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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. E N G . , U n i v e r s i t y of L o u v a i n , 1 9 6 7  A THESIS SUBMITTED  IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF FORESTRY i n the Department of FORESTRY  We accept t h i s t h e s i s as conforming' t o the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA July,  1971  In p r e s e n t i n g  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  r e q u i r e m e n t s f o r an advanced degree a t the of B r i t i s h Columbia,I agree t h a t the it  f r e e l y a v a i l a b l e f o r reference  I f u r t h e r agree t h a t  and  study.  p e r m i s s i o n f o r e x t e n s i v e copying  Department or by  I t i s understood  that  his  written  Department The  permission.  of  University  Vancouver 8,  of B r i t i s h  Canada  be granted  by  representatives.  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 g a i n s h a l l not my  University  L i b r a r y s h a l l make  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 the Head of my  the  Columbia  be allowed  without  ABSTRACT A new t a p e r e q u a t i o n log d = b  Q  + b  1  l a presented,  log D + b  2  l o g 1 + b<j l o g H where  d l a t h e diameter I n s i d e bark In Inches a t any g i v e n 1 i n feet,D  i s t h e diameter b r e a s t h e i g h t o u t s i d e bark i n Inches,  1 Is the d i s t a n c e from the t i p of the t r e e In f e e t , H i s t h e t o t a l height of the t r e e i n f e e t and b b ^ , b 0 f  2  and bj a r e  the r e g r e s s i o n 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 t a p e r and volume e q u a t i o n s  system of t r e e  a r e dlscussed.One method  ves c o n v e r s i o n of the l o g a r i t h m i c t a p e r e q u a t i o n  invol-  into a  l o g a r i t h m i c volume equation.The other i n v o l v e s the d e r i v a t i o n of t h e l o g a r i t h m i c t a p e r e q u a t i o n from an e x i s t i n g l o g a r i t h m i c volume e q u a t i o n t o p r o v i d e c o m p a t a b l l i t y i n volume e s t i m a t i o n and a t the same time ensure as a good fit  as p o s s i b l e f o r the e s t i m a t i o n of upper bole  diameters  (taper).  T e s t s f o r p r e c i s i o n and b i a s of volume oarried  estimates,  out on t h e B r i t i s h Columbia F o r e s t S e r v i c e t a p e r  c u r v e s and l o g a r i t h m i c volume e q u a t i o n s , I n d i c a t e t h a t the l a t t e r approach i s p r e f e r a b l e t o the f o r m e r .  Page TITLE PAGE  i 11  ABSTRACT TABLE OF CONTENTS  H i  LIST OF TABLES  v  LIST OF FIGURES  v i i  ACKNOWLEDGEMENTS  viii  INTRODUCTION  1  LITERATURE REVIEW  4 13  DERIVATIONS OF THE EQUATIONS AND TESTS The new t a p e r  equation  F i t t i n g t h e taper Columbia F o r e s t  equation  taper equation  taper equation Columbia F o r e s t  volume e q u a t i o n  16  logarithmic  • •  . . . . . . .  20  logarithmic  from t h e l o g a r i t h m i c f i t t e d on t h e B r i t i s h Service taper  D e r i v a t i o n of a compatible taper equation  curves • • • • •  from the l o g a r i t h m i c  D e r i v a t i o n of a compatible volume e q u a t i o n  13  on t h e B r i t i s h  S e r v i c e taper  D e r i v a t i o n of a compatible volume e q u a t i o n  •  curves  22  logarithmic  from a l o g a r i t h m i c •  •  D e r i v a t i o n of compatible l o g a r i t h m i c e q u a t i o n s from t h e B r i t i s h  26  taper  Columbia F o r e s t  S e r v i c e l o g a r i t h m i c volume e q u a t i o n s • . . • •  2?  Page IMPROVEMENT OF THE ACCURACY AND THE PRECISION 36  OF THE LOGARITHMIC TAPER EQUATION DISCUSSION,SUMMARY AND SUGGESTIONS  40  LITERATURE CITED  42  APPENDIXES  47  1.  Common Names and L a t i n Names of t h e 47  Tree S p e c i e s 2.  D e r i v a t i o n of a Volume E q u a t i o n the Taper E q u a t i o n of Hfljer  3»  D e r i v a t i o n of a Volume E q u a t i o n  from • • • • • • • from 49  the Taper E q u a t i o n of Behre 4,  D e r i v a t i o n of a Volume E q u a t i o n t h e Taper E q u a t i o n of Matte  5«  from • • • • • • •  D e r i v a t i o n of t h e Height E q u a t i o n  50  from  t h e L o g a r i t h m i c Taper E q u a t i o n . . . . . . 6.  48  51  D e r i v a t i o n of a L o g a r i t h m i c Volume E q u a t i o n from t h e L o g a r i t h m i c Taper 52  Equation 7«  D e r i v a t i o n of t h e Formula t o E s t i m a t e Volumes of Logs between S p e c i f i c D i s t a n c e s from t h e T i p of the T r e e , from t h e L o g a r i t h m i c Taper E q u a t i o n  8.  D e r i v a t i o n of a Compatible  . . .  53  Logarithmic  Taper E q u a t i o n from a L o g a r i t h m i c Volume E q u a t i o n  . . . . . .  54  Table I.  Page Summary of Taper Equations British  Fitted  on t h e  Columbia F o r e s t S e r v i c e Taper  Curves II.  Test  18  of the Taper E q u a t i o n s  Fitted  on the  B r i t i s h Columbia F o r e s t S e r v i c e Taper Curves III.  • • • • • •  Summary of t h e L o g a r i t h m i c Derived  Volume  from t h e L o g a r i t h m i c  Equations,Fitted  • Equations  Taper  on t h e B r i t i s h  Columbia 23  F o r e s t S e r v i c e Taper Curves IV.  Comparison of t h e B r i t i s h Service Logarithmic  Columbia F o r e s t  Volume  w i t h the Volume Equations V.  Equations from Table  Summary of t h e Taper Equations  VI.  VII.  29  Derived  the B r i t i s h Columbia F o r e s t  from  Service  Volume Equations  . . . . . . .  30  D i s t r i b u t i o n of t h e B i a s over the D i f f e r e n t Height  C l a s s e s w i t h i n t h e Same  Species  ( f o r Mature C o a s t a l D o u g l a s - f i r ) VIII.  25  Service  Volume Equations  T e s t of t h e Taper Equations  Logarithmic  III •  Derived  from t h e B r i t i s h Columbia F o r e s t Logarithmic  19  • • • • •  32  Comparison of Standard E r r o r s of Estimate of S e v e r a l Methods  . . . . . .  33  Table IX.  Page A b s o l u t e Frequency D i s t r i b u t i o n of the Standard  X.  E r r o r s of E s t i m a t e  • • •  A b s o l u t e Frequency D i s t r i b u t i o n of the Differences  i n Standard  E r r o r s of  Estimate XI.  34  34  Maximum Decrease i n Standard E s t i m a t e t o Be Expected  E r r o r of  from Using  the  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  Douglas-fir)  ( f o r Mature •  Coastal ••  39  LIST OF FIGURES Figure 1.  Page The standard e r r o r of e s t i m a t e as a f u n c t i o n of the v a l u e of p i n d e r i v i n g the  logarithmic  logarithmic  t a p e r e q u a t i o n from the  volume e q u a t i o n ( f o r mature  coastal Douglas-fir) 2.  28  Optimum v a l u e of p as a f u n c t i o n of t o t a l height  ( f o r mature c o a s t a l D o u g l a s - f i r ) .  •  38  vlii ACKNOWLEDGEMENTS The  author i s Indebted t o a l l I n d i v i d u a l s and agen-  c i e s concerned w i t h The  support of h i s s t u d i e s and r e s e a r c h .  author wishes t o express? h i s g r a t i t u d e t o  D r . D. D. Munro who suggested t h e problem and under whose d i r e c t i o n t h i s study was undertaken. A d v i c e on d a t a p r o c e s s i n g  was provided by  Dr. A. Kozak. D r s . D. D. Munro,A. Kozak and J . H. G. Smith a r e g r a t e f u l l y acknowledged f o r t h e i r h e l p , u s e f u l and  criticism  review of t h e t h e s i s . D e r i v a t i o n s of t h e f u n c t i o n s , g i v e n as Appendixes 2 - 8 ,  were a l s o reviewed by Mr. G. G. Y o u n g , A s s i s t a n t  Professor,  whose h e l p i s a p p r e c i a t e d . Thanks a r e due t o Mrs. Lambden,technician,for drawing the  figures. The  opportunity  t o use t h e t a p e r curves and equations  derived  by t h e F o r e s t  Service  i s acknowledged.  The for  Inventory D i v i s i o n of t h e B.C. F o r e s t  U n i v e r s i t y of B r i t i s h Columbia i s acknowledged  the computing  facilities.  F i n a n c i a l support was provided  i n t h e form of a  F a c u l t y of F o r e s t r y Teaching A s s i s t a n t s h l p and a MacMlllan Bloedel L t d . Fellowship Some a s s i s t a n c e  In F o r e s t  i n computing was provided  Research C o u n s l l of Canada g r a n t s  Mensuration.  by t h e N a t i o n a l  A-2077 and A-3253 In  support of s t u d i e s of t r e e shape and form.  "We  must d e v e l o p a mathematical t r e e volume expres-  s i o n which can be e f f i c i e n t l y  programmed f o r g e n e r a l l y  a v a i l a b l e e l e c t r o n i c computing and  equipment  to y i e l d  tree  stand volumes from i n p u t s of t r e e diameter o u t s i d e  bark and t o t a l  h e i g h t (form e s t i m a t e s o p t i o n a l ) and f o r  any demanded stump h e i g h t and t o p d i a m e t e r " . (Honer and  Sayn-Wittgensteln,1963)  T h i s t h e s i s demonstrates t h a t a mathematical  stem  p r o f i l e e q u a t i o n whloh oan be I n t e g r a t e d t o volume can meet t h e s e requirements.When  Y i s the diameter  inside  bark a t any g i v e n h e i g h t i n f e e t and X i s the h e i g h t above the ground  i n f e e t , t h e n volume can be c a l c u l a t e d  by r e v o l v i n g the e q u a t i o n of the stem curve about the X - a x i s and  I n t e g r a t i n g f o r v a l u e s of X from base t o t i p .  Merchantable volume t o any standard of u t i l i z a t i o n  can  be c a l c u l a t e d by u s i n g the a p p r o p r i a t e X - v a l u e s . S e c t i o n diameter can be e s t i m a t e d a t any h e i g h t or the s e c t i o n h e i g h t 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 d e f e c t can be determined precisely. Taper f u 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 p r e v i o u s s t u d i e s are s i m i l a r i n t h a t a  t a p e r e q u a t i o n l a c a l c u l a t e d from the d a t a t o g i v e the best f i t f o r t a p e r and t h e r e f r o m volume i a c a l c u l a t e d . T h i s o f t e n r e s u l t s i n a good e s t i m a t e of t a p e r but a l e s s than s a t i s f a c t o r y e s t i m a t e f o r volume. In most p r a c t i c a l cases a l o c a l or r e g i o n a l volume e q u a t i o n a l r e a d y e x i s t s , h a s been used w i d e l y f o r a l o n g time and w i l l c e r t a i n l y c o n t i n u e t o be used  In t h e f u t u r e .  I t i s c l e a r t h a t f o r such a s i t u a t i o n a t a p e r e q u a t i o n has t o be d e r i v e d which on the one hand g i v e s t h e best p o s s i b l e f i t f o r t a p e r but on t h e other hand i s c o m p a t i b l e * w i t h the e x i s t i n g volume e q u a t i o n . T h i s study d e a l s w i t h both approaches.The new t a p e r e q u a t i o n presented  i s a l o g a r i t h m i c one.Two methods a r e  derived,illustrated  and t e s t e d :  a) D e r l v a t i o n of a compatible  l o g a r i t h m i c volume  e q u a t i o n from an e x i s t i n g l o g a r i t h m i c t a p e r e q u a t i o n . b) D e r i v a t i o n of a compatible  logarithmic taper  e q u a t i o n from an e x i s t i n g l o g a r i t h m i c volume equation. Both t e c h n i q u e s a r e t e s t e d on t h e B r i t i s h Columbia F o r e s t Service(B.C.F.S.)  t a p e r curves  l o g a r i t h m i c volume e q u a t i o n s The  (B.C.F.S.,1968) and on the  (Browne,1962).  standard e r r o r s of e s t i m a t e  (SEg) are compared  those g i v e n by Kozak,Munro and Smith  Compatible  (1969b,TableI)  means here t h a t both equations ( f o r  t a p e r and volume) g i v e 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.  with  which a r e baaed on t h e same t a p e r  curves.  Some attempt i s made t o f i n d a u s e f u l  relationship  between t h e t a p e r e q u a t i o n c o e f f i c i e n t s and some t r e e characteristics height  such as t o t a l h e i g h t . d i a m e t e r  (dbh) and t h e r a t i o of b o t h .  breast  In b a s i c  " F o r e s t M e n s u r a t i o n " textbooks  (Chapman  and Meyer,1949;Bruce and Schumacher,1950;Spurr,1952; Meyer,1953;Loetsch and H a l l e r , 1 9 6 4 ; P r o d a n , 1 9 6 5 ; A v e r y , I967) almost no comments a r e made about the d e s i r a b i l i t y of d e v e l o p i n g compatible t a p e r and volume e q u a t i o n s which would be u s e f u l f o r the e s t i m a t i o n of merchantable volume t o any standard  of u t i l i z a t i o n .  P e t t e r s o n (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 f o r m - c l a s s e s would be g i v e n by d i f f e r e n t p a r t s of t h i s curve.A t a n g e n t i a l f u n c t i o n was used by H e i j b e l  (1928)  t o d e s c r i b e the main p a r t of the s t e m . D i f f e r e n t  equations  were used f o r the t o p p r o f i l e and the stem below 10$ of the h e i g h t . Volume and t a p e r were more or l e s s combined i n one system by t h e G i r a r d f o r m - c l a s s t a b l e s  (Mesavage and  Girard,1946). A o c o r d i n g t o Spurr  (195 ) 2  a possible solution f o r  merchantable c u b i c - f e e t t r e e volume t a b l e s i s t o c a l c u l a t e d i f f e r e n t e q u a t i o n s f o r stump and t o p volumes and then t o s u b t r a c t these volumes from t o t a l volume. Another s o l u t i o n could be t o c a l c u l a t e the r e g r e s s i o n between merchantable volume and t o t a l volume. Meyer  (1953)  stated  t h a t the c o n s t r u c t i o n of a t a p e r  curve (or e q u a t i o n ) f o r a c e r t a i n s p e c i e s or group of species i s s t i l l a d i f f i c u l t  task.  G r a p h i c a l t e c h n i q u e s were used by D u f f and B u r s t a l l ( 1 9 5 5 ) t o d e v e l o p t a p e r and volume t a b l e s showing  mer-  c h a n t a b l e volumes f o r each t e n - f o o t - h e i g h t c l a s s w i t h i n each dbh and t o t a l h e i g h t class.Volume and t a p e r t a b l e s were f i r s t  tables  prepared i n d e p e n d e n t l y .  To make them compatible the t a p e r d a t a were adjusted t o fit  t h e i n d e p e n d e n t l y c a l c u l a t e d volumes and t h e  d i a m e t e r s were made t o agree w i t h t h e a l r e a d y known volumes. Speidel  (1957)  used g r a p h i o a l t e c h n i q u e s t o r e l a t e  t h e percentage of t o t a l volume t o t h e percentage of t o t a l tree height. I t was shown by Newnham ( 1 9 5 8 ) t h a t a q u a d r a t i c p a r a b o l a gave a good f i t t o a l a r g e p a r t of the b o l e shape. Three models were developed  (1964;1965a,  by Honer  b ; 1 9 6 7 ) t o express t h e d i s t r i b u t i o n of volume over the t r e e stem: a) v / V = b  Q  + b  x  h / H + b  b) v / V = b* + b[ d  2  / D  2  c) v / V = b Q + b J d / D +  2  h  2  / H  + bg(d  2  2  /D ) 2  bg(l - h / H )  where v i s t h e volume below t h e merchantable V i s the t o t a l volume,h i s t h e merchantable base,d  2  2  limit, h e i g h t from the  i s t h e merchantable diameter and D i s t h e dbh.  These models d e s c r i b e w e l l the d i s t r i b u t i o n of volume over the t r e e stem and to  any  standard  can be used  of u t i l i z a t i o n when a p p l i e d t o an  mate of t o t a l volume.They cannot diameter  t o estimate volume  be used  esti-  t o estimate  a t a g i v e n h e i g h t or h e i g h t 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 T u r n b u l l and  Hoyer  (1965),  w i l l not be d i s o u s s e d here because they don't g i v e a compatible  system of volume and  taper.  Heger ( 1 9 6 5 ) r e p o r t e d a t r i a l on l o d g e p o l e p i n e trated  2  of Hohenadl's approach  grown i n A l b e r t a . S t a n e k  the method f o r l o d g e p o l e p i n e and  in British New  (1966)  illus-  Engelmann spruce  Columbia.  tree-measurement concepts were i n t r o d u c e d by  Grosenbaugh ( 1 9 5 4 ; 1 9 6 6 ) . Some work has been c a r r i e d  out on t r e e t a p e r curves  u s i n g m u l t i v a r i a t e methods ( F r i e s , 1 9 6 5 ; F r l e s and  Matern,  1 9 6 5 ) . H o w e v e r , a f t e r comparison of m u l t i v a r i a t e and  other  methods f o r a n a l y s i s of t r e e taper,Kozak  (1966)  oonoluded  and  t h a t the use of s i m p l e r methods i s b e s t .  While many a u t h o r s have made i t c l e a r and  Smith  (Kozak,Munro  Smith,1969a;Munro,1970) t h a t no p r a c t i c a l advan-  tage can be gained  The  from any measurement of  common t r e e names used  throughout  form  this  are g i v e n w i t h the c o r r e s p o n d i n g L a t i n names i n Append i s  1 •  thesis  in  a d d i t i o n t o dbh  Mlngard and  and  total  height,Schmid,Holko-Jokela,  ( 1 9 7 1 ) have shown t h a t the measure-  Zobelry  at 6 - 9  ment of d b h , t o t a l h e i g h t and diameter  meters  h e i g h t i s the best method of volume d e t e r m i n a t i o n . Some important detailed was  t a p e r f u n c t i o n s are worthy of more engineer,Hfljer ( 1 9 0 3 ) »  review.A Swedish c i v i l  the f i r s t  t o propose a mathematical e q u a t i o n t o  d e s c r i b e the stem p r o f i l e : d / D = c where d was  x  In ( ( c  2  + 1  the diameter  t a n c e from the t l p , D was  100  / H) /  c ) 2  i n s i d e bark a t any g i v e n d i s the dbh  I n s i d e b a r k , l was  the  d i s t a n c e from the t i p , H the t o t a l t r e e h e i g h t above b r e a s t h e i g h t and  c^ and  c  2  were the c o n s t a n t s t o be  d e f i n e d f o r eaoh f o r m - c l a s s . Jonson ( C l a u g h t o n - V i a l l i n , 1 9 1 8 ) cal  f o r m u l a as c o m p l e t e l y  applied in  d e s c r i b e d t h i s mathemati-*  conforming  w i t h nature when  t o spruce of a l l f o r m - c l a s s e s , b u t  some stands,which  overeatlmations meter at any  had  occurred  i n the upper s e c t i o n s . T h e  seeds, dia-  i n e s t i m a t i n g volume.A volume t a b l e can  a l s o be c a l c u l a t e d  by d e r i v i n g a volume e q u a t i o n from  the t a p e r e q u a t i o n by 2  been grown from Imported  h e i g h t of the stem being known t h e r e i s  no d i f f i c u l t y  V = D  stated that  integration:  H O.OO5454 c f (K ( I n E (In K - 2) (for  where K = 1 + 100  /  c  2  + 2 ) -  2)  proof aee Appendix  2)  (1910;19H»  In order t o o b t a i n b e t t e r r e s u l t s , J o n s o n 1926-27)  introduced a new  "biological  a  constant":  d / D = c  In ( ( c  x  where  constant which he c a l l e d  was  the new  + 1 100  2  / H - c^)/ c ) 2  constant.Equations  were computed  f o r each form-class.W1th the i n t r o d u c t i o n of t h i s " b i o l o g i c a l c o n s t a n t " an I n c o n s i s t e n c y was  introduced  because t h i s t a p e r e q u a t i o n d i d n ' t g i v e a r e s u l t f o r a p o r t i o n e q u a l t o c^ on the upper stem.A volume e q u a t i o n oan be d e r i v e d  i n the Bame way  as f o r the formula  of  H8jer. The  t a p e r equations  of Jonson and  composite t a p e r equations.They of t r e e species.The was  Hfljer are In f a c t  are complied  f o r m - c l a s s which had  u s u a l l y measured or estimated  approach.Claughton-VJallln  and  about t h i s t h a t the d i f f i c u l t y  Independently  t o be known  by the "form p o i n t "  V l c k e r (1920) reported Is t o estimate the form-  c l a s s of a s t a n d i n g t r e e or the average f o r m - c l a s s of a stand  but they b e l i e v e d t h a t a l i t t l e p r a c t i s e would  overcome t h i s . Wlckenden (1921) claimed of any type of f o r e s t regions.Wright  of  does not vary much even f o r l a r g e  (1923) b e l i e v e d however t h a t t h e r e was  considerable v a r i a t i o n i n a stand  t h a t the form q u o t i e n t  i n the form of i n d i v i d u a l  a  trees  timber.  As a r e s u l t  of h i s I n v e s t i g a t i o n s on many s p e c i e s ,  Behre (1923;1927;1935) presented  a new  equation f o r  t h e stem curve which seemed t o be more c o n s i s t e n t w i t h nature: d / D = ( l / H ) / ( b  0  + b  1  l / H )  where t h e symbols have t h e same meaning as i n the Hfljer's equation.The c o e f f i c i e n t s b fitting  Q  and b^ can be c a l c u l a t e d by  the r e g r e s s i o n l i n e :  ( l / H ) / ( d / D ) = b  0  + b  ( l / H )  1  t h i s f u n c t i o n i s i d e n t i c a l t o the equation: ( D / d  ) = b j + b j ( H / l )  Behre's t a p e r equation,when i n t e g r a t e d t o v o l u m e , y i e l d s t h e f o l l o w i n g compatible volume V = D  H O.OO5454  2  equation:  ( 1 / b\ ) ( 1 - b ) + 2 b 2  0  In b  ( f o r proof see Appendix Matte ( 1 9 4 9 )  2  / D  2  = b  0  l  2  / H  + b! l 3 /  2  H  3 + b  2  1^ / H  4  where t h e symbols have t h e same meaning as In the e q u a t i o n of H 8 j e r . l t  i s w o r t h w i l e t o mention t h a t t h e  t a p e r e q u a t i o n c o e f f i c i e n t s a r e p a r t i a l l y d e f i n e d by a c o n d i t i o n about volume. The f o l l o w i n g volume e q u a t i o n can be d e r i v e d by i n t e gration: V = 0.005454  D  2  H ( b  0  / 3 + b  x  / 4 + b  2  / 5 )  ( f o r proof see Appendix bg and b^ were found height.  ) 3)  d e s c r i b e d the stem p r o f i l e above b r e a s t  h e i g h t by t h e f u n o t i o n : d  0  t o be r e l a t e d  4)  t o dbh and t o t a l  (1959)  A q u i t e s i m i l a r e q u a t i o n was t e s t e d by Osuml d / D = b  0  1 / H + b  x  l  / H  2  + b  2  2  l  / H  3  3  from which a l s o a volume e q u a t i o n can be d e r i v e d . The  (1963)  t a p e r e q u a t i o n p r e f e r r e d by G i u r g i u  was a 1 5 t h degree  polynomial:  d / D = 1 5 t h degree p o l y n o m i a l of 1 / H where D was t h e d i a m e t e r i n s i d e bark a t . 1 height  of t o t a l  and was f u r t h e r expressed as a f u n c t i o n of dbh  outside  bark and t o t a l h e i g h t . T h i s  f u n c t i o n can a l s o  be i n t e g r a t e d t o volume. Prodan ( 1 9 6 5 ) found t h e f o l l o w i n g t a p e r  function  satisfactory: d / D = ( h / H ) where h i s t h e h e i g h t With r e s p e o t  2  / ( b  Q  + b  x  h / H + b  2  h  / H  2  )  2  above t h e ground.  t o t h e t a p e r e q u a t i o n of Osumi,he s t r e s s e d  t h a t a 4 t h degree p o l y n o m i a l w i t h I n t e r c e p t would be muoh b e t t e r . As an e x t e n s i o n Osumi and G i u r g i u , a n  of t h e methods used by Matte, integrated  system of t a p e r and  volume e q u a t i o n f o r r e d a l d e r was provided C u r t i s and Vanooeverlng d  2  / D  2  = b  0  X  3 / 2  by Bruce,  (1968): + ( X  3 / 2  + ( X  3 / 2  + ( X  3 / 2  - X? ) ( b - X  3 2  )  D + b  ±  ( b  3  H )  H D + b  - X*°) ( b- H  L  2  where X i s 1 / ( H - 4 . 5 ) and D i s dbh o u t s i d e V e r y h i g h powers of X were r e q u i r e d  2  ) bark.  t o describe the  b u t t s w e l l . T h e a u t h o r s expected t h a t the use of some  H  1 / 2  )  measure of form would  Improve the f i t of t h i s t a p e r  equation.In t h e i r opinion,the p r i n c i p a l encountered to  by Hfljer,Jonson,Behre and  oversimplified  difficulties  o t h e r s were due  e q u a t i o n s which d i d not  satisfactory  d e s c r i b e the b u t t s w e l l and t i p . A f t e r Munro ( 1 9 6 8 ) found t h a t upper stem d i a m e t e r s c o u l d be e s t i m a t e d w i t h r e a s o n a b l e SE from a f u n c t i o n I n v o l v i n g dbh, e q u a t i o n was d  2  / D  2  h / H and  proposed = b  0  + b  where D i s the dbh  b For  0  / H  ,the f o l l o w i n g t a p e r  by Kozak,Munro and Smith t  h / H + b  2  h  2  2  H  i n feet.The least  c o n d i t i o n e d by Imposing  + bj + b  /  (1969a,b):  2  o u t s i d e bark i n Inches and  h e i g h t above the ground s o l u t i o n was  h  h i s the  squares  the r e s t r a i n t :  = 0  spruoe and redoedar a d d i t i o n a l c o n d i t i o n s were  n e c e s s a r y t o prevent n e g a t i v e d i a m e t e r s near the t o p . These t a p e r f u n c t i o n s were oomputed f o r 23 s p e c i e s or s p e o l e s g r o u p s from B.C.F.S. t a p e r c u r v e s 1968)  to f a c i l i t a t e  (B.C.F.S.,  e f f i c i e n t a n a l y s i s w i t h modern e l e c -  t r o n i c computers.Several t e s t s on these e q u a t i o n s Munro and  Smith, 1 9 6 9 a ; S m i t h and Kozak,1971) suggested! a  s t a b l e e s t i m a t i n g system.It appeared advantage  resulted  as i f l i t t l e  real  from the use of more complex powers,  l i k e those used by B r u c e , C u r t i s and Vancoevering to  (Kozak,  (1968),  e s t i m a t e t r e e taper.These t a p e r e q u a t i o n s w e r e , l a t e r  on,converted factors  i n t o volume e q u a t i o n s and p o i n t  (Demaersohalk,1971)•  sampling  Awareness of the d e s i r a b i l i t y of development of comprehensive systems f o r e s t i m a t i o n of net merchantable volumes of t r e e s by l o g s i z e and u t i l i z a t i o n c l a s s e s i s growing.The need has been f e l t f i r s t research analyses  i n operations  of l o g g i n g systems i n Sweden and i n  s t u d i e s t o d e v e l o p improved methods of i n v e n t o r y i n Austria.However,no p u b l i c a t i o n s i n c o r p o r a t i n g the features described  h e r e i n have come t o t h e a u t h o r ' s  attention. No review w i l l be g i v e n about the d i f f e r e n t form t h e o r i e s ( n u t r i t i o n a l , m e c h a n i s t i c , w a t e r hormonal and p i p e m o d e l ) . I n t e r e s t i n g  et  al.(1965).  (1963),Heger  conductive,  d i s c u s s i o n s about  the d i f f e r e n t a l t e r n a t i v e s were g i v e n by Gray Newnham ( 1 9 5 8 ) , L a r s o n  tree  U965)  a n < a  (1956), Shinozaki  DERIVATIONS OP THE EQUATIONS AND TESTS The  new t a p e r  The  equation  l o g a r i t h m i c taper equation tested i n t h i s  study i s : log d = b  + b  0  logD + b  1  2  l o g 1 + bj l o g H  (1)  where d i s t h e diameter i n s i d e bark i n i n c h e s a t any given 1 i n feet,D 1  Is the dbh o u t s i d e bark i n i n c h e s ,  i s t h e d i s t a n c e from the t i p of the t r e e In f e e t ,  H I s t h e t o t a l h e i g h t of the t r e e i n f e e t and b , b i , n  b  2  The  and b^ a r e t h e r e g r e s s i o n c o e f f i c i e n t s . same t a p e r e q u a t i o n can be expressed d = 10b0 D l l b  b  i n other ways:  2 Hb3  (2)  or d  w  / D  = K 1*  v  where w = 1. v = b y = b  / H  (3)  Z  z = - b^ K = 10b°  x  2  J u s t as t h e l o g a r i t h m i c volume e q u a t i o n V = 10a D  b  H  c  i s the unconditioned  form (with r e s p e c t t o the powers  of D and H) of the combined v a r i a b l e volume e q u a t i o n  V = b  H  Q  (without I n t e r c e p t )  1  where the power of D l a c o n d i t i o n e d t o 2 and the power of H t o l , t h l s taper e q u a t i o n (the aquare of formula  3)  i s the unconditioned form of the w e l l known g e n e r a l f o r m u l a f o r the p r o f i l e of c e r t a i n a o l l d a of r e v o l u t i o n ( c o n e , p a r a b o l o i d and d  / D  2  2  nelloid):  = ( 1 / H )  v  where the powera of d and 1 a r e c o n d i t i o n e d t o be e q u a l t o r e s p e c t i v e l y the powera of D and H. T h l a t a p e r e q u a t i o n i s v e r y slmple.No c o n d i t i o n i n g i s necessary t o ensure t h a t the estimated diameter a t the t o p i s z e r o and t h a t no n e g a t i v e e s t i m a t e s of diameter  occur.Prom f o r m u l a 2 i t can be seen e a s i l y t h a t  d can never be n e g a t i v e and becomes z e r o when 1 i s z e r o ( a t t h e t i p of the t r e e ) . Formula 2 can be used  t o e s t i m a t e diameter  inside  bark a t any s e l e c t e d d i s t a n c e (1) from the t i p . D i s t a n c e t o any s p e c i f i c t o p diameter  (d) can be e s t i -  mated by t r a n s f o r m a t i o n of the b a s i c e q u a t i o n t o the form: 1 = ( l ( T 0 d D" b  b l  H" 3 b  )  1 / b  2  (4)  ( f o r proof see Appendix  5)  The l o g a r i t h m i c t a p e r e q u a t i o n can be d e r i v e d i n two b a a l c a l l y d i f f e r e n t waya: a)The t a p e r e q u a t i o n can be f i t t e d  on t a p e r d a t a  by the l e a a t squares method.This f u n c t i o n can e a s i l y be converted  subsequently t o a compatible  l o g a r i t h m i c volume e q u a t i o n . b)The t a p e r e q u a t i o n oan be d e r i v e d from an e x i s t i n g l o g a r i t h m i o volume e q u a t i o n when some d a t a about taper are a v a i l a b l e . T h i s taper equation w i l l  be  compatible w i t h the e x i s t i n g volume e q u a t i o n . Both ways w i l l be e x p l a i n e d and t e s t e d on the B.C.F.S. taper curves equations  (B.C.F.S.,1968)  (Browne,1962).  and l o g a r i t h m i c volume  F i t t i n g the t a p e r e q u a t i o n on t h e B r i t i s h  Columbia  F o r e s t S e r v i c e t a p e r curves  The l o g a r i t h m i c t a p e r f u n c t i o n (formula 1) was computed f o r 23 s p e c i e s o r s p e c i e s g r o u p s on t h e B.C.F.S. t a p e r curves (B.C.F.S.,1968) by the l e a s t squares method.Diameters  i n s i d e bark had been taken from each  t a p e r curve a t the h e i g h t o f 1 f t . , 4 . 5 f t . o f t o t a l h e i g h t and punched  on computer  and a t d e c i l e s cards f o r  Kozak,Munro and Smith (1969,).In the c a l c u l a t i o n s ,  b dbh o u t s i d e bark was used as t h e measure o f diameter i n s i d e bark a t 1 f t .  height2  The assumptions o f the r e g r e s s i o n a n a l y s i s were t e s t e d by p l o t t i n g f o r each s p e c i e s l o g d over l o g D,log 1, and l o g H.For every s p e c i e s and f o r every v a r i a b l e , t h e r e was almost a p e r f e c t 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 v a r i a b l e . V a r i a n c e s were homo-  geneous. Because r e l a t i v e s t a n d a r d e r r o r s a r e sometimes g r e a t l y a f f e c t e d by t h e s i z e o f the mean and comparisons i n terms o f r e a l d*s were d e s i r e d , t h e f o l l o w i n g a p p r o x i mation was used: SE  = (( S ( da - de ) ) / ( n - m - 1 ) ) 2  E  1  /  (5)  2  where da i s the a c t u a l diameter i n s i d e bark.de i s the e s t i m a t e d diameter i n s i d e bark,n i s the number o f -'Except f o r mature c o a s t a l D o u g l a s - f i r , w h i c h a much b e t t e r f i t w i t h o u t a d j u s t i n g .  gave  observations  used f o r t h e l e a 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  r e g r e s s i o n c o n s t a n t s of the t a p e r e q u a t i o n and the  SE£s a r e summarized  i n t a b l e I and the average b i a s of  d i a m e t e r i n s i d e bark a t d i f f e r e n t h e i g h t s i s g i v e n in table I I . The  SE£s ranged from .245 t o 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 g i v e n i n  t a b l e IX.Large SE£s,however,do not n e c e s s a r i l y i n d i c a t e a poor f i t , b u t more l i k e l y r e p r e s e n t a wider range of taper curves  (Hejjas,1967).  A l l t h e s p e c i e s f o l l o w almost t h e same trend of average b i a s . T h e r e  i s u s u a l l y an u n d e r e s t i m a t i o n  base of t h e t r e e , a n o v e r e s t i m a t i o n from .1 u n t i l or . 5 of t h e t o t a l h e i g h t , a s l i g h t .4 or . 5 u n t i l  a t the .4  underestimation  from  .8 of the t o t a l h e i g h t and a s m a l l  o v e r e s t i m a t i o n a t t h e top.For e l e v e n s p e c i e s the average b i a s a t any h e i g h t i s l e s s than one i n o h .  Summary of Taper E q u a t i o n s P i t t e d  on t h e B r i t i s h  Columbia F o r e s t S e r v i c e Taper Curves Species 4 R M group  Equation b  0  coefficients b 2  SEr  t>3  No*  (inches)  Alder  C  M -0.071459  0.802730  0.795203 -0.637059 0.587  92  Aspen  I  M  0.014506  0.944389  0 . 7 6 6 6 5 5 - 0 . 7 4 2 5 7 6 0.448  53  Balsam  C  M  0.369223  1.064119  O.656O8O -0.878118 0 . 8 1 3  85  n  I  M  0.025923  0.925263  0 . 7 2 9 9 6 3 - 0 . 6 9 7 5 2 5 0.420  85  Birch  I  M  0.051560  0.979513  0.899947 -0.913041  0.245  55  Ced a r  C  M  0.448191  O.968O76  0.812954  -I.032109 2.431  114  C  I  0.195945  0.759688  0 . 8 2 4 2 5 4 - 0 . 7 8 3 3 5 6 1.128  134  I  M  0.379992  1.011860  0.799019  -1.012500 1.379  127  C o t t o n - CI M - 0 . 2 6 2 8 4 3 wood Douglas c M 0 . 2 0 4 3 8 9 i ir c I O.092707  0.865273  0 . 8 2 7 0 2 3 - 0 . 6 1 3 2 3 3 0.755  0.984578  0 . 7 0 1 1 6 5 -0.821202 1.431 114  0.826471  0.680451 -0.637352  1.032  174  I I  »  I I  it  92  I  M  0.004827  0.892425  0.741884 - 0 . 6 9 0 8 2 1 1 . 2 5 1  160  Hemlock c  M  0.299130  1.016430  0.746148 -0.908821 0.871  118  c  I  0.065941  0.857932  0.829013 -0.776866  0.691  128  I  M  0.036873  0.999704  0.716169 -0.736237  0.740 104  Lodgep. CI M 0 . 4 7 2 7 0 2 pine I M -0.012680 Larch  1.044069  0.634633 -0.909768 0.774  0.843926  0.696431 -0.618374  1.230  148  0.316  48  I I  M  Maple  c  M -0.010447  0.863337  0.909104 -0.822074  Spruce  c  M  0.294001  0.978388  0.783387  n  I  M  0.100700  0.915903  0.742631 -0.744977 0.526  -0.912494  65  2 . 3 7 6 378 93  White CI M 0 . 6 9 0 0 4 4 1.215400 81 0.707159 -1.185360 1.159 pine Yellow CI M 0 . 1 3 0 2 6 0 0.89H70 0.762970 -0.762321 0.574 50 oedar Yellow CI M 0.044221 0 . 6 7 4 2 4 7 - 0 . 8 1 0 4 1 5 1 . 2 0 5 124 1.148219 —pine p . ; C i s Coast M i s Mature Number of t a p e r l i n e s s c a l e d from I i s I n t e r i o r I i s Immature t h e B.C.F.S. t a p e r c u r v e s 5  r  Teat of the Taper Equations F i t t e d Columbia F o r e s t S e r v i c e Taper Species group  Curves  Average b i a s ( i n i n c h e s ) of diameter I n s i d e bark a t R  C I C I Birch I c Cedar c it I n Cottonwood CI C Douglas-fir c it I •i Hemlock c ti c I ti Lodgepole p i n e CI Larch I Maple c Spruce c I ti White p i n e CI Yellow cedar CI Yellow p i n e CI Alder Aspen Balaam  on the B r i t i s h  w M M M M M M I M M M I M M I M M M M M M M M M  1» 0.06 -0.12 -0.43 -0.31 -0.03 -4.21 -1.32 -2.47 -0.43 -3.24 -2.01 -2.74 -1.01 -0.48 -0.33 -0.12 -2.73 -0.16 -3.25 -0.78 -0.42 -0.28 -1.13  0.34 0.44 0.17 0.35 O.25 -3.33 -1.10 -1.45 0.94 0.13 -0.01 0.66 0.11 0.01 0.74 0.06 0.31 -0.05 -2.77 -0.18 0.21 -0.23 1.34  see f n . 4 and 5 i n t a b l e I  0.1H  0.2H  0.3H  0.4H  0.5H  0.6H  0.7H  0.8H  0.9H  0.42 0.55 0.86 0.51 0.29 0.48 0.19 0.75 1.04 1.05 0.42 1.48 0.95 0.60 1.07 0.37 1.27 0.16 1.66 0.48 0.85 0.52 1.47  0.17 0.34 0.61 0.29 0.09 1.60 0.75 1.30 0.51 1.27 0.54 1.22 0.74 0.45 0.76 0.50 1.34 0.16 2.11 0.56 0.65 0.73 0.99  -0.14 0.07 0.14 0.02 -0.06 1.22 0.55 0.92 0.00 0.81 0.41 0.46 0.24 0.10 0.32 0.29 0.82 0.03 1.14 0.26 0.24 0.44 0.37  -0.36 -0.14 -0.23 -0.17 -0.14 0.73 0.24 0.42 -O.38 0.23 0.19 -0.23 -0.18 -0.18 -0.12 0.07 0.16 -0.05 0.24 -0.05 -0.14 0.04 -0.18  -0.45 -0.29 -0.45 -0.25 -0.16 0.37 0.01 0.03 -0.57 -0.25 -0.02 -0.67 -0.43 -0.34 -O.45 -0.12 -0.37 -0.11 -0.35 -0.23 -0.39 -0.34 -0.57  -0.39 -0.32 -0.45 -0.23 -0.14 0.20 -0.11 -0.17 -O.56 -0.46 -0.16 -0.71 -0.48 -0.35 -0.57 -0.28 -O.63 -0.11 -0.55 -0.25 -0.45 -0.54 -0.67  -0.22 -0.24 -0.28 -0.16 -0.08 0.10 -0.13 -0.20 -O.36 -0.36 -0.17 -0.45 -0.32 -0.18 -0.51 -0.35 -0.56 -0.07 -0.39 -0.16 -0.3? -0.44 -0.53  0.01 -0.06 -0.06 -0.05 -0.01 0.01 -0.04 -0.11 -0.07 -0.13 -0.04 -0.03 -0.06 0.03 -0.24 -0.26 -0.23 -0.01 0.01 -0.02 -0.19 -0.09 -0.29  0.23 0.15 0.11 0.08 0.04 -0.08 0.10 0.03 0.18 0.15 0.16 0.35 0.19 0.21 0.14 -0.10 0.28 0.05 0.13 0.14 -0.16 0.19 0.02  1.0H 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  D e r i v a t i o n of a compatible  l o g a r i t h m i c volume e q u a t i o n  from t h e l o g a r i t h m i c t a p e r e q u a t i o n  The  l o g a r i t h m i c t a p e r e q u a t i o n can be converted  a compatible  into  l o g a r i t h m i c volume e q u a t i o n : (6)  log V = a + b log D + c log H where a = l o g ( O.OO5454 1 0 0 / ( 2 b 2b  2  + D)  b = 2 b  1  c = 2 b  + 2  2  where t>n» i» 2 b  b  a  n  d  b  +1 3  a  r  e  c o e f  iolenta  from t h e  logarithmic taper equation. ( f o r proof see Appendix 6 ) T h i s volume e q u a t i o n i s t h e f o r m u l a t o be used t o e s t i mate t o t a l volume of t h e t r e e i n c u b i c - f e e t . A n a l t e r n a t i v e form of t h i s e q u a t i o n i s : V = 10 D a  b  H  (?)  c  To e s t i m a t e volumes of l o g s between s p e c i f i c d i s t a n c e s from t h e t i p of t h e t r e e , t h e f o l l o w i n g e q u a t i o n has t o be  used: V = K D  I!* ( 1* - i f )  v  (8)  where v = 2y = 2 b  3  2 = 2  2  b  + 1  2b K = O.OO5454 10 and  1  1  and 1  2  0  / (2 b  2  + 1 )  a r e r e s p e c t i v e l y t h e lower and upper  d i s t a n c e from t h e t i p of t h e t r e e . ( f o r proof see Appendix 7)  I f the l i m i t  s i z e s of the l o g a r e g i v e n as d i a m e t e r s  i n s i d e bark,the same f o r m u l a 8 can be used  after  c o r r e s p o n d i n g d i s t a n c e s from the t i p of the t r e e have been c a l c u l a t e d  w i t h f o r m u l a 4.  D e r i v a t i o n of a compatible l o g a r i t h m i c volume e q u a t i o n from the l o g a r i t h m i c  taper e q u a t i o n f i t t e d  B r i t i s h Columbia F o r e s t  Service taper  on the  curves  The l o g a r i t h m i c volume f u n c t i o n ( 6 )  was d e r i v e d  the t a p e r e q u a t i o n s i n t a b l e I f o r the 23 B.C. groups and a r e summarized  from  species-  i n t a b l e III.Because of the  f a c t t h a t the B.C.F.S. l o g a r i t h m i c volume e q u a t i o n s and t a p e r c u r v e s a r e based on the same sample t r e e s (B.C.F.S., 1 9 6 8 ) , w e would expect t h a t the volume from the t a p e r functions,would logarlthmlo  equations,derived  be s i m i l a r t o the B.C.F.S.  volume e q u a t l o n s . A l t h o u g h i t i s t r u e f o r  c e r t a i n s p e c i e s , f o r others deviations.This  t h e r e a r e some r a t h e r  suggests t h a t a good t a p e r  large  equation i s  no guarantee f o r a good volume e q u a t i o n i f only the p r e c i s i o n of t h i s t a p e r f u n c t i o n i s i n d i o a t e d by the SEpon diameter.A SEgof 1 i n c h , f o r example,has no meaning f o r volume when one knows nothing effect  of b i a s v a r i e s c o n s i d e r a b l y  about the b i a s . T h e w i t h the p o s i t i o n  on the t r e e and w i t h the s i z e of the t r e e . T h e r e f o r e best  the  check of a t a p e r t a b l e , w h i c h i s t o be used t o  c a l c u l a t e v o l u m e , i s a check of a volume t a b l e d e r i v e d therefrom,as was r e c o g n i z e d  by Bruce and Schumacher  The f a c t t h a t i n the B.C.F.S. l o g a r i t h m i c  (1950).  volume  e q u a t i o n s t h e sum of squares of the r e s i d u a l s of the l o g a r i t h m of volume i s minimized,while i n the l o g a r i t h m i c t a p e r e q u a t i o n the sum of squares of the r e s i d u a l s of  Summary of the L o g a r i t h m i c Volume E q u a t i o n s Derived from the L o g a r i t h m i c Taper E q u a t i o n s , F i t t e d on the B r i t i s h Columbia F o r e s t S e r v i c e Taper Curves Species group  B  M  Alder  C  M  -2.81955?  1.605459  1.316288  Aspen  I  M  -2.637947  1.888778  1.048159  Balsam  C  M  -1.888843  2.128239  0.555924  I  M  -2.602346  1.850526  1.064877  Birch  I  M  -2.607292  1.959025  0.973812  Ced a r  c  M  -1.786168  1.936152  O.56I689  c  I  -2.294382  1.519376  1.081796  I  M  -1.917933  2.023720  0.573038  Cottonwood  CI M  -3.212865  1.730546  1.427580  Douglas-fir  c  M  -2.235126  1.969155  0.759926  c  I  -2.450935  1.652942  1.086198  I  M  -2.648728  1.784849  1.102126  c  M  -2.061610  2.032860  0.674654  it  c  I  -2.555949  1.715863  1.104294  it  I  M  -2.5755^9  1.999408  0.959865  Lodgepole pine CI M  -1.673752  2.088139  0.449730  I I  I I  •t Hemlock  Equation 8  a  coefficients b c  Larch  I  M  -2.667549  1.687852  1.156114  Maple  c  M  -2.734138  1.726673  1.174060  Spruce  c  M  -2.084657  1.956776  0.741786  •t  I  M  -2.457243  1.831805  0.995308  White pine  CI M  -1.265978  2.430799  0.043597  Yellow  cedar  CI M  -2.405174  1.782339  1.001298  Yellow  pine  CI M  -2.545619  2.296438  0.727664  ^ s e e f n . 4 and 5 i n t a b l e I .  the l o g a r i t h m  of diameter l a minimized  Is one  of  the  r e a s o n s f o r these apparent c o n t r a d l o t I o n s . A n o t h e r oan  be the f a c t  t h a t the b a s i c d a t a used f o r  reason  the  c a l c u l a t i o n s of the B.C.F.S. l o g a r i t h m i c volume e q u a t i o n s lnoluded forked  f o r c e r t a i n s p e c i e s deformed t r e e s , f o r example  t r e e s f o r r e d c e d a r (Browne,1962),while  B.C.F.S. t a p e r trees.Table two  c u r v e s are p r o b a b l y only based  IV g i v e s an example of t h r e e  where the s i m i l a r i t y  d e v i a t i o n s are r a t h e r  i s high and  large.  the on normal  species,  one where the  Comparison of the B r i t i s h Columbia Forest Service Logarithmic Volume Equations with the Volume Equations from Table I I I Species group(R  M)" Differences as a percentage of B.C.F.S. volume  Spruoe (I M) T o t a l 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 50 -2.45 -3.14 - 3 . 7 2 -4.21 - 4 . 6 5 -5.04 60 - 2 . 6 2 - 3 . 3 1 -3.88 -4.38 -4.81 -5.20 Hemlock (I M) dbh 20 40 60 (inch.) 10 +0.80 - 0 . 3 9 -1.08 20 +1.68 +0.98 0 +2.20 0 50 60  T o t a l height (feet) 80 100 120 140 -1.56 +0.48 +1.70 +2.57  -1.94 +0.10 +1.31 +2.18 +2.86 +3.42  -0.21 -0.48 +1.00 + 0 . 7 3 +1.86 +1.59 +2.54 +2.27 +3.10 +2.82  160  +0.50 +1.36 +2.04 +2.59  180  200  +0.30 +1.16 +0.97 +1.83 +1.64 +2.38 +2.20  white pine (CI M) -----------------T o t a l 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 50 +35.24+19.12 +6.50 -3.65 60 +49.88+32.01+18.02 +6.77  see f n . 4 and 5 i n table I.  D e r i v a t i o n of a compatible l o g a r i t h m i c t a p e r from a l o g a r i t h m i c volume  Any  equation  equation  l o g a r i t h m i c volume  equation  l o g V .= a + b l o g D + c l o g H can be converted log d = b where b  Q  i n t o a logarithmic taper + b  1  logD + b  = l o g ( ( 4 144 1 0  Q  b  1  = b / 2  b  2  = ( p c - 1 ) / 2  b  3  a  2  equation  l o g 1 + bj l o g H  p c / 3.1416 J ' ) 1  2  = ( l - p ) c / 2  where"a,b and o a r e t h e c o e f f i c i e n t s from t h e l o g a r i t h m i c volume equation.The v a l u e has  of p which i s not y e t d e f i n e d ,  t o be chosen so as t o minimize t h e S E o f d i a m e t e r .  Therefore  E  some d a t a about t a p e r a r e needed.  T h i s t a p e r equation,when i n t e g r a t e d t o t o t a l volume, w i l l f o r any v a l u e  of p y i e l d  e x a c t l y the same volume  as g i v e n by t h e l o g a r i t h m i c volume e q u a t i o n  from  which i t i s d e r i v e d . ( f o r 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 F o r e s t S e r v i c e l o g a r i t h m i c volume  equations  The B.C.F.S. l o g a r l t h m l o volume equations 1962)  are f o r 23  compatible  s p e c i e s or s p e c i e s g r o u p s  l o g a r i t h m i c t a p e r equations  (Browne,  converted  by  to  selecting  the v a l u e of p so as t o minimize the S E o f diameter  on  £  the B.C.F.S. t a p e r curves  (B.C.F.S.,1968).  F i g u r e 1 shows f o r mature c o a s t a l D o u g l a s - f i r the  SE  E  of diameter as a f u n c t i o n of the v a l u e of p.The v a l u e f o r which the S E i s minimized i s the optimum v a l u e  to  E  be adopted f o r p i n d e r i v i n g the t a p e r e q u a t i o n the volume  from  equation.  A summary of the t a p e r e q u a t i o n  coefficients,  the optimum p v a l u e s as w e l l as the SE^s t a b l e V.These t a p e r equations  i s given i n  g i v e by i n t e g r a t i n g the  same t o t a l volume as g i v e n by the B.C.F.S. l o g a r i t h m i c volume equations.The optimum v a l u e of p ranged from 2.03  to 2.85  and  had  a mean v a l u e of  2.32.  The average b i a s of diameter I n s i d e bark a t the d i f f e r e n t h e i g h t s i s g i v e n i n t a b l e VI.For these taper equations and  most of the s p e c i e s ,  have the same p a t t e r n of under-  o v e r e s t i m a t i o n as i n t a b l e I I . F o r some s p e c i e s ,  however,such as a l d e r , b i r c h , i m m a t u r e hemlock and  c o a s t a l Western  maple,there i s a s l i g h t o v e r e s t i m a t i o n  the e n t i r e stem of the t r e e . T h i s a g a i n can be due  along t o 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  50  40  30  20  10  I 1-0  20 Value of p  30  Summary of the Taper Equations Derived from the B r i t i s h Columbia F o r e s t S e r v i c e L o g a r i t h m i c Volume Equations Species group R Alder  C  S E optimum (Inches) p value M -O.OO7438 O.960308 0.740496 - 0 . 7 0 3 4 8 5 0.746 2 . 3 1  Aspen  I  M  0.004808  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  "  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  2.85  "  C  I  0.130780 0.860380  1.264  2.62  M  I  M  0.120068 0.850996 0.881813 -0.848294 1.340  2.59  M  b  Q  Equation c o e f f i c i e n t s bj^ b g  b^  E  0.973017 0.704131 - 0 . 6 9 1 7 3 4 0 . 5 3 2 2 . 3 5  0.875467 - 0 . 8 5 0 4 8 0  2.09  C o t t o n - 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 0.829506 0.743543 -0.645686 1.348 2.08 fir 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 . 7 6 5 1 4 5 -0.682128 1.381  2.17  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  2.21  I - 0 . 0 1 4 5 9 0 0.921340 0.786591 - 0 . 7 2 4 7 6 1 1.182  2.29  Hemlock C "  C  " 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 . 9 2 3 5 6 I 0.684947 - 0 . 6 6 2 9 4 3 1.281 2 . 2 7 Maple  C  M - O . 0 3 3 6 3 5 0.942906 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  2.32  "  I  M  0.061738 0 . 9 2 0 6 1 3 0.756372 -0.739346 O . 5 2 2  2.43  white CI M 0 . 0 7 6 7 8 9 pine Yellow CI M 0.101402 cedar Yellow CI M -O.O57496 •—-pine-"—--—" lOaee f n . 4 and 5  O . 9 3 3 6 4 3 0.673334 -O.676I58 O.938  2.36  O.870522 O.73837O -O.709152 O . 5 8 7  2.34  0.95^739 0.623679 - 0 . 5 8 0 8 3 9 -• . 1 1 i n table I.  2.07  I.O63  T e s t of the Taper Equations Derived from t h e B r i t i s h F o r e s t S e r v i c e Logarithmlo Volume Species group Alder Aspen Balsam  C I  M M M M M M I M M M I M M I M M M M M M M M M  I I  Birch Cedar  c c  ti ii  Cottonwood Douglas-fir  •t  I CI  c c  I  I I  Hemlock  c c  Lodgepole Larch Maple Spruce  tt  M  c  •i  •i it  R  I p i n e CI I  White pine Yellow cedar Yellow pine  c c  I CI CI CI  11  Equations  Average b i a s ( i n i n c h e s ) of diameter  i n s i d e bark a t  1«  4.5'  0.1H  0.2H  0.4H  0.5H  0.6H  0.7H  0.8H  0.9H  0.48 -0.13 -0.76 -0.44 0.26 -1.90 -0.13 -1.98 -1.09 -2.96 -1.58 -3.12 -1.10 0.31 -0.87 -0.39 -2.33 0.24 -2.69 -0.55 -0.74 -0.27 -1.89  0.77 0.47 -0.15 0.24 0.58 -1.21 0.01 -1.03 0.33 0.38 O.38 0.28 0.04 0.81 0.25 -0.17 0.70 O.36 -2.26 0.04 -0.07 -0.20 0.66  0.88 0.60 0.57 0.42 O.63 2.18 1.21 1.03 0.50 1.22 0.74 1.09 0.88 1.39 0.68 0.15 1.66 O.56 1.93 O.67 0.61 O.56 O.85  O.67 0.19 0.39 0.08 0.45 0.25 0.37 -0.05 -0.38 0.24 0.01 -0.14 0.48 0.32 0.37 2.68 1.72 0.71 1.22 0.74 1.59 0.07 0.77 1.35 0.10 -0.28 - 0 . 5 3 0.09 0.77 1.33 0.16 0.74 0.49 0.80 0.03 -0.66 0.68 0 . 1 9 -0.21 1.25 0.89 0.59 0.52 0 . 2 3 -0.07 0.02 0.34 0.19 1.18 1.72 0.51 O.56 0.43 0.33 2.09 0.85 -0.29 0.02 0.71 0.37 0.49 0.15 -0.15 0.81 O.56 0.19 0.52 0.05 - 0 . 3 5  0.12 -0.01 -0.54 -0.20 0.32 -0.11 0.35 -0.47 -0.61 -0.48 -0.15 -1.10  0.17 0.00 -0.50 -0.15 0.35 -0.65 0.08 -0.79 -0.50 -0.76 -0.37 -1.14 -0.50 0.34 -0.26 -0.23 -0.31 0.23  0.32 0.09 -0.28 -0.05 0.39 -1.02 -0.07 -0.91 -0.22 -0.73  0.50 0.27 -O.O3 0.08 0.40 -1.22 -0.09 -0.83 0.13 -0.52 -0.38 -0.39 -0.05 O.56 0.24 -0.12 0.01 0.24 -1.01 -0.07 0.02 0.10 0.03  0.61 0.44 0.17 0.20 0.35 -1.16 -0.01 -0.57 0.39 -0.21 -0.17 0.07 0.21 0.59 0.62 0.06  see f n . 4 and 5 i n t a b l e I  0.3H  Columbia  -0.46  -0.85 -0.32 0.40 0.45 -0.26 -0.09 -0.12 -0.24 -0.28 -0.03 0.26 0.23 - 1 . 0 9 -1.46 -1.40 -0.20 - O . 2 5 - 0 . 1 9 - 0 . 3 4 - 0 . 3 3 -0.17 -0.18 - O . 3 6 - 0 . 2 5 -0.60 - 0 . 5 6 - 0 . 3 1  -0.46  0.46  0.22 -0.74 0.08 0.06 O.36 0.39  1.0H 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  above mentioned  I n c o n s i s t e n c y f o r c e r t a i n s p e c i e s between  t h e B.C.F.S. l o g a r i t h m i c volume equations curves.Therefore,it and  and  the  taper  i s d o u b t f u l i f the r e s u l t s f o r coast  i n t e r i o r redcedar,yellow  cedar and  the deciduous  s p e c i e s , f o r which f o r k i n g of the stem i s a common abnorm a l i t y , can be used as  such.  An example f o r mature c o a s t a l D o u g l a s - f i r i n t a b l e V I I shows how  the b i a s i s d i s t r i b u t e d  over the v a r i o u s h e i g h t  c l a s s e s w i t h i n the same s p e c i e s . E x c e p t  f o r the  three  s m a l l e s t h e i g h t c l a s s e s , t h e o v e r - a l l S E g i s a f a i r l y good r e p r e s e n t a t i v e f o r a l l the h e i g h t c l a s s e s . T a b l e V I I I g i v e s a summary of the SE|s t a b l e V and  those  of Kozak,Munro and  Smith ( 1 9 6 9 b , t a b l e I ) .  F o r the l o g a r i t h m i c t a p e r e q u a t i o n s , d e r i v e d B.C.F.S. l o g a r i t h m l o volume e q u a t i o n s , t h e f r o m .421  to 2 . 3 2 2  frequency  d i s t r i b u t i o n of the SE|s  inches.Table  An a b s o l u t e frequency  of t a b l e I,  from the  SE^s  IX g i v e s the f o r each  ranged  absolute case.  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 t a b l e X. Because they  i n c l u d e the e r r o r s i n h e r e n t i n both bark and  wood,these SE£s are r e l a t i v e l y s m a l l compared w i t h SE£s of s e c t i o n double bark t h i c k n e s s , e s t i m a t e d  from  diameter outside b a r k , t o t a l h e i g h t , s e c t i o n height s e c t i o n h e i g h t as a percentage of t r e e from .111 The  t o .842  inches  (Smith and  the  and  height,ranging  Kozak,1967).  S E o f the l o g a r i t h m i c t a p e r e q u a t i o n , f i t t e d on E  t h e t a p e r c u r v e s or d e r i v e d from the l o g a r i t h m i c volume  D i s t r i b u t i o n of the B i a s over the D i f f e r e n t Height C l a s s e s w i t h i n the Same S p e c i e s ( f o r Mature C o a s t a l D o u g l a s - f i r ) Height class (feet) 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 Total  Average b i a s 1( i n Inches) 1  diameter  i n s i d e bark a t  SE (Inches) C  0.1H  0.2H  0.3H  0.4H  0.5H  0.6H  0.7H  0.8H  0.9H  0.61 -0.20 0.65 -0.26 0.71 0.77 -0.04 0.82 0.73 0.47 -1.07 0.85 -1.83 0.70 1.23 -2.12 0.71 1.32 1.47 -2.69 0.59 1.46 -3.60 0.50 1.42 0.46 -4.13 0.08 1.62 -*.55 1.44 0.13 -5.03 0.10 -3-98 1.43 -5.10 - 0 . 0 6 1.27 1.88 -4.46 0.16 0 . 0 9 -0.21 -3.57 0.23 I.69 -3.63  0.47 0.65 O.56 0.95 1.21 1.28 1.29 1.34 1.37 1.52 1.57 1.63 1.77 1.70 1.48 1.94  0.19 0.29 0.16 0.48 O.63 0.65 O.65 0.66 0.74 1.00 1.02 1.07 1.08 1.03 1.01 1.24  -0.14 -0.03 -0.14 -0.15 -0.08 0.04 0.04 -0.10 -0.03 0.20 0.32 0.48 0.38 0.29 0.16 0.21  -0.42 -0.28 -0.26 -0.73 -0.73 -0.44 -0.48 -0.76 -0.61 -O.39 -0.12 -0.07 -0.13 -0.37 -0.77 -0.61  -0.68 -0.47 -0.28 -1.11 -1.23 -0.69 -0.82 -1.08 -O.76 -0.61 -0.31 -0.48 -0.44 -0.54 -1.09 -0.78  -0.85 -0.59 -0.21 -1.24 -1.65 -0.90 -0.82 -1.13 -0.60 -0.53 -0.30 -0.40 -0.24 -0.07 -O.63 -0.40  -0.83 -0.61 0.00 -1.24 -1.80 -0.90 -0.49 -1.11 -0.42 -0.40 -0.28 -0.07 0.30 0.45 -0.10 0.23  -0.59 -0.39 0.21 -0.88 -1.42 -0.49 -0.13 -0.99 -0.24 -0.21 -0.13 0.27 0.64 0.70 0.28 0.81  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  0.610 0.560 0.458 1.168 1.348 1.109 1.3*5 1.449 1.452 1.564 I.667 1.426 1.701 1.613 I.658 1.502  -2.96  1.33  0.77  -0.73  - 0 . 5 2 -0.21  0.0  1.348  1»  *.5'  O.38  1.22  0.09 -0.48 -O.76  l.OH  Comparison of Standard  E r r o r s of E s t i m a t e of S e v e r a l Methods  S p e c i e s group  R  M«  SEd) ^  SE£(2)14  SE(3)15  Alder  C  M  0,84  0.587  0.746  Aspen  I  M  0.59  0.448  0.532  Balsam  C  M  0.90  0.813  0.752  I I  I  M  0.58  0.420  0.421  Birch  I  M  0.32  0.245  0.505  Ced a r  C  M  2.13  2.431  2.002  it  c  I  1.61  1.128  1.264  I  M  1.30  1.379  1.340  Cottonwood  CI M  0.84  0.755  0.684  Douglas-fir  c  M  1.5*  1.431  1.348  n  c  I  1.35  1.032  0.987  •i  I  M  1.33  1.251  1.381  c  M  0.98  0.871  0.795  c  I  1.16  0.691  1.182  I  M  0.73  0.740  0.610  p i n e CI M  0.72  0.774  O.76O  M  Hemlook  •t I I  Lodgepole  1  E  Larch  I  M  1.33  1.230  1.281  Maple  c  M  0.41  0.316  0.483  Spruce  c  M  2.34  2.376  2.322  I  M  0.71  0.526  0.522  White pine  CI M  1.01  1.159  0.938  Yellow  cedar  CI M  0.78  0.57*  0.587  Yellow  pine  CI M  1.02  1.205  I.O63  I I  aee f n . 4 and 5 i n t a b l e I . l 3 f r o m t a b l e I of Kozak,Munro and Smith ( 1 9 6 9 b ) I n e q u a t i o n s f i t t e d on the t a p e r curves ( t a b l e I) ^ e q u a t i o n s d e r i v e d from the volume e q u a t i o n s ( t a b l e V) xc  Standard  E r r o r s of E s t i m a t e  Number of s p e c i e s g r o u p s SE (1) (2) (3)16 (Inches) 2 2 .0 < - < -5 4 E  •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 t a b l e V I I I .  TABLE X A b s o l u t e 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 Standard Difference (inches)  (2) BT (1)  E r r o r s of Estimate  Number of s p e c i e s g r o u p s 1 7 (1) (3) (1) (3) ( 2 ) BT BT BT BT BT (2) ( 3 ) (1) (3) (2) 1 8  .0  —  .1  6  4  5  6  9  4  .1  —  .2  6  2  9  1  3  4  .2  —  .3  2  1  -  .3  —  .4  2  -  -  .4 -  .5  -  1  1  17 see 1 8  fn.  -  2 mm  13 ,14 and 15 i n t a b l e V I I I .  B T ]means B e t t e r  Than.  1  e q u a t i o n , i s f o r s i x t e e n speolesgroups out of twenty-three s m a l l e r than the S E g i v e n by Kozak,Munro and Smith E  (1969b).  The S % o f the t a p e r f u n c t i o n d e r i v e d from the volume e q u a t i o n i s f o r t h i r t e e n speolesgroups s m a l l e r , b u t f o r t e n s p e o l e s g r o u p s l a r g e r than t h e S E o f E  fitted  on t h e t a p e r c u r v e s .  the t a p e r e q u a t i o n  IMPROVEMENT OF THE PRECISION OF THE  ACCURACY AND  THE  LOGARITHMIC  TAPER EQUATION A w e l l known technique  f o r improving  the accuracy  p r e c i s i o n of a t a p e r e q u a t i o n c o n s i s t s of r e l a t i n g  and  the  t a p e r e q u a t i o n c o e f f i c i e n t s t o some known t r e e c h a r a c t e r i s t i c s . In most i n v e n t o r y work only dbh  o u t s i d e bark and  t o t a l h e i g h t are measured.Although the number of p o s s i b l e r e l a t i o n s h i p s to i n v e s t i g a t e i s l a r g e , a c l o s e look t a k e n only Into some v e r y simple  approaches.  To improve the t a p e r e q u a t i o n f i t t e d curves,the  on the  r e l a t i o n s h i p s between the t a p e r  coefficients  (b ,b ,b  equation and  the r a t i o of both were i n v e s t i g a t e d . T h e c o r r e l a t i o n  was  1  2  b-^) and  taper  dbh,total height  0  and  was  v e r y poor ( a p p a r e n t l y a second or t h i r d degree m i a l ) or n o n - e x i s t e n t , e x c e p t  for b  2  polyno-  where a good  s h i p w i t h the r a t i o of t o t a l h e i g h t over dbh was  relatloalways  present.However,this r e l a t i o n s h i p d i d not have the same p a t t e r n f o r the d i f f e r e n t s p e c i e s . A  t r i a l of a t a p e r  e q u a t i o n model i n which each c o e f f i c i e n t was as a second degree p o l y n o m i a l f u n c t i o n of the of t o t a l h e i g h t over dbh  was  ratio  s u c c e s s f u l only f o r f i v e  s p e c i e s , d e c r e a s i n g the S E b y from . 1 E  expressed  to . 3  inches.  To improve the t a p e r e q u a t i o n d e r i v e d from the volume e q u a t i o n the r e l a t i o n s h i p between the optimum  v a l u e of p and d b h , t o t a l h e i g h t and the r a t i o of both was  i n v e s t i g a t e d . T h i s was attempted f o r mature c o a s t a l  D o u g l a s - f i r . A good c o r r e l a t i o n was found between the optimum v a l u e of p and t o t a l h e i g h t I t i s expected optimum v a l u e  (see f i g u r e 2 ) .  t h a t using t h i s r e l a t i o n s h i p between the of p and t o t a l h e i g h t , i n s t e a d  o v e r - a l l optimum v a l u e , c a n  of only the  Improve the p r e c i s i o n and  acouracy.But even i f the optimum v a l u e of p f o r each t o t a l h e i g h t c l a s s can be p r e d i c t e d from t o t a l without  height  e r r o r , t h e d e c r e a s e i n S E w l l l only be moderately E  Important f o r the s m a l l e s t h e i g h t c l a s s e s (see t a b l e X I ) . The o v e r - a l l S E w i l l l i k e l y change o n l y E  little.  Figure 2- Optimum value of p as o function of total height (for mature coastal Douglas-fir)-  Optimum Value of p  Number of observations = 16 o R X 100 = 9 0 % for a parabolic relationship  2-2  20  16  1-6  LA I V50  I I I I  I 100  I I I I  I  I I I I  1  5  Total Height (feet)  0  2  0  l 0  Maximum Decrease In Standard E r r o r of Estimate Expected from Using  t h e R e l a t i o n s h i p between the  Optimum V a l u e of p and T o t a l Height Coastal  t o Be  ( f o r Mature  Douglas-fir) Optimum v a l u e of P  in  50  1.75  0.610  0.350  60  1.80  0.560  0.365  70  1.92  0.458  0.383  80  1.84  1.168  0.957  90  1.85  1.348  1.117  100  1.97  1.109  1.046  110  2.01  1.3*5  1.321  120  2.01  1.449  1.409  130  2.09  1.452  1.441  140  2.11  1.56*  1.550  150  2.15  1.66?  1.626  160  2.13  1.426  1.396  170  2.18  1.701  1.618  180  2.15  1.613  1.562  190  2.13  1.658  1.633  200  2.12  1.502  1.481  Height class (feet)  19  . using the  o v e r - a l l optimum  using f o r each t o t a l height optimum v a l u e of p  SEg (inches) 1 9  o  (2)  n 2 0  v a l u e of p c l a s s the a p p r o p r i a t e  T h i s proposed derived  system of t a p e r and  from each o t h e r and  can meet the requirements Sayn-Wlttgenatein The  volume f u n c t i o n s  compatible w i t h each other  s t a t e d by Honer and  (1963).  taper f u n c t i o n f i t t e d  on the t a p e r curves as  w e l l aa the e q u a t i o n d e r i v e d from the volume e q u a t i o n d e s c r i b e s w e l l the stem p r o f i l e of the most  important  s p e c i e s of B r i t i s h Columbia.However,lt should be that a taper equation f i t t e d guarantee  realized  on diameter d a t a g i v e s no  of a good volume e q u a t i o n . T e s t s on t r e e  measurements should be c a r r i e d  out i n t h i s  field.  Whenever a t a p e r e q u a t i o n i s f i t t e d  on d a t a , t h e f u n c t i o n  should be t e s t e d both f o r diameter  and volume t o know f o r  both the p r e c i s i o n and T h i s was  r e c o g n i z e d by Bruce and  done by D u f f and graphical  the accuracy of the system.  Burstall  (1955)  Schumacher ( 1 9 5 0 )  i n the a p p l i c a t i o n of  techniques.  G i v i n g more weight t o l a r g e d i a m e t e r s would ensure b e t t e r f i t f o r volume,probably r e s u l t i n g fit  and  a  In a b e t t e r  a t the b u t t of the t r e e but a worse f i t h i g h e r on the  stem.This  would make the e q u a t i o n l e s s s u i t a b l e f o r  p r e d i c t i o n of s e c t i o n diameters  or h e i g h t s . I n s t e a d of 2 w e i g h t i n g , t h e 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 t h e t a p e r e q u a t i o n could be done  by a n o n - l i n e a r l e a s t would probably The  squares procedure.But t h i s a g a i n  have the same disadvantages  as w e i g h t i n g .  t r i a l s t o Improve the t a p e r e q u a t i o n by r e l a t i n g  the t a p e r e q u a t i o n c o e f f i c i e n t s t o some known t r e e c h a r a c t e r i s t i c s were not comprehensive enough t o draw f i n a l conclusions.The  p r e l i m i n a r y I n v e s t i g a t i o n s In  t h a t d i r e c t i o n were d i s c o u r a g i n g . Preference taper equation equation.In and  i s g i v e n t o the system i n which the i s d e r i v e d from t h e l o g a r i t h m i c volume  t h i s way the best f i t i s achieved  t h e f i t f o r diameter i s optimized  f o r volume  by t h e c h o i c e of  the optimum v a l u e of p.Moreover, i t i s the only p o s s i b l e way t o c r e a t e a t r u l y compatible volume i n those  system of t a p e r and  i n s t a n c e s where a l o g a r i t h m i c volume  e q u a t i o n a l r e a d y e x i s t s and probably w i l l be used i n t h e f u t u r e .  continue t o  Avery, T. E . 1 9 6 7 . F o r e s t Measurements. McGraw-Hill Book Co.,Inc.,N.Y. 290 p. B r i t i s h Columbia F o r e s t S e r v i c e , 1 9 6 8 . B a s i c t a p e r curves f o r the commercial s p e c i e s of B r i t i s h Columbia. F o r e s t Inventory D i v i s i o n , B.C.F.S., Dept. of Lands, F o r e s t s and Water Resources, V i c t o r i a , B.C.,unpaged graphs• Behre, C. E . 1 9 2 3 . P r e l i m i n a r y notes on s t u d i e s of t r e e form. J o u r . F o r . 2 1 : 5 0 7 - 5 1 1 . • 1 9 2 7 » F o r m - c l a s s t a p e r curves and volume t a b l e s and t h e i r a p p l i c a t i o n . J o u r . A g r . Res. 3 5 ( 8 ) : 6 7 3 - 7 4 3 . — — — — 1 9 3 5 . F a c t o r s i n v o l v e d i n the a p p l i c a t i o n of f o r m - c l a s s volume t a b l e s . J o u r . A g r . Res. 5 1 ( 8 ) : 6 6 9 - 7 1 3 . Browne, J . E. 1 9 6 2 . Standard c u b i c - f o o t volume t a b l e s f o r the commercial t r e e s p e c i e s of B r i t i s h Columbia,1962. 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 . McGraw-Hill Book Co., I n c . , N.Y.,  F o r e s t Mensuration. 4 8 3 P»  B r u c e , D., C u r t i s , R. 0 . and C. V a n c o e v e r l n g , 1 9 6 8 . Development of a system of t a p e r and volume t a b l e s f o r red a l d e r . F o r . S c . 1 4 ( 3 ) 0 3 9 - 3 5 0 . Chapman, H. H. and W. H. Meyer, 1 9 4 9 . F o r e s t Mensuration. McGraw-Hill Book Co.,Inc., N . Y . , 5 2 2 p. C l a u g h t o n - W a l l l n , H. 1 9 1 8 . The a b s o l u t e form q u o t i e n t . Jour. For. 1 6 : 5 2 3 - 5 3 * — — — — — and F . McVlcker, 1 9 2 0 . The Jonson*absolute form q u o t i e n t as an e x p r e s s i o n of t a p e r . J o u r . F o r . 18: 346-357. Demaerschalk, J . P. 1 9 7 1 * Taper equations can be converted t o volume e q u a t i o n s and p o i n t sampling f a c t o r s , (submitted t o the F o r . Chron.),typed, 6 p. D u f f , G. and S. W. B u r s t a l l , 1 9 5 5 * Combined t a p e r and volume t a b l e s . F o r e s t Research I n s t i t u t e . Note No. New Zealand F o r e s t S e r v i c e , 73 p.  1.  P r i e s , J . 1 9 6 5 « E i g e n v e c t o r a n a l y s e s show t h a t b i r c h and p i n e have s i m i l a r form In Sweden and B r i t i s h Columbia, For. Chron. 4 1 ( 1 ) : 1 3 5 - 1 3 9 . ————— and B. Matern, 1 9 6 5 * On the use of m u l t i v a r i a t e methods f o r the c o n s t r u c t i o n of t r e e t a p e r c u r v e s , I.U.F.R.O. S e c t i o n 2 5 . Paper No. 9 , S t o c k h o l m Conf e r e n c e , October, 1 9 6 5 * 32 p. G i u r g i u , V. 1 9 6 3 . (An a n a l y t i c a l method of c o n s t r u c t i n g d e n d r o m e t r l c a l t a b l e s w i t h the a i d of e l e c t r o n i c computers). Rev. P a d u r l l o r 7 8 ( 7 ) ' 3 6 9 - 3 7 4 ( i n Ruman i a n ) , see B r u c e , C u r t I s and V a n c o e v e r i n g ( 1 9 6 8 ) . Gray, H. R. 1 9 5 6 . The form and t a p e r of f o r e s t - t r e e stems. Imp. F o r . I n s t . Oxford, I n s t . Paper No. 3 2 , 79 p. Grosenbaugh, L. R. 1 9 5 ^ * New t r e e measurement c o n c e p t s : h e i g h t a c c u m u l a t i o n , g i a n t t r e e , t a p e r and shape. U.S.F.S. South. F o r . Exp. S t a . O c c a s i o n a l 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 . F o r . Chron. 4 2 ( 4 ) : 4 4 3 - 4 5 6 .  Heger, L. 1 9 6 5 a . Morphogenesis of stems of D o u g l a s - f i r . U n i v . of B.C., F a c u l t y of F o r e s t r y , 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 e s t i m a t i o n . F o r . Chron. 4 1 ( 4 ) : 4 6 6 - 4 7 5 * H e i j b e l , I . 1 9 2 8 . (A system of e q u a t i o n s f o r d e t e r m i n i n g stem form i n p i n e ) . S v e n s k . SkogsvFfiren. T l d s k r . 3 - 4 : 3 9 3 - 4 2 2 . ( i n Swedish, summary i n E n g l i s h ) . HeJJas, J . 1 9 6 7 . Comparison of a b s o l u t e and r e l a t i v e standard e r r o r s and e s t i m a t e s of t r e e volumes. U n i v . of B.C., F a c u l t y of F o r e s t r y , M.P. t h e s i s , t y p e d , 58 p. Hcjjer, A. G. 1 9 0 3 . T a l l e n s och granens t i l l v f i x t . Blhang t i l l F r . Loven. Om v a r a b a r r s k o g a r . Stockholm, 1 9 0 3 * (In Swedish), see Behre ( 1 9 2 3 ) . Honer, T. G. 1 9 6 4 . The use of h e i g h t and squared diameter r a t i o s f o r the e s t i m a t i o n of merchantable c u b i c - f o o t volume. F o r . Chron. 4 0 ( 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 S e c t i o n , Pulp and Paper Magazine of Canada. Woodlands S e c t i o n . 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 c u b i c - f o o t volume f u n c t i o n . F o r . Chron. 4 1 ( 4 ) : 4 7 6 - 4 9 3 « • '  1 9 6 7 * Standard volume t a b l e s and merchantable conv e r s i o n f a c t o r s f o r the commercial t r e e s p e c i e s of c e n t r a l and e a s t e r n Canada. F o r e s t Management r e s e a r c h and s e r v i c e s i n s t i t u t e , Ottawa, O n t a r i o , I n f o r m a t i o n r e p o r t FMR-X-5, 153 p.  — — — — — — and L. S a y n - W i t t g e n s t e i n , 1 9 6 3 * Report of the committee on f o r e s t mensuration problems. J o u r . F o r . 6l(9):663-667. Jonson, T. 1 9 1 0 . T a s a t o r i s k a undersflkningar om skogstrfldens form.I.Granens stamform. Skogsvardsfflreningens T i d s k r . 1 1 : 2 8 5 - 3 2 8 . ( i n Swedish), see Behre ( 1 9 2 3 ) . 1 9 H • T a x a t o r l s k a undersfikningar om skogstrSdens f o r m . I I . T a l l e n s stamform. SkogsvardsfBrenlngens T i d s k r . 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 . S t a t e n s S k o g s f f i r . 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 a n a l y s i s of m u l t i v a r i a t e techniques f o r estimating tree taper suggests t h a t s i m p l e r methods are b e s t . F o r . Chron. 42(4):458-463. —  — Munro, D. D. and J . H. G. Smith, 1 9 6 9 a . Taper f u n c t i o n s 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. F o r . Chron. 4 5 ( 4 ) : 1 - 6 . Munro, D. D. and J . H. G. Smith, 1 9 6 9 b . More accuracy r e q u i r e d . Truck Logger. D e c e m b e r : 2 0 - 2 1 .  L a r s o n , P. R. 1 9 6 3 . Stem form development of f o r e s t t r e e s . F o r . Sc. Monograph No. 5» 42 p. L o e t s c h , F. and K. E. H a l l e r , 1 9 6 4 . F o r e s t Inventory. V o l . I . S t a t i s t i c s of f o r e s t i n v e n t o r y and informat i o n from a e r i a l photographs. BLV V e r l a g g e s e l l s c h a f t • Mttnchen. ( t r a n s , by E. F. B r t i n i g ) . Matte, L. 1 9 4 9 . The t a p e r of c o n i f e r o u s s p e c i e s w i t h s p e c i a l r e f e r e n c e t o l o b l o l l y p i n e . F o r . Chron. 25:21-31. Mesavage, C. and J . VI. G i r a r d , 1 9 4 6 . Tables f o r estimat i n g b o a r d - f o o t content of timber. U.S.F.S. Washington D.C., 94 p.  Meyer, H. A. 1 9 5 3 . F o r e s t Mensuration. Penns V a l l e y P u b l i s h e r s , Inc., S t a t e C o l l e g e , P e n n s y l v a n i a , 357  p.  Munro, D. D. 1 9 6 8 . Methods f o r d e s c r i b i n g d i s t r i b u t i o n of soundwood i n mature western hemlock t r e e s . U n i v . of B.C., F a c u l t y of F o r e s t r y , Ph.D. thesis,mlmeo, 188 p. 1970. The u s e f u l n e s s of form measures i n the e s t i m a t i o n of volume and t a p e r of the commercial t r e e s p e c i e s of B r i t i s h Columbia. Paper presented at a meeting of the working group " E s t i m a t i o n of Increment". I.U.F.R.O., S e c t i o n 2 5 , Birmensdorf Conference, September, 1 9 7 0 . 14 p. Newnham, R. M. 1 9 5 8 . A study of form and t a p e r of stems of D o u g l a s - f i r , W e s t e r n hemlock and Western redcedar on the U n i v e r s i t y r e s e a r c h forest,Haney,B.C. U n i v . of B.C., F a c u l t y of F o r e s t r y , M.F. t h e s i s , t y p e d , 71 P« Osumi, S. 1 9 5 9 . S t u d i e s 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. J o u r . J a p . F o r . Soc. 4 1 ( 1 2 ) : 4 7 1 - 4 7 9 . ( i n J a p a n e s e , a b s t r a c t i n E n g l i s h ) P e t t e r s o n , H. 1 9 2 7 * S t u d i e r over Stamformen. Medd. S t a t e n s S k o g f 8 r s 8 k s a n s t a l t . 2 3 : 6 3 - 1 8 9 . ( i n Swedish). Prodan, M. 1 9 6 5 . H o l z m e s s l e h r e . J.D.SauerlUnder's V e r l a g , F r a n k f u r t am Mein. 644 p. Schmld, P., R o l k o - J o k e l a , P., Mingard, P. and M. Z o b e i r y , 1971. The o p t i m a l d e t e r m i n a t i o n of the volume of s t a n d i n g t r e e s . M i t t e i l u n g e n der F o r s t l i c h e n BundesV e r s u c h s a n h a l t , Wlen. 9 1 : 3 3 - 5 * • S h l n o z a k l , K., Yoda K y o j i , Hozumi, K. and T. K i r a , 1964. A q u a n t i t a t i v e a n a l y s i s of p l a n t form.The pipe model theory. Jap. Jour. E c o l . 14(3):97-104. Smith, J . H. G. and A. Kozak, 1 9 6 7 . T h i c k n e s s and p e r c e n t age of bark of the commercial t r e e s of B r i t i s h Columbia. U n i v . of B.C., F a c u l t y of F o r e s t r y , mlmeo. 33 p. — — — — — and A. Kozak, 1971• F u r t h e r a n a l y s e s of form and t a p e r of young D o u g l a s - f i r , W e s t e r n hemlock,Western r e d c e d a r and S i l v e r f i r on the U n i v e r s i t y of B r i t i s h Columbia r e s e a r c h f o r e s t . Paper presented a t the Northwest S c i e n t i f i c A s s o c i a t i o n Annual Meeting, U n i v . of Idaho, A p r i l , 1971• mlmeo. 8 p.  S p e l d e l , G. 1957* D i e r e c h n e r i a c h e n grundlagen d e r l e i s t u n g s k o n t r o l l e und l h r e p r a k t i s c h e durchftirung i n d e r f o r a t e i n r i c h t u n g . S c h r l f t e n r e l h e der F o r s t l l c h e n F a k u l t U t , U n l v e r s l t f l t G f l t t l n g e n . No. 1 9 , 118 p . Spurr, S. H. 1 9 5 2 . F o r e s t 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 v a l u e of l o d g e p o l e pine and Engelmann spruce l n t h e i n t e r i o r of B r i t i s h Columbia. Univ. of B.C., F a c u l t y of F o r e s t r y . Ph.D. t h e s i s , t y p e d . 252 p. T u r n b u l l , K. J . and G. E . Hoyer, I 9 6 5 . C o n s t r u c t i o n and a n a l y s i s of comprehensive tree-volume t a r i f t a b l e s . Resource management r e p o r t . No. 8. Department of N a t u r a l Resources, S t a t e of Washington. 63 p. Wickenden, H. R. 1 9 2 1 . The Jonson a b s o l u t e form q u o t i e n t : how i t i s used i n timber e s t i m a t i n g . J o u r . F o r . 19:584-593. W r i g h t , W. G. 1 9 2 3 . I n v e s t i g a t i o n of t a p e r as a f a c t o r l n measurement of s t a n d i n g t i m b e r . J o u r . F o r . 2 1 : 5 6 9 - 5 8 1 .  Common Names and L a t i n Names of  21 the Tree l  a  2.  Species  Red A l d e r (Alnus r u b r a Bong.). Trembling  Aspen (Populus t r e m u l o l d e s  3. Coast Balsam S p e c i e s and A.grand i s  Michx.).  (Abies a m a b i l i s (Dougl.)  (Dougl.)  Forbes  Lindl.).  4 . I n t e r i o r Balsam S p e c i e s (Abies l a s i o c a r p a  (Hook.)  N u t t . and A . g r a n d i s ) . 5.  White B i r c h S p e c i e s  (Betula p a p y r i f e r a v a r i e t i e s ) .  6. Western Red Cedar (Thuja p l i c a t a Donn). 7.  B l a c k Cottonwood  (Populus t r l c h o c a r p a T o r r . and G r a y ) .  8. Douglas F i r (Pseudotsuga  m e n z i e s i i (Mlrb.) F r a n c o ) .  9. Western Hemlock (Tsuga h e t e r o p h y l l a (Raf.) S a r g . ) . 10.  Lodgepole  11.  Western L a r c h  12.  B r o a d l e a f Maple (Acer macrophyllum P u r s h ) .  13.  C o a s t a l Spruce ( P l c e a s i t c h e n s i s  Pine  (Pinus c o n t o r t a D o u g l . ) . (Larlx o c c i d e n t a l l s Nutt.).  (Bong.) C a r r . ) .  14. I n t e r i o r Spruce S p e c i e s ( P i c e a g l a u c a (Moench) Voss, P.Engelmanni Parry,and 15»  Western White Pine  16. Yellow  P.marlana ( M i l l . ) B.S.P.).  (Pinus m o n t i c o l a D o u g l . ) .  Cedar (Chamaecyparis n o o t k a t e n s i s (D.Don)  Spach)• 17.  Western Yellow  Pine  Based on Appendix  (Pinus ponderosa Laws.).  I from Browne  (I962).  APPENDIX 2 D e r i v a t i o n of a Volume E q u a t i o n  from  the Taper E q u a t i o n of Hfljer  V = DH  0.005454  d  = c  2  2  / D  2  2  substituting  / (a  (ln((o  (2)  / D ) d (1  2  2  + 1  2  (1)  / H)  100 / H)/ o ) )  (2)  2  2  (1)  in  A V = D H 2  = DH 2  0.005454  0  / (o (ln((o 2  0.005454  c  (1  2  (1  )(i (i 1  2  = DH 2  (1  0.005454 o  100 where K = 1 + —— Co  2  c  )(ln(l 2  1  1 0 0  100  +  ) H c  2  +  2  100 +  „  2  )) + 2 (1 H o  c  0  ))2  100  +  2  1 0 0 ? 2/ = D H 0.005454 c | ( ( l + — ) ( l n ( l  \  100 H c  2  )(ln(l H o  1  +  n  100  +  / H)  2  1 100 H c  1 2  +  + 100 1 / H)/ c ) ) ) d (1 2  2  .  J  >> )) 2  °2 100 + — ^ ) ) + 2 (1 o  100 +  2  ( K ( l n K ( l n K - 2)  + 2)  -  2)  o  2  ) - 2  2  APPENDIX 3 D e r i v a t i o n of a Volume E q u a t i o n from the  V = DH  0.005*5* j 0  d  = (1  2  2  / D  2  substituting  2  (d  / H) /(b (2)  2 = D^H  2  + b  Q  1 / H)  x  / ((1 O-'  / H) /(b 2  (  2  Q  + b  1 / H ) ) d (1 2  x  (b  0  1 / H) - 2 b  +  b  0.005454—  (b  b  If b  + b  Q  1  = 1  (1927)  Behre  V = DH 2  x  0  0  1 / H) -  + b  / ( b  1  0  Q  + "°  2  0  (b  - 2 b  +  In ( b  0  b  = D^H  / H)  r  1  0.005*5*—r  (  (1)  in  l  d (1 / H)  / D )  2  2  0.005*5*  V = D H  Taper E q u a t i o n of Behre  V  + b j 1 / H)  0  In ( b  0  +2  b  0  l n  0  + bj) -  V  what was u s u a l l y the c a s e , a c c o r d i n g t o  then  1 0.005454— b  l  (1  - bp + 2 b  Q  ln b ) Q  APPENDIX 4 D e r i v a t i o n of a Volume E q u a t i o n from the Taper E q u a t i o n of Matte  H  v = 0 . 0 0 5 * 5 * / (a ) d(i)  (1)  2  d  2  = b  0  D  2  (l  substituting  /  2  H  (2)  + b  )  D  2  (l3  H  /  3  + b  )  D  2  2  (l  4  /  H  4  )  H  / (b  2  x  (1)  in  ,  V = 0.005*5* D  2  Q  (l  2  / H ) 2  + b  (l3 / H3) +  x  0-> bp U  4  0  H  = 0.005*5*  = 0.005454  b D'  D  d  H  /  l3  0  3  H (b  H<  Q  b  l  *  / 3 + b  l H3  x  4  b  2  5  i5 H  / 4 + b  4  2  / 5)  4  ) )  d  (1) =  (2)  APPENDIX 5 D e r i v a t i o n of the Height E q u a t i o n the L o g a r i t h m i c Taper E q u a t i o n  The  taper equation  (formula 2) i s :  d = 10 ° D l l b  b  b  H 3  2  b  thus l  b  2  = d / ( 1 0 0 D l H 3) b  b  b  1 = (d / ( 1 0 0 D l H 3 ) ) b  b  b  1 / b  or 1 = (10'  b 0  d D " l H" 3) b  b  1 / b  2  2  from  D e r i v a t i o n of a L o g a r i t h m i c Volume E q u a t i o n  from  the L o g a r i t h m i c Taper E q u a t i o n  V = O.OO5454 / (d*) d ( 1 )  d  2  = 10 0 D 2 b  substituting  2 b  l  l  2  b  H  2  2 b  (1)  3  (2)  (2) i n (1)  V = 0.005454 1 0 0 / ( D OJ 2 b  H  2 b  l  - 2b! D  = 0.005454  10  2 b  l  2 b  2 H 3 ) d (1) = 2 b  ^bgfl  io °  2bi  R  2b  2  a f t e r taking the logarithm  log V = a + b log D + c log H '0.005454 (2 b  2  10  2 b  + 1)  b = 2 b.  2  2  0  + 1)  + 2b  (2 bo + 1)  c = 2 b  n 3  2 b  D  where a = l o g  2b  ° (2 b  0.005454  R  + 2 b + l  °  3  + 1  D e r i v a t i o n of the Formula t o E s t i m a t e Volumes of Logs between S p e c i f i c D i s t a n c e s from the T i p of the Tree,from the L o g a r i t h m i c Taper E q u a t i o n  When the lower and upper d i s t a n c e from the t i p of the  t r e e a r e r e s p e c t i v e l y 1^ and l , t h e n the volume of  the  l o g between t h e s e two d i s t a n c e s i s :  2  .U  V = O.OO545* / ( d ) d (1) = 2  0.005454 1 0  °  2 b  /(D  2 b  io^2 bo r . 9  D  l  2b!  H  2  (2 b  2 b  2  1 1  1  D  0.005454  l l  b  H 3 ) d (1) 2 b  _  2  2b2+l  9 H  .  -  =  l 2  2b3  + 1)  2  0.005454 1 0 0 (2 b + 1) 2 b  D  2b  3  ( 1  2b  2 +  l _ ^bz+lj  _  2  =  where K =  K  %  D  v  v  „V  /.Z  al  R-y  0.005454 (2 b  v = 2 b.  2  -  if)  io o 2 b  + 1)  7 = 2 b,  z = 2 b  2  + 1  D e r i v a t i o n of a Compatible L o g a r i t h m i c Taper from a L o g a r i t h m i c Volume  The proof  log d = b  + b  b  x  Q  log D + b  1  /10a where b  Equation  i s g i v e n t h a t the t a p e r  Q  = log I —  p c  \  —  J  2  Equation  equation  l o g 1 + b^ l o g H  1 / 2  b  = b / 2  b  2  3  = (p c - 1 ) /  = (1  2  - p)c / 2  y i e l d s f o r any v a l u e of p , a f t e r i n t e g r a t i n g , t h e same volume as g i v e n by the l o g a r i t h m i c volume  equation  log V = a + b log D + c l o g H  Proof:  V = 0.005454  /  •H (1)  / ( d ) d (1) 2  )J  2 from the t a p e r equation,d 2  i s d e f i n e d as  10 p c b «(P c - 1) = ^ D 1 0.005454 D  (.1 - p) c H  v  v  (2)  substituting  (2) i n (1) Hn  v = 0.005454  /10 /10  p c  a  D  b  X  ( P c - 1)  0.005454 1 0  a  D  b  H  = 10  a  D  b  H°  P  )  _ 0  Hr-  p 0  D  b  1  P  O.OO5454 p c a  P  005454  /  = 10  ( i -  H  ° H° "  P  °  a f t e r taking the logarithm  log V = a + b log D + o log H  t h i s completes t h e p r o o f .  ° H  ( 1  "  p  )  0  d (1) =  

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