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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An integrated system for the estimation of tree taper and volume Demaerschalk, Julien Pierre 1971
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Title | An integrated system for the estimation of tree taper and volume |
Creator |
Demaerschalk, Julien Pierre |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | A new taper equation is presented, log d = b₀ + b₁ log D + b₂ log 1 + b₃ log H where d is the diameter inside bark in inches at any given 1 in feet,D is the diameter breast height outside bark in inches, 1 is the distance from the tip of the tree in feet,H is the total height of the tree in feet and b₀,b₁,b₂ and b₃ are the regression coefficients. Two methods of deriving a compatible system of tree taper and volume equations are discussed.One method involves conversion of the logarithmic taper equation into a logarithmic volume equation.The other involves the derivation of the logarithmic taper equation from an existing logarithmic volume equation to provide compatabllity in volume estimation and at the same time ensure as a good fit as possible for the estimation of upper bole diameters (taper). Tests for precision and bias of volume estimates, carried out on the British Columbia Forest Service taper curves and logarithmic volume equations,indicate that the latter approach is preferable to the former. |
Subject |
Forests and forestry -- Measurement. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-04-27 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0075435 |
URI | http://hdl.handle.net/2429/34021 |
Degree |
Master of Forestry - MF |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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