design was clearly the best. The large designunderestimated volume by the smallest amount as evidenced by the lowest overall rankings5.0 Discussion 78in MDVOL and MDVOLMAX, and had the lowest MADVOL in almost every class andsample size.There are at least two reasons for the success of the large design. First, there is likelygreater variation among the larger dbh trees in most of the piots. The large designincreases the number of samples taken from the more variable stratum. The second reasonhas to do with the shape of the chosen height-diameter model. If large samples are omittedwhen fitting the regression, larger trees can be seriously underestimated because the modelmay reach its maximum early and begin to decrease. The estimates for the largest treeswill be extrapolations beyond the range of the data, resulting in serious underestimation.This problem also existed for the height estimates. However, errors in large trees have amuch greater effect on volume estimation than on height estimation, because volume is acubic measure while height is in linear units.Overall, the extreme design performed well in estimating MDVOLMAX, MDVOLMINand MADVOL, although it did show some large underestimations in MDVOL. As withthe large design, the good results were likely due to the increased sampling of larger, morevariable trees.The small design was very poor in estimating volume for all criteria. This is likely due inpart to the shape of the model. The uniform design performed well in MDVOL, but was5.0 Discussion 79not much better than the random design in MDVOLMAX and was poorer inMDVOLMIN. However, it did perform better in MADVOL than the random design.With volume estimates, the average standard deviation and the maximum mean deviationdemonstrated generally good results at sample size 16 for plots in age classes 1 and 2. Inage class 1, sampling beyond 12 showed very little improvement in MADVOL. In ageclass 3, a sample size of 16 or 20 usually gave good results for these criteria and largersample sizes generally produced diminishing benefits.5.2.4 Application in the fieldOverall, the uniform design performed well in estimating height. The uniform designwould improve if there were more dbh classes as this would ensure more even samplingacross the dbh range. Three classes were chosen for this study because it could be easilyand quickly applied in the field.There may be several ways to apply any of the designs from this study in the field. Thesampling design of choice could be applied in a strict fashion, or in a more flexible manner.For example, a strict implementation of the uniform design would begin with themeasuring and recording of all dbh’s in the plot. On a separate tally sheet the trees wouldbe transcribed from the original sheet, in order of ascending dbh size. The trees wouldthen be divided into strata, and trees selected for sampling from the tally sheet. If damageand pathological comments are recorded while measuring dbh’s there would be no danger5.0 DiscussIon 80of selecting damaged trees as sample trees. If a data recorder or a field computer is beingused, it may be possible to program the sorting and stratifying capability into the system. Ifthe plot has been previously measured, and it is not required to remeasure heights takenpreviously (or, if tree heights have not yet been taken), the stratification could beperformed in the office using the dbh’s from the previous measurement.This application does have some drawbacks. Mainly, it would add to the time and cost ofthe sample, especially in plots with a large number of trees. It may be extremely difficultand not at all cost effective to sort by dbh in the field if there are a large number of treesand the dbh range is small. In many cases, this application may not be practical unlessmeasurements have been recorded on a field computer or data recorder with sorting andstratifying capabilities, or, if the stratification can be performed in the office based onprevious dbh measurements.The advantage of a strict application like this is the ability to select a veiy uniform sample.If the effort is already being put into sorting and stratifying, it may also not add much costto increase the number of strata and further improve the sample.A flexible application would not require a formal stratification, but simply require that thefield crew make an effort to identify trees to be sampled for height while measuring dbh’sand noting these trees on the tally sheet or data recorder. If the target number of samplesis 15 trees, the crew would make an effort to mark 5 suitable trees that, in their estimation,5.0 Discussion 81fall into each of the three classes of small, medium and large. This need not be carried outin a strict sense, as long as an effort is made to get a reasonably uniform distribution. Aftermeasuring all dbh’s, the crew could examine their selections on the field sheet todetermine if a reasonable range has been selected. If necessary, some minor adjustmentscould be made in the selections and the trees could then be sampled for height.This flexible approach has the advantage of being quick, easily implemented, and notadding significantly to the cost of a normal sample. However, it will not likely achieve theresults that a strict implementation would. Mistakes could easily be made and treesincorrectly stratified. It is quite likely that field crew experience could play a critical role inthe success of this flexible application.5.2.5 Measurement errors and costsWhile the cost of measuring tree heights is not likely to vary widely with changes in treesize, it is quite likely that the measurement of large trees will be somewhat more costlythan small trees. If trees are small enough to be measured with a height pole then the costsof measurement will be much lower.Many other factors can affect the cost of sampling. In very dense stands it may be difficultto see tree tops and bottoms, adding to the time, and therefore, the cost, of the sample.Stands of mixed species may also be more costly to measure than relatively pure stands,especially if there are indistinct canopy layers that interfere with crown visibility. Higher5.0 Discussion 82site stands will tend to have greater density and larger trees, and will therefore be moreexpensive to measure than lower site stands. Older stands will have larger trees and willexhibit greater variability than younger stands, but they may be less dense.Some of the factors that affect costs will also affect measurement error. In general, therewill be larger errors associated with measuring larger trees, especially if measurement isperformed with a cinometer. This means that old stands, and stands of higher site qualitymay require more samples than younger, lower site stands. Anything which affectsvisibility in the stand can add to errors in measurement.6M CONCLUSIONS AND RECOMMENDATIONSGiven the importance and relative high cost of tree height measurement it is desirable tosample in such a way that achieves satisfactory precision at the lowest possible cost. Thisthesis has explored several different inexpensive ways of designing a sample, andsimulated the effects of those designs at different sample sizes in fixed area plots of secondgrowth Douglas-fir. The results of this study indicate that the uniform design is a gooddesign for estimating tree height, and that it shows improvement over purely randomsampling. The problems which became evident in volume estimation were likely caused inpart by the shape of the curve. Much of this could likely be alleviated either by choosing amodel which does not have a peak, or by ensuring that sampling does not exclude thelargest trees in the plot, and that the remaining samples are not concentrated in smalldiameter classes. If one of the objectives of the sample is to determine site index, it islikely that one or two of the largest diameter trees will be sampled. The simple addition ofthese largest dbh trees would likely improve the uniform sample by preventing the curvefrom reaching its maximum too early, and would therefore minimize the level ofunderestimation in large trees.In this study it was found that there was often a diminishing benefit to using sample sizeslarger than 16 for both height estimation and volume estimation for young plots (from 10to 50 years old) of Douglas-fir. If plots are young and relatively uniform, sampling morethan 16 trees will probably be wasteful unless precision requirement are high. For plots836.0 Conclusions and Recommendations 84older than 50 years, the benefits of sampling more trees was greater because the variationwas larger than in the younger plots, and it probably would be desirable to sample closerto 20 trees. Given these results, the current B.C. ministry of forests recommendation of 15trees plus top height trees is likely sufficient for Douglas-fir.Deciding on an appropriate sample size in practice will have to include the costs ofsampling. Since a formal stratification may be expensive to implement, a flexibleimplementation of the uniform design with a sample size of 15 plus 1 or 2 top height treesshould give good results in both height and volume estimation at reasonable cost.Given the pressures to manage forests more intensively, it is essential that sampling of anykind be as efficient as possible. This study has suggested ways in which height samplingcould be carried out in a more efficient manner than the commonly applied random sampleand quantified the impact of different designs. It would be desirable to further this workwith a larger range of age and site index classes, and to look at other tree species whosecharacteristics and sampling requirements may be different than those of Douglas-fir. Itwould also be useful to explore the benefits and costs of using more than 3 strata as thiswould almost certainly improve the precision of the results. Given the rising pressures anddemands on forest management, the issue of efficiency in sampling will be very costly toignore.7.0 LITERATURE CITEDAlexander, J. L. 1945. A mathematical method for construction of diameter total heightcurves. B. C. Forest Service, File Report. 3 pp.Arney, 3. D. 1985. A modeling strategy for the growth projection of managed stands. Can.J. For. Res. 15: 5 11-518.Atkinson, A.C. 1982. Developments in the design of experiments. International StatisticalReview. 50: 161-177.Baskerville, G. L. 1972. Use of logarithmic regression in the estimation of plant biomass.Can. 3. For. 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Minimum Standards for theestablishment and remeasurement of permanent sample piots in British Columbia.Secretariat, Forest Productivity Councils of British Columbia, Inventory Br. Mm. ofForests, Victoria. 27 pp.Furnival, G. M. 1961. An index for comparing equations used in constructing volumetables. For. Sci. 7: 337-341.Johnson, R. R. and F. Romero. 1991. Estimating tree heights: summary of techniques andapplications. Presented at the Western Mensurationists convention, Orcas Island,Wa. June 27 - 28. (Not published).Ker, 3. W. And J. H. G. Smith. 1955. Advantages of the parabolic expression of height-diameter relationships. For. Chron. 31: 236-246.Ker, 3. W. And J. H. G. Smith. 1957. Sampling for height-diameter relationships. J. ofFor. 55: 205-207.King, 3. E. 1966. Site index curves for Douglas-fir in the Pacific Northwest. WeyerhauserFor. Pap. No. 8.Kozak, A. 1973. Notes on the uses of conditioned regressions in forestry. 4th Conferenceof the Advanced Group of Forest Statisticians. Sect. 25, I.U.F.R.O. Vancouver,B.C. Aug. 20-24. pp. 113-125.Larsen, D.R, and D.W. Hann. 1987. Height diameter equations for 17 tree species. For.Res. Lab, School Of Forestry, Oregon State Univ. Res. Paper: No 49, 16 ppMeyer, H. A. 1940. A mathematical expression for height curves. 3. For. 38: 415-420Murphy, P. A. and R. M. Farrar, Jr. 1987. Tree height characterization in uneven-agedforest stands. Presented at the IUFRO Forest Growth Modelling And PredictionConference, Minneapolis, MN.7.0 Literature Cited 87Penner, M. 1989. Optimal design with variable cost and precision requirements. Can. 3.For. Res. 19: 1591-1597.Payendah, B. 1983. Some applications of nonlinear regression models in forestry research.For. Chron. 59: 244-248.Schreuder, H. T., H. T. Bhattachaiyya and 3. P. McClure. 1982. Towards a unifieddistribution theory for stand variables using the Sbbb distribution. Biometrics. 38:137-142.Snowdon, P. 1981. Estimation of height and diameter measurements in fertilizer trials.Aust. For. Res. 11: 223-230.Snowdon, P. 1991. A ratio estimator for bias correction in logarithmic regressions. Can. 3.For. Res. 2 1:720-724.Steinberg, D. M. andW. G. Hunter 1984. Experimental design: review and comment.Technometrics 26:71-97.Stiell, W. M. 1965. Height sampling in red pine and white spruce plantations. For. Chron.41: 175-181.Trorey, L. G. 1932. A mathematical method for the construction of diameter height curvesbased on site. For. Chron. 8: 121-132.Van Deusen, P. C. and T. B. Lynch. 1987. Efficient unbiased tree-volume estimation For.Sci. 33:583-590.Wang, C. H. and D. W. Hann. 1988. Height-diameter equations for sixteen tree species inthe central western Wilamette Valley of Oregon. Forest Research Laboratory,College Of Forestry, Oregon State University, Corvallis, OR. Research Paper 51.7p.Watts, S. B. 1983. (ecL). Forestry handbook for British Columbia - Fourth Edition. TheForestry Undergraduate Society, Faculty of Forestry, University of BritishColumbia, 611 p.Weisberg, Sanford. 1985. Applied linear regression. John Wiley & Sons, Inc. 324 pp.West, P. W. 1979. Estimation of height, bark thickness and plot volume in regrowtheucalypt forest. Aust. For. Res. 9: 295-308.Wilkinson, L. 1989. SYSTAT: The system for statistics. Evanston, IL: SYSTAT, Inc. 638pp.7.0 Literature Cited 88Wykoff, W. R., N. L. Crookston, and A. R. Stage. 1982. User1sguide to the StandPrognosis Model. USDA-For. Ser., Intermountain Forest And Range ExperimentStation, Ogden, UT. General Technical Report TNT-133. 112 pp.Zakrzewski, W. T. and L E. Bella. 1988. Two new height models for volume estimation oflodgepole pine stands. Can. 3. For. Res. 18: 195-201.Ziegel, E.R. 1984. Discussion of Steinberg and Hunter. 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In each table, samplerefers to the sampling distribution method where:E = Extreme designL = Large designR = Random designS = Small designU = uniform designSize refers to the sample size while the remaining column headings refer to the age - siteindex classes.Table 18. Average mean deviation in heightSize Sample AGE1SI2 AOE1SI3 AGE2SI1 AGE2SI2 ACE2SI3 AGE3SI1 AGE3SI28 E 0.02132 0.17692 0.11477 —0.13642 —0.04609 0.41061 0.028618 L 0.12576 0.09416 0.07129 0.09620 —0.15591 0.29823 —0.060998 R 0.17445 0.09512 0.06793 -0.06120 —0.00084 0.17595 0.118688 S 0.03963 —0.05761 0.09971 0.01804 0.01487 0.25017 0.266768 U 0.14397 0.18552 0.07852 0.04839 0.11634 0.25503 —0.0061412 E —0.00358 0.15712 0.34184 0.13520 —0.05020 0.18857 0.0397812 L 0.11185 0.01921 -0.02911 0.13351 0.07496 0.24263 0.0490012 R 0.07390 0.06101 0.09326 0.12986 —0.10990 0.13810 0.2045412 S 0.01270 0.05692 0.16562 0.20670 —0.12986 0.25034 0.2352212 U 0.07002 —0.01508 —0.07659 0.04545 —0.06065 0.30031 0.0685016 E -0.03497 0.11398 0.18141 0.09119 -0.23597 0.07641 —0.1088416 L —0.00183 0.11256 0.08652 —0.09369 —0.00227 0.02321 —0.2290516 R 0.09427 —0.00004 0.00661 0.02000 0.10101 0.15858 0.2636916 S 0.08559 0.02887 0.05617 —0.06769 0.00766 0.13197 0.2252316 U —0.03398 0.05432 —0.05890 0.12188 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N 0 1 C) L1 H H C) I cI H w C) I.3 0) H I C) L’3 0) H C) b3 0) H U C) 0) U 0) H C) 0) U Cl) H bJ0 0 0Cl)a)a)a)a)a)0 Cl)(CtH 0CDCDCDCDCDCDCDCDC)CDC)CDCl)CDCDC)CDC’CDHN)N)ClN)N)01b301HN)(C)ClHCDCDCDCDCDCDDC)CDCDCDCDCDCDHCDCDCDCDH01(C(0.011.)u01N)(CHN)????CDCDCDCDCDCDCDCDCDCDCDCDWHN)CDCDHHCla).(CN)Ha)CD0)a)HClCDCDCDCDCDCDCDCDCDCDCDCD£HHN)CDCDHHN)a)a)WJClJa)—)ClCDCDCDCDCDCDCDCDCDCDCDCDCl)H’H(C)CDWHH01-)N)WHHuN)J.—)N)a)‘CDCDCDCDCDCDCDCDHCDCDHCl)(C(C)a)Cl1TIH(C),(C)ClN)HCla)N)N)(0-J?D:)?CDCDCDCDCDCDCDCDHCDCDCD(C)(C)HCDN)ClHa)HJCD(C)C)C I C) II0)N)N)N)N)N)N)N)N)N)N)N)N)N)N)N)HHHHHHHHHHa)a)a)a)a)CDCDCDCDCDClClClClClN)N)N)N)N)N 00101010101(TICl01010101JClCl01(TIClClClCl0101(C-Ja)3(CC)DJ(TI(ON)wj,.O1ClN)a)-JW(0ClJ0)H’a)(CHN)a)a)HCD01a)N)HWW(C)ClN)(C-JCD-.J(C(C)(CCDCD(CCl-JC))(CN)N)H0101HHN)Cl(C)(0(C).Ja)N)a)i,(CCD(C(C(C)ClHN)HCDCl(C01-J‘-.J-JClHCDa)N)Cl(C)(CHN),(0a)a)N)ClCD(C)H(CCDH0)N)CDCD(C)H(C)ClNJCDCD(CCl(C)(C)CD01(C)(Ila)N).3a)IIIIIIIIIIIIIIIIIIIIICl01010101-J0101(7)01(71-J-J(C0)01Cla)a)(7)ClCl-J01(0a)(Ca)(C—JHa)CDJ(7)-N)-)a)(C)N)-)0(CN)(0(C-J(C--JN)01(C)-(CI(CCDN)(CHCD(7)Cla)(7)010101Cl(C)H(001CD(C)(C(C)-CD01-J01HCDCDN)-Ja)H01N)CDa)01(C)(C01(CHN)CDCl,-.301-.3-01(C)ClCl.JCD-0)(C-J(C(CN)ClClH-.JCl01(C(C)(C)a)N)(C).JCDCDCl-.301a)w(Ca)Cl(71(C)CDCl(C)N)CDCD-.JClClCl(C(C)(C)Cl-.3-J01ClH(Ca)N)HIIIIIIIIIIIIIIIIIIIIIIIIIIII01(C)(C)(C)(C)Cl01W(C)(71(C)-.J-.3-J(31(CCla)0)-J01a).JN)(C)0)-.3(CN)a)01(71(CCDI-’-JN)N)(0(C)-.3H-.3(C)(C)HCl(7)CD-)(C)a)CD01(CN)Cl(71Cl-)-.3CD01(0Cl(71HHCD01ClN)(CCDa)CDa)a)CDN)-.3(C)01CDCDiWN)(Ca)N)-.3(C-.3H-3(C(C(C(CCDHCD.ClCl-3CD(71-3a)a)CDN)01(CClCDa)a)ClCD01(C(C)N)(C)Cl‘.CDClH01(71H(0a)(71CD(C),.(C(001H(C)(C)(C)CD01(CHCD--CDN)IIIIIIIIIIIIIIIIIIIIIIIa)Cla)a)(71a)a)a)(C(Ca)(Ca)a)a)a)(0a)(Ca)(C(Ca)(0a)(C(0(0(C(CCl-ClCl01ClHN)0101a)N)HN)(CH01CD(C)(TI010101(C)NJ-.3CDWClCl)N)N)(C)-1CDCD(7)N)ClN)HN)(CN)(C(CH(C.a)(C)CD(Ca)a)(C)ClHa)H01CDa)N)a)N)(Ca)H(C)(CH-.34).Cl—I-301CDHa)(C)-)N)-3(C)H1.31a)ClHCD-.3-.3(71(C)(CCDHN)HN)CD-01(C)(C(C)(C)01(TICla)H(C)-.3IIIIIIIIIIIIIIIIIIIIIIII-.3ClCl(C-.3--(C(C-)0)(C-.3a)a)-.3--.30)-)a)3-‘.JCl0)(C(C0)01HN)-JCDC1HWCl(C(Cl-.3(7)a)Cla)Cl(CClClHCDWN)a)a)4a)H(C(C(C)N),01(C)a)H(31,-a)(CCDa)a)-30101HCi)a)a)ClCDCD(Cl(CCla)3(C)01-.3(51Cl(C(C(CHN)(Ca)(C(31(C-)CDH(C)CD0101Cla)N)N)-.3CDH(C)HClCl(Ca)-N)ClH(CCDa)a)(C)(C)HwH-.3N)CDCl(CN)-.3CD4).-.34).(CN)(Ca)01CDCDCD-(C)ClCDCDCl—3CDIIIIIIIIIIIIIIIIIIIIIIIII-.3-)-3.(TI-.3-)a)Ja)a)(Ca)—.3a)(C(Ca)a)a)Cl(Ca)(C(C(Ca)(CC)N)(CIN)-CDClCD(11CDHClH-.3HN)010)a)0)—.3WN)(CCl-.3(C(TI01CD(C)-.3HN)01N)01CDN)401HN)a)ClCD(C(CHN)Cl(C,,a)Ha)a)Cl(C)(CN)N)4H01Cla)-)(C)a)a)-3Ha)N)H01a)a)CDClClN)Ha)01CDClCDH(C(CCD01N)01CDCl(CCDCD(Ca)ClN)—3-3(C)01N)H(CWIIIIIIIIIIIIIIIIIIIIIIII(CCla)a)(71(CCl(C(CCl(C(C(C(C(C(Ca)(C(Ca)(C(C(C(Ca)(C(Ca)(0(CN)-.Ja)01ClWClN)a)CD(0ClClClN)01(CCD(0W(0Cla)CD01a)CD(C)01Cl-.3(CCl0101-.3CDa)(C-(C)-.301N)(CClHN)ClH-.3CI)CD01a)-)Wa)0101H-01(C)4HN)(CCDCDWCD(C)N)-.3(C(C)01H(C)(C)a)HHa)(C)N)H(CH-.3(Ca)ClClCl(C)H-JiCDCD4).-.J-.301N).a)CDN)(C)(C)01(C)(CN)(C)HCD01(CN)ClCl(C)Ha)CDCliCl-3ClClH (TIa)(C H Cl(C CD (11H 01 (C)a)(C a)(C)-J —I:1).(C)CD (C N.)CD a)(C (C H -J H a)4).Appendixi Analysis ofVariance 98Size Sample AGE1SI2 AGE1SI3 ACE2SI1 AGE2SI2 AGE2SI3 AGE3SI1 AGE3SI212 L 0.00026 —0.00046 0.00030 0.00243 0.00144 0.00774 0.0008912 R 0.00005 0.00081 0.00184 0.00186 —0.00396 0.00633 0.0012312 S 0.00093 0.00032 0.00186 0.00350 —0.00372 0.01224 0.0044712 U 0.00054 —0.00074 —0.00022 0.00145 —0.00171 0.01109 0.0054416 E 0.00019 0.00084 0.00154 0.00122 —0.00656 0.00415 -0.0008816 L —0.00019 0.00090 0.00137 0.00004 —0.00061 0.00362 —0.0051416 R 0.00069 —0.00016 —0.00021 0.00173 0.00032 0.00527 0.0061416 S 0.00039 0.00017 0.00092 —0.00131 0.00013 0.00486 0.0035716 U —0.00052 0.00042 0.00017 0.00336 0.00158 0.00160 —0.0014820 E —0.00036 0.00142 0.00149 0.00353 —0.00591 0.00227 0.0000920 L —0.00008 0.00001 0.00060 0.00287 —0.00339 0.00339 —0.0015620 R 0.00010 0.00049 0.00115 —0.00010 0.00140 0.00114 —0.0043220 S 0.00021 0.00021 -0.00029 —0.00056 —0.00029 0.00585 —0.0068420 U 0.00060 —0.00007 —0.00014 —0.00023 0.00021 0.00300 0.0021724 E —0.00065 0.00099 0.00151 0.00226 —0.00120 0.00563 —0.0035124 L 0.00030 —0.00026 0.00022 —0.00044 0.00050 0.00178 0.0007424 R 0.00037 0.00004 0.00035 0.00151 0.00160 0.00377 —0.0000624 S 0.00004 0.00106 —0.00008 0.00025 —0.00083 0.00535 0.0023824 U 0.00041 0.00030 0.00028 0.00086 —0.00253 0.00349 —0.0007028 E —0.00042 0.00053 0.00034 0.00254 —0.00102 0.00394 0.0016928 L —0.00001 0.00036 —0.00006 0.00055 0.00031 0.00174 —0.0012428 R —0.00029 0.00067 0.00072 —0.00079 0.00155 0.00214 —0.0021628 S 0.00009 0.00017 0.00064 0.00223 0.00333 0.00573 —0.0013928 U 0.00032 0.00028 0.00027 0.00033 0.00018 —0.00040 0.00113Table 26. Average maximum deviation in volumeSize Sample AGE1SI2 AGE1SI3 ?GE2SI1 AGE2SI2 PGE2SI3 ?.G3SI1 ?.GE3SI28 E 0.02571 0.04140 0.04679 0.14752 0.09523 0.18562 0.190248 L 0.01933 0.02571 0.04727 0.09132 0.09960 0.13850 0.205968 R 0.02836 0.02647 0.08072 0.11989 0.10730 0.15046 0.158018 S 0.04719 0.03040 0.08016 0.14691 0.12063 0.19176 0.247378 U 0.02326 0.03003 0.06105 0.15343 0.12129 0.17698 0.1603112 E 0.01443 0.02616 0.06095 0.10123 0.07746 0.11467 0.1736712 L 0.01796 0.02271 0.04056 0.11807 0.10768 0.14211 0.1429512 R 0.02337 0.04633 0.07219 0.13333 0.07773 0.12714 0.1722512 S 0.05031 0.02629 0.06470 0.13611 0.07596 0.20330 0.1699012 U 0.02359 0.02605 0.04961 0.11889 0.10482 0.20920 0.1788016 E 0.03436 0.02296 0.03717 0.09165 0.05352 0.09122 0.1717716 L 0.01870 0.03026 0.04431 0.08702 0.07422 0.08940 0.1287616 R 0.02882 0.02562 0.03896 0.14650 0.09755 0.10964 0.1835716 S 0.02754 0.02426 0.06374 0.11394 0.11664 0.13247 0.1688016 U 0.01446 0.03245 0.04877 0.12064 0.07615 0.09677 0.1446720 E 0.03178 0.02292 0.03488 0.10061 0.05547 0.06651 0.1358520 L 0.01447 0.02103 0.03867 0.08601 0.04917 0.06507 0.1148820 R 0.01389 0.02925 0.03745 0.07936 0.09066 0.09622 0.14418Appendixi Analysis ofVariance 99Size Sample AGE1SI2 AGE1SI3 AGE2SII. AGE2SI2 AGE2SI3 AGE3SI1 AGE3SI220 S 0.02495 0.01904 0.04866 0.12868 0.08023 0.11656 0.1342120 U 0.02055 0.02042 0.04001 0.09471 0.08062 0.10780 0.1528524 E 0.02330 0.02214 0.03962 0.05701 0.05059 0.08496 0.1002124 L 0.01873 0.01713 0.03494 0.04815 0.05865 0.06857 0.0899224 R 0.01666 0.03259 0.04798 0.08355 0.07813 0.11150 0.1039824 S 0.01923 0.03807 0.05373 0.09332 0.09712 0.12715 0.1314224 U 0.01929 0.02251 0.03763 0.08916 0.06862 0.08546 0.1143628 E 0.01114 0.01772 0.02123 0.05663 0.04026 0.06792 0.0703328 L 0.01250 0.01708 0.01927 0.06993 0.03366 0.05629 0.0336928 R 0.01337 0.01977 0.02970 0.05399 0.08651 0.07473 0.0378728 S 0.01956 0.02334 0.03870 0.07122 0.10636 0.11142 0.0719628 U 0.01740 0.02036 0.03169 0.08881 0.05591 0.06592 0.09439Table 27. Average minimum deviation in volumeSize Saniple AGE1SX2 ACE1SI3 AGE2SI1 AG2SI2 AGE2SI3 ACE3SI1 AQR3SI28 E —0.03069 —0.03198 —0.05653 —0.09615 —0.12974 —0.15032 —0.201758 L —0.03400 —0.04192 —0.06067 —0.11370 —0.12510 —0.20369 —0.220058 R —0.03880 —0.05533 —0.05704 —0.12066 —0.12231 —0.15778 —0.237538 S —0.03525 —0.04687 —0.04836 —0.08155 —0.15519 —0.18110 —0.240698 U —0.03386 —0.03448 —0.05834 —0.12970 —0.12020 —0.15060 —0.2155612 E —0.02505 —0.02933 -0.03466 —0.09290 —0.11699 -0.14591 —0.1981212 L —0.02451 —0.04671 —0.04095 —0.09440 —0.10585 —0.12238 —0.1620712 R —0.03176 —0.03380 —0.03472 —0.10113 —0.13663 —0.10285 —0.2045012 S —0.02551 —0.03411 —0.03936 —0.09301 —0.12182 —0.14799 —0.2116012 U —0.02401 —0.03930 —0.05142 —0.12788 —0.16877 —0.13652 —0.1734416 E —0.01932 —0.02989 —0.03546 —0.09000 —0.13094 —0.14955 —0.1702616 L —0.02133 —0.02620 —0.03438 —0.08945 —0.11025 —0.07846 —0.1831116 R —0.02403 —0.03542 —0.04924 —0.09463 —0.13330 —0.11751 —0.1895916 S —0.02596 —0.02999 —0.06306 —0.14973 —0.12138 —0.17023 —0.1967416 U —0.02830 —0.03486 —0.04688 —0.09054 —0.08911 —0.16720 —0.1974020 E —0.01804 —0.02128 —0.02397 —0.06388 —0.11445 —0.11105 —0.1431220 L —0.02038 —0.03690 —0.02669 —0.05125 —0.09637 —0.06300 —0.1361520 R —0.02207 —0.02606 —0.03160 —0.08304 —0.09258 —0.08299 —0.1856620 S —0.01763 —0.02673 —0.04325 —0.11919 —0.16300 —0.15996 —0.2167620 U —0.02229 —0.02951 —0.04111 —0.09893 —0.11333 —0.08600 —0.1414024 E —0.01665 —0.02449 —0.02001 —0.09330 —0.08423 —0.05359 —0.1463924 L —0.01319 —0.02353 —0.02319 —0.07361 —0.07588 —0.08656 —0.1116924 R —0.02228 —0.02552 —0.03902 —0.06830 —0.08579 —0.09261 —0.1377724 5 —0.02231 —0.02462 —0.04323 —0.08280 —0.09922 —0.14150 —0.1472824 U —0.01686 —0.02564 —0.02702 —0.07662 —0.12031 —0.06441 —0.1280028 E —0.01568 —0.03062 —0.01925 —0.05205 —0.04640 —0.04108 —0.0450028 L —0.01621 -0.02049 —0.02330 —0.07914 —0.03960 —0.03939 —0.0569728 R —0.01955 —0.02146 —0.01540 —0.08425 —0.04845 —0.05592 —0.1206528 S —0.01527 —0.02289 —0.02686 —0.05898 —0.04289 —0.08141 -0.1038028 U —0.01620 —0.02054 —0.03041 —0.06480 -0.08073 -0.12109 —0.07688Appendz:xl Analysis ofVariance 100Table 28. Average mean absolute deviation in volumeSize Sample AGE1SI2 AGE1SI3 AGR2SI1 ACE2SI2 ACE2SI3 AGE3SI1 AGE3SI28 Extreme 0.00759 0.00870 0.01031 0.02167 0.02930 0.02718 0.045528 Large 0.00627 0.00818 0.00909 0.01852 0.02747 0.02877 0.045028 Random 0.00809 0.00861 0.01262 0.02469 0.02938 0.03522 0.049148 Small 0.00623 0.00847 0.01388 0.02337 0.03301 0.03204 0.060698 Uniform 0.00501 0.00986 0.00914 0.02116 0.02979 0.02696 0.0441712 Extreme 0.00424 0.00660 0.00805 0.01635 0.02276 0.02243 0.0363012 Large 0.00470 0.00700 0.00707 0.01661 0.02240 0.01923 0.0352212 Random 0.00454 0.00955 0.00912 0.01794 0.02391 0.02544 0.0383412 Small 0.00453 0.00763 0.00969 0.02125 0.02468 0.02494 0.0527612 Uniform 0.00429 0.00768 0.00816 0.01590 0.02458 0.02190 0.0379716 Extreme 0.00351 0.00526 0.00588 0.01275 0.01598 0.01974 0.0298016 Large 0.00306 0.00552 0.00596 0.01206 0.01615 0.01414 0.0225216 Random 0.00343 0.00560 0.00729 0.01358 0.01875 0.02079 0.0301916 Small 0.00439 0.00699 0.00779 0.01668 0.02136 0.02056 0.0354816 Uniform 0.00402 0.00675 0.00671 0.01523 0.01911 0.01849 0.0287920 Extreme 0.00299 0.00480 0.00472 0.01153 0.01425 0.01588 0.0218820 Large 0.00252 0.00508 0.00393 0.01009 0.01270 0.01131 0.0188820 Random 0.00285 0.00511 0.00668 0.01196 0.01443 0.01659 0.0220720 Small 0.00351 0.00540 0.00647 0.01433 0.01648 0.01904 0.0305320 Uniform 0.00288 0.00497 0.00575 0.01213 0.01526 0.01333 0.0240724 Extreme 0.00241 0.00424 0.00368 0.00865 0.01007 0.01096 0.0138624 Large 0.00200 0.00393 0.00327 0.00692 0.00867 0.00789 0.0112224 Random 0.00253 0.00525 0.00413 0.01093 0.00965 0.01237 0.0163424 Small 0.00292 0.00453 0.00482 0.01178 0.01389 0.01500 0.0212824 Uniform 0.00244 0.00458 0.00433 0.01070 0.01140 0.01181 0.0183028 Extreme 0.00226 0.00354 0.00303 0.00652 0.00722 0.00916 0.0081828 Large 0.00156 0.00321 0.00259 0.00479 0.00540 0.00682 0.0077328 Random 0.00224 0.00358 0.00291 0.00758 0.00703 0.00900 0.0104028 Small 0.00237 0.00375 0.00389 0.00856 0.00822 0.00984 0.0118028 Uniform 0.00178 0.00346 0.00352 0.00757 0.00680 0.00761 0.01102Table 29. Average standard deviation of volume differencesSize Sample AGE1SI2 AGE1SI3 AGE2SI1 AGE2SI2 AGE2SI3 ACE3SI1 AGE3SI28 E 0.00903 0.01216 0.01608 0.03825 0.04213 0.06062 0.072818 L 0.00919 0.01097 0.01753 0.03407 0.04149 0.05919 0.078758 R 0.01151 0.01252 0.02232 0.03782 0.04160 0.05643 0.077948 5 0.01357 0.01291 0.02332 0.04021 0.04870 0.06625 0.095958 U 0.00978 0.01117 0.01950 0.04285 0.04573 0.05449 0.0717512 E 0.00650 0.00898 0.01444 0.03100 0.03586 0.04481 0.0645612 L 0.00672 0.01105 0.01215 0.03159 0.03725 0.04221 0.0575112 R 0.00863 0.01199 0.01687 0.03600 0.03895 0.04008 0.06675c)‘-4--4CLOOC)(N.)C)C)m(N,-LOCONm(NHmLONC)C)Nm(NCO‘)C)LflC)C)C)NLOLOLflLt)LOLOLOIf)LOIf)mr(N(N(NC)C)C)C)0C)0C)C)C)C)C)C)0C)0C)C)C)0C)CC)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)NIf)C)C)N(NC)C’COIf)mqLOC)COC)tOriC)If)C)COCOtoCOC)NCOtOC)LD(N(NNCOC)C)NtOHtOC)NNC)NI’)C)rIf)(NI(NCOC)C)V)(NCtom(N(N(N(N(Nm(N,-i,-,-imC)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)COmNCO-1NtON))C)toN0)mN-40)N0)COCOC)COCOr)tO(NNNC))mNmto-imN‘toHCOtO(NC)NC)C)CO10C)COC),-1C)toC)C)(NC)COC)COc-cqm(N(N(Nm(N(N(N‘,m—lrH,4(N,-C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)CC)CC)C)C)C)tOc)toC)CONtom,-1,-C)10toto(NtoCOCO,—4COtototoCOLOC)NC)N(Nto,-LOC)(NCOHtototoIf)N,toC)NC)C)tomtoItoC)C)O-lC1(N()m(N(N(N0)(N(N(N(N(N,-l-1,-(N0C)0C)CC)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)CC)C)C)C)C)0C)0C)C)C)(N(N0)CO(NtoC)COCOtOtO0),-0)CO(Nto0)CO‘N,-—4C)C)tON0)tON(N(NCOC)C)toIf)to(NH,-l,-0)NCOC),-(NCON-10)C)If)tOtOCOCOCOC)C)C)C)C)C)C)C)C)C)C)C)C0C)C)C)C)C)C)C)C)C)C)C)00C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)COCO-1C)CO(N0),-Nto(NNtO(NtO(N0)(NC)C)(‘1NtOtOCOIf)NN‘toNN,-10)N0)0)COCO0)C)C)COCOC)COC)NCOCONNNtOCOC)NtOLUtoNtoC),—4C)C)C)C)‘-C)C)C)C)C)C)C)C)C)0C)C)C)C)C)NC)C)C)C)C)C)C)C)C)C)C)C)0C)C)C)CC)CC)0C)C)C)CC)CC)CC),-,—4to,-COCO0)NCO0)0)tOC)COtoC)NC),-,-1CO,-4C)N0)to0)NtOLU,-1(N0)C)CONHCOCOtoCOCOtototOU)tOtOtOtotoLU40)toCOC)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)0C)0C)C)C)C)C)C)C)C)C)C)0000C)0C)0C)C)C)C)CCCCCCC)C)CCCCCCCCCC)C)wI-)J)-(1)rCl)a,N(N(NtOtOtOtOtOC)C)C)0C)COCOCOCOCO.rl,1,-1,-4,1H-1,-1(N(N(N(N(N(N(N(N(N(N(N(N(N(N(NriCOtoNC),-1(N0)N-iCOtoN(NC)tototoC)C)CO(NIf)‘ItoqH,-toC),-1toC)NNtONC)LOH0)U)C)10)0)Nto,-toto,-4tO(NN(NC)0)toC)(NC)0)(NtONC)(NNtO(NC)C),—N(NC)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)If)(N,tONH(NNCOCOC)0)(NtoC)C)C)C)COC)N0)r-4tO0)0)C)C)tO0)tOCOC)NHC)C)NC)r-1(N,-10),-0)(Nto,-0)(NLO0)C)tOU)LUCOU)If)0)(Ntoto(NC),-IC)(NCOr-4C)CCCCCC)CC)C)C)C)C)C)C)C)C)C)C)C)C)NCOCO(NH(NC)(Nto-i0)tON0)0)(N0)C)(NC)C)C)N(NC)0)r—NN—lN(N,-0)tOH(NC)(N0)C)COC)C)C)(NtONtoHrto(N0)C)0)rtOtoCOCOC)toC)C)0C)(N(N(N(N0)(N(N,-(N(N-1,-4m0)4r,—(N,-I,-C)C)CC)C)C)C)C)CC)C)C)C)C)C)C)C)C)C)C)tOC)U)COC)C)COCOC)N(NC’)(NCONNCOtoNto(Nrq(NC)0),-lC’)(NCO(N‘,-4tor’)‘N(N—4H,-C)COIf)(N(NIf)C)C)CO-.4C),-(N,-tOtO0)C)CO0)C)tONC)0)CO,-NCOCOI’)mtoCO(NCOC)C)C)C)C)C)CC)0C)0C)C)C)C)C)C)C)C)C),-1N‘to-1N‘totoC)0)C)NNC’)CO,-C)C)(N(NNi(N-10)(NNto(NCOU)U)C)COCO0)HC’)0)N(NC)COCO(No)C’)NC)LOtoC)C)C)If)tOC)(N-1COto(NCOCOC)LI),-4(N(NC)CO(N0)(N(NNC(NHC)r1C)CC)C)C)CC)C)C)C)C)CC)C)C)CC)C)C)C)C)NNC)‘tON0)If)to,-i‘C’)C)CO(NC)(Nto0)C)C)NNN(NC’)LUU)C)C)C’)tOC)toC)C)If)HNNIC)(NN(NNC)to(NNC),-(N(NtoLI)C)C)C)0)NCOC’)NtoCOC)toC’CONtotoN‘‘C)C)C’)C)C)C)(NCC)(NC)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)C)NrHC)(Nr1NCOtotoC)C)C’)COIf)C)C)‘NC)NIf)(NC),-tO,-4COIf)(NIf)10CO‘toC)C)COCOHIf)tONIf)(N(NC)C)C)(NNCOC’)(NN‘(NC)N,--1NN(N(NC)rHC)(N)N0)Cr-4C)(N0(NC)-C)rHC)C)--C)C)CC)CC)C)C)CC)C)C)CC)C)C)CC)C)C)ai.-1Ic-Cl)I11C)))Cl)ICCl)Cl)(C/)COa,N(N(N(NC’)(NtototototooC)C)C)C).r-4r-1,-1r—-1,-,-,-1(N<N(N(N(NCOiILr(NOCNO0(NLflNCoHNQ(NLULO(N(NCl)cCDCDr1CDCDCDCDCDCDCDCDCDCDCD0LU(NLOr1CDLOLULUCOrimOLOLONLULU(NHH(flNCOLULULU(COLOU)O(NLOCDNCDLU(N‘—orn-icr-cCDC’CDCDCDCDCDCDCDCDCONLOmQ(N(N(N‘LOLOCD(Nr-lLOrqc’)OCOmHLONLULUCOQLUCDLUCO(N(NLULOLOmLOCOCDCDCDCDCDCDCDCDCDCDaLO(NmCONNLULUCCOLOCO(N‘COLULO,-HmLUCDN-4LUNLUN-1U)COCCOCD-4LULU(‘.-lCrnCLCDCDCDCDCDCDCDCDCDCD(N(NCO-1CO(NLU(LU‘Co-4C’O‘CDNC’,NCOHLULUc‘LU-imCDLUCD‘NLU‘LUCOCD(N(NCDCDCDCDCDCDCDCDCDCDCDCDCDLU(N‘‘LUrHOLU(NLULU‘(NOLUCD(NH(LULULf‘NU)COLUC3NmLULUNNLUCDCDc(N-lCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDLU-4UN‘(NLUO‘‘(N(NNCD‘‘NONH(NLUmCOCD(NU-4CD-lU)-1NCOCOUNCO(N0CDCDCDCDr-1CDCDCDCDCDCDCDCDCDw-I,-10)cx),-)0)Cl)0N’’’’COCOCOCOCO.(N(N(N(N(N(N(N(N(N(NU)‘COLUNLUCOCOOLUCO‘LU(N(‘)LUCDOmLUCDLU-lLUN‘LUmCOm(NC)mLULUc)-1‘-lNCDmNCDNmOOLULUCOLU(NLU(N‘O‘‘H0NCO(NCOUOLU‘(NNCDOLUNCCONm‘LULUCONC)(NmU)LUNCDCDNmqN(N(NcmLULU(N-4HCDNNcrqc’N‘Nm0CDCDCD-1CDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCD.IIIIIIIIIIIIIIIIIIIIIIIIIIICDCDLULU(NCDNCDr-l(NmLUCOCOCD‘CD-1(NCOLULUCO(NCOLU-4(NCOCOCDCDLU-4m‘COCOCDLUCOLUNN(N(NNNNmCO‘CO(NCONCDHCD(NCO(NCO(NCD(NCOLUCDN—4CD‘mCDN()LULUm(NCOLUCOCOU)‘COCDLUICDCOLULUCD,-CD(N‘CO(NCOLUCOCDLUNCDCOCOr-NNNLUCONLUN,mLULULUcN,-I,-(NLULU0CDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDIIIIIIIIIIIIIIIIIIIIIIILUrHLUNLU(Nm‘LULU(NLU(NN-l(NCD(N‘mc)r’CO‘LUCONCOCOLUCDNLU(NCO(NmLUCDCDCOCOLUm-1COI)NCOLUICOCOCD-4CDCONNLUHCOcqNCOCON‘HLUCDCON‘(NNNCDNLUCOr-lNNNCDCOLU‘LULU(N(N-ICOCOmNr1LUNCOH(NLULULUCDLULU‘CDLU(N0CDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCD,IIIIIIIIIIIIIIIIIIIIIIIIIIINLU(NCDLU(NCOCO-I(NLULULUNCO-1CO‘COCOCDmr’CO-iNLU‘(NNLUN‘(NNCO-lLU—lNmLUqm—ICOLUNCD(NILUNNCOLULUHH(N1CO‘C’)LUCOLU-lI”r-CDr-ICDmLUCO(NCOCD‘‘‘(‘H(N(NLUNLULUCOLULUCONCONCOLUCD(NCO(NNCDN‘(Nm(NCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDIIIIIIIIIIIIIIIIIIIIIIIIIIIIIr-LUHILUN‘‘COCDCD‘LU(NCO‘(NLU‘LUCOCDCONLUCO‘(NCON‘1I(NCOHr-rHCOCD(N(N(NCOLUCOLULUCDCDCOCOCO(N‘1i‘CO()NCDNCDCO‘CO—lCOCOCO(NCOCDCDCD(N‘NNCDCDCD(NNCDCONCOCOCOCDr-CDmLULULUCO-ICO(NHNCD‘LUmCONr-(N1CDIrH14CDCDCDrCDCDCDCDCDØCDCDCDC)CDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDC)‘IIIIIIIIIIIIIIIIIIIIIIIINLUCO(N‘()mCOCOCDN-l‘N-lLULUCD-ICD(NLUCO‘COCOCOmLUNCO-ICOLUCOCOLUCDLU‘COCONCOCOC’)COC’)C’)NCOC’)CD‘(NLUHLUNCDH(NCONLULUCDrHLULUCOCO-4(N1LUHCOl‘(N(N‘‘U)(NmC’NCDN-1CON‘COCOmC’)COLU,-4CONCOCONCOCOLUmNC3COC’)(Ns—I-1C’sHs-Is-Is-ICDs-IsH-IC)(Ns-ICDs-.Is-ICDC’s-Is-ICDCD0CDCDCD00CDCDCDCDCDCDCDCDCDCDCDCDCDCD0C)0CDCDCDCDCDCD00‘IIIIIIIIIIIIIIIIIIIIIII‘mNCDV‘LUCDLULU(N,-ILUC’)NCDLUs-ICONmLUmNLUs-INCOCOCOLUs-I‘(NCDILUs-Is-ICOCDCDLUCDC’)s-Is-ICDC’)(NLUCDLUCOC’)‘LU(NCOH(Ns-INLUCOCOCON(NLU(N‘-Ic’)C’)(NLU‘NNLUCOLULULUNLULU‘U)LUNCOCOLULUCO‘LUNLUNLUNNLUNLULUN5x4‘LULULU‘LULU‘LUCDCDrCDCDs-ICDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCDCD0CDCDCDCD0CDCDCDCDCDCDCD00CDCDCDCDCDCD00CDCDCDCDCDCDCDCDuIIIIIIIIIIIIIIIIIIII0Icx),0)x),)Cl)0),Cl)0),-U)0),-U)0),-1C!)0N(N(N(N(N(NLULULULULUCDCDCDCDCD‘‘‘‘‘COCOCOCOCO,fs-Is-Is-Is-Iv-1s-Is-Is-Is-Is-I(N(N(N(N(N(N(N(N(N(N(N(N(N(N(NCOAppendixi Analysis ofVariance 103Tables 34 to 39 show sample design rankings for each age-site class and sample size. Thefollowing criteria is ranked:the average mean deviation in height (MDHT);the average of the largest positive mean deviations in height (MDHTMAX);the average of the largest negative mean deviations in height (MDHTMIN);the average mean absolute deviation in heightthe average mean deviation in tree volume (MDVOL);the average of the largest positive mean deviations in volume (MDVOLMAX);the average mean absolute deviation in volume; and,the average of the largest negative mean deviations in volume (MDVOLMIN).Table 32. Sample rankings for sample size 8Class Sample MDHT MDHTKAX KDHTMIN MADHT MDVOL MDVOLAX MDVOLMXN MADVOLAGE1SI2 Extreme 1 1 3 3 4 3 2 4Large 3 3 2 4 2 1 4 3Random 5 5 1 5 3 4 5 5Small 2 2 4 1 5 5 3 2Uniform 4 4 5 2 1 2 1 1AGE1SI3 Extreme 4 2 1 4 5 5 2 4Large 2 4 2 3 2 1 4 1Random 3 3 5 5 1 2 5 3Small 1 1 4 1 3 4 3 2Uniform 5 5 3 2 4 3 1 5AGE2SI1 Extreme 5 1 3 4 3 1 4 3Large 2 3 4 1 1 2 1 1Random 1 4 2 3 2 5 3 4Small 4 5 1 5 5 4 5 5Uniform 3 2 5 2 4 3 2 2AGE2SI2 Extreme 5 3 5 3 3 4 5 3Large 4 2 3 4 1 1 1 1Random 3 1 2 5 2 2 4 5Small 1 5 1 1 5 3 3 4Uniform 2 4 4 2 4 5 2 2AGE2SI3 Extreme 3 1 4 1 2 1 1 2Large 5 3 5 5 4 2 5 1I.I I I.r C)C)C)C)C)00CliCliCliCliHWS.)HCl)Cl)Cl)Cl)Cl)CI)HHHHHHH(JHt)CcCli<ClCl)ClCI)t<ClrtH-D)rrH-p)rtH-JrrH-)rtH-i)c-tDl11MiD)111MiD)11MiD)11MiD)-‘-1HiD)11HQLO(DO(DOH0LQ(DOHo((DDOCDHHW)01).JHH01t’3WtJ,WH01HW01W)01H‘-3DlH)L)01‘l4Ht)01DlWHHb.)W01WDl01t’-)HW01)DlHWH01Dl011’)l.)HiDl01Ht’-)W01W)DlH)Dl(11WHW014HH01DlWWt)Dl01HHDl01t-H01)DlWDl3H01Wb-.)H01‘-301HW)WDl01HE’J‘)01HH01WHDl)01Dl01HWD-)—————————————————————————————IWHDlH(-.J01)W01Dl)Ht’JDl01HW.JDl01HWDl01WH001H&-lDl01WDlH)01DlL.iH01(.JCOH4WCO01HH01COWz.Dl01H()Dl01WHCOH01Dl)CO(J01DlHCO.DlW01COHCOJDl01H0 lC)C)C)H(ij()WCl)Cl)HHCOHDlD)ClDlD)>Sq)H-D)Dl1CtH-DDlCtH-DDlHiH(DLO1MiHQ)OII1-hHO0HOCD(DOHOCD(DOHOH‘1SSSQSOSOSHCT)WCOWCOHDl01DlCOH3CO(71.DlbiH01biCOHDlDl01COCi01DlHCOb)01DlbiHCOHWCO1-3biDl(71HCOHbi(71DlCODl1-)‘-3bi01HCODlU)COH01U)01H< 0—————————————CO01HU)U)01COHDl01U)0HU)Dl01COH01U)DlCODlU)COH01DlCOU)HDl01U)CO401U)0 1:1Cj I Ip 00C)000OHLiLiLiLiLiLiLiC..)C..)S)C’.)HCl)Cl)Cl)Cl)Cl)Cl)CI)HHHHHHHC’)HC..)C’)HC..)C’)Ccl)rLiCCl)CLiC’MiHØCO11MiH0.10-1I-tIH0.101-1I-hH0.10tiI-tiH0.10I-IMiH(1(0IiMiH0.10II0HOCD(DOHOCD(DOHOCD(DOHOCDCDOHOCD(DOHOCD(DOHOCDCDt-ISSSSSSHI..)0’C’)C..)0’HC’.)I..)C’)HQ(71C’.)HC..)C..)C’)H1)1WC’)H01C’.)01HC..)(.)01HC’),011..)HC’)W.01C’)HtC..)(71HC’)(71HC’.)C..)HC’)C.flC..)H.01C’)C..)C..)C’)H01C’)W01HHC’.)W01C’)tC..)01H01C.)C’)1-’C)’C’)C..)H01H1.)C’)1)1C’)HC..)C’)HC.)01C’)1.)C.)’HC.)’C’.)C..)H01C’.)C.)HW(i.C’.)1J1HtHC..)01C’)C’)C.)01HH01C’)C..).HC’)C.)01(JiC..)HC’)HC..)C’)i01C.)C’)H1)1C..)C.)’C’)H< gC’.)C.)0’HC.)01HC’)C..)(31C’)H,C.)1)1HC’)C.)’C’)C.)H01C’)C..)HHC.)C’)010 <LI’i.C.)C’)H(ClC’)HC..)HC..)01C’)C.)01JHC’)C.)C.)’HC’)C..)01HC’)LilC..)C’)H> 0C)LiHC.) C/)H C’.)CtiC01—5P)Ct1-hip)tillHi0H(0(DOIiHCDC.)01C’)H01H01C’)C.)01iC..)C’)H01C’)I-I01C’)HC.)<CuC’)C..)H01<C’)01,HC.)(.)0C..)01.HC’)C’)CC) H I3I IC)H (I)H x rt ‘-1 CD CD U’ U’ U,0 U,U,w F-’C-’CD w U,H I-’I I Ix’ C)C)C)C)C)C)t,iIzitiCiCiCit1WHH(I)(I)Cl)Ci)Cl)Cl)HHHHHHHC’3HHU)HHlOI-CMHLOI-CMIHLOI-CHHIPI-CMIHC)CO1I-liHCOI-CI-tiHØC10HOCD(DOHOCD(DOHO(C)(DOHOCD(DOHOCDCDOHOCD(DOHOCDCDt-’U’WHI-JWU’H,)HWU’WD-HU’)H,U’Ht%)WU’U’H‘.3HU’WU’HJWWU’3HJ,HWU’U’HWI--)HiU’WQHWI-S)1.3HU]WI-3HU’tiC.)I-.)C.)HU’U’C.).)HU’C)C.)Hl\)C.)01HC.)HI-iU’4WU’HHI-.)C.)U’H..)U’WC.)HC’iU’,t’JU’C.)HC.)HU’C)U’L.)H1..)1.3C.)U’I-.)HC.)U’HC’.)H‘)C.)U’C’.)C.)I-’U’HC’.)C.)U’MC.)HU,U’C.’C’)H0 t.4 <U’C’.)4HC.)U’C.)HC’.)C.)U’HC’.)C.)U’HC’.)01C’.)C.)HC’)HU’C.)C.)iHC’.)U’0EN.)U’HC.)WU’C’)HiC.)U’HC’)4U’C.)HC’)U’C.)C’)HC.)C’)U’HU’HC.)C’)ii 1. CI I.> C) t’ H 11)HHwIiiI-rri.D1rtt-tH(QIIfrtHQ)II0H0IDID0H0IDIDHJ.HU’U’HJ.WWt’)HU’U’t’iC.)HU’Ht’fHWU’U’WHU.H.3HU’WHU’< 0—--——1.U’WH‘JViWH0)WHUiWHU’‘J0 H z‘iU’.HWE’JU’WH0C)C)C)C)C)C)HtihihihihiWWHIf)Cl)Cl)CliHHHHHHt\)HWHWhH0,10lMH0,1lH0,1I-IH0,101r-1H1-hH0,101i-hI-’0H0IDID0H0IDD0H0IDcD0H0IDID0H0IDID0H0IDID0HH0N.).HWUiWU’HN.)U’L..)N)HHN)U’()HN)U’HN)WViN)H3WN)U’HC..)N)HQiN)U’WHU’.C..)HN)HWN)U’HU’C..)N)C..)HC..)HN)UiN)wU,HU’C..)HN)C..)N)U’HC..).U’N.)HU’HWN)C..)N)HUiN)WI-’HC..)N)U’C..)HU’N)1.)N)U’HU’N)C..)HHU’C..)N)HN)HC..)C.),N)C..)HU’U’N)H1.)C...)HN)U’C..)HN)U’C..)U’HN)H0--------------------—-----1ViC..)HN)C..)U’,HN)C..)U’N)HU’C..)HN)1-3U’HC..)C.)U’HN.)w0N)ViC..)HN)U’C..)HU’WH1-.)C..)HN)U’C..)U’N)F-’U’C..)HN)C..)U’zU’C..)F-’N)WU’HN)C.),N)HC.)C..)U’HN)U’C..)I-’N)C..)U’HN)C..)U’0 t.iItx”x,’0000OF’tiltititIltip,wL\)tJtI(I)Ci(12(i2HHHHHI-’wF’HI-’fri-hF’IIMiF’III-hF’frMiF’P(121F’.)(iiF’Qc..)WF’UiI’.)(.)‘JF’WUiUiF’WtiW,F’b)UiF’(Tit’3W)UiWF’W.F’UibJUi(.JF’t’3(11WF’Uih)F’WUiF’.(.)WUiF’UiWF’I’)HiU,F’W)WU,F’3Ui3F’WU,t)WF’F’.)C)Ui,3F’WUi‘JF’UiWF’UiWF’WUi.)U’F’W.< 0 t. <U,itJI-’W&iU,F’WWUi,F’MUi,F’WL’JUiWF’0 c::WUi3F’UiWF’U’F’WW)UiF’U,F’WUiWF’L)U,WF’t.3UiWF’iWUi,F’)Uit’JF’W0 t.4C 00


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