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An integrated system for the estimation of tree taper and volume Demaerschalk, Julien Pierre 1971

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AN INTEGRATED SYSTEM FOR THE ESTIMATION OF TREE TAPER AND VOLUME by JULIEN PIERRE DEMAERSCHALK FOR. ENG.,University of Louvain , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY in the Department of FORESTRY We accept t h i s t h e s i s as conforming' to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Ju l y , 1971 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia,I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT A new taper equation l a presented, log d = b Q + b 1 log D + b 2 log 1 + b<j log H where d l a the diameter Inside bark In Inches at any given 1 i n feet,D i s the diameter breast height outside bark i n Inches, 1 Is the distance from the t i p of the tree In feet,H i s the t o t a l height of the tree i n feet and b 0 fb^,b 2 and bj are the regression c o e f f i c i e n t s . Two methods of d e r i v i n g a compatible system of tree taper and volume equations are dlscussed.One method i n v o l -ves conversion of the logarithmic taper equation into a logarithmic volume equation.The other involves the d e r i -v a t i o n of the logarithmic taper equation from an e x i s t i n g logarithmic volume equation to provide compatabllity i n volume estimation and at the same time ensure as a good f i t as possible f o r the estimation of upper bole diameters (taper). Tests f o r p r e c i s i o n and bias of volume estimates, oarried out on the B r i t i s h Columbia Forest Service taper curves and logarithmic volume equations,Indicate that the l a t t e r approach i s preferable to the former. Page TITLE PAGE i ABSTRACT 11 TABLE OF CONTENTS H i LIST OF TABLES v LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i INTRODUCTION 1 LITERATURE REVIEW 4 DERIVATIONS OF THE EQUATIONS AND TESTS 13 The new taper equation • 13 F i t t i n g the taper equation on the B r i t i s h Columbia Forest Service taper curves • • • • • 16 Derivation of a compatible logarithmic volume equation from the logarithmic taper equation • • . . . . . . . 20 Deri v a t i o n of a compatible logarithmic volume equation from the logarithmic taper equation f i t t e d on the B r i t i s h Columbia Forest Service taper curves 22 Derivation of a compatible logarithmic taper equation from a logarithmic volume equation • • 26 Derivat i o n of compatible logarithmic taper equations from the B r i t i s h Columbia Forest Service logarithmic volume equations • . . • • 2? Page IMPROVEMENT OF THE ACCURACY AND THE PRECISION OF THE LOGARITHMIC TAPER EQUATION 36 DISCUSSION,SUMMARY AND SUGGESTIONS 40 LITERATURE CITED 42 APPENDIXES 47 1 . Common Names and L a t i n Names of the Tree Species 47 2. Derivation of a Volume Equation from the Taper Equation of Hfljer • • • • • • • 48 3» Der i v a t i o n of a Volume Equation from the Taper Equation of Behre 4 9 4 , Derivation of a Volume Equation from the Taper Equation of Matte • • • • • • • 50 5 « Derivation of the Height Equation from the Logarithmic Taper Equation . . . . . . 51 6 . Derivation of a Logarithmic Volume Equation from the Logarithmic Taper Equation 52 7 « Derivation of the Formula to Estimate Volumes of Logs between S p e c i f i c Distances from the Tip of the Tree, from the Logarithmic Taper Equation . . . 53 8 . Derivation of a Compatible Logarithmic Taper Equation from a Logarithmic Volume Equation . . . . . . 54 Table Page I. Summary of Taper Equations F i t t e d on the B r i t i s h Columbia Forest Service Taper Curves 18 I I . Test of the Taper Equations F i t t e d on the B r i t i s h Columbia Forest Service Taper Curves • • • • • • • 19 I I I . Summary of the Logarithmic Volume Equations Derived from the Logarithmic Taper Equations,Fitted on the B r i t i s h Columbia Forest Service Taper Curves 23 IV. Comparison of the B r i t i s h Columbia Forest Service Logarithmic Volume Equations with the Volume Equations from Table III • 25 V. Summary of the Taper Equations Derived from the B r i t i s h Columbia Forest Service Logarithmic Volume Equations 29 VI. Test of the Taper Equations Derived from the B r i t i s h Columbia Forest Service Logarithmic Volume Equations . . . . . . . 30 VII. D i s t r i b u t i o n of the Bias over the D i f f e r e n t Height Classes within the Same Species (for Mature Coastal Douglas-fir) • • • • • 32 V I I I . Comparison of Standard Errors of Estimate of Several Methods . . . . . . 33 Table Page IX. Absolute Frequency D i s t r i b u t i o n of the Standard Errors of Estimate • • • 34 X. Absolute Frequency D i s t r i b u t i o n of the Differences i n Standard Errors of Estimate 34 XI. Maximum Decrease i n Standard Error of Estimate to Be Expected from Using the Relationship between the Optimum Value of p and T o t a l Height (for Mature Coastal Douglas-fir) • • • 39 LIST OF FIGURES Figure Page 1. The standard error of estimate as a fu n c t i o n of the value of p i n deriving the logarithmic taper equation from the logarithmic volume equation (for mature coast a l Douglas-fir) 28 2. Optimum value of p as a function of t o t a l height (for mature coastal D o u g l a s - f i r ) . • 38 ACKNOWLEDGEMENTS v l i i The author i s Indebted to a l l Individuals and agen-c i e s concerned with support of his studies and research. The author wishes to express? his gratitude to Dr. D. D. Munro who suggested the problem and under whose d i r e c t i o n t h i s study was undertaken. Advice on data processing was provided by Dr. A. Kozak. Drs. D. D. Munro,A. Kozak and J . H. G. Smith are g r a t e f u l l y acknowledged f o r t h e i r help,useful c r i t i c i s m and review of the t h e s i s . Derivations of the functions,given as Appendixes 2 -8 , were also reviewed by Mr. G. G. Young,Assistant Professor, whose help i s appreciated. Thanks are due to Mrs. Lambden,technician,for drawing the f i g u r e s . The opportunity to use the taper curves and equations derived by the Forest Inventory D i v i s i o n of the B.C. Forest Service i s acknowledged. The University of B r i t i s h Columbia i s acknowledged f o r the computing f a c i l i t i e s . F i n a n c i a l support was provided i n the form of a Faculty of Forestry Teaching Assistantshlp and a MacMlllan Bloedel Ltd. Fellowship In Forest Mensuration. Some assistance i n computing was provided by the National Research Counsll of Canada grants A-2077 and A-3253 In support of studies of tree shape and form. "We must develop a mathematical tree volume expres-sion which can be e f f i c i e n t l y programmed f o r generally a v a i l a b l e e l e c t r o n i c computing equipment to y i e l d tree and stand volumes from inputs of tree diameter outside bark and t o t a l height (form estimates optional) and f o r any demanded stump height and top diameter". (Honer and Sayn-Wittgensteln,1963) This t h e s i s demonstrates that a mathematical stem p r o f i l e equation whloh oan be Integrated to volume can meet these requirements.When Y i s the diameter inside bark at any given height i n feet and X i s the height above the ground i n feet,then volume can be calculated by revolving the equation of the stem curve about the X-axis and Integrating f o r values of X from base to t i p . Merchantable volume to any standard of u t i l i z a t i o n can be calculated by using the appropriate X-values. Section diameter can be estimated at any height or the se c t i o n height f o r any diameter.The maximum volume a v a i l a b l e i n c e r t a i n s i z e s and q u a l i t i e s and the l o s s of wood by breakage and defect can be determined p r e c i s e l y . Taper fu n c t i o n s , s u i t a b l e f o r a l l these purposes, have been proposed by many mensurationists.However a l l these previous studies are s i m i l a r i n that a taper equation l a calculated from the data to give the best f i t f o r taper and therefrom volume i a ca l c u l a t e d . This often r e s u l t s i n a good estimate of taper but a l e s s than s a t i s f a c t o r y estimate f o r volume. In most p r a c t i c a l cases a l o c a l or regional volume equation already exists,has been used widely f o r a long time and w i l l c e r t a i n l y continue to be used In the future. It i s c l e a r that f o r such a s i t u a t i o n a taper equation has to be derived which on the one hand gives the best possible f i t f o r taper but on the other hand i s compatible* with the e x i s t i n g volume equation. This study deals with both approaches.The new taper equation presented i s a logarithmic one.Two methods are d e r i v e d , i l l u s t r a t e d and tested: a) Der l v a t i o n of a compatible logarithmic volume equation from an e x i s t i n g logarithmic taper equation. b) Der i v a t i o n of a compatible logarithmic taper equation from an e x i s t i n g logarithmic volume equation. Both techniques are tested on the B r i t i s h Columbia Forest Service(B.C.F.S.) taper curves (B.C.F.S. , 1968) and on the logarithmic volume equations (Browne,1962). The standard errors of estimate (SEg) are compared with those given by Kozak,Munro and Smith ( 1 9 6 9 b ,TableI) Compatible means here that both equations (for taper and volume) give l d e n t i o a l r e s u l t s f o r t o t a l volume. which are baaed on the same taper curves. Some attempt i s made to f i n d a useful r e l a t i o n s h i p between the taper equation c o e f f i c i e n t s and some tree c h a r a c t e r i s t i c s such as t o t a l height.diameter breast height (dbh) and the r a t i o of both. In basic "Forest Mensuration" textbooks (Chapman and Meyer,1949;Bruce and Schumacher,1950;Spurr,1952; Meyer,1953;Loetsch and Haller ,1964;Prodan ,1965;Avery, I967) almost no comments are made about the d e s i r a b i l i t y of developing compatible taper and volume equations which would be useful f o r the estimation of merchantable volume to any standard of u t i l i z a t i o n . Petterson (1927) suggested the use of a l o g a r i t h -mic curve f o r the main stem.Taper of the d i f f e r e n t form-classes would be given by d i f f e r e n t parts of t h i s curve.A tangential function was used by H e i j b e l (1928) to describe the main part of the stem.Different equations were used f o r the top p r o f i l e and the stem below 10$ of the height. Volume and taper were more or l e s s combined i n one system by the Girard form-class tables (Mesavage and Girard , 1 9 4 6 ) . Aocording to Spurr (195 2 ) a possible s o l u t i o n f o r merchantable cubic-feet tree volume tables i s to calcu-l a t e d i f f e r e n t equations f o r stump and top volumes and then to subtract these volumes from t o t a l volume. Another s o l u t i o n could be to calc u l a t e the regression between merchantable volume and t o t a l volume. Meyer (1953) stated that the construction of a taper curve (or equation) f o r a c e r t a i n species or group of species i s s t i l l a d i f f i c u l t task. Graphical techniques were used by Duff and B u r s t a l l ( 1 9 5 5 ) to develop taper and volume tables showing mer-chantable volumes f o r each ten-foot-height class within each dbh and t o t a l height class.Volume tables and taper tables were f i r s t prepared independently. To make them compatible the taper data were adjusted to f i t the independently calculated volumes and the diameters were made to agree with the already known volumes. Speidel ( 1 9 5 7 ) used graphioal techniques to r e l a t e the percentage of t o t a l volume to the percentage of t o t a l tree height. It was shown by Newnham ( 1 9 5 8 ) that a quadratic parabola gave a good f i t to a large part of the bole shape. Three models were developed by Honer ( 1 9 6 4 ; 1 9 6 5 a , b ; 1 9 6 7 ) to express the d i s t r i b u t i o n of volume over the tree stem: a) v / V = b Q + b x h / H + b 2 h 2 / H 2 b) v / V = b* + b[ d 2 / D 2 + bg(d 2 / D 2 ) 2 c) v / V = b Q + b J d / D + bg(l - h / H) 2 where v i s the volume below the merchantable l i m i t , V i s the t o t a l volume,h i s the merchantable height from the base,d i s the merchantable diameter and D i s the dbh. These models describe well the d i s t r i b u t i o n of volume over the tree stem and can be used to estimate volume to any standard of u t i l i z a t i o n when applied to an e s t i -mate of t o t a l volume.They cannot be used to estimate diameter at a given height or height of a c e r t a i n diameter. T a r l f t a b l e s , l i k e those of Turnbull and Hoyer ( 1 9 6 5 ) , w i l l not be disoussed here because they don't give a compatible system of volume and taper. Heger ( 1 9 6 5 ) reported a t r i a l of Hohenadl's approach on lodgepole p i n e 2 grown i n Alberta.Stanek ( 1 9 6 6 ) i l l u s -trated the method f o r lodgepole pine and Engelmann spruce i n B r i t i s h Columbia. New tree-measurement concepts were introduced by Grosenbaugh ( 1 9 5 4 ; 1 9 6 6 ) . Some work has been car r i e d out on tree taper curves using m u l t i v a r i a t e methods (Fries, 1 9 6 5;Frles and Matern, 1 9 6 5 ).However,after comparison of multivariate and other methods f o r a n a l y s i s of tree taper,Kozak and Smith ( 1 9 6 6 ) oonoluded that the use of simpler methods i s best. While many authors have made i t cl e a r (Kozak,Munro and Smith,1969a;Munro,1970) that no p r a c t i c a l advan-tage can be gained from any measurement of form The common tree names used throughout t h i s t h e s i s are given with the corresponding L a t i n names i n Append is 1 • i n a d d i t i o n to dbh and t o t a l height,Schmid,Holko-Jokela, Mlngard and Zobelry ( 1 9 7 1 ) have shown that the measure-ment of dbh,total height and diameter at 6 - 9 meters height i s the best method of volume determination. Some important taper functions are worthy of more detailed review.A Swedish c i v i l engineer,Hfljer ( 1 9 0 3 ) » was the f i r s t to propose a mathematical equation to describe the stem p r o f i l e : d / D = c x In ( ( c 2 + 1 100 / H) / c 2 ) where d was the diameter inside bark at any given d i s -tance from the tlp,D was the dbh Inside bark,l was the distance from the tip,H the t o t a l tree height above breast height and c^ and c 2 were the constants to be defined f o r eaoh form-class. Jonson (Claughton-Viallin , 1 9 1 8 ) described t h i s mathemati-* c a l formula as completely conforming with nature when applied to spruce of a l l form-classes,but stated that i n some stands,which had been grown from Imported seeds, overeatlmations occurred i n the upper sections.The d i a -meter at any height of the stem being known there i s no d i f f i c u l t y i n estimating volume.A volume table can a l s o be calculated by deriving a volume equation from the taper equation by i n t e g r a t i o n : V = D 2 H O.OO5454 cf (K (In E (In K - 2) + 2 ) - 2) (for proof aee Appendix 2) where K = 1 + 100 / c 2 In order to obtain better results,Jonson ( 1 9 1 0 ; 1 9 H » 1926-27) introduced a new constant which he called a " b i o l o g i c a l constant": d / D = cx In ( ( c 2 + 1 100 / H - c ^ ) / c 2 ) where was the new constant.Equations were computed fo r each form-class.W1th the introduction of t h i s " b i o l o g i c a l constant" an Inconsistency was introduced because t h i s taper equation didn't give a r e s u l t f o r a portion equal to c^ on the upper stem.A volume equation oan be derived i n the Bame way as f o r the formula of H8jer. The taper equations of Jonson and Hfljer are In f a c t composite taper equations.They are complied Independently of tree species.The form-class which had to be known was usually measured or estimated by the "form point" approach.Claughton-VJallln and Vlcker (1920) reported about t h i s that the d i f f i c u l t y Is to estimate the form-c l a s s of a standing tree or the average form-class of a stand but they believed that a l i t t l e p r actise would overcome t h i s . Wlckenden (1921) claimed that the form quotient of any type of f o r e s t does not vary much even f o r large regions.Wright (1923) believed however that there was a considerable v a r i a t i o n i n the form of i n d i v i d u a l trees i n a stand of timber. As a r e s u l t of his Investigations on many species, Behre (1923;1927;1935) presented a new equation f o r the stem curve which seemed to be more consistent with nature: d / D = ( l / H ) / ( b 0 + b 1 l / H ) where the symbols have the same meaning as i n the Hfljer's equation.The c o e f f i c i e n t s b Q and b^ can be calculated by f i t t i n g the regression l i n e : ( l / H ) / ( d / D ) = b 0 + b 1 ( l / H ) t h i s f u n c t i o n i s i d e n t i c a l to the equation: ( D / d ) = b j + b j ( H / l ) Behre's taper equation,when integrated to volume,yields the following compatible volume equation: V = D 2 H O .OO5454 ( 1 / b\ ) ( 1 - b2) + 2 b 0 In b 0 ) (for proof see Appendix 3 ) Matte ( 1 9 4 9 ) described the stem p r o f i l e above breast height by the funotion: d 2 / D 2 = b 0 l 2 / H 2 + b! l 3 / H 3 + b 2 1^ / H 4 where the symbols have the same meaning as In the equation of H8jer. l t i s worthwile to mention that the taper equation c o e f f i c i e n t s are p a r t i a l l y defined by a condition about volume. The following volume equation can be derived by i n t e -g r a t i o n : V = 0 . 0 0 5 4 5 4 D 2 H ( b 0 / 3 + b x / 4 + b 2 / 5 ) (for proof see Appendix 4 ) bg and b^ were found to be related to dbh and t o t a l height. A quite s i m i l a r equation was tested by Osuml ( 1 9 5 9 ) d / D = b 0 1 / H + b x l 2 / H 2 + b 2 l 3 / H 3 from which also a volume equation can be derived. The taper equation preferred by Giurgiu ( 1 9 6 3 ) was a 1 5 t h degree polynomial: d / D = 1 5 t h degree polynomial of 1 / H where D was the diameter inside bark at . 1 of t o t a l height and was fur t h e r expressed as a function of dbh outside bark and t o t a l height.This function can also be integrated to volume. Prodan ( 1 9 6 5 ) found the following taper function s a t i s f a c t o r y : d / D = ( h / H ) 2 / ( b Q + b x h / H + b 2 h 2 / H 2 ) where h i s the height above the ground. With respeot to the taper equation of Osumi,he stressed that a 4 t h degree polynomial with Intercept would be muoh be t t e r . As an extension of the methods used by Matte, Osumi and Giurgiu,an integrated system of taper and volume equation f o r red alder was provided by Bruce, Cu r t i s and Vanooeverlng ( 1 9 6 8 ) : d 2 / D 2 = b 0 X 3 / 2 + ( X 3 / 2 - X? ) ( b± D + b 2 H ) + ( X 3 / 2 - X 3 2 ) ( b 3 H D + bL H 1 / 2) + ( X 3 / 2 - X*°) ( b- H 2 ) where X i s 1 / ( H - 4 . 5 ) and D i s dbh outside bark. Very high powers of X were required to describe the butt swell.The authors expected that the use of some measure of form would Improve the f i t of t h i s taper equation.In t h e i r opinion,the p r i n c i p a l d i f f i c u l t i e s encountered by Hfljer,Jonson,Behre and others were due to oversimplified equations which did not s a t i s f a c t o r y describe the butt swell and t i p . A f t e r Munro ( 1 9 6 8 ) found that upper stem diameters could be estimated with reasonable SE from a function Involving dbh, h / H and h / H ,the following taper equation was proposed by Kozak,Munro and Smith ( 1 9 6 9 a ,b): d 2 / D 2 = b 0 + b t h / H + b 2 h 2 / H 2 where D i s the dbh outside bark i n Inches and h i s the height above the ground i n feet.The least squares s o l u t i o n was conditioned by Imposing the r e s t r a i n t : b 0 + bj + b 2 = 0 For spruoe and redoedar a d d i t i o n a l conditions were necessary to prevent negative diameters near the top. These taper functions were oomputed f o r 2 3 species or speolesgroups from B.C.F.S. taper curves (B.C.F.S., 1 9 6 8 ) to f a c i l i t a t e e f f i c i e n t a n a l ysis with modern ele c -t r o n i c computers.Several t e s t s on these equations (Kozak, Munro and Smith, 1 9 6 9 a ;Smith and Kozak,1971) suggested! a stable estimating system.It appeared as i f l i t t l e r e a l advantage resulted from the use of more complex powers, l i k e those used by Bruce,Curtis and Vancoevering ( 1 9 6 8 ) , to estimate tree taper.These taper equations were,later on,converted into volume equations and point sampling f a c t o r s (Demaersohalk , 1 9 7 1 ) • Awareness of the d e s i r a b i l i t y of development of comprehensive systems f o r estimation of net merchantable volumes of trees by log s i z e and u t i l i z a t i o n classes i s growing.The need has been f e l t f i r s t i n operations research analyses of logging systems i n Sweden and i n studies to develop improved methods of inventory i n Austria.However,no publications incorporating the features described herein have come to the author's a t t e n t i o n . No review w i l l be given about the d i f f e r e n t tree form theories (nutritional,mechanistic,water conductive, hormonal and pipe model).Interesting discussions about the d i f f e r e n t a l t e r n a t i v e s were given by Gray ( 1 9 5 6 ) , Newnham ( 1 9 5 8 ),Larson (1963),Heger U 9 6 5 ) a n < a Shinozaki et a l . ( 1 9 6 5 ) . DERIVATIONS OP THE EQUATIONS AND TESTS The new taper equation The logarithmic taper equation tested i n t h i s study i s : log d = b 0 + b1 log D + b 2 log 1 + bj log H (1) where d i s the diameter inside bark i n inches at any given 1 i n feet,D Is the dbh outside bark i n inches, 1 i s the distance from the t i p of the tree In f e e t , H Is the t o t a l height of the tree i n feet and b n , b i , b 2 and b^ are the regression c o e f f i c i e n t s . The same taper equation can be expressed i n other ways: d = 1 0 b 0 D b l l b 2 H b 3 ( 2 ) or d w / D v = K 1 * / H Z ( 3 ) where w = 1. z = - b^ v = b x K = 1 0 b ° y = b 2 Just as the logarithmic volume equation V = 1 0 a D b H c i s the unconditioned form (with respect to the powers of D and H) of the combined va r i a b l e volume equation V = b Q H 1 (without Intercept) where the power of D l a conditioned to 2 and the power of H to l , t h l s taper equation (the aquare of formula 3) i s the unconditioned form of the well known general formula f o r the p r o f i l e of c e r t a i n aollda of revolution (cone,paraboloid and n e l l o i d ) : d 2 / D 2 = ( 1 / H ) v where the powera of d and 1 are conditioned to be equal to r e s p e c t i v e l y the powera of D and H. Thla taper equation i s very slmple.No conditioning i s necessary to ensure that the estimated diameter at the top i s zero and that no negative estimates of diameter occur.Prom formula 2 i t can be seen e a s i l y that d can never be negative and becomes zero when 1 i s zero (at the t i p of the t r e e ) . Formula 2 can be used to estimate diameter inside bark at any selected distance (1) from the t i p . Distance to any s p e c i f i c top diameter (d) can be e s t i -mated by transformation of the basic equation to the form: 1 = ( l ( T b 0 d D" b l H"b3 ) 1 / b 2 (4) (for proof see Appendix 5) The logarithmic taper equation can be derived i n two b a a l c a l l y d i f f e r e n t waya: a)The taper equation can be f i t t e d on taper data by the leaat squares method.This function can e a s i l y be converted subsequently to a compatible logarithmic volume equation. b)The taper equation oan be derived from an e x i s t i n g logarithmio volume equation when some data about taper are av a i l a b l e . T h i s taper equation w i l l be compatible with the e x i s t i n g volume equation. Both ways w i l l be explained and tested on the B.C.F.S. taper curves (B.C.F.S. , 1968) and logarithmic volume equations (Browne,1 9 6 2 ) . F i t t i n g the taper equation on the B r i t i s h Columbia Forest Service taper curves The logarithmic taper function (formula 1) was computed f o r 23 species or speciesgroups on the B.C.F.S. taper curves (B.C.F.S.,1968) by the l e a s t squares method.Diameters insid e bark had been taken from each taper curve at the height of 1 f t . , 4 . 5 f t . and at de c i l e s of t o t a l height and punched on computer cards f o r Kozak,Munro and Smith (1969,).In the ca l c u l a t i o n s , b dbh outside bark was used as the measure of diameter i n s i d e bark at 1 f t . height2 The assumptions of the regression analysis were tested by p l o t t i n g f or each species l o g d over l o g D,log 1, and l o g H.For every species and f o r every variable,there was almost a perfect s t r a i g h t l i n e r e l a t i o n s h i p between dependent and independent variable.Variances were homo-geneous. Because r e l a t i v e standard errors are sometimes greatly a f f e c t e d by the s i z e of the mean and comparisons i n terms of r e a l d*s were desired,the following approxi-mation was used: SE E = (( S ( da - de ) 2) / ( n - m - 1 ) ) 1 / 2 (5) where da i s the actual diameter i n s i d e bark.de i s the estimated diameter i n s i d e bark,n i s the number of -'Except for mature coastal Douglas-fir,which gave a much better f i t without adjusting. observations used f o r the lea s t squares f i t , m i s the number of independent v a r i a b l e s and S i s the sum. The regression constants of the taper equation and the SE£s are summarized i n table I and the average bias of diameter inside bark at d i f f e r e n t heights i s given i n table I I . The SE£s ranged from .245 to 2.431 Inches.An abso-l u t e frequency d i s t r i b u t i o n of the SEjjs i s given i n table IX.Large SE£s,however,do not necessarily indicate a poor f i t , b u t more l i k e l y represent a wider range of taper curves (Hejjas,1967). A l l the species follow almost the same trend of average bias.There i s usually an underestimation at the base of the tree,an overestimation from .1 u n t i l .4 or . 5 of the t o t a l height,a s l i g h t underestimation from .4 or . 5 u n t i l .8 of the t o t a l height and a small overestimation at the top.For eleven species the average bias at any height i s l e s s than one inoh. Summary of Taper Equations Pit t e d on the B r i t i s h Columbia Forest Service Taper Curves Species 4 Equation c o e f f i c i e n t s SEr No* group R M b 0 b 2 t>3 (inches) Alder C M -0.071459 0 . 8 0 2 7 3 0 0 . 7 9 5 2 0 3 - 0 . 6 3 7 0 5 9 0 . 5 8 7 92 Aspen I M 0 . 0 1 4 5 0 6 0 . 9 4 4 3 8 9 0 . 7 6 6 6 5 5 - 0 . 7 4 2 5 7 6 0.448 5 3 Balsam C M 0 . 3 6 9 2 2 3 1 . 0 6 4 1 1 9 O .656O8O -0.878118 0 . 8 1 3 8 5 n I M 0 . 0 2 5 9 2 3 0 . 9 2 5 2 6 3 0 . 7 2 9 9 6 3 - 0 . 6 9 7 5 2 5 0.420 8 5 B i r c h I M 0 . 0 5 1 5 6 0 0 . 9 7 9 5 1 3 0 . 8 9 9 9 4 7 - 0 . 9 1 3 0 4 1 0 . 2 4 5 55 Ced ar C M 0 . 4 4 8 1 9 1 O .968O76 0 . 8 1 2 9 5 4 -I . 0 3 2 1 0 9 2 . 4 3 1 114 I I C I 0 . 1 9 5 9 4 5 0 . 7 5 9 6 8 8 0 . 8 2 4 2 5 4 - 0 . 7 8 3 3 5 6 1.128 134 » I M 0 . 3 7 9 9 9 2 1 . 0 1 1 8 6 0 0 . 7 9 9 0 1 9 - 1 . 0 1 2 5 0 0 1 . 3 7 9 127 Cotton- CI M - 0 . 2 6 2 8 4 3 0 . 8 6 5 2 7 3 0 . 8 2 7 0 2 3 - 0 . 6 1 3 2 3 3 0.755 92 wood Douglas c M 0 . 2 0 4 3 8 9 0 . 9 8 4 5 7 8 0 . 7 0 1 1 6 5 -0.821202 1.431 114 i i r I I c I O . 0 9 2 7 0 7 0 . 8 2 6 4 7 1 0 . 6 8 0 4 5 1 - 0 . 6 3 7 3 5 2 1 . 0 3 2 174 it I M 0 . 0 0 4 8 2 7 0 . 8 9 2 4 2 5 0.741884 - 0 . 6 9 0 8 2 1 1 . 2 5 1 160 Hemlock c M 0 . 2 9 9 1 3 0 1 . 0 1 6 4 3 0 0 . 7 4 6 1 4 8 - 0 . 9 0 8 8 2 1 0 . 8 7 1 118 I I c I 0 . 0 6 5 9 4 1 0 . 8 5 7 9 3 2 0 . 8 2 9 0 1 3 - 0 . 7 7 6 8 6 6 0 . 6 9 1 128 M I M 0 . 0 3 6 8 7 3 0 . 9 9 9 7 0 4 0 . 7 1 6 1 6 9 - 0 . 7 3 6 2 3 7 0.740 104 Lodgep. CI M 0 . 4 7 2 7 0 2 1 . 0 4 4 0 6 9 0 . 6 3 4 6 3 3 - 0 . 9 0 9 7 6 8 0 . 7 7 4 65 pine Larch I M - 0 . 0 1 2 6 8 0 0 . 8 4 3 9 2 6 0 . 6 9 6 4 3 1 - 0 . 6 1 8 3 7 4 1 . 2 3 0 148 Maple c M -0.010447 0 . 8 6 3 3 3 7 0 . 9 0 9 1 0 4 - 0 . 8 2 2 0 7 4 0 . 3 1 6 48 Spruce c M 0 . 2 9 4 0 0 1 0 . 9 7 8 3 8 8 0 . 7 8 3 3 8 7 - 0 . 9 1 2 4 9 4 2 . 3 7 6 378 n I M 0 . 1 0 0 7 0 0 0 . 9 1 5 9 0 3 0 . 7 4 2 6 3 1 - 0 . 7 4 4 9 7 7 0 . 5 2 6 9 3 White CI M 0 . 6 9 0 0 4 4 1 . 2 1 5 4 0 0 0 . 7 0 7 1 5 9 - 1 . 1 8 5 3 6 0 1 . 1 5 9 81 pine Yellow CI M 0 . 1 3 0 2 6 0 0 . 8 9 H 7 0 0 . 7 6 2 9 7 0 - 0 . 7 6 2 3 2 1 0 . 5 7 4 50 oedar Yellow CI M 0.044221 1 . 1 4 8 2 1 9 0 . 6 7 4 2 4 7 - 0 . 8 1 0 4 1 5 1 . 2 0 5 124 — p i n e p . 5 ; rC i s Coast M i s Mature Number of taper l i n e s scaled from I i s I n t e r i o r I i s Immature the B.C.F.S. taper curves Teat of the Taper Equations F i t t e d on the B r i t i s h Columbia Forest Service Taper Curves Average bias ( i n inches) of diameter Inside bark at Species group R w 1» 0.1H 0.2H 0 . 3 H 0.4H 0 . 5 H 0 . 6 H 0.7H 0.8H 0.9H 1.0H Alder C M 0.06 0 . 3 4 0.42 0.17 -0.14 -0 . 3 6 -0 . 4 5 -0 . 3 9 -0.22 0.01 0 . 2 3 0.0 Aspen I M -0.12 0 . 4 4 0 . 5 5 0 . 3 4 0 . 0 7 -0.14 -0 . 2 9 -0 . 3 2 -0.24 -0.06 0 . 1 5 0.0 Balaam C M -0 . 4 3 0.17 0.86 0.61 0.14 -0 . 2 3 -0 . 4 5 -0 . 4 5 -0.28 -0.06 0.11 0.0 I M - 0 . 3 1 0 . 3 5 0 . 5 1 0 . 2 9 0.02 -0 . 1 7 -0 . 2 5 -0.23 -0.16 -0 . 0 5 0.08 0.0 Birch I M - 0 . 0 3 O . 2 5 0 . 2 9 0 . 0 9 -0 . 0 6 -0.14 -0.16 -0.14 -0.08 -0.01 0.04 0.0 Cedar c M -4.21 - 3 . 3 3 0.48 1.60 1.22 0 . 7 3 0 . 3 7 0.20 0.10 0.01 -0.08 0.0 it c I - 1 . 3 2 -1.10 0.19 0 . 7 5 0 . 5 5 0.24 0.01 -0.11 -0 . 1 3 -0.04 0.10 0.0 n I M - 2 . 4 7 -1 . 4 5 0 . 7 5 1 . 3 0 0 . 9 2 0.42 0 . 0 3 -0.17 -0.20 -0.11 0 . 0 3 0.0 Cottonwood CI M - 0 . 4 3 0 . 9 4 1.04 0 . 5 1 0.00 -O .38 -0 . 5 7 -O . 5 6 -O . 3 6 -0 . 0 7 0.18 0.0 Douglas-fir C M -3.24 0 . 1 3 1 . 0 5 1 . 2 7 0.81 0 . 2 3 -0 . 2 5 -0.46 -0 . 3 6 -0 . 1 3 0 . 1 5 0.0 it c I -2.01 -0.01 0.42 0 . 5 4 0.41 0 . 1 9 -0.02 -0.16 -0 . 17 -0.04 0.16 0.0 •i I M -2 . 7 4 0 . 6 6 1.48 1.22 0.46 -0 . 2 3 -0 . 6 7 -0 . 71 -0 . 4 5 -0 . 0 3 0 . 3 5 0.0 Hemlock c M -1.01 0.11 0 . 9 5 0 . 7 4 0.24 -0.18 -0 . 4 3 -0.48 -0 . 3 2 -0 . 0 6 0 . 1 9 0.0 ti c I -0.48 0.01 0.60 0 . 4 5 0.10 -0.18 -0.34 - 0 . 3 5 -0.18 0.03 0.21 0.0 ti I M - 0 . 3 3 0.74 1.07 0 . 7 6 0 . 3 2 -0.12 -O . 4 5 -0 . 5 7 -0 . 5 1 -0.24 0.14 0.0 Lodgepole pine CI M -0.12 0.06 0 . 3 7 0 . 5 0 0 . 2 9 0 . 0 7 -0.12 -0.28 -0 . 3 5 -0 . 2 6 -0.10 0.0 Larch I M -2 . 7 3 0 . 3 1 1 . 2 7 1 . 3 4 0.82 0.16 -0 . 3 7 -O . 6 3 -0 . 5 6 -0 . 2 3 0.28 0.0 Maple c M -0.16 -0 . 0 5 0.16 0.16 0 . 0 3 -0 . 0 5 -0.11 -0.11 -0.07 -0.01 0 . 0 5 0.0 Spruce c M - 3 . 2 5 - 2 . 7 7 1 . 6 6 2.11 1.14 0.24 -0 . 3 5 - 0 . 5 5 -0 . 3 9 0.01 0 . 1 3 0.0 ti I M -0.78 -0.18 0.48 0 . 5 6 0.26 -0 . 0 5 -0 . 2 3 -0 . 2 5 -0.16 -0.02 0.14 0.0 White pine CI M -0.42 0.21 0 . 8 5 0 . 6 5 0.24 -0.14 - 0 . 3 9 -0 . 4 5 - 0 . 3 ? -0.19 -0.16 0.0 Yellow cedar CI M -0.28 -0 . 2 3 0 . 5 2 0 . 7 3 0.44 0.04 -0 . 3 4 -0 . 5 4 -0.44 -0 . 0 9 0 . 1 9 0.0 Yellow pine CI M -1 . 1 3 1 . 3 4 1 . 4 7 0 . 9 9 0 . 3 7 -0.18 - 0 . 5 7 -0 . 6 7 - 0 . 5 3 -0 . 2 9 0.02 0.0 see f n . 4 and 5 i n table I Derivation of a compatible logarithmic volume equation from the logarithmic taper equation The logarithmic taper equation can be converted into a compatible logarithmic volume equation: l o g V = a + b log D + c log H ( 6 ) where a = log ( O.OO5454 10 2 b0 / ( 2 b 2 + D ) b = 2 b1 c = 2 b 2 + 2 + 1 where t>n» bi» b2 a n d b3 a r e c o e f i o l e n t a from the logarithmic taper equation. (for proof see Appendix 6 ) This volume equation i s the formula to be used to e s t i -mate t o t a l volume of the tree i n cubic-feet.An a l t e r n a -t i v e form of t h i s equation i s : V = 10 a D b H c (?) To estimate volumes of logs between s p e c i f i c distances from the t i p of the tree,the following equation has to be used: V = K D v I!* ( 1* - i f ) (8) where v = 2-y = 2 b 3 2 = 2 b 2 + 1 2b K = O.OO5454 10 0 / ( 2 b 2 + 1 ) and 1 1 and 1 2 are re s p e c t i v e l y the lower and upper distance from the t i p of the t r e e . (for proof see Appendix 7) If the l i m i t s i z e s of the log are given as diameters inside bark,the same formula 8 can be used a f t e r corresponding distances from the t i p of the tree have been calculated with formula 4. D e r i v a t i o n of a compatible logarithmic volume equation from the logarithmic taper equation f i t t e d on the B r i t i s h Columbia Forest Service taper curves The logarithmic volume function ( 6 ) was derived from the taper equations i n table I f o r the 2 3 B.C. species-groups and are summarized i n table III.Because of the f a c t that the B.C.F.S. logarithmic volume equations and taper curves are based on the same sample trees (B.C.F.S., 1 9 6 8 ) ,we would expect that the volume equations,derived from the taper functions,would be s i m i l a r to the B.C.F.S. logarlthmlo volume equatlons.Although i t i s true f o r c e r t a i n species,for others there are some rather large deviations.This suggests that a good taper equation i s no guarantee f o r a good volume equation i f only the p r e c i s i o n of t h i s taper function i s indioated by the SEpon diameter.A SEgof 1 inch,for example,has no meaning f o r volume when one knows nothing about the bias.The e f f e c t of bias v a r i e s considerably with the p o s i t i o n on the tree and with the siz e of the tree.Therefore the best check of a taper table,which i s to be used to ca l c u l a t e volume,is a check of a volume table derived therefrom,as was recognized by Bruce and Schumacher ( 1 9 5 0 ) . The f a c t that i n the B.C.F.S. logarithmic volume equations the sum of squares of the resi d u a l s of the loga-rithm of volume i s minimized,while i n the logarithmic taper equation the sum of squares of the residuals of Summary of the Logarithmic Volume Equations Derived from the Logarithmic Taper Equations,Fitted on the B r i t i s h Columbia Forest Service Taper Curves Species group B M 8 Equation c o e f f i c i e n t s a b c Alder C M - 2 . 8 1 9 5 5 ? 1 . 6 0 5 4 5 9 1 . 3 1 6 2 8 8 Aspen I M - 2 . 6 3 7 9 4 7 1 . 8 8 8 7 7 8 1 . 0 4 8 1 5 9 Balsam C M -1.888843 2 . 1 2 8 2 3 9 0 . 5 5 5 9 2 4 I M - 2 . 6 0 2 3 4 6 1 . 8 5 0 5 2 6 1 . 0 6 4 8 7 7 Birch I M - 2 . 6 0 7 2 9 2 1 . 9 5 9 0 2 5 0 . 9 7 3 8 1 2 Ced ar c M - 1 . 7 8 6 1 6 8 1 . 9 3 6 1 5 2 O . 5 6 I 6 8 9 c I - 2 . 2 9 4 3 8 2 1 . 5 1 9 3 7 6 1 . 0 8 1 7 9 6 I I I M - 1 . 9 1 7 9 3 3 2 . 0 2 3 7 2 0 0 . 5 7 3 0 3 8 Cottonwood CI M -3 . 2 1 2 8 6 5 1 . 7 3 0 5 4 6 1 . 4 2 7 5 8 0 Douglas-fir c M - 2 . 2 3 5 1 2 6 1 . 9 6 9 1 5 5 0 . 7 5 9 9 2 6 I I c I - 2 . 4 5 0 9 3 5 1 . 6 5 2 9 4 2 1 . 0 8 6 1 9 8 •t I M - 2 . 6 4 8 7 2 8 1 . 7 8 4 8 4 9 1 . 1 0 2 1 2 6 Hemlock c M - 2 . 0 6 1 6 1 0 2 . 0 3 2 8 6 0 0 . 6 7 4 6 5 4 it c I - 2 . 5 5 5 9 4 9 1 . 7 1 5 8 6 3 1 . 1 0 4 2 9 4 it I M - 2 . 5 7 5 5 ^ 9 1.999408 0 . 9 5 9 8 6 5 Lodgepole pine CI M - 1 . 6 7 3 7 5 2 2 . 0 8 8 1 3 9 0 . 4 4 9 7 3 0 Larch I M - 2 . 6 6 7 5 4 9 1 . 6 8 7 8 5 2 1 . 1 5 6 1 1 4 Maple c M - 2 . 7 3 4 1 3 8 1 . 7 2 6 6 7 3 1 . 1 7 4 0 6 0 Spruce c M - 2 . 0 8 4 6 5 7 1 . 9 5 6 7 7 6 0 . 7 4 1 7 8 6 •t I M - 2 . 4 5 7 2 4 3 1 . 8 3 1 8 0 5 0 . 9 9 5 3 0 8 White pine CI M - 1 . 2 6 5 9 7 8 2 . 4 3 0 7 9 9 0 . 0 4 3 5 9 7 Yellow cedar CI M - 2 . 4 0 5 1 7 4 1 . 7 8 2 3 3 9 1 . 0 0 1 2 9 8 Yellow pine CI M -2.545619 2 . 2 9 6 4 3 8 0 . 7 2 7 6 6 4 ^see f n . 4 and 5 i n table I. the logarithm of diameter l a minimized Is one of the reasons f o r these apparent contradlotIons.Another reason oan be the f a c t that the basic data used f o r the c a l c u l a t i o n s of the B.C.F.S. logarithmic volume equations lnoluded f o r c e r t a i n species deformed t r e e s , f o r example forked trees f o r redcedar (Browne,1962),while the B.C.F.S. taper curves are probably only based on normal trees.Table IV gives an example of three species, two where the s i m i l a r i t y i s high and one where the deviations are rather l a r g e . Comparison of the British Columbia Forest Service Logarithmic Volume Equations with the Volume Equations from Table III Species group(R M)" Differences as a percentage of B.C.F.S. volume Spruoe (I M) Total height (feet) dbh 20 40 60 80 100 120 140 160 180 200 (inch.) 10 +5.41 +2.62 +1.02 -0.10 -0.96 -1.66 20 +1 .95 +0 .36 -0.75 -1.61 -2 . 3 0 -2.88 30 -0.02 -1 . 1 3 -1.98 -2 . 6 7 -3 . 2 5 -3.75 40 -1.40 -2 . 2 5 -2.94 -3 . 1 5 -4.01 -4.45 5 0 -2.45 -3.14 -3 . 72 -4.21 -4 . 6 5 -5.04 6 0 -2 . 6 2 -3 . 3 1 -3.88 -4.38 -4.81 -5.20 Hemlock (I M) Total height dbh 20 40 60 80 100 (inch.) 10 +0.80 - 0 . 3 9 -1.08 - 1 . 5 6 - 1 . 9 4 2 0 + 1 . 6 8 + 0 . 9 8 + 0 . 4 8 + 0 . 1 0 0 + 2 . 2 0 + 1 . 7 0 + 1 . 3 1 0 + 2 . 5 7 +2.18 5 0 + 2 . 8 6 6 0 +3.42 (feet) 120 140 160 180 200 -0.21 -0.48 + 1 . 0 0 + 0 . 7 3 + 0 . 5 0 + 0 . 3 0 +1.86 +1.59 +1.36 +1.16 +0.97 +2.54 +2.27 +2.04 +1.83 +1.64 +3.10 +2.82 +2.59 +2.38 +2.20 white pine (CI M) ------------------ Total height (feet) dbh 20 40 60 80 100 120 140 160 180 200 (lnoh.) 10 +79.69+22.21 -7.04-24.81 20 +37-39+11.12 -6.56-19.30 30 +39.65+17.42 +1.42-10.68 40 +38 .09+19.26 +5.04 -6.08 5 0 +35.24+19.12 +6 .50 -3.65 60 +49.88+32.01+18.02 +6.77 see fn. 4 and 5 in table I. Derivation of a compatible logarithmic taper equation from a logarithmic volume equation Any logarithmic volume equation log V .= a + b log D + c log H can be converted into a logarithmic taper equation log d = b Q + b1 log D + b 2 log 1 + bj log H where b Q = log ( ( 4 144 10 a p c / 3.1416 J 1 ' 2 ) b 1 = b / 2 b 2 = ( p c - 1 ) / 2 b 3 = ( l - p ) c / 2 where"a,b and o are the c o e f f i c i e n t s from the logarithmic volume equation.The value of p which i s not yet defined, has to be chosen so as to minimize the SE Eof diameter. Therefore some data about taper are needed. This taper equation,when integrated to t o t a l volume, w i l l f o r any value of p y i e l d exactly the same volume as given by the logarithmic volume equation from which i t i s derived. (for proof see Appendix 8) D e r i v a t i o n of compatible logarithmic taper equations from the B r i t i s h Columbia Forest Service logarithmic volume equations The B.C.F.S. logarlthmlo volume equations (Browne, 1962) are f o r 23 species or speciesgroups converted to compatible logarithmic taper equations by s e l e c t i n g the value of p so as to minimize the SE £of diameter on the B.C.F.S. taper curves (B.C.F.S. , 1 9 6 8 ) . Figure 1 shows f o r mature coastal Douglas-fir the SE E of diameter as a f u n c t i o n of the value of p.The value f o r which the S E E i s minimized i s the optimum value to be adopted f o r p i n d e r i v i n g the taper equation from the volume equation. A summary of the taper equation c o e f f i c i e n t s , the optimum p values as well as the SE^s i s given i n table V.These taper equations give by integrating the same t o t a l volume as given by the B.C.F.S. logarithmic volume equations.The optimum value of p ranged from 2 . 0 3 to 2 . 8 5 and had a mean value of 2 . 3 2 . The average bias of diameter Inside bark at the d i f f e r e n t heights i s given i n table VI.For most of the species, these taper equations have the same pattern of under-and overestimation as i n table II.For some species, however,such as alder,birch,immature coastal Western hemlock and maple,there i s a s l i g h t overestimation along the e n t i r e stem of the tree.This again can be due to the Figure I- The standard error of estimate as a function of the value of p in deriving the logarithmic taper equation from the logarithmic volume equation (for mature coastal Douglas-fir )• S Ef[inches) / 1 5 0 4 0 3 0 2 0 1 0 1-0 I 2 0 Value of p 3 0 Summary of the Taper Equations Derived from the B r i t i s h Columbia Forest Service Logarithmic Volume Equations Species Equation c o e f f i c i e n t s S E E optimum group R M b Q bj^ b g b^ (Inches) p value Alder C M -O.OO7438 O .960308 0.740496 - 0 . 7 0 3 4 8 5 0.746 2 . 3 1 Aspen I M 0 .004808 0.973017 0.704131 - 0 . 6 9 1 7 3 4 0 . 5 3 2 2 . 3 5 Balsam C M 0 .023528 O .903387 0 . 6 4 3 9 2 4 -O . 5 9 6 5 9 I 0 . 7 5 2 2 . 0 9 " I M 0.072540 0 .932481 0.710907 - 0 . 7 0 8 4 5 6 0.421 2.41 Birch I M -O . 0 3 5 4 0 5 0.955840 0.826483 - 0 . 7 7 3 7 8 2 O . 5 0 5 2.40 Cedar C M 0 . 1 7 7 6 9 ^ 0.841150 0.981589 -0.961733 2 . 0 0 2 2 . 8 5 " C I 0.130780 0 .860380 0.875467 - 0 . 8 5 0 4 8 0 1 . 2 6 4 2.62 M I M 0.120068 0.850996 0.881813 -0.848294 1.340 2 . 5 9 Cotton- CI M - 0 . 1 3 7 4 4 4 0.901986 0.776018 -O .65659I 0.684 2 . 0 6 wood Douglas C M -0.026595 0 . 8 2 9 5 0 6 0.743543 -0.645686 1.348 2.08 f i r M C I -0.000998 0.869962 0 . 7 3 5 1 7 2 -0.668579 0.987 2.18 " I M -0.034045 0 .869709 0.765145 -0.682128 1 .381 2 . 1 7 Hemlock C M - 0 . 0 0 2 5 3 3 0.895115 0 . 7 4 2 9 8 3 - 0 . 6 8 0 5 4 7 0.795 2 . 2 1 " C I - 0 . 0 1 4 5 9 0 0.921340 0.786591 - 0 . 7 2 4 7 6 1 1.182 2 . 2 9 " I M 0 . 0 2 7 2 3 1 0.984855 0 . 6 5 2 8 6 3 -0.664361 0.610 2 . 3 6 Lodgep. CI M - 0 . 0 0 4 5 4 2 0.923752 0.602058 -0.559172 0 . 7 6 0 2 . 0 3 pine Larch I M O.OO6838 O .92356I 0.684947 - 0 . 6 6 2 9 4 3 1.281 2 . 2 7 Maple C M -O . 0 3 3 6 3 5 0 . 9 4 2 9 0 6 0.876421 - 0 . 8 1 6 9 0 0 0 . 4 8 3 2.46 Spruce C M -0.002832 0.877085 0.850855 - 0 . ? 6 8 5 9 0 2 . 3 2 2 2 . 3 2 " I M 0.061738 0 . 9 2 0 6 1 3 0.756372 -0.739346 O . 5 2 2 2 . 4 3 white CI M 0 .076789 O . 9 3 3 6 4 3 0.673334 -O .676I58 O .938 2 . 3 6 pine Yellow CI M 0.101402 O .870522 O.73837O -O.709152 O . 5 8 7 2 . 3 4 cedar Yellow CI M -O.O57496 0.95^739 0 .623679 - 0 . 5 8 0 8 3 9 I . O 6 3 2 . 0 7 •—-pine-"—--—" -• . 1 1 lOaee f n . 4 and 5 i n table I. Test of the Taper Equations Derived from the B r i t i s h Columbia Forest Service Logarithmlo Volume Equations Average bias ( i n inches) of diameter inside bark at Species 11 0.4H group R M 1« 4 . 5 ' 0.1H 0.2H 0 . 3 H 0 . 5 H 0.6H 0.7H 0.8H 0.9H 1.0H Alder C M 0.48 0 . 7 7 0.88 O . 6 7 0 . 3 9 0.19 0.12 0.17 0 . 3 2 0 . 5 0 0.61 0.0 Aspen I M -0 . 1 3 0 . 4 7 0.60 0 . 4 5 0 . 2 5 0.08 -0.01 0.00 0 . 0 9 0 . 2 7 0 . 4 4 0.0 Balsam c M -0 . 7 6 -0 . 1 5 0 . 5 7 0 . 3 7 -0 . 0 5 -0 . 3 8 -0 . 5 4 -0 . 5 0 -0.28 - O . O 3 0.17 0.0 •i I M -0.44 0.24 0.42 0.24 0.01 -0.14 -0.20 -0 . 1 5 -0 . 0 5 0.08 0.20 0.0 Birch I M 0.26 0 . 5 8 O . 6 3 0.48 0 . 3 7 0 . 3 2 0 . 3 2 0 . 3 5 0 . 3 9 0.40 0 . 3 5 0.0 Cedar c M -1 . 9 0 -1.21 2.18 2.68 1 . 7 2 0 . 7 1 -0.11 -0 . 6 5 -1.02 -1.22 -1.16 0.0 ti c I -0 . 1 3 0.01 1.21 1 . 5 9 1.22 0 . 7 4 0 . 3 5 0.08 -0 . 0 7 -0 . 0 9 -0.01 0.0 ii I M -1.98 -1 . 0 3 1 . 0 3 1 . 3 5 0 . 7 7 0 . 0 7 -0 . 4 7 -0 . 7 9 -0 . 9 1 -0 . 8 3 -0 . 5 7 0.0 Cottonwood CI M -1 . 0 9 0 . 3 3 0 . 5 0 0.10 -0.28 - 0 . 5 3 -0.61 -0 . 5 0 -0.22 0 . 1 3 0 . 3 9 0.0 Douglas-fir c M - 2 . 9 6 0 . 3 8 1.22 1 . 3 3 0 . 7 7 0 . 0 9 -0.48 -0 . 7 6 -0 . 7 3 -0 . 5 2 -0.21 0.0 •t c I -1 . 5 8 O . 3 8 0 . 7 4 0 . 7 4 0 . 4 9 0.16 -0 . 1 5 -0 . 3 7 -0.46 -0 . 3 8 -0 .17 0.0 I I I M -3.12 0.28 1.09 0.80 0 . 0 3 -0 . 6 6 -1.10 -1.14 -0 . 8 5 -0 . 3 9 0.07 0.0 Hemlock c M -1.10 0.04 0.88 0.68 0 . 1 9 -0.21 -0.46 -0 . 5 0 -0 . 3 2 -0 . 0 5 0.21 0.0 •i c I 0 . 3 1 0.81 1 . 3 9 1 . 2 5 0.89 0 . 5 9 0.40 0 . 3 4 0 . 4 5 O . 5 6 0 . 5 9 0.0 it I M -0.87 0 . 2 5 0.68 0 . 5 2 0 . 2 3 -0 . 0 7 -0 . 2 6 -0.26 -0 . 0 9 0.24 0 . 6 2 0.0 Lodgepole pine CI M -0 . 3 9 -0 .17 0 . 1 5 0 . 3 4 0 . 1 9 0.02 -0.12 -0 . 2 3 -0.24 -0.12 0.06 0.0 Larch I M - 2 . 3 3 0 . 7 0 1 . 6 6 1 . 7 2 1.18 0 . 5 1 -0 . 0 3 -0 . 3 1 -0.28 0.01 0.46 0.0 Maple c M 0.24 O . 3 6 O . 5 6 O . 5 6 0 . 4 3 0 . 3 3 0.26 0 . 2 3 0 . 2 3 0.24 0.22 0.0 Spruce c M -2 . 6 9 -2.26 1 . 9 3 2 . 0 9 0 . 8 5 -0 . 2 9 -1 . 0 9 -1.46 -1.40 -1.01 -0 . 7 4 0.0 tt I M - 0 . 5 5 0.04 O . 6 7 0 . 7 1 0 . 3 7 0.02 -0.20 - O . 2 5 -0 . 1 9 -0 . 0 7 0.08 0.0 White pine CI M -0 . 7 4 -0 . 0 7 0.61 0 . 4 9 0 . 1 5 -0 . 1 5 -0 . 3 4 - 0 . 3 3 -0 .17 0.02 0 . 0 6 0.0 Yellow cedar CI M -0 . 2 7 -0.20 O . 5 6 0.81 O . 5 6 0 . 1 9 -0.18 - O . 3 6 -0 . 2 5 0.10 O . 3 6 0.0 Yellow pine CI M -1 . 8 9 0 . 6 6 O . 8 5 0 . 5 2 0 . 0 5 - 0 . 3 5 -0.60 - 0 . 5 6 -0 . 3 1 0 . 0 3 0 . 3 9 0.0 see f n . 4 and 5 i n table I above mentioned Inconsistency f o r c e r t a i n species between the B.C.F.S. logarithmic volume equations and the taper curves.Therefore,it i s doubtful i f the r e s u l t s f o r coast and i n t e r i o r redcedar,yellow cedar and the deciduous species,for which f o r k i n g of the stem i s a common abnor-mality, can be used as such. An example f o r mature coastal Douglas-fir i n table VII shows how the bias i s d i s t r i b u t e d over the various height classes within the same species.Except f o r the three smallest height classes,the o v e r - a l l SEgis a f a i r l y good representative f o r a l l the height c l a s s e s . Table VIII gives a summary of the SE|s of table I, table V and those of Kozak,Munro and Smith ( 1 9 6 9 b ,table I ) . For the logarithmic taper equations,derived from the B.C.F.S. logarithmlo volume equations,the SE^s ranged from .421 to 2 . 3 2 2 inches.Table IX gives the absolute frequency d i s t r i b u t i o n of the SE|s f o r each case. An absolute frequency d i s t r i b u t i o n of the d i f f e r e n c e s i n SE£s i s shown i n table X. Because they include the errors inherent i n both bark and wood,these SE£s are r e l a t i v e l y small compared with the SE£s of section double bark thickness,estimated from diameter outside bark,total height,section height and sec t i o n height as a percentage of tree height,ranging from . 1 1 1 to .842 inches (Smith and Kozak,1 9 6 7 ) . The SE Eof the logarithmic taper equation,fitted on the taper curves or derived from the logarithmic volume D i s t r i b u t i o n of the Bias over the D i f f e r e n t Height Classes within the Same Species (for Mature Coastal Douglas-fir) Height Average bias 1 (in Inches) 1 class (feet) 1» * . 5 ' 0.1H 0.2H 0.3H 0.4H 50 -0.20 0 . 6 5 0 . 6 1 0.47 0 . 1 9 -0.14 60 -0.26 0 . 7 1 0 . 7 7 0 . 6 5 0 . 2 9 -0 . 0 3 70 -0.04 0 . 7 3 0.82 O . 5 6 0.16 -0.14 80 -1 . 0 7 0 . 4 7 0 . 8 5 0 . 9 5 0.48 -0 . 1 5 90 -1 . 8 3 0 . 7 0 1 . 2 3 1.21 O . 6 3 -0.08 100 -2.12 0 . 7 1 1 . 3 2 1.28 0 . 6 5 0.04 110 -2 . 6 9 0 . 5 9 1 . 4 7 1 . 2 9 O . 6 5 0.04 120 - 3 . 6 0 0 . 5 0 1.46 1 . 3 4 0 . 6 6 -0.10 130 -4 . 1 3 0.46 1.42 1 . 3 7 0 . 7 4 -0 . 0 3 140 - * . 5 5 0.08 1 . 6 2 1 . 5 2 1.00 0.20 150 - 5 . 0 3 0 . 1 3 1 . 4 4 1 . 5 7 1.02 0 . 3 2 160 - 3 - 9 8 0.10 1 . 4 3 1 . 6 3 1 . 0 7 0.48 170 -5.10 -0.06 1 . 2 7 1 . 7 7 1.08 0.38 180 - 4.46 0 . 1 6 1.88 1.70 1 . 0 3 0 . 2 9 190 - 3 . 5 7 0 . 0 9 -0.21 1.48 1.01 0 . 1 6 200 - 3 . 6 3 0 . 2 3 I . 6 9 1 . 9 4 1.24 0.21 Total -2 . 9 6 O . 3 8 1.22 1 . 3 3 0 . 7 7 0.09 diameter inside bark at SE C (Inches) 0 . 5 H 0.6H 0.7H 0.8H 0.9H l.OH -0.42 -0 . 6 8 -0 . 8 5 -0 . 8 3 -0.59 0.0 0 . 6 1 0 -0.28 -0 . 4 7 -0 . 5 9 -0.61 -0.39 0.0 0 . 5 6 0 -0.26 -0.28 -0.21 0.00 0.21 0.0 0 . 4 5 8 -0 . 7 3 -1.11 -1.24 -1.24 -0.88 0.0 1.168 -0 . 7 3 -1 . 2 3 -1 . 6 5 -1.80 -1.42 0.0 1.348 -0 . 4 4 -0 . 6 9 -0.90 -0.90 -0.49 0.0 1 . 1 0 9 -0.48 -0.82 -0.82 -0 . 4 9 -0 . 1 3 0.0 1 . 3 * 5 -0 . 7 6 -1.08 -1 . 1 3 -1.11 -0 . 9 9 0.0 1 . 4 4 9 -0.61 -O .76 -0.60 -0.42 -0.24 0.0 1 . 4 5 2 -O . 3 9 -0.61 -0 . 5 3 -0.40 -0.21 0.0 1.564 -0.12 -0 . 3 1 -0 . 3 0 -0.28 -0 . 1 3 0.0 I . 6 6 7 -0.07 -0.48 -0.40 -0 . 0 7 0 . 2 7 0.0 1.426 -0 . 1 3 -0 . 4 4 -0.24 0 . 3 0 0.64 0.0 1.701 -0 . 3 7 -0 . 5 4 -0.07 0 . 4 5 0.70 0.0 1.613 -0 .77 -1.09 -O . 6 3 -0.10 0.28 0.0 I . 6 5 8 -0.61 -0.78 -0.40 0 . 2 3 0.81 0.0 1 . 5 0 2 -0.48 -O .76 -0 . 7 3 -0 . 5 2 -0.21 0.0 1.348 Comparison of Standard Errors of Estimate of Several Methods Species group R M « S E d ) 1 ^ S E £ ( 2 ) 1 4 S E ( 3 ) 1 5 E Alder C M 0,84 0 . 5 8 7 0.746 Aspen I M 0 . 5 9 0.448 0 . 5 3 2 Balsam C M 0 . 9 0 0 . 8 1 3 0 . 7 5 2 I I I M 0 . 5 8 0.420 0.421 Birch I M 0 . 3 2 0.245 0 . 5 0 5 Ced ar C M 2 . 1 3 2 . 4 3 1 2 . 0 0 2 it c I 1 . 6 1 1.128 1.264 M I M 1 . 3 0 1 . 3 7 9 1.340 Cottonwood CI M 0.84 0 . 7 5 5 0.684 Douglas-fir c M 1.5* 1 . 4 3 1 1.348 n c I 1 . 3 5 1 . 0 3 2 0 . 9 8 7 •i I M 1 . 3 3 1 . 2 5 1 1.381 Hemlook c M 0 . 9 8 0 . 8 7 1 0 . 7 9 5 •t c I 1 . 1 6 0 . 6 9 1 1.182 I I I M 0 . 7 3 0.740 0 . 6 1 0 Lodgepole pine CI M 0 . 7 2 0 . 7 7 4 O.76O Larch I M 1 . 3 3 1 . 2 3 0 1.281 Maple c M 0.41 0 . 3 1 6 0 . 4 8 3 Spruce c M 2 . 3 4 2 . 3 7 6 2 . 3 2 2 I I I M 0 . 7 1 0 . 5 2 6 0 . 5 2 2 White pine CI M 1 . 0 1 1 . 1 5 9 0 . 9 3 8 Yellow cedar CI M 0 . 7 8 0 . 5 7 * 0 . 5 8 7 Yellow pine CI M 1 . 0 2 1 . 2 0 5 I . O 6 3 xcaee f n . 4 and 5 i n table I. l 3 f r o m table I of Kozak,Munro and Smith ( 1 9 6 9 b ) Inequations f i t t e d on the taper curves (table I) ^eq u a t i o n s derived from the volume equations (table V) Standard Errors of Estimate Number of speciesgroups SE E ( 1 ) ( 2 ) ( 3 ) 1 6 (Inches) . 0 < - < - 5 2 4 2 • 5 < - < 1 . 0 10 9 12 1 . 0 < - < 1 . 5 7 8 7 1 . 5 < - < 2 . 0 2 -2 . 0 < - < 2 . 5 2 2 2 see f n . 13»14 and 15 i n table V I I I . TABLE X Absolute Frequency D i s t r i b u t i o n of the Differences i n Standard Errors of Estimate Difference (inches) ( 2 ) BT ( 1 ) Number (1) BT ( 2 ) of speciesgroups ( 3 ) ( 1 ) ( 3 ) BT BT BT ( 1 ) ( 3 ) ( 2 ) ( 2 ) 1 7 B T 1 8 ( 3 ) . 0 — . 1 6 4 5 6 9 4 . 1 — . 2 6 2 9 1 3 4 . 2 — . 3 2 1 - - - 1 . 3 — .4 2 - 2 - - -.4 - . 5 - mm - 1 1 17 see f n . 13 ,14 and 15 i n table V I I I . 1 8BT ] means Better Than. equation,is f o r sixteen speolesgroups out of twenty-three smaller than the SE Egiven by Kozak,Munro and Smith ( 1 9 6 9 b ) . The S % o f the taper function derived from the volume equation i s f o r t h i r t e e n speolesgroups smaller,but f o r ten speolesgroups la r g e r than the SE Eof the taper equation f i t t e d on the taper curves. IMPROVEMENT OF THE ACCURACY AND THE PRECISION OF THE LOGARITHMIC TAPER EQUATION A well known technique f o r improving the accuracy and p r e c i s i o n of a taper equation consists of r e l a t i n g the taper equation c o e f f i c i e n t s to some known tree characte-r i s t i c s . In most inventory work only dbh outside bark and t o t a l height are measured.Although the number of possible r e l a t i o n s h i p s to investigate i s large,a close look was taken only Into some very simple approaches. To improve the taper equation f i t t e d on the taper curves,the r e l a t i o n s h i p s between the taper equation c o e f f i c i e n t s ( b 0 , b 1 , b 2 and b-^ ) and dbh,total height and the r a t i o of both were investigated.The c o r r e l a t i o n was very poor (apparently a second or t h i r d degree polyno-mial) or non-existent,except f o r b 2 where a good r e l a t l o -ship with the r a t i o of t o t a l height over dbh was always present.However,this r e l a t i o n s h i p did not have the same pattern f o r the d i f f e r e n t species.A t r i a l of a taper equation model i n which each c o e f f i c i e n t was expressed as a second degree polynomial function of the r a t i o of t o t a l height over dbh was successful only f o r f i v e species,decreasing the SE Eby from . 1 to . 3 inches. To improve the taper equation derived from the volume equation the r e l a t i o n s h i p between the optimum value of p and dbh,total height and the r a t i o of both was investigated.This was attempted f o r mature coastal Douglas-fir.A good c o r r e l a t i o n was found between the optimum value of p and t o t a l height (see f i g u r e 2). It i s expected that using t h i s r e l a t i o n s h i p between the optimum value of p and t o t a l height,instead of only the o v e r - a l l optimum value,can Improve the p r e c i s i o n and acouracy.But even i f the optimum value of p f o r each t o t a l height class can be predicted from t o t a l height without error,the decrease i n S E E w l l l only be moderately Important f o r the smallest height classes (see table XI). The o v e r - a l l S E E w i l l l i k e l y change only l i t t l e . Figure 2- Optimum value of p as o function of total height (for mature coastal Douglas-fir)-Number of observations = 16 Optimum o Value of p R X 100 = 9 0 % for a parabolic relationship 2-2 2 0 16 1-6 L A I I I I I I I I I I I I I I I l V50 100 1 5 0 2 0 0 Total Height (feet) Maximum Decrease In Standard Error of Estimate to Be Expected from Using the Relationship between the Optimum Value of p and Tot a l Height (for Mature Coastal Douglas-fir) Height c l a s s (feet) Optimum value of P i n 1 9 SEg (inches) o n ( 2 ) 2 0 5 0 1.75 0.610 0 . 3 5 0 60 1.80 0 . 5 6 0 0 . 3 6 5 70 1 . 9 2 0.458 0.383 80 1.84 1.168 0.957 90 1.85 1.348 1 . 1 1 7 100 1.97 1.109 1.046 110 2.01 1.3*5 1 . 3 2 1 120 2.01 1.449 1.409 130 2 . 0 9 1.452 1.441 140 2.11 1.56* 1 . 5 5 0 150 2 . 1 5 1.66? 1.626 160 2 . 1 3 1.426 1.396 170 2.18 1.701 1.618 180 2 . 1 5 1.613 1 . 5 6 2 190 2 . 1 3 1.658 1 . 6 3 3 200 2.12 1 . 5 0 2 1.481 19 . using the o v e r - a l l optimum value of p using f o r each t o t a l height c l a s s the appropriate optimum value of p This proposed system of taper and volume functions derived from each other and compatible with each other can meet the requirements stated by Honer and Sayn-Wlttgenatein ( 1 9 6 3 ) . The taper function f i t t e d on the taper curves as w e l l aa the equation derived from the volume equation describes well the stem p r o f i l e of the most important species of B r i t i s h Columbia.However,lt should be realized that a taper equation f i t t e d on diameter data gives no guarantee of a good volume equation.Tests on tree measurements should be carried out i n t h i s f i e l d . Whenever a taper equation i s f i t t e d on data,the function should be tested both f o r diameter and volume to know for both the p r e c i s i o n and the accuracy of the system. This was recognized by Bruce and Schumacher ( 1 9 5 0 ) and done by Duff and B u r s t a l l ( 1 9 5 5 ) i n the a p p l i c a t i o n of graphical techniques. Giving more weight to large diameters would ensure a better f i t f o r volume,probably r e s u l t i n g In a better f i t at the butt of the tree but a worse f i t higher on the stem.This would make the equation l e s s suitable f o r p r e d i c t i o n of section diameters or heights.Instead of 2 weighting,the dependent v a r i a b l e could be taken as d and the c a l c u l a t i o n of the taper equation could be done by a non-linear l e a s t squares procedure.But t h i s again would probably have the same disadvantages as weighting. The t r i a l s to Improve the taper equation by r e l a t i n g the taper equation c o e f f i c i e n t s to some known tree c h a r a c t e r i s t i c s were not comprehensive enough to draw f i n a l conclusions.The preliminary Investigations In that d i r e c t i o n were discouraging. Preference i s given to the system i n which the taper equation i s derived from the logarithmic volume equation.In t h i s way the best f i t i s achieved f o r volume and the f i t f o r diameter i s optimized by the choice of the optimum value of p.Moreover, i t i s the only possible way to create a t r u l y compatible system of taper and volume i n those instances where a logarithmic volume equation already e x i s t s and probably w i l l continue to be used i n the fu t u r e . Avery, T. E. 1 9 6 7 . Forest Measurements. McGraw-Hill Book Co.,Inc.,N.Y. 2 9 0 p. B r i t i s h Columbia Forest Service, 1 9 6 8 . Basic taper curves f o r the commercial species of B r i t i s h Columbia. Forest Inventory D i v i s i o n , B.C.F.S., Dept. of Lands, Forests and Water Resources, V i c t o r i a , B.C.,unpaged graphs• Behre, C. E. 1 9 2 3 . Preliminary notes on studies of tree form. Jour. For. 2 1 : 5 0 7 - 5 1 1 . • 1 9 2 7 » Form-class taper curves and volume tables and t h e i r a p p l i c a t i o n . Jour. Agr. Res. 3 5 ( 8 ) : 6 7 3 - 7 4 3 . — — — — 1 9 3 5 . Factors involved i n the a p p l i c a t i o n of form-class volume t a b l e s . Jour. Agr. Res. 5 1 ( 8 ) : 6 6 9 - 7 1 3 . Browne, J . E. 1 9 6 2 . Standard cubic-foot volume tables f o r the commercial tree species of B r i t i s h Columbia,1 9 6 2 . B.C.F.S., V i c t o r i a , B.C., 107 p. Bruce, D. and F. X. Schumacher, 1 9 5 0 . Forest Mensuration. McGraw-Hill Book Co., Inc., N.Y., 4 8 3 P» Bruce, D., C u r t i s , R. 0 . and C. Vancoeverlng, 1 9 6 8 . Deve-lopment of a system of taper and volume tables f o r red a l d e r . For. Sc. 1 4 ( 3 ) 0 3 9 - 3 5 0 . Chapman, H. H. and W. H. Meyer, 1 9 4 9 . Forest Mensuration. McGraw-Hill Book Co.,Inc., N.Y . , 5 2 2 p. Claughton-Wallln, H. 1 9 1 8 . The absolute form quotient. Jour. For. 1 6 : 5 2 3 - 5 3 * -— — — — — and F. McVlcker, 1 9 2 0 . The Jonson*absolute form quotient as an expression of taper. Jour. For. 18: 346 - 3 5 7 . Demaerschalk, J . P. 1 9 7 1 * Taper equations can be converted to volume equations and point sampling f a c t o r s , (sub-mitted to the For. Chron.),typed, 6 p. Duff, G. and S. W. B u r s t a l l , 1 9 5 5 * Combined taper and volume t a b l e s . Forest Research I n s t i t u t e . Note No. 1 . New Zealand Forest Service, 73 p. P r i e s , J . 1 9 6 5 « Eigenvector analyses show that b i r c h and pine have s i m i l a r form In Sweden and B r i t i s h Columbia, For. Chron. 41( 1 ) : 1 3 5 - 1 3 9 . — — — — — and B. Matern, 1 9 6 5 * On the use of multivariate methods f o r the construction of tree taper curves, I.U.F.R.O. Section 2 5 . Paper No. 9 ,Stockholm Con-ference, October, 1 9 6 5 * 32 p. Giurgiu, V. 1 9 6 3 . (An a n a l y t i c a l method of constructing dendrometrlcal tables with the aid of e l e c t r o n i c computers). Rev. Padurllor 7 8 ( 7 ) ' 3 6 9 - 3 7 4 ( i n Ruma-nian), see Bruce,CurtIs and Vancoevering ( 1 9 6 8 ) . Gray, H. R. 1 9 5 6 . The form and taper of f o r e s t - t r e e stems. Imp. For. Inst. Oxford, Inst. Paper No. 3 2 , 79 p. Grosenbaugh, L. R. 1 9 5 ^ * New tree measurement concepts: height accumulation,giant tree,taper and shape. U.S.F.S. South. For. Exp. Sta. Occasional Paper No 1 3 4 , 32 P. — " 1 9 6 6 . Tree form: d e f i n i t i o n , I n t e r p o l a t i o n , e x t r a -p o l a t i o n . For. Chron. 42 ( 4 ) : 4 4 3 - 4 5 6 . Heger, L. 1 9 6 5 a . Morphogenesis of stems of Douglas-fir. Univ. of B.C., Faculty of Forestry, Ph.D. t h e s i s , L l t h o . 176 p. 1 9 6 5 b . A t r i a l of Hohenadl's method of stem form and stem volume estimation. For. Chron. 41 ( 4 ):466 - 4 7 5 * H e i j b e l , I. 1 9 2 8 . (A system of equations f o r determining stem form i n pine).Svensk. SkogsvFfiren. Tldskr. 3 - 4 : 3 9 3-422.(in Swedish, summary i n E n g l i s h ) . HeJJas, J . 1 9 6 7 . Comparison of absolute and r e l a t i v e standard errors and estimates of tree volumes. Univ. of B.C., Faculty of Forestry, M.P. thesis,typed, 58 p. Hcjjer, A. G. 1 9 0 3. Tallens och granens t i l l v f i x t . Blhang t i l l F r . Loven. Om vara barrskogar. Stockholm, 1 9 0 3 * (In Swedish), see Behre ( 1 9 2 3 ) . Honer, T. G. 1 9 6 4 . The use of height and squared diameter r a t i o s f o r the estimation of merchantable cubic-foot volume. For. Chron. 40 ( 3 ) : 3 2 4 - 3 3 1 . 1 9 6 5 a . Volume d i s t r i b u t i o n i n i n d i v i d u a l t r e e s . Woodlands Review Section, Pulp and Paper Magazine of Canada. Woodlands Section. Index 2349 ( F - 2 ) : 4 9 9 - 5 0 8 . Honer, T. G. 1 9 6 5 b , A new t o t a l cubic-foot volume function. For. Chron. 4 1 ( 4 ) : 4 7 6 - 4 9 3 « • ' 1 9 6 7 * Standard volume tables and merchantable con-version f a c t o r s f o r the commercial tree species of c e n t r a l and eastern Canada. Forest Management re-search and services i n s t i t u t e , Ottawa, Ontario, Information report FMR-X-5, 153 p. — — — — — — and L. Sayn-Wittgenstein, 1 9 6 3 * Report of the committee on f o r e s t mensuration problems. Jour. For. 6 l ( 9 ) : 6 6 3 - 6 6 7 . Jonson, T. 1 9 1 0 . Tasatoriska undersflkningar om skogstrfldens form.I.Granens stamform. Skogsvardsfflreningens Tidskr. 1 1 : 2 8 5 - 3 2 8 . ( i n Swedish), see Behre ( 1 9 2 3 ) . 1 9 H • Taxatorlska undersfikningar om skogstrSdens form.II.Tallens stamform. SkogsvardsfBrenlngens Tidskr. 9 - 1 0 : 2 8 5 - 3 2 9 . ( i n Swedish), see Behre ( 1 9 2 3 ) . — — — 1 9 2 6 - 1 9 2 7 . Stamformsproblemet. Medd. f . Statens Skogsffir. 2 3 : 4 9 5 - 5 8 6 . ( i n Swedish). Kozak, A. and J . H. G. Smith, I 9 6 6 . C r i t i c a l analysis of multivariate techniques f o r estimating tree taper suggests that simpler methods are best. For. Chron. 4 2 ( 4 ) : 4 5 8 - 4 6 3 . — — Munro, D. D. and J . H. G. Smith, 1 9 6 9 a . Taper functions and t h e i r a p p l i c a t i o n i n f o r e s t Inventory. For. Chron. 4 5 ( 4 ) : 1 - 6 . Munro, D. D. and J . H. G. Smith, 1 9 6 9 b . More accuracy required. Truck Logger. December : 2 0 - 2 1 . Larson, P. R. 1 9 6 3 . Stem form development of forest t r e e s . For. Sc. Monograph No. 5 » 42 p. Loetsch, F. and K. E. H a l l e r , 1 9 6 4 . Forest Inventory. V o l . I. S t a t i s t i c s of f o r e s t inventory and informa-t i o n from a e r i a l photographs. BLV Verlaggesellschaft• Mttnchen. (trans, by E. F. Brtinig). Matte, L. 1 9 4 9 . The taper of coniferous species with s p e c i a l reference to l o b l o l l y pine. For. Chron. 2 5 : 2 1 - 3 1 . Mesavage, C. and J . VI. Girard, 1 9 4 6 . Tables f o r estima-t i n g board-foot content of timber. U.S.F.S. Washington D.C., 94 p. Meyer, H. A. 1 9 5 3 . Forest Mensuration. Penns Valley Publishers, Inc., State College, Pennsylvania, 357 p. Munro, D. D. 1 9 6 8 . Methods f o r describing d i s t r i b u t i o n of soundwood i n mature western hemlock tre e s . Univ. of B.C., Faculty of Forestry, Ph.D. thesis,mlmeo, 188 p. 1 9 7 0 . The usefulness of form measures i n the estimation of volume and taper of the commercial tree species of B r i t i s h Columbia. Paper presented at a meeting of the working group "Estimation of Increment". I.U.F.R.O., Section 2 5 , Birmensdorf Conference, September, 1 9 7 0 . 14 p. Newnham, R. M. 1 9 5 8 . 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., Faculty of Forestry, M.F. t h e s i s , typed, 71 P« Osumi, S. 1 9 5 9 . Studies on the stem form of the f o r e s t trees ( 1 ) . On the r e l a t i v e stem form. Jour. Jap. For. Soc. 41(12) : 4 7 1 - 4 7 9 . ( i n Japanese,abstract i n English) Petterson, H. 1 9 2 7 * Studier over Stamformen. Medd. Statens Skogf8rs8ksanstalt. 2 3 : 6 3 - 1 8 9 . ( i n Swedish). Prodan, M. 1 9 6 5 . Holzmesslehre. J.D.SauerlUnder's Verlag, Frankfurt am Mein. 644 p. Schmld, P., Rolko-Jokela, P., Mingard, P. and M. Zobeiry, 1 9 7 1 . The optimal determination of the volume of stan-ding t r e e s . Mitteilungen der F o r s t l i c h e n Bundes-Versuchsanhalt, Wlen. 9 1 : 3 3 - 5 * • Shlnozakl, K., Yoda K y o j i , Hozumi, K. and T. K i r a , 1 9 6 4 . A quantitative analysis of plant form.The pipe model theory. Jap. Jour. E c o l . 14 ( 3 ) : 9 7-104. Smith, J . H. G. and A. Kozak, 1 9 6 7 . Thickness and percent-age of bark of the commercial trees of B r i t i s h Columbia. Univ. of B.C., Faculty of Forestry, mlmeo. 33 p. — — — — — and A. Kozak, 1971• Further analyses of form and taper of young Douglas-fir,Western hemlock,Western redcedar and S i l v e r f i r on the University of B r i t i s h Columbia research f o r e s t . Paper presented at the Northwest S c i e n t i f i c Association Annual Meeting, Univ. of Idaho, A p r i l , 1971• mlmeo. 8 p. Speldel, G. 1957* Die rechneriachen grundlagen der le i s t u n g s k o n t r o l l e und lhre praktische durchftirung i n der f o r a t e i n r i c h t u n g . Schrlftenrelhe der F o r s t l l c h e n FakultUt, Unlversltflt Gflttlngen. No. 1 9 , 118 p. Spurr, S. H. 1 9 5 2 . Forest Inventory. Ronald Preas Co., N.Y. 476 p. Stanek, W. 1 9 6 6 . Occurrence,growth and r e l a t i v e value of lodgepole pine and Engelmann spruce l n the i n t e r i o r of B r i t i s h Columbia. Univ. of B.C., Faculty of Forestry. Ph.D. thesis,typed. 2 5 2 p. Turnbull, K. J . and G. E. Hoyer, I 9 6 5 . Construction and ana l y s i s of comprehensive tree-volume t a r i f t a b l e s . Resource management report. No. 8. Department of Natural Resources, State of Washington. 63 p. Wickenden, H. R. 1 9 2 1 . The Jonson absolute form quotient: how i t i s used i n timber estimating. Jour. For. 1 9 : 5 8 4 - 5 9 3 . Wright, W. G. 1 9 2 3 . Investigation of taper as a f a c t o r l n measurement of standing timber. Jour. For. 2 1 : 5 6 9 - 5 8 1 . Common Names and L a t i n Names of 21 the Tree Species l a Red Alder (Alnus rubra Bong.). 2 . Trembling Aspen (Populus tremuloldes Michx.). 3. Coast Balsam Species (Abies amabilis (Dougl.) Forbes and A.grand i s (Dougl.) L i n d l . ) . 4. I n t e r i o r Balsam Species (Abies l a s i o c a r p a (Hook.) Nutt. and A.grandis). 5 . White Birch Species (Betula papyrifera v a r i e t i e s ) . 6. Western Red Cedar (Thuja p l i c a t a Donn). 7 . Black Cottonwood (Populus trlchocarpa Torr. and Gray). 8. Douglas F i r (Pseudotsuga menziesii (Mlrb.) Franco). 9. Western Hemlock (Tsuga heterophylla (Raf.) Sarg.). 1 0 . Lodgepole Pine (Pinus contorta Dougl.). 1 1 . Western Larch (Larlx o c c i d e n t a l l s Nutt.). 1 2 . Broadleaf Maple (Acer macrophyllum Pursh). 1 3 . Coastal Spruce (Plcea s i t c h e n s i s (Bong.) Carr.). 14. I n t e r i o r Spruce Species (Picea glauca (Moench) Voss, P.Engelmanni Parry,and P.marlana ( M i l l . ) B.S.P.). 1 5 » Western White Pine (Pinus monticola Dougl.). 16. Yellow Cedar (Chamaecyparis nootkatensis (D.Don) Spach)• 1 7 . Western Yellow Pine (Pinus ponderosa Laws.). Based on Appendix I from Browne ( I 9 6 2 ) . APPENDIX 2 Derivation of a Volume Equation from the Taper Equation of Hfljer V = D 2H 0 . 0 0 5 4 5 4 / ( a 2 / D 2) d (1 / H) ( 1 ) d 2 / D 2 = c 2 ( l n ( ( o 2 + 1 100 / H)/ o 2 ) ) 2 ( 2 ) s u b s t i t u t i n g ( 2 ) i n ( 1 ) A V = D 2H 0 . 0 0 5 4 5 4 / ( o 2 ( l n ( ( o 2 + 100 1 / H)/ c 2 ) ) 2 ) d ( 1 / H) 0J . = D 2H 0 . 0 0 5 4 5 4 c 2 1 100 1 100 0 (1 + ) ( i n ( i + ) ) 2 „ H c 2 H c 2 1 100 1 100 1 100 2 (1 + ) ( l n ( l + )) + 2 (1 + ) H o 2 H o 2 H c 2 ? 2 / 1 0 0 1 0 0 >> = D H 0 . 0 0 5 4 5 4 c | ( ( l + — ) ( l n ( l + ) ) 2 -\ c 2 ° 2 100 100 100 2 (1 + ) ( l n ( l + — ^ ) ) + 2 (1 + ) - 2 c 2 o 2 o 2 = D 2H 0 . 0 0 5 4 5 4 o 2 (K(ln K (ln K - 2 ) + 2 ) - 2 ) 100 where K = 1 + —— Co APPENDIX 3 Derivation of a Volume Equation from the Taper Equation of Behre 0 V = D 2H 0 . 0 0 5 * 5 * j ( d 2 / D 2) d (1 / H) ( d 2 / D 2 = (1 / H ) 2 / ( b Q + b x 1 / H) 2 ( su b s t i t u t i n g ( 2 ) i n (1) V = D 2H 0 . 0 0 5 * 5 * / ( ( 1 / H ) 2 / ( b Q + b x 1 / H) 2) d ( 1 / H) O-' l r ( b 0 + 1 / H) - 2 b 0 In ( b Q + 2 1 = D^H 0 . 0 0 5 * 5 * — r b 2 "° b x 1 / H) - 0 ( b 0 + bj 1 / H) = D^H 0 . 0 0 5 4 5 4 — ( b 0 + b 1 - 2 b 0 In ( b 0 + bj) -b 0 / ( b 0 + V + 2 b 0 l n V If b Q + b 1 = 1 what was usually the case,according to Behre ( 1 9 2 7 ) then 1 V = D 2H 0 . 0 0 5 4 5 4 — (1 - bp + 2 b Q l n b Q) b l APPENDIX 4 Derivation of a Volume Equation from the Taper Equation of Matte H v = 0 . 0 0 5 * 5 * / (a 2) d ( i ) (1) d 2 = b 0 D 2 ( l 2 / H 2 ) + b x D 2 ( l 3 / H 3 ) + b 2 D 2 ( l 4 / H 4 ) ( 2 ) s u b s t i t u t i n g ( 2 ) i n ( 1 ) V , H = 0 . 0 0 5 * 5 * D 2 / ( b Q ( l 2 / H 2 ) + b x ( l 3 / H 3 ) + 0-> bp U 4 / H 4 ) ) d (1) = = 0 . 0 0 5 * 5 * D' H 0 b 0 l 3 b l l 4 b 2 i 5 3 H < * H 3 5 H 4 = 0 . 0 0 5 4 5 4 Dd H (b Q / 3 + b x / 4 + b 2 / 5 ) APPENDIX 5 Derivation of the Height Equation from the Logarithmic Taper Equation The taper equation (formula 2) i s : d = 10 b° D b l l b 2 H b3 thus l b 2 = d / (10 b0 D b l H b3) 1 = (d / (10 b0 D b l H b 3 ) ) 1 / b 2 or 1 = ( 1 0 ' b 0 d D " b l H " b 3 ) 1 / b 2 Derivation of a Logarithmic Volume Equation from the Logarithmic Taper Equation V = O.OO5454 / (d*) d ( 1 ) d 2 = 1 0 2 b 0 D 2 b l l 2 b 2 H 2 b 3 (1) (2) s u b s t i t u t i n g (2) i n (1) V = 0.005454 10 2 b0 / ( D 2 b l l 2 b 2 H 2 b3) d (1) = OJ = 0.005454 1 0 2 b ° 0.005454 i o 2 b ° (2 bo + 1) H - D 2 b ! ^ b g f l R 2 b 3 n 0 (2 b 2 + 1) D 2 b i R 2 b 2 + 2b 3 + 1 a f t e r taking the logarithm log V = a + b log D + c log H where a = log '0.005454 1 0 2 b ° (2 b 2 + 1) b = 2 b. c = 2 b 2 + 2 b + l Derivation of the Formula to Estimate Volumes of Logs between S p e c i f i c Distances from the Tip of the Tree,from the Logarithmic Taper Equation When the lower and upper distance from the t i p of the tree are r e s p e c t i v e l y 1^  and l 2 , t h e n the volume of the log between these two distances i s : .U V = O.OO545* / (d 2) d (1) = 0.005454 1 0 2 b ° / ( D 2 b l l 2 b 2 H 2 b 3 ) d (1) = 2 b 1 r 1 1 - l 2 0.005454 i o^ D o 9. _ 9. D2 b l 2 2b2+l H 2b3 (2 b 2 + 1) 0.005454 1 0 2 b 0 (2 b 2 + 1) %v „ V /.Z D 2 b ! H 2 b 3 ( 1 2 b 2 + l _ ^ b z + l j _ where K = = K D v R-y al - i f ) 0.005454 i o 2 b o (2 b 2 + 1) 7 = 2 b, v = 2 b. z = 2 b 2 + 1 Derivation of a Compatible Logarithmic Taper Equation from a Logarithmic Volume Equation The proof i s given that the taper equation log d = b Q + b 1 log D + b 2 log 1 + b^ log H / 1 0 a p c \ 1 / 2 where b Q = log I — — J b 2 = (p c - 1 ) / 2 bx = b / 2 b 3 = (1 - p)c / 2 y i e l d s f o r any value of p,after integrating,the same volume as given by the logarithmic volume equation log V = a + b log D + c log H Proof: • H V = 0 . 0 0 5 4 5 4 / (d 2) d (1) (1) / )J 2 from the taper equation,d i s defined as 2 10 p c b «(P c - 1) (.1 - p) c = ^ D D 1 H v v ( 2 ) 0 . 0 0 5 4 5 4 s u b s t i t u t i n g (2) i n (1) v = 0 . 0 0 5 4 5 4 H / 1 0 a p c /n/10 005454 D b X ( P c - 1) H ( i - P ) d (1) = 0.005454 10 a p 0 O.OO5454 p c H r -D b 1 P ° H ( 1 " p ) 0 _ 0 = 10 a D b H P ° H° " P ° = 10 a D b H° a f t e r taking the logarithm log V = a + b log D + o log H t h i s completes the proof. 

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