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Derivation and analysis of compatible tree taper and volume estimating systems Demaerschalk, Julien Pierre 1973

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c.l DERIVATION AND ANALYSIS OF COMPATIBLE TREE TAPER AND VOLUME ESTIMATING SYSTEMS by JULIEN PIERRE DEMAERSCHALK M. F.,University of British Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Forestry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Forestry The University of British Columbia Vancouver 8, Canada extensive copying of this thesis for scholarly purposes may be Supervisor: Dr. D. D. Munro i i ABSTRACT Compatible taper and volume equations give identical estimates of total volume of trees. Two basically opposite techniques for the construction of compa-tible systems of estimating tree taper (decrease in diameter with increase in height) and volume were derived and examined statistically. In the first method compatible taper equations are derived from volume equations fitted on tree volume data.In the second method compatible volume equations are derived from taper equations fitted on tree taper data. Both systems have been tested for bias in the estimation of diameter inside bark at any height,height for any diameter,section volume and total tree volume.In addition to conventional estimates for 2 a l l trees,classes representing each fi f t h of the D H range were used. No method gave completely satisfactory results for the equations tested. However,a few equations in both systems appear to be sufficiently unbiased to be useful for many purposes. All tests were repeated on data where butt flare measurements were eliminated.Taper equations on these adjusted data showed much less bias over most of the length of the tree bole. Weighting techniques did not produce any significant improvement. Use of non-linear techniques made a small difference in some cases. Meyer's correction factor of the logarithmic volume equation was tested and found to be unnecessary. A good relationship which existed between coefficients from taper and volume equations and form is thought to be useful in certain applications. i i i TABLE OF CONTENTS Page TITLE PAGE i ABSTRACT . . . . . . . . . . . . . . . . i i TABLE OF CONTENTS . . . . . . . . . i i i LIST OF TABLES vi LIST OF FIGURES ix ACKNOWLEDGEMENTS '. x SELECTED SYMBOLS xi Chapter 1. Introduction . . . . . . . . 1 2. Literature review 5 3. Data 15 4. Volume-based systems of tree taper and volume estimation. 17 4.1. Volume equations 17 4.1.1. Fitting volume equations . . . . 17 4.1.2. Tests of total volume estimation . . . . . . . 19 4.1.3. Non-linear fitting of volume equations . . . . 29 4.1.4. Correction factor for the logarithmic equation. 31 4.1.5. Volume equations for combinations of species . 33 4.1.6. Volume equations for data adjusted for butt flare 35 4.2. Volume-based taper equations 37 4.2.1. Derivation of compatible taper equations from volume equations 37 iv Page 4.2.2. Tests of diameter estimation 45 4.2.3. Tests of section volume estimation with known heights 52 4.2.4. Tests of height estimation 56 4.2.5. Tests of section volume estimation with unknown heights 59 4.2.6. Tests of volume-based taper equations for data adjusted for butt flare 61 5. Taper-based systems of tree taper and volume estimation . 64 5.1. Taper equations 64 5.1.1. Fitting taper equations 64 5.1.2. Tests of diameter estimation 67 5.1.3. Tests of section volume estimation with known heights 73 5.1.4. Tests of height estimation 75 5.1.5. Tests of section volume estimation with unknown heights 76 5.1.6. Taper-based taper equations for data adjusted for butt flare . 77 5.2. Taper-based volume equations . 79 5.2.1. Derivation of compatible volume equations from taper equations 79 5.2.2. Tests of total volume estimation 82 5.2.3. Tests of total volume estimation for data adjusted for butt flare 84 V Page 6. Additional aspects of both systems and possible ways to improve them 86 6.1. Taper equations on data above breast height 86 6.2. Relation between coefficients and form . . . . . . . 87 6.2.1. Relation between taper-based taper equation coefficients and form 87 6.2.2. Relation between volume-based taper equation parameters and form 90 6.2.3. Relation between volume-based volume equation coefficients and form 92 6.2.4. Use of the relation between coefficients and form 93 7. Discussion,summary and suggestions 96 LITERATURE CITED . . . . . 100 APPENDIXES . 107 1. Common Names and Latin Names of the Tree Species and Species Groups . . . . . . . . . . . . 107 2. Numbering of the Volume and Taper Equations 108 3. Summary of the Volume-Based Volume Equations 109 4. Derivation of Compatible Taper Equations from Volume Equations and the Functions to Estimate Height and Section Volume 115 5. Summary of the Taper-Based Taper Equations 122 6. Derivation of Compatible Volume Equations from Taper Equations and the Functions to Estimate Diameter, Height and Section Volume 127 vi LIST OF TABLES Table Page I. Averages and Range of Data 16 II. Number of Trees and Mean Volume for All Size Classes and for All Species 20 III. Total Volume Estimation Tests of Volume-Based Linear Volume Equations 23 IV. Total Volume Estimation Tests of Volume-Based Non-Linear Volume Equations . . 30 V. Meyer's Correction Factors for the Logarithmic Volume Equation 1 32 VI. Total Volume Estimation Bias after Application of Meyer's Correction Factor . 32 VII. Total Volume Estimation Bias for Combinations of Species. 34 VIII. Total Volume Estimation Tests of Volume-Based Volume Equations for Data Adjusted for Butt Flare 36 IX. Taper Equations Derived from the Linear Volume Equations and Their Standard Errors of Estimate 40 X. Taper Equations Derived from the Non-Linear Volume Equations and Their Standard Errors of Estimate . . . . 41 XI. Taper Equation 1 Derived with Different Dependent Variables 44 XII. Example of a Diameter Estimation Test of a Volume-Based Taper Equation . . . . . . . . . . . . 46 VX1 Table Page XIII. Total Diameter Bias of Volume-Based Taper Equations for Douglas-fir and Aspen . . . . . 47 XIV. Total Diameter Bias of Equations l f c and 8** . . . . . . . 48 XV. Pattern of Diameter Bias of Equation 3*" for Cottonwood „ 52 XVI. Example of a Section Volume Estimation Test with Known Heights of a Volume-Based Taper Equation . . 54 XVII. Bias of Section Volume Estimation with Known Heights of Volume-Based Taper Equations for Douglas-fir and Aspen . . . . . . . . . . . . 55 XVIII. Section Volume Bias of Largest Size Class of Several Taper Equations for Douglas-fir . . . . . . . 55 XIX. Example of a Distance Estimation Test of a Volume-Based Taper Equation . . . 57 XX. Bias of Distance Estimation of Some Volume-Based Taper Equations . . . . . . . . . . . . . . . . . . . . . . . 58 XXI. Bias of Section Volume Estimation with Unknown Heights of a Volume-Based Taper Equation . . . . . 60 XXII. Volume-Based Taper Equations for Data Adjusted for Butt Flare and Their Standard Errors of Estimate . . . . . . 62 XXIII. Tests of Volume-Based Taper Equation l * 1 for Data Adjusted for Butt Flare . . . . . . . . . . . . . . . . 62 XXIV. Diameter Estimation Test of Taper-Based Taper Equations . 69 XXV. Comparison of Diameter Bias of Volume-Based and Taper-Based Taper Equations for Douglas-fir . . . . . . . . . 72 XXVI. Diameter Bias of Equation I Fitted with Different Dependent Variables . . . . . . 72 v i i i Table Page XXVII. Bias of Section Volume Estimation with Known Heights of Taper-Based Taper Equations 74 XXVIII. Comparison of Bias of Section Volume Estimation with Known Heights of Volume-Based and Taper-Based Taper Equations for Douglas-fir 75 XXIX. Bias of Distance Estimation of Some Taper-Based Taper Equations 76 XXX. Bias of Section Volume Estimation with Unknown Heights of a Taper-Based Taper Equation 77 XXXI. Tests of Taper-Based Taper Equations for Data Adjusted for Butt Flare 78 XXXII. Taper-Based Volume Equation Coefficients 81 XXXIII. Bias in Total Volume Estimation of Taper-Based Volume Equations 83 XXXIV. Bias in Total Volume Estimation of Taper-Based Volume Equations for Data Adjusted for Butt Flare 85 XXXV. Diameter Bias of Equations I and V for Reduced Data . . 86 XXXVI. Relation between Taper-Based Taper Equation Coefficients and Form 88 "XXXVII. Relation between Volume-Based Taper Equation Parameters and Form 91 . XXXVIII. Relation between Volume-Based Volume Equation Coefficients and Form . 94 ix LIST OF FIGURES Figure Page 1. Flowchart of the Derivation of the Equations . . . . . . 4 2. Relationships between Dependent and Independent Variables for Some Volume Equations . . . . . . . . . 22 3. Pattern of Bias in Diameter and Height Estimation for Volume-Based Taper Equations 50 4. Patterns of Bias in Diameter Estimation of Taper-Based Taper Equations 68 X ACKNOWLEDGEMENTS I first wish to express my deepest gratitude to my supervisor, Dr. D. D. Munro, for his constant help and encouragement during my studies at the University of British Columbia. Working under his guidance was a great privilege and a pleasure. I must pay a tribute to Drs. A. Kozak, S. W. Nash, M. Schulzer and J. V. Zidek who generated my enthusiasm for statistics through their marvellous lectures on the subject. I gratefully acknowledge the helpful suggestions of Drs. D. D. Munro, A. Kozak and J. H. G. Smith, their constructive comments and attentive review of the thesis. The opportunity to use tree taper data, collected by the Forest Inventory Division of the B.C. Forest Service, is appreciated. The University of British Columbia is acknowledged for the computing facilities. I am grateful for the financial support which was provided in the form of a MacMillan Bloedel Ltd. Fellowship in Forest Mensuration and a National Research Council Scholarship.Additional financial support was received from the Faculty of Forestry, University of British Columbia, in the form of teaching assistantships. xi SELECTED SYMBOLS Symbol Page Meaning first used <C 54 less thanje.g. 2 <3 y] 19 s u m °f 37 integral notation TJ 37 3.14159 F 16 Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) C 16 western redcedar (Thuja plicata Donn.) S 16 spruce (Picea glauca (Moench) Voss, P. Engelmanni Parry and P. mariana (Mill.) B.S.P.) B 16 balsam (Abies amabilis (Dougl.) Forbes and A. grandis (Dougl.) Lindl.) A 16 aspen (Populus tremuloides Michx.) Cot 16 cottonwood (Populus trichocarpa Torr. and Gray) Pl 16 lodgepole pine (Pinus contorta Dougl.) Pw 16 white pine (Pinus monticola Dougl.) D 5 diameter breast height,outside bark, in inches H 5 total height of tree,in feet V 5 total volume,inside bark,in cubic feet x i i Symbol Page Meaning first used V..V. . 19 individual volume observation / \ / \ V.,V. . 19 individual predicted volume J J-J B* 6 basal area,outside bark,of tree in square feet at 4.5 feet above the ground = 3.14159 D2/(4 (144)) D, , 7 diameter,in inches,at the small end of lb the first 16 foot log D^,. 7 diameter,in inches,at 25 feet height d 8 diameter inside bark,in inches,at any given height or distance from the tip d^ 35 diameter inside bark,in inches,at one foot height d, 8 diameter inside bark,in inches,at breast height. h 11 height above the ground,in feet 1 8 distance from the tip,in feet 1^,1^ 52 lower and upper limit of tree section expressed as distances from the tip AFQ 15 absolute form quotient = ratio of diameter at half the height between breast height and total height and diameter at breast height,all measured inside bark CFF 7 cylindrical form factor = V /(BH) x i i i Symbol a,b,c,e,f ,g bo ' ^ l ' ' *' C1'C2 p,q,...,u k log In m N R MB MB. SE Page Meaning first used 7 volume or taper equation coefficients 5 volume or taper equation coefficients 8 taper equation constants 37 free parameters of volume-based taper equations 52 4(144)/II = 183.3466 5 logarithm to base ten 8 logarithm to base e (e = 2.71828) 19 number of independent variables in a regression 19 number of observations in a given size class 19 total number of observations for a given species 18 multiple correlation coefficient 31 population variance of the dependent variable in regression analysis 19 approximated mean bias of the variable of interest for a given size class 21 approximated mean bias of the variable of interest over a l l size classes 19 approximated standard error of estimate of the variable of interest for a given size class xiv Page Meaning first used 19 approximated standard error of estimate of the variable of interest over a l l size classes 14 standard error of estimate of the variable used as dependent variable in fitting the regression equation 1 1.Introduction A taper and volume system is here defined as compatible when integration of the taper equation yields the same total volume as given by the volume equation.The most important benefit of a compatible model is that consistent results are obtained in taper and volume analyses.The user of a taper equation is confused when confronted by a taper analysis that,upon summation of the section volumes,yields a different total volume from that obtained in a volume analysis. Taper and volume data should not be considered independently but should be analysed as mathematically dependent quantities. This leads us to another advantage of compatible systems namely that appropriate models are suggested through the existing knowledge of volume models. There are two basically different techniques which can be used to obtain compatible systems of taper and volume.One technique involves fitting a taper equation on taper data and deriving from it,by integration,a volume prediction system.The other technique is more or less the opposite;a volume equation is fitted on the volume data and from i t a compatible taper equation is derived.One is a taper-based system while the other is volume-based. A good tree taper and volume estimating system should be unbiased in the prediction of diameter at any height,height for any diameter,volume of any section and total volume. 2 The objectives of this study are to: 1 . demonstrate with many examples how volume equations can be converted to taper equations and vice versa. 2. find out how equations should be tested and analysed to detect the biases involved in diameter estimation and study their effects on height,section volume and total volume predictions. 3. compare bpth techniques for several species in order to see which technique is best. A flowchart of the derivation of the equation systems is given in figure 1 . Taper and volume equations can be fitted by several statistical methods.Conventional least squares,weighted least squares and non-linear least squares procedures are a l l examined in this study. Butt flare often causes significant bias in taper and volume estimation.How much this bias can be reduced by fitting the equations on data adjusted for butt flare is investigated.Adjusting for butt flare is compared with the results obtained from ignoring the observations below breast height. Some species are apparently similar in form.The possibility of combining species of similar form is examined and the resulting loss in precision and accuracy is assessed. For some equations the use of theoretically derived correction factors has been advised to correct for bias.The usefulness of these correction factors is evaluated. Possible relationships between some equation coefficients and tree form are examined.Some practical applications are discussed. Theoretical derivations of a l l volume-based taper equations and a l l taper-based volume equations as well as the formulae for the computation of heights and section volumes are explained in detail. Figure 1. Flowchart of the Derivation of the Equations Section volumes Volume data Comparisons and tests of estimation of diameter,height, section volume and total volume I Fitting of volume equations 1 Volume equations I Derivation of Compatible volume-based taper equations 5 2. Literature review There are few topics in forestry which have been discussed for so many years and by so many authors as have taper,form and volume of trees.Extensive reviews are given in most forest mensuration books (Chapman and Meyer,1953;Loetsch and Haller,1964;Prodan,1965;Avery,1967; Husch et al.,1972). Only the contributions most important to this thesis are reviewed here.Significant aspects of some publications are discussed in more detail in later sections of the thesis. No review is made of the different tree form theories (nutritional, mechanistic,water conductive,hormonal and pipe model) and biological relationships between tree or stand characteristics and tree form are not considered here.Interesting discussions about these topics were given by Gray (1943;1944;1956),Newnham (1958),Larson (1963),Heger (1965a), Shinozaki et. al_. (1965) and Doerner (1965). Tree volume equations have been discussed in the literature since the early 19th century.Since then many different functions have been proposed. Schumacher and Hall (1933) proposed the following logarithmic volume equation: log V = b Q + b1 log D + b 2 log H This equation is tested extensively in later sections. Meyer (1944) suggested a volume-diameter ratio equation: V / D = b + b. D + b„ H + b 0 DH o 1 2 3 A similar equation was preferred by Stoate (1945): V = b + b, D2 + b 0 H + b_ o 1 2 3 2 but he argued that using only D H was almost as good as any other equation. A comprehensive comparison of volume equations was made by Spurr (1952) who decided that the combined variable volume equation: V = b + b, D2H o 1 was one of the most promising.Since then the fact that the relationship 2 between V and D H could not be expressed by a single linear regression over the fu l l range of data has been recognized often (Smith and Ker, 1957;Smith and Breadon,1964;Myers and Edminster,1972). Honer (1965) deyeloped a new volume equation : V = D2 / (b + bj H) o 1 which was fitted by a linear least squares procedure as : D2/ V = b + bj H o 1 Tarif tables,developed in Britain by Hummel (1955),which provide a "local volume table" for each particular stand,were extensively discussed by Turnbull and Hoyer (1965).They also have developed a tarif-based procedure for estimation of diameters and volumes of 16 foot logs to facilitate studies of growth and yield. Zaharov (1965) studied the linear relationship between form height and total height.This is basically the same as the equation: V/B* = b + b, H o 1 which was tested by Smith and Munro (1965) and Christie (1970). After a tr i a l of Hohenadl's method,Heger (1965b) concluded that i t was an efficient and accurate means of both stem form and stem volume estimation. 7 Newnham (1967) proposed a modification to the combined variable volume equation : V = b + b, o 1 where a and b were the coefficients of log D and log H in the logarith-mic volume equation. A suitable form factor equation to predict volume was given by Van Laar (1968): CFF = b Q + bl log D + b 2 log H + b 3 log D 2 5 From their studies on Ponderosa pine''",Hazard and Berger (1972) concluded that direct calculation by use of the optical dendrometer (Barr and Stroud,model FP-15) appears to be an improvement over volume tables. Van Laar was not the first one to include a third independent factor.Mesavage and Girard (1946) used as a third measure the ratio D^ g/D. Naslund (1947) introduced the crown ratio as a third explanatory variable in the prediction of volume,as a substitution for the form point. Ilvessalo (1947) and Van Soest (1959) preferred as a third measure the diameter at twenty feet above the ground.Van Soest concluded, however,that the diameter at thirty percent of the height was best. The same idea was shared by Pollanschutz (1966). Schmid e_t al. (1971) used the diameter outside bark at 6-9 meters as an additional variable. Not only total volume prediction but also the distribution of volume within the tree were studied. ''The tree names are given with the corresponding Latin names in Appendix 1. 8 Speidel (1957) used graphical techniques to relate the percentage of total volume to the percentage of total tree height. Log position volume tables,based on hand drawn harmonized taper curves,were developed by Fligg and Breadon (1959). Honer and Sayn-Wittgenstein (1963) stressed the need for a mathe-matical tree volume expression which would yield tree and stand volumes based on D and H (form estimates optional) for any demanded stump height and top diameter.Fulfilling this need Honer (1964;1965b) proposed three mathematical models to express the volume distribution over the tree stem.These models describe well the distribution of volume and can be used to estimate volume to any standard of utilization when applied to an estimate of total volume. ' Burkhart et al.(1971) developed a technique to predict proportions of tree volume by log positions.Separate prediction equations were fitted for each peeler log. Because the most complete information concerning the form of a tree can be given by means of a taper equation,taper curve or a taper table (Meyer,1953) many authors have concentrated their efforts on this problem.Taper equations, describing the tree profile,have been developed since the beginning of this century. Hojer (1903) was the first to propose a mathematical equation to describe the stem profile: d/d4 5 = cx In ((c 2 + l(100/H))/c2) The constants c^ and were defined for each form class. Jonson (1910;1911;1926-27) introduced a new constant into the equation of Hojer in order to obtain better results.These taper equa-tions were compiled independently of tree species.The form class* which had to be known,was usually measured or estimated by the "form point" approach.A good description of how these taper equations are constructed was given by Claughton-Wallin (1918). Jonson's "absolute form quotient" was mentioned as an excellent expression of taper or stem form (Claughton-Wallin and McVicker,1920). His investigations on many species led Behre (1923 ; 1927 ; 1935) to present a new taper equation which seemed to be more consistent with nature: d/D = (l/H)/(b + b 1/H) This equation is discussed in more detail in.later sections. Some transformations of the Behre equations were given by Bruce (1972). i Petterson (1927) suggested two separate logarithmic curves to describe the stem profile,one for the main stem and the other for the top portion.Heijbel (1928) tried a combination of three different equations for different portions of the stem. Matte (1949) described the stem profile above breast height by the function: (d/D)2 = b o(l/H) 2 + b 1(l/H) 3 +-b 2(l/H) 4 The coefficients were found to be related to D and H. Bruce and Schumacher (1950) mentioned that the best check of a taper table is a check of a volume table derived therefrom. Graphical techniques were used by Duff and Burstall (1955) to develop an integrated system of taper and volume. In order to get a prediction system in which D could be estimated from diameter measurements at different heights,Breadon (1957) fitted butt-taper equations on plotted averages. 10 The following taper equation was tested by Osumi (1959): d/D = b (1/H) + bri/H) 2 + b„(l/H)3 o - 1 — l — This equation is tested in later sections. Giurgiu (1963) proposed as a taper equation a 15th degree poly-nomial of d/D as a function of l/H. Prodan (1965) suggested the taper equation: d/D = (l/H) 2/(b + b.(l/H) + b 0(l/H) 2) — o 1 - z — Some work has been carried out on tree taper curves using multi-variate methods (Fries,1965).Fries and Matern (1965) agreed that models expressed in mathematical functions have a considerable advantage over those given only as tables or graphs.They found polynomials to be the appropriate expressions i f certain restrictions were imposed.After comparison of multivariate and other methods for analysis of tree taper,Kozak and Smith (1966) concluded that the use of simple functions, sorting and graphical methods is adequate for many uses in operations research. Kuusela (1965) used diameter at 0.1 of the height as the basic diameter in a system in which nine regression equations were used for each form factor to predict proportional diameters at different heights. According to Grosenbaugh (1954;1966),who introduced some new tree measurement concepts,polynomial analysis may rationalize observed variation in form after measurement but i t does not promise more efficient estimation procedures. An extensive study of thickness and percentage of bark (Smith and Kozak,1967) made it possible to convert outside bark measurements of form and taper to inside bark values. An integrated system of taper and volume equations for red alder 11 was provided by Bruce e_t al. (1968). This taper equation,which contained very high powers of the term l/(H - 4.5),was conditioned so as to provide a constant double bark thickness ratio at breast height.This equation is also tested later.Referring to the dependence of the taper measurements taken on the same tree,they mentioned that no widely applicable solution is available.They gave also some good reasons why prediction of the square of diameter might be preferred. After Munro (1968) had discussed the estimation of upper stem 2 diameters from a function involving D,h/H and (h/H) ,the following taper equation was proposed by Kozak e_t al. (1969a,b): (d/D)2 = b + b. (h/H) + b„ (h/H)2 o 1 l. To make the diameters inside bark zero at the top,the least squares solution was conditioned by imposing the restraint b^ + b^ + b^ = 0. For spruce and western redcedar an additional condition was necessary to prevent negative diameters near the top.These taper equations were later converted into volume equations and point sampling factors (Demaerschalk,1971a).In their forest inventory program,Kozak and Munro (personal communications) use a system of separately fitted taper and volume equations.Volumes computed from the taper equation are adjusted to the volumes given by the volume function,by application of a proportional percentage correction on each tree section. After i t was demonstrated (Demaerschalk,1971b;1972a) that the following logarithmic taper equation : log d = b Q + b log D + b 2 log 1 + b 3 log H could be derived from a logarithmic volume equation and vice versa, it soon became clear that this technique of deriving taper equations from existing volume equations could be applied to many volume 12 functions (Demaerschalk,1973). According to Smith and Kozak (1971) best results will come from use of locally derived equations for estimation of upper bole diameters. The studies of slash pine by Bennett and Swindel (19 72) included the development of a new interesting equation for the prediction of taper above breast height: d = bQD(l/(H - 4.5)) + b (1 (h - 4.5)) + b H(l (h - 4.5)) + b (1 (h - 4.5))(H + h + 4.5) This equation is discussed in more detail in later sections, A general method to convert taper and volume equations from one unit system to another was described in detail (Demaerschalk,1972b). In fitting taper and volume equations,the assumptions of the regression analysis often are not met.Heterogeneity of variances seems to be one of the most serious problems.Being more compatible with the homogeneity of variance requirement,the logarithmic volume equation was favoured by many investigators.A correction factor for the bias,introduced by the equation,was developed (Meyer,1938;1944; Baskerville,1972). Weighting,another way to correct for heterogeneous variances, has been studied by many authors,Meyer (1953) described the form factor method of preparing volume tables as the proper method of weighting the tree volume residuals.This approach was followed by Evert (1969) who developed and tested several form factor equations,one of which is described in later sections. Cunia (1964;1965;1968) proposed weighted least squares to overcome the difficulty of obtaining an equally good f i t for a l l sections of the tree volume curve. Many authors came to the conclusion that the variance of volume 2 is directly related to the square of the quantity D H (Munro,1964; Haack,1963;Gregory and Haack,1964;Evert,1969;Smalley and Beck,1971). This would agree with the form factor approach. Gerrard (1966) settled on an exponential relationship between the variance of volume and D and H. Smith and Munro (1965) found no wholly satisfactory method of weighting or transformation. Non-linear fitting with weighting of the volume equation : V = b D bl Hb2 o was recommended by Moser and Beers (1969) who also found that the 2 variance of volume was exponentially related to D H.This equation would retain the statistical advantages of lack of bias and over-come the shortcomings of the logarithmic volume equation by weighting (Husch et al.,1972). Comparison and testing the functions for precision and accuracy is s t i l l a difficult task.Freese (1960) recommended the chi-square test to check the accuracy. To compare different volume equations,Furnival (1961) developed an index employing the concept of maximum likelihood. After comparison of several absolute and relative standard errors of estimate of tree volume,Hejjas (1967) concluded that none could by itself indicate the best equation.The standard error of estimate (SE_) and possibly the sum or mean of absolute deviations should always be calculated. Williams (1972) studied the effects some violations of the assumptions might have on the outcome of the regression analysis. He concluded that i f errors are involved in the measurement of the independent variables,these might seriously bias the regression coefficients and SE,, . 15 3.Data The data used in this study consists of a sample of 752 tree records for eight species or species groups taken from the B r i t i s h Columbia Forest Service (B.C.F.S.) data bank.It represents a sub-sample of the data used by Kozak et al.(1969a) in their taper studies.These eight species or species groups were chosen because of their homogeneous distribution of trees over the D-H range. For each "species" the sample was taken so as to give a good repre-sentation of the widest possible range of D, H and absolute form quotient (AFQ).Combinations of D,H and form,represented by only a few trees or near the extremes of the data bank records were avoided. The observations from each tree are: -diameters inside bark and outside bark at one foot,4.5 feet and at each tenth of the height above breast height,to the nearest tenth of an inch. -total height to the nearest tenth of a foot. A summary of the data is given for each species in table I.Only infor-mation about D,H,form and double bark thickness has been studied.lt has been shown that average effects of age,site,crown class and similar factors are small in relation to those of D,H,form and bark thickness (Demaerschalk and Smith,1972). Table I Averages and Range of Data species or species groups number of trees D (inches ) H (feet) D2H (AFQ)2 < 5 D2 V (cubic feet) min ave max min ave max min max min max mean min ave max F - coastal 65 5.3 15.2 25.8 52.4 104.4 159.9 1640 85096 .36 .62 .74 3.4 62.5 166.9 C - coastal 63 5.9 14.8 28.2 31.1 76.4 127.3 1083 78803 .19 .60 .88 3.3 . 44.2 139.2 S - interior 91 5.1 13.1 21.3 37.4 89.4 134.5 1215 57165 .25 .59 .88 2.9 43.9 120.9 B - coastal 71 5.9 18.1 33.6 32.5 107.6 164.5 1131 174824 .30 .62 .91 2.7 110.3 369.2 A 111 4.7 9.8 1.6.3 41.5 76.2 101.8 1085 25247 .24 .64 .84 2.4 21.1 53.3 Cot 109 5.1 8.9 15.1 44.8 73.2 106.9 1306 20880 .19 .57 .82 2.4 14.6 43.9 Pl 152 5.1 12.3 20.4 42.6 79.4 123.8 1340 43668 .26 .72 .91 3.6 34.3 96.2 Pw 90 5.8 15.7 23.4 45.4 105.4 147.1 •1802 71855 .28 .57 .90 4.5 68.9 157.4 17 4.Volume-based systems of tree taper and volume estimation 4.1.Volume equations 4.1.1.Fitting volume equations A selection of volume equations was made. Some of these are used extensively,others rarely.The following equations were considered for testing: 1.logarithmic volume equation (Schumacher and Hall,1933) log V = b Q + b log D + b 2 log H 2.logarithmic combined variable volume equation (Spurr,1952) log V = b Q + bl log (D2H) , 3. Honer1s volume equation (Honer,1965a) D2/V = b + b„ / H o 1 4. combined variable volume equation (Spurr,1952) V = b + b, D2H o 1 5. weighted combined variable volume equation V /(D2H) = b + b1/(D2H) o 1 6. comprehensive combined variable equation (Spurr,1952;Gerrard, 1966) V = b + b, D + b. H + b, DH + b, D2 + b, D^ H o 1 2 3 4 5 7. weighted comprehensive equation V /(D2H) = b Q + b1/(DH) + \> / D2 + b^/ D + hj H + b5/(D2H) 8. combined variable equation without intercept V = b D2H o 9. volume over basal area as a function of height (Smith and Munro, 1965;Christie,19 70) V / B* = b + b.H o 1 18 10. V / B as a f unction of H and H (Christie,1970;Demaerschalk and Smith, 1972) V / B' = b + b, H + b„ H 2 o 1 2 11. Meyer's volume-diameter ratio equation (Meyer,1944) V / D = b + b, H + b. D + b, DH o 1 2 3 12. cylindrical form factor equation (Evert,1969) V /(D2H) = b + b.(H /(H - 4.5))2 o 1 The numbering system of a l l taper and volume equations is given in Appendix 2.The tree volumes were computed by Smalian's formula,except the tip section which was considered having a form factor 0.4 and the section below one foot which was considered as a cylinder. All these volume equations were fi r s t fitted by a linear least i squares procedure. Plottings of the variances of volume showed that there was a 2 2 linear relationship between the variance of volume and (D H) ,favouring the use of weighted volume equations 5,7 and 12. In fitting these equations,the underlying assumptions of the regression analysis were usually not met.For many of these equations the assumed linear model is not correct.Variances are seldom homo-geneous. In some cases,the violations of the assumptions are minor. The coefficients of the equations as well as the standard errors 2 of estimate (SE ) and the coefficients of determination (R 100) are hi given for a l l species in Appendix 3. 4.1.2.Tests'of total volume estimation The results in Appendix 3 are far from sufficient to judge the effectiveness of each equation in estimating the total volume.The SE 1s are merely an overall measure of variation of the data and are of l i t t l e use when variances are not homogeneous or when the linear model is biased.Moreover they cannot be compared directly i f the dependent variables differ.The volume estimation was therefore tested in a more detailed manner.For each equation an approximated standard error of estimate (SE ) and mean bias (MB ) was computed for c c 2 2 each fi f t h of the range of D H within each species,D H is more or less linearly related to volume and was therefore used as a measure to 2 break down the data into size classes.In order to have in each D H class a minimum number of five trees,only five size classes could be used.A summary of the number of trees and the mean volume for each size class and for a l l species is given in table II. For each size class the SE^ was computed in the following way: SEC - <E<v. -fy2/ n>% C j=l J J What is considered as mean bias (MB ) is a measure of lack of f i t and c computed for each size class as the mean of the residuals: n /\ MB = V (V. - V.)/ n C J-l J J An approximation' of the overall standard error of estimate in terms of volume was computed as: SEt " (ZSKi -^ n>2/.(M " m - * »% i=l j=l J J An overall measure of bias was computed for each equation as the Table II Number of Trees and Mean Volume for All Size Classes and for All Species Number of trees for the species Size D C S B A Cot PI Pw class 1 28 35 36 36 50 54 68 25 2 10 11 21 18 20 22 33 18 3 11 6 19 6 15 15 26 22 4 5 6 8 5 17 12 17 12 5 11 5 7 6 9 6 8 13 Total 65 63 91 71 111 109 152 90 Mean volume for the species (in cubic feet) Size D C S B A Cot PI Pw class 1 16 17 13 35 7 6 12 19 2 49 48 40 121 20 15 34 50 3 84 70 64 196 33 23 54 76 4 119 101 92 254 41 31 68 111 5 145 124 109 328 47 39 88 140 Total 63 44 44 110 21 15 . 34 69 21 mean of a l l residuals: 5 n. MB • V* Y\V. . - V. .) / N i=l j - l A tabulation of a l l standard errors of estimate and biases for a l l volume equations and a l l species is given in table III. Some conclusions which can be drawn from a comparison of these results will now be discussed. For most species the best volume equations seem to be 1,2,6,7 and 11.The logarithmic volume equation 1 is slightly negatively biased when calculated without adjustments.However,the bias is very small for most species (-1.50 to +1.04 cu. ft. for Douglas-fir) and the total bias is negligible (-0.16 cu. ft. for Douglas-fir). The combined variable logarithmic volume equation 2 comes very close to equation l,but is definitely not as good because of the conditioning of the powers of D and H.Honer's volume equation 3 is good for some species but for others the SE 's and MB 's are for some sizes much c c larger.The combined variable volume equation 4 always has an overall zero bias because i t was fitted with V as dependent variable. This is,however,a good example of an equation in which the linear model does not hold (see figure 2).The real relationship between V 2 and D H is a curve.A straight line overestimates the smallest,under-estimates the middle and overestimates again the largest size classes. The comprehensive combined variable volume equation 6,which assumes 2 an additional significant effect of D,H,D and interaction D-H, gives good results for a l l the species.Although the difference between the overall SE^_ and MBfc for equations 4 and 6 may not seem important,the differences for the individual classes are substantial. Figure 2 Relationships between Dependent and Independent Variables for Some Volume Equations volume equation real relationship Table III Total Volume Estimation Tests of Volume-Based Linear Volume Equations Douglas-fir size SEc(in cubic feet) of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 1.46 1.70 1.86 2.04 1.71 1.41 1.42 2.07 1. 77 1.84 1.57 1.81 2 4.35 3.93 3.91 3.93 4.00 4.54 4.50 4.43 4.00 4.12 4.67 3.98 3 7.31 9.29 10.54 9.55 9.24 6.90 6.97 9.85 9.49 8.89 6.94 9.69 4 9.64 8.19 8.47 8.19 7.98 10.37 10.09 7.99 8. 24 8.40 9.91 8.09 5 13. 74 16.51 18.74 16.51 18.03 13.38 13.48 16.68 18.82 17.76 13.63 18.49 SEt 7.38 8.47 9.51 8.55 8.97 7.47 7.48 8. 70 9.32 8.94 7.41 9.23 MB c (in cubic feet) of volume 1 -0.02 0.07 -0.03 0.95 -0.20 0.00 0.06 -1.13 -0. 16 -0.25 0.01 -0.04 2 -0.36 -0.23 -0.18 -0.71 -0.36 0.10 -0.05 -2.06 0.13 -0. 89 -0.30 0.03 3 -0.95 -2.55 -3.31 -3.37 -1.43 0.10 -0.26 -3.94 -1.03 -1.51 -0.39 -1.67 4 -1.50 -1.20 -1.07 -1.74 2.05 -1.32 -1.34 -1.41 2. 77 2. 72 -1.37 1.75 5 1.04 2.36 2.86 2.41 7.65 0.41 0.90 3.45 8.46 9.42 1.26 7.05 MBt -0.16 -0.13 -0.20 0.00 1.07 0.00 0.02 -1.00 1.42 1.31 0.00 1.03 Western redcedar size SE c(in cubic feet) of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 1.62 1.65 2.12 2.45 1.87 1.78 1. 69 4.09 2.09 2.09 1.79 2.03 2 6.16 6.00 5.57 6.28 6.57 6.64 6.19 7.76 5.74 5.79 6.39 5.97 3. 11.90 12.18 13.52 13.19 11.98 10.81 11. 61 13.94 12.99 13.03 11.50 12.67 4 9.16 9.60 13.67 9.49 17.39 8.50 8.80 9.82 13.26 13.63 8. 79 14.35 5 11.12 11.68 16.12 12.37 25.50 9.75 10.46 14.28 17.16 17.16 10. 61 19.50 SEfc 6.43 6.55 8.11 7.00 10.34 6.22 6.41 8.11 8.15 8.29 6.35 8. 72 MB (in cubic feet) of volume c 1 0.01 -0.01 -0.21 1.19 -0.70 0.23 0.31 -3.65 -0.13 -0.09 0.31 -0.29 2 -0.27 -0.13 0.89 -2.19 1. 13 0.65 0.32 -4.81 1.08 1.36 -0.19 1.40 3 -3. 78 -3.89 -3.84 -6.36 0.36 -2.47 -3.43 -7.54 -2. 72 -2.49 -3.60 -1.52 4 1.64 1.60 2.08 0.64 14.54 0.62 0.76 2.52 6.11 5.74 1.10 9.06 5 2.23 2.17 2.67 3.33 22.02 -0.83 0.39 7.25 8.85 7.93 1.25 13.22 MBfc -0.07 -0.07 0.06 0.00 2.98 0.00 0.01 -2.77 1.14 1.13 0.00 1.85 Table III (continued) 24 Spruce size SE„(in cubic feet) of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 1.36 1.36 1.38 1. 70 1.43 1.40 1.33 1.81 1.38 1.31 1.33 1.39 2 3.05 3.12 3.30 3.25 3.07 3.07 3.02 3.62 3.27 3.29 3.10 3.18 3 6.88 6. 75 6.56 6.84 6. 79 6. 66 6.84 6.85 6.61 6.53 6.69 6.67 4 5.50 5.52 5.73 5.73 5.99 5. 78 5.67 5.66 5.66 5.92 5.76 5.78 5 8. 78 8.95 9.18 8.58 11.78 7.50 7.57 9.37 9.99 8.90 7.83 10.90 SEt 4. 70 4.67 4. 70 4.73 5.20 4.57 4. 60 4.92 4.83 4. 69 4.56 5.00 MB (in cubic feet) of c volume 1 -0.21 -0.22 -0.31 0.74 -0.40 -0.21 -0.08 -1.09 -0.22 -0.14 0.05 -0.32 2 -0.55 -0.59 -0.64 -0.95 -0.51 0.38 0.26 -1.94 -0.57 -0.06 0.09 -0.46 3 -0.21 -0.16 -0.02 -1.02 0.89 0.59 0.19 -1.23 0.25 0.55 -0.09 0.68 4 -0. 73 -0. 70 -0. 75 -1. 68 1.94 -2.08 -1.93 -0.98 0.15 -0.86 -2.01 1.12 5 4.33 4.59 4.92 3.76 8.76 0.72 1.32 5.19 6.25 4.42 2.03 7.58 MBfc 0.02 0.03 0.04 0.00 0.76 0.00 0.00 -0.82 0.33 0.31 0.00 0.59 Balsam size SE c(in cubic feet) of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 3.71 4.17 4.40 5.40 4.49 3.89 3.57 7.02 4.65 4.20 2 9.64 10. 79 13.01 10.64 13.01 9.36 9.47 12.09 12.40 13.62 3 16.86 17.23 20.18 15.89 24.53 16.13 16.58 15.73 18.44 18.31 4 6.32 6.98 12.98 6.62 23.33 6.95 6.53 6.62 10.34 8.91 5 14.04 17.62 27.17 17.60 38.87 12.17 15.70 18.75 24.28 24.47 3.69 4.51 9.47 13.00 16.48 22.53 6.35 18.99 14.22 33.85 SEfc 8.79 9.79 12.88 9.84 16.66 8.61 9.13 10.91 11.83 12.10 8.76 14.99 MBc(in cubic feet) of volume 1 -0.64 -1.17 -1.53 0.73 -1.88 0.29 -0.11 -5.07 -1.48 -0.77 -0.27 -1.74 2 1.05 1.28 3.77 -2.49 4.81 -1.06 1.38 -5.40 1.89 3.79 0.16 4.28 3 6.26 7.00 12.51 2.41 19.22 3.96 5.17 2.27 9.53 7.54 5.15 16.40 4 0.38 2.57 10. 74 -1.21 22.16 -0.21 -1.86 0.57 7.62 1.12 -0.32 17.69 5 -4.84 3.06 17.54 1.69 34.27 -2.34 -8.43 6.15 14.02 2.63 -3.76 28.00 MBt 0.09 0.76 3.48 0.00 6.35 0.00 -0.11 -3.19 2.26 1.51 0.00 5.20 25 Table III (continued) Aspen size SE c(in cubic feet) of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 0.54 0.51 0.54 0.60 0.51 0.62 0.53 0. 60 0.51 0.50 0.57 0.51 2 2.02 2.11 2.09 2.13 2.09 2.04 1.95 2.14 2.11 2.11 1.97 2.09 3 1.83 2.01 1.78 2.20 1.77 1.59 1.76 2.22 1.71 1. 73 1.79 1.65 4 4.57 4.43 4.38 4.49 4.38 4.61 4.55 4.45 4.38 4.39 4.54 4.38 5 2.91 5.13 5.82 4.82 5.79 2.27 3.32 5.07 5.99 6.01 3.36 6.10 SEt 2.31 2.59 2.67 2.59 2.67 2.28 2.36 2.63 2.70 2. 72 2.34 2.71 MB ( c in cubic feet) of volume 1 -0.04 -0.09 -0.19 0.25 -0.11 -•0.11 -0.02 -0.30 -0.11 -0.10 -0.01 -0.13 2 0.40 -0.03 0.02 -0. 08 0.07 0.63 0.25 -0.35 0.13 0.17 0.32 0.16 3 -1.20 -1.45 -1.14 -1. 70 -1.13 -•0.41 -0.94 -1.74 -1.03 -1.05 -0.99 -0.94 4 -0.99 -0.58 0.00 -0.94 -0.02 -•0.63 -0.59 -0.78 0.13 0.07 -0. 75 0.26 5 1.76 3.83 4. 74 3.38 4. 69 1.09 2.54 3.76 4.91 4.84 2.41 5.07 MB -0.09 -0.02 0.15 0.00 0.19 0.00 0.03 -0.25 0.25 0.25 0.00 0.29 Cottonwood size SE c(in cubic feet) of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 0.57 0.59 0.59 0.67 0.57 0.58 0.56 0.62 0.56 0.57 0.65 0.56 2 0.90 1.11 1.00 1.14 1.06 0.93 0.89 1.22 0.99 0.98 0.93 1.00 3 1.14 1.97 1.75 2.02 1.95 0.99 1.08 2.05 1.86 1.77 1.11 1.87 4 3.35 4.00 4.06 3.89 4.09 3.12 3.24 3.92 4.14 4. 12 3.32 4.14 5 3.06 4.43 4.95 3.91 4.68 2.69 2.67 4.07 5.01 5.07 2.95 4.96 SEt 1.52 1.97 2.01 1.90 2.01 1.43 1.46 1.93 2.05 2.04 1.52 2.04 MB (in cubic feet) of c volume 1 -0.04 -0.09 -0.20 0.25 -0.08 -0.05 -0.03 -0.17 -0. 12 -0.13 0.03 -0.12 2 -0.24 -0.39 -0.32 -0.42 -0.25 0.14 0.08 -0.62 -0.17 -0.24 -0.16 -0.17 3 -0.15 -0.36 0.08 -0.75 -0.12 0.23 0.12 -0. 75 0.17 0.17 -0.18 0.14 4 0.10 0. 76 1.45 -0.01 1.09 -0.28 -0.17 0.20 1.52 1.58 -0.10 1.47 5 1.31 2.32 3.60 1.14 2.76 -0.14 0.13 1.57 3.46 3. 75 0.96 3.37 MBfc -0.01 0.04 0.21 0.00 0.17 0.00 0.01 -0.20 0.29 0.29 0.00 0.27 26 Table III (continued) Lodgepole pine size SE c(in cubic feet) of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 1.06 1.07 1.18 1.28 1.12 1.07 1.07 1.66 1.22 1.17 1.08 1.20 2 1.95 2.08 2.27 2.28 2.11 1.88 1.90 2.80 2. 67 2.26 1.94 2.20 3 3.82 4.18 4.82 4.28 4.30 3.71 3.69 4.32 4.71 4.93 3.90 4. 70 4 5.92 6.02 7.08 5.98 7.02 5.87 5.85 6.19 7.16 7.27 5.89 7.39 5 5.76 5.72 6.41 5.84 7.23 5.87 6.05 6.26 6. 79 6.51 5.75 7.29 SEt 3.11 3.22 3.70 3.31 3.62 3.11 3.12 3.55 3. 73 3.78 3.14 3.82 MB (in cubic feet) of c volume 1 -0.01 -0.01 -0.24 0.47 -0.23 0.09 0.03 -1.23 -0.11 -0.11 0.09 -0.18 2 -0.47 -0.61 -0.48 • -1.11 -0.68 -0.11 0.09 -1.96 -0.46 -0. 26 -0.40 -0.37 3 -0.40 -0.40 0.60 • -0.92 0.69 -0.36 -0.17 -0.88 0. 76 0.86 -0.44 1.09 4 1.43 1.56 3.31 1.45 3.96 1.01 0.92 2.17 3.79 3.55 1.34 4.33 5 -0.35 -0.18 2.40 0.50 4.09 -1.31 -2.15 2.03 3.28 2.59 -0.51 4.16 MBt -0.04 -0.04 0.39 0.00 0.53 0.00 -0.01 -0. 78 0.58 0.58 0.00 0. 73 White pine size SE c(in cubic feet) . of volume for volume equations class 1 2 3 4 5 6 7 8 9 10 11 12 1 1.89 1.93 2.03 2.20 2.04 1.87 1.85 2.58 2.06 2.01 1.91 2.07 2 3.73 3.69 3.76 3.83 3.90 3.67 3. 70 4.61 3.78 3.71 3.65 3.80 3 4.19 4.19 4.50 4.16 4.15 4.25 4. 20 4.19 4.40 4.68 4.23 4.36 4 9.15 8.95 8.84 8.96 10.05 8.84 8.98 9.08 9.18 8.93 8.94 9.59 5 9.65 9.48 9.37 9.48 10.13 9.44 9.50 9.59 9.30 9.34 9.47 9.50 SEt 5.81 5.70 5.72 5.74 6.13 5.80 5.84 5.95 5.76 5.79 5.76 5.90 MB (in cubic feet) of volume 1 0.09 -0.03 -0.31 0.74 -0.52 0.31 0.15 -1.64 -0.24 -0.14 0.12 -0.41 2 -0.68 -0.78 -0.85 -1.36 -1.30 -0.81 -0.56 -2.82 -0.97 -0.42 -0.57 -0.99 3 0.15 0.32 0. 75 -0.33 0.93 0.11 0.43 -0.96 0.80 1.28 0.46 1.08 4 1.71 2.03 2.61 2.19 5.10 2.21 2.07 2. 70 3.42 2.93 2.12 4.36 5 -2.34 -2.18 -2.14 -1.01 3.07 1.70 -2.34 0.31 -0.31 -2.22 -2.17 1.27 MBfc -0.19 -0.13 -0.03 0.00 0.95 0.00 -0.03 -0.85 0.35 0.26 0.00 0.72 27 In the case of Douglas-fir,for example,difference in size class bias is as high as -3.47 cu, ft. Volume equation 8,which is identical to equation 4 but with zero intercept,assumes a constant CFF for a l l trees: CFF = 183.3466 b o 2 Because of the real relationship between V and D H,this equation underestimates always the smaller sizes and overestimates the larger. The overall bias is always negative (see figure 2). i 2 The volume equations 9 and 10 with V/B as a function of H and H are mostly negatively biased for the small,and positively biased for the larger,size classes. The bias can be large. The linear model j clearly does not hold because V/B1,which normally decreases when H decreases,increases again when H becomes small (see figure 2). i The functions' 9 and 10 assume a linear relationship between CFF and 1/H or between CFF and l/H and H. Meyer's volume equation 11 performs very well.Overall bias is always zero and bias of the individual size classes is very small (-1.37 cu. ft. maximum for Douglas-fir). The cylindrical form factor volume equation 12 has the same pattern of bias as equation 9,due to a wrong assumption of the linear model (see figure 2). Weighting was tested in equations 5 and 7. Equation 5,which is the weighted form of equation 4,has an intercept which is always smaller and a slope which is always steeper than in 4 (this comparison of the coefficients is made after transformation of 5 to the same function as 4XThis shifting is caused by giving more weight to the smaller trees and results in an overestimation ,sometimes very large 28 (e.g. +7.65 cu. ft. for Douglas-fir),of the larger size classes and an underestimation of the small sizes (see figure 2).Overall bias is always positive.Weighting has a negative effect here because the assumption of a linear model is not met.Weighting has,generally speaking,no effect in 7,compared with 6. It can be seen in these tests that i f two equations differ in SEc for a given size class i t usually is due to a difference in bias,MB^.The square of SE^ consists of two components.One component is a measure of the variation of the data (pure error),the other component is a measure of the square of the bias (lack of f i t ) . Although various methods of data collection and analysis can result in errors which appear to be very large,in relation to the biases discussed here,such errors should be reducible by further sampling. Bias is of great importance because i t can lead to consistently high or low estimates which may be undetectable in conventional inventory methods. Because in these studies of taper and volume estimation it's most important to minimize any systematic bias,most emphasis in the following tests will be put on the amount of bias. 29 4.1.3. Non-linear fitting of volume equations Volume equations 1 and 3 were transformed into a non-linear form: 13. V = 10b° D bl Hb2 14. V = D2/(b + b,/ H) o 1 and fitted by a non-linear least squares procedure (UBC BMDX85 computer program) with and without weighting.The same weights were used as in equations 5 and 7.The coefficients of these equations are given in Appendix 3.2. and the results of the total volume estimation tests in table IV. i To start the iteration in the non-linear least squares procedure, the values of the corresponding linear equations were used as first approximations of the coefficients. ! In case of weighting,the coefficient b^ in equation 13 is usually larger and b^ usually smaller than the corresponding coefficients in 1. Without weighting i t is just the reverse.Although there is a systematic difference between the coefficients of 1 and 13,the differences are very small in the case of weighting.This is because taking the loga-rithm is in itself a weighting. Generally,the non-linear equation 13 gives higher estimates than the linear,and weighting gives higher estimates than non-weighting. For some species,e.g.western redcedar and cottonwood,the unweighted form of 13 seems to be superior,for others both weighted and unweighted are as good as l,but not better. In case of no weighting,the coefficient b in equation 14 is o usually smaller and b^ much bigger than the corresponding coefficients in 3.With weighting 14 and 3 are very similar.This could be expected Table IV Total Volume Estimation Tests of Volume-Based Non-Linear Volume Equations equation 13 size MB (in cubic feet) of volume for the species class D C ° S B A Cot PI Pw 1 0.15 0. 77 0.57 -1.30 -0.03 0.16 0.25 -0.08 2 -0.07 -0.15 0.14 0.05 0.62 -0.04 -0.23 -0.80 3 -0.09 -3.37 -0.18 6.21 -0.71 -0.08 -0.45 0.48 4 -1.44 0.54 -2.17 0.93 -0.83 -0.36 1.06 2.30 5 0.61 0.11 1.87 -4. 63 1.58 0.44 -1.14 -2.00 MBfc 0.03 0.14 0.17 -0.45 0.01 0.04 0.05 -0.05 >quation 13( w) MB (in cubic feet) c of volume 1 0.02 0.08 -0.17 -0.56 -0.02 -0.03 0.02 0.17 2 -0.17 0.26 -0.40 1.59 0.43 -0.19 -0.38 -0.52 3 -0.62 -3.06 0.09 7.34 -0.93 -0.06 -0.22 0.29 4 -0.84 3.08 -0.27 1.90 -0. 77 0.27 1.68 1.89 5 1.97 4.14 4.96 -2.58 2.21 1.55 -0.02 -2.08 MBfc 0.15 0.42 0.22 0.66 0.00 0.06 0.08 -0.04 equation 14 MB (in cubic feet) c of volume 1 -2.32 2 -4.18 3 -4.53 4 -1.44 5 4.77 -2.08 -2.17 -5.94 1.64 3.83 0.02 -0.46 -0.33 -2.15 2.81 -3.72 -3.03 2.84 -0.64 3.28 -0.56 -0.58 -1.72 -0. 66 3.88 -0.47 -0.93 -0. 78 0.32 2.25 -0.91 -1.62 -0.75 1.98 1.30 -0.59 -1.26 0.38 2.57 -1.75 MBfc -1.71 -1.64 -0.14 -2.18 -0.38 -0.37 -0.60 -0.23 equation 14( w) MB c (in cubic feet) of volume . 1 -0.00 2 0.42 3 -1.41 4 1.93 5 6.96 0.04 1.77 -2.52 4. 29 5.18 -0.21 -0.33 0.48 -0.05 5.78 -1.20 3.45 10.24 6.94 11.98 -0.12 0.14 -1.00 0.16 4.95 -0.13 -0.22 0.17 1.55 3.65 -0.11 -0.26 0.87 3.60 2.67 -0.19 -0.56 1.18 3.19 -1.48 MBt 1.15 0.91 0.38 2.63 0.26 0.29 0.59 0.33 because 3 is nearly completely homogeneous in variance.The weighted equation gives for a l l species higher estimates than the unweighted. 4.1.4.Correction factor for the logarithmic equation Because the antilog of the mean of the logarithms of some values is smaller than the arithmetic mean of these values,it is generally assumed that the logarithmic volume equation 1 gives an overall under-estimation of volume. If the normality and homogeneous variance assumptions are assumed to be correct in the linear logarithmic volume equation l,then i t can be shown that the correction factor,to be applied to correct for the underestimation,is 2 10 " (for logarithm base 10) 2 a is estimated by the square of SE„ of log V (see table V).This CJ factor is often called Meyer's correction factor (Meyer,1938;1944). Although the logarithmic volume equation 1 gives for most species (6 out of 8) an overall underestimation,this negative bias is extremely small.The largest negative overall bias is -0.19 cu. ft.For two species (spruce and balsam) there is an average overestimation.Application of the correction factor gives,of course,higher estimates but results for most species ( 5 out of 8 ) in a larger absolute bias.This means that the correction factor is too big.This may be explained by the fact that the assumptions,under which the correction factor is derived,do not hold in practise.lt is doubtful i f the normality assumption holds and there may be slight departures from the linear model. (See table VI for the results of the volume estimation test , ) Table V Meyer's Correction Factors for the Logarithmic Volume Equation 1 species S E E of log V Meyer1s correction factor Douglas-fir Western redcedar Spruce Balsam Aspen Cottonwood Lodgepole pine White pine 0.037144 0.049756 0.038673 0.035998 0.037248 0.038152 0.033634 0.032335 1.0037 1.0066 1.0040 1.0035 1.0037 1.0039 1.0030 1.0028 Table VI Total Volume Estimation Bias after Application of Meyer's Correction Factor equation 1 size MB (in cubic feet) of volume for the species class D C u S B A Cot Pl Pw 1 0.04 0.12 -0.16 -0.52 -0.01 -0.02 0.02 0.14 2 -0.18 0.05 -0.40 1.47 0.48 -0.18 -0.37 -0.54 3 -0.65 -3.35 0.04 6.95 -0.90 -0.06 -0.24 0.36 4 -1.07 2.31 -0.36 1.26 -0.84 0.22 1.64 2.03 5 1.58 3.06 4.78 -3.73 1.94 1.46 -0.08 -1.96 MB 0.06 0.22 0.19 0.47 -0.01 0.05 0.07 0.01 33 4.1.5.Volume equations for combinations of species Covariance tests were carried out to check i f several species could be combined in one equation without a loss in precision and accuracy.The tests were done with volume equation 1 and using the following combinations of species: combination combined species number 1. a l l eight species 2. Douglas-fir,spruce and balsam 3. Douglas-fir,spruce,balsam and western redcedar 4. aspen and cottonwood 5. lodgepole pine and white pine The coefficients of the equations are in Appendix 3.3. and the biases for individual and combined species are given in table VII. Because of the differences in bark thickness and form,the combination of species in the same equation was unsuccessful and results in large bias except for the combination of the two broad-leaved species and the combination of the two pines,for which only the intercept is significantly different. 34 Table VII Total Volume Estimation Bias for Combinations of Species equation 1 combination 1 size class 1 1.84 2. 5.17 3 7.01 D MB (in cubic feet) of volume for the species C C S B 1.95 2.09 -0.37 -0.29 -0.46 -0.33 -0.18 5.08 4.71 3.60 2.48 -1.26 -1.20 -1.31 6.38 5.44 3.91 1.87 -1.24 -1.39 -2.11 4 12.40 11.19 8.36 19.19 14.41 -2.13 -2.65 -4.09 -6.62 -10.08 -18.57 5 19.25 17.47 13.13 27.06 20.27 2.68 1.87 -0.51 -7.26 -12.27 -25.53 MBfc 6.99 6.52 5.41 4. 77 3.43 -0.71 -0.79 -1.21 1 2 3 -3.88 -3.85 -3.96 -5.50 -6.37 -8.98 -1.46 -3.79 -9.61 -4.57 -5.64 -8.56 MB (in cubic feet) of volume for the species Cot P l Pw combination 1 size class 1 0.00 -0.49 0. 75 0.32 -1.08 -0.11 -0.84 0.55 2 -0.13 -0.65 1.20 0.57 -2.54 -0.80 -2.19 0.05 3 -1.91 -2.37 1.69 1.03 -2.84 -0.89 -1.33 0.89 4 -1.50 -2.44 3.04 1.81 -1.10 0.82 0.47 2.29 5 2.26 0.32 4.83 3.47 -2.89 -1.13 -3.40 -1.99 MBfc -0.33 -1.01 1.45 0.81 -1.80 -0.34 -1.42 0.40 35 4.1.6.Volume equations for data adjusted for butt flare The previous fittings and tests were carried out on the data exactly as recorded,Later on,however,taper equations will be fitted on both the original data and data adjusted for butt flare.The adjusting consists of replacing the original observation of diameter inside bark at one foot by the diameter at breast height outside bark. This results in a strongly reduced butt flare which might allow a much better f i t of the taper equations.To make i t possible to compare these taper equations fitted on adjusted taper data with volume-based taper equations,these volume equations too should be fitted on volumes adjusted for butt flare.Therefore a l l volume equations have been fitted on the adjusted data as well.The coefficients for equations 1 and 4 are in Appendix 3.4. and the results from the total volume estimation tests are in table VIII. Volumes of the adjusted data are smaller for most species.This results in volume equations giving lower estimates than the non-adjusted equations.Adjusting the data does not mean very much for Douglas-fir and cottonwood where about half of the trees have a D bigger than the d^.It means much more for species with an important butt flare,like cedar,where adjusting may reduce the total volume by as much as 14 cu. ft. 36 Table VIII Total Volume Estimation Tests of Volume-Based Volume Equations for Data Adjusted for Butt Flare equation 1 equation 4 SE (in cubic feet) of volume for the species size D C A Pl D C A Pl class 1 1.45 • 1.54 0.49 1.03 2.02 2.32 0.62 1.26 2 4.39 5.63 1.80 1.93 4.01 5.62 1.97 2.30 3 7.02 9.94 1. 78 3.75 9.24 11.20 2.31 4. 24 4 9.33 7.85 3.74 5.60 8.45 8. 72 3.55 5.84 5 13.45 12.32 2.68 5.71 16.19 13.64 4.35 5. 78 SEfc 7.20 5.99 1.99 3.02 8.41 6.62 2.67 3.26 equation 1 equation 4 MB (in cubic feet) of volume for the species size D C A Pl D C A Pl class 1 -0.05 -0.07 -0.05 -0.02 0.91 0.95 0. 27. 0.46 2 -0.34 0.03 0.36 -0.40 -0.77 -1.67 -0.20 -1.09 3 -0.90 -3.32 -1.10 -0.48 -3.32 -5.62 -1.89 -0.99 4 -0. 77 1.60 -0.34 1.57 -1.06 0.98 -0.39 1.66 5 0.86 1.16 1.28 -0.74 2.18 2.62 2.82 0.25 MB -0.14 -0.10 -0.06 -0.04 0.00 0.00 0.00 0.00 I 37 4.2.Volume-based taper equations 4.2.1.Derivation of compatible taper equations from volume equations The reasoning process by which a compatible taper equation is derived from a volume equation is based on the premise that total volume estimates,based on integration of the taper equation,must be identical to those given by the existing volume equation. This means that : / ( I l d2/(4(144))) dl = Volume Function (let's call i t VF) c r or alternatively : IId2H/(4(144)) = VF ! 2 The value of d can be calculated specifically as: d 2 = 4(144) VF /(LTH) From here a more generalized taper function can be derived.Using the taper data,the values of the unknown parameters (free parameters) of these taper functions can be derived by a least squares procedure so as to minimize the SE of diameter inside bark. E The taper functions derived from the volume equations are the following: l\ d --a. Db 1 C He l\ d = a Db 1C H6 3C. d = (a D 2l P/(b HP+1 + c H P)) % 4fc. d = (a 1P/ H P + 1 + b D 2 i q / H q) % 5fc. d - (a ijl HP"*"1 + b D 2 l q / Hq)* 6. d = (a 1P/ H P + 1 + b D I q/ H q + 1 f D V / H t + 1 + g D21U/ H U) % 38 d (a 1 ?l HP+1 + b D Iq/ Hq + 1 + c l V Hr + e D 1S/ HS + f DV/ H1 t+1 + g D21U/ H U) % d d d a D(l / H) P / 2 (a DV/ H^ 1 + b D2iq/ H q) % (a D21P/ H P + 1 + b D2iq/ Hq + c D2lr/ H'"1^ d d (a D 1P/ H^1 + b D Iq/ Hq + c D2lr/ Hr + 1 + e D218/ HS)% (a D21P/ HP + b D2iqH2 _ q/(H - 4.5)2)% Throughout the text and appendixes these volume-based taper equations have the same number as the volume equations from which they are derived,except that a subscript "t" is added to distinguish them as taper equations (see Appendix 2). ; i The derivation of these taper equations,the formulae to compute height and section volume as well as the meaning of a l l these coefficients is given in Appendix 4. I The coefficients p,q,...,u are called here "free parameters" and a,b,c,e,f and g are coefficients whose value is based on the volume equation coefficients b ,b,,...,b.- and on the values of the free o 1 5 parameters. It was first attempted to f i t a l l these taper equations on the taper data by a non-linear least squares procedure in order to minimize the SE„ of d.This was carried out for most functions and E most species with satisfactory results.For some equations,however, t t t the derivation became troublesome (equations 10 ,11 and 12 ) and sometimes practically impossible as was the case for equations 6*" and 7*". Difficulties are caused by too many negative coefficients in the volume equations resulting in negative diameters.No suitable set of values for the free parameters can be found to make d positive 39 for a l l heights.After removing these troublesome coefficients,the taper equations were often conditioned such that they become useless for estimating d. Also,conditioning of these taper equations meant that they lost their compatability with the volume equations from which they are derived.Thus,although the derivation of a taper equation from a given volume function may be theoretically possible,practically a meaningful derivation may be impossible. The fact that no useful taper equation can be derived can be considered a weak point for a volume function i f a compatible system of taper and volume is desired. It was also tested for taper equations,with two or more free parameters to be estimated,how many of these parameters could be kept constant without any significant loss.These tests showed clearly that . optimizing only one parameter,while the others are kept constant with appropriate values,does not result in any significant loss in precision and accuracy.This greatly facilitates the estimation procedure. A summary of the values of the free parameters of these volume-based taper equations as well as the SE of d is given in table IX. t t t t The taper equations with only one free parameter (1 ,2 ,3 and 8 ) had a p-value usually ranging from 1.3 to 2. O.lt may seem unusual that each species has almost exactly the same p-value for each of the four equations.This may be explained by the fact that,after the unimportant equation terms are eliminated,they have almost identical forms. t t t t For the equations with two free parameters (4 ,5 ,9 and 12 ) the parameter p was usually kept constant with value l.The estimated values of the other parameter q are closely related to the parameter values in Table IX Taper Equations Derived from the Linear Volume Equations and Their Standard Errors of Estimate Parameter values of the taper equations species l f c 2 t 3t 4 t 5 t 8fc 9 t P P P P q P q P P q D 1.3 1.3 1.3 1.0 1.3 1.0 1.3 1.3 1.0 1.3 C 2.0 2.0 2.0 1.0 2.1 1.0 2.0 2.0 1.0 3.4 S 1.6 1.6 1.6 1.0 1.7 1.0 1.6 1.7 1.0 1.8 B 1.6 1.6 1.6 1.0 1.7 1.0 1.6 1.6 1.0 1.8 A 1.5 1.5 1.5 1.0 1.5 1.0 1.5 1.5 1.0 1.5 Cot 1.6 1.6 1.6 1.0 1.6 1.0 1.6 1.6 -1.0 1.5 Pl 1.4 1.4 1.4 1.0 1.5 1.0 1.4 1.4 1.0 1.5 Pw 1.5 1.5 1.5 1.0 1.6 1.0 1.5 1.5 1.0 1.7 Parameter values of the taper equations t t t species 10 11 12 P q r P q r s P q D 1.0 1.8 1.0 -1.0 1.0 -1.0 1.2 1.0 1.3, C 1.0 2.5 1.0 -1.0 1.0 -1.0 5.6 (- - )2 S -1.0 1.5 -1.0 -1.0 1.0 -1.0 2.2 1.0 1.9 B -1.0 1.4 -1.0 -1.0 1.0 -1.0 1.7 -1.0 1.5 A -1.0 1.4 -1.0 -1.0 1.0 -1.0 1.4 1.5 1.5, Cot 1.0 1.8 1.0 -1.0 1.0 -1.0 1.6 (- - ) 2 Pl 1.0 1.4 -1.0 -1.0 1.0 -1.0 1.6 -1.0 1.4, Pw -1.0 1.4 -1.0 -1.0 1.0 -1.0 1. 7 (- - ) 2 no useful taper equation could be derived 4 1 Table IX (continued) SE (in inches) of d for the taper equations species l f c 2t 3t 4 ^ 5 f c 8 * 9 f c 1 0 * l l ' 1 2 f c D 1 . 0 4 1 . 0 6 1 . 0 9 1 . 1 0 1 . 0 6 1 . 0 9 1 . 0 7 1 . 0 7 1 . 7 4 1 . 0 7 , C 1 . 9 7 1 . 9 8 2 . 0 0 2 . 0 1 1 . 9 5 2 . 1 4 1 . 8 0 1 . 8 9 1 . 7 0 ( ) S 1 . 4 3 1 . 4 3 1 . 4 2 1 . 4 5 1 . 4 1 1 . 4 5 1 . 4 1 2 . 1 0 1 . 5 6 1 . 4 0 B 2 . 0 9 2 . 0 8 2 . 0 7 2 . 2 1 2 . 0 6 2 . 1 7 2 . 0 6 3 . 3 2 2 . 3 0 2 . 1 7 A 1 . 1 0 1 . 1 1 1 . 1 1 1 . 1 2 1 . 1 1 1 . 1 1 1 . 1 1 1 . 2 2 1 . 6 0 1 . 1 1 , Cot 0 . 5 1 0 . 5 3 0 . 5 4 0 . 5 5 0 . 5 4 0 . 5 4 0 . 6 4 0 . 5 5 1 . 5 8 ( ) PI 0 . 8 9 0 . 8 9 0 . 8 9 0 . 9 1 0 . 8 8 0 . 9 3 0 . 8 9 1 . 1 3 0 . 9 1 0 . 9 4 , Pw 1 . 2 4 1 . 2 4 1 . 2 4 1 . 2 5 1 . 2 2 1 . 2 6 1 . 2 2 1. 78 1 . 3 6 ( ) 3 no useful taper equation could be derived Table X Taper Equations Derived from the Non-Linear Volume Equations and Their Standard Errors of Estimate Parameter values p SE^(in inches) of d of the equations for the equations species 1 3 * 13{w) 1 4 * 1 4 v > ) 1 3 t 1 3 v > ) 1 4 * l^w) D 1 . 3 1 . 3 1 . 3 1 . 3 1 . 0 5 1 . 0 4 1 . 1 3 1 . 0 8 C 2 . 0 2 . 0 2 . 0 2 . 0 1 . 9 9 1 . 9 7 2 . 0 4 1 . 9 9 S 1 . 6 1 . 6 1 . 6 1 . 6 1 . 4 4 1 . 4 2 1 . 4 3 1 . 4 2 B 1 . 6 1 . 6 1 . 6 1 . 6 2 . 1 0 2 . 0 9 2 . 1 3 2 . 0 8 A 1 . 5 1 . 5 1 . 5 1 . 5 1 . 1 0 1 . 1 0 1 . 1 1 1 . 1 1 Cot 1 . 6 1 . 6 1 . 6 1 . 6 0 . 5 2 0 . 5 1 0 . 5 4 0 . 5 4 PI 1 . 4 1 . 4 1 . 4 1 . 4 0 . 8 9 0 . 8 8 0 . 9 1 0 . 8 9 Pw 1 . 5 1 . 5 1 . 5 1 . 5 1 . 2 4 1 . 2 3 1 . 2 4 1 . 2 3 42 the equations with only one parameter. The derivation of taper equation 9*" was a problem in the case of cottonwood which had a negative b Q value in volume equation 9. For equation IO*",useful functions could be derived for four species. A meaningful derivation was impossible for the other species, having negative b and b„ coefficients in volume equation 10. o 2 In equation ll t,two terms had to be eliminated for a l l species. Except for Douglas-fir and cottonwood,the results were s t i l l reasonable. The derivation of 12*" was useful for some species but did not work for others (western redcedar,cottonwood and white pine). Looking at the SE 1s of d,the results are almost identical for E most taper equations (ranging,for Douglas-fir,from 1.04.to 1.10 inches for nine equations out of ten). Taper equations were also derived from the non-linear volume equations 13,13(w),14 and 14(w),whose results are given in table X. The values of the parameters of the weighted non-linear volume equations are the same as for the unweighted equations and the SE 's of d are identical or differ only by insignificant amounts (maximum 0.05 inches). In a l l previous taper equations d was used as dependent variable. To check what differences occur when other dependent variables are used,taper equation l f c was fitted in four different ways for Douglas-f i r and aspen. As can be seen in table XI the values of the parameters for different fittings sometimes differ (maximum difference is 0.3), but the differences in standard error of estimate are minor (maximum 0.08 inches).This,however,does not mean that i t is immaterial which p-value is used. 43 From comparisons of the SE 's of these taper equations,it would seem as i f more or less identical taper equations can be derived from volume equations which differ substantially.This may seem contradictory because,after a l l , i f a volume equation is biased then the taper equation,which integrates to the same volume,should be also biased. And there are large differences in bias of the different volume equations.This illustrates the fact that the SE of d is a poor measure hi of comparison in those cases where there is bias and variances are heterogeneous.Different equations may have an identical SE of d,but hi their pattern of bias of under- and overestimation for the different size classes may be quite different depending on the functional form of the equation or which p-value is used. The taper equations are also used to estimate total tree volumes and volumes of particular tree sections.Different patterns of bias in diameter estimation will result in a different bias of volume depending on the size of the tree and the height where the bias occurs. Volumes of tree sections are obtained by integrating the taper equation between the appropriate limits.These limits of integration may be defined in different ways.They can be expressed as section heights in which case the integration is straightforward.If the limits are given as section diameters,then the corresponding section heights should first be estimated by solving the taper equation for the heights,before the integration can be carried out.This last procedure introduces another source of error section volume estimation.If the diameter for a given height is biased,then so will be the height.for a given diameter. 44 To determine the overall performance of the taper equations, they will not only be tested for bias in diameter estimation but also for bias in section volume estimation with known heights,for bias in section heights and for bias in section volume estimation with unknown heights.The bias in total volume estimation has already been tested while testing the volume equations. Table XI Taper Equation l f c Derived with Different Dependent Variables Douglas - f i r Aspen dependent para- SE t para- SEt variable meter (inches) meter (inches) P of d P of d d 1.30 1.04 1.50 1.10 log d 1.30 1.04 1.55 1.11 d 2 1.40 1.08 1.80 1.18 d 2/ D bl 1.35 1.06 1.65 1.13 c 45 4. 2. 2.Tests of diameter estimation To test the diameter estimation of the taper equations, approximate standard errors of estimate SE^ and estimations of the bias MB^  of d are computed for a l l size classes and for the different heights within each size class.The heights of interest are 1 foot, 4.5 feet and each tenth of the height between breast height and tree top.The standard errors of estimate and the estimations of bias are computed in the same way as was done for the total volume estimation of the volume equations. A comparison of the overall SE 's of d has been made already in the previous section. As an example of a complete diameter estimation test,all results for one equation and one species are given in Table XII,It would be too lengthy to reproduce here a l l the results for a l l equations and al l species.Some tables of results and examples of output,useful for comparison,are given and a l l important features are discussed. For reasons mentioned in a previous section,most emphasis will be on bias rather than on standard.error of estimate. To compare the performance of different taper equations on the same species,all taper equations were tested on Douglas-fir,aspen and cottonwood.The results concerning the total bias for Douglas-fir are presented in table XIII with some results for aspen. To compare the performance of the same taper equations on different species,four equations were tested on a l l species.Results for equations l t and 8fc are in table XIV. Table XII Example of a Diameter Estimation Test of a Volume-Based Taper Equation Test of taper equation 1 for Douglas-fir Mean d (in inches) at heights size class 1' 4.5' ' .IH .2H .3H .4H .5H . 6H . 7H ,8H .9H 1 H 1 9.4 8.2 7.6 7.2 6.8 6.3 5.8 5.0 4.3 3.2 1.9 0.0 2 15.2 13.0 12.0 11.1 10.6 9.9 9.0 8.0 6.8 5.5 3.2 0.0 3 18.5 15.6 14.3 13.5 12.5 11.7 10.8 9.6 8.0 6.1 3.4 0.0 4 23.1 18.5 16. 6 15.6 14.8 14.0 12.9 11.5 9.5 7.2 3.8 0.0 5 24.2 20.4 18.1 17.1 16.0 14.9 13.9 12.3 10.1 7.8 4.3 0.0 tot. 15.4 13.0 11.9 11.2 10.5 9.8 9.0 8.0 6.7 5.1 2.9 0.0 size class SE (in inches) c of diameter 1 0.9 0.5 0. 6 0.5 0.4 0.5 0.5 0.5 0.5 0.5 0.4 0.0 2 2.0 0.8 0.8 0.8 0. 6 0. 6 0.7 0.8 0.8 1.1 0.3 0.0 3 2.8 0.6 0.8 0.8 0.8 0.8 1.0 1.1 1.1 0.9 0. 6 0.0 4 4.3 0.7 1.3 1.2 0.8 1.0 0.9 1.3 1.1 1.1 0.8 0.0 5 3.5 0.6 1.5 1.3 1.1 0.9 1.1 1.3 1.0 1.1 1.3 0.0 SEfc 2.4 0.6 0.9 0.8 0.7 0.7 0.8 0.9 0.8 0.9 0.7 0.0 size class MB (in inches) c of diameter 1 -0.5 0.4 0.4 0.2 0.0 -0.1 -0.3 -0.3 -0.3 -0.1 0.1 0.0 2 -1.4 0.6 0.6 0.5 0.1 -0.2 -0.4 -0.5 -0.6 -0.8 -0.1 0.0 3 -2.1 0.5 0.7 0.5 0.2 -0.1 -0.5 -0.7 -0.6 -0.4 0.2 0.0 4 -3.8 0.5 1.2 0.8 0.2 -0.3 -0.8 -1.0 -0.8 -0.5 0.6 0.0 5 -3.1 0.4 1.3 0.9 0.5 -0.0 -0.6 -0.8 -0. 6 -0.4 0.4 0.0 MBfc -1.6 0.5 0.7 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 47 Table XIII Total Diameter Bias of Volume-Based Taper Equations for Douglas-fir and Aspen Douglas-fir t a p e r MB^  of diameter (in inches) at heights eq. 1' 4.5" .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H l!;(p=1.3) -1.6 0.5 0.7 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 1 (Prl.4) -1.3 0.7 0.9 0.6 0.2 -0.2 -0.6 -0.7 -0.7 -0.6 -0.1 0.0 2 -1.6 0.5 0.7 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 3 -1.6 0.4 0.7 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 4^  -1.5 0.5 0.8 0.6 0.3 -0.0 -0.3 -0.4 -0.4 -0.2 0.3 0.0 5^  -1.5 0.5 0.8 0.5 0.2 -0.1 -0.4 -0.5 -0.4 -0.3 0.2 0.0 8^  -1.8 0.3 0.5 0.3 0.0 -0.3 -0.5 -0.6 -0.6 -0.4 0.1 0.0 9 -1.5 0.5 0.8 0.5 0.3 -0.1 -0.3 -0.4 -0.4 -0.2 0.3 0.0 10^ -1.5 0.6 0.8 0.5 0.2 -0.1 -0.4 -0.5 -0.4 -0.1 0.5 0.0 11 -0.0 2.0 2.3 2.1 1.8 1.5 1.1 1.0 1.0 1.0 1.3 0.0 12^ -1.5 0.5 0.8 0.5 0.3 -0.1 -0.4 -0.5 -0.4 -0.3 0.2 0.0 13^ , -1.6 0.5 0.8 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 13^ (w) -1.6 0.5 0.7 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 14); -2.0 0.1 0.4 0.1 -0.1 -0.4 -0. 7 -0.7 -0.7 -0.5 0.1 0.0 14 (w) -1.5 0.6 0.8 0.6 0.3 -0.1 -0.4 -0.5 -0.4 -0.3 0.2 0.0 Aspen M"R n f H i a m p f p v I taper eq. 1* 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H lt(p=1.5) -1.4 0.6 0.6 0.3 0.1 -0.1 -0.3 -0.4 -0.4 -0.2 0.2 0.0 " ' " ' v " " ~ " 0.7 0.4 0.1 -0.2 -0.4 -0.6 -0.5 -0.3 0.0 0.0 lf(p=1.7) " L I 0.9 0.8 0.4 0.1 -0.2 -0.5 -0.7 -0.7 -0.5 -0.1 0.0 1 (p=1.8) -0.9 1.1 0.9 0.5 0.1 -0.3 -0.6 -0.9 -0.9 -0.7 -0.3 0.0  1  6 B t 1' 4.5 -1.2 0.8 -1.148 Table XIV Total Diameter Bias of Equations 1^ and 8*" species Equation 1 MBfc of diameter (in inches) at heights 1' 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H D -1.6 0.5 0.7 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 C -3.7 0.9 1.7 1.1 0.4 -0.1 -0.6 -0.9 -1.2 -1.0 -0.6 0.0 S -2.8 0.7 0.9 0.6 0.2 -0.2 -0.4 -0.5 -0.4 -0.3 -0.2 0.0 B -3.5 0.9 1.5 0.8 0.2 -0.3 -0.6 -0.9 -1.0 -0.8 -0.4 0.0 A -1.4 0. 6 0.6 0.3 0.1 -0.1 -0.3 -0.4 -0.4 -0.2 0.2 0.0 Cot -0.3 0.4 0.3 0.1 -0.1 -0.2 -0.2 -0.2 -0.0 0.1 0.2 0.0 PI -1.5 0.4 0.5 0.4 0.2 -0.0 -0.2 -0.4 -0.3 -0.2 0.1 0.0 Pw -2.7 0.5 1.0 0.6 0.1 -0.3 -0.5 -0.5 -0.5 -0.3 -0.0 0.0 species MB. Equation 8 of diameter (in inches) at heights V 4.5' .IH ,2H .3H .4H ..5H . 6H .7H .8H • 9H 1 H D -1.8 0.3 0.5 0.3 0.0 -0.3 -0.5 -0.6 -0.6 -0.4 0.1 0.0 C -4.7 0.0 0.9 0.3 -0.2 -0.7 -1.1 -1.3 -1.5 -1.2 -0.7 0.0 S -2.7 0. 7 0.9 0.4 0.0 -0.4 -0.7 -0.7 -0.7 -0.6 -0.4 0.0 B -4.1 0.3 1.0 0.3 -0.3 -0.7 -0.9 -1.2 -1.2 -0.9 -0.5 0.0 A -1.5 0.5 0.6 0.3 0.0 -0.2 -0.3 -0.5 -0.4 -0.2 0.2 0.0 Cot -0.4 0.3 0.2 0.0 -0.2 -0.2 -0.2 -0.2 -0.1 0.1 0.2 0.0 PI -1.7 0.1 0.3 0.2 -0.0 -0.2 -0.4 -0.5 -0.4 -0.3 0.0 0.0 Pw -2.9 0.3 0.8 0.5 -0.0 -0.4 -0. 6 -0.6 -0.5 -0.3 -0.1 0.0 49 Most equations have basically the same pattern of bias (see figure 3).The stump diameter (at one foot) is always underestimated. The lower 30 to 40% of the tree is overestimated and the upper part is again underestimated,except for the extreme top section. The pattern of bias is more or less the same for a l l species.The size of the bias differs slightly from one equation to another.Equations which seem to give less underestimation in the upper part usually have more overestimation in the lower part and vice versa. • These results do not explain by themselves why there are these large differences in bias of total volume in the volume equations from which these taper equations are derived.lt is therefore necessary to examine the distribution of the bias over the different size classes.lt is here that the differences among the taper equations become apparent. Taper equation 1*" has more or less the same pattern of bias for all size classes.Equation 4*" overestimates almost the entire profile of the smallest and largest trees,but gives much lower estimates than 1*" for the middle sized trees.This is due to the fact that the combined variable volume equation 4 overestimates total volume of smallest and largest trees and underestimates the middle sized trees.Equation 5*" is more negatively biased for the smaller trees and more positively biased for the larger trees.This could be expected,knowing the bias of the weighted combined variable volume equation 5. The way the bias of taper estimation differs from one equation to another and from one size class to another can be explained by the bias of total volume of the corresponding volume equations. The bias of diameter,added over a l l size classes,is fairly 50 Figure 3 Pattern of Bias in Diameter and Height Estimation for Volume-Based Taper Equations diameter d j, W2 '£ distance from tip of tree real tree profile taper equation profile d. , i = 1,2 a given diameter i = 1,2 bias of distance for d, constant,but there is a substantial difference among the taper equations in the way the pattern of bias changes from one size class to another.In table XII,the pattern of bias for the given example is constant for a l l size classes,Table XV shows an example where the pattern of bias changes from one size class to another.In this case, the equation looks excellent,total bias being nearly zero,however, there is substantial bias in each size class.The positive bias in one class is eliminated by the negative bias in another. The absolute bias at the different heights is for most species fairly small (seldom larger than one inch,except at one foot). The bias at one foot may be larger due to butt flare. Results are usually poor for the taper equations which are conditioned (e.g. equation ll*"). The taper equations,derived from the non-linear volume equations, are sometimes similar to the linear equations (equation 13fc),sometimes different (equation 14fc).Weighting of the non-linear equations makes no difference for equation 13t(w),but increases the bias in the lower part and decreases the bias in the upper part of the tree for equation 14fc(w). Fitting taper equation l f c with d 2 or d 2/ D bl as dependent variable causes more overestimation in the lower tree and more underestimation 2 in the upper tree.Using d gives more weight to diameters which were already overestimated. 52 Table XV Pattern of Diameter Bias of Equation 3 for Cottonwood MB (in inches) of diameter at heights size :lass l 1 4.5' .IH .2H .3H .4H ,5H. . 6H ..7H .8H .9H 1 H 1 -0.2 0.2 0.1 -0.0 -0.2 -0.2 -0.2 -0.1 -0.0 0.1 0. 2 0.0 2 -0.3 0.3 0.2 -0.1 -0.2 -0.2 -0.3 -0.3 -0.1 0.2 0.2 0.0 3 -0.3 0.7 0.6 0.1 -0.1 -0.2 -0.3 -0.3 -0.2 0.0 0.1 0.0 4 -0.5 0.7 0.7 0.3 0.1 0.0 0.0 -0.0 0.1 0.4 0.5 0.0 5 -0.8 1.1 0.9 0.7 0.5 0.4 0.3 0.3 0.3 0.3 -0.0 0.0 MB -0.3 0.4 0.3 0.1 -0.1 -0.1 -0.2 -0.1 -0.0 0.1 0.2 0.0 4.2.3.Tests of section volume estimation with known heights When the heights of the section are known,the volume is computed by a straightforward integration of the taper equation between these two heights: , The section volume equations are worked out for a l l volume-based taper functions in Appendix 4. In this case the pattern of bias of section volume will be the same as for diameter.The absolute value of the bias will be more or less linearly related to the bias of squared diameter and section length. Tests,identical to those for diameter,were carried out for section volumes.The section volumes of interest are the section below 4.5 feet and each tenth of the bole above breast height. V = (d 2/ k) dl 53 There was no need to repeat these tests for a l l equations and species since the way equations and species compare with each other remains the same as for diameter estimation. To get an indication of absolute values of bias which may be involved,a complete example of a test is given in table XVI.Several equations are compared for the same species in table XVII. Between most equations,the differences in bias,added over a l l size classes,are minor (usually much less than 0.5 cu. ft.),but differences in bias for individual size classes may be important (1 cu. ft. and more).They lead to the large differences in bias of total volume.The bias for the largest size class is compared for a few equations for Douglas-fir in table XVIII. Part of the bias is due to the different ways volumes are computed. The volume observations are based on Smalian's formula,which assumes a paraboloid form.The top section was assumed to have a cylindrical form factor of 0.4 while the stump was taken as a cylinder.Even i f the taper equation would give an exact estimation of the tree profile, there would be a discrepancy between observed and predicted value. This discrepancy would only be important for top and bottom. The actual values of the bias are fairly small (usually less than 1 cu. ft.).Bias increases towards the lower part of the tree. 54 Table XVI Example, of a Section Volume Estimation Test with Known Heights of a Volume-Based Taper Equation Test of taper equation l*" for Douglas-fir Mean V (in cubic feet) at heights size class <4.5' ' .IH .2H .3H .4H .5H ,6H .7H .8H .9H 1 H 1 2.11 2.78 2.44 2.20 1.94 1. 66 1.34 1.00 '0. 66 0.32 0.08 2 5.16 8.56 7.35 6.47 5. 78 4.93 3.97 3.02 2. 12 1.13 0.23 3 7.52 15.15 13.05 11.48 9.96 8.57 7.09 5.33 3.64 1.69 0.33 4 11.40 21.32 17.98 16.04 14.34 12.51 10.33 7.73 5.00 2.35 0.43 5 12.84 26.76 22.33 19.72 17.26 14.98 12.42 9.24 5.95 2.93 0.59 tot. 6.02 11.25 9.55 8.45 7.43 6.42 5.28 3.95 2.59 1.28 0.26 size class SE (in cubic feet) of section volume c 1 0.26 0.42 0.40 0.27 0.27 0.29 0.29 0.30 0.17 0.10 0.04 2 0.94 1.07 1.01 0.87 0. 65 0.64 0.67 0.61 0.61 0.41 0.04 3 1.51 1.33 1.41 1.25 1.21 1.31 1.39 1.26 0.93 0.49 0.12 4 2.72 2.22 2.51 1.92 1.61 1.73 1.85 1.75 1.32 0.69 0.22 5 2.38 2.75 3.35 2.78 2.11 1.94 2.20 1.94 1.38 1.01 0.38 SEfc 1.44 1.49 1.72 1.42 1.14 1.12 1.23 1.11 0.82 0.53 0.18 size class MB (in cubic feet) of c section volume 1 -0.10 0.31 0.25 0.09 -0.04 -0.13 -0.15 -0.14 -0.10 -0.02 0.01 2 -0.52 0.83 0. 78 0.44 -0.04 -0.30 -0.40 -0.44 -0.44 -0.26 0.01 3 -0.98 1.23 1.12 0.57 0.05 -0.51 -0.88 -0.85 -0.57 -0.19 0.06 4 -2.31 2.01 2.22 1.14 -0.06 -1.00 -1.44 -1.29 -0.81 -0.15 0.17 5 -1.97 2.26 2.80 1. 65 0.50 -0. 68 -1.38 -1.26 -0. 77 -0.23 0.13 MB t -0.80 1.01 1.06 0.57 0.06 -0.38 -0.62 -0.58 -0.40 -0.13 0.05 55 Table XVII Bias of Section Volume Estimation with Known Heights of Volume-Based Taper Equations for Douglas-fir and Aspen Douglas-fir taper MB^  (in cubic feet) of section volume at heights eq. <4.5' ,1H .2H .3H . 4H .5H .6H .7H .8H .9H I H lj(p=1.3) -0.80 1.01 1.06 1.57 0.06 -0.38 -0.62 -0.58 -0.40 -0.13 0.05 1 (P=L4) -0.58 1.43 1.30 0.66 0.03 -0.50 -0.80 -0.79 -0.60 -0.29 -0.01 2 -0.80 1.01 1.06 0.57 0.07 -0.38 -0.62 -0.58 -0.40 -0.13 0.05 3* -0.77 0.99 1.05 0.56 0.06 -0.39 -0.63 -0.59 -0.40 -0.13 0.05 4* -0.77 0.98 1.05 0.57 0.08 -0.36 -0.59 -0.55 -0.37 -0.11 0.06 5^  -0.70 1.24 1.26 0.74 0.21 -0.26 -0.52 -0.51 -0.35 -0.10 0.06 8; -0.88 0.84 0.92 0.45 -0.04 -0.46 -0.68 -0.63 -0.43 -0.15 0.05 9*; -0.69 1.25 1.29 0.78 0.25 -0.21 -0.48 -0.46 -0.31 -0.07 0.07 10^ -0.66 1.28 1.23 0.68 0.14 -0.31 -0.54 -0.47 -0.24 0.04 0.16 11^ 0.34 3.63 3.55 2.90 2.21 1.55 1.07 0.83 0.69 0.58 0.32 12;; -0.70 1.20 1.24 0.73 0.21 -0.26 -0.52 -0.50 -0.34 -0.09 0.06 13^ -0.79 1.04 1.09 0.60 0.09 -0.36 -0.61 -0.57 -0.39 -0.13 0.05 13^ (w) -0.78 1.07 1.11 0.61 0.10 -0.35 -0.60 -0.57 -0.39 -0.13 0.05 lh -0.98 0.71 0.81 0.35 -0.12 -0.53 -0.73 -0.66 -0.45 -0.16 0.04 14 (w) -0.67 1.26 1.28 0.76 0.22 -0.26 -0.52-0.51 -0.35 -0.11 0.06 Aspen taper MB^  (in cubic feet) of section volume at heights eq. <4. .IH .2H .3H .4H . 5H . 6H . 7H . 8H .9H 1 H l![(p=1.5) -0.55 0.57 0.44 0.19 -0.01 -0.14 -0.21 -0.21 -0.14 -0.04 0.01 1 (p=1.8) -0.26 0.93 0.62 0.23 -0.08 -0.28 -0.39 -0.40 -0.29 -0.14 -0.03 taper eq. I Table XVIII Section Volume Bias of Largest Size Class of Several Taper Equations for Douglas-fir MB (in cubic feet) of section volume c of largest size class at heights <4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H -1.97 2.26 2.80 1.65 0.50 -0.68 -1.38 -1.26 -0.77 -0.23 0.13 -1.74 2.60 3.10 1.90 0.70 -0.51 -1.25 -1.16 -0.71 -0.20 0.13 -1.83 2.47 2.99 1.82 0.65 -0.54 -1.26 -1.16 -0.70 -0.18 0.14 -1.42 3.55 3.92 2.60 1.29 -0.04 -0. 88 -0.89 -0.53 -0.10 0.16 -1.73 2.73 3.20 1.99 0.78 -0.45 -1.21 -1.13 -0.69 -0.19 0.14 -1.41 3.58 3.98 2.68 1.39 0.07 -0.77 -0.78 -0.43 -0.03 0.19 56 4.2.4.Tests of height estimation The height h of a given diameter is computed by estimating first its distance from the tip, l,and then subtracting this from the total height H of the tree.So,although the title of this section is about the height,the following tests and discussions will center around the distance 1 from the tip. Some taper equations (e.g.l*",3*" and 8*") can simply be transformed into an equation predicting 1 as a function of d,D and H.These examples were described in Appendix 4.Volume-based taper equations with more than one free parameter usually can not be transformed this way. In these cases,distances must be estimated by an iteration procedure. Two methods of iteration were tested,the "Binary chop" and the Newton-Raphson method. A cone approach was used to give a first approximation.The Newton-Raphson method was found to be the most appropriate. The pattern of bias for distance estimation is exactly the opposite of the pattern for diameter.Distances from the tip of the tree will be underestimated when diameters are overestimated and vice versa (see figure 3). Of interest in these tests were the distances for the diameters at one foot,4.5 feet and for the diameters at each tenth of the height. Tables XIX and XX contain a summary of some of the results.They indi-cate that bias can be very large.Biases of more than five feet are common.Bias of the distance for the diameter at one foot is very large (in many cases more than 20 feet) due to butt flare. 57 Table XIX Example of a Distance Estimation Test of a Volume-Based Taper Equation Test of taper equation 1*" for Douglas - f i r Mean 1 (in feet) for diameters at . heights size class 1" 4.5' .IH .2H .3H .4H .5H .6H .7H ,8H .9H 1 H 1 75.2 71.7 64.6 57.5 50.4 43.3 36.2 29.1 22.0 14.9 7.8 0.0 2 104.5 101.0 90.9 80.9 70.8 60.7 50.7 40.6 30.5 20.5 10.4 0.0 3 128.3 124.8 112.3 99.8 87.3 74.9 62.4 49.9 37.5 25.0 12.5 0.0 4 131.7 128.2 115.5 102.7 89.9 77.2 64.4 51.7 38.9 26.1 13.4 0.0 5 136.3 132.8 119.5 106.3 93.0 79.8 66. 6 53.3 40.1 26.8 13.6 0.0 tot. 103.4 99.9 89.9 80.0 70.1 60.1 50.2 40.2 30.3 20.4 10.4 0.0 SE /-» (in feet) of distance size \~class 1 11. 6 6.1 7.2 5.4 4.5 5.1 5.0 5.2 5.0 3.5 2.3 0.0 2 23.1 9.6 8. 6 8. 7 7.1 5.9 6.8 7.4 6.6 9.2 1.6 0.0 3 33.3 6.9 8.9 8.2 7.8 7.5 9.7 9.7 8.8 6.3 2.8 0.0 4 42.8 6.7 11.6 11.9 7.2 8.5 7.3 9.4 7.8 7.6 4,0 0.0 5 34.8 6.2 13.1 11.3 9.3 7.4 8.6 9.9 7.0 6.1 5.1 0.0 SEt 25.9 6.9 9.3 8.3 6.8 6.4 7.4 7.7 6.6 6.0 3.1 0.0 MB /•» (in feet) of distance size class 1 7.1 -5.2 -5.3 -2.4 -0.3 1.5 3.0 2. 6 2.7 0.8 -0. 6 0.0 2 16.3 -6.7 -6.9 -5.9 -1.5 2.1 3.3 4.5 4.6 5.7 0.5 0.0 3 25.8 -5.7 -8.2 -4.6 -1.9 1.9 5.5 7.1 5.6 3.4 -0.7 0.0 4 38.8 -5.1 -11.1 -8.3 -3.0 2.0 6.0 7.1 5.3 2.8 -2.6 0.0 5 30.7 -3.6 -11.6 -7.6 -4.1 0.5 4.8 6.5 4.4 2.7 -1.5 0.0 MBfc 18.1 -5.2 -7.5 -4.6 -1.6 1.5 4.0 4.7 4.0 2.5 -0.8 0.0 58 A bias in diameter estimation of one tenth of an inch causes a bias of approximately one foot in distance estimation.This simple rule is sufficient to get an indication of the bias involved. Table XX Bias of Distance Estimation of Some Volume-Based Taper Equations Douglas-fir MtWin feet) of the distance for diameters at heights taper t eq. 1' 4.5' .IH ,2H ,3H ,4H .5H .6H .7H .8H ,9H 1 H l ' 18.1 -5.2 -7.5 -4.6 -1.6 1.5 4.0 4.7 4.0 2.0 -0.8 0.0 3^  19.1 -4.3 -6.7 -4.0 -1.0 2.1 4.5 5.1 4.3 2.7 -0.6 0.0 8 21.7 -2.3 -4.9 -2.2 0.6 3.5 5.8 6.1 5.1 3.2 -0.4 0.0 Taper equation 1*" MB (in feet) of the distance for diameters at heights species t 1* 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H D A Cot 18.1 -5.2 -7.5 -4.6 -1.6 1.5 4.0 14.4 -6.2 -6.0 -3.1 -0.8 1.1 2.3 3.1 -3.8 -2.6 -0.7 1.2 1.5 1.6 4.7 4.0 3.7 3.1 1.5 0.4 2.0 -0.8 0.0 1.4 -0.7 0.0 -0.7 -1.1 0.0 59 4.2.5.Tests of section volume estimation with unknown heights Section limits may be given in terms of diameter.In that case, the section heights must be estimated first before the integration can be carried out.This procedure of section volume estimation is subject to two sources of error,namely bias in diameter estimation and bias in the estimation of both distances from the tip. The same tests as for section volume estimation with known heights were repeated here and some results are given in table XXI. Bias in diameter and bias in distance are always of the opposite sign.They may partially eliminate each others'effect on section volume,thus producing a fairly good final result. Bias in distance usually has more weight than bias in diameter. This results in a pattern of bias similar to the one for distance estimation and opposite to the one of section volume with known heights. The absolute bias in the upper parts of the tree is comparable with the bias which occurred when heights were known.Bias is larger in the lower parts of the tree.Results are useless for the lower 10% of the tree because of the large bias in distance estimation at butt flare. One other reason why this kind of estimation is s t i l l reasonable, in spite of the large bias in section distances,is the fact that usually both section distances are biased in the same way and more or less by the same amounts. Table XXI Bias of Section Volume Estimation with Unknown Heights of a Volume-Based Taper Equation Test of taper equation 1 for Douglas-fir MB (in cubic feet) of section volume at heights size class .2H ,3H .4H .5H .6H .7H .8H .9H 1 H 1 -0.95 -0.85 -0.53 -0.36 0.08 -0.04 0.18 0.08 0.01 2 -0.66 -2.89 -2.13 -0.49 -0.45 0.03 -0.32 0.68 0.04 3 -3.91 -2.45 -2.92 -2.50 -0.90 0.62 0.60 0.60 0.03 4 -5.32 -6.34 -6.08 -3.17 -1.14 1.18 0.93 1.12 -0.02 5 -7.10 -5.29 -5.98 -4.88 -1.59 1.62 0.65 0.83 • 0.15 MB -2.78 -2.60 -2.53 -1.72 -0.54 0.46 0.31 0.47 0.02 Taper equation 1 MBt(in cubic feet) of section volume at heights species . 2H . 3H .. 4H . 5H . 6H . 7H . 8H . 9H 1 H D -2.78 -2.60 -2.53 -1.72 -0.54 0.46 0.31 0.47 0.02 A -1.38 -0.96 -0.79 -0.33 -0.33 0.10 0.14 0.13 0.01 Cot -0.79 -0.47 -0.09 -0.04 -0.01 0.09 0.06 0.00 0.01 61 4.2.6.Tests of volume-based taper equations for data adjusted for butt flare Taper equations were also derived from some of the volume'equations fitted on the adjusted data.The values of the free parameters for equations 1*" and 4*" are given in table XXII, together with their SE ' s E of d.The parameter values are smaller than for the non-adjusted data. Smaller parameter values are related to a better tree form.In later sections this is investigated further. All previously described tests were repeated and the results for equation 1*" are given in table XXIII. The partial elimination of butt flare improves the estimation of taper.The overall SE 's of d are decreased by about 50% and the equations are much less biased.There is less overestimation in the lower part of the tree and less underestimation in the upper part. For some species there is a slight increase in bias in the top section (maximum 0.3 inches). Because of the improvement in the estimation of diameter,there is also an improvement in section volume and height estimation. Only for Douglas-fir and cottonwood are results similar to the unad-justed data,for reasons mentioned in section 4.1.6. 62 Table XXII Volume-Based Taper Equations for Data Adjusted for Butt Flare and Their Standard Errors of Estimate parameter values SE of d of the equations (in inches) for equations t , t ,t , t 1 4 1 4 jecies P P q D 1.3 1.0 1.3 0.92 0.98 C 1.6 1.0 1.8 1.01 1.11 S 1.4 1.0 1.4 0.65 0.75 B 1.4 1.0 1.4 0.89 1.16 A 1.3 1.0 1.4 0.59 0.63 Cot 1.6 1.0 1.6 0.47 0.51 Pl 1.3 1.0 1.3 0.56 0.61 Pw 1.4 1.0 1.4 0.66 0.69 Table XXIIi:. Tests of Volume-Based Taper Equation 1*" for Data Adjusted for Butt Flare MB (in inches)of diameter at heights species 1» 4.5* .IH .2H .3H ,4H .5H ,6H .7H ,8H .9H 1 H D -1.5 0.4 0.7 0.5 0.2 -0.1 -0.4 -0.5 -0.5 -0.3 0.2 0.0 C -0.7 -0.4 0.8 0.5 0.2 -0.1 -0.3 -0.4 -0.5 -0.3 0.0 0.0 S -0.4 0.0 0.4 0.2 -0.0 -0.2 -0.3 -0.2 -0.1 0.1 0.2 0.0 B -0.3 0.0 0.9 0.4 -0.0 -0.3 -0.4 -0.5 -0.5 -0.2 0.2 0.0 A -0.4 0.1 0.3 0.1 -0.0 -0.1 -0.2 -0.2 -0.1 0.2 0.5 0.0 Cot -0.2 0.3 0.3 0.0 -0.1 -0.2 -0.2 -0.2 -0.0 0.1 0.2 0.0 Pl -0.2 0.0 .0.2 0.2 0.0 -0.1 -0.2 -0.2 -0.2 -0.0 0.3 0.0 Pw -0.4 0.1 0. 6 0.4 -0.0 -0.3 -0.4 -0.4 -0.3 -0.0 0.2 0.0 63 Table XXIII (continued) MB (in cu. ft.) of section volume (with known heights)at height species 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H D -0. 66 0.98 1.04 0.55 0.05 -0.39 -0. 63 -0.59 -0.40 -0.13 0.05 C -0.67 -0.07 0. 75 0.41 0.14 -0.06 -0.17 -0.20 -0.16 -0.06 -0.00 A -0.11 0.18 0.18 0.05 -0.06 -0.11 -0.14 -0.11 -0.03 0.04 0.04 Cot 0.02 0.23 0.11 -0.02 -0.08 -0.09 -0.08 -0.06 -0.02 0.00 0.00 MB (in feet) of distance for diameters at height species 1' 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H D 16.5 -5.2 -7.5 -4.6 -1.6 1.6 4.0 4.7 4.0 2.5 -0.7 0.0 C 4.6 2.2 -5.0 -3.0 -1.0 0.6 1.9 2.3 2.8 1.6 0.1 0.0 A 4.3 -1.9 -3.3 -1.2 0.3 1.3 1.6 2.3 1.1 -0.8 -2.3 0.0 Cot 1.5 -3.6 -2.5 -0.3 1.4 1.6 1.7 1.6 0.6 -0.7 -1.1 0.0 MB (in cu. ft.) of section volume (with unknown heights) at heights species .2H .3H .4H .5H . 6H .7H .8H .9H 1 H D -2. 78 -2.60 -2.53 -1.72 -0.53 0.47 0.32 0.47 0.04 C -1.65 -1.22 -0.85 -0.63 -0.13 -0.15 0.12 0.16 0.01 A -1.23 -0.60 -0.52 -0.07 -0.17 0.22 0.17 0.11 0.00 Cot -0.79 -0.48 -0.09 -0.04 -0.01 0.09 0.06 0.00 0.01 64 5.Taper-based systems of tree taper and volume estimation 5.1.Taper equations 5.1.1.Fitting taper equations The taper equations,which were derived from the volume equations, can a l l be fitted on the taper data without being conditioned by the volume functions.Until recently (Demaerschalk,1971b;1972a;1973),taper and volume studies have been considered as two essentially separate fields.Therefore most of these taper equations have never been tried in the past.Yet,many other promising forms of taper functions were developed.Some of these are tested in this study,together with a few functions whose form was derived from the previous volume equations. The following selection of taper equations was made: I. logarithmic taper equation (Demaerschalk,1971b) log d = b Q + b^ log D + log 1 + b^ log H (compares with 1*") II.equation developed by Kozak,Munro and Smith (1969a) (d / D) 2 = b Q + b (h / H) + b 2(h / H) 2 III.Bennett and Swindel's taper equation (1972) d = b D 1 /(H - 4.5) + b.(l(h - 4.5)) + b0H 1 (h - 4.5) + o — 1 — z -b 3(l(h - 4.5))(H + h + 4.5) IV.equation with same form as equation 4*" (d / D) 2 = b Q 1 P/(D 2H P + 1) + b (1 / H) q V.Matte's taper equation (1949) (d / D) 2 = b o ( l / H) 2 + b ^ l / H) 3 + b 2(l / H) 4 VI.equation proposed by Osumi (1959) (d / D) = b (1 / H) + b (1 / H) 2 + b 2(l / H) 3 65 VII.taper equation developed by Bruce,Curtis and Vancoevering (1968) (d / D) 2 = t^X 3 7 2 + b L(X 3 / 2- X3) D (10"2) + b 2(X 3 / 2- X3) H (IO"3) + b 3(X 3 / 2- X 3 2) H D (IO"5) + b 4(X 3 / 2- X 3 2) H l / 2 (10~3) + b 5(X 3 / 2- X 4 0) H2 (IO - 6) where X = (1 /(H - 4.5)) VIII. equation with same form as equation 8*" (d / D) 2 = b (1 / H) P o — IX. equation with same form as equation 9*" (d / D) 2 = b 1P/ H P + 1 + b.(l / H) q o — 1 — X.Behre''s taper equation (1923) in conditioned form (d / D) = (1 / H)/(b + b. (1 / H)) — o 1 — XI.Behre's taper equation with condition b + b. = 1 o 1 ((1 / H)/(d / D) - 1) = b x ((1 / H) - 1) All taper equations are conditioned such that the diameter at the top of the tree is zero.Taper equation II had to be severely conditioned for western redcedar in order to avoid negative diameters. In equation VII,the b^ coefficient is conditioned to be equal to the 2 mean (d, / D) for each species.Plottings indicated this ratio to be constant throughout the range of observations. Plottings of dependent over independent variables showed again that often the assumptions of the regression analysis were not met (wrong model,heterogeneous variances).Observations from the same tree are not independent. Taper equation I,the analogue of l t , w i l l later on be fitted in several different ways. The coefficients of the equations,the SE 's and the coefficients of determination are summarized in Appendix 5.1. 66 For some equations,the coefficients of determination vary considerably (from 38% up to 80% for equation VII) from one species to another. All independent variables are not always significant (equations IV and VII).The condition which Matte (1949) posed on his taper equation V,namely that b + b, + b„ = 1, to make d = D for o 1 I 1 = H,seems acceptable for Douglas-fir and cottonwood.For the other species,double bark thickness and butt flare are such that this condition does not hold.The condition in XI that b + b„ = 1 looks o 1 reasonable,except for aspen and Douglas-fir. The p and q values in equations IV,VIII and IX are the ones obtained in the derivation of the corresponding equations from the volume functions. There are some consistent differences between the coefficients of these taper equations and these derived from the volume functions. In taper equation I>b^  is always smaller and b^ always larger than t t the corresponding coefficients in 1 and 2 .In taper equation IV, b. is smaller and b, is larger than in 4fc and 5t.The b coefficient o 1 o in equation VIII is always larger than in 8*".^  equation I X> D 0 is usually smaller and b^ larger than in 9*".These differences seem minor but will prove to be very important in the estimation of total volume. The b, coefficient in equation IV,b coefficient in VIII and the 1 o b^ coefficient in IX are very similar and closely related to double bark thickness.This will be discussed in more detail in a later section. These taper equations will now be tested for the estimation of diameter,section volume with known heights,height and section volume with unknown heights.Volume equations are derived from them and tested 67 for total volume estimation. The formulae to compute diameter,height,section volume and total volume are derived for each taper equation in Appendix 6. In future discussions,these taper equations I to XI will be called the "taper-based" taper equations while the taper equations l f c to 14fc,derived from the volume functions, are called the "volume-based" taper equations,unless i t is clear from the context which equations are meant. 5.1.2.Tests of diameter estimation These taper-based taper equations were subjected to the same tests as were carried out on the volume-based taper equations.Some results of the tests of diameter estimation appear in tables XXIV and XXV. Although the standard errors of d are similar for most equations, some seem to give consistently better results than others.The better ones seem to be equation III,V,VI and VII.But before the bias has been considered,no conclusion can be drawn from this. The volume-based taper equations had a l l the same pattern of bias.This is not the case here (see figure 4).Equations I,II,IV,VIII and IX have essentially the same pattern as before,although equations II and VIII overestimate a larger portion of the bigger trees. Figure 4 gives only an idea about the kind of bias (positive or negative),not about the size of the bias. Equation X has the same pattern for the smaller trees,but over-estimates the complete tree profile of the bigger trees.Equation XI, the conditioned form of X,has a bias even worse than X. 68 Figure 4 Patterns of Bias in Diameter Estimation of Taper-Based Taper Equations small trees Taper equations I,II,IV,VIII and IX large trees small trees Taper equation III A d large trees Taper equations V and VI k d large trees Taper equation VII real tree profile • taper equation profile diameter I distance from the tip of the tree 69 Table XXIV Diameter Estimation Test of Taper-Based Taper Equations species D C S B A Cot Pl Pw SEfc (in inches)' of diameter for equations I II III IV V VI VII VIII IX X XI taper eq. I II III IV V VI VII VIII IX X XI species 1.04 1.08 2.06 2.08 1.44 1.40 2.11 2.05 1.12 1.11 0.55 0.57 0.90 0.89 1.22 1.21 0. 97 1.07 1.67 2.01 1.24 1.41 1. 76 2. 10 1.02 1.11 0.49 0.53 0.74 0.89 1.04 1.23 0.93 0.94 0.90 1.88 1.73 1.31 1.25 1.23 1.16 1.86 1.86 1.57 1.03 1.03 1.01 0.53 0.52 0.67 0. 79 0. 79 0. 71 1.07 1.05 0.89 1.08 1.07 2.08 1.85 1.41 1.42 2.08 2.10 1.11 1.11 0.53 0.54 0.90 0.89 1.23 1.24 1.16 1.54 2.20 2.17 1.68 1.51 2.28 2.24 1.78 1.13 0.55 0.54 1.05 0.99 1.42 1.32 Douglas-fir MB^  (in inches) of diameter at heights 1' 4.5' .IH ,2H .3H . 4H .5H . 6H . 7H . 8H .9H I H -1.4 -1.2 •1.1 -1.4 -0.6 -0.9 •0.2 •1.3 •1.4 •2.2 •0.2 0. 0. 0. 0. 0. 0. 0. 0. 0. •0. 1. 0.9 1.0 0.6 0.9 0.3 0.5 -0.2 1.0 0.9 0.5 2.1 0.6 0.6 0.2 0.7 -0.1 0.1 -0.1 0.7 0.7 0.5 1.8 0.3 0.3 0.0 0.4 -0.1 -0.0 -0.1 0.4 0.4 0.5 1.4 -0.0 •0.0 •0.1 0.0 0.1 •0.1 •0.1 0.1 0.1 0.3 0.9 -0.3 -0.3 -0.2 -0.3 0.2 -0.0 -0.1 -0.3 -0.2 0.1 0.4 -0.4 -0.4 -0.1 -0.4 0.3 0.1 0.0 -0.4 -0.4 0.0 0.0 •0.4 -0.3 •0.1 •0.4 0.3 0.2 0.3 •0.4 •0.3 •0.0 •0.3 -0.3 -0.0 -0.2 •0.3 0.1 0.1 0.6 -0.2 -0.2 -0.2 -0.6 0.2 0.6 -0.1 0.3 0.0 0.1 1.2 0.2 0.3 -0.1 -0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Taper equation I ies) of diameter j 4.5' .IH .2H .3H . 4H .5H . 6H . 7H . 8H .9H 1 H MBfc (in inches at heights D -1.4 0.6 0.9 0. 6 0.3 -0.0 -0.3 -0.4 -0.4 -0.3 0.2 0.0 C -4.5 0.3 1.4 1.0 0.6 0.3 -0.0 -0.2 -0.4 -0.2 0.0 0.0 S -2.8 0.7 1.0 0. 6 0.3 0.0 -0.2 -0.2 -0.2 -0.1 0.1 0.0 B -3.8 0.7 1.4 0.9 0.4 0.1 -0.2 -0.4 -0.4 -0.1 0.1 0.0 A -1.2 0.8 0.8 0.5 0.2 -0.1 -0.3 -0.5 -0.4 -0.2 0.1 0.0 Cot -0.1 0. 6 0.4 0.1 -0.1 -0.2 -0.3 -0.3 -0.2 -0.1 0.0 0.0 Pl -1.4 0.4 0. 6 0.5 0.2 0.0 -0.1 -0.3 -0.3 -0.2 0.1 0.0 Pw -2.6 0.7 1.1 0.8 0.3 -0.1 -0.3 -0.3 -0.3 -0.1 0.1 0.0 70 Equations V and VI show a new pattern of bias.This bias is very large for some species (e.g. for western redcedar there is for several ' sections a bias of more than 3 inches).The smaller trees have some underestimation in the lower half and near the top. Taper equation III underestimates the total top half of the smaller trees.For the bigger trees the underestimated area shifts to the middle,leaving top and bottom overestimated. Equation VII,which has the smallest standard error of estimate of d of a l l equations,has quite a surprising pattern of bias.The profile of the smaller trees is overestimated.Other size classes have 60 to 70% of their lower profile underestimated and have a large overestima-tion near the top.The f i t near the base of the tree is fairly good, because the taper curve is forced through the diameter inside bark at breast height. When only total bias is considered,some taper equations (e.g. V, III,VI,VII and X),whose pattern of bias is of the opposite sign for different size classes,look very attractive,although they may be worse than others. The equations III,V,VI and VII have a sigmoid form.This characte-ristic does not seem to be sufficient to produce a good taper function. The results for the logarithmic taper equation I do not differ very much from the ones for equation l^jthe taper equation derived from the logarithmic volume equation (e.g. for Douglas-fir maximum difference is 0.2 inches).This suggests that two different procedures,to f i t basically the same taper equation,may produce similar final results with regard to the diameter estimation.The same could be said for t t t taper equations VIII and 8 .Equations 1 and 8 have for most species a better f i t in the lower part of the tree,while I and VIII are better in the upper part. For example, for Douglas-fir equation 8*" is for some lower sections 0.5 inches less biased but for some upper sections 0.2 inches more biased than equation VIII. Most taper equations show a similar performance for a l l species. Some equations (e.g. equations V and VI) are fairly good for some species and unacceptable for others (e.g. for western redcedar an overestimation of 4 inches in the upper part of the larger size classes). To check i f the kind of dependent variable used could make any significant difference,taper equation I was fitted in three different 2 ways.The three dependent variables are log d,d and d .The regression coefficients are given in Appendix 5.2. and results from the diameter estimation tests are in table XXVI.These results .indicate that,by giving more weight to d,bQ and b^ increases and b^ decreases.This was the case for both species for which the test was done (Douglas-fir and cottonwood).Although the SEt's of d are almost identical,the different methods cause some differences in bias.For Douglas-fir, using log d or d as dependent variable is better for diameter esti-2 mation (for some sections the bias for d is 0.3 inches larger than 2 for log d or d).Using d as dependent variable causes more overesti-mation in the lower tree and more underestimation in the upper tree. 2 2 For cottonwood,d or d seems to be best (for some sections d or d is 0.3 inches less biased than log d).So,the effect of the kind of dependent variable used may differ from one species to another. 72 Table XXV Comparison of Diameter Bias of Volume-Based and Taper-Based Taper Equations for Douglas-fir MB (in inches) of diameter at heights taper t eq. 1' 4.5' .IH ,2H .3H ,4H .5H .6H .7H ,8H .9H 1 H l ' I -1.6 -1.4 0.5 0.6 0.7 0.9 0.5 0.6 0.2 0.3 -0.1 -0.0 -0.4 -0.3 -0.5 -0.4 -0.5 -0.4 -0.3 -0.3 0.2 0.2 0.0 0.0 IV -1.5 -1.4 0.5 0.7 0.8 0.9 0.6 0.7 0.3 0.4 -0.0 0.0 -0.3 -0.3 -0.4 -0.4 -0.4 -0.4 -0.2 -0.3 0.3 0.3 0.0 0.0 8fc VIII -1.8 -1.3 0.3 0.7 0.5 1.0 0.3 0.7 0.0 0.4 -0.3 0.1 -0.5 -0.3 -0.6 -0.4 -0. 6 -0.2 -0.4 -0.2 0.1 0.2 0.0 0.0 Table XXVI Diameter Bias of Equation I Fitted with Different Dependent Variables Douglas-fir MB (in inches) of diameter at heights dependent t variable V 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H log d \* -1.4 -1.5 -1.2 0.6 0.5 0.8 0.9 0.8 1.0 0.6 0.6 0.6 0.3 0.3 0.2 -0.0 -0.3 0.0 -0.3 -0.2 -0.5 -0.4 -0.4 -0.7 -0.4 -0.3 -0.7 -0.3 -0.2 -0.6 0.2 0.3 -0.1 0.0 0.0 0.0 Cottonwood log d \* -0.1 -0.3 -0.3 0.6 0.3 0.4 0.4 0.3 0.3 0.1 0.1 0.1 -0.1 -0.1 -0.1 -0.2 -0.3 -0.1 -0.2 -0.1 -0.2 -0.3 -0.1 -0.2 -0.2 -0.0 -0.0 -0.1 0.1 0.1 0.0 0.2 0.2. 0.0 0.0 0.0 73 5.1.3.Tests of section volume estimation with known heights Section volume estimation was only tested for equations which proved reasonable for diameter estimation.Osumi1s taper equation VI and equation IX were not tested because of their similarity to Matte's equation V and equation IV respectively.Some of the results of the tests on equations I,II,III,IV,V and VIII are presented in table XXVII.The functions to compute section volumes are given in Appendix 6. It was easy to find a good relationship between the diameter bias and the bias in height (0.1 inch of bias in diameter corresponds roughly with 1 foot bias in height).Such a good relationship does not exist for section volume because the bias of a section is based on the bias of two diameters and depends on the size of these diameters themselves. The pattern of bias and a l l the differences between the equations and the species are,of course,similar to the ones for diameter. Table XXVIII features a comparison between volume-based and corres-ponding taper-based taper equations.This again shows how the volume-based equations f i t the tree better in the lower part but are more biased than the taper-based equations in the upper part of the tree. Some taper equations (e.g. equation V) are positively biased over most of the length of the tree profile.This will result in a large overestimation of total volume. 74 Table XXVII Bias of Section Volume Estimation with Known Heights of Taper-Based Taper Equations Douglas-fir i>EB (in cubic feet) of section volume at heights taper t eq. 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H I (log d) -0.64 1.41 1.40 0.85 0.29 -0.20 -0.49 -0.49 -0.34 -0.11 0.05 I ( •d ) -0.73 1.23 1.30 0.81 0.30 -0.15 -0.41 -0.40 -0.25 -0.04 0.09 I ( d 2 ) -0.51 1.61 1.45 0.78 0.13 -0.43 -0.75 -0.76 -0.58 -0.28 -0.01 II -0.45 1.74 1.59 0.96 0.36 -0.14 -0.39 -0.34 -0.14 0.12 0.20 III -0.41 1.07 0.49 0.00 -0.19 -0.23 -0.16 -0.03 -0.00 -0.04 -0.02 IV -0.58 1.52 1.50 0.95 0.38 -0.12 -0.42 -0.43 -0.30 -0.08 0.07 V -0.08 1.27 0.36 0.00 0.11 0.28 0.38 0.38 0.20 -0.00 -0.03 VIII -0.51 1.70 1.66 1.08 0.49 -0.04 -0.36 -0.39 -0.27 -0.07 0.07 Taper equation I MBt (in cubic feet) of section volume at heights species 4.5' .IH .2H .3H • .4H .5H „6H ,7H ,8H .9H 1 H D -0. 64 1.41 1.40 0.85 0.29 -0.20 -0.49 -0.49 -0.34 -0.11 0.05 C -2.96 0.68 1.33 0.85 0.46 0.16 -0.03 -0.13 -0.13 -0.06 -0.01 S -1.50 1.15 1.12 0.63 0.20 -0.10 -0.20 -0.18 -0.10 -0.03 0.01 B -3.28 2.39 3.07 1.64 0. 60 0.02 -0.38 -0.50 -0.30 -0.07 0.03 A -0.43 0.77 0.58 0.28 0.05 -0.11 -0.21 -0.23 -0.15 -0.06 -0.00 Cot 0.07 0.40 0.21 0.03 • -0.07 -0.11 -0.12 -0.10 -0.06 -0.03 -0.01 PI -0.63 0.55 0.63 0.42 0.17 -0.02 -0.14 -0.18 -0.13 -0.05 0.01 Pw -1.37 1.60 1.68 0.86 0.13 -0.24 -0.34 -0.28 -0.16 -0.04 0.02 taper Douglas-fir MB (in cu. ft.) of section volume of largest size class c at heights eq. 4.5* .IH .2H .3H „4H .5H .6H • 7H .8H .9H 1 H I -1.59 3.31 3.69 2.39 1.09 -0.21 -1.04 -1.02 -0.63 -0.16 0.14 II -0.82 4.92 4.85 3.23 1.74 0.33 -0.50 -0.44 0.01 0.46 0.51 III -0.76 2. 75 1.24 -0.14 -0.56 -0.69 -0.51 -0.00 0.22 0.05 -0.03 IV -1.15 4.26 4.53 3.12 1.72 0.31 -0.62 -0.70 -0.41 -0.04 0.18 V 0.05 3.92 1.86 0.88 1.08 1.31 1.37 1.32 0.84 0.17 -0.04 VIII -0.94 4.82 5.01 3.53 2.06 0.58 -0.41 -0.55 -0.31 0.01 0.19 75 Table XXVIII Comparison of Bias of Section Volume Estimation with Known Heights of Volume-Based and Taper-Based Taper Equations for Douglas-fir taper eq. MB t 4.5' ( i n cubic feet) of ' -.1H .2H .3H section volume at heights .4H .5H ,6H „7H .8H .9H 1 H lf c I -0.80 -0.64 1.01 1.41 1.06 1.40 0.57 0.85 0.06 0.29 -0.38 -0.20 -0.62 -0.58 -0.40 -0.49 -0.49 -0.34 -0.13 -0.11 0.05 0.05 4* IV -0.77 -0,58 0.98 1.52 1.05 1.50 0.57 0.95 0.08 0.38 -0.36 -0.12 -0.59 -0.55 -0.37 -0.42 -0.43 -0.30 -0.11 -0.08 0.06 0.07 8fc VIII -0.88 -0.51 0.84 1.70 0.92 1.66 0.45 • 1.08 •0.04 0.49 -0.46 -0.04 -0.68 -0.63 -0o43 -0.36 -0.39-0.27 -0o15 -0.07 0.05 0.07 5.1.4.Tests of height estimation There is no need to repeat a l l tests for a l l the equations and a l l the species since i t is known that 0.1 inch of bias in diameter corresponds with 1 foot bias i n height. The results of equations I,II and VIII for Douglas-fir,western redcedar,aspen and cottonwood are summarized in table XXIX. As could be expected from the results of the diameter estimation, these equations are less biased in the upper portion of the tree,but more biased in the lower portion. Differences in bias of 6 feet or more may exist between the different derivation methods. 76 Table XXIX Bias of Distance Estimation of Some Taper-Based Taper Equations Douglas-fir MB (in feet) of the distance for diameters at heights taper t v eq. 1' 4.5' .IH . 2H ,3H ,4H .5H .6H .7H ,8H .9H 1 H I 15.4 -7.3 -9.2 -6.2 -3.0 0.4 3.0 3.9 3.4 2.2 -0.9 0.0 II 12.7 -7.3 -8.7 -8.7 -5.4 -2.2 1.0 3.5 3.9 2.7 -2.9 0.0 VIII 15.3 -7.3 -9.2 -6.2 -3.0 0.3 2.9 3.8 3.3 2.0 -0.9 0.0 species Taper equation I D 15.4 -7.3 -9.2 -6.2 -3.0 0.4 3.0 3.9 3.4 2.2 -0.9 0.0 C 27.6 -2.1 -8.3 -5.9 -3.4 -1.4 0.2 1.1 2.0 1.3 0.0 0.0 A 11.0 -7.8 -7.3 -3.4 -1.5 0.6 2.1 3.6 3.3 1.8 -0.4 0.0 Cot 0.3 -5.5 -3.9 -1.3 0.9 1.6 2.2 2.4 1.7 0.5 -0.1 0.0 5.1.5.Tests of section volume estimation with unknown heights The results of equation I for Douglas-fir,western redcedar,aspen and cottonwood are given in table XXX. The kind of bias in section volume estimation,with unknown heights, is the same as for the estimation of distance from the tip of the tree. The results are very similar to those for the volume-based taper equations. 77 Table XXX Bias of Section Volume Estimation with Unknown Heights of a Taper-Based Taper Equation Taper equation I MBt(in cubic feet) of section volume at heights species .2H .3H .4H ,5H .6H .7H .8H .9H 1 H D -2.99 -2.77 -2.67 -1.85 -0.66 0.34 0.24 0.43 0.03 C -1.97 -1.50 -1.06 -0.79 -0.28 -0.24 0.06 0.12 0.01 A -1.49 -1.04 -0.88 -0.40 -0.38 0.06 0.18 0.12 0.01 Cot -0.94 -0.62 -0.22 -0.13 -0.07 0.06 0.05 0.00 0.01 5.1.6.Taper-based taper equations for data adjusted for butt flare All taper equations I to XI were fitted on the adjusted data as well,for four species.Coefficients,SE's and coefficients of determi-nation for some equations are summarized in Appendix 5.3. Because butt flare is largely eliminated,the SE 's are usually hi much smaller and the coefficients of determination larger.This is also true for the so called sigmoid taper curves of which some are expected to account for butt flare. The same tests on diameter,height and section volume estimation were repeated on these adjusted equations.Results are given in table XXXI.Despite a slight occasional increase in bias in the top section,all equations have greatly improved after butt flare was eliminated.The greatest improvement is,as expected,in the lower part of the tree and for those species with the largest butt swell. These results are very similar to the ones for the volume-based taper equations for adjusted data. 78 Table XXXI Tests of Taper-Based Taper Equations for Data Adjusted for Butt Flare SE t(in inches) of diameter for the equations species I II III IV V VI VII VIII IX X XI D 0.91 0.97 0.85 0.95 0.82 0.83 0.78 6.97 0.96 1.04 1.48 C 1.03 1.27 0.95 1.25 1.22 1.18 0.89 1.32 1.09 1.27 1.39 A 0.69 0.66 0.53 0.65 0.57 0.58.0.67 0.66 0.64 1.72 0.66 Cot 0.52 0.54 0.44 0.49 0.49 0.48 0.64 0.49 0.50 0.50 0.51 Taper equation I MBt(in inches) of diameter at heights species 1' 4.5' .IH .2H ,3H .4H ,5H .6H .7H .8H .9H 1 H D -1.3 0.6 0.9 0.6 0.3 -0.0 -0.3 -0.4 -0.4 -0.3 0.2 0.0 C -0.8 -0.4 0.8 0.5 0.2 -0.0 -0.2 -0.3 -0.4 -0.2 0.1 0.0 A 0.1 0.6 0.6 0.3 0.0 -0.2 -0.3 -0.5 -0.4 -0.2 0.1 0.0 Cot 0.1 0.6 0.4 0.1 -0.1 -0.2 -0.3 -0.3 -0.2 -0.1 0.0 0.0 Taper equation I MB^Cin cu. ft.) of section volume (with known heights) at heights species 4.5' .IH .2H .3H .4H .5H .6H „ 7H .8H .9H 1 H D -0.50 1.39 1.38 0.84 0.28 -0.22 -0.50 -0.50 -0.35 -0.11 0.05 C -0.74 -0.18 0.68 0.37 0.13 -0.05 -0.14 -0.17 -0.14 -0.04 0.00 A 0.16 0.55 0.41 0.16 -0.04 -0.17 -0.24 -0.24 -0.16 -0.06 0.00 Cot 0.13 0.38 0.20 0.01 -0.08 -0.11 -0.12 -0.10 -0.06 -0.03 -0.01 Douglas-fir MB (in feet) of the distance for diameters at heights eq. 1' 4.5* .IH ,2H .3H .4H .5H .6H .7H .8H .9H 1 H I 13.8 -7.2 -9.2 -6.1 -3.0 0.4 3.0 3.9 3.4 2.2 -0.8 0.0 II 12.1 -7.0 -8.5 -5.3 -2.2 1.1 3.4 3.8 2.6 0.4 -3.0 0.0 VIII 14.0 -7.1 -9.0 -6.0 -2.9 0.4 3.0 3.9 3.4 2.1 -0.9 0.0 Taper equation I MB (in cu. ft.)of section volume (with unknown heights)at height species .2H .3H .4H .5H .6H .7H .8H .9H 1 H D -2.98 -2.76 -2.66 -1.84 -0.66 0.35 0.25 0.43 0.03 C -1.52 -1.12 -0.78 -0.58 -0.10 -0.13 0.13 0.15 0.01 A -1.41 -0.95 -0.81 -0.33 -0.33 0.11 0.14 0.13 0.01 Cot -0.93 -0.48 -0.21 -0.12 -0.06 0.06 0.06 0.00 0.02 79 5.2.Taper-based volume equations 5.2.1.Derivation of compatible volume equations from taper equations Volume equations were derived from taper equations I to XI by inte-gration of the taper equation over the total length of the tree: r H 2 V = / (d / k) dl The integrations are shown for each equation in Appendix 6.The volume functions derived from the taper equations have the same number as the taper equations from which they are derived,except that a subscript "v" is added to distinguish them as volume equations.They are the i following: ; I V. log V = a + b log D + c log H v 2 i II . V = a D H 1 III V. V = (a 2 H3/3 + b 2 H5/5 + c 2 H?/7 + 2ab H4/4 + 2ac H5/5 + 2bc H6/6)/k where a, b and c are functions of D,H and the coefficients of equation III. r V V . V = a + b D H VV. V = a D2!! VI V. V = a D H VII V. V = D2(a X 1 + b X 2 + c X 3 + e X4)/k where 3^  = H 5 / 2/((H - 4 . 5 ) 3 / 2 5/2 ) X3 = H 3 3/((H - 4 .5) 3 2 33) X2 = H4/((H - 4.5) 34) X 4 = H 4 1/((H - 4.5)4°41) and a,b,c and e are functions of D,H and the coefficients of equation VII. VIIlY. V = a D^ 80 IXV. V / B' = a + b H XV. V = a D2H X IV. V = a D2H The coefficients a,b,c and e can be computed directly from the coefficients of the taper-based taper equations according to the formulae given in Appendix 6. The equations I V to X I V are called "taper-based" volume equations while the volume equations 1 to 14,fitted on the volume data,are called "volume-based" volume equations,unless i t is clear from the context which equations are meant. v v All taper-based volume equations,except I I I and V I I ,have a functional form identical to some of the volume-based volume equations. To compare these corresponding equations,the coefficients of some taper-based volume equations are given in table XXXII. The functional forms of these volume equations are very important as they reveal the built-in assumptions about the cylindrical form factors. Not less than six taper equations (equations II,V,VI,VIII,X and XI) integrate to the combined variable volume equation without inter-cept (compares with volume equation 8).These taper equations assume a constant cylindrical form factor for a l l trees which is computed as: CFF = 183,3466 a All volume equations,derived from these six taper equations,have coefficients which are larger than the coefficient in equation 8. The coefficients of I I V , V V , V I I I V and X I V are very similar.Equation V I V is always bigger than equation 8,but consistently smaller than the other four equations. Table XXXII Taper-Based Volume Equation Coefficients Coefficients of the volume equations I V IV V species a b c a 2 b 10 D -2.950510 1.736276 1.281513 0.436084 0.200044 C -2.346608 1.493320 1.144144 0.989094 0.216252 S -2.752611 1.732854 1.214090 0.036790 0.236367 B -2.864758 1.518873 1.416063 0.169288 0.236882 A -3.606290 1.431840 -1.817881 -0.127365 0.241152 Cot -3.676927 1.563550 1.757194 -0.026213 1.208321 PI -2.822723 1.580697 1.350776 0.464814 0.236496 Pw -2.553785 1.779 785 1.090640 0.448075 0.230738 Coefficients of the volume equations I I V v v v i v V I I I V x v 7 2 2 species a 10 a 10 a 10 a 10 a 10 D 0.204565 0.202589 0.201616 0.204169 0.201624 C 0.229419 0.228051 0.220935 0.229425 0.214935 S 0.234827 0.236009 0.232074 0.236797 0.218765 B 0.244619 0.244346 0.240604 0.243075 0.240822 A 0.236419 0.236834 0.232532 0.238427 0.311075 Cot 0.211062 0.210420 0.208322 0.207725 0.200125 PI 0.246868 0.245626 0.243868 0.247618 0.237819 Pw 0.233643 0.233513 0.230895 0.238621 0.227153 82 Comparing the coefficients of I V with equation l,the intercept is smaller and the other coefficients are larger.Differences are for some species important. The intercepts of equation IVV are for a l l species smaller than in equation 4 and the slope coefficients are bigger.This will result in lower total volume estimates for the smallest trees and higher estimates for the larger size classes.The same can be said for v equations IX and 9. Some taper equations (equations III and VII) result in fairly complicated volume equations whose functional form can not be compared with any other volume equation. 5.2.2.Tests of total volume estimation Total volume estimation was tested for a l l taper-based volume equations I V to XI V and for a l l species.These tests are important as they will show what kind of bias may be expected i f volume equations are derived from taper equations. A summary of the bias of the equations is given for some species in table XXXIII. From the comparison of the coefficients in the previous section i t could be expected that most of these taper-based volume equations would be more biased than the volume-based volume equations 1 to 14. Most equations have an overall overestimation for most species. Equation VII V largely underestimates volume,due to the fact that 60 to 707. of the lower bole is underestimated by taper equation VII. 83 Table XXXIII Bias iri Total Volume Estimation of Taper-Based Volume Equations Douglas-fir MB (in cubic feet) of volume for equations s ize class IV IV(d)IV(d2) IIV III* IVV VV VIV VIIV VIIIV IXV XV XIV 1 0.2 0.2 -0.1 -0.0 0.3 0.1 -0.2 -0.2 -2.3 -0.0 0.2 -0.2 2.5 2 0.8 0.8 -0.1 1.4 0.9 0.7 0.9 0.6 -8.1 1.3 1.2 0.6 8.9 3 2.0 1.7 -0.2 1.8 0.1 0.4 1.0 0.6-•18.0 1.7 0.8 0.7 14.8 4 2.4 2.5 0.6 7.2 -0.5 4.9 6.0 5.4-•26.2 7.0 5.5 5.4 26.4 5 6.0 6.3 4.2 14.3 1.6 11.2 12.7 12.0-•31.2 14.0 11.9 12.0 38.4 MB 1.7 1.8 0.6 3.5 0.5 2.5 2.9 2.5-•12.6 3.4 2.9 2.5 13.5 Western redcedar MB (in cubic feet) of volume for equations — — c size class Iv I IV IIIV IVV vv v iv v VII VIIIV IXV xv XIV 1 0.6 -0.1 -0.4 -0.1 -0.2 -0.7 -1.4 -0.1 -0.4 -1.2 -1.0 2-0.2 6.5 2.0 4.4 6.2 4.5 -4.5 6.5 1.0 3.1 3.6 3 -3.3 8.9 -1.8 5.3 8.4 5.9 -13.0 8.9 -2.3 3.9 4.7 4 1.1 29.6 5.0 23.1 28.8 24.8 -20. 7 29.6 7.5 21.4 22.7 5 1.0 41.5 6.7 33.0 40.5 35.4 -31.3 41.5 11.1 31.1 32.8 MB. 0.2 8.1 1.0 6.0 7.7 6.1 -7.3 8.1 1.3 4.8 5.3 Balsam size class I I I MB (in cubic feet) of volume for equations c rv r r T „ V III IV V VI VII VIII IX X .v XIv 1 1.0 -0.8 0.1 -1.7 -0.8 -1.3 -5.5 -1.0 -1.9 -1.3. -1.0 2 5.0 11.4 5.1 7.4 11.3 9.2 -20.7 10.6 4.9 9.4 10.7 3 12.5 31.1 9.1 24.1 30.8 27.4 -42.2 29.6 18.4 27.6 -29.6 4 7.1 37.7 1.8 28.6 37.3 32.9 -69.0 35.8 20.7 33.1 35.8 5 -1.3 54.9 2.8 42.9 54.4 48.6 -96.4 52.4 32.3 48.9 52.4 MBfc 3.2 12.4 2.5 8.7 12.3 10.4 -24.6 11.6 6.0 10.5 11.6 84 Most equations have more or less the same pattern of bias. The smaller trees are underestimated and the larger size classes are overestimated.Most taper-based taper equations overestimated a large portion of the tree profile of the larger size classes.This causes an overestimation in total volume which is so important that results often become useless. While the logarithmic volume equation 1 underestimates the v volume of most sizes,equation I overestimates most size classes. In the diameter estimation,fitting equation I with d as dependent variable was superior for Douglas-fir.For total volume estimation 2 d seems to be slightly better. v v Only equations I and III give reasonable results for volume estimation for a l l species and can compete in performance with the best volume-based volume equations. 5.2.3.Tests of total volume estimation for data adjusted for butt flare Taper equations,fitted on adjusted data,were converted in the same way to taper-based volume equations.Total volume estimation tests are given in table XXXIV. Although adjusting results in some improvement,bias for most equations is s t i l l considerable.Adjusting for butt flare is not sufficient to produce relatively unbiased taper-based volume equations, v v except for I and III . 85 Table XXXIV Bias in Total Volume Estimation of Taper-Based Volume Equations for Data Adjusted for Butt Flare Douglas-fir MB (in cubic feet) of volume for equations class I V I I V III V V IV vv v iv VII V v VIII IXV xv x iv 1 0.2 -0.1 0.1 -0.0 -0.2 -0.3 -2.5 -0.1 0.2 -0.3 2.4 2 0.8 1.3 0.8 0.5 0.8 0.6 -8.3 1.1 1.1 0.7 9.0 3 2.0 2.1 0.2 0.2 1.2 0.9 -17.4 1.7 0.3 1.0 15.1 4 3.1 8.1 0.1 5.1 6.8 6.4 -24.3 7.6 5.4 6.5 27.4 5 5.7 14.4 1.2 10.4 12.8 12.2 -29.5 13.7 10.4 12.3 38.6 MB 1.8 3.6 0.4 2.2 2.9 2.7 -12.2 3.3 2.5 2.8 13.6 Western redcedar MB (in cubic feet) of volume for equations size class I I I V c III V i v v v v VI V VII V VIII V IX V x v XI V 1 -0.4 -2.2 -0.8 -2.0 -0.7 -0.9 -1.8 -2.2 -0.7 -0.8 0.0 2 -0.2 -0.1 1.1 -3.0 4.6 3.9 -3.2 -0.1 -2.2 4.4 7.0 3 -3.5 -1.0 -1.9 -5.9 5.8 4.8 -8.0 -1.0 -8.8 5.5 9.3 4 -0.1 12.1 3.0 2.9 23.2 21.7 -6.9 12.1 -6.3 22.8 29.1 5 -1.4 18.0 2.9 6.0 32.2 30.2 -11.3 18.0 -8.9 31.7 39.5 MBfc-0.3 1.2 0.1 -1.5 5.7 5.1 -3.8 1.2 -2.9 5.5 8.0 Aspen MB (in cubic feet) of volume ; for equations size class I I I V III V i vv vv v iv v VII v VIII i xv xv XI V 1 0.1 -0.1 -0.1 -0.3 -0.2 -0.2 -0.8 -0.4 -0.2 2.0 -0.0 2 1.3 • 0.4 0.3 -0.9 0.2 0.1 -2.9 -0.6 -0.7 7.0 0.7 3 0.2 -0.5 -1.3 -2.6 -0.8 -1.0 -6.4 -2.0 -2.5 9.9 -0.0 4 0.3 1.7 -0.3 -1.1 1.3 1.1 -6.6 -0.3 -1.0 15.2 2.3 5 0.3 5.7 1.9 2.0 5.2 4.9 -5.5 3.2 2.3 22.7 6.4 MBt 0.4 0.7 -0.1 -0.7 0.5 0.4 -3.2 -0.4 -0.5 7.7 1.0 86 6.Additional aspects of both systems and possible ways to improve them 6.1.Taper equations on data above breast height Adjusting the observation at one foot does not completely eliminate the butt flare.To check i f taper equations could be further improved by eliminating the observation at one foot,two taper equations (equation I and V) were fitted on the data of Douglas-fir and western redcedar. I is a non-sigmoid and V is a sigmoid taper equation. The data with the observation at one foot eliminated are called the "reduced data". Results of some diameter estimation tests are given in table XXXV. Compared with the taper equations on adjusted data,there is no signi-ficant improvement. Adjusting, as applied before,seems to be as efficient in reducing the bias as eliminating the observation below breast height. Table XXXV Diameter Bias of Equations I and V for Reduced Data Taper equation I MBfc (in inches) of diameter at heights species 4.5' .IH .2H .3H .4H .5H .6H .7H .8H .9H 1 H D 0.4 0.7 0.4 0.2 -0.1 -0.4 -0.5 -0.4 -0.3 C -0.5 0.7 0.4 0.2 -0.1 -0.3 -0.3 -0.4 -0.2 0.2 0.1 0.0 0.0 Taper equation V MBt (in inches) of diameter at heights species 4.5' .IH .2H ,3H .4H .5H .6H .7H .8H .9H 1 H D C 0.1 0.2 0.1 0.2 0.2 0.2 0.1 0.0 -0.2 -0.1 0.6 0.3 0.3 0.4 0.5 0.6 0.5 0.5 0.2 0.3 0.0 0.0 87 6.2.Relation between coefficients and form Because of their functional form,some taper and volume equations assume a constant cylindrical form factor for a l l trees and therefore do not account for variation in form.Even the so called "variable form" taper and volume equations seldom account for the whole range of variation in form.The equation coefficients define a mean profile for all trees or a mean profile for.all trees of a given D and H class and vary according to this mean form.This introduces a bias for all trees having a form different from the mean form. If there exists a relationship between the form and the values of the coefficients,then this relationship could be used to account for a wider variation in form and as such reduce the bias of the equations. In the following sections,the relationship between the coefficients and the form of the trees is examined for taper-based taper equations and for taper-based and volume-based volume functions. 6.2.1.Relation between taper-based taper equation coefficients and form The following two taper equations I. log d = b + h log D + b log 1 + b„ log H o i /. — J VIII. log(d / D) = b Q + b log(l / H) were fitted on the different classes of squared absolute form quotient for western redcedar and cottonwood.These two species have a wide range of form classes represented by a sufficient number of trees. Coefficients are given in table XXXVI. 88 T a b l e X X X V I R e l a t i o n b e t w e e n T a p e r - B a s e d T a p e r E q u a t i o n C o e f f i c i e n t s and F o r m C o e f f i c i e n t s o f e q u a t i o n I s p e c i e s AFQ n u m b e r c l a s s o f t r e e s C 0 . 2 8 0 . 1 4 5 0 8 0 0 . 7 9 2 2 7 2 0 . 9 3 6 9 6 8 - 0 . 8 8 7 7 3 9 0 . 3 20 0 . 1 5 8 1 4 5 0 . 8 6 9 9 5 3 0 . 8 5 8 1 0 6 - 0 . 8 5 6 0 1 3 0 . 4 22 - 0 . 0 8 5 4 5 3 0 . 8 6 6 0 6 9 0 . 7 9 2 2 9 4 - 0 . 6 5 5 5 2 4 0 . 5 13 0 . 0 9 2 5 3 5 0 . 9 0 0 5 0 8 0 . 7 4 0 0 6 2 - 0 . 7 0 9 8 7 0 C o t 0 . 3 31 - 0 . 4 3 5 4 4 7 0 . 7 7 8 0 6 5 0 . 9 4 0 8 4 5 - 0 . 5 8 7 4 0 0 0 . 4 61 - 0 . 2 4 7 9 7 5 0 . 8 8 4 0 4 5 0 . 8 4 8 0 5 7 - 0 . 6 5 0 6 8 7 0 . 5 17 - 0 . 0 8 4 7 3 0 0 . 9 5 5 4 9 0 0 . 7 8 4 3 7 5 - 0 . 7 0 6 6 6 9 s p e c i e s AFQ c l a s s C o e f f i c i e n t s o f e q u a t i o n V I I I b 1 0 2 b. o 1 0 . 2 0 . 3 0 . 4 0 . 5 • 2 . 6 4 8 9 4 0 0 . 2 8 1 3 5 4 2 . 2 1 2 0 2 0 4 . 7 1 3 3 9 0 0 . 9 3 5 3 8 7 0 . 8 5 7 2 5 7 0 . 7 9 1 8 3 0 0 . 7 3 9 2 0 4 C o t 0 . 3 0 . 4 0 . 5 0 . 5 4 7 3 1 6 1 . 0 1 8 1 3 0 2 . 1 9 0 3 2 0 0 . 9 4 1 2 9 2 0 . 8 4 8 1 7 0 0 . 7 8 4 5 6 9 s p e c i e s AFQ c l a s s r e a l mean,. AFQ" M e a n AFQ o f e a c h f o r m c l a s s e s t i m a t e d b y o v e r a l l e q u a t i o n I s e p a r a t e e q u a t i o n s I 0 . 2 0 . 3 0 . 4 0 . 5 0 . 2 1 9 1 0 . 3 0 6 6 0 . 4 0 2 1 0 . 5 1 1 4 . 0 . 3 1 0 1 0 . 3 2 7 7 0 . 3 6 7 8 0 . 3 8 9 9 0 . 2 4 8 1 0 . 3 1 6 4 0 . 3 7 6 9 0 . 4 4 2 8 For both equations and both species there is a good relationship between coefficients and form,although the relationship for equation VIII is better than for equation I. To check what kind of improvement may be expected from the use 2 of this relationship,the mean (AFQ) was computed for the trees in 2 each (AFQ) class.The computation was done once with the overall taper equation I and once with the separate taper equations for each 2 (AFQ) class.The results are in table XXXVI. Taper equation I,which may be considered a variable form taper equation,accounts for only part of the variation in form.Use of the relationship between the taper-based taper equation coefficients and form improves the estimation system significantly. In equation VIII,the relationship is even better and identical for both species.As the squared absolute form quotient increases, b increases and b. decreases.This relationship can be justified in o 1 a theoretical way.Taper equation VIII is nothing else than (d / D ) 2 = 10 2 bo (1 / H ) 2 b l If double bark thickness would be zero and D measured at ground level we would have the following equation (d / D ) 2 = q / H ) 2 b i For a neiloid form,b^ would be 1.5,for a cone 1.0 and for a paraboloid 0.5.The b^ value decreases with improving form.This is exactly what happens with the b^ coefficient in equation VIII for the different form classes.For western redcedar,b^ went down from 0.94 to 0.74 with increasing form. The cylindrical form factor of a tree as defined by equation VIII is computed as CFF = 102bo /(2 b + 1) which again shows that an increase of b and a decrease of b, means o 1 an improvement of the tree form. 6.2.2.Relation between volume-based taper equation parameters and form In most volume-based taper equations,the free parameters p and q can be compared with the 2b ^  values in equation VIII (as defined in the previous section).This explains why for most species p and q ranged from 1.3 to 2.0.A paraboloid would be represented by 1.0 and a cone by 2.0. The relationship found in the previous section suggests the existence of a similar relationship for the free parameters.This has been tested for taper equations 1*" and 8*" for the same species (see table XXXVII).The results show a strong negative correlation between the parameter value and the form.The parameter values for each form class are almost identical for both equations. 2 The mean AFQ has been computed for each form class using once an overall and second separate parameter values (see table XXXVII). Although the use of the relationship between form and parameter value does not make the system perfect.it improves i t a great deal. Table XXXVII Relation between Volume-Based Taper Equation Parameters and Form Optimum parameter values for western redcedar cottonwood t t 2 equation 1 equation 8 AFQ class p p class AFQ2 *" fc Q2 equation 1*" equation S*" 0.2 2.5 2.5 0.3 1.8 1.8 0.3 2.1 2.1 0.4 1.6 1.6 0.4 1.7 1.8 0.5 1.4 1.4 0.5 1.4 1.5 Western redcedar Mean AFQ of each form class estimated by AFQ2 real equation 1 equation 8 class mean„ overall separate overall separate AFQ P P's . P P's 0.2 0.2191 0.2713 0.2175 0.2530 0.2030 0.3 0.3066 0.2813 0.2696 0.2519 0.2414 0.4 0.4021 0.3053 0.3451 0.2481 0.2694 0.5 0.5114 0.3224 0.4125 0.2372 0.2920 92 6.2.3.Relation between volume-based volume equation coefficients and form Some volume equations,like the logarithmic volume equation 1 and the combined variable volume equation 4 are considered to be variable form equations because the cylindrical form factors of the trees,as defined by these equations,change with D and H. The CFF,as defined by equation l,is equal to CFF = 10b° k D b l " 2 H^" 1 where b^ is usually less than 2 and b^ usually larger than 1 (except for western redcedar) such that the CFF decreases with increasing D and increases with increasing H. The CFF,as defined by equation 4,is equal to CFF = b k/(D2H) + b, k o 1 where both,bQ and b^,are usually positive,such that the CFF decreases with both increasing D and H. Notice that the assumptions which the volume functions make about the CFF may be contradictory.The fact that a variable form volume equation defines different CFF values for different values of D and H does not necessarily mean that they account for the fu l l range of variation in form factors;neither does i t mean that these CFF-D-H relationships are the real ones.Volume equations are often signifi-2 cantly biased.Because V-D H is not a linear relationship,the intercept of equation 4 is more a measure of the range of the observations. But i t is this intercept which will define how the CFF will vary according to D and H. 93 Furthermore,these equations s t i l l assume that the form within the same D and H class is constant.This is not true. To check i f there is a clearcut relationship between volume equation coefficients and form,equations 1 and 4 were fitted on the different CFF classes of western redcedar (see table XXXVIII). In both equations,coefficients and form are highly correlated. For each form factor class the mean CFF was computed,first by using only the overall volume equation and second by using separate volume equations for each form class.The results of this test are in table XXXVIII. In case only one overall volume equation is used,equation 1 accounts only for part of the variation in form while equation 4 gives highly biased results.Only by using the relationship between coefficients and form factor can the whole range of form be accounted for in an unbiased way. 6.2.4.Use of the relation between coefficients and form Is i t of any use to know these relationships between coefficients and form i f form factors or form quotients are impossible or,for many reasons,not practical to measure on each tree? First of a l l a distinction must be made between selecting an appropriate set of coefficients for each individual tree and selecting an appropriate set for a group of trees.The word "group" must be inter-preted in a broad sense.It could be an age group,a provenance or trees subject to a particular thinning or fertilization regime or trees in a particular forest region etc. 94 Table XXXVIII Relation between Volume-Based Volume Equation Coefficients and Form CFF number class of trees 0.3 17 0.4 31 0.5 15 -Western redcedar Coefficients of equation 1 equation 4 b± 10 -2.550800 1.751960 1.056900 3.963690 0.161048 -2.465400 1.880370 0.968841 3.113030 0.194225 -2.273020 1.973080 0.849701 0.824420 0.254421 CFF real class mean CFF overall Mean CFF of each form class estimated by overall separate separate equation 1 equations 1 equation 4 equations 4 0.3 0.318 0.4 0.405 0.5 0.501 0.348 0.317 0.407 0.404 0.452 0.501 0.343 0.318 0.441 0.426 0.578 0.504 As far as individual trees are concerned,the taper and volume estimation might be improved without taking any more measurements,if a good correlation could be found between form factor and a function of D and H (the only two variables measured).The relationship between 2 2 AFQ and the variables D,H,D / H and D H was tested (by plotting) for Douglas-fir,western redcedar and cottonwood.A relationship exists between form and some of these variables,but only a small amount of the variation is accounted for.Some relationships are of the opposite kind for different species.A good correlation is lacking. Therefore,if one wants to select the appropriate set of coefficient for each individual tree,one will reduce the bias of taper and volume by only a small amount i f form factor is predicted from D and H functioi Taking additional measurements for a better prediction of form will ensure a greatly reduced bias but may only be of importance in special circumstances,e.g. research studies of thinning,fertilization,et Selecting the appropriate coefficients may prove to be more valuable for groups of trees i f there really exists an important diffe-rence in mean form.Form may,for some species,be closely related to age. A better form may result from particular climatic conditions.Some prove-nances may have a significantly different bark thickness.Thinning and fertilization may result in a different tree profile. Even i f these factors are responsible for only a small amount of variation in form in the population,then,selecting the appropriate coefficients based on the mean form of the group,will reduce the individual tree bias only slightly,but may nearly eliminate the overall bias of the group. 96 7. Discussion,summary and suggestions After having seen al l these results some conclusions may be drawn.They are an expression of how one's own degrees of belief about particular assumptions may have been changed,or have remained unchanged,and are therefore subjective. Some people would have had different opinions about most assump-tions because of different past experiences.Others would have used different criteria to select the data or might have used the same criteria but would have selected other data.Different people formulate the problems or assumptions in a different way and test them differently. People with the same prior opinions might have changed their minds in a different way after having seen the same results. Therefore, people,interested in these problems,should know why the study was done,how i t was done,what the results look like and what the current opinion of the investigator is.They are then free to make up their own mind. The objectives of this study were explained in the introduction. How i t was done and what the results are can be found throughout the whole text.The way the degrees of belief of the investigator have been modified or have - remained unchanged will now be summarized. Many of the volume equations tested give reasonable results for some species but only a few (equations 1,2,6,7 and 11) are relatively unbiased for a l l species. All the volume equations studied could be converted theoretically into compatible volume-based taper equations,but the conversion was 97 impracticable for some of them (equations 6,7 and 11). Most volume-based taper equations give very similar results but differ in the way the pattern of bias changes from one size class to another.Usually,the taper equations derived from the best volume equations also perform best for diameter estimation. ' None of the taper-based taper equations is without any systematic bias. Sigmoid taper curves may look more attractive but s t i l l f a i l in solving the problem of bias. All the taper-based taper equations could be transformed into compatible taper-based volume equations.Many of these volume equations are simple and similar in form.Most are very much biased for total volume estimation,especially for the larger size classes. The volume-based system can be fairly unbiased for total volume and good for diameter estimation.The best performing equation here seems to be the logarithmic volume equation 1. For the taper-based systems,the equations which gave best results for most species are the logarithmic equation I and Bennett-Swindel1s taper equation III. The taper-based systems are usually more biased for total volume and not better than the volume-based systems for diameter estimation. Therefore,if a compatible estimating system for tree taper and volume is desired,a volume-based system looks more promising. In selecting a particular system or a particular function,one also has to take into account the costs of analysis and introduction of the new system.There may be no need to start off with a completely new system i f one already has a satisfactory volume equation from which a suitable taper equation can be derived. 98 Transformation of a taper equation into a height estimating system may be theoretically correct but often involves large biases. Although biased,section volume estimation is adequate for several equations.If the bias is small and similar in a l l size classes,a correc-tion may be applied.The correction w i l l be positive for some sections and negative for others according to the kind of bias involved.When the heights are unknown,section volume estimation should be avoided in the lower parts of the tree. 2 Conventional methods of equation testing (SE ,R ,etc) often are E . inadequate to compare the effectiveness of different functions. It is recommended that extensive tests of bias of diameter,height, section volume and total volume be carried out before adopting any particular system of tree taper and volume. Taper equations' which do not account for butt flare should not use the observations below breast height. Application of weighting procedures in f i t t i n g taper or volume equations seldom seems to be e f f i c i e n t . l t has l i t t l e or no effect on bias i f the model is correct and i t often makes things worse i f the model is wrong. Non-linear f i t t i n g makes a small difference in some cases and none in other cases.It should be tested in each particular application. For the data analysed herein,in which extreme values have been eliminated,there does not seem to be any need for using Meyer's correc-tion factor for logarithmic equations.However,more investigation is needed to find out how violations of the assumptions affect the effectiveness of the correction factor. 99 In fitting the equations,some species,similar in form,may be combined in the same system without too much loss. However,not too much will be gained unless adjustments are made for bark thickness and form. Whether or not the findings in this study apply to unusually shaped and large trees requires further investigation. Some of these taper and volume systems may be improved further. Separate equations for small and large trees may be effective in case the linear model assumption is not met.An improvement may also result from different conditioning of the coefficients. Addition of other variables may reduce the bias significantly.Use of different taper equations for lower and upper tree bole could be an alternative to sigmoid taper equations. There exists a good relationship between coefficients of some equations and form.This relationship is interesting from both a theoretical and a practical point of view. 100 LITERATURE CITED Avery, T.E. 1967. Forest Measurements. McGraw-Hill Book Co. ,Inc.,N.Y. 290 p. Baskerville, G. L. .1972. Use of logarithmic regression in the estimation of plant biomass. Can. J. Forest Res. 2:49-53. British Columbia Forest Service, 1968. 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Woodlands Section. Index 2349(F-2):499-508. and L. Sayn-Wittgenstein, 1963. Report of the committee on forest mensuration problems. Jour. For. 61:663-667. Hummel, F. C. 1955. The volume / basal area line;a study in forest mensuration. U.K. For. Comm. Bull. No. 24. Husch, B.,Miller, C. I. and T. W. Beers. Forest Mensuration (second ed.). Ronald Press Co., N.Y., 410 p. Ilvessalo, Y. 1947. (Volume tables for standing trees.) Comm. Inst. Forestalls Fenniae. 34:1-149. Jonson, T. 1910. Taxatoriska unders8kningar om skogstrHdens form. I. Granens stamform. SkogsvardsfBreningens Tidskr. 11:285-328. (in Swedish),see Behre (1923). Kozak, A. and J. H. G. Smith, 1966. Critical analysis of multivariate techniques for estimating tree taper suggests that simpler methods are best. For. Chron. 42:458-463. Munro, D. D. and J. H. G. Smith,1969a. Taper functions and their application in forest inventory. For. Chron. 45:1-6. Munro, D. D. and J. H. G. Smith,1969b. More accuracy required. Truck Logger. December:20-21. Kullervo Kuusela. 1965. A method for estimating the volume and taper of tree stems and for preparing volume functions and tables.Comm. Inst. For. Fenniae. 20:1-18. Larson, P. R. 1963. Stem form development of forest trees. For. Sc. monograph No. 5, 42 p. Loetsch, F. and K. E. Haller, 1964. Forest Inventory.Vol. I.,BLV Verlag-gesellschaft. Mllnchen. (trans, by E. F. Brllnig). 436 p. Matte, L. 1949. The taper of coniferous species with special reference to loblolly pine. For. Chron. 25:21-31. 104 Mesavage, C. and J. W. Girard, 1946. Tables for estimating board-foot content of timber.USDA Forest Service. Washington D.C, 94 p. Meyer, H. A. 1938. The standard error of estimate of tree volume from the logarithmic volume equation. Jour. For. 36:340-341. 1944. A correction for a systematic error occurring in the application of the logarithmic volume equation.The Pennsylvania State Forest School,State College,Penna.,Research paper No. 7. 3 p. 1953. Forest Mensuration. Penns Valley Publishers,Inc.,State College,Pennsylvania. 357 p. Meyer, W. H. 1944. A method of volume-diameter ratios for board-foot volume tables.Jour. For. 42:185-194. Moser, J. W. Jr. and T. W. Beers, 1969. Parameter estimation in non-linear volume equations. Jour. For. 67:878-879. Munro, D. D. 1964. Weighted least squares solutions improve precision of tree volume estimates. For. Chron. 40:400-401. 1968. Methods for describing distribution of soundwood in mature western hemlock trees.Univ. of B.C.,Fac. For.,Ph. D. thesis, 188 p. Myers, C. A. and C. B. Edminster, 1972. Volume tables and point-sampling factors for Engelmann spruce in Colorado and Wyoming.USDA Forest Service,Research paper RM-95, 23 p. Naslund, M. 1947. (Formulae and tables for use in determining the volume of standing trees). Medd. Statens Skogsftir. 36:1-81.(in Swedish). Newnham, R. M. 1958. A study of form and taper of stems of Douglas-fir, western hemlock and western redcedar on the University research forest,Haney,B.C.,Univ. of B.C.,Fac. For.,M.F. thesis, 71 p. 1967. A modification to the combined variable formula for computing tree volumes. Jour. For. 65:719-720. Osumi, . S. 1959. Studies on the stem form of the forest trees (1). Jour. Jap. For. Soc. 41:471-479. (in Japanese,abstract in English). Petterson, H. 1927. Studier over Stamformen. Medd. Statens SkogfbrsBks-anstalt. 23:63-189. (in Swedish). Pollanschutz, J. 1966. A new method of determining stem form factor of standing trees. IUFRO Advisory group of forest statisticians confe-rence, Sec. 25. Stockholm,1965. Dept. forest biometry. Roy. Coll. For. Res. Notes,9:266-276. 105 Prodan, M. 1965. Holzmesslehre. J. D. SauerlUnder's Verlag,Frankfurt am Mein, 644 p. Schmid, P.,Roiko-Jokela, P.,Mingard, P. and M. Zobeiry, 1971. The optimal determination of the volume of standing trees. Mitteilungen der Forstlichen Bundes-Versuchsanhalt, Wien. 91:33-54. Schumacher, F. X. and F. S. Hall, 1933. Logarithmic expression of timber-tree volume. Jour. Agric. Res. 47:719-734. Shinozaki, K.Yoda Kyoji,Hozumi, K. and T. Kira, 1964. A quantitative analysis of plant form. The pipe model theory.Jap. Jour. Ecol.14:97-104. Smalley, G. W. and D. E. Beck, 1971. Cubic-foot volume table and point-sampling factors for white pine plantations in the southern Appalachians.USDA Forest Service,Research note S0-118. 2 p. Smith, J. H. G, and R. E. Breadon, 1964. Combined variable equations and volume-basal area ratios for total cubic-foot volumes of the commercial trees of B.C. For. Chron. 40:258-261. 2 and J. W. Ker, 1957. Timber volume depends on D H. British Columbia Lumberman,September,1957. Reprint, 2 p. and A. Kozak, 1967. Thickness and percentage of bark of the commercial trees of B.C.,Univ. ofB.C.,Fac. For.,mimeo, 33 p. and A. Kozak, 1971. Further analyses of form and taper of young Douglas-fir,western hemlock,western redcedar and silver f i r on the Univ. of B.C. research forest.Paper presented at the NWSA annual meeting, Idaho, 1971. mimeo, 8 p. and D, D. Munro, 1965. Point sampling and merchantable volume factors for the commercial trees of B.C.,Univ. of B.C.,Fac. For., mimeo, 41 p. Speidel, G. 1957. Die rechnerischen grundlagen der leistungskontrolle und ihre praktische durchfUrung in der forsteinrichtung.Schriften-reihe der Forstlichen FakultUt,UniversitMt Gtfttingen. No. 19, 118 p. Spurr, S. H. 1952. Forest Inventory. Ronald Press Co.,N.Y. , 476 p. Stoate, T. N. 1945. The use of a volume equation in pine stands. Australian forestry. 9:48-52. . Turnbull, K. J. and G. E. Hoyer, 1965. Construction and analysis of comprehensive tree volume tarif tables.Resource Management Report No. 8. Dept. Nat. Res.,Washington. 63 p. Van Laar, A. 1968. The measurement of out-of-reach diameters for the estimation of tree volumes from volume tables. South African For. Jour. 64:19-23. Van Soest, J. 1959. Stem form and volume of Japanese larch in the Netherlands. Wageningen, Williams, D. H. 1972. Bias in least squares regression.Univ. of B.C., Fac. For.,M.Sc. thesis, 44 p. Zaharov, V. K. 1965. (Form height of trees and stands). Lesn. Hoz. 18:26-28. (in Russian). 107 APPENDIX 1 Common Names and Latin Names of the Tree Species and Species Groups ^  1. Trembling Aspen (Populus tremuloides Michx.) 2. Coast Balsam Species (Abies amabilis (Dougl.) Forbes and A. grandis (Dougl.) Lindl.) 3. Western Redcedar (Thuja plicata Donn.) 4. Black Cottonwood (Populus trichocarpa Torr. and Gray) 5. Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) 6. Lodgepole Pine (Pinus contorta Dougl.) 7. Ponderosa Pine (Pinus ponderosa Laws.) 8. Slash Pine (Pinus e l l i o t t i i Engelm. var. e l l i o t t i i ) 9. Western White Pine (Pinus monticola Dougl.) 10.Interior Spruce Species (Picea glauca (Moench) Voss,P. Engelmanni Parry and P. mariana (Mill.) B 0S„P 0) Based on Appendix I from Browne (1962),except numbers 7 and 8. 108 Appendix 2 Numbering of the Volume and Taper Equations 1.Volume-based systems Name Number of volume volume-based equation taper equation logarithmic volume equation logarithmic combined variable Honer's volume equation combined variable volume equation weighted combined variable comprehensive combined variable weighted comprehensive combined variable (zero intercept) V / B1as function of H ^ V / B1 as function of H and H Meyer's volume-diameter ratio equation cylindrical form factor volume equation non-linear form of logarithmic,no weighting non-linear form of logarithmic,with weighting non-linear form of Honer's equation,no weighting non-linear form of Honer's equation,with weighting 1 l ! 2 2' 3 3 4 < 5 6 6* 7, 7* 8 8* 9 9t 10 1 0 t 11 U t 12 1 2 t 13 1 3 t 13(w) 13^(w) 14 l 4 t 14(w) 14 (w) 2.Taper-based systems Name Number of taper equation taper-based volume equation logarithmic taper equation I taper equation of Kozak,Munro and Smith II Bennett and Swindel's taper equation III equation with same form as 4*- IV Matte's taper equation V Osumi's taper equation VI taper equation of Bruce,Curtis and Vancoevering VII equation with same form as 8^  VIII equation with same form as 9 IX unconditioned Behre' taper equation X conditioned Behre"s taper equation XI II III" IVV VV VII VIII IXV xv XIV v v Appendix 3 Summary of the Volume-Based Volume Equations 3.1.Linear volume equations 1. log V = b + b, log D + b log H o 1 I species b o b l b2 SEE 10 2 2 i r io D -2.83276 1.76116 1.20365 0.37144 99.4 C -2.20396 1.68684 0.94454 0.49756 98.6 S -2.54178 1.88818 1.01215 0.38673 99.3 B -2.60213 1.69759 1.17051 0.35998 99.6 A -3.00096 1.72263 1.33892 0.37248 99.2 Cot -3.25913 1.73206 1.44513 0.38152 98.9 PI -2.48880 1.78377 1.05510 0.33634 99.2 Pw -2.51801 1.82359 1.04051 0.32335 99.3 2. log V = b + b. log (D2H) o 1 species b o b l SE E 10 2 2 IT 10-D -2.52827 0.960198 0.40079 99.3 C -2.12342 0.869511 0.49560 98.6 S -2.48746 0.962877 0. 3854.2 99.3 B -2.34958 0.937439 0.38904 99.5 A -2.57462 0.983941 0.40855 99.0 Cot -2.69274 1.002760 0.43874 98.5 PI -2.35261 0.934576 0.34784 99.2 Pw -2.40040 0.944235 0.32372 99.3 3. D2 / V = b + b, / H o 1 species b o S E E 2 2 R 10 D 0. 697910 430.855 0.48836 90.2 C 2.830720 252.549 0.82766 66.3 S 0.616650 383.407 0.49396 91.7 B 0.635343 355.833 0.45566 93.6 A -0.879354 547.995 0.84125 80.9 Cot -0.033198 436.402 0.62789 87.8 PI 0.814179 349.554 0.56406 86.0 Pw 0.911824 345.288 0.34874 90.4 110 4. V = b + b, D2H o 1 species b o bj_ 102 2 2 R 10 D 2.43215 0.186177 8.55 97.2 C 5.85424 0.168317 7.00 96.5 S 2.19154 0.211675 4.73 98.0 B 6.79594 0.206315 9.84 99.0 A 0.68262 0.221711 2.59 97.3 Cot 0.55679 0.199651 1.90 97.0 Pl 2.15687 0.218430 3.31 98.2 Pw 2.88724 0.211811 5.74 98.3 5. V /(D2H) = b + b, /(D2H) o 1 species 2 b 10 o b l SEE 103 2 2 R 10 D 0.195342 0.544671 0.18985 14.1 C 0.200187 1.578370 0.30334 41.1 S 0.224735 0.371562 0.21057 10.6 B 0.230976 0.770271 0.26654 15.4 A 0.230173 0.089940 0.21312 1.0 Cot 0.211162 • -0.090401 0.20058 0.9 Pl 0.230832 0.855657 0.20327 31.1 Pw 0.221252 0.890530 0.18189 26.2 + b D + b 2 H + b 3 D H + b. D2 4 + b 5 D2H species b o b l b2 b 3 10 b. 10 4 2 b 5 10 S E E 2 2 R 10 D 8.57299 -0.324102 -0.228254 0.308613 -0.712229 0.149274 7.47 98.0 C 10.73580 -0.858671 -0.371924 0.562510 -0.051280 0.026351 6.22 97.4 S 15.28650 -3.747810 -0.172540 0.439281 2.052750 -0.011020 4.57 98.2 B -34.39410 6.715050 0.108075 -0.203816 -2.960250 0.339907 8.60 99.3 A 9.44231 -0.533039 -0.298294 0.474012 -1.007390 0.135440 2.28 98.0 Cot 10.37870 -1.430180 -0.268870 0.535666 -0.537087 0.064625 1.43 98.3 Pl 2.13527 -0.114786 -0.109543 0.226315 -0.166353 0.146681 3.11 98.4 Pw -6.58185 -0.121157 0.185972 -0.110171 0.555434 0.199308 5.80 98.3 I l l 7. V /(D2H) = b + b/(D H) +-b_/ D2 + b- / D + b./ H + b /(D2H) o 1 I j 4 5 species b 10 r o D 0.177184 C 0.079447 S 0.079118 B 0.166709 A 0.349691 Cot 0.045153 Pl 0.060537 Pw 0.138884 0.262031 -0.074431 -1.685210 0.305371 1.478920 -1.844980 -1.628170 -1.399970 -0.139823 -0.130385 -0.123082 -0.137910 0.018987 -0.202850 -0.190995 0.008645 b 3 10 0.204433 0.272067 0.293750 0.266846 -0.055435 0.470614 0.411507 0.129167 b. 10 4 -0.854369 0.068316 0.876817 -0.594730 •1.728240 0.114538 0.541317 0.759510 3.33407 1.63112 7.20132 0.63753 -4.03959 9.39339 8.89101 3.57925 2 2 SE,, ET 10 0.1684 36.7 0.2500 62.6 0.2031 20.6 0.1968 56.6 0.1954 19.8 0.1716 30.2 0.1827 45.8 0.1683 39.7 8. V = b D2H o 2 S E E 2 JI species b 10 o R 10 D 0.190636 8.63 97. 1 C 0.181876 8.05 95.3 S 0.218613 4.89 97.8 B 0.213506 10.84 98.8 A 0.226414 2.61 97.3 Cot 0.204669 1.92 96.9 Pl 0.227784 3.54 97.9 Pw 0.218347 5.92 98.1 9. V / B* = b "+ b. H o 1 species b o b l SE E 2 2 R 10 D 2.09803 0.345823 4.06856 87.0 C 12.52330 0.225925 3.95751 61.8 S 4.44776 0.368000 3.45596 87. 7 B 9.38033 0.339157 4.62709 86.2 A 0.50554 0.419175 3.00959 86.7 Cot -2.86449 0.423589 2.71700 84.8 Pl 5.62314 0.371020 3.03212 86.4 Pw 8.14278 0.337024 3.44886 83.3 10. V / B* = b + b, H + b„ H2 o 1 2 species b o b l b 2 103 2 2 R 10 D 12.01110 0.141068 0.971455 4.00404 87.6 C 10.01420 0.293838 -0.424102 3.98295 61.9 S -3.85863 0.578065 -1.216960 3.39840 88.2 B -2.63995 0.607556 -1.326650 4.38988 87.8 A -2.08107 0.494368 -0.512147 3.02087 86.8 Cot 1.93972 0.286102 0.942181 2.72106 84.8 PI 0.92035 0.498520 -0.806094 •3.02151 86.6 Pw -0.37319 0.525672 -0.978412 3.40892 83.8 = b + b, H + h D + b„ D H o 1 2 3 species b 0 b 102 b2 10 b3 i o2 2 2 R 10 D -0.294880 1.27424 -0.491841 0.179870 3. 56637 96.8 C -0.551858 2.09796 0.257830 0.086887 3. 27701 94.1 S -0.560030 0.98764 0.592112 0.138466 2. 74893 96.7 B -0.260299 1.75939 -0.425959 0.185516 3. 84807 98.3 A 0.032511 0.649 67 -0.729600 0.254262 1. 82375 96.4 Cot -0.201713 0.97272 -0.637007 0.217405 1. 32613 96.5 PI -0.268547 1.02856 0.121876 0.169301 1. 99489 97.1 Pw -0.265554 0.66074 0.340717 0.167891 3. 06261 96.7 = b + o b x (H /(H - 4.5))2 3 2 A 2 2 species b 10 o b1 10 E 1 0 R 10 D 0.054422 0.177392 1.94184 10.1 C -3.713480 0.517358 2.85293 47.9 S 0.520494 0.158726 2.09424 11.6 B -0.273878 0.240028 2.58145 20. 7 A 2.274120 0.004383 2.14215 0.0 Cot 3.568640 -0.129582 1.96237 5.0 PI -0.167796 0.229431 2.22406 17.5 Pw -1.247120 0.321505 1.82124 26.0 3.2.Non-linear volume equations 13 and 13 (w). V = 10b° D bl Hb2 without weighting with weighting species b o b o \ b2 D -2.9G79 1.6948 1.2809 -2.8271 1.7712 1.1958 C -2.3220 1.4743 1.1437 -2.1874 1.7196 0.9173 S -2.2681 1.8669 0.8898 -2.5375 1.8954 1.0067 B -2.7860 1.6615 1.2797 -2.5969 1.7079 1.1624 A -3.1407 1.6608 1.4462 -2.9562 1.7507 1.3013 Cot -3.3495 1.5947 1.5667 -3.2324 1.7481 1.4235 PI -2.4479 1.7385 1.0616 -2.4746 1.7936 1.0427 Pw -2.3236 1.9296 0.8823 -2.5434 1.8071 1.0633 14 and 14 (w). V = D2/(b + b / H) o 1 without weighting with weighting species b o bi b o bi D -0.90764 645.36 0.44330 451.51 C 1.53950 391.04 2.69910 254.86 S 1.10420 343.61 0.61799 379.79 B 0.45065 403.54 0.84351 331.44 A -0.64754 501.53 0.07047 424.91 Cot -1.22320 595.87 -0.69806 530.35 PI 0.38200 401.47 0.88161 341.01 Pw 0.74006 366.73 0.92371 341.49 3.3.Volume equation 1 for combinations of species combination number coefficients of equation 1 1. 2. 3. 4. 5. •2.55388 •2.50380 •2.50416 •3.06473 •2.45284 1.89858 1.88952 1.82087 1.78035 1.80574 1.00835 0.98854 1.02751 1.32984 1.02162 3.4.Volume equations for data adjusted for butt flare 1. log V = b Q + b log D + b 2 log H species b o b l b2 SEE 10 2 2 R 10 D -2.80967 1. 76364 1.19068 0.37131 99.4 C -2.26455 1. 66945 0.97222 0.48639 98.7 S -2.63154 1. 81916 1.08910 0.40659 99.2 B -2.57132 1. 67844 1.16149 0.34646 99.6 A -2.99624 1. 69457 1.34460 0.33833 99.3 Cot -3.23852 1. 73268 1.43303 0.35945 99.0 PI -2.50795 1. 75606 1.07620 0.33217 99.2 Pw. -2.50698 1. 83438 1.02347 0.32528 99.3 4. V = b + b, D2H o 1 2 2 2 species b o b x 10 R 10 D 2.50881 0.185490 8.41 97.3 C 5.30481 0.158568 6.62 96.5 S 2.57052 0.200707 4.80 97.7 B 7.06665 0.198745 10.08 98.9 A 0.85918 0.211975 2.27 97.7 Cot 0.60635 0.197921 1.88 97.0 PI 2.21598 0.212704 3.26 98.1 Pw 2.77376 0.206884 5.80 98.1 115 Appendix 4 Derivation of Compatible Taper Equations from Volume Equations arid the Functions to Estimate Height and Section Volume 1*".Derivation of taper equation from volume equation 1 Integration of the taper equation over the total length of the tree must yield volume equation 1 H f (d 2/ k) dl = 10b° D bl Hb2 0 ~ or d2H / k = 10bo D bl Hb2 d 2 - k 10 b ° D bl H^" 1 where b ,b- and b„ are the coefficients of volume equation 1. o 1 2. A more general and useful taper equation is d 2 = k (p + 1) 10b° D bl l P/(H p- b2 + 1) or d - (k 10bo(p + 1))* D V 2 1 P / 2 H(b2-P"l>/2 which is the same as d = a Db 1 C HS ( l f c) . b J-where a = (k 10 o (p + 1)) 2 c = p / 2 b = bj/2 e = (b 2 - p - 1)/ 2 and where p is the only free parameter. Other ways of writing this taper equation are b c "* 6 d/D = a l / H which looks very familiar and log d = log a + b log D + c log 1 + e log H which is the logarithmic form of equation 1*". The height of a given diameter can be computed by the following transformation of equation l * " : 1 - (d /(a D V ) 1 / C 116 where a,b,c and e have same meaning as in equation 1 . Volume of a given section can be computed as )/(k (2 c + 1)) = ) where 1^  and 1^  are respectively the lower and upper distances from the tip of the tree. 2*". Taper equation from volume equation 2. Integration of the taper equation over total tree length must yield volume equation 2 This equation leads to the same type of taper equation as 1 ,however, some coefficients will be different: t d = a D b 1 C H6 where a = (10b° k (p + 1)) c P / 2 b = b 1 e (b x - p - 1)/ 2 and p is the free parameter. Height for a given diameter is estimated by 1 = (d /(a D b H G)) and section volume can be computed as = i o b ° D 2 b i H ( br p- X ) ( i f 1 - i f 1 ) 3 .Taper equation from volume equation 3 Integration of taper equation must yield volume equation 3 f (d 2/ k) dl = D2/(b + b./ H) 0 J " o 1 or d2H / k = D2/(b + b,/ H) — d 2 = D2 k /(b H.+ b,) o 1 o 1 A more general equation is d 2 = k (p + 1) D2 l P/(b H P + 1 + b, HP) — o 1 or d - (a D2 l P/(b H P + 1 + c H P)) % ( a' where a = (k (p + 1)) b = b c = b. o 1 and p is the free parameter. Height of a given diameter is computed as 1 = (d 2(b H P + 1 + c H P)/(a D 2 ) ) 1 / p and section volume as V = D 2(1 P + 1 - l P + 1 ) / ( b H P + 1 + c HP) s -I -I 4*". Taper equation from volume equation 4 Integration of taper equation yields volume equation 4 f H 2 2 / (d / k) dl = b + b, D H J — o 1 H or d 2 H / k = b + b, D2H — d 2 = k b / H + k b, D2 o 1 o 1 or more generally as d 2 = k b (p + 1) 1P/ H1*"1 + k b 1 ( q + 1) D2 I q / Hq o — 1 — or d = (a 1P/ H P + 1 + b D2 I q/ H q) % where a = k b (p + 1) b = k b,(q + 1) o 1 and p and q are the two free parameters. Volume of a tree section is computed as V = b ( 1 P + 1 - i f 1 ) / H^ 1 + b D 2 ( i q + 1 - i f 1 ) / Hq s o —1 —2 1 -1 -L 5 .Taper equation from volume equation 5 The only difference with equation 4 is that the intercept b , o in equation 5,corresponds with the slope coefficient in equation 4 and vice versa,so that d = (a 1P/ HP + 1 + b D 2 lq/ Hq)% ( 5t ) with a - k b ( p +1) b = k bQ(q + 1) and p and q are the free parameters. The section volume is given by V = b M f1 - i f1) / H^1 + b D2 ( i f1 - i f1) / Hq s 1 — 1 — l o —1 —I 6*".Taper equation from volume equation 6 T H 2 2 2 / (d / k ) dl = b + b , D + b H + b_ D H + b. D + b D H Q J — o 1 2 J 4 D d2 = k ( b + b, D + b n H + b n D H + b. D2 + b c D2H)/ H O 1 2. 3 "4 J A general taper equation is d = (a 1P/ H^1 + b D Iq/ Hq + 1 + c lr/ Hr + e D f / HS + f D2 i V Ht + 1 + g D2 1U/ HU where a = k (pr+ 1) b Q c = k (r + 1) b 2 f = k (t + 1) b = k ( q + 1) b e = k (s + 1) b 3 g = k (u + 1) b $ and p,q,...,g are the free parameters. Section volume is computed as V = b ( i f 1 - i f 1 ) / H^1 + h D ( i f 1 - i f 1 ) / Hq + 1 + S O — 1 -2. 1 — 1 -Z b 2 ( l f 1 - i f 1 ) / Hr + b 3 D ( i f 1 - i f 1 ) / HS + \ ^ i r 1 - i 2 + i > / - H t + - + b 5 ° 2 (ii + i - - 2 + l ) / r u " 7 .Taper equation from volume equation 7 119 2 After multiplication by D H, equation 7 has exactly the same form t as volume equation 6.Therefore,taper equation 7 has the same form as equation 6 t -the only differences will be in the coefficients which are: a = k (p + 1) b 5 c = k (r + 1) b 2 f = k (t + 1) b b = k (q + 1) b. e = k (s + 1) b„ g = k (u + 1) b 1 J o t t To make the section volume equation of 6 applicable for 7 ,replace only b by b r and vice versa. J o J 5 8*".Taper equation from volume equation 8 f (d 2/ k) dl = b D2H 0 J - o d 2 = k b D2 and more general d 2 = k (p + 1) b D21P/ HP d 2 H / k = b D2H o or d = a D (1 / H ) p / 2 ( 8* ) where a = (k (p + 1) bj^ The height equation is 1 = (d 2 H P/(a 2 D 2 ) ) 1 / P and the section volume equation is V = b D2 ( 1 P + 1 - i f 1 ) / H P s o -1 -2 9*".Taper equation from volume equation 9 / (d 2/ k) dl = b B ' + K H B' —>• d 2 H / k = b B* + b H B* 0 J - o 1 o 1 d 2 = k (b Q B1 / H + b^ B') and a more general equation is d = (a D2 1P/ H^ 1 + b D2 l q / Hq)^ -• ( 9fc ) where a = (p + 1) b b = (q + 1) b, o i and p and q are the two free parameters. 120 Section volume is given by V = b B'(1 P + 1 - i f 1 ) / H^ 1 + h B 'Clf 1 - i f 1 ) / Hq 10*".Taper equation from volume equation 10 0 / (d / k) dl = b B'+b, B'H + b„ B1 H ./ — o ' l 2 ,2 d = k (b B* / H + b B* + b 2 B1 H) or more general d = (a D2 fl H P + 1 + b D2 I q / Hq + c D2 l r / H r _ 1 ) % ( 10* ) where a = (p + 1) b b = (q + 1) b, c = (r + 1) b„ o 1 I The section volume equation is V s = b Q B' ( i f 1 - i f 1 ) / H P + 1 + b x B 'Clf 1 - i f 1 ) / Hq + b 2 B ' q f 1 - i f 1 ) / ^ " 1 ll t.Taper equation from volume equation 11 H2 2 2 (d / k) dl = b D + b, D H + b, D + b, D H d 2 = k (b D / H + b D + b D2/ H + b D2) and more general o 1 z 3 d = (a D 1P/ H P + 1 + b D I q / Hq + c D2 l r / H r + 1 + e D2 1S/ H S) % ( l l ' ) where a = k (p + 1) b Q c = k ( r + l ) b 2 b = k (q + 1).b e = k (s + l)-b and p,q,r and s are four free parameters. Section volume is given by • V s = b o D ( i f 1 - i f 1 ) / H P + 1 + b x D ( i f 1 - i f 1 ) / Hq + 0 b 2 D 2 ( l f X - i f 1 ) / H r + 1 + b 3 D 2 ( l f 1 - i f 1 ) / HS 12 .Taper equation from volume equation 12 /H (d2/ k) dl = b D2H + b, D2H (H /(H - 4.5))2 u - o 1 d2 = k (b D2 + b„ D2(H /(H - 4.5))2) or more general o 1 d = (a D2 1P/ HP + b D2 Iq H2"q/(H - 4.5)2)% ( 12*) where a -= k (p + 1) .b b = k (q + 1) b and p and q are the two free parameters. The section volume equation is V = b D2(1P + 1 - i f1) / HP + b. D2 H2-q( iq + 1 - i f V C H - 4.1 s o -1 —c 1 -1 122 Appendix 5 Summary of the Taper-Based Taper Equations 5.1.Linear taper equations I. log d = b + b o 1 log D + b 2 log 1 + b 3 log H species b o b l b2 b3 SE^ 102 E 2 2 R 10 D -0.162157 0.868138 0.653171 -0.512414 4.71777 97.0 C 0.169313 0.746660 0.821086 -0.749014 7.55771 93.6 S -0.044339 0.866427 0.757857 -0.650812 5.35571 96.6 B -0.106822 0.759437 0.721269 -0.513237 5.60663 96.4 A -0.467402 0.715920 0.779922 -0.370981 7.05527 94.2 Cot -0.488623 0.781775 0.865773 -0.487176 6.23952 95.8 PI -0.090332 0.790349 0.696068 -0.520680 4.89761 96.4 Pw 0.051129 0.889893 0.735208 -0.689888 4.54512 97.2 II. (d / D)2= b Q + b 1(h / H) + b 2(h / H) 2 species b o b l b2 SEE 10 2 2 R 10^ D 0.871680 -1. 23628 0.364600 0.950599 88.6 C 1.261930 -2. 52386 1.261930 2.320710 75.3 S 1.193263 -2. 18970 0.996437 1.788320 81.7 B 1.168674 -1. 98361 0.814936 1.926460 78.4 A 1.119859 -1. 87857 0.758711 2.137020 72.3 Cot 0.989716 -1. 63695 0.647234 0.724694 94.9 PI 1.111171 -1. 72886 0.617689 1.119800 90.2 Pw 1.129284 -1. 94681 0.817526 1.342570 87.6 III. d = b D 1 /(H - 4.5) + b, 1 (h - 4.5) + b H 1 (h - 4.5) + o — 1 — I — b 3 1 (h - 4.5)(H + h + 4.5) species b o b x 103 4 b 2 10 H 4 b 3 10 S E E 2 2 R 10 D 0.90353 1.33319 -0.276904 0.145493 0.97 96.5 C 1.04891 0.46416 -0.851405 0.495464 1.67 92.1 s 1.05401 0.92629 -0.446678 0.249172 1.24 94.1 B 1.06210 1.24688 -0.400974 0.214041 1.76 94.7 A 1.00922 0.89985 -0.522060 0.312830 1.02 92.5 Cot 0.93994 0.35499 -0.109357 0.089113 0.49 97.5 PI 1.01177 1.97007 -0.618876 0.315387 0.74 97.1 Pw 1.029 71 1.21326 -0.367131 0.192799 1.04 96.5 IV. (d / D) 2 = b Q 1P/(D2 H P + 1) + b (1 / H) q species b o b l P q SEE 10 2 2 R 10 D 159.909 0.84360 1.0 1.3 0.954973 88.5 C 362.694 1.22912 1.0 2.1 2.287740 76.2 S 13.491 1.17010 1.0 1.7 1.810280 81.2 B 62.077 1.17265 1.0 1.7 1.948000 77.9 A -46.704 1.10536 1.0 1.5 2.153190 71.9 Cot -9.612 0.99307 1.0 1.6 0.727191 94.8 Pl 170.444 1.08402 1.0 1.5 1.132630 90.0 Pw 164.306 1.09993 1.0 1.6 1.361990 87.2 / D) 2 = b (1 / H ) 2 + b.(l o — 1 — / H ) 3 + b 2 ( l / H ) 4 species b o b l b2 SE_ 10 R2 102 h. D 4.62922 -8.7816 5.11886 0.825042 91.5 C 6.22618 -14.6201 9.98878 1.862160 84.3 S 5.44968 -11.4127 7.34664 1.539780 86.4 B 5.67746 -11.4947 7.14594 1.742120 82.4 A 5.35872 -10.7451 6.67131 2.004280 75.7 Cot 2.92868 -4.1791 2.27177 0.691878 95.3 Pl 5.11474 -9.5667 5.68553 0.976763 92.6 Pw 5.30943 -10.7156 6.68614 1.094080 91.8 / D) = b (1 / H ) + b,(l / o — 1 - H ) 2 + b 2 q , / H ) 3 species b o b l b2 SE,, 10 R2 102 D 2.28893 -3.021.5 1.69490 0.539458 94.5 C 2.30077 -3.64133 2.54321 1.027900 89.2 S 2.22228 -2.96840 1.87881 0.785964 92.7 B 2.34849 -3.10336 1.86728 0.863131 90.8 A 2.21314 -2.75188 1.62288 0.891086 89.9 Cot 1.56226 -1.02407 0.45500 0.522427 96.3 Pl 2.34731 -2.95912 1.70004 0.601989 94.9 Pw 2.25931 -2.95878 1. 79786 0.628728 94.9 124 VII. (d / D) b X 3 / 2 + b,(X 3 / 2 - X3) D (IO - 2) + b 0 ( X 3 / 2 o 1 I X3) H (IO - 3) 3/2 32 -5 3/2 32 1/2 -3 + b (3T - XJ ) H D (10 ) + b4(X ' - X ) H ' (10 ) + b 5(X 3 / 2- X 4 0) H 2(10" 6) species b^ b i b 2 D 0.73450 -1.99546 3.53279 C 0.88447 -2.78067 2.79508 S 0.88363 -0.64805 1. 74957 B 0.90870 -2.00471 4.04452 A 0.84443 -0.90189 0.89915 Cot 0.81910 -3.03294 0.63127 Pl 0.90531 -3.59780 4.66746 Pw 0.89597 -2.03845 1.85461 2 2 SE R 10 0.51737 0.70021 -8.91048 3.34638 -3.71850 5.09090 -3.49117 11.03750 2.36775 15.68650 8.93185 14.21590 14.50930 4.15988 8.41048 13.88770 -9.0379 -37.1231 -15.1479 -27.8347 -24.5373 -11.8752 -13.4578 -33.4201 0.073476 0.137140 0.125641 0.139566 0.187546 0.077595 0.078120 0.075477 52.9 77.5 67.8 60.4 37.8 40.9 66.6 80.2 VIII. (d / D) = b (1 / H ) p o — species b o P SEE 10 2 : R 10 D 0.86098 1.3 0.964611 88.3 C 1.26193 2.0 2.330710 75.3 S 1.17223 1.7 . . 1.809440 81.2 B 1.15874 1.6 1.944430 78.0 A 1.09287 1.5 2.153450 71.9 Cot 0.99023 1.6 0.727003 94.8 Pl 1.08960 1.4 1.142300 89.8 Pw 1.09376 1.5 1.379860 86.9 IX. (d / D) 2 = b 1P/ H P + 1 + b o — species b o b l D 4.19968 0.81415 C 21.08250 1.09720 S 2.32421 1. 15975 B 5.52793 1.13070 A -4.40072 1.16083 Cot -3.96910 1.03407 Pl 6.41902 1.01592 Pw 7.39517 1.04899 q / H ) q 2 2 p q SEE 10 R 10 1.0 1.3 0.960627 88.4 1.0 3.4 2.134400 79.3 1.0 1.8 1.802380 81.4 1.0 1.8 1.951150 77.8 1.0 1.5 2.151000 72.0 1.0 1.5 0. 719214 94.9 1.0 1.5 1.131260 90.0 1.0 1.7 1.357030 87.3 I 125 0 = ( l / H)/(b + o y i / H)) fitted as D / d = species b o b l 2 2 R 10 D 0.465224 0.690940 0.367501 91.4 C 0.662119 0.354711 0.862773 80.9 S 0.516681 Q.556082 0.423853 91.8 B 0.504741 0.511545 0.517858 86.9 A 0.726016 0.051026 0.992652 81.1 Cot 0.740751 0.290296 0.958412 81.5 Pl 0.457406 0.591305 0.453220 88.1 Pw 0.509568 0.541935 0.364708 92.8 XI. ((1 / H)/(d / D) - 1) = b ((1 / H) - 1) 2 2 species b l R 10 D 0.431930 0.113473 69.8 C 0.314941 0.160535 24.4 S 0.423523 0.098748 60.9 B 0.473939 0.102240 65.2 A 0.410182 0.120581 50.1 Cot 0.254496 0.116287 31.6 Pl 0.501599 0.087244 74.4 Pw 0.446054 0.087263 70.7 5.2.Taper equation I fitted with different dependent variables dependent variable log d •0.162157 •0.136230 -0.096128 Douglas-fir 0.868138 0.883840 0.906790 0.653171 0.632780 0.701590 •0.512414 -0.515890 -0.613660 SE„ of d t (inches) 1.04 1.03 1.08 dependent variable log d d„ Cottonwood -0.488623 -0.276700 -0.175560 0.781775 0.863640 0.904020 0.865773 0.792930 0.800010 -0.487176 -0.576870 •0.657910 SE of d (inches) 0.55 0.51 0.51 5.3.Taper equations for data adjusted for butt flare I. log d = b Q + b log D + b 2 log 1 + b 3 log H species b o b2 b3 SEE 10 2 2 R 10 D -0.450252 0.868583 0.653324 -0.518731 0.460612 97.1 C 0.157124 0.747335 0.785345 -0.718886 0.661530 94.7 A -0.462614 0.704369 0.763453 -0.356531 0.666486 94.6 Cot -0.481989 0.780574 0.863673 -0.488747 0.619915 95.8 II. (d / D) 2 = b + b.(h / H) + b 0(h / H) 2 O i L species b o V 'SE E 10 2 R 10' D 0.86687 -1. 21924 0.352370 0.780893 92.0 C 1.03041 -2. 06082 1.030410 1.065230 88.2 A 0.96139 -1. 32943 0.368035 0.642552 95.4. Cot 0.97231 -1. 57660 0.604290 0.589740 96.4 IV. (d / D) 2 = b 1P/(D2 H P + 1) + b. (1 / H) q o — 1 -species b b o 1 p q SE E 10 R^  D 190.755 0.837508 1.0 1.3 0.779648 92.0 C 594.564 0.957025 1.0 2.1 0.949124 90.7 A 108.710 0.966915 1.0 1.5 0.697680 94.6 Cot 11.225 0.975905 1.0 1.6 0.595081 96.3 V. (d / D) 2 = b (1 / H) 2 + o — b (1 / H)3 + b 2 ( l / H) 4 species b^ b l b2 SE E 10 2 2 R 10* D 4.57828 -8.62166 5.00122 0.636191 94.7 C 3.64827 -6.28611 3.67434 0.845534 92.6 A 4.02690 -6.45690 3.43559 0.593059 96.1 Cot 2.77992 -3.70044 1.91081 0.559366 96.8 127 Appendix 6 Derivation of Compatible Volume Equations from Taper Equations and the Functions to Estimate Diameter,Height and Section Volume IV.Derivation of a volume equation from taper equation I Diameter is estimated from taper equation I. as d = 10b° Dbl lb2 Hb3 and distance from the tip for any diameter is computed as 1 = (10"bo D_ bl d H"b3)l/b2 If 1^  and 1^ are respectively the lower and upper distance from the tip,then the volume of that section is obtained by • V = / (d2/ k) dl = 1 02 b° D2 bl H2b3 ( l2 b2+ 1 - l2b2+l) / ( k (2b0 + 1)) s , J — -1 - Z Z -2 Total volume is derived from section volume by taking 1^  = H and 12 = 0 ,then V = 1 02 b° D2 bl H2 b2+ 2 b3+ 1 / (k (2 b2 + 1)) or log V = a + b log D + c log H ( IV ) where a = log (102 b° /(k (2 b£ + 1))) b = 2 b c = 2 b2+ 2 b3+ l v II .Volume equation from taper equation II The diameter equation is d = D (b + b.(h / H) + b„(h / H)2)% o 1 Z and the height equation is h = H (-bx -(b2 - 4 b2(bQ - (d / D)2))%)/(2 b2) 128 Section volume is computed as Vs = D 2(b Q(h 2 - h±) + b^h 2 - h2)/(2 H) + b 2(h 3 - h3)/(3 H 2))/ k where h^ and h 2 are respectively the lower and upper height of the section. The total volume is V = a D H ( I I V ) where a = (b + b,/ 2 + b„/ 3)/ k o 1 / Taking into account the first conditioning,"a" can be written as a = (-b1-/2 -2b2/3)/ k After the second conditioning: a = b3/(3 k) v III .Volume equation from taper equation III The diameter equation is the taper equation itself. Section volume is obtained after a lengthy,but straightforward,integration V s = ( a 2 ( l 3 - 1 2)/ 3 + b2(l* - 1*)/. 5 + c2(lJ - 1 2)/ 7 + 2 a b (1* - 1 2)/ 4 + 2a c (l/j - 1*)/ 5 + 2 b c ( i f - l\)/ 6)/ k and the volume equation is V = (a 2 H3/ 3 + b 2 H5/ 5 + c 2 H7/ 7 + 2ab H4/ 4 + 2ac H5/ 5 + 2bc H6/ 6)/ k ( III V ) where a = b D /(H - 4.5) + b H - 4.5 b + b 2H 2 -4.5 b2H + b 3(2H 2 -.4.5 H -(4.5)2) b = -b - b2H - 3b3H 129 v IV .Volume equation from taper equation IV d = D (bolP/(D2HP + 1) + bx(l / H)q)% Vs = D2( bQ( l fX" ifVCCP + 1)D2HP+1) + b l( lq + 1- lq + 1) / ( ( q + l)Hq))/k V = a + b D2H (1VV) where a = bQ/((p + 1) k) b = b /((q + 1) k) V.Volume equation from taper equation V d = D (b (1 / H)2 + b.(l / H)3 + b.(l / H)4)% , o - 1 - I -Vs = D2(bQ(l3 - 13)/(3H2) + b ^ l4 - 14)/(4H3) + b2(l* - l2)/(5H4))/k V = a D2H (VV) where a = (b / 3 + b, / 4 + hj 5)/ k o 1 2 VI.Volume equation from equation VI d = D (bQl / H + b (1 / H)2 + b (1 / H)3) Vg = D2(b2(l3 - 13)/(3H2) + b2(lj - 12)/(5H4) + b2U7 - 12)/(7H6) + 2bQb1(l4 - 14)/(4H3) + 2bQb2(l^ - 1^)/(5H4) + 2b1b2(lJ - 1^)/(6H5))/ k V = a D2H ' (VIV) where a = (b2/ 3 +' b2/ 5 + b2/ 7 + 2b b_/ 4 + 2b b_/ 5 + 2b.b./ 6)/ k o 1 2 ol o2 12 130 v VII .Volume equation from equation VII 3/2 3/2 3 -2 3/2 3 -3 d = D (b XJ/ + b.,(XJ/ - JT) D (10 ) + b„ ( XJ / - X ) H (10 ) + o 1 2 b3(X3 / 2- X3 2) H D (IO"5) + b^X3^- X3 2) Hl / 2(10"3) + b^X3 7 2 - X4 0) H2(10-6))% 5 where X = 1 /(H - 4.5) V = D2( ( ( l f2- i;j/ 2)/((H - 4.5)3 / 25/2)(b 4- b.D 10_2+ b.H 10"3 + s -1 -2 o 1 2 b3D H 10"5 + b4Hl / 2 10'3 + b5H210"6) + ( ( l j - 12)/(4(H - 4.5)3)) (-b D 10"2 - b2H 10"3) + ( ( l3 3- 13 3)/(33(H - 4.5)32))(-b3HD 10"5-b4Hl/210"3) + ( ( l4 1- 14 1)/(41(H - 4 . 5 )4°))(- b5H2 10"6))/ k Volume equation VIIV is derived from here by substituting H for 1^  and 0 for 1^. v VIII .Volume equation from equation VIII d = D (b (1 / H )P)% o — 1 = H (d2/(b D2) )1 / P — o V = D2(b ( 1P + 1- lP + 1) / ( ( p + 1) HP))/ k s o —I -z V = a D2H where a = b /(k (p + 1)) V = a D2H (VIIIV) i 131 v IX .Volume equation from taper equation IX d = D (b 1P/ H P + 1 + b(l I H) q) % o — 1 — V < ? - D 2 ( b ( l P + 1 - l P + 1 ) / ( ( p + 1) H P + 1) + b d f 1 - I q + 1 ) / ( ( q + l)H q))/ k s o —1 —2 1 - 1 . - 2 V / B = a + b H ( I X V ) where a = b 7(p + 1) b = b,/(q + 1) o 1 v X .Volume equation from taper equation X d = D (1 / H)/(b + b_(l / H)) — o 1 — 1 = b d H /(D - b,d) — o 1 V. = D 2 ( ( l 1 - 1„) - 2b H ln((b + b.l./ H)/(b + b.l./ H))/ b. + s —1 —2 o o 1-1 o 1—2 1 b 2 H (l/(b + b , l 0 / H) - l/(b + b 1 / H))/ b ))/(k b 2) o o 1-2 o 1-1 1 1 V = a D2H \ ( XV ) where a = (1 + 2b ln(b /(b + b,))/ b. + b 2(-l/(b + b.) + l/b )/b1)/(kb2) o o o l l o o l o l 1 XIV.Volume equation from taper equation XI The same equations as for equation X can be applied here,using the relationship b Q = 1 - b^ 

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