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Volume and taper estimation systems pinus patula and cupressus lusitanica growing in Kenya forest plantations Gor-Kesiah, John Odhiambo 1978

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VOLUME AND TAPER ESTIMATION SYSTEMS FOR PINUS PATULA AND CUPRESSUS LUSITANICA GROWING IN KENYA FOREST PLANTATIONS by John OdhiamboJGor-Kesiah B.ScV For.s (Hons-.) Maker ere University, i'075' A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF ' THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Forestry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1978 (c) John Odhiambo Gor-Kesiah In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y 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 r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f 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 be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Forestry. The U n i v e r s i t y o f B r i t i s h Co lumbia 207 S Wesbrook Place Vancouver, Canada V6T 1WS - i -ABSTRACT Volume and taper studies i n Plnus patula and Cupressus  lusltanica trees growing in Kenya forest plantations are discussed. The ultimate objective of the study was to find suitable models for estimating volumes and taper rates using two approaches for each parameter (i.e. volume and taper). These included producing volume models i n the traditional way and by integrating taper models. Taper models derived from the best of the volume models were compared to the taper models produced from taper data in the traditional way to find out the efficiency of each approach. Data from the two species were used to f i t a few popular volume and taper models. Tri a l s were also made to develop new models. Of the popular volume models tested the logarithmic volume model was found to be giving very good estimates. Weighted models conditioned through the origin, by a technique proposed in the study, were also giving similar good estimates. Models weighted by current approach were, however, giving relatively poor estimates. Volume-based taper models were found to be giving biased diameter estimates along the tree pro f i l e s . However, when integrated for total volume, the volume estimates given seemed to be better than those given by the other taper models tested. Because of their bias in estimating the diameters and other points along the tree profile, volume-based taper models are not recommended for estimating volumes to any other points along the tree profiles. Other popular taper models were also giving biased diameter estimates. They lacked inflection points. When - i i -integrated for volumes, they were giving very poor estimates. Two f a i r l y simple taper models have been proposed which have most of the characteristics needed in a taper model to give proper profile description. They have been recommended for constructing the inside volume tables and taper tables for the two species. They should als< apply well i n other conifers. One model describes profiles of trees with butt swell while the other describes trees with smooth stem form - i i i -TABLE OF CONTENTS Page ABSTRACT ± TABLE OF CONTENTS . . . i i i LIST OF TABLES v i LIST OF FIGURES ix ACKNOWLEDGEMENTS .xi 1. INTRODUCTION 1 1.1 General 1 .1.2 Objectives of the Study ' 2 2 LITERATURE REVIEW 3 3 THE DATA ". .16 3.1 Source 16 3.2 Size Range and Distribution of Sample Trees 16 '3.3 Sample-Measurements 22 4 ANALYSIS OF PRELIMINARY RESULTS 24 4.1 Volume Models Fitted by Traditional Approaches 24 4.1.1 Fitting of Popular Volume Models 24 4.1.2 Selection of Best Models .31 4.2 Taper Models Derived from the Best of the Volume Models. • 40 4.2.1 Testing Volume-Based Models 50 4.2.1.1 Testing the Models for Diameter Estimation- • • .50 4.2.1.2 Testing the Volume-Based Taper Models for Sectional Volume Estimation 55 - i v -4.3 Taper models produced from taper data 60 4.3.1 Results of Fi t t i n g Common Taper Models 60 4.3.1.1 Testing the Models for Diameter, Sectional Volume, and Distance from the Top Estimations . . 63 4.3.2 New Taper Models 71 4.3.2.1 General Approach 71 4.3.2.2 The New Developed Taper Models 76 4.3.2.3 Tests on the New Taper Models for Sectional Diameter, and Volume Estimations 101 4.4 Other Interesting Relationships 109 4.4.1 Approximate Sectional Diameter Estimates 109 4.4.2 Approximate Utilizable Tree Volumes 110 5. DISCUSSIONS ON THE RESULTS OF THE VARIOUS SECTIONS 115 6. SUMMARY OF CONCLUSIONS 129 LITERATURE CITED 132 APPENDICES 139 Appendix 1 Unweighted volume equations 139 2(a) Plots of residuals of volume estimates for model 1 - specific for P_. patula volume estimates. 140 2(b) Plots of residuals of volume estimates for model 2 - specific for T?. patula volume estimates. 141 2(c) Plots of residuals of volume estimates for models 3 and 4 - specific for P_. patula volume estimates . . . .142 2(d) Plots of residuals of volume estimates for models 13, 14 and 15 - specific for _P. patula volume estimates 143 - V -3 Sample programs for estimating the free-parameter values in volume-based taper models 144 4 P_. pa tula; Taper tables for data used in taper studies - given by diameter classes 145 5 _C_. lusitanica: Taper tables for data used in taper studies - given by diameter classes 146 6 jP. patula: Volume distribution along the profiles of trees used in taper studies- •' * • 147 7 C_. lusitanica: Volume distribution along the profiles of trees used i n taper studies . . 148 8 Derivation of volume models from the proposed taper models 149 9(a) P_. patula: Plot of diameter residuals for model 31 • • 154 9(b) C_. lusitanica: Plot of diameter residuals fpr model 33 155 9(c) P_. patula: Plot of residuals of distance from top of trees for model 27 156 9(d) iC. lusitanica: Plot residuals of distances from top of trees for model. 26 157 10 P_. patula: Total volume overbark 158 11 P_. patula: Total volume underbark . . 159 12 C_. lusitanica: Total volume overbark 160 13 lusitanica: Total volume underbark . .161 14 Extract pages of comprehensive volume-and-taper table -162 - v i -LIST OF TABLES P a g e I Minimum, Maximum and Average Sizes of the Data Sampled • 16 11(a) CJ. lusitanica; Height and'.DBH Distribution of Sample Trees Used in Volume Study 18 11(b) P. patula; Height and DBH Distributions of Sample Trees Used in Volume Study 19 11(c) (J. lusitanica: Height and DBH distributions of Sample Trees Used in Taper Study . .20 11(d) P. patula: Height and DBH Distributions of Sample Trees Used in Taper Study 21 III Volume Models Produced or Weighted by Current Approach 25 IV Examples of values used in f i t t i n g some volume equations . .26 V Volume ModelsFitted by a New Proposed Weighting 32 VI Efficiency of Various Volume Models on Estimating P_. patula Volumes 35 VII Efficiency of Various Volume Models on Estimating lusitanica Volumes 36 VIII Tests on Adopted Volume Models Including Standard Errors of Estimate at Each Diameter Class 39 IX C_. lusitanica: Efficiency of Volume-Based Taper Model 18 for Diameter Estimation 52 X P_. patula: Efficiency of Volume-Based Taper Model 19 for Diameter Estimation .53 XI P^. patula: Efficiency of Volume-Based Taper Model 20 for Diameter Estimation 54 XII C. lusitanica: Efficiency of Volume-Based Taper Model 18 for Volume Estimation 57 XIII \P. patula: Efficiency of Volume-Based Taper Model 19 for Volume Estimation 58 - v i i -» A i V £• patul a; Efficiency of Volume-Based Taper Model 20 for Volume Estimation 59 XV Regression Coefficients and Standard Errors of Estimates for Three Popular Taper Models Tested 62 XVI P_. patula; Efficiency of Taper Model 24 for Diameter Estimation. 67 XVII C_. lusitanica: Efficiency of Taper Model 24 for Diameter Estimation 68 XVIII P. patula: Efficiency of Taper Model 24 for Volume Estimation 69 XIX C. lusitanica: Efficiency of Taper Model 24 for Volume Estimation 70 XX C. lusitanica: Factors for Converting Diameters at DEI] (overbark) to diameter (inside-bark) at various points at the tree profiles 75 XXI patula: Factors for Converting Diameters at DBH (overbark) to Diameter (inside-bark) at Various Points at the Tree Profiles 75 XXII P. patula: Efficiency of the New Taper Model 31 for Diameter Estimation 102 XXIII C. lusitanica: Efficiency of Taper Model 33 for Diameter Estimation 103 XXIV P. patula: Efficiency of Taper Model 31 for Volume Estimation 104 OCV C. lusitanica: Efficiency of Taper Model 33 for Volume Estimation 105 DCVI Proportions of Total Volumes That Are Available Up to Various Sections Along the Tree Profiles For Trees of Various Sizes for P_. patula .111 [XVII Proportions of Total Volumes That Are Available Up to Various Sections Along the Tree Profiles For Trees of Various Sizes for C. lusitanica . . . . . . . . 112 - v i i i -XXVIII Factors for Converting Total Volumes to Volumes at the Different Limits 113 XXIX(a) P.patula: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for. the Estimates Using Models 15, 22 and 35 for a small tree 117 XXIX(b) P.patula: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 15, 22 and 35 for a medium-sized tree 118 XXIX(c) P.patula: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 15, 22 and 35 for a large tree 119 XXX(a) C.lusitanica: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 15, 22 and 34 for a small tree 120 XXX(b) C.lusitanica: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 15, 22 and 34 for a medium-sized tree 121 XXX(c) C.lusitanica: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 15, 22 and 34 122 - i x -LIST OF FIGURES P a R e 1 Points where samples were taken along the felled tree profiles 23 2(a) Changes in the values of the standard error of estimate as the 'free parameter' values change 45 2(b) Changes in average diameter bias as the 'free parameter' values change 46 2(c) Changes in average diameter bias corresponding to changes in standard error of estimate 47 3 P.patula: Effect of f i t t i n g model 25 for the whole tree or above dbh only 65 4 C.lusitanica: Effect of f i t t i n g model 25 for the whole tree or above dbh only 66 5 P.patula: General tree profile (form) scattergram 73 6 lusitanica: General tree profile (form) scattergram . . . . . 74 7 P. patula: Tree forms predicted by new tested models of varying complexities 80 8 C. lusitanica: Tree forms predicted by new tested models of varying complexities 81 9 P_. patula: The performance of the new adopted taper model (model 31) 84 10 C^. lusitanica: The performance of the new taper model later discarded (model 32) 85 11 C_. lusitanica: The performance of the new adopted taper model ( model 33) 87 12 P_. patula: The true profile given by model 31 when more points are generated 89 13 C_. lusitanica: The true profile given by model 33 when more points are generated 90 14 P_. patula: Graphical representation of tree forms and diameter estimates given by a l l the 5 taper models tested; shown for a small tree 91 P_. patula: Graphical representation of tree forms and diameter estimates given by a l l the 5 taper models tested; shown for a medium-sized tree 92 P_. patula: Graphical representation of tree forms and diameter estimates given by a l l the 5 taper models tested; shown for a large tree • 93 (J. lusitanica: Graphical representation of tree forms and diameter estimates given by a l l the 5 taper models tested; shown for a small tree • 94 (3. lusitanica: Graphical representation of tree forms and diameter estimates given by a l l the 5 taper models tested; shown for a medium-sized tree 95 f j . lusitanica: Graphical representation of tree forms and diameter estimates given by a l l the 5 taper models tested; shown for a large tree 96 J?. patula: Graphical representation of volume estimates given by the adopted taper model 107 C^. lusitanica: Graphical representation of volume estimates given by the adopted taper model. 108 - x i -ACKNOWLEDGEMENTS The opportunity to take a Master's degree programme, of which t h i s Thesis forms a part, was made poss i b l e by a scholarship o f f e r e d by CIDA (Canadian In t e r n a t i o n a l Development Agency) to the Kenya Government. My sincere gratitude goes to CIDA and through CIDA to the Canadian c i t i z e n s f o r t h e i r support. I am also extremely g r a t e f u l to the Kenya Government f o r o f f e r i n g the scholarship to me. For the success of my study at UBC, I f e e l indebted to the continuous encouragement and cooperation I obtained from a l l the professors under whom I took the various courses. In p a r t i c u l a r , I f e e l indebted to the s t a f f of the Biometrics Group f o r t h e i r i n c r e a s i n g help and openness i n discu s s i o n while introducing me to many aspects of computer programming and s t a t i s t i c s . For the preparation of t h i s t h e s i s , I am p a r t i c u l a r l y g r a t e f u l f o r the assistance, encouragement and constructive c r i t i c i s m s I c o n t i n u a l l y received from Drs. A. Kozak, D.D. Munro, J.P. Demaerschalk and P.A. Murtha. Special word of gratitude goes to Dr. Demaerschalk who was my supervisor throughout the study. He suggested a number of problems which acted as back-bone to the study. His o f f i c e door was always open to me at a l l times f o r various kinds of consultations. May God bless him, h i s house and h i s career even more. I also owe s p e c i a l thanks to the Chief Conservator of Forests, Kenya, Mr. 0 . Mburu; the Conservator of Management Services, Mr. B.R.K. Shuma; the D.F.O., Elburgon, Mr. Njuguna Ndatho; Asst. D.F.O., Mr. Gikonyo Gichohi and a l l the s t a f f at the - x i i -Forest Department, Inventory Division, for a l l the precious assistance afforded me during the data collection. Special gratitude goes to the UBC Computing Centre for making available to me their computing f a c i l i t i e s during the data analysis. The real heart of the success of my study l i e s in the encouragement given, and the patience shown by my wife and children during the two years I had to stay away from them to undertake the study overseas. Thanks. 1. INTRODUCTION 1.1 General Plantation forestry in Kenya i s f a i r l y new. The f i r s t active tree planting started in the early 1940's. Plantations have since been established using mainly three exotic softwood species: Cupressus  lusitanica M i l l . (Mexican Cypress) (hereafter referred to simply as C_. lusitanica or Cypress) and Pinus patula Schiede and Deppe (hereafter referred to simply as P_. patula or Pine) and Pinus radiata D. Don. These species now comprise over 80% of the total managed plantations area in the country - the f i r s t two being the most important. The cool tropical climate in the Kenya Highlands, with f a i r l y heavy r a i n f a l l , i s extremely suitable for these species.- They are ready for harvesting for sawlogs at an age of 25 to 30 years. - - - -A particular concern at this time is the lack of acceptable tree or stand volume tables to f a c i l i t a t e quick and f a i r l y accurate determination of volumes in the plantations. Early efforts saw the production of a few volume tables in the 1950's and early 1960's. However, they have not been put to use because sample trees were used which were much younger than would be exploited during the f e l l i n g stages. Over the years, i t has become clear that without a system for giving precise volume estimates of standing trees, i t i s d i f f i c u l t to make reasonable short and long term plans for wood-based industries. Standard tree volume - 2 -and/or s tand volume models a r e e s s e n t i a l . Good models should be a b l e to min imize the p resent t e d i o u s l o g measurement t e c h n i q u e s , and a l s o be reasonab ly f l e x i b l e i n d e t e r m i n i n g the volume es t imates to any u t i l i z a t i o n l i m i t s . Because a l l p l a n t a t i o n s i n the country r e c e i v e (or a t l e a s t they are r e q u i r e d to r e c e i v e ) r e g u l a r and s i m i l a r s i l v i c u l t u r a l t reatments throughout t h e i r a c t i v e growth p e r i o d s , and hence shou ld have s i m i l a r growth c h a r a c t e r i s t i c s , the models produced shou ld be a b l e t o g i v e p r e c i s e e s t i m a t e s of any s tand volumes. 1 .2 O b j e c t i v e s o f the Study a) To produce s tandard volume e s t i m a t i o n models tha t are reasonably p r e c i s e t o be used f o r q u i c k e s t i m a t i o n of volumes of s t a n d i n g t r e e s of C^ . l u s i t a n i c a and P_. p a t u l a . b) To produce w h o l e - b o l e taper e s t i m a t i o n models t h a t can be used to p r e d i c t the upper stem d i a m e t e r s , or h e i g h t s to any predetermined upper d i a m e t e r s , f o r any t r e e b e l o n g i n g to any g i v e n d iameter (dbh) c l a s s . c) To examine the p o s s i b i l i t y of p roduc ing compat ib le - volume and taper models from the best volume models and taper models produced i n the f i r s t two p a r t s o f t h i s s t u d y . T h i s would g i v e a l t e r n a t i v e approaches to the volume e s t i m a t i o n - thereby p r o v i d i n g a l t e r n a t i v e cho ices of methods. I n an attempt t o meet these o b j e c t i v e s v a r i o u s known models and new models w i l l be t e s t e d . Because computers a re not expected to be used i n the Kenya F o r e s t Inventory i n the f o r e s e e a b l e f u t u r e , ve ry complex models w i l l not be pursued . - 3 -LITERATURE REVIEW In the young and developing nations of the third world the use of volume or taper estimation equations alone i s of limited importance. Therefore, any volume or taper models must be presented in tabular forms to be of practical use. This implies that as u t i l i z a t i o n limits' change with advancement in technology new tables w i l l need to be produced. If the models used to construct current tables are flexible the process of revision can be done quickly and at an extremely low cost. Cupressus lusitanica, Pinus patula and Pinus radiata have been planted in Kenya since the early 1940's. Early efforts to produce volume estimation models for the species did not start u n t i l the 1950s and 1960s. By 1967 the available tables included those produced by Davis (1953 and 1956), Pudden (1958), Dyson (1961), Fry (1961 and 1963), Pearman (1963), Dyson et_ a l . (1964) and Paterson (1967). However, none of these tables have ever been put into use, partly because they were considered inefficient because the samples from the young plantations used were not representative of trie populations" whose volumes would be estimated, and partly because even the f a i r l y good ones (e.g. Paterson, 1967) were not revised from imperial to metric systems when Kenya adopted the metric system in 1968. Like many early volume tables t they were a l l constructed by graphical methods (Pearman, 1963, possibly exception). In a recent publication, Wanene and Wachiori (1975) have referred to some revised models produced by H.L. Wright at Oxford University for these species. Tables produced from the models - 4 -have not come to the author's notice. In recent years, the use of least squares techniques has been preferred over the graphical methods because i t is said to eliminate subjectivity in determining where the line should pass. Detailed discussions of the different volume construction techniques and models can be found in Bruce and Schumacher (1950); Chapman and Meyer (1953); Spurr (1952); Avery (1967); Husch et a l . (1972); Loetsch, Zohrer and Haller (Vol. 2, 1973). The great number of formulae that are now available and can be used simply indicate that there i s no volume equation which is generally applicable and valid for a l l species (Loetsch et a l . , 1973). The more common among them are the following: 1) ' Log V = bQ + b1logD + b 2logH (Schumacher and Hall, 1933) 2) V = b Q + bjD + b2DH + b ^ 2 + bH + b^ D^ H (Comprehensive) 3) V = b Q + bjD 2 + b2H + b3D2H (Australian) 2 4) V = b Q + b-jD H (combined variable; Spurr 1952) where V, D and H are respectively, Volume, Diameter Overbark at breast-height (dbh) and Total height b 0 ' b l e t C ' a r e r e S r e s s i o n coefficients. - 5 -Of the models above, the one that has been most used is the logarithmic volume model (1). Many foresters, however, have come out . strongly against i t s use claiming that i t is too simple and that i t gives biased volume estimates (Spurr, 1952; Cunia 1964; Gerrard, 1966). Evert (1969) gave an example of a case where the volume of a small tree was overestimated about 65 times i t s actual volume by assuming homogeneity of volume variances. After several studies Meyers (1953) found that the logarithmic volume model (1) was able to minimize this weakness. The model was, however, giving slightly biased estimates. He proposed a factor for correcting the bias (1944). Baskerville (1972) also proposed the same factor. More recently the model has been fitted by the non-linear estimation approach to overcome the bias limitation. In an effort to cope with limitations in the use of least squares procedure, caused by increasing residual volume variances as the tree sizes increase, the use of weighted regression has been recommended and widely used. A general review of the technique can be found in Draper and Smith (1966). More specific examples in the application of -the techniques in the construction of volume tables can be found in Cunia (1964); Gerrard (1966); Smalley and Beck (1971); and Smalley (1973). Although the general concensus of foresters i s that the volume 2 2 variances are proportional to the term (D H) (Munro, 1964; Cunia, 1964) weighting the individual observations by — ^ — has not always proved D H satisfactory (Cunia, 1964; Smalley and Beck, 1971; and Smalley, 1973). Gerrard (1966) however, found that the volume variance was not completely 2 2 proportional to (D H) • He estimated the volume variance as: - 6 -cr" = Exp (a + bD + cH) In practice, different terms have been used as the weight depending on whichever appears most suitable for the case being investigated and the model f i t t e d (Cunia, 1964). A review of the different techniques for constructing volume tables i s not complete without mentioning the methods used to obtain the total volumes of the sample trees. Volume table data are often obtained from felled trees using one of the three log volume estimation formulae below: 1) Ruber's formula 2) Smalian's formula 3) Newton's formula volume of log sectional area at log midpoint sectional area at large end of log sectional area at small end of log log length. V = SmL V = (8 X +'gs)L V = ( g ] L + 4g m + g s)L where: V = 8m H 8 s - 7 -Because trees assume different shapes at different points along the stem profile these formulae only give good and f a i r l y identical estimates i f the log lengths are small. . In general, Smalian's formula w i l l give greater volume error i f used over longer log lengths. Generally, however, Smalian's formula i s the cheapest and perhaps the most reliable because the measurements (on felled logs) are taken at the log ends; bark and diameter measurements are taken at the same points - thus making i t the most adaptable for computing both inside and outside volumes of logs. Parts of trees with greater tapering demand shorter sectioning to get reliable estimates (Husch et a l . , 1972; Bruce and Schumacher, 1950; Spurr, 1952). The use of relascope and other dendrometers permit measuring arbitrary diameters at upper stem heights. From these measurements cubic contents of standing trees can be determined either as total or merchantable volumes to any desired minimum upper diameter. Because no bark measurements are taken and because measurement errors are larger on standing trees, the method is less reliable (Loetsch e_t _al., 1973). In volume studies, totals of sectional volumes per tree computed by any one of these approaches is always taken as the standard. As in volume studies, many taper models have been proposed over the years. Grosenbaugh (1966) commented on these by stating that many mensurationists have sought to discover a single simple two-variable function involving only a few parameters which could be used to specify the entire tree profile. Unfortunately, trees seem capable of assuming an i n f i n i t e variety of shapes. Polynomials (or quotients of polynomials) - 8 -with degrees at least two greater than the observed number of inflections are needed to specify various inflected forms. He went further to explain that in realization of this problem, foresters have tended to seek uniformity in their volume studies by stratifying the samples by species, size and form to derive regression models. Although the models may give reliable aggregates of volume estimates they do not give any indication as to the distribution of the volumes along the stem profiles which would depict the form (shape) of the trees under study. Because measurements of sectional diameter at different heights along the tree profile are not a direct composite of these models the total volume estimates given to some specific upper diameters are not necessarily the sums of the sectional volumes computed using similar models below the point of interest. Husch £rt a_l. (1972) stated that the applicability of standard volume tables depend on the form (shape) of the tree to which i t i s applied rather than on species or loc a l i t y . They state that for each diameter-height class the form of the trees to which the table i s applied should agree with the form of the trees from which the table was prepared. This implies that i f a model can be produced which i s capable of giving good description of the shape of the tree species under study i t should, i f integrated for volume, be able to give good estimates in any other l o c a l i t y , provided the shapes of trees do not change from one locality to another. This would further imply that extensive distribution of sample trees with regard to locality as currently applied for volume table construction would not be necessary. The d i f f i c u l t y with putting this approach into practice has been the d i f f i c u l t y of expressing the longitudinal profile of a tree, relating height and diameter in a functional form (Husch et aj^., 1972 and Grosenbattgh, 1966). If height-- 9 -diameter measurements are enough to locate a l l the points of inflection^the concavity and convexity at different parts of the tree become relatively unimportant. By this approach the woody component (or volume) of single trees,both small and large;would be integrated the same way without need for weighting. . The volume integral, in particular, i s of interest in that the effect of convexity or concavity has been isolated into a single taper-dependent term (Grosenbaugh, 1966). Honer and Sayn-Wittgenstein (1963) stated that "we must develop a mathematical tree volume expression which can be effi c i e n t l y programmed for generally available electronic computing equipment to yield tree and stand volumes from inputs of tree diameter outside bark and total height (form estimates optional) and for any demanded stump height and top diameter". This requirement does not seem practicable. It i s impossible to g«t such a flexible volume model, i f the shape (or form) of the trees are not put into•consideration as this i s the only sure way the volumes to the different u t i l i z a t i o n limits can be reliably computed. .In an attempt to describe the tree profile many models of varying complexities have been developed over the years. It i s not intended to give a detailed review of this literature. Any interested persons could get a good review of some of the most important past work in Demaerschalk (1973b). A review w i l l , however, be given on three or four taper models which have gained popular recognition in the past few years. - 10 -These include:-d2 _ h h2 X ) ~Z2 ~ b0 + b l ~ l T + b 2 ~ Kozak « al. (1969) " H where: h i s the distance from the ground to a point up the tree where underbark diameter i s to be estimated. D i s diameter at breast-height (dbh) overbark H i s t o t a l height bp, b ,^ are regression c o e f f i c i e n t s . Working with a number of tree species i n B r i t i s h Columbia (Canada) Kozak, et a l . (1969) found that for most species the model, though good, was greatly overestimating the diameter close to the t i p . They introduced a condition that:-bQ + b x + b 2 = 0 b Q = - b 1 - b 2 From t h i s the o r i g i n a l model changed to 4 - *x c - g - - « + (-4 - 1 ) D H - 11 -For most species this conditioning seemed to minimize the original bias. However, for two of the species, Coastal Spruce and Cedar spp, further conditioning that b^ = b^ was necessary resulting in the use of a linear model: 2 9 D H H 2 2) log d = b Q + b1logD + b logH + b logL Demaerschalk (1971) where: L i s distance from the top of the tree to point where insidebark diameter i s to be estimated. Other symbols similar to the above equations. . In effect, this model is similar to the logarithmic volume equation and changes to a similar model when integrated for total volume, One of the f i r s t taper models produced that seemed to have a number of characteristics of a tree profile was that produced by Curtis, Bruce and Van coevering (1968). Their taper model (also reported in Bruce et a l . (1968))was a high degree polynomial with powers of up to 40. Bennett and Swindel (1972) produced a simpler model (which seems to be a follow-up of the technique used by Curtis e_t a l . (1968)) to estimate the taper of slash pine above breast-height. Their model was given as: 3) d = bxDL/(H - 1.3) + b 2L(h - 1.3) + b 3L(h - 1.3)(H) + b 4L(h - 1.3)(H + h + 1.3) - 12 -Although the model was reported to give good tree profile description above breast-height i t s use does not seem to have been reported elsewhere. In a series of papers Demaerschalk (1972, 1971b, 1973a, 1973b) and Munro and Demaerschalk (1974) introduced and showed that taper equations could also be obtained from existing volume equations and that such taper equations could be integrated to get volume estimates which are compatible with the estimates given by the volume models. Munro and Demaerschalk (1974) concluded^ after several investigations, that very few taper-based systems provide reasonably satisfactory results for both taper and volume estimation. They recommended volume-based taper models although they cautioned that volume-based taper models have no inflection points. Of the sets of taper and volume models tested, however, volume-based models seemed to be less biased. Since then Goulding and Murray (1975) and, Ek and Kaltenberg (1975) have found that volume-based models could be made more capable of describing the shapes of the trees by performing some modification on the basic taper models. In a recent publication, however, Demaerschalk and Kozak (1977) were convinced that good taper-based models are far much better and less biased compared to volume-based models. Their model, which is rather complex, was produced by matching the shapes of trees with the various Matchacurve (Jensen 1973, 1976 and Jensen and Homeyer, 1970, 1971). The characteristics in the model which produced the best matching curve was used to determine the functional characteristics needed in a taper model to get similar profiles in taper curves. The model they - 13 -so developed proved to describe the shapes of the trees better than any other models ever produced before and gave the smallest bias i n both diameter and volumes of the various sections of the trees. The complexity of the model makes i t only useful with appropriate computing f a c i l i t y . Because several models would need to be tested to determine which model i s the most appropriate, i t i t important to review some of the c r i t e r i a that have been used i n the past. However, i t has been noted that they do not always select the same model (Hejjas 1967). In regression studies i f a l l the models have the same dependent variable the decisions have often been based on the standard error of estimate or the co r r e l a t i o n c o e f f i c i e n t . These s t a t i s t i c s lose their c r e d i b i l i t y as soon as a transformation i s done on the dependent v a r i a b l The appropriate standard error of estimate can, however, always be recomputed from the re l a t i o n s h i p : (Yi - ¥i) Z n - p i = 1 where: p = number of parameters n = number of .observations A Y i and Y i are the actual and predicted i n d i v i d u a l values respectively. SEE = - 14 -Sometimes the decision i s based simply on observing the bias of the models either as s i z e class bias or as t o t a l bias: n BIAS = *y ^ (Yi - Y i ) ; ^ ^ (Yi - Y i ) / n i = 1 i = 1 (Total bias) (Average bias) Honer (1965b) proposed a modified version of these two to compute the sum of squares of the r a t i o of the BIAS/observed, that i s , n i = 1 A l t e r n a t i v e l y t h i s can be computed without squaring which gives a r e l a t i v e bias ( i . e . the bias per value of the observed): n - YD i = 1 n 2> i = 1 - 15 -Furnival (1961) however proposed an index which i s based on the theory of maximum l i k e l i h o o d . Freese (I960) suggested that a decision be based on the accuracy of the model. He suggested that a Chi-square test has a l l the desirable features to make t h i s a desirable approach. Hejjas (1967) af t e r using several of these indices, and others developed, concluded that none i s i n i t s e l f s a tisfactory. At least two methods should be used. Both r e l a t i v e and absolute measurements must be considered. 3. THE DATA 3.1 S ource The data used i n t h i s study was obtained from two sources. Most of i t was collected by me personally during the period between June and August 1977 while I was on a summer vacation. Due to the d i f f i c u l t y i n arranging accommodation and transportation to the d i f f e r e n t regions i n the country a l l my data was collected within Elburgon Forestry D i v i s i o n . As such the sampling scheme was not completely to the standards required i n most volume studies - This set being the bulk of the data used i n the study^ the re s u l t s of the study may be more applicable i n that D i v i s i o n . The second set of data was provided to me by the Inventory D i v i s i o n of the Forest Department. I t was part of data collected by Inventory D i v i s i o n i n 1967 to produce log volume tables currently i n use. Different regions i n the country were represented i n t h i s set. Because measurements were not collected at the same points i n both sets of data the added data was only used i n volume study and not i n the taper study. 3 , 2 Size Range and D i s t r i b u t i o n of Sample Tr ees The trees used i n t h i s study ranged i n sizes as indicated below: Table I. Minimum,1 Maximum and Average Sizes of Trees Sampled. Diameter Average Height Average Species Range (cm) Diameter Range (m) Height Pinus patula 4.9-52.3(62.2)* 24.8 4.5-34.0 21.8m Cupressus 6.1-86.7(92.2) 33.8 5.3-37.8 22.1m l u s i t a n i c a - 17 -* The figures i n brackets are the upper l i m i t s for the trees used i n volume study. The averages are for the trees used i n taper i study only. The o r i g i n a l aim of the study during the-data c o l l e c t i o n was to approach the volume study through taper study. As such the sample size and sample d i s t r i b u t i o n was determined with only t h i s i n mind. Later, i t was decided to use the same data for volume study also ( i n the t r a d i t i o n a l way). The sample size might not be altogether satisfactory for volume study but was considered s u f f i c i e n t for taper study. For each species, diameter classes were created, each 8 cm wide. This was done as a s l i g h t adaptation of the log size classes used i n Kenya. In each cla s s , 15 to 20 trees were sampled and as many available size classes as possible were covered. During the data c o l l e c t i o n , however, i t was not possible to get sample trees for large trees of Pw patula. As sawmillers prefer (3. l u s i t a n i c a to P_. patula, i t was not possible to get a plantation where pine trees larger than 52.3 cm at breast-height were being cut. In some diameter classes the samples appeared to be too small. This i s why i t was considered necessary to add a few extra data from the Inventory D i v i s i o n . In my sampling scheme, samples were collected from 95 and 133 trees of P_. patula and Cypress respectively. For volume study, I added 25 and 32 trees of_P. natula and Cypress respectively (from the Forest Inventory D i v i s i o n ) . The height and diameter d i s t r i b u t i o n for the trees used i n this study i s given i n TablesII(a) to 11(d). These tables show that although the DBH CLASS (in cm) 2 4 6 8 4.0 0 0 0 0 8.0 0 0 2 8 12.0 0 0 1 8 16.0 0 0 0 0 20.0 0 0 0 0 24.0 0 0 0 0 28.0 0 0 0 0 32.0 0 0 0 0 36.0 0 0 .0 0 40.0 0 0 0 0 44.0 0 0 0 0 48.0 0 0 0 0 52.0 0 0 0 0 56.0 0 0 0 0 60.0 0 0 0 0 64.0 0 0 0 0 68.0 0 0 0 0 72.0 0 0 0 0 76.0 0 0 0 0 80.0 0 0 c 0 84.0 0 0 0 0 88.0 0 0 0 0 92.0 0 0 0 0 96.0 0 0 0 0 tal 0 0 3 16 Number of 10 12 14 16 18 20 0 0 0 0 0 0 1 0 c 0 0 0 4 2 1 0 0 0 2 2 3 2 0 0 0 0 2 5 5 0 0 0 1 2 0 2 0 0 0 0 0 4 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 " 0 0 1 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 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 7 4 : 7 9 7 ; 8 .mple trees HEIGHT CLASS(in m) 22 24 2o 28 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 c 0 0 0 0 0 0 0 0 2 0 0 0 0 2 1 0 0 2 4 3 0 0 0 2 2 2 0 0 0 3 1 2 0 0 2 3 1 0 0 I 0 0 0 0 0 3 2 0 0 0 0 1 0 0 1 1 0 0 0 0 1 2 0 0 J 1 1 0 0 u 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 1 0 0 0 0 0 2 10 14 ' 13 11 32 34 36 38 40 Total 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 16 0 0 0 0 0 9 0 0 0 0 . 0 12 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 12 0 1 0 0 0 7 2 0 0 0 0 9 3 2 0 0 0 11 2 0 4 0 0 7 1 i 0 0 0 7 3 5 3 0 0 12 2 5 2 0 0 11 1 1 2 0 0 7 3 2 0 0 0 8 I 0 0 0 0 1 1 0 0 1 0 3 0 0 0 0 0 0 1 0 0 0 0 1 3 i 0 0 0 5 0 0 0 0 0 1 0 0 1 0 0 1 23 18 12 1 0 165 Table 11(a). C. l u s i t a n i c a : Height and DBH d i s t r i b u t i o n of s a m p i e trees used i n volume study. Number of sample trees O B H HEIGHT CLASS CLASS( ( i n cm) 2 4 6 3 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Total 4.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8.C 0 0 12 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 12.0 0 0 0 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 16.0 0 0 0 0 6 6 0 1 0 0 0 0 0 0 0 0 0 0 0 0 13 20.0 0 0 0 0 C 0 0 1 0 0 0 3 2 0 0 0 0 0 0 0 6 24.0 0 0 0 0 0 0 1 0 1 0 1 5 3 1 2 0 0 0 0 0 14 2«,0 0 0 0 0 ' C • 0 0 0 0 0 0 0 3 0 2 2 0 0 0 0 7 32.0 0 0 0 0 0 0 0 0 0 0 0 1 3 2 2 0 1 0 0 0 9 36.C 0 0 0 0 0 0 0 0 0 0 1 0 1 2 5 2 2 1 0 0 14 40.0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 2 0 3 0 0 0 7 44.0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 1 3 4 1 0 0 0 9 48.0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 2 4 0 0 0 10 52.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 56.0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 3 60.0 0 0 0 0 0 0 0 0 0 0 0 1 0 l n n ~ " 64.0 0 0 0 0 0 n n « . - u o 0 3 u u O l 0 1 0 0 0 0 0 0 2 „ u u 0 0 0 0 0 0 ' 0 0 0 0 0 0 0 1 0 0 0 0 1 68.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 72.0 O C O O O O O O O O O O O O O O O O O O 0 £ 76.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 80.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 84.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 88.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 92.0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 96.0 o o o o o o o o o o o o o o o o o o o o 0 Total 0 0 12 3 10 6 1 2 1 0   26 28 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 0 0 1 5 3 1 2 0 0 3 0 2 0 1 3 2 2 1 0 1 2 5 0 1 1 0 2 0 0 0 1 3 0 0 2 0 2 0 0 0 0 0 0 0 0 1 0 0 1 0 1 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 2 11 15 8 18 Table 11(b) 13 12 1 P. £ a t u l a 1 Height and DBH d i s t r i b u t i o n of sample trees used i n volume study. 115 CBH CLASS (i n cm) 2 4 6 8 10 12 4.0 0 0 0 0 0 0 8. C 0 0 2 8 1 0 12.0 0 0 1 8 4 2 16 .0 0 0 0 c 2 2 20.0 0 0 0 0 0 0 24.0 0 0 0 0 0 0 28.0 0 0 0 0 0 0 32.0 0 0 0 0 0 0 36. C 0 0 0 0 0 0 40.0 0 0 0 0 0 0 44.0 0 0 0 0 0 0 4S.0 0 0 0 0 0 0 52.0 0 0 0 0 0 0 56.0 0 0 0 0 0 0 60.0 0 0 0 0 0 0 64.0 0 0 0 0 0 0 68.0 0 0 0 0 0 0 72.0 0 0 0 0 0 0 76.0 0 0 0 0 0 0 SO.O 0 0 0 0 0 0 84.0 0 0 0 0 0 0 88.0 0 0 0 0 0 0 92.0 0 0 0 0 0 0 96.0 0 0 0 0 0 0 Total 0 0 3 16, 7 4 Table 1 1 ( c ) . C. l u s i t a n i c a : Number of sample trees HEIGHT CLASS(in n) 14 16 18 20 22 24 26 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 2 0 o x 0 0 0 0 2 5 5 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 4 0 2 0 0 0 0 2 1 : l 2 3 0 0 0 0 0 0 1 2 2 0 0 0 0 0 0 3 1 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 1 0 0 , o 0 0 0 0 0 0 0 0 0 0 0 1 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 c 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 7 9 7 5 1 5 10. 6 Height and DBH d i s t r i b u t i o n of sample 30 32 34 36 38 40 Total 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 16 0 0 0 0 0 . 0 9. 0 0 0 0 0 0 12 0 0 0 0 0 0 3 0 0 0 0 0 0 6. 0 0 0 0 0 0 9 0 0 1 0 0 0 6 2 2 0 0 0 0 8 1 3 2 0 0 0 9 0 2 0 4 0 0 6 2 1 1 0 0 0 5 1 2 4 3 0 0 10 0 2 5 2 0 0 10 2 0 1 2 0 0 5 1 2 2 0 0 0 5 0 1 0 0 0 0 1 0 0 0 0 1 0 1 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 15 16 11 1 0 133 trees used i n taper study. Number of sample trees (i n cm) 2 4 6 8 10 12 14 4.0 0 0 0 0 0 0 0 8.0 0 0 12 2 0 0 0 12.0 0 0 0 1 4 0 0 16.C 0 0 0 . 0 6 5 0 20.0 0 0 0 0 0 0 0 24.0 0 0 0 0 0 ' o o 28.C 0 0 0 0 0 0 0 32.0 0 0 0 0 0 0 0 36.0 0 0 0 0 0 0 0 40.0 0 0 0 0 0 0 0 44.0 0 0 0 0 0 0 0 43. 0 0 0 0 0 0 0 o 5 2.0 0 0 • 0 0 c 0 0 56.0 0 0 0 0 0 . 0 o 60.0 0 0 0 0 0 0 0 64.0 0 0 0 0 0 0 o 68.0 0 0 0 0 0 0 0 72.0 0 0 0 0 0 0 0 76.0 0 0 0 0 0 0 0 80.0 0 0 0 0 0 0 o 84.0 0 0 0 0 0 0 o 88.0 0 0 0 0 0 0 0 92.0 0 0 0 0 c 0 o 96.0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 18 0 0 0 0 c 0 0 0 0 0 0 0 0 0 .0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T o t a l 12 10 HEIGHT CLASS (in m) 24 0 0 0 0 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 26 28 30 32 34 36 33 40 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 2 0 0 0 0 0 0 0 3 1 2 0 0 0 0 0 3 0 2 2 0 0 0 0 3 2 2 0 1 0 0 0 1 I 5 2 2 1 0 0 0. 0 2 0 3 0 0 0 0 1 2 4 1 0 0 0 0 0 1 1 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 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 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 12 5 16 10 12 1 0 0 T o t a l 0 14 • 5 11 5 12 7 8 12 5 8 6 1 1 0 0 0 0 0 0 0 0 0 0 95 Cable 11(d) P. £atula: Height and DBH d i s t r i b u t i o n of sample trees used i n taper study. - 22 -samples were very well distributed in the available diameter classes the distribution with respect to height was not a l l that good for P_. patula. This seems to suggest that sampling could perhaps have been better done i f sample distribution were based on age classes as well. This would have taken care of both diameter and height distribution. 3.3 Sample Measurements In taper studies, the parameters usually measured are the sectional diameters at various points along the tree profile. At the same points bark measurements are also taken. In the past, the points where the samples have been taken on the sample trees have been, in many cases, defined as some specific distances along the stem, from the base to the top. Taper models produced in this way would always be biased against small trees because fewer samples, some specific lengths apart, w i l l be taken from a small (short) tree compared to a t a l l (big) tree. Moreover this does not give f a i r description at the points where trees taper fastest (e.g. the below dbh) and does not fu l l y express the inflection points along the profile of each tree included as a sample. In this study, 15 samples were taken from each tree at the same relative point. More samples were taken below dbh to allow for f u l l expression of tapering at this region. The other samples were taken at the relative points above and including the dbh by simply dividing the section of the tree above dbh into ten equal parts and taking samples at each l/10th of the distance. The points where the samples were taken are shown in the following diagram: x - 23 -Figure 1 . Points where samples were taken along the f e l l e d tree p r o f i l e s . At each of the points, the samples taken included: sectional diameter, and bark thickness. Total height of the tree was also recorded (including stump). The distances below dbh were measured from ground l e v e l . A diameter tape was used for taking the sectional diameters. The bark-thickness was i n i t i a l l y taken from a cut notch. Later the measurements so taken were compared with those obtained by using a bark gauge. Although the bark-gauge s l i g h t l y overestimated the bark thickness i t was found to be faste r . Measurements taken using the bark gauge were lat e r corrected for the bias. - 24 -4. THE ANALYSIS OF PRELIMINARY RESULTS The results of the study are reported under four sections: les 4.1 Volume models f i t t e d by t r a d i t i o n a l approach^ 4.2 Taper models derived from the best of the volume models 4.3 Taper models produced from the taper data 4.4 Some inter e s t i n g relationships 4.1 Volume Models F i t t e d by Tr a d i t i o n a l Approaches 4.1.1 F i t t i n g of Popular Volume Models The following models were f i t t e d on the basic data to fi n d out which one would best estimate the volumes of the tree species under study: Log V = b Q + b 1logD + b logH (1) V = b 0 + b l D 2 H ( 2 ) V = b Q + bjD 2 + b 2H + b 3D 2H ( 3 ) V = b Q + bjD + b2DH + b 3D 2H ( 4 ) Table III. Volume Models Produced or Weighted by Current Approach. TRADITIONAL WEIGHTING  Pinus patula (1) (2) (3) (4) LogVb = -4.1859385 + 1.73650661ogD + 1.11514711ogH -4.4797125 + 1.75663861ogD + 1.2566107IogH LogVI Vo VI Vo VI Vo VI = 0.3018 + 0.00003021DZH = 0.2351 + 0.00002767D2H = -0.44296203 + 0.028758415H + 0.000028514182D2H = -0.36439172 + 0.023149574H + 0.00002630507D2H = -0.40904244 + 0.00011637803D2 + 0.0000082683284D2H + 0.0013832198DH = -0.38572466 + 0.00016113533D2 + 0.0000066574801D2H + 0.0012007055DH Cupressus lusitanica LogVo = -4.2119844 + 1.78605511ogD + 1.05851411ogH -4.2834363 + 1.80663061ogD + 1.060l8571ogH (1) (2) (3) (4) LogVI Vo VI Vo VI Vo VI • 0.4636 + O.00002644D^ H = 0.3863 + 0.00002490D2H - -0.79376764 +0.042593051H+ 0.00043754992D2 + 0.000012113449D2H = -0.64809649 + 0.034348235H+ 0.00044121595D2 + 0.000010688384D2H - -0.30066708 + 0.00033194454D2 + 0.0000091035119D2H + 0.0008514628DH - -0.24904328 + 0.00035570293D2 + 0.0000082857323D2H + 0.00068500636DH SEE and R For Weighted Models True SEE SEE R2 0.04486 0.9968 0. 124312 0.05914 0.9950 0. 128544 33.58 0.9394 0. 224622 32.69 0.9321 0. 198672 29.40 0.9540 . 0. 184357 30.02 0.9432 0. 168686 26.81 0.9620 0. 205279 28.25 0.9502 0. 1988579 0.03860 0.9978 0. ,203721 0.0434 '.' 0.9976 0-196169 73.02 0.9737 0. ,337771 68.74 0.9736 0. ,293619 66.99 0.9782 0. .281064 62.94 0.9783 0. .244749 66.35 0.9786 0.216925 62.52 0.9786 0. .197730 ro - 26 -where: V, D and H are respective Volume, Diameter overbark at breast-height (dbh) and Total height, bp, b^, etc. are regression c o e f f i c i e n t s . Models 2 to 4 were f i t t e d with and without weighting. 2 Weights were 1/D H. Because unweighted models are of limited p r a c t i c a l value i n volume estimation, they w i l l not be discussed much i n this study. They are, however, given i n Appendix 1 for information. The other models of interest are given i n Table I I I . I t w i l l be noted by examining Table I I I that some models are not completely si m i l a r to th e i r corresponding forms above. Some of the variables shown above were not s i g n i f i c a n t at 5% significance l e v e l during the f i t t i n g of the models. Table IV below shows an example of the values from a sampled tree that would be entered i n the model to f i t , for example, models (1) and (3) by weighting each observation by 1/D2H. Table IV. Examples" of values used i n f i t t i n g some volume equations. V 3.6 V -5 o = 6.2806 x 10 D = 43.0 D H H 31.0 D -4 2 9 = 7.50188 x 10 D 1849 D H 2 D H 57319 = H -4 o = 5.40833 x 10 Log 1 QV 0.5563 D H Log 1 QD 1.6335 D 2 = 3.2258065 x 10' o Log 1 ( )H — 1.4914 D H -2 - 27 -By current weighted regression model (3) i s actually f i t t e d as shown i n model (5) below. V b 0 V 2 b 2 H _ = _ + -A + — | - + b ' (5) D H D H D H D H 3 What t h i s actually means i s that the variables used i n the regression are - / 1 D 2 H . 2 = f ( ~2~ ~2~ ~T~~ ) (6) D~H D"H D"H D"H 2 Although variable 1/D H i s not one of the variables obtained d i r e c t l y by any of the transformations shown i n Table IV, i t i s simply introduced i n the model as a variable of convenience. I t s purpose i s to make i t possible to obtain a constant term (or intercept) i n the f i n a l form of the model (after both sides of the equation have been multiplied by the weight v a r i a b l e ) . For example, model (5) changes i n i t s f i n a l form to V = b Q + bjD 2 + b 2H + b 3D 2H (7) - 28 -Although i n the f i n a l form model (5) changes to a form (shown i n equation (7)), which i s si m i l a r to the unweighted model (3), i t i s important to r e a l i z e that the basic weighted equation i s equation (5)• Table IV shows that the variable used as weight i s given a unit weight i n every set of observations. As such the function f i t t e d could perhaps better be represented as 2 - T — - f Hh . -§- ' ~~T~ > , , 1 M> (6*> D H D H D H D H where "1" s i g n i f i e s a constant, 2 D H 2 " D H :ion is Becuase a constant 1 cannot be used d i r e c t l y i n f i t t i n g the equatj (because i t i s not a v a r i a b l e ) , i t i s only accounted f o r . I t t h i s constant that i s represented by the constant term i n equation (5). In the past, i t has been assumed that including the variable, — \ , i n model (5) r e s u l t s i n the best estimates from the D H weighted model. In practice, however, better estimates may s t i l l be obtained by using fewer variables. By excluding t h i s variable, for example, we may need to f i t a weighted equation for a conditioned model V = bjD 2 + b 2H + b 3D 2H (8) - 29 -By using symbols similar to those given i n equation (6') we would then f i t the following function: ~Y~ = f ( - f 2 - , - f - , "1") <9). D H D H D^ H D 2 H When f i t t e d with — a n d — — as the only variable we would get D H D H the model In i t s f i n a l form t h i s model would change to V = b QD 2H + b ^ 2 + b 2H (10-) I t i s a common knowledge that ordinary models when conditioned through the o r i g i n usually give worse estimates than their corresponding models with intercepts. At best, they can only give si m i l a r estimates. However, there lacks knowledge on the ef f i c i e n c y of weighted models when they are conditioned through the o r i g i n . I t was thought that models so f i t t e d might eliminate any i n e f f i c i e n c y i n models weighted by current approach - which might be due to including the 'variable of convenience' as a variable i n f i t t i n g the basic weighted models. I f the weighted model - 30 -could be l e f t i n the form given i n equation (10), i t could be seen that i t i s not s t r i c t l y conditioned through the o r i g i n . I t appears thatmodel (10) and (10') are the s t r i c t weighted forms of model (3). What i s currently c a l l e d the weighted form "appears to be a completely d i f f e r e n t model. The theory behind which i t i s produced i s only applicable i n cases where the variable used as weight was not one of the variables o r i g i n a l l y present i n the unweighted model. In an e f f o r t to maintain s i m i l a r i t y i n form (symbol-wise) of an unweighted model, foresters (and possibly other s c i e n t i s t s ) have erred by f i t t i n g a completely new equation. By the new approach the weighted regression coefficient for the combined variable model (model 2) would be obtained as the average of the weighted r a t i o s : b. = l V i ~2 (ID D H. l n i = 1 — n V = bD 2H (12) - 31 -where: and b are r e s p e c t i v e l y i n d i v i d u a l and average r a t i o . 2 and D H^ , are i n d i v i d u a l observed volume and product of square diameter overbark a t br e a s t - h e i g h t (dbh) and t o t a l h e i g h t . 2 2 V and D H are g e n e r a l i z e d forms of v\ and D H.. i i Using t h i s new approach to weig h t i n g models (3) and (4) were r e - f i t t e d . Other models were a l s o t r i e d . In these more v a r i a b l e s were t r i e d as p o t e n t i a l v a r i a b l e s . The models produced by t h i s approach are given i n Table V. 4.1.2 S e l e c t i o n of the Best Models A l l the models f i t t e d were f i r s t judged by the s i z e s of the standard e r r o r s of estimates. Since a l l the models were a t r a n s -f o r m a t i o n of some s o r t (except those f i t t e d without weighting) the c o r r e l a t i o n c o e f f i c i e n t could not be used. Even the standard e r r o r s of estimates given by the models were f o r the transformed data and not f o r the a c t u a l data. As such the t r u e (or a c t u a l ) standard e r r o r s of estimates had to be computed from the r e l a t i o n s h i p below: n A 9 (Y. - Y . ) 2 SEE 1 ~ 1 n - p Table V. volume Models Fitted by a New Proposed Weighting. NEW EXPERIMENTED WEIGHTING  Pinus patula (13) . (14) (15) Vo VI Vo Vo VI 0.000032406D H Cupressus lusitanica (13) Vo (U) (15) 0.0000305786D2H + 0.00021052H + 0.0000756269D2 VI = O.O000285061D2H + 0.000134333H + 0.0000624554D2 Vo = 0.00003073946D2H - n nnnm'--'--'-^ VI Vo 0.000014261026D2 + 0.00013143623DH 0.000028793327D2H- 0.000015092939D2 + 0.00010640115DH 0.0000307881D2H + 0.00163759D + 0.0000851862H2 VI = 0.0000286625D2H + 0.00132956D + 0.0000711725H2 0.004123311ogD2H 0.003452061ogD2H SEE and R FOR WEIGHTED MODELS „2 0.0000389084D2H + 0.00084093H - 0.00037755D2 0.0000373313D2H + 0.000949151H- 0.000131663D2 0.000034343803D2H - 0.00009824331D2 + 0.00023027317DH VI = 0.000032331001D2H - 0.00015632482D2 + 0.00020483114DH 0.00827566H + 0.0000594097D2 + 0.00034446H2 + 0.02484871ogH2 0.0000317707D2H - 0.00828071H+ 0.0000315392D2 + 0.000312889H2+ 0.02566081ogH2 SEE 0.000003950 0.6290 0.000003733 0.5555 0.000003967 0.6259 0.000004390 0.1889 0.000003544 0.7267 0.000003734 0.4238 True SEE 0.177801 0.152913 0.136718 0.133337 0.144584 0.136978 0. .000003753 0.5684 0. ,350545 0. 000003521 0.4726 0. 311810 0. 000003493 0.6262 0. 293679 0. 000003333 0.5273 0. 269395 0. 000003486 0.6300 0. 285247 0. 000003325 0.5327 0. 259915 - 33 -where: Y^ were the actual observation volume A Y^ were their predicted volume n number of observation p number of regression parameters (or c o e f f i c i e n t s ) i n the model. . By th i s c r i t e r i a , i t was only possible to observe, i n general terms, which models might be better than others. I t was not possible to have a complete ranking of the models. Further tests were carried out to f i n d out how w e l l each model was predicting the volumes of the trees i n each diameter c l a s s . The s t a t i s t i c s used here were the mean bias per tree and the bias per cubic metre. Mean class bias was computed as: n k 1 = 1  n k where: k i s the diameter class. The bias per cubic metre per class was computed as: n k i = 1 k k nk i = 1 - 34 -Tables VI and VII give mean class bias and the bias per cubic metre for the models 1, 2, 3, 4, 13, 14 and 15 fit t e d on P. patula and overbark and VI" refers to volume underbark (or insidebark). The f i r s t row under each category i s the mean bias per tree and the second row is the bias per cubic metre. Note that a negative bias means that the model overestimates the volume and a positive bias means that the model i s underestimating the volume. Under model (1) '*' refers to the bias after Meyer's Bias,Correction factor had been applied. The factor was computed as: lusitanica data respectively. Under each model "VO" refers to volume f = 101.1513 (S 2) where: f = correction factor S is the Variance or Mean Squares Residual of the logarithmic model (in log volume form) For the equations f i t t e d the factor had the following values: Volume overbark Volume insidebark Pinus patula 1.005349 1.009315 Cupressus lusitanica 1.003958 1.004323 Table VI. E f f i c i e n c y o f Various Volume Models i D E s t i m a t i n g P. p a t u l a Volumes. DBH C l a s s i 2 3 4 5 6 7 8 Model No. (1) Vo 0.001 -0 .005 -0 .016 1 0.02! i 0 .032 ! 0 .076 > 0.128 -0 .146 0.069 .-0.080 -0 .04C 1 0.029 1 0 .022 ! 0 .035 > 0.043 -0 .049 *Vo 0.001 -0 .006 -0 .018 0.021 . 0 .024 • 0 .064 . 0.113 -0 .163 0.064 -0 .086 -0 .046 0.024 • 0 .017 ' 0 .030 • 0.038 -0 .055 VI 0.001 -0 .006 -0 .010 0.018 0 .037 0 .069 0.151 -0 .030 0.108 -0 .127 -0 .028 0.024 0 .030 0 .036 0.056 -0 .011 *VI ' 0.001 -0 .006 -0.013 0.011 0.026 0 .052 0.127 -0 .056 0.100 -0 .138 -0 .039 0.015 0 .021 0 .027 0.047 -0 .020 (2) Vo -0.297 -0 .286 -0, .202 -0.075 -0, .003 0 .090 0.139 -0 .229 •25.085 -4, .349 -0. .519 -0.086 -0. .002 0 .042 0.046 -0 .077 VI -0.233 -0. .234 -0. 168 -0.072 -0. ,000 0 .069 0.134 -0 .170 •28.115 -4. .985 -0. 498 -0.095 -0. 000 0, .036 0.050 -0 .063 (3) Vo 0.296 0. 178 -0. 098 -0.082 -0. 036 0. .062 0.156 -0, .100 25.012 2. 714 -0. 247 -0.094 -0.026 0. .029 0.051 -0. .034 VI 0.244 0. 140 -0. 084 -0.078 -0. 027 0.047 0.148 -0. .066 29.437 2. 983 -0. 249 -0.103 -0. 022 - 0. 024 0.055 -0. ,024 (4) VI 0.344 0. 244 0. 025 -0.004 -0. 050 0. 011 0.164 -0. 034 41.490 5. 212 0. 073 -0.058 -0. 041 0. 006 0.060 -0. 012 Vo 0.366 0. 268 0. 023 -0.041 -0. 061 0. 020 0.169 -0. 049 30.936 4. 079 0. 059 -0.047 -0. 044 0. 010 0.056 -0. 017 (13)Vo 0.000 -0. 000 0. 012 0.048 0. 006 -0. 069 -0.214 -0. 652 0.005 -0. 004 0. 030 0.055 0. 005 -0. 032 -0.072 -0. 219 VI 0.000 -0. 002 0. 007 0.033 0. 021 -0. 035 -0.107 -0. 425 0.037 -0. 052 0. 021 0.044 0. 017 -0. 018 -0.040 -0. 156 (14)Vo 0.000 -0. 004 -0. 007 0.040 0. 035 0. 034 -0.001 . -0. 349 0.014 -0. 058 -0. 017 0.046 0. 025 0. 016 -0.000 -0. 117 VI 0.000 -0. 007 -O . i 008 0.029 OJ 050 0. 061 0.086- -0. 146 0.053 -0. 144 0 .1 324 0.034 0.1 341 0. 032 0.032 -0.054 (15)Vo 0.000 O J 301 -0.1 310 0.028 0.1 317 0. 008 -0.051 -0. 449 0.005 0.1 317 -0.1 326 0.032 0.1 312 0. 004 -0.017 -0. 151 VI 0.000 -0.( 300 -0.( 308 0.016 0.1 322 O . i 011 -0.002 -0. 299 0.035 -0.( 308 -0.( 324 0.022 0.1 318 0 .1 006 -0.001 -0.. 110 Note: - 1st row i s average bias ;: 2nd row i s bias per m3 * Meyer's bias correction factor applied on the estimates - N a t i v e bias means o v e r e s t i m a t e while positive means und Overall. 0.019 0.019 0.014 0.014 0.023 0.025 0.015 0.015 -0.114 -0.114 -0.102 -0.102 0.042 0.042 I 0.036 0.036 0.084 1 0.084 0.082 0.082 0.026 0.026 -0.013 -0.013 0.007 0.007 0.022 0.022 0.007 0.007 0.000 0.000 stimation DBH C l a s s Model No. T a b l e V I I . E f f i c i e n c y o f Va r i ou s Volume Models i n E s t i m a t i n g C. l u s i t a n i c a Volumes. 2 3 V 5 6 7 8 9 10 11 12 (1) Vo -0.000 -0-.001 0.001 -0 .012 0.023 0. .051 0.031 -0.063 0 .191 -0:108 -0 .259 -0 .159 -0.009 -0.011 0.005 -0 .020 0.019 0, .026 . 0.011 -0.019 0 .045 -0.021 -0 .042 -0 .021 VI • -0.000 -0.001 0.005 -0, .006 0.017 0. .043 0.014 -0.085 0, .179 -0.110 -0 .191 -0. .203 -0.014 -0.021 0.021 -0, .010 0.016 0. ,024 0.005 -0.027 0. .045 -0.023 -0, .033 -0, .029 *Vo -0.000 -0.000 0.005 -0. .003 0.042 • 0. 083 0.081 -0.000 0. .269 -0.128 -0, .284 -0. .188 -0.002 -0.001 0.018 -0. ,005 0.034 °-043 0.029 -0.000 0. ,063 -0.025 -0. .046 -0. .025 *VI -0.000 -0.001 0.004 -0. ,008 0.012 0. 035 0.003 -0.098 0. 162 -0.131 -0. ,217 -0. ,234 -0.019 -0.025 -0.017 -0. 014 0.011 0. 020 0.001 -0.032 0. 041 -0.028 -0. 037 -0. 033 (2) Vo -0.458 -0.444 -0.385 -0. 313 -0.161 -0. 013 0.062 0.015 0. 275 0.043 -0. 158 -0. 023 -31.146 -7.807 -1.517 -0. 507 -0.132 -0. 007 0.021 0.004 0. 064 0.008 -0. 026 -0. 003 0.000 0.000 0.006 0.006 0.013 0.013 -0.002 -0.002 -0.099 -0.099 VI -0.382 -0 .371 -0 .318 -0.255 -0.133 -0.005 -0 .049 -0.009 0 .260 0 .032 -0 .090 -0 .068 -0 .089 -29.470 -7 .332 -1 .358 -0 .448 -0.120 -0.003 0 .019 -0 .003 0 .065 0 .007 -0 .016 -0 .010 -0 .089 (3) VI 0.405 0 .299 0 .068 -0 .061 -0.094 -0.082 0 .025 -0 .038 0 .205 0 .050 -0 .148 0 .028 0 .038 31.321 5 .919 0 .292 -0 .107 -0.085 -0.046 0 .010 -0 .012 0 .052 0 .011 -0 .025 0 .004 0 .038 Vo 0.497 0.372 0 .090 -0 .064 -0.109 -0.047 0 .024 -0 .026 0 .226 ' 0 .061 -0 .194 0 .090 0.047 33.844 6 .536 0 .354 -0 .104 -0.090 -0.024 0 .009 -0 .008 0 .053 0 .012 -0 .031 0 .012 0 .047 (4) Vo 0.256 0 .204 0 .082 -0.035 -0.043 . 0.001 0 .014 -0 .060 0 .179 0 .017 -0 .170 0 .152 0 .028 17.459 3 .594 0.324 -0 .057 -,0.035 0.001 0 .005 -0 .018 0 .042 0 .003 -0. .028 0, .020 0 .028 VI 0.210 0 .163 0, .061 -0 .038 -0.041 0.013 0. .314 -0. .066 0, .167 0 .015 -0. .129 0, .078 0, .024 16.224 3. .228 0, .263 -0 .067 -0.037 0.007 0, .006 -0, .021 0, .042 • 0, .003 -0, .022 0. .012 0, .024 (13)Vo -0.0000 0, .002 0. ,017 0. .011 0.056 0.067 -0. ,052 -0. .245 -0. .160 -0. .622 -1. .154 -1. .320 0. .053 -0.027 0, .033 0. ,069 0. .018 0.047 0.034 -0. .018 -0. .073 -0. .037 -0. .122 -0. ,187 -0. ,173 0. ,053 VI -0.001 0. .000 0. ,014 0. ,010 0.036 0.048 -0. ,036 -0. ,193 -0. .024 -0. ,426 -0. .956 -1. 194 -0. .051 -0.024 0. ,019 0. 076 0. 023 0.039 0.031 -0. 021 -0. 077 -0. 030 -0. .116 -0. ,165 -0. 168 -0. ,051 (14)Vo -0.001 0. 001 0. 012 0. 008 0.047 0.061 -0. 024 -0.186 -0. 041 -0. ,472 -0. 885 -1. 004 -0. 036 -0.056 0. 015 0. 049 0. 012 0.039 0.031 -0. 009 -0. 055 -0. 009 -0. 093 • -.0. 144 -0. 131 -0. 036 VI -0.001 0. 002 0. 014 0. 010 0.036 0.048 -0. 036 -0. 193 -0. 024 -0. 426 -o: 733 -0. 934 -0. 037 -0.056 0. 004 0. 059 0. 018 .0.032 0.027 -0. 014 . -0. 062 -0. 006 -0. 090 -0. 126 -0. 131 -0. 037 (15)Vo -0.000 -0. 001 0. 009 0. 006 0.051 0.073 -0. 001 -0. 094 -0. 012 -0. 435 -0. 853 -0. 963 -0. 027 -0.025 -0. 010 0. 035 0. 010 0.042 0.037 -0. 000 -0. 028 -0. 003 -0. 085 -0. 139 -0. 125 -0. 027 VI -0.000 -0. 001 0. 011 0. 009 0.063 0.060 -0. 014 -0. 166 0. 004 -0. 388 -0. 700 -0. 591 -0. 031 -0.024 -0. 022 0. 045 0. 016 0.036 0.033 -0. 005 -0. 054 0. 001 -0. 082 -0. 120 -0. 125 -0. 031 Note: - 1st row i s average b i a s ; 2nd row i s b i a s per n 3 - * Heyer's b i a s c o r r e c t i o n f a c t o r a p p l i e d on the e s t i m a t e s - Negative b i a s means o v e r e s t i m a t e w h i l e p o s i t i v e b i a s means u n d e r e s t i m a t i o n - 37 -Because the ultimate performance of a model is judged on i t s volume predictive capability, the bias-per-cubic-metre c r i t e r i a was adopted as the standard for judging a l l the models. The st a t i s t i c indicates how much bias can be expected i f a model i s used to estimate one cubic metre of timber. In other words, i t gives the impression on how large the bias would be when the model's volume estimate i s 1 cubic metre. Thus i t gives a relative measure of bias for a l l models and a l l tree sizes. According to this s t a t i s t i c there are only 4 models which are giving good estimates in a l l diameter classes. These are models 1, 13 14 and 15. Their biases are not totally identical in a l l diameter classes. One or two of them may give slightly better estimates in some diameter classes. The differences are actually so small that in practice, no differences may be observed. As such i t becomes d i f f i c u l t to decide which model should be adopted for which "species. The differences being so small any personal bias in adopting any one of the models, should have no practical significance .For me, however, model (1) seemed slightly better for both insidebark and outsidebark volume estimation in C_. lusitanica. Models 14 and 15 also seemed slightly better for estimating the volumes of P_. patula trees - model 14 for overbark volumes and model 15 for underbark volumes. The application of Meyer's correction factor on model 1 seemed to increase the accuracy on the model just slightly. Because the actual improvement is very small the importance of applying the factor was not noticed. - 38 -I t can be seen from the selected models that they are predicting the volumes to within 10% of the actual volume. The only outstanding exception i s that of trees i n class 8 for P_. patula. This can be explained by the fact that the number of samples i n that class was too small (only 3). As such they may not-be giving the true picture on the bias of the estimates at that diameter class. To allow for the computation of the confidence l i m i t s of the estimates at each diameter c l a s s , the standard error of estimates were computed for each of the adopted models. These are given i n Table V I I I . In the table rows 1 and 2 have the same meanings as i n Tables VI and VII. I t i s the t h i r d row that refers to the standard error of estimate. To summarize, model 1 was adopted for estimating the volumes (both inside and outside bark) for,C_. l u s i t a n i c a . Models 14 and 15 were adopted for estimating the outside- and i n s i d e - volumes, respectively of P_. patula trees. As . s u c h a l l comparison with the models produced by other techniques i n the next sections were based on these models as representation of volume models produced by t r a d i t i o n a l approaches. Appendices 2(a), 2(b), 2(c) and 2(d) show t y p i c a l plots of the residuals of the volume estimates for models 1, 2, 3 and 4, and 13 to 15, respectively. Although they were a l l plotted from inside bark volume estimates for P_. patula they were also considered f a i r l y good representative plots not only on the outside bark estimate for P_. patula but also for both inside- and outside- bark estimates i n C.. l u s i t a n i c a . V i s u a l l y the plots were sim i l a r i n both species. However, the true precision can only be judged on the width of the confidence l i m i t s of any volume estimates made. The standard Table VIII. DBH Class Model No. Pinus patula (14)Vo (15)VI 0.000 0.014 0.001 0.000 0.035 0.001 -0.004 -0.058 0.007 -0.000 -0.008 0.010 Cupressus lusitanica (1) Vo VI -0.000 -0.001 -0.009 -0.011 0.002 0.004 -0.000 -0.001 -0.014 -0.021 0.002 0.005 -0.007 -0.017 0.045 -0.008 -0.024 0.038 0.040 0.046 0.101 0.016 0.022 0.099 0.035 0.025 0.157 0.022 0.018 0.157 0.034 -0.001 0.016 -0.000 0.200 « 0.011 -0.002 0.006 -0.001 0.227 -0.349 -0.117 -0.299 -0.110 0.001 -0.012 0.005 -0.020 0.025 0.084 0.005 -0.006 0.021 -0.010 0.024 0.076 0.023 0.019 0.105 0.017 0.016 0.094 0.051 0.026 0.222 0.043 0.024 0.222 0.031 -0.063 0.011 -0.019 0.237 0.287 0.191 -0.108 -0.259 -0 159 0.045 -0.021 -0.042 -0.021 0.461 ._• 0 > 6 3 1  0.014 0.005 0,231 -0.085 -0.027 0.278 0.179 0.045 0.439 -0.110 -0.023 -0.191 -0.203 -0.033 -0.029 0.577 N.B. The first row; refers to bias per tree 2nd row refers to bias per cubic metre NldgativerebfirS £° St3ndard ; Err°r °f Negative bias means overestimate while positive M , ' mxe positive bias means underestimation Overall 0.007 0.007 0.137 0.000 0.000 0.137 0.000 0.000 0.204 0.006 0.006 0.196 o - AO -errors of estimates given in Table VIII (by diameter classes) and in Table III and V (last columns) would be very useful to that end. 4.2 Taper Models Derived From the Best of the Volume Models The principle by which taper models are derived from volume equations are f u l l y discussed in Demaerschalk (1972a, 1972b, 1973a, 1973b) and Munro and Demaerschalk (1974). In short, i t i s based on the need to produce taper models which when integrated for total volume w i l l give exactly the same volume estimates as an existing volume equation. The volume estimates from such a model w i l l , therefore, be compatible with the volume estimates from an existing volume model. The advantage of this new approach is that whereas the existing volume model is only useful for predicting total volumes the volume-based taper model can be integrated to any desired upper diameter or height to give volume estimates to such diameter or height. Thus volume estimates to the different u t i l i z a t i o n limits can easily be obtained from such a model. Since taper studies are always based on inside bark diameters, only inside bark volume models have been used in deriving volume-based taper models. As such volume-based taper models derived from the models selected as the best for inside bark volume estimations for P_. patula and Cypress can theoretically be given as: - 41 -Plnus patula VI - b ^ H + b 2H + b 3 D 2 + b 4 H 2 + b 5 l o g H 2 H / K d 2 dL = VI 0 where, K = ^ =0.00007854 4 x 10000 d= diameter at any point L = metres from the top of the tree d 2=pb 1D 2H 1-P LP- 1 + q ^ H 1 " ^ " 1 + r b 3 D 2 H - r L r - 1 + s b 4 H 2 S L S 1 + t b 5 l o g H 2 H " t L t - 1 ( 1 6) where: b.^ b ^ b 3 , b^, b 5 are regression c o e f f i c i e n t s from volume equation. p>q» r»s,t are free parameter values. - 42 -Note that t h i s i s the form of the model before integrating i t for volume, I t has been conditioned such that when integrated for t o t a l volume a l l the respective L's with their corresponding powers disappear from the model thus leaving the o r i g i n a l variables only. The diameter estimates at the various L distances from the top of the tree are obtained as the square root of the equation. Cupressus l u s i t a n i c a Log VI = b Q + bjlogD + b 2logH V b l b2 VI = 10 T) H H / 2 kd dL = VI 2 b0 b l V n n _ 1 A = nlO UD XH Z L (17) where: b Q, b ^ b 2 are regression c o e f f i c i e n t s from volume equation n = f r e e parameter value - 43 -Taper data i s needed to f i t these equations. The founder of the technique suggested that the model could be fi t t e d using the non-linear regression programs to get the powers of "L" which would best predict the different diameter changes along the tree profile. For simpler models, however the powers, (or as they were called, "free parameters"), can be obtained by iterating the different values u n t i l a value is obtained that would minimize the standard error of estimate of the diameter. An attempt to f i t the above models using a non-linear regression program was met with very frustrating results. The values of the free parameters obtained were very small. This would result in tree profiles which are almost parallel from the tip to the base. It was therefore decided to use the other approach where the parameters have to be guessed. Because not more than one parameter could be guessed at the same time i t appeared unrealistic to try to f i t model 16. The only way this model could be produced was by assigning the same free parameter value to a l l the terms in the model. Although this approach was attempted, I was s t i l l suspicious of i t s results. As such model (17) was f i t t e d on both Cypress and Pine data. Because running such a program is f a i r l y expensive the process was carried out in two stages. In the f i r s t stage, three trees were randomly selected from data. Different parameter values were assigned in the model to get the parameter which would give diameter estimates, for the three trees, reasonably close to the observed. Parameter values of 2, 3, 4 and 5 were used. The second stage involved starting from the parameter that seemed most reasonable and increasing and decreasing i t s value by steps - 44 -of 0.05, 20 times both i n the positive and i n the negative directions^ Each time the value of the parameter was increased or decreased, the corresponding standard error of estimate and t o t a l bias of estimate was recorded together with the value of the parameter. From this the parameter value that gave the smallest standard error of estimate was selected. The f o r t r a n programs for this i t e r a t i o n are given i n Appendix 3(a) and (b). I t was i n i t i a l l y intended to f i n d the parameter that gave the smallest bias. This value corresponded well to the value at which the standard error of estimate was also least. Figure 2(a) shows how the standard error of estimate changes with changes i n the values of the "free parameter". Figure 2(b) shows the average bias change with changes i n the "free parameter". Figure 2(c) shows the changes i n the average bias corresponding to the d i f f e r e n t changes i n the standard error of estimate. These plots were for Cypress. Although for Pine the shapes were sim i l a r they were more shifted downwards ind i c a t i n g that the best parameter values are obtained at the point where the actual diameter i s a b i t overestimated - compared to the i l l u s t r a t e d case where the reverse i s the case. Figure 2(c) shows that as the standard error of estimate goes down the bias likewise decreases. However, because the standard error of estimate w i l l only go down to the. point at optimum parameter value i t w i l l begin to increase again, as the optimal point i s passed. The bias, however, continues to drop (even to the negative side) u n t i l the largest single observation or group of observations have the smallest bias magnitude ( p a r t i c u l a r l y near the ground l e v e l ) . The 8.000 V 6.000 2.000 « « • » •* •> ** •» 4.000 . . - . . * • • • **»« « 0.41667 1.2500 » n o n " • » „ ? , . » * • • • • ° - " 3 " » • " « 2 . " o o - 2 - ' 1 6 7 3 - 7 5 0 0 4 l 6 6 7 * • » > » Value of Free Parameter " 5- 0 0 a° Figure 2(a). Changes in the values of the-standard error of estimate as the free parameter values change a 0 . 9 iCO » « ** » • ** • •« • « •» 6 • • »• » • : . **• <3 0.4200 . *** M . *. . **• a • o . E t8 -H « . -a . 0 > -0.6600 - 1 . 2 0 0 0 . 4 1 6 6 7 . , ; n » • • . . °«0 o B v » v » 1-2500 2.0833 3 * ° - 8 3 3 " » • « « 2.5000 2 , 9 1 6 7 3 „ „ 3 ' 7 « 0 4.S8H Value of Free Parameter ** 1 6 6 7 5- a 3 0 a Figure 2(b).. Changes in average diameter bias a, rh. f 8 lameter bias as the free parameter values change. I. 500 • 4 * . * * * * ' 0.9630 . * • • « ** • . 2 c ' 2 * ° 3 , ^ « . • RS • • . *2 , •H . * • * W . *»** p 0.42C0 . *• * o • •* * u . * o e •H Q * a o o . • a a -o.iioo . ^ . . -0.6603 -1.230 0-83333 2.5000 *.1667 5.8333 7.5000 9.1667 ° - ° 1.6667 3.3333 „. , ,5^)000- r _ . 6.6667 . 8.3333 13.000 Standard Error of Estimate(cm). Figure 2(c). Changes in average diameter bias corresponding t& changes in standard error of estimate. - 48 -"free parameter" values obtained from the logarithmic volume models were respectively 3.05 and 2.10 for Cupressus lusitanica and Pinus patula. By assuming the same parameter value for the model originally accepted as the best for estimating the volume of Pinus patula trees the corresponding parameter value was 2.05. The following models were, therefore, obtained as the volume-based taper models: Cypress 9 b b. b -p p-1 d 2 = pl0_°D LH 2 L (18. k SEE = 4.0611 where: p = 3.05 b Q =-4.283463 Pine b^ = 1.8066306 | These values were obtained from the volume equation. b 2 = 1.0601857 L = i s the distance from the top of the tree k = TT 4 x"l0000 9 b n b l b 9 _ P P"1 d 2 = plO °D XH 2 L (19) SEE - 1.6778 - 49 -where: p = 2.10 b Q = -4.4797125 b 1 = 1.7566386 b 2 = 1.25-6107 4 x 10000 In p ractice, however, the d i f f e r e n t u t i l i z a t i o n l i m i t s are usually specified i n terms of some s p e c i f i c upper diameters. To get the corresponding distances from the top of the tree the model 18 (or 19) i s transformed to the form: t> 0 b -fa -p l / ( p - l ) ,..„»/(P "Y° ».• The diameter estimates for pine can also be given by a transformed form of model (16) thus: d 2 - p b . D V - P l P " 1 .+ p b . H 1 - ^ - 1 + pb D V P I T 1 + *• _3_ s pb 4H 2 P L P 1 + pb 5logH 2H P L P 1 (20) SEE = 1.6652 - 50 -where: p = 2.05 b1 = 0.0000317707 t>2 = 0.00828071 t>3 = -0.0000315392 J From the volume equation t>4 = 0.000312889 b 5 = 0.0256608 k = * 4 x 10000 L = distance from the top, 4.2.1 Testing Volume-based Models A l l tests concerning the accuracy of a model w i l l be based on bias. The standard error of estimate is also used to give an indication of the precision of such estimates. 4.2.1.1 Testing the Models for Diameter Estimat es To test the diameter estimation of the models, the data were divided into several diameter classes (similar to those used in Section 4.1.2). Because each tree had 15 measurements along the stem profile from the base to the t i p , and because these measurements were at the same relative points, the estimates at the different relative pointswere also tested separately for trees in each diameter class. - 51 -Also because trees vary a lot in heights i t was also decided to show how well the models would perform on trees of different total heights. Tables IX, X and XI show the performance.of models 18, 19 and 20 respectively. In each table, and at each measurement point, the f i r s t row shows the bias per tree (in cm). Second row shows the bias per metre (measure of effect of total height of trees), and third row i s the standard error of estimate at that point. From the three tables i t can be noticed that there does not seem to be any pattern of bias with respect to total height (2nd row) This result was based on the assumption that the tree heights increase with increase in tree diameters. In practice, however, i t is known that above some diameter size, trees can attain any heights (e.g. trees taller than 18 metres can belong to any of the larger diameter classes). A more r e a l i s t i c analysis on effect of tree height i s discussed in Chapter 5. The other two rows (i.e. row 1 and 3) indicate that the standard error of estimate (3rd row) generally increase as the tree size increase. Generally, the worst estimates are at the ground level, and on trees larger than class 5 (for most measurement points). Models 19 and 20 seem to give f a i r l y similar estimates although model 20 gives a l i t t l e better estimates. The systematic bias observed in the three tables is more marked in the Cypress (model 18). Table IX. fj. l u s i t a n i c a : E f f i c i e n c y of Volume-based Taper Model 18 for Diameter Estimation. DBH Class 1 0.951 0.0 0.14? 1.492 -0.190 0.3 -0.C28 0.926 -0.496 0.6 -0.074 0.922 -0.557 0.9 -C.083 0.906 -0.312 1.3 -0.046 .0.551 -0.095 I -0.014 0.494 0.001 II 0.003 0.613 0.205 III 0.030 ' C.606 C.160 IV 0.024 0.497 0.231 V 0.034 0.657 0.171 VI 0.025 0.555 C.152 VII C.023 0.638 C.218 VIII 0.032 0.510 0.093 IX 0.014 0.328 -0.011 X -0.002 0.014 2 3 1.252 0.132 1.654 3.132 0.204 4.288 -0.776 -0.082 1. 158 -0.901 -0.059 1.793 -1.197 -0.125 1.478 -1.554 - 0 . 101 2.380 -1.197 -0.126 1.486 -1.968 -0.128 2.349 -0.951 - 0 . 100 1.281 -1.878 -0.123 2.222 -0.374 -0.039 0.707 -0.804 -0.052 1.145 0.025 0.003 0.532 0.530 0.035 0.948 0. 30S 0.032 0.653 1.211 0.079 1.726 0.490 0. 052 1. 128 1.731 0.113 2.277 0.356 0. 038 1. 141 2.010 0.131 2.518 0.400 0.042 1.044 1.898 0.124 2.465 0.300 0.03? 0.839 1.253 0.C82 1.838 0. 267 0. 028 0.613 0.551 0.036 1.011 -0.046 -0.005 0.387 0.116 0.008 0.616 -0.013 -0.001 0.014 -0.C12 -0.001 0.015 4 5 4.833 9.545 0.233 0.348 6.646 12.234 0.342 1.027 0.016 0.037 2.047 2.456 -1.589 -1.342 -0.077 -0.049 2.319 2.077 -2.320 -2.367 -0.112. -0.086 2.852 3.0C7 -2.623 -3.521 -0.126 -0.128 3.120 4.212 -2.277 -3.985 -0.110 -0.145 2.999 4.871 -0.825 -1.746 -0.040 -0.064 1.698 2.396 -0.036 0.138 -0.002 0.005 1.866 1 .444 1.361 2.102 0.066 0.077 2. 502 2.874 1.846 3.042 0.089 0.111 3.517 3.894 1.623 3.991 0.078 0.145 3.399 5.133 1.153 3.788 0.056 0.138 2.938 5.042 0.559 3.016 0.027 0.1 10 2.212 - 4.448 0.046 1.3 39 0.002 0.049 1.117 2.343 -0.013 -0.012 -0.001 -0.000 0.015 0.014 6 7 10.372 12.165 0.333 0.384 13.269 16.419 2.802 3.596 0.090 0.113 3.849 10.118 -0.275 0.803 -0.009 0.025 1.979 2.250 - 2 . 165 -1.561 -0.069 -0.049 2.817 2.200 -3.765 ' -3.413 -0.121 -0.108 4.442 4.014 -4.902 -5.147 -0.157 -0.162 6.126 6.309 -2.471 -3.318 -0.079 -0.105 3.942 4.338 0.085 -0.685 0.003 -0.022 2.637 2.254 1.532 1.767 0.049 0.056 3.314 2.997 3.513 3.730 0.113 0.118 4.888 4.922 4.624 4.792 0.148 0.151 6. 081 5.983 3.626 4.200 0.116 0.132 5.354 5.716 2.350 2.082 0.075 0.066 3.384 3.216 0.603 0.894 0.019 0.028 1.527 2.180 -0.013 -0.015 -0.00.0 -0.000 0.015 0.017 8 9 15.299 . 18.188 0.477 0.577 19.488 34.804 5.070 6.638 0.158 0.211 11.658 14.652 -0.239 2.037 -0.007 0.065 8.407 6.205 -1.736 -1.147 -0.054 -0.036 3.749 3.176 -3.204 -2.570 -0.100 -0.082 3.847 4.5 18 -6.288 -4.775 -0.196 -0.151 8.330 8.775 -3.862 -3.132 -0.120 -0.099 5.299 6.699 -1.429 0.890 -0.045 0.028 2.896 4.011 1 .229 3.420 0.038 0. 109 2.566 6.607 2.988 5.073 0.093 0.161 4.113. 9.736 4.431 5.241 0.138 0.166 6.177 10.308 3.542 3.667 0.110 0.116 5.051 7.562 1.430 2.953 0.045 0.094 2.917 7.753 -0.133 0.537 -0.004 0.017 1.139 3.373 -0.016 -0.018 -0.001 -0.001 0.019 0.032 10 11 12.362 49.034 0.3?7 1.662 4 .203 17.258 0.111 0.585 1.545 6.182 0.041 0.210 -0.714 3.605 -0.019 0.122 -2.860 0.536 -0.076 0.018 -9.274 1.003 -0.245 0.034 -9.709 0.346 -0.257 0.012 - 4 . 5 6 9 4.762 -0.121 0.161 -4.157 -3.154 -0.110 -0.107 1.223 -7.305 0.032 -0.248 -0.635 -8.099 -0.017 -0.275 0.559 -4.146 0.015 -0.141 -0.412 2.735 -0.011 0.093 1.119 3.707 0.030 0.126 -0.017 -0.025 - 0 . 0 0 0 -0.001 By rows! 1-bias/tree; 2-Mas/metre ( t o t a l height e f f e c t ) ; 3"standard error of estimate. - 53 -Table X. P. patula: Efficiency of Volume-Based Taper Model 19 for Diameter Estimation. By rows: l=bias/tree; 2=bias/metre (total height effect); 3=standard error of estimate. BH Class t 2 3 4 5 6 7 0.0 2 . 2 3 6 . 0 . 4 2 3 2 . 0 04 1 . 5 8 7 0 . 160 2. 160 1 . 1 6 3 0.04(1 2. 176 0 . 009 0 . 0 0 3 1 .613 0 . 9 7 2 0 . 0 3 2 1 .7 80 2 . 5 1 1 0 . 0 0 2 3 . 5 7 7 3.6 04 O.I 14 0.3 l.l<?0 0 . 2 2 6 1.492 0 . 8 1 4 0 . 006 1. 302 0 . 6 6 5 0 . 0 2 7 1 . 9 6 4 - 0 . C 3 1 - 0 . 0 3 0 1 . 9 5 7 0 . 4 6 4 0 . 0 1 5 1 .3 50 1 .675 0 . 0 5 4 2 . 0 6 9 3 . 5 4 3 0 . 1 1 2 0.6 0 . 8 7 9 0 . 166 1 .134 0 . 3 4 4 0 . 0 3 6 o.aoo 0 . 0 2 1 0 . 0 0 1 1 . 5 7 4 - 1 . 1 7 7 - 0 . 0 4 3 2 . 1 G 9 - 0 . 1 6 7 - O . 0 C 5 1 . 2 C 3 1.290. 0 . 0 4 2 2 . 5 6 0 2 . 7 3 3 0 . 0 0 6 0.9 0 . 6 5 9 0 . 1 2 4 0 . 9 2 4 - 0 . 0 4 B - 0 . 0 0 5 0 . 6 2 9 - 0 . 5 3 4 . - 0 . 0 2 2 1 . 1 3 5 - 1 . 4 4 9 - 0 . 0 5 2 2 . 3 9 7 • - 0 . 7 4 4 - 0 . 0 2 4 1 . 3 0 8 0 . 5 5 0 0 . 01n 1. H 10 1 .724 0 . 0 5 4 1.3 0 . 3 3 7 0 . 0 6 4 C.457 - 0 . 4 3 4 - 0 . 0 4 6 0 . 6 7 7 - 1 . 1 9 5 - 0 . 0 4 9 ' 1 . 5 0 2 - 1 . 7 1 3 - 0 . 0 6 2 2 . 3 8 5 - 1 . 3 3 6 - 0 . 0 4 4 1 . 6 9 5 - 0 . 6 7 3 - 0 . 0 2 2 1.44 5 0 . 9 9 7 0 . 0 3 1 . I 0 . 1 4 1 0 . 0 2 7 0 . 481 - 0 . 7 P 2 - 0 . 0 8 3 1 .017 - 0 . 9 1 2 - 0 . 0 3 7 . 1 . 3 4 0 - 1 . 1 5 8 - 0 . 0 4 2 1 .895 - 0 . 6 9 7 - 0 . 0 2 3 1 . 7 2 5 - 1 . 0 9 4 - 0 . 0 3 5 1 . 9 6 4 0 . 2 1 7 0 . 0 0 7 II 0 . 0 1 0 0 . 0 0 2 0 . 4 1 7 - 0 . 8 4 9 - 0 . 0 9 0 1 . 1 2 6 - 0 . 4 9 1 - 0 . 0 2 0 1 . 1 5 4 - 0 . 1 2 1 - 0 . 0 0 4 1 . 2 1 5 - 0 . 1 4 2 - 0 . 0 0 5 1 . 7 4 3 - 0 . 3 3 0 - 0 . 0 1 1 1 . 7 3 7 1 . 5 6 6 0 . 0 4 9 III - 0 . 2 0 3 - 0 . 0 3 8 0 . 3 9 7 - 0 . 9 1 8 - 0 . 0 9 ? 1 . 2 5 6 - 0 . 5 0 S - 0 . 0 2 1 1 . 5 4 0 0 . 3 1 2 0 . 0 1 1 1 . 4 4 9 0 . 6 C 6 0 . 0 2 0 1 . 9 7 3 0 . 3 6 2 0 . 0 1 2 1 . 9 0 L 2 . 4 6 9 0 . 0 7 8 IV - 0 . 3 2 3 - 0 . 0 6 1 0 . 5 5 ? - 1 . 2 3 7 - 0 . 1 3 1 1 - 5 7 7 0 . 2 3 9 0 . 0 1 0 1. C 8 0 0 . 7 4 6 0 . 02-7 1 . 8 2 0 1 . 0 C 7 0 . 0 3 3 2 . 4 2 3 0 . 6 « 3 0 . 0 2 2 2 . 0 2 0 I.R 59 0 . 0 5 9 . V - 0 . 3 5 ? - 0 . 0 6 7 0 . 5 4 6 - 1 . 4 3 1 - 0 . 1 5 i 1 . 7 5 6 0 . 2 1 1 0 . C C 9 1 . 0 2 4 1 . 1 4 8 0 . 0 4 1 2 . 3 5 3 1.6 59 0 . 0 54 2 . 7 1 3 1 . 6 1 7 0 . 0 5 2 . 2 . 9 7 0 2 . 5 8 3 0 . 0 0 1 VI - 0 . 5 5 3 -O. 105 0 . 7 2 2 - 1 . 4 7 1 - 0 . 155 1 . 8 8 4 0 . 3 8 8 0 . C 1 6 1 . 4 3 5 1.481 0 . 0 5 4 2 . 3 4 4 2 . 0 6 7 0 . 0 6 0 2 . 8 4 5 1 . 4 2 8 0 . 0 4 6 2 . 3 7 0 4 . 4 1 3 0 . 1 3 9 VII' - 0 ^ 5 4 1 - 0 . 1 0 2 C.702 - 1 . 6 3 6 - 0 . 1 7 3 2 . 0 0 9 C .210 0 . C 0 9 1 . 6 2 7 1 . 8 5 4 0 . 0 6 7 3.11.5 1 . 0 6 4 0 . 0 6 1 3 . 0 2 7 1.241 0 . 0 4 0 2 . 9 0 9 3 . 2 7 0 0 . 1 0 3 VIII - 0 . 4 5 3 - 0 . 0 U 6 0 . 6 2 5 - 1 . 5 7 1 - 0 . 1 6 6 1 . 9 5 3 - 0 . 2 5 0 - 0 . 0 1 1 1 . 9 0 8 0 . 3 3 9 0 . 0 1 2 1 .837 0 . 3 4 0 0 . 0 1 1 3 . 0 8 4 - 0 . 4 8 2 - 0 . 0 1 6 2 . 9 9 2 I . 6 8 3 0 . 0 53 IX " - 0 . 2 4 4 - 0 . 0 4 6 0 . 3 9 V - 0 . 9 5 6 - 0 . 1 0 1 1 . 2 1 H - 1 . 1 4 3 - 0 . 0 4 7 I.HO* - 1 . 193 - 0 . 0 4 3 2 . 3 9 2 - 1 . 3 2 1 - 0 . 0 4 3 . 3 . 0 2 9 - 3 . 2 5 1 - ' 0 . 1 0 5 4 . 5 0 4 . - 2 . 5 6 3 - 0 . 0 0 1 X - ' 0 . 1 9 1 - C . C 3 6 0 . 2 2 7 - 0 . 2 6 5 . - 0 . 0 2 0 0 . 3 0 6 • 0 .277 - 0 . 0 1 1 0. 310 - 0 . 340 - 0 . 0 1 2 C . 3 9 0 - 0 . 4 0 1 - 0 . 0 1 3 0 . 4 60 - 0 . 4 04 - 0 . 0 1 6 0 . 37 3 - 0 . 5 43 - 0 . 0 1 7 Note: Negative bias means overestimation while positive means'-underestimation. Table XI. - 54 -P. patula: Efficiency of Volume-Based Taper Model 20 for By rows: l=blas/tree; 2=bias/metre (total height effect); Diameter Estimation. 3=standaird error of estimate. 1 4 5 6 7 0.0 2.004 0.379 2.67 3 2.352 0. 2 48 3. 065 1 .439 .0.059 2.467 0.39 3 0.014 1 .750 1 .222 0.040 2.058 2.4*61 0.079 3.700 3.156 0.100 0.3 0.965 0. 18? 1.332 1. 557 0. 164 2.065 0.933 0.030 2.153 -0.536 -0.019 1.88? 0.705 0.023 1.521 I .595 0.051 2.950 3.006 0.097 0.6 0.646 0.122 0.991 1 . 065 0.112 1.436 0.201 0.012 1.620 -0.091 -0.032 2.077 0.064 0.002 1.226 1.209 0.039 2.614 2.268 0.072 0.9 0.426 o .oe i 0.763 0.651 0. 069 0.979 -0.202 -0.012 1.026 -1. 173 -0.042 2.271 -0.522 -0.017 1.291 0.452 0.015 1.819 1.250 0.039 1.3 0.106 0.020 0.437 0. 235 0.025 0.468 -0. 954 -0.039 1 .308 -1.448 -0.052 2.229 -1.127 -0.037 1.530 -0.784 -0.025 1.606 0.511 0.016 I -0.091 -0.017 0.565 -0.175 -0.01.8 0. 484 -0.733 -0.030 1.240 -0.971 -0.035 1.022 -0.580 -0.019 1.741 -1.297 -0.042 2.237 -0.354 -0.011 II -0.221 -0.042 0.537 -0.307 -.0. 03? 0.635 -0.373 -0.015 1.125 -0.013 -0.000 1.264 -0.115 -0 .004 1.823 -0.623 -0.020 1.988 0.912 0.029 III -0.433 -0.082 0. 706 -0.441 -0.047 0. 864 -0.452 -0.019 1.558 0.341 0. 012 1. 510 0.543 0.018 2.061 -0.019 -0.001 2.000 1.737 0.055 IV -0.550 -0.104 0.819 -0.8?8 -0.CC7 1.206 0.234 0.010 1.105 0.698 0.025 1.661 0.854 0.028 2.458 0.217 0.007 2.031 1.053 0.033 V -0.575 -0.109 0.810 -1.092 -0.J 15 1.446 0.146 0.006 1.028 1.024 0.037 2.363 1.419 0.047 2.641 1.071 0.034 2.700 1.711 0.054 VI -0.768 -0.145 0.994 -1.206 -0.127 1 .673 0.264 0.011 1.419 1.283 0.046 2.266 1.745 0.057 2.662 0.808 0. 026 2.714 3.487 0.110 VII -0.74 5 -0.141 0.969 -1.447 -0.153 1.081 0.030 0.001 1.650 1.587 0.057 3.034 1.467 0.0 40 2.8 46 0.562 0.010 2.798 2.306 0.073 VIII -0.640 -0.121 0.856 -1.463 -0.154 1.916 -0.485 -0.020 2.01 5 0.014 0.001 1.860 -0.117 -0.0 04 3.206 -1. 196 -0.039 3.507 0.717 0.023 IX -0.399 -0.075 0. 573 -0.933 -0.099 1 .2 47 -1.397 -0.057 2.163 -1.546 -0.056 2.7 70 -1.756 -0.059 3.493 -3.941 -0.127 5.529 -3.455 -0.109 X -0.233 -0.044 -0.293 - o . m i -0.332 -0.0!4 -0.410 - o . o r . 0. 503 -0.487 -0.593 -0.671 0.291 0. 355 0. 39 6 -O.Olo 0.5 00 -0.019 0.740 -0.021 Note: Negative means overestimation while positive bias means underestimation. - 55 -4.2.1.2 Testing the Volume-based Taper Models for Sectional Volume Estimation For volume estimation the sectional diameters are integrated as H ltd" d L = V For models 18, 19 and 20, the volume integrals are obtained by 2 replacing d i n the above model by the taper model to be integrated. For models 18 and 19 t h i s becomes: V ^ I O S ^ 2 ( L P - L P ) (21) where, and are respectively lower and upper distances from the top of the tree. I f t o t a l volume i s to be estimated = 0 and = H. Thus the t o t a l volume i s given by b 0 b l b2 V = 10 D H as i n our st a r t i n g volume-model. - 56 -The i n t e g r a l of model 20 i s given as: V = I ^ D V - P ( LP - L P ) + b ^ P ( LP - LP) + b 3D 2H-P( LP - LP) + b 4H 2"P ( LP - LP) + b 5logH 2H P ( LP - LJ) ( 2 2 ) The standard errors of volume estimate for model 21 for Cypress and 3 3 Pine were respectively 0.12867 m and 0.06980 m and for model 22 was 3 0.06826 m ( i . e . Pine only). Tables X I I , X I I I and XIV respectively show the performance of models 18, 19 and 20 when integrated for volume estimation. Each row i n each set of three s t a t i s t i c s have the same meanings as i n Table V I I I . The bias shown by Table XII indicates that the model 18 generally overestimates the various sectional volumes. However, the magnitudes of such bias are very small, especially above dbh. Therefore, the model generally gives reasonably good estimates for most p r a c t i c a l purposes. At most points the magnitude of the bias i s less than 10% (second rows). Tables X I I I and XIV show that the corresponding models give even better estimates. Although both models have less systematic bias compared to the previous table (Cypress)^model 20 i s a b i t superior to model 19 for estimating the volumes of pine trees. This conforms - 57 -c o • H rt 6 •rH 4J W W 0) o > o oo o S u <u p. to H T J <U (0 rt I QJ O > o [>* o QJ • H C J * H AW * H W o o o o o o o o o o o o o o o • • • o o o o o o o o o o o o • • • o o o o o o • • • o o o o> — f - CO — vf f l st m t o o — o o m O N O in IA CD f i r - st O N O * f l o e g st m O N O O N O • • • o o o O N O • • • o o o st f -o in o o — o e g O r g t o N f t st cr • o o •0 tn O — — vt 0> t n st N M f t m O O so o t n o r g O CO - i pg — 0 o 1 t oo tn f t st f t f t o o ~< c r m o t o f t o N g> w CO tn - 4 t • 0 o 1 I cr- f t r -st r ~ o — O f i o o o o o o o o o o o o o o o o o o f l vf O st c r so o — o f t rg f i r - r-o o o O st f t CO f t O O — o m m — f t 03 N — N ~< cr f t r - st O — O e CO cc m —< -r o — o — f> rg N O vT O O O O O O O O O o o o o o o o o o o c o o o o o o o •i rg m — o sO co * c f t — vO O - — ' f i —' O vO —« o o -o o o o o o o o o •st c CO — O CO O O O O O O o o o o o c • • • o o o st c c r o * r o o c o o o o co c r i n f- CC o o o o o o I 1 J) C1 H 0 s r - vO p- eo in c> eo r> m co o — m N —< m r g - t r N o n - r - r\j c r o —• o o o o - - — -c o o N eo st i n f l y0 O O O I I t n m r o r j M n . o o o o o o t I O rg st . - - - \t <0 CD o — o — — — o o o o o o I I r g r n t o CO f t o o t> f l o r - eg • • 0 o 1 I vt !- st r» <0 CO — O f i o o o I I so f i r-o o o o o o o o o o o o o o o o o o o o o — i — — rg N — N o o o o o o I I p o o — r-st o st c o * o co r g • • O O I I r- — f t r g st * o — O f t • • • O O O I I t o gj- r -x0 CO r g — f i r > tn f » c r co — • • 0 o 1 I f t CO c r c r m — O rg o « y f t co r g o •O -J" IM st _j rr\ CD f> ff1 r - r » f t i n — i n c o m o f i r g c o O — o — — o — — O —< ~-• •• • • • • • • o o o o o o o o o c o o — o c c n m * ^ h -f i f i r n st o m f i c o m o — o o — o o o o o o o I I o o o I I CO r g f i co r -N O f i O CO CO i n ^0 r g — O r g o o o o o o — vt r ~ O O O O O O I I o r -r g i n i n o o o — sO — f t r g O f> r> O r-t p — st rg m r g — o O r n r>- r - o %o st — f t f t ^ r -O O c - i O O f i CO rg so r -c o — 0 o 1 I st st t n C IM -t o o st r--.N0 M ^ O — o m in in m r g — o C* s0 in s0 CO — 0 o 1 I — st O St f l st n O i n o I st m so st f l vt — o m o o o o o o o o o o o o o o o CO sO vt — O f l O O O o o o o o o st O- sO c r f i c r O O -O O O ' o o o o o o rg vt s0 r - vt O 0 O rg o o o 1 I o o o I I vf vf — f l st f l — O f l • • • O O O I I o o o I I • ga st i f t r g • O f t f t f i r g O W N — O f i o o o o o o or f* st — o c r o o — O st CO — O CT O O -O O O O O O st O — N r - 0 s f t r g O — o c r O O r g o O — • • • • • • O O O o o o I I I I o o o o o o o o o I I o o o o o o o o o • • • o o o r r- — r - r g g y f t v n o — * N vD O — v0 O O f - — > o o o o o o o o o o o o o o . m r g o o r~ o o o 0 o o 1 I r j f i — o f i rg — O f i r » f t co O O O -O O — rg t o o o c r st rg c o O O cr „ w w-o o — o o — • • • • • • o o o o o o f i r j f i f - r - f t — c r r g c r c c r — f t f i c o o o c o — o c r — — c r r g — rg rg o o r -0 o o • • • o o o 1 I O cd • H CO 3 o o o o o o o o o o o o — C r f i o r g f i _ i c o vt O O O O — O O vT O o — O O s-> o O O O o o o o o o O st o o o o o o o • • • o o o o f t — o i n o 0 c o o o o 1 1 — f l N o o o o — o o o o o o o st sO <C O CO o o o o r g f i f . O ro o o — o C T s o r g O s O f i s O v O — N o o m c r r g c o O s O - - f t c r O — — — C O — O v t ^ * O — — O — — O s t — — u-i — O — O C O O O O O c o o o o o o o o o o o o o o I I f i c c m O st o 0 — o • • • o o o 1 I r g eo o —« i n r g o o o f l O O f i O o — vO rg - 4 gj rg o o o o o o o o o o o o st r - s0 O rg O 0 — o • • * o o o 1 I st O sO O O O c — o o o o I i o o o I I f i rg i n o r - o O O C o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o I I r g r - m — c o m — f i us, O f i O O N O O N O O O O o o o o o o 0 o o 1 I o o o I I o o o o o o o o o I I I H M X 0) o o o . o o o o m ' o O c r O o o o O r - — O — O o o o o CO ~t O N O o o o o o o I I O st — 0 m o o o o o o o 1 I o — — O sC o o o o 0 c o 1 I — CO N o m o o o o o o o t I — O N O m o o o o O O N O st o o o o o o o I I O N N O f i o o o o 0 i n N O N O o o o o o o 1 I O r ~ N o — o o o o u c O I I O st N 0 — o o o o o o o 1 I o o o -JO cd H - 58 -u i d s , / L r e e , / Dias/m ,3=standard error of estimate. DBH Class 0.0 0.3 0.6 0.9 1 0.0 0.0 0.0 0.001 C.40? 0.001. 0.00! 0.35P 0.001 C.001 0.316 0.001 0.0 0.0 o.o 0.001 0. ! 82 0.001 0.001 0. 141 0.002 0.001 0. 1C8 0.002 0.0 0.0 0. 0 o.ooi 0.0<71 0.C02 0.001 .0. 065 0. 003 0.001 0.C3R 0.004 0.0 0.0 0.0 -0.000 -0.025 0.002 -0.002 -0.049 C.004 -0.003 -0.066 0.007 0. 0 0.0 0.0 0.001 0.043 0.0C? 0.001 0.027 0.004 0.001 0.010 0.0C5 6 0.0 0.0 0.0 0.0 04 0.00ft 0.006 0.007 0.005 0.012 0. 009 0.073 0.016 0.0 0. 0 0.0 0!! 0.1 3V 0.0 15 0.131 0.020 0.113 1.3 0.001 0.280 0.002 0.001 0.068 0.002 0.000 0.C02 0.005 -0.005 -0.084 0.010 -0.001 -0.012 0.006 0.009 0.051 0.018 0.023 0.101 I 0.001 0.248 C.002 0. 000 0.005 0.002 -0.C07 -0.C71 0.011 -0.019 -0.106 0.023 -0.015 -0.04 6 0.022 -0.007 -0.014 0.031 0.035 0.050 II 0.001 0.222 C.002 -0. 001 -0.036 0.003 -0.012 -0.075 0.018 -0.025 -0.085 0.039 -0.019 -0.037 0.030 -0.019 -0.024 0.058 0.050 0.045 III 0.001 0.198 C.002 -0.002 -0.065 0. 004 -0.C14 -0.069 0.023 -0.024 -0.059 0.046 -0.014 -0.021 0.054 -0.019 -0.018 0.082 0.08S 0.057 IV 0.001 0.1 76 0.002 -0.003 -0.091 0.0C5 -0.014 -0.057 0.026 -0.018 -0.037 0.053 -0.003 -0 .0 04 0.074 -0. 012 -0.009 0.104 0.122 0.066 V 0.001 0. 157 0.002 -0.004 -0.119 0.006 -0.012 -0.044 0.029 ,-0.010 -0.017 0. 062 0.013 0.013 0.096 0.003 0.002 0.127 0.1 56 0.073 VI 0.001 0. 140 -0.005 -0. 144 -o.ou -0.033 0.032 0.001 0.001 0.073 0.032 0.021 0.205 0.002 0.000 0.029 0.117 0.013 0. 152 0.087 VII 0.001 0.125 0.002 -0.006 -0.166 0.009 -0.009 -0.025 0.C35 0.012 0.018 0.005 0.050 0.042 0.139 0.035 0.020 0.176 0.255 0.099 VIII 0.001 0. 115 0.002 -0.007 -0.185 0.010 -O.OCfl -0.022 0. 03 8 0.020 0.027 0.096 0.060 0.047 0. 1 57 0.040 0.021 0.196 0.2 8? 0. 104 IX 0.001 0. 110 -0.008 -0. 197 -0.009 - 0.C24 0.041 0.020 C.02o 0. 100 0.060 0.032 0.283 0.002 o.ou 0.046 0. 163 .0.016 0.205 0. 1 ll,' X 0.001 0.1 Oil -0.00(3 -0.200 -0.009 -0.026 0.042 0.019 0. 0? 5 0.101 0 .059 0.02 5 <j. i;;, 0.001 0.011 0. 045 0 .1 64 0.013 0.206 0.1 Note: undlrelJ^atLr"8 ° V e r e S t i D l a t i° n -d positive'bias means - 59 -Table XIV. DBH Class 0.0 0.3 0.6 0.9 1.3 P. patula: Efficiency of Volume-Based Tap Volume Estimation. By rows: l=bias/tree; 3=standard error of estimate. er Model^O for 2=bias/m ; I 0.0 0.0 0.0 0.001 0.360 0.001 0.001 0. 30'. . O . O O I 0.001 0.267 0.001 0.001 0.227 0.001 0.0 0.0 0. 0 0.001 0. 207 0.001 0.002 0. 251 0.002 0.002 0. 220 0.003 0.003 0.195 0.003 3 4 *; 6 7 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 0.001 0.1 15 0.002 -0.000 -0.002 0.002 0.002 0.0 57 0.003 0.004 0.095 0.006 0.007 0.122 0.002 C.090 0.004 -0.001 -0.026 0.004 0.002 0.041 0.005 0.007 0.002 0.012 0.013 0.1 14 0.002 0.063 0.005 -0.002 -0.042 0.007 0.002 • 0.024 0.006 0.008 ' 0.069 0.016 0.017. 0.100 0.001 O.C20 0.005 -0.004 -0.060 0.010 0.000 0.0 02 0.006 0.008 0.047 0.018 0.019 0.082 I 0.001 0.194 C.002 0.003 0. 129 0.004 -0.005 -0.046 0.009 -0.016 -0.085 0.026 -0.011 -0.034 0.020 -0.011 -0.021 0.034 0.019 0.027 II 0.001" 0.165 0.002 0.002 0.090 0. 004 -0. CCfl -0.053 0.015 -0.020 -0.067 0.037 -0.014 -0.027 0.037 -0.027 -0.034 0.066 0.022 0.020 III 0.001 0.139 0.002 0.002 0. 063 0.003 -0.010 -0.050 0.020 -0.018 -0.045, 0.044 -0.010 -0.014 0.056 -0.032 -0.031 0.094 0.045 0.030 IV 0.001 0.115 0.002 0.001 O. 038 0.004 -0.010 -0.041 0.024 -0.013 -0.026 . 0.052 0. 000 0.000 0. 070 -0.031 -0.024 0.118 0.068 0.037 V 0.001 0.09J 0.002 0.000 0. Oil 0. 004 -0.009 . -0.031 0.027 -0.005 -0.008 0. 064 0.014 0.014 0. 102 -0.023 -0.015 0. 141 0.089 0.042 VI • 0.001 0.074 0.001 -0.001 -0.014 0.004 -0.007 -0.023 0.031 0.004 0.007 0.076 0.031 0.027 0.123 -0.012 -0.007 0. 164 0.126 0.053 VII C.OOO 0.056 0.001 -0.00) -0.036 0.005 -0.006 -0.018 0.034 0.015 0.021 0. 089 0.046 0.030 0.143 -C.005 -0.003 0.188 0.164 0.064 VIII o.ooo 0.044 0.C01 -0.002 -0.055 0.005 -0.006 -0.017 0.038 0.021 0.028 0.099 0.053 0.042 0.160 -0.006 -0.003 0.209 0.101 0.067 IX 0.000 0.037 0.001 -0.003 -0.067 0.006 -0.007 -0.020 0.040 0.019 0.026 0. 103 0. 050 0. 039 0.166 -0.019 -0.010 0.223 0.175 0.063 X o.ooo 0.OH5 o.ooi -0.003 -0.070 0. 006 -o.con -0.023 0.041 0.010 0.023 0. 104 0.04 7 0.036 0.1 66 -0.02H -0.014 0.227 0.) 66 0.060 Note: Negative bias means overestimation means underestimation. and positive bias - 60 -to the conclusion drawn when their corresponding t r a d i t i o n a l volume models were compared i n section 4.1.2. I t also indicates that a s i n g l parameter could be used to give the optimum sectional diameter and volume estimates. 4.3 Taper Models Produced From Taper Data 4.3.1 Results of F i t t i n g Common Taper Models Three models were selected for testing using the data from the two species to f i n d out how w e l l they would describe the p r o f i l e of the two species. They were selected either due to their popularity or because they had been recommended elsewhere as being f a i r l y good for describing the p r o f i l e s .of trees i n general. The following were the models tested: D2 0 + H + - J T (23) log d = b Q + b 1logD + b 2logH + b 3logL (24) b ^ L d =(H-1.3) + b 2 L(h-1.3) + b3L0'-1.3) (K) + b 4L(h -1.3) (H + h + 1.3) (25) - 61 -where: h i s the distance from the ground to a point up the tree where the diameter i s to be estimated (in metres) L i s the distance from the top of the tree to the same point ( i n metres) H i s t o t a l height ( i n metres) bp, b^, b^ etc. are regression c o e f f i c i e n t s , d i s diameter inside-bark at any point along the tree p r o f i l e ( i n cm) D i s diameter at breast-height (dbh) overbark (in cm) Other models such as proposed by Bruce et al. (1968) and Demaerschalk and Kozak (1977) which have been put into actual inventory work were not t r i e d , either because they were considered to be too complex to be of p r a c t i c a l value i n Kenya or because their use (e.g. Demaerschalk and Kozak 1977) are possible only when integrated into computerized inventory. The tested models were some of the simple ones which could, i f sat i s f a c t o r y , be used i n Kenya. The respective regression c o e f f i c i e n t s of the tested model together with t h e i r respective standard errors of estimate are given i n Table XV. Usually, u t i l i z a t i o n l i m i t s are specified i n terms of upper sectional diameters. As such the corresponding distances (from ground or top) have to be estimated. For the tested models these are estimated Table XV. Regression Coefficients and Standard Errors of Estimates for Three Popular T*per Models Tested. Model No. Pinus patula (23) Equation — - 0.96912 - 1.26523 — + 0.294517 — R & SEE From Model Direct SEE 0.1348 0.8621 True SEE 1.8829 (24) (25) - -1.26831 (-£-- 1) + 0.298989 (rK - 1) (adopted) 0.1348 Logd - 0.0677668 + 0.7677071ogD - 0.7288531bgH + 0.1208 0.9662851ogL d - 0.91752955DL/(H - 1.3) + 0.0339276L (h - 1.3) -0.0032047249HL(h - 1.3) + 0.0020912969L (h - 1.3)(H + h+1.3) 1.433 0.8621 0.9644 0.9859 1.8215 4.151 1.433 ON tN> Cupressus lusitanica ,2 (23) (24) (25) " 1-30415 - 2.62774 -2- + 1.35793 2 2 -1.321(1--^  + -^) (adopted) logd = 0.107742 + 0.7332781ogD - 0.7985081ogH + 1.071241ogL d - 0.936251DL/(H - 1.3) + 0.057839L(h - 1.3) -0.00267612HL (h-1.3) + 0.000940734L(h-1.3)(H+h+1.3) 0.2012 0.8516 4.575 0.2041 0.9364 4.290 0.0744 0.9886 4.376 1.7544 0.9868 1.7544 - 63 -For Model (23) v (bH) 2 - 4c(aH 2 - ) h = -bh - V _ _ _ _ _ _ D 2 (26) 2c For Model (24) -b -b -b 1/b, L = (10 UdD H ) - (27) The complexity of model 25 makes the derivation of a similar function very d i f f i c u l t . As such no corresponding function i s given for height estimation. 4.3.1.1 Testing the Models for^ Diameter, Sectional Volume and Distance From the Top Estimations When the models were tested using the same approach as i n Section 4.2.1 for diameter estimation,it was not immediately clear which one was superior. In general, they were not too good although model 25 was f a i r for some sizes of trees. Some gave better estimates near the ground and worse at other sections. Others would give better estimates i n the middle sections and others near the top. A l l the models had systematic bias either depending on the tree sizes or up the trees. They were, however, giving s l i g h t l y better estimates for P. patula. - 6 4 -Because the pine trees had very l i t t l e butt f l a r e i t seemed that the models could only give better estimates for trees with uniform p r o f i l e s but could not account for the marked butt swell i n Cypress trees. After many tests on the d i f f e r e n t forms (conditioned forms) of model (23) for precision and accuracy, the f i r s t and second conditioned forms (see l i t e r a t u r e review) were a reasonable compromise for Pine and Cypress, respectively. For the Pines the diameters near the top of the trees were s t i l l very much overestimated. Attempts to f i t model (25) for the whole tree instead of above dbh alone as was proposed by Bennett & Swindel (1972) showed that the model could s a t i s f a c t o r i l y be used for the whole tree i n Pine. The sectional diameter tests on such a modification compared very w e l l with the test for upper stem (above dbh) only. The res u l t s were, however, worse for Cypress. Using H and h instead of (H-1.3) and (h-1.3) as given i n the model gave s l i g h t l y worse re s u l t s . Figures 3 and 4 show the effects of having one model for the whole p r o f i l e . Because sectional diameter tests could not reveal which of the three was superior i t was decided to test their respective sectional volume estimation c a p a b i l i t y . From t h i s test, i t was concluded that model (24) was the best of the three. Because model 24 was giving worse diameter estimates near the base while the others were giving worse estimates above dbh, the res u l t s seemed to imply that a taper model can be reasonably good for volume estimation even i f i t does not wel l take care of the butt swell, provided i t gives f a i r l y precise and f a i r l y accurate estimates for most of the upper stem, especially near the t i p . Tables XVI to XIX show the res u l t s of the tests on model (24). Figure 3. _ P. patula: Effect of f i t t i n g model 25 for the whole tree or above dbh only. Key F i t t e d above dbh F i t t e d for whole tree The true p r o f i l e - A — A ~ G— e -Figure 4. C. lusitanica: Effect of f i t t i n g model 25 for the whole tree or above dbh only. Key Fitted above dbh Fitted for whole tree The true prof i l e O S T 8.0 1 12.0 i 1 r 16.0 20.0 24.0 DISTANCE FROM TCP (m) 28.0 J2.0 36.0 40.0 - 67 -Table XVI. _. patula: Efficiency of Tapcjj Model 24 for Diameter Estimation. By rows: l=blas/tree; 2=bias/m ; 3=standard error of estimate. 1 Class 1 2 3 0.0 1.534 0.021 2.041 0.855 0. 006 1. 512 0.695 0.002 1.925 0.3 0.551 0. 007 0.005 0.129 0.001 0.919 0.223 0.001 1.813 0.6 C.288 0.004 0.571 -0.293 -0.002 0.790 -0.395 -0.001 1.627 0.9 0.124 0.002 0.52 7 -0.630 -0.004 0.970 -0.925 -0.002 1.423 1.3 -0.122 -0.002 0.271 -0.960 -0.006 1.207 -1.551 -0.004 1.073 I -0.243 -0.003 0.537 -1.182 -0.008 1.445 -1.C74 -0.003 1.477 II -0.299 - 0 . 004 0.549 -1.123 -0.007 1.412 -0.461 -0.001 1.128 III -0.437 -0.006 0.604 -1.067 -0.007 1.402 -0.292 -0.001 1.540 IV -0.4C3 -0.007 0.700 -1.263 -0.003 1.597 0.635 0.002 1.330 V -0.438 -0.006 0.625 -1.339 -0.009 1.647 0.778 0.002 1.394 VI -0.567 -0.008 0.732 - 1. 267 -0.000 1.654 1.1 11 0 . 00 3 1.913 VII -0.406 -0.007 0.636 -1.329 -0.009 1.665 1.065 0.003 2.033 VIII -0.336 -0.005 0.501 -1.100 -0.000 1.515 0.607 0.002 2.061 IX -0.003 -0.001 0.287 -0.523 -0.003 0. 761 -0.209 -0.001 1 .369 X -O.C2n -0.000 0.033 •0.035 -0.000 0.041 -0.033 -0.000 0.037 4 5 6 7 0.0 51 0.000 1.604 I .4 04 0.003 2. 162 3.737 0.009 4.889 5.407 D.087 -O.044 -0.002 1.996 1.002 0.002 1.703 2.957 0.007 4.082 5.4 52 0.086 -1.164 -0.003 2.222 0.397 0.001 1.317 2.607 0.006 3.725 4.669 0.074 -1.410 -0.003 2.398 -0.155 -0.000 1.170 1.886 0.004 2.813 3.687 0.058 -1.640 -0.004 2.341 -0.713 -0.001 1 .1>88 0.698 0.002 1.448 2.995 0.047 -0.862 -0.002 1.677 0.168 0.000 1.621 0.533 0.001 1.546 2.478 0.039 0.391 0.001 1.290 0.957 0.002 2.135 1.540 0.004 2.461 4.074 0.064 1.032 0.002 1.859 1.927 0.004 2.900 2.460 0.006 3.404 5.205 0.082 1.663 0.004 2.530 2.534 0.005 3.622 2.988 0.007 3.934 4.797 0.076 2.248 0.005 3.282 3.371 0.007 4.332 4.099 0.009 5.297 5.688 0.090 2.740 0.007 3.594 3.933 0.008 4.779 4.045 0.009 5.241 7.636 0.120 3.237 0.008 4.426 3.B35 0.007 4.898 3.930 0.009 5.215 6.532 0.103 1.783 0.004 2.769 2.331 0 .005 4 .02R 2.166 0.005 3.705 4.853 0.077 0. 1 6<? 0.000 1.978 0.499 0.001 2.62? -0.890 -0.002 2.454 0.224 0.0 04 ••0.035 -0.000 0.04 5 -0.045 -0.000 0.051 -0.053 -0.000 0.063 -0.0'iv -0.001 Table XVII. CJ. lusitanica: Efficiency of Taper Model 24 for Diameter Estimation. DBH Claaa 1 2 3 4 5 6 7 0.0 1.9R2 0.027 2.6*2 3. 20* 0.01* 3.682 6.61* 0.029 8.213 10.32* 0.033 12.596 16.193 0.042 19.722 18.618 0.0*0 22.**5 23.606 0.050 28.746 28.7B9 0.060 3*.558 3*.667 0.1 83 61.98* 30.**5 0.805 73.9*4 2.537 0.3 0.657 0.009 1,308 0.970 0.00* 1.321 2.373 0.010 3.*76 5.606 0.018 6.863 7.*6* 0.019 9.0** 10.826 0.023 12.817 14.767 0.031 19.451 18.259 0.038 23.751 22.767 0.120 40.61* 21.961 0.581 41.670 1.413 0.6 0.100 0.002 0.820 0.3*7 0.0J1 0.815 1.513 0.007 2.787 3.* 50 0.011 *.283 *.B85 0.013 5.891 7.529 0.016 9.030 11.710 0.025 13.785 12.6*9 0.026 17.2*5 17.819 0.09* 31.466 18.979 0.502 33.397 1.023 0.9 -0 .0*7 -0.001 0.66* 0.1*7 0.001 0.639 0.895 0 . 00* 1. t6* 2.496 0.008 3.083 3.650 0.009 *.533 5 .MS 0.012 6.500 9.07* 0.019 10.6*7 10.85* 0.023 13.205 1*.289 0.076 25.005 16.397 0.43* 27.326 0.916 1.3 -0 .015 -0 .000 0.*71 0.135 0.001 0.638 0.716 0.003 1.3*8 1.896 0.006 2.468 2.219 0.006 2.967 3.526 0.008 *.235 6.867 0.01* 8.056 8.988 0.019 10.670 12.*06 0.066 21.586 13.822 0.366 23.303 0.790 I -0 .06* -0 .001 0.52* 0.227 0.001 0.*97 0.889 0. 00* 1. *89 0.871 0.003 1.778 0.01* 0.000 1.466 0.291 0.001 2. 135 2.511 0.005 3.503 2.959 0.006 5.*28 6.818 0.0 36 12.219 3.58* 0.095 19.230 0.6 52 II -0 .215 -0 .003 0.720 0. 171 0.001 0.*67 1.376 0.006 1.970 1.026 0.003 1.872 0.607 0.002 1.320 0.735 0.002 2.792 1.8*9 0.00* 2.886 2.580 0.005 4.39* 5.232 0.329 9.976 - 0 . 5 0 0 - 0 . 0 1 3 14.221 3.*<J2 III -0 .236 -0 .003 0.66* 0.035 0.000 0.*85 1.271 0.006 1.870 0.605 0.002 1.9*9 0.957 0.002 1.835 l.*3* 0.003 3.135 2.1*2 0.005 3.218 2.374 0.005 3.711 6.206 0.333 11.45* 1.199 0.032 14.502 0.*92 IV -0 .*79 -0.006 0.781 - 0 . 1 5 9 -0 .031 0.899 1.077 0.005 1.590 0.898 0.003 2.017 1.518 0.00* 2.285 1.180 0.003 3.075 2.437 0.005 3.560 2.589 0.005 3.780 5.902 0.031 10.608 - 1 . 5 8 5 -0 .0*2 2.710 0.092 V -0 .573 -C .008 0.92* -0 .617 -3 .003 1.156 0.730 0.003 1.162 O.*07 0.001 2.5*7 1.218 0.003 2.026 1.651 0.00* 3.165 2.46* 0.005 3.626 . 2.1*6 0.00* 3.377 • .98* 3.026 9.*9* 0.900 0.02* -5 .032 - 0 . 1 7 0 VI -0 .756 -0 .010 1.072 -0 .831 - 0 . 0 0 * 1.23* 0. 107 0.000 1.000 -0.631 -0 .002 2.619 1.135 0.003 2.295 l .*9* 0.003 3.175 1.873 0.00* 2.981 1.692 0.00* 3.88* 2.916 0.015 7.128 -3 .472 - 0 . 0 9 2 -8 .956 - 0 . 3 0 * VII -0 .8*2 -0 .011 1.198 -1 .098 - 0 . 0 0 5 1.376 -0 .896 - 0 . 0 0 * 1.48 3 -1 .694 -0.005 3.056 0.1 77 0.000 2.13* - 0 . * 4 3 -0 .001 3.068 0.008 0.000 2.789 -0 .680 -0 .001 2.960 - 0 . * * 3 - 0 . 0 0 2 4.161 - 4 . 2 7 5 - 0 . 1 1 3 -7 .622 - 0 . 2 5 8 VIII - 0 . 7 5 9 -0 .010 1.02* -1 .165 - 0 . 0 0 5 1.369 -1 .731 -0 .008 2.169 -2 .557 -0 .008 3.511 -0 .935 -0 .002 2.733 -2 .175 , -0 .005 3.069 -2 .819 -0 .006 3.861 - 3 . 6 6 6 -0 .008 . 4.833 - 2 . 2 7 5 -0 .012 7.036 - 6 . * 8 l -0 .171 - 2 . 5 6 0 -0 .087 IZ -0 .719 - 0 . 0 1 0 0.936 - 1 . 2 7 5 -0 .005 l.*83 - 1 . 8 9 6 -0 .008 2.308 -2 .761 -0 .009 3.358 -2 .221 -0 .006 3.095 -3 .518 -o.ooa 4.266 - 3 . 716 -0 .308 4.7*3 -5 .018 -0 .010 5.9*8 -4 .62 7 -0 .324 8.687 - 4 . 8 3 2 - 0 . 1 2 8 - 1 . 9 9 1 -3 .067 z - 0 . 0 2 * -0 .000 0.030 - 0 . 029 -0 .000 0.032 - 0 . 0 3 * -0 .000 0.0*0 -0 .0*0 -0 .000 0.0*6 - 0 . 0 4 1 -0 .000 0.0*9 - 0 . 0 4 * - 0 . 0 0 0 0.052 -0 .051 - 0 . 0 0 0 0.059 -0 .055 - 0 . 0 0 0 0.064 - 0 . 0 6 0 - 0 . 0 3 0 0.104 - 0 . 0 6 1 -0 .002 - 0 . 0 7 5 -0 .003 By row*I l - b l a a / t r a a | 2-fclaa/aatra; 3>ataaaara arror of a a t l a a t a . - 69 Table XVIII. P „ f i c i e n c y of Taper Model 24 for Volume Estimation. By rows: l=bias/tree; 2=bia S/ m3; 3=standard error of estimate. DBH Class 1 ?. 3 4 5 6 7 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 0.3 0.000 0.1','. 0.000 -0.001 -0.163 0.001 -0.005 -0.432 0.005 -0.009 -0.557 0.011 -0.011 -0.411 0.013 -0.011 -0.271 0.014 - o . m o -0.102 0.6 0.000 0.091 0.000 -0.001 -0.205 0.002 -0 .C09 -0.465 0.011 • -0.010 -0.587 0.022 -0.022 -0.420 0.026 -0.024 -0.204 0.029 -0.021 - o.in o 0.9 0.000 0.067 0.001 -0.002 -0.236 0. 003 -0.015 -0.500 0.C17 -0.020 -0.605 0.033 -0.034 -0.4 4 7 0.040 -0.036 -0.296 0.045 -0.033 -0.201 1.3 0.000 0.040 0.001 -0.004 -0.269 0.005 t -0.022 -0.545 0 .026 -0.040 -0.622 0.048 -0.051 -0.471 0.059 -0.055 -0.319 0.067 -0.051 -0.218 I 0.000 0 . C37 0.001 -0.006 -0.306 0.008 -0 .059 -0.59 5 0.069 -0.103 -0.591 0.129 -0.140 -0.462 0 . 1 7 1 -0.177 -0.355 0.215 -0.165 -C.236 II 0.000 0.033 0.001 -0.008 -0.312 0.010 -0.083 -0.538 0.097 -0.147 -0.500 0. 175 -0.202 -0.391 0.234 -0.249 -0.315 0.305 -0 .215 -0.193 III 0.000 0.034 0.001 -0.009 -0.300 0.011 '-0.095 -0.470 0.112 -0.162 -0.407 0.194 -0^220 -0.316 0.255. -0.269 -0.255 • 0.335 -0.197 -0.131 0.000 0.037 0.001 -O.0O9 -0.23 7 0.012 -0.098 -0.393 0.115 -0.159 '-0.324 0.193 -0.209 -0.243 0.246 -0.252 - 0.195 0.323 -0.140 -0.C76 y 0.000 0.042 0.001 . -0.010 -0.276 0.013 -0.093 -0.329 O . U O -0.143 -0.250 0.177 -0.177 -0.177 0.210 -0.205 -0.138 0.284 -0.059 -0.023 • v i 0.000 0.047 0.001 -0.010 -0.264 .0.013 -O.CR5 -0.270 0.102 ' -0.118 -0.185 0.154 -0.131 -0.118 0.179 -0.143 -0.006 0.236 0.053 0.022 VII 0.000 0.052 0.001 -0.010 -0 . 255 0.013 -0.075 -0.222 0.09 J -0.090 -0.129 0.131 -0.083 -0.06P 0. 147 -0.080 -0.04 5 0.205 0.1 69 0.066 VIII 0.000 0. 057 0.001 -0.010 -0.240 0.013 -0.067 -0.107 0.003 -0.066 -0.090 0.115 -0.044 -0.035 0.139 -0.031 -0.016 0.190 0.256 0.095 IX 0.001 0.062 O.OOf' -0.010 -0 .2 4? 6.01 3 -0.061 -0.169 0.C79 -0.054 -0.072 0. 109 -0.0 24 -0.010 0.141 -0.007 -0.004 0.20? 0.300 0.109 X 0.001 0.064 0.00) -0.010 - 0 . ? 3 f i 0.013 -0.060 - 0 . 164 0.077 -0.051 -0.067 0. 1 00 -0.010 -0.013 0. 1 42 -0.001 -0.001 0.203 0.310 0.1 1.' Table XIX. C. lusitanica; Efficiency ofvTaper Model 24 for Volume Estimation. DBH C l a a a 1 2 3 * 5 6 T 8 9 13 11 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 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.000 0.044 0.000 ' -0.000 -0.028 0.001 0.000 0.013 0.003 0.001 0.043 0.004 0.004 0.083 0.008 0.006 0.3R8 0.012 0.014 0.145 0.028 0.026 0.208 0.041 0.042 0.262 0.088 0.020 0.112 0.185 0.523 0.6 -0.000 -0.036 0.001 -0.001 -0.127 0.002 -0.002 -0.093 0.005 -0.003 -0.053 0.008 -0.003 ' -0.038 0.008 -0.001 -0.009 0.013 0.012 0.064 0.036 0.027 0.120 0.059 0.053 0.184 0.119 0.018 0.056 0.251 0.430 0.9 -0.000 -0.C82 0.001 -0.002 -0.180 0.003 -0.005 -0.152 0.008 -0.008 -0.118 0.014 -0.013 -0.114 0.018 -0.014 -0.079 0.023 • 0.001 0.006 0.035 L 0.017 0.055 0.066 0.050 0.126 0.126 0.008 0.017 0.286 0.367 1.3 -0.001 - 0 . 107 0.002 -0.003 -0.212 0.005 -0.009 -0.198 0.012 -0.017 -0.174 0.023 -0.029 -0.185 0.036 -0.034 -0.152 0.046 -0.019 -0.056 0.043 -0.002 -0.006 0.070 0.038 0.070 0.120 -0.016 -0.025 0.317 0.310 I -0.001 -0.115 0.002 -0.005 -0.233 0.009 -0.018 -0.230 0.024 -0.054 -0.271 0.073 -0.133 -0.356 0.165 -0.202 -0.348 0.252 -0.196 -0.224 0.245 -0.195 -0.181 0.269 -0.097 -0.070 0.233 -0.362 -0.189 0.472 0.190 I I -0.001 -0.110 0.003 -0.006 -0.211 0.011 -0.022 -0.193 0.029 -0.079 -0.273 0.107 -0.208 -0.376 0.259 -0.331 -0.383 0.417 -0.341 -0.263 0.422 -0.363 -0.231 0.489 -0.215 -0.105 0.444 -0.765 -0.275 0.608 0.167 I I I -0.001 -0.100 0.003 -0.007 -0.180 0.011 -0.020 -0.145 0.029 -0.090 -0.245 0.123 -0.244 -0.344 0.305 -0.396 -0.356 0.507 -0.417 -0.2 52 0.513 -0.452 -0.226 0.612 -0.246 -0.094 0.531 -1.035 -0.297 0.7 75 0.168 IV -0.001 -0.087 0.003 -0.006 -0.148 0.011 -0.017 -0.103 0.027 -0.090 -0.2 09 0.125 -0.251 -0.297 0.317 -0.416 -0.313 0.541 -0.430 -0.220 0.534 -0.473 -0.201 0.656 -0.198 -0.064 0.500 -1.178 -0.290 0.887 0.168 V -0.001 -0.077 0.003 -0.006 -0.122 0. 010 -0.012 -0.068 0.025 -0.082 -0.171 0.117 -0.239 -0.249 0.306 -0.403 -0.269 0.538 -0.401 -0.182 0.506 -0.447 -0.169 0.648 -0.112 -0.032 0.434 -1.226 -0.272 0.8 56 0.151 V I -0.001 -0.067 0.003 -0.005 -0.105 0.009 -0.008 -0.042 0.022 -0.074 -0.142 0.108 -0.217 -0.208 0.283 -0 . 369 -0.225 0.507 -0.352 -0.146 0.455 -0.398 -0.139 0.614 -0.021 -0.006 0.426 -1.218 -0.251 0.791 0.135 V I I -0.001 -0.061 0.003 -0.005 -0.093 0. 009 -0.005 -0.026 0.021 -0.067 -0.123 0.101 -0.194 -0.176 0.258 -0.336 -0.194 0.474 -0.306 -0.120 0.408 -0.350 -0.116 0.581 0.044 0.01 I 0.474 -1.214 -0.241 .0.752 0. 126 V I I I -0.001 -0.056 0.003 -0.004 -0.0B7 0.008 -0.004 -0.018 0.020 -0.063 -0.114 0.097 -0.177 -0.155 0.238 -0.316 -0.177 0.451 -0.278 -0.107 0.384 -0.324 -0.105 0.563 0.082 0.020 0.524 -1.205 -0.234 0.763 0. 126 IX -0.001 -0.053 0.003 -0.004 -0.084 0.008 -0.003 -0.016 0.020 -0.062 - 0 . 110 0.096 -0.169 -0.146 0.229 -0.308 -0.171 0.441 -0.269 -0.102 0.375 -0.316 -0.101 0.557 0.099 0.024 0.554 -1.198 -0.231 0.792 0.129 X -0.001 -0.052 0.003, -0.004 -0.084 0.008 -0.003 -0.016 0.020 -0.061 -0.109 0.096 -0.167 -0.144 0.227 -0.306 -0.170 0.439 -0.266 -0.101 0.372 -0.314 -0.101 0.555 0.102 0.02S 0.559 -1.193 -0.230 0.804 0.131 - 71 -Tests of height estimation using the models revealed that model 27 was giving very good estimates for Cypress sectional heights. It was, however, not too impressive for pine. Model 26 was definitely too poor for any practical use. 4.3.2 New Taper Models 4.3.2.1 General Approach After the popular taper models discussed in Section 4.3.1 failed to give satisfactory estimates i t was decided to investigate the possib i l i t y of developing models that would specifically take into account the characteristics of the data. It was necessary to follow a systematic approach by thoroughly examining any possible relationships the data could reveal that would lead to the development of an appropriate model (or appropriate models). Some of the relationships investigated included the examination of the general trend of the profile changes, the distribution of volumes along the tree profiles for trees in the various diameter classes, and differences that might be evident between classes. Thus an approach almost similar to that followed by W.G. Wright (1923) was used. Although Wright's findings were seriously c r i t i c i s e d by Baker (1924) and Behre (1924), their criticisms were based purely on the then knowledge of form quotients and form factors. These statistics simply express the percentage taper of the trees without giving detailed description on the distribution of the woody material along the profiles S - 72 -of such t r e e s . As s u c h , W r i g h t ' s approach was cons ide red a p p r o p r i a t e . By t h i s approach a number of r e l a t i o n s h i p s , t h a t might not o n l y be of i n t e r e s t to t h i s study but o ther s i m i l a r s t u d i e s , were r e v e a l e d . An i n i t i a l s tep was to show i n f i g u r e s the shapes of t r e e s b e i n g s t u d i e d . F i g u r e s 5 and 6 show the g e n e r a l c o n f i g u r a t i o n of P. p a t u l a and C. l u s i t a n i c a t r e e s . From these f i g u r e s , i t seems that the l u s i t a n i c a t r e e s have more p r e d i c a b l e p r o f i l e s than P_. p a t u l a t r e e s . However,, because of t h e i r marked b u t t f l a r e i t might be more d i f f i c u l t t o d e s c r i b e t h e i r g e n e r a l p r o f i l e than would be f o r P_. p a t u l a which have f a i r l y smooth p r o f i l e s ( i . e . i n a s i n g l e g e n e r a l d i r e c t i o n ) . Appendices 4 and 5 show the r a t e s of d iameter changes a long the p r o f i l e s of the t r e e s used i n the taper s t u d y . Appendices 6 and 7 show the volume d i s t r i b u t i o n a long the p r o f i l e s of the same t r e e s . A l t h o u g h Appendices 4 and 5 show the r a t e s of d iameter changes a l o n g the t r e e p r o f i l e s i n the d i f f e r e n t c l a s s e s , the a c t u a l t r e n d i s d i f f i c u l t t o g r a s p . T a b l e s XX and XXI d e r i v e d from Appendices 4 and 5 g i v e a p i c t u r e o f the d iameter d i s t r i b u t i o n w i t h g r e a t e r i m p a c t . The t a b l e s show the f a c t o r s f o r c o n v e r t i n g d iameters a t b r e a s t - h e i g h t (dbh) overbark a t v a r i o u s h e i g h t s ( r e l a t i v e ) a long the t r e e p r o f i l e s . Look ing a c r o s s the d iameter c l a s s e s a t the same measurement p o i n t s i t w i l l be n o t i c e d t h a t the c o n v e r s i o n f a c t o r s do not seem to be dependent on the s i z e c l a s s e s of the t r e e s . The f a c t o r s a r e reasonab ly i d e n t i c a l a t the same measurement p o i n t s f o r a l l d iameter c l a s s e s . T h i s i s more so i n CJ. l u s i t a n i c a . In _P. p a t u l a , however, the re seems to be a s l i g h t d i f f e r e n c e c o o u CJ u a e C3 4) I.7S7 I.IC6 0.7550 0.5C4C 0.2530 C.23CCF-C26. Figure 5. P. patula; General tree profile (form) scattergram. • »22 *» • 7 « * ** «* 0 » u> 0.8333JE-01 O.2S0C0 r r « ..... 0.41667 0.58333 0.75C00 0.91667 f « c 0.16667 0.33333 0.50000 - w.vioer 0.66667 0.83333 1.0C00 Relative Distance from the Top Relative Diameter of Section i o 2 so n> < o H -03 rr 0> 3 r> * Mi § 3* (0 i-3 O •a fs» II M 14 • Table X X . Factors for converting diameters overbark at breast-height to the various insidebark diameters along the tree profile. cuss* C. lusitanica 4 5 H* i.07 ,-:of ?:rr • 1-33 °- 5 ! - 0 5 I - O L I'.'cj llll . ? • » I-" 1.1* « c.ii s:i: °;H !•;« -x-°7 ^ 1 2 3 1-3* 1.31 1-32 1.16 1.09 1.0? 1.05 1.01 1.030.59 0.97 0.59 C.S5 0.94 0.96 C.33 0.T? 0.39 o.e2 0.91 C • B 7 0.72 0. 74 0. 30 0 - i I 0.65 0.72 0. 52 0.54 . 0.62 0. 41 0.4* C. 51 0.31 0.33 0.37 0.22 0.22 0.23 0.11 C.C9 0. 11 0.0 0.0 0.0 1: 25 15 1-30 1.37 1.00 0.97 0.81 n T I " u * ' a 0. 76 0.75 C 67 2-J* ° ' 7 3 0.68 0. 87 0.84 0-81 0.79 t : o-'-i 4* -0M I'll O.'M ois's 0*55 C - 3 1 0.13 0 37 o"3l 2\S? • J ' " 0.47 0:2 o°: 5' 0°1t 211 c i o o.,4 o.u 0 0 a i f . U 0.0 0.0 «-n « n - . 15 14 6 7 1.34 1.31.' 1.15 1-13 1-07 1.07 1.01 1.01 0.56 0.97 0.S3 0.83 0. 78 0. 76 0.73 0.70 0.66 0.64 0. 60 0.580.52 0.49 0. 39 0.38 0.26 0.24 O.U 0.11 0.0 0.0 15 15 8' 0.0 15 9 1 0 1 1 1.33 1.21 1-59 1. 14 1. 09 1.21 1.05 1.05 l - 07 1.00 1.01 1-03 0.97 0.57 C.9S 0.33 0.7J 0-89 0.75 0.67 0.73 0.71 0.64 C. 73 0.65 0.54 0.54 0.57 0.51 0.40 0.47 0. 39 0.29 0.35 0.30 . 0.24 0.24 0. 19 0.22 0.10 0.11 0. 13 0.0 0.0 0.0 6 I 1 Table XXI. . Factors for converting diameters overbark at breast-height to the various insidebark diameters along the tree profile. 2SH Class* O l d P_. patula 1.27 1.05 1.04 „ u , t . u - 0.97 0.97 0-93 C.98 0.3 1.03 0.97 1.0 1 0.93 0.95 0 . 9 6 0 . 98 . O.S 1.00 C.92 0.97 0.91 0-93 0.94 0.56. 0.9 at, 0.87 0.93 ' 0.89 0.91 0.92 0.93 1.3 C.85 O.S2 0. 89 0.87 C-89 0.89 0.91. I 0.77 0.74 0.85 0. 34 C.85 0.83 0.64 I I 0.70 0.68 0.82 0.82 C 9 1 0.79 0.82 I I I 0.62 0.63 0. 76 C. 78 0.78 0.T5 0.78 IV 0 . 5 5 0.54 0.73 0.73 0-73 0. 70 0. 71 v 0.'9 0.47 C.66 0.68 0.68 0.65 0.66 VI 0 . 3 9 0.40 0.59 0. 62 0.62 0. 50 0.63 VII 0.33 0.31 0.50 0.55 0.53 0.49 0.52 VIII 0.26 0.22 0.38 0.40 0.39 0.36 0-40 IX 0.19 0.16 0.21 0. 22 0.22 0.18 0.20 X 0.0 0.0 0.0 0.0 0.0 0.0 -0.0 Trees 14 16 17 15 17 1 4 2 Each diameter class is 8cm wide, (smallest tree-being less than 8cm) general conversion factor; .SEE - standard error of estimate of F. - 76 -between rates of taper f o r trees below and including c l a s s 3 on one hand and trees l a r g e r than c l a s s 3 on the other; p a r t i c u l a r l y below dbh (1.3 m). The fa c t o r s are, however, markedly d i f f e r e n t between the two species. These find i n g s seem to give some sort of i n d i c a t i o n that trees of P_. patula have d i f f e r e n t stem forms (shapes) at d i f f e r e n t ages (diameter classes) but only up to a c e r t a i n age (size) c l a s s , a f t e r which a l l trees have a s i m i l a r form. As was observed i n f i g u r e s 5 and 6 , these tables f u r t h e r -point to the fac t that the two species have d i f f e r e n t stem forms. 4.3.2.2 The New Developed Taper Models The systematic arrangement of the conversion factors in Tables XX and XXI seem to imply that i t i s poss i b l e to develop a s i n g l e taper model that would reasonably describe the form of trees, in a l l diameter c l a s s e s , from the base to the top. The model must e s t a b l i s h r e l a t i v e r e l a t i o n s h i p s (not absolute) so as to describe the forms of both short and t a l l trees a l i k e . Since t r e e forms generally narrow from the base to the t i p , the model that would allow the rates of form changes to be located at the same r e l a t i v e points should e s t a b l i s h the r e l a t i v e points i n terms of distance from the top, or distance from the ground or both ( i n c l u d i n g p o s s i b l e i n t e r a c t i o n of the two). Because the conversion f a c t o r s (used as a guidance) were obtained from the r e l a t i o n s h i p - 77 -di, /D. f - i ~ l k \ where: , k = size class n = number of trees in the size class the tentative model would then be d - f t 1 h Lh "D~ " f ("IT • ~ H ~ ' O R ~Y— ) (28) H where: h i s the distance from the ground to a point up the tree where the diameter i s to be estimated (in metres) L i s the distance from the top of the tree to the same point (in metres) H i s total height (in metres) d is diameter insidebark at any point along the tree p r o f i l e (in cm) D i s diameter at breast-height (dbh) overbark (in cm) The changes in diameters (or form) along the tree profile being not s t r i c t l y linear, i t was considered that simple linear model would not sufficiently account for the variations. It was considered that the f i t t i n g of polynomials of varying degrees would reveal those degrees - 78 -which would best describe the profile of each species. Because tables XX and XXI also revealed that changes in diameter along the tree profiles do not quite depend on the sizes of the trees, i t was decided that any polynomials would only be on the variables to the right of the equal sign (in equation 28). Since the conversion factors as such are less f l e x i b l e for sectional diameter estimation i t was decided to f i t polynomials of the functions. d = Df ( -|- , - | — , o r ^ | - ) (28') The above equation was f i t t e d on the taper data with polynomials of 'h' or !L' up to 7 degrees using a step-wise regression program. From the equations i t was evident that the f i r s t of the variables accounted for most of the variations i n tree sectional diameter changes. Other variables seem to be necessary to give specific description of the configuration at various levels up the tree. As many as 5 variables were contributing significantly to the models (for P_. patula and CJ. lusitanica) . Apart from the variables that accounted for most of the variations (discussed above) the significant variables were different for the two species. Some of the variables were later excluded from the model. Upon testing for sectional diameters, i t was found that their inclusion did not result in substantial improvement in sectional diameter estimates. In some cases, the estimates in most parts of the upper stem were worse. The estimates close to the ground were, however, better. Moreover, the more variables there were in the - 79 -model, the more d i f f i c u l t i t was, sometimes impossible, to integrate the models f o r s e c t i o n a l volume estimation. I t was decided, therefore to p l o t the various predicted shapes and comparing the predicted forms with a standard form (from a mean t r e e ) . By so doing, i t was possible to get the simplest model that gives the best possible, compromise i n estimating the shapes of the trees species being studied. Figures 7 and 8 show the various shapes of the standard trees that were being predicted by the various models f o r P_. patula and C_. l u s i t a n i c a , r e s p e c t i v e l y . I t could be seen that most models seem to be g i v i n g the same shapes. The simplest models which had the best shapes to the required were: P_. p a t u l a DL DLh 2 d - f (• H ) (29) C. l u s i t a n i c a d = f ( H DL DL h ) (30) patula: Tree forms predicted by new tested models of varying complexities Key 1 variable 2 variables 3 variables A variables 5 variables True profile -4—<£V-DBH lusitanica: Tree forms predicted by new tested models of varying complexities. Key 1 variable 2 variables 3 variables 4 variables 5 variables True profile - 82 -For C_. l u s i t a n i c a , i n p a r t i c u l a r , the model which had most v a r i a b l e s , a l t h o u g h showed ve ry s m a l l s tandard e r r o r of e s t i m a t e s t a t i s t i c , gave the poores t form i n the upper stem. T h i s seemed to c a s t doubt on the v a l i d i t y of s e l e c t i n g taper models on the c r i t e r i a of t h i s s t a t i s t i c . S i n c e s e c t i o n a l d iameters v a r y most c l o s e to the ground (base) i t seems t h a t the b e s t m o d e l , on t h i s c r i t e r i a , would t r y to m i n i m i z e the e r r o r s a t t h i s r e g i o n of the t r e e s . The r e s u l t of t h i s i n v e s t i g a t i o n suggested t h a t t h i s c o u l d l e a d to a c c e p t i n g models t h a t g i v e more p r e c i s e (or ' a c c u r a t e ' ) e s t i m a t e s c l o s e to the ground, at the expense of the more impor tant upper s e c t i o n s of the t r e e . A t f i r s t the models were c o n d i t i o n e d to go through the } o r i g i n by f i t t i n g them w i t h o u t i n t e r c e p t . T h i s was done by i m i t a t i n g model 2 5 . A l t h o u g h such a model gave r e a s o n a b l y good e s t i m a t e s of s e c t i o n a l d i a m e t e r s , when i n t e g r a t e d for" volume e s t i m a t i o n , the performance f o r the P_. p a t u l a model was f a i r l y good but the model f o r f j . l u s i t a n i c a volume was too b i a s e d . T e s t s performed on the models showed t h a t the i n t e r c e p t was s i g n i f i c a n t l y d i f f e r e n t from zero f o r /J. l u s i t a n i c a . As s u c h , i t was d e c i d e d to f i t the models w i t h i n t e r c e p t s f o r both s p e c i e s . These models are g i v e n be low. P_. p a t u l a 2 d = - 0 . 1 0 1 7 6 3 1 8 + 0.96059032 - — - + 1.5697385 DL1? (31) H R 3 R 2 = 0.9836 SEE = 1.543 cm - 83 -C. l u s i t a n i c a 3 d = 0.58088952 + 1.1794184 - + 3.6090853 D L h 4 - 7.0252433 D L h + J .6090853 ~~ " (32) H H 5 R 2 = 0.9766 SEE = 3 .051 cm F i g u r e s 9 and 10 show how the models per form i n p r e d i c t i n g the s e c t i o n a l d iameters of t y p i c a l s tandard t r e e s . I t w i l l be noted how w e l l the models t r y to f o l l o w the p r o f i l e s of the t r e e s . Model f o r C. l u s i t a n i c a "(model 32). . t r i e s to f o l l o w the t r e e stem p r o f i l e t rend but the i n f l e c t i o n p o i n t s are always a t d i f f e r e n t p o i n t s . A l s o worth o b s e r v i n g i s the f a c t t h a t the models g i v e g r e a t e r d iameters below dbh than t h e s t a n d a r d . For P_. p a t u l a , examinat ion of the a c t u a l f i g u r e s showed t h a t t h e e s t i m a t e s were o n l y s l i g h t l y h i g h e r . The a c t u a l v a l u e s were l e s s than the o u t s i d e bark s e c t i o n a l d i a m e t e r s , which was encourag ing . F o r . C. l u s i t a n i c a , however, est imates were reasonab ly good f o r s m a l l t r e e s . Fo r l a r g e t r e e s , however, the b r e a s t - h e i g h t d iameters were even 4 cm more than the dbh overbark. ' The model was cons idered u n s a t i s f a c t o r y f o r s e c t i o n a l d iameter e s t i m a t i o n . As has been mentioned e a r l i e r poor e s t i m a t e s may r e s u l t from the f a c t that the model developed was a t t e m p t i n g to min imize e s t i m a t e s c l o s e to ground l e v e l and tha t by so d o i n g i t may be g i v i n g poor e s t i m a t e s above t h i s l e v e l . The upper - 86 -stem e s t i m a t e s b e i n g more important than the ground l e v e l e s t i m a t e s , i t was dec ided to e x c l u d e , f o r CJ. l u s i t a n i c a model , a l l the o b s e r v a t i o n s at ground l e v e l i n f i t t i n g a new model . The f o l l o w i n g model was adopted as my f i n a l s e c t i o n a l d iameter e s t i m a t i o n model f o r C. l u s i t a n i c a . C. l u s i t a n i c a d = 0.789673 + 1.10804 + 2.25910 D L ^ h - 5.56962 ^ h (33) A J . J U 3 U i — H H R 2 = 0 .9849 SEE = 2 .2591 F i g u r e 11 shows the performance of t h i s model f o r s e c t i o n a l d iameter e s t i m a t i o n . A l though i t was s t i l l s l i g h t l y o v e r e s t i m a t i n g the s e c t i o n s below dbh i t was now g i v i n g v e r y good e s t i m a t e s f o r most p a r t s of the t r e e . However, t h i s new v e r s i o n now s l i g h t l y ove res t imated d iameters c l o s e to the t i p of the t r e e as w e l l . The s e c t i o n a l d iameters f o r IP. p a t u l a c o u l d perhaps a l s o have been improved i f o b s e r v a t i o n s a t ground l e v e l were not i n c l u d e d i n f i t t i n g the m o d e l . Rut t h i s cou ld perhaps have a l s o r e s u l t e d i n s l i g h t o v e r e s t i m a t i o n a t the r e g i o n s near the t i p o f the t r e e s . The performance of the P_. p a t u l a model , produced by u s i n g a l l the d a t a , was c o n s i d e r e d good f o r p r a c t i c a l purposes . - 88 -Because the rate of tapering is highest close and below dbh, figures 9 and 11, which were plotted from the adopted models, may give an impression that the models do not give smooth diameter changes at these points. As such i t was decided to generate more points at these regions to see how exactly the diameters change in the region. Figures 12 and 13 show that the models actually give smooth and continuous pr o f i l e changes. The predicted dimensions are actually very close to the true values. The models were developed, as has been shown above, after a long, systematic approach. Many systematically increasing polynomials of the variables i n equation 28 were tried as potential variables. It should be remembered that attempts to add even one other variable to the adopted models not only worsened the diameter estimates (in some cases) but also made the models impracticable to integrate for volume. These models could possibly be standard taper models - one model describing the profiles of tree species with marked butt swell while the other takes care of tree species with smooth profile changes throughout the stem. Any intermediate forms could be predicted by either of the two. When both models are tried on any given species, whichever one shows the lower standard error of estimate would be the more appropriate for that particular case. To get a feeling on the performance of a l l the taper models studied, let us examine figures 14 to 19. The figures show the performance of the various taper models in predicting (or estimating) the various sectional diameters of the standard mean trees (plotted in dashes) for three different sizes of trees. Figures -14 to 16 are for _P. patula and 17 to 19 are for C_. lusitanica. They were a l l plotted on the same scale. - 68 -Figure 14. patula: Graphical representation of tree forms and diameter estimates given by all the 5 taper models tested; shown for a small tree. Key Model 20 Model 23 !*s—ki— Model 24 1 1 ' Model 31 ^ Model 25 Q ^ Actual profile i VO Figure 15. p_. patula: Graphical representation of tree forms and diameter estimates given by all the 5 taper models tested; shown for a medium-sized tree. Key. Model 20 Model 23 Model 24 Model 31 Model 25 Actual profile f---*r-Figure 16. P_. patula: Graphical representation of tree forms and diameter estimates given by a l l the 5 taper models tested; shown for a large tree. Key Model 20 Q Qr-Model 23 — Model 24 ——-+-Actual p r o f i l e Model 31 — Model 25 Q Q VO C J L U CO C3 Ll • or U J r— UJ g° •—.lO. Q 1 D Figure 17. £. lusitanica: Graphical representation of tree forms and diameter estimates given by all the 5 taper models tested; shown for a small tree. Key K £ Y . Model 18 f $ ® — A £r Model 23 Model 24 Model 33 Model 25 4-Actual profile — — ^ — — -1 1 1 1 1 1 1 1 1 1 0.0 4.0 8.0 12.0 1S.0 20.0 24.0 28.0 32.0 35.0 40.0 DISTANCE FROM TOP (m) o I F ± g U r e 1 8' ^ l u s i t a n l c a : Graphical representation of tree forms and diameter 4j estimates given by a l l the 5 taper models tested; shown for a — m e d i u m - s i z e d tree. Ke Model 18 ©—Q~ Model 23 Model 24 Model 33 2 o Model 25 v/- v 2""" Actual profile — 4""" ~ vo Figure 19. £ lusitanica: Graphical representation of tree forms and diameter estimates given by all the 5 taper models tested; shown for a large tree vo T 16.0 2 0 . 0 2 4 . 0 DISTANCE FROM TOP (m) 4 0 . 0 - 97 -From the figures i t w i l l be possible to judge at a glance which models gave closest sectional diameter estimates to the standard. Such a model would also give the best volume estimates of the standard upon integration. Although for small trees a l l the models gave f a i r l y similar profiles, except models 25, 31 and 33, the differences become more apparent as we consider larger and larger trees. For a l l cases, models 31 and 33 continue to follow the general trend of the tree profile changes. At most points, the two models are giving estimates which are almost identical to the standard. Although model 25 also attempts to follow the tree profile i t seems to lack sufficient inflection and at the right points for i t to give true estimates of the sectional diameter. Note that for this model the part below dbh was plotted using a separate model below dbh. It i s the part above dbh that was of interest to this study. The new models can also be integrated for volume estimation. The models are integrated with respect to L. The following volume estimation models are obtained from the new taper models (i.e. model 33 and 31, respectively). C. lusitanica 2 2 3 A V s = 0.60007854 ( b 0 2 • b o b l Sj}- + bj i - i - + b Q b 2 2 i j + J r i J.H 5 2 5 6 ' (2b b - 2b b ) ~~~J7 + 2bb 5_L _ DL + 0 3 0 2 5 R4 1 2 5 R4 0 3 3 H5 D 2L 6 2 D 2L 7 <bib3 - biV + ( b2 - 2 bi b3>. ^ r- + - 98 -b2 A " 3 nr ; (34) 11H u SEE = 0.111053 m3 + patula V s = 0.00007854 (b 2 + ( b ^ + b ^ ) , -f~ + ( b 2 + b 2 + 2 b l b 2 ) ° 2 o , ^ 3H2 - 4 b 0 b 2 ^ + b 0 b 4 - ( b 2 + W 7 + ( 6 b 2 + 2 b l b 2 ) - 2b2 ^ + b 2 ) ( 3 5 2 3H 5 2 7H6 SEE = 0.073113 m3 The coefficients b^, b^, etc. are as they are given i n the relevant taper models. Although these models may appear too long and complex their development i s simple and straight forward. They also indicate that trees are not a l l that simple to describe (volume-wise). Models which merely use a few variables to predict the tree volumes may not give a true picture of the variations in the tree growth characteristics that must properly be accounted for to obtain r e a l i s t i c volume contents of such trees. The procedure for deriving these models is shown in detail in Appendix 8. - 99 -Since u t i l i z a t i o n limits are always specified in terms of upper stem sectional diameters, an integral part of this study was to get models that give good estimates of distances from the top of the tree to these points. Most authors have been obtaining this value from the respective taper models by deriving i t mathematically, by simply making the 'L' term in the taper model the dependent variable and using the coefficients in taper model. Poor volume estimates may be obtained simply because this distance has not been properly estimated. It was considered that the development of such a model i s just as much important as the development of a taper model just discussed. As much use as possible was made of the relationship noted in the taper study. The variables did not necessarily have to be complete derivatives of the taper models. Other variables were also considered. As observed earlier the most important relationship in describing the tree profile i s the linear relationship between diameter at any given point and the distance from the top of the tree to that point. By making this distance the dependent variable the simple linear taper model could change to: L = b, + b d H 0 ul D (36) This model was f i t t e d for both C. lusitanica and P. patula. The model accounted for over 95% of variation in the values of in C. lusitanica but only 92% in P. patula. As such i t was decided to f i t more variables, - 100 -tha t i s , p o l y n o m i a l s of the independent v a r i a b l e i n the above model . No s i g n i f i c a n t improvement was ach ieved f o r C. l u s i t a n i c a . However, 2 i n P_. p a t u l a ^by i n c r e a s i n g the number of v a r i a b l e s to 4, the R v a l u e i n c r e a s e d to j u s t over 95%. As was done i n taper s t u d y , i t was dec ided to exc lude a l l the o b s e r v a t i o n s a t ground l e v e l i n f i t t i n g the model f o r C. l u s i t a n i c a . At tempts to do the same f o r P_. p a t u l a data produced worse e s t i m a t e s . The models f i n a l l y adopted a r e g i v e n below: C. l u s i t a n i c a L = - 0 . 6 0 2 1 6 + 0 .948391- dH "D (36') R 2 = 0.9857 SEE = 1.7164 m JP. p a t u l a L = - 0 . 8 3 3 7 3 + 1.00088 - 0.0007275586 + 2 2 3 0.0009925553 - 0.0000002889 - ( 3 7 ) R 2 = 0 .9521 SEE = 2.2890 m - 101 -When the residual 'L' estimates were plotted against, the observed 'L' i t was observed that model 36' gives reasonably good estimates. But a similar model (i.e. model 36) when fitted for P_. patula showed serious hunch at the middle sections indicating that the model was seriously overestimating the mid-section distances from the top of the tree. Although model 37 was a substantial improvement i t did not completely eliminate the mid-section overestimations, for some trees. The plots of residuals for sectional diameter and distances from the top of the tree for these new proposed models are given in Appendix 9(a) to 9(d). The plots of residuals for volume estimation showed very good resemblance to the plots for model 15 given in Appendix 2(d). 4.3.2.3 Tests on the New Taper Models for Sectional Diameter and Volume Estimations Tests as described in Section 4.2.1 were carried out to find out the bias and the standard errors of estimates at the various measurement points for trees in the different diameter classes. Tables XXII and XXIII give the results of the tests with respect to diameter show results of estimation; tables XXIV and XXV similar tests with respect to volume estimation. It w i l l be noticed from tables XXIII in particular that the sectional diameters for (J. lusitanica are overestimated at a number of points. The overestimates are, however, not too large for a l l diameter classesexcept for trees in classes 10 to 11 where the estimates may not be good because the sample sizes in these diameter classes were Table XXII. - 102 P. patula^ Efficiency of the New Taper Model 31 for Diameter Estimation By rows: l=bias/tree; 2=bias/m ; 3=standard error of estimate. DBH Class I 2 3 4 5 6 7 0.0 2.129 C.402 2.416 1.235 0. 130 1.686 1.682 0.069 2.450 0.2 30 0.008 1.421 o. 5 e o 0.019 1.4 65 1 .C27 0.03 3 2.130 1.184 0.037 0.3 1.222 0.231 1.386 0.620 0. C65 1.054 1.286 0.C53 2.263 -0.566 -0.020 1.688 0.226 O.0C7 1.210 0.355 0.011 1.935 1.334 0.042 0.6 C.?75 C. 184 1.131 0.268 0.C28 0.701 0.733 C.C30 1.775 -0.799 -0.029 ' 1.790 -0.2 70 -0.CC9 1.189 0.149 0.005 1.834 0.721 0.023 0.9 0.778 0.147 C.955 -0.04 0 -0.004 0.533 0.259 0.011 1.017 -0.968 -C.035 1.910 -0.723 -0.C24 1 .3 34 -0.441 -0.014 1.543 -0.106 -0.003 1.3 0.429 0.C81 C.5C6 -0.364 -0.038 0. 554 -C.308 -0.013 0.729 - I .110 -C.040 1.703 -1.166 -0.038 I *422 -1 .474 -0.047 1.960 -0.612. -0.019 I 0.165 0.031 C.467 -0.716 -0.076 0.866 0.226 0.009 0.844 -0.146 -0.005 1.389 0.055 0.002 1 .490 -1.102 -0.035 1.781 -0.423 -0.014 II -0.053 -C.010 0.405 -0.915 -0.097 1.C95 0.521 0.021 1.132 0.774 0.028 1.468 0.514 0.017 1.813 -0.395 -0.013 1.698 0.899 0.028 III -C.350 -C.C66 C.494 - 1 . 172 -0.124 1.404 0.154 0.C06 1 .367 C.775 0*028 1.652 0.757 0 .025 2.051 -0.272 -0.009 1.809 1.189 0.038 i -IV -0.528 -C.100 0.697 -1.668 -C.176 1.905 0.4 76 0.020 1.112 0.639 0.C24 ' 1.726 0.507 0.017 2. 140 -C.699 -0.023 1.961 -0.241 -0.00S V -0.567 -0.107 0.705 -1.966 - C . 208 2. 192 0.093 0.004 0.933 0.620 C.022 1.963 0.618 0.020 1.960 -0.373 -0.012 2.215 -0.174 -0.005 VI -0.713 -C.13'5 0.833 -1.978 -C.2C9 2.271 0.116 0.005 1.302 0.774 0.028 1.714 0.830 0.C27 1.784 -0.733 -0.024 2.405 1.509 0.043 VII -C.569 -0.107 C.682 -1.936 -0.204 2. 179 0.C97 C.C04 1 .546 1.395 C.050 2.560 0.9 79 0. 022 2.293 -0.395 -0.013 2.452 1.025 0.032 VIII -0.271 -0.051 0.422 -1.464 -0.155 1 .710 C. 164 0.1C7 1.813 0.648 C.023 1.047 0.476 0.016 2.925 -0.74 3 -0.024 2.007 1.093 0.034 IX 0. 184 C.035 0.316 -0.305 -0.032 C.54 0 0.C99 0.004 1.299. 0.291 0.011 1.071 0. 3 74 0.012 2.4 73 -1.436 -0.046 2.623 -0.671 -0.021 X 0.071 C.013 0.C76 0. 068 0.CC7 0.013 c.cno 0.003 C.CU6 0.076 0.003 0.002 0.073 0.002 0.C77 0.066 0.002 0.071 o.of.;: 0.C-C2 . Table XXIII. C. lusitanica: Efficiency of Taper Model 33 for Diameter Estimation. DBH Claaa 0.0 1.1*3 0.170 1.676 1.515 0. 160 1.899 3.320 0.217 * .«*5 *.563 0.220 6.373 0.3 0.518 0.077 1.110 0.163 0.017 0.781 0. 072 . 0.005 1. **5 0.970 0. 0*7 2.33* 0.6 . 0.457 0.068 0.88* 0.150 0.016 0.650 0.031 0.002 1.588 -0.209 -0.010 l . * 5 7 0.9 0.*36 0.065 0. 756 0.3*9 0.037 0.628 0.077 0.005 0.606 -0.320 -0.015 1.020 1.3 0.529 0.079 0. 706 0.625 0.066 0. 769 0.5 75 0.0 38 0.918 0.015 0.001 0.711 I 0.368 0.055 0.558 0.86* 0.091 0.986 1.76* 0.115 2. 191 1.278 0.062 1.916 I I 0.030 0.00* 0.*72 0.593 0.063 0.895 2. 113 0.138 2.587 1.5*7 0.075 2.260 I I I -0.138 -0.021 0.515 0.206 0.0?2 0.723 1.565 0. 102 2.093 0.562 0.027 2.053 - 0 . * 3 9 -0.107 1 .098 0.*6& 6.8*1 9.050 0.322 0.290 l l . * 7 9 11.851 1.202 2.*17 0. 0** 0.078 2.619 3.*83 -0.392 0.179 - 0 . 0 1 * 0.006 1. *93 1.971 -0.7*3 -0.965 -0.027 -0.031 l . * * 2 1.752 -1.139 -1.706 -0.0*1 -0.055 1.532 2.158 0.296 -0.173 0.011 -0.006 1.226 2.077 1.311 0.970 0.048 0.031 2.001 2.795 0.988 0.9*6 0.036 0.030 1.88* 2.84* 1.3 12 0.093 0.303 2.860 IV -0.065 -0.011 0.072 0.022 0.037 0.704 1.069 1.707 2.088 1.926 -0.503 -0.509 0.823 0.103 3.839 V -0.075 -0.054 0.054 0.005 0.031 „.„«., 0.875 1.223 1.410 2.819 1.818 2.877 -0.603 -0.535 0. 572 -0.301 1 .5*8 1.616 VI -0.090 -0.056 0.037 - 0 . 0 1 * 0.056 0.052 0.758 0.02* 0.927 1.102 1.36* 2.865 2.743 -0.613 -0.589 0.056 -0.512 1.707 V T T -0.091 -0.162 0.00* -0.025 0.062 0.98* 0.991 1.203 2.685 3.085 -0.534 -0.549 -0.439 -0.707 1.503 VIII -0.079 -0.058 -0.029 -0.034 0.055 0.798 0.800 0.992 2.262 3.208 -0.670 -0.832 . -0.739 -0.925 0.265 -0.587 T , - 0 . 103 -0.088 -0.048 -0.345 3.0 10 -0.019 0.896 0.987 1.062 1.551 1.766 1.506 -0.301 -0.803 - 0 . 8 0 * -0.805 - 0 . 8 0 * -0.805 T - 0 119 -0.095 -0.052 -0.039 -0.029 -0.026 1.005 0.876 0.939 0.940 0.952 0.940 3.423 1.105 0.035 3.504 0.571 0.018 2.059 7 8 9 10 11 9.648 0.304 13.964 11.962 0.373 15.966 13.635 0.433 27.785 6.753 0.179 40.3*3 1.385 2.212 0.070 9.6 76 2.992 0.093 10.559 3.535 0.112 10.909 -0.318 - 0 . 0 0 0 11.104 0.376 0.4*2 0.014 2.236 -1.18* -3.037 8.462 0.239 0.008 5.036 -1.405 -0.037 1.850 0.363 -1.0 17 -0.032 1.836 -1.667 -0.052 3.709 -1.778 -3.355 3.95* -2.502 -0.366 0.89* 0.330 -1.315 -0.057 2.228 -1.960 -0.06 1 2.515 -1.8*7 -3.359 3. 3*1 -3.261 -0.085 -0.309 -0.310 -0.118 -0.004 1.864 -I.146 -0.036 4.073 0.404 0.013 2.995 -3.833 -0.131 5.890 0.200 0.2 6* 0.008 1 .99* -0.262 - 0 . 0 0 8 2.641 0.373 0.012 3.976 -5.826 - 0 . 154 3.174 0.108 - 0 . 152 -0.005 2.171 -1 . 158 -0.036 2.734 0. 706 0.022 3.976 -4.814' -0.127 3.318 0. 102 - 0 . * 1 * -0.013 2.23* -1.461 -0.O46 2.760 -0.027 -0.301 2.983 -8.147 -0.216 -8.877 -0.301 0 .037 0.001 2.235 -1.32* -0.0*1 2.721 - 0 . 123 - 0 . 0 0 * * . 1 7* - 4 . 793 -0.127 -15.029 -0.539 0.883 0.028 2.311 -0.058 -0.002 3.*20 -0.066 -0.332 *.760 -6.782 -0.179 -15.706 -3.532 0.991 0.031 3. 107 -0.113 -0.034 2.910 - 0 . 6 11 -3.319 4. 322 -4.383 -3.115 - 1 0 . 167 -0.345 -0.117 - 0 . 0 0 * 2.097 -1.043 -0.033 2.693 0.097 0.033 5.868 -3.675 -0.097 -I.199 -3.341 -0.*83 -0.015 2.00* -1 .632 -0.051 2.225 -1.135 -0.036 3.833 -0.731 -0.019 1.5*5 0.052 •0.808 •0.025 0 . 9 * * -0.810 -0.025 0.945 -0.813 -0.026 1.438 -0.812 -0.021 -0.922 -0.028 By rows! l * b i a 8 / t r e e ; 2-blas/raetre; 3-standard error of estimate. - 104 -Table XXIV. P. patula: Efficiency of Taper Model 31 for Volume Estimation. By rows: l=bias/tree; 2=bias/mJ; 3=standard error of estimate. DBH Class 1 2 3 4 5 6 7 0.0 -0.000 0.0 c .coo - 0 . 0 0 0 0.0 C. 000 - o . c o o 0.0 o.coo - 0 . 0 00 0.0 o.oco -0.OCO 0.0 0.000 - 0 . 100 0.0 0.000 -0 .000 0.0 0.3 0.001 C.393 0.001 0.001 0. IA? 0.0C1 0.C01 0.140 0.C02 - 0 . 0 00 -0 .010 0.002 0.001 0 .024 0.0C2 0.001 0.033 0.004 0.003 0.049 0.6 0.001 0.351 0.001 0.001 0. 109 0.001 0.002 C. 121 0.C04 -0.001 - 0 . 0 2 9 0.004 O.OCI 0.012 0.0C4 0.00? 0.024 0.0C8 0.005 0.045 0.9 0.001 0.325 0.001 0.001 0. OF/i 0.001 0.003 C.C99 0.005 -0 .002 -0.041 0.006 - 0 . 0 00 -0.001 0. 005 0.C02 0.014 0.011 0.006 0.034 1.3 C.001 0.29 5 • 0.002 0.001 0.052 0.002 0.003 C.C71 0.006 - 0 . 003 - 0 . 053 0.009 - 0 . 0 02 - 0 .018 0.007 -0.001 - 0 . 0C4 0.013 0.005 0.020 I C.001 0.265 0.002 - 0 . 000 - 0 . o c o 0.002 0.CC3 0.C23 0.009 -C .008 - 0 . 0 4 6 0 .0 20 -o .oon -0 .C24 0.010 - 0 .022 - 0 . 045 0.039 - 0 .006 -0 .009 I I 0.001 0.235 ' C.C02 - o . o c i - 0 . 040 0.002 0.CC5 0.035 0.014 - 0 . 0 0 4 - 0 . 015 0.029 - 0 . 002 - 0 . 0 0 3 0.035 - 0 . 034 - 0 . 043 0.065 -0 .002 -0 .002 I I I ' 0.001 0.206 C.C02 - 0 . 0 0 2 - 0 . 074 0. CC4 0.C06 0.039 C.C20 0.004 0.009 0.041 0.CC8 0.012 0.058 - 0 . 039 - 0 . 0 3 7 0.090 0.016 O .OU IV 0.001 C.177 0.C02 - 0 . 0C4 - O . l l l 0.CC5 0.010 0-042 0.C26 C.011 0.02? 0.055 0.018 0.021 0.C82 -.0. 046 - 0 . 036 0.117 0.023 0.012 V 0.001 C. 151 0.002 - 0 . 005 - 0 .152 0.CC7 0.012 0.04 3 0.C31 0.017 0".029 C. 069 0. 025 0.025 0. 104 - 0 . 0 5 5 - 0 . 037 0.144 0.017 0.008 V I 0.001 C. 1 29 0.002 - 0 . 0 07 - 0 . 189 0.C09 0.01 3 C.C42 0.C36 0.023 0.035 0.083 0.033 0.030 0. 122 - 0 . 06 3 - 0 .038 0. 171 0.025 0.011 V I I C.C01 . 0.111 0.001 - 0 . 009 - 0 . 2 IB 0.011 0.014 0.042 0.041 0.030 0. 044 0.096 0 .042 0 .0 34 0.138 - 0 . 0 70 - 0 . 039 0.196 0.041 0.016 V I I I C.CC1 0.102 C.001 - 0 . 009 - 0 . 239 0.012 0.015 C.043 0.045 0.037 0. 051 0. 106 0.049 0.038 0.154 - 0 . 0 7 5 - 0 . 040 0.217 0.0 52 0.019 IX C.001 0 . 10 I C.COl - 0 . 0 1 0 - 0 . 2 45 0.012 0. CIf. 0.045 0.040 0.04 0 0. 054 0.112 0.053 0.040 0.162 -0 .079 -0.041 0.229 0.057 0.021 X 0.001 C.10? 0.001 - 0 . 0 1 0 0.012 C.C17 0.G4 7 C. C49 0.042 0.056 0.115 0. 056 0.0 4? 0.165 - 0 . 079 -0 .041 0.231 0.0 r*9 0.021 Table XXV. 'C, lusitanica; Efficiency -of Taper Model 33 for Volume Estimation-DBH Class 1 2 3 '4 •5 * 7 18 •9 10 11 0.0 •0.000 0.0 0. 000 -0.000 0.0 0.000 -0.000 0.0 o.coo - C . 0 0 0 0.0 0.000 -0.000 0 .0 0.000 - 0 . 0 0 0 0.0 0.000 -0.000 0.0 0.000 - 0 . 0 0 0 0.0 0.000 - 0 . 0 0 0 0.0 0.000 - 0 . 0 0 0 0.0 - 0 . 0 0 0 0.0 0.3 0.000 C.174 0.001 0.001 0.118 0.001 0.002 0.152 0.003 0.005 0.162 0.007 0.011 0.2 21 0.015 0.014 0.210 '0.019 0.019 0.189 0.031 0.026 0.210 0.040 0.034 0.209 0.075 O.014 0.078 0.139 0.393 0.6 0.001 0. 160 0.001 0.001 0.078 0.001 0.002 0.093 0.C04 0.005 0.1 02 0.009 0.012 0.136 10.016 0.017 0.142 , 0.024 0.023 0.128 0.043 0.031 0.138 0.059 0.041 0.141 0.099 O . O U 0.034 0.169 0.289 0 .9 0.001 0. 155 0.001 0.001 0.068 0.001 '0.002 0.071 0.005 0.005 0.069 0.010 0.0 11 0.090 0.016 0.017 0.098 0.024 0.022 0.089 0.043 0.028 0.090 0.065 0.038 0.096 0.104 0.0 04 0.009 0.175 0.224 1.3 0.001 0.155 0.001 0.001 0.072 0.002 0.003 0.C63 0.006 0.005 0.048 0.010 0.009 0.054 0.014 0.013 0.058 0.022 0.018 0.052 0.041 0.021 0.051 0.065 0.031 0.058 0.101 - 0 . 0 0 9 -0.315 0.177 0.173 I 0.001 0.154 0.002 0.002 0.095 0.003 0.008 0.C99 0.011 0.011 0.056 0.017 ' 0.0 08 0.021 0.017 0.004 0.007 0.032 0.0 10 0.011 0.048 -0 .000 -0.000 0.074 0.034 0.024 0.113 -0.101 -0.053 0.308 0.124 II 0.001 0.141 0.002 0.003 0.111 O.004 0.015 0.135 0.019 0.022 0.076 0.032 0.020 0.036 0.035 0.013 0.015 0.066 0.017 0.013 0.072 -0.010 -0.007 0.126 0.053 0.026 0.177 -0.242 -0.087 0.464 0.127 III 0.001 0.124 0.002 0.004 0.113 0. 005 0.022 0.154 0.027 0.030 0.082 0.045 0.034 0.047 0.056 0.027 0.0 24 0.103 0.021 0.013 0.097 -0.024 - 0 . 012 0. 169 0.069 0.026 0.254 -0.399 -0.114 0.554 0.121 IV o.om C.104 0.002 0.004 0. 108 0.006 0.026 0.158 0.033 0.034 0.080 0.057 0.044 0.052 0.075 0.034 0.026 0.137 0.017 0.009 0.122 -0.050 -0.021 0.211 0.077 0.025 0.310 -0.576 -0.142 0.492 0.093 V 0.001 C. CR7 0.002 0.005 0. 100 0.007 0.029 0.1 57 0.036 0.038 0.079 0.069 0.053 0.055 0.091 0.040 0.027 0. 166 0.015 0.007 0.147 -0.075 -0.02 8 0.250 0.076 0.022 0.350 -0.740 -0.164 0.249 0.044 VI 0.001 0.073 0.002 0.004 0.092 0.008 0.030 0.155 0.033 0.040 0.077 0.080 0.064 0.061 0.107 0.055 0.033 0.192 0.022 0.009 0.168 -0.084 -0.029 0.281 0.077 0.020 0.390 -0.860 -0.177 -0.004 -0.001 VII 0.C01 0.064 0.002 0.004 0.086 0.008 0.031 0. 153 0.040 - 0.041 0.075 0.086 0.076 0.069 0.126 0.071 0.041 0.212 0.036 0.014 0. 1 88 -0.082 -0.027 0.300 0.077 0.019 0.429 -0.952 -0.189 -0.173 -0.029 VIII 0.001 0.058 O.C02 O.004 0.082 0.008 0.031 0.151 0.040 0.041 0. 074 0.090 0.087 0.076 0.142 0.080 0.045 0.220 0.044 0.017 0.203 -0.082 -0.026 0.308 0.078 0.019 0.470 -0.995 -0.193 -0.232 -0.038 IX 0.001 0.055 0.002 0.004 0.079 0.008 0.031 0.149 0.040 0.041 0.073 0.092 0.092 0.080 0.151 0.083 0.046 0.222 0.045 0.017 0.208 -0.084 -0.027 0.311 0.081 0.020 0.497 -1.010 -0.195 -0.229 -0.037 X 0.001 C.054 0.002 0.004 0.078 0.008 0.031 0.148 0.039 0.041 0.073 0.092 0.093 0.080 .0.153 0.084 0.046 0.222 0.046 0.018 0.209 -0.085 -0.027 0.311 0.031 0.020 0.502 -1.008 -0.194 -0.222 -0.036 O , <J1 By rows: 1 -blas/tree; . ' - b i a s / a 3 ; 3-standard e r r o r of estimate. - 106 -too s m a l l . As was e x p e c t e d , the d iameters at the ground l e v e l are s e r i o u s l y u n d e r e s t i m a t e d . Tab le XXV, produced by i n t e g r a t i n g the same m o d e l , g i v e s a c o m p l e t e l y d i f f e r e n t p i c t u r e from what would have been e x p e c t e d . I t shows t h a t the volumes are underes t imated a t most p o i n t s . Because the taper model adopted f o r l u s i t a n i c a i s shown ( i n t a b l e X X I I I ) to be g i v i n g s l i g h t l y more s e c t i o n a l d iameter e s t i m a t e s at a number of p o i n t s i n some d iameter c l a s s e s , i t should be expected to g i v e s l i g h t l y more volume e s t i m a t e s a t those p o i n t s as w e l l . However, T a b l e XXV shows t h a t the volume e s t i m a t e s were a c t u a l l y l e s s than the e s t i m a t e s expected (by the s t a n d a r d ) . The anomaly i s e x p l a i n e d i n Chapter 5 . F i g u r e s 20 and 21 show i n g r a p h i c a l forms t y p i c a l volume e s t i m a t e s up the t r e e s o b t a i n e d by i n t e g a t i n g the proposed taper mode ls . The t h r e e p a i r s o f c u r v e s r e p r e s e n t t r e e s o f d i f f e r e n t s i z e s . The curve marked a t i n t e r v a l s a l o n g the p r o f i l e r e p r e s e n t the t r u e or mean t r e e volume e s t i m a t e s (obta ined by the s t a n d a r d ) . The f i g u r e s show examples of the volume e s t i m a t e s to be e x p e c t e d . I n d i v i d u a l t r e e volume e s t i m a t e s w i l l v a r y around the l i n e s ( i . e . f o r t r e e s w i t h the same d i m e n s i o n s ) . As d i s c u s s e d l a t e r i n Chapter 5 each of the p a i r s a r e a c t u a l l y s t a t i s t i c a l l y i d e n t i c a l by computing 95% c o n f i d e n c e l i m i t s of the e s t i m a t e s . The 95% c o n f i d e n c e l i m i t l i n e s o v e r l a p both the t r u e and the p r e d i c t e d vo lumes. F u r t h e r d i s c u s s i o n on the w i d t h s of such i n t e r v a l s i s found i n Chapter 5 . Figure 21. C_. lusitanica: Graphical representation of volume estimates given by the adopted taper model. 0.0 4.0 8.0 12.0 1S.0 20.0 24. C 28.0 Z2.Q 35.0 40.0 DISTANCE FROM GROUND (m) - 1 0 9 -4,4 Other Interesting Relationships 4.4.1 Approximate Sectional Diameter Estimates As observed i n Tables XX and XXI there i s a very good c o r r e l a t i o n between diameter at breast-height (overbark) and other sectional diameters up the tree. These tables could, therefore, be used to get good estimates of sectional diameters at any of the referenced points f o r any tree within the same species. Because i n C. l u s i t a n i c a the conversion factors are b a s i c a l l y the same at a l l corresponding points along the tree p r o f i l e for a l l diameter classes, the o v e r a l l conversion factor at each section (given i n the second to l a s t column) could be used for general purposes to determine sectional diameters at the various points. In P_. -patula where the size class of the tree seems to have some s l i g h t effect on the rate of sectional diameter changes, generalization i s not recommended. Although the variances of the conversion factors were not s t a t i s t i c a l l y tested, to find out i f they are r e a l l y d i f f e r e n t i t was evident, by v i s u a l inspection, that they change d i f f e r e n t l y for trees i n d i f f e r e n t diameter classes. As such the conversion factors corresponding to the various diameter classes have to be used. For trees larger than the sizes given, the factors for the largest given s i z e class should give f a i r l y good estimates, When int e r p o l a t i o n i s done between the reference 1 points the estimates w i l l not be very accurate. The two tables can also be used a l o t i n practice instead of taper tables. - 110 -4.4.2 Approximate Utlllzable Tree Volumes Tables XXVI and XXVII below show the distributions of total tree volumes along the stem from the base to the tip. The tables show the proportions of total volumes that are available at the various levels from the ground for P_. patula and C_. lusitanica respectively. These tables were constructed from the information in tables in Appendices 6 and 7 . Using also the table in Appendices .4 and 5 i t was possible to determine, for trees i n each diameter class, how far from the ground one has to go up the trees to reach the various u t i l i z a t i o n limits. From tables XXVI and XXVII i t was possible to approximate the proportions of the total tree volumes that correspond to the various u t i l i z a t i o n limits. Table XXVIII shows this information for both C. lusitanica and P^. patula. The stump volume proportions, which had to be removed from the various proportions obtained from tables XXVI and XXVII, was assumed to be half the volume at 30 cm. In other words, the stump height was assumed to be 15 cm. If, while using Table XXVIII, trees are encountered which are larger than the maximum classes, the proportion values for the largest diameter classes would be applicable. Although not s t a t i s t i c a l l y tested this table should give f a i r l y good estimates of u t i l i z a b l e volumes. S t a t i s t i c a l testing would have required that the data be divided into several groups and tested whether each group was giving similar proportions. The total number of observations in each diameter class was not large enough for this to be done. These tables are to be used with standard total tree volume tables. Table XXVI. Proportions of Total Volumes That Are Available Up to Various Sections Along the Tree Profiles For Trees of Various Sizes for P. patula. DBH CLASS Measurement Point 0.0 m 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.30 m 0.13 0.10 0.03 0.02 0.02 0.02 0.02 0.60 m 0.25 0.18 0.05 0.04 0.04 0.04 0.04 0.90m 0.38 0.25 0.08 0.06 0.06 0.06 0.06 1.30m 0.63 0.35 0.11 • 0.09 0.08 0.09 0.08 I 0.63 0.50 0.27 0.24 0.25 0.26 0.25 II 0.75 0.63 0.42 0.39 0.39 0.41 0.40 III 0.88 0.75 0.56 0.53 0.53 0.55 0.54 IV 0.88 0.83 0.67 0.65 0.65 0.67 0.66 V 1.00 0.90 0.78 0.76 0.76 0.78 0.76 VI 1.00 0.95 0.86 0.85 0.85 0.87 0.85 VII 1.00 0.98 0.93 0.92 0.92 0.93 0.93 VIII 1.00 1.00 0.97 0.97 0.97 0.98 0.97 IX 1.00 1.00 0.99 0.99 0.99 1.00 1.00 X 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Table XXVII. Proportions of Total Volumes That Are Available Up to Various Sections Along the Tree Profiles For Trees of Various Sizes for C. lusitanica. DBH CLASS 1 2 3 4 5 Measurement Point ' 0.00 0.00 0.0m 0.00 0.00 0.00 0.30m 0.15 0.10 0.06 0.05 0.04 0.60m 0.23 0.16 0.11 0.09 0.07 0.90m 0.31 0.22 0.15 0.13 0.10 1.30m 0.46 0.29 0.20 0.17 0.14 I 0.62 0.45 0.38 0.36 0.32 II 0.69 0.59 0.53 0.52 0.48 III 0.77 0.71 0.67 0.65 0.61 IV 0.85 0.80 0.78 0.77 0.73 V 0.92 0.88 0.87 0.86 0.83 VI 0.92 0.94 0.93 0.93 0.90 VII 1.00 0.96 0.98 0.97 0.95 VIII 1.00 0.98 1.00 0.99 0.99 IX 1.00 1.00 1.00 1.00 1.00 X 1.00 1.00 1.00 1.00 1.00 6 7 8 9 10 11 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.04 0.04 0.04 0.03 0.06 0.07 0.07 0.07 0.07 0.06 0.10 0.09 0.10 0.10 0.10 0.09 0.13 0.13 0.13 0.13 0.13 0.12 0.17 0.32 0.33 0.34 0.34 0.37 0.40 0.48 0.49 0.50 0.50 0.54 0.59 0.62 0.63 0.64 0.64 0.67 0.75 0.73 0.74 0.76 0.76 0.78 0.85 0.83 0.84 0.85 0.85 0.87 0.92 0.91 0.91 0.92 0.92 0.93 0.96 0.96 0.96 0.97 0.97 0.97 0.97 0.99 0.99 0.99 0.99 C.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Table XXVIII. Utilization Limit (cm) Total 5 Diameter Class Class Size (cm) 1 < 7.9 0.93 0.70 2 8-15.9 0.95 0.89 3 16-23.9 0.97 0.96 4 24-31.9 0.98 0.97 5 32-39.9 0.98 0.98 6 40-47.9 0.98 0.98 7 48-55.9 0.98 0.98 8 56-63.9 0.98 0.98 9 64-71.9 0.98 0.98 10 72-79.9 0.98 0.98 11 80-87.9 0.98 0.98 12 88+ 0.98 0.98 Factors For Converting Total Volumes to Volumes to the Different Limits 10 15 20 25 Total 5 10 15 20 25 Cupressus lusitanica Pinus patula 0.94 0.57 0.40 0.95 0.89 0.31 0.90 0.65 0.08 0.99 0.97 0.87 0.66 0.04 0.95 0.88 0.69 0.25 0.99 0.99 0.97 0.93 0.67 0.04 0.97 0.93 0.85 0.65 0.99 0.99 0.99 0.94 0.88 0.55 0.97 0.96 0.92 0.84 0.99 0.99 0.98 0.97 0.95 0.88 0.97 0.96 0.94 0.90 0.99 0.99 0.99 0.98 0.96 0.94 0.97 0.97 0.96 0.93 0.99 0.99 0.99 0.99 0.97 0.95 0.98 0.97 0.96 0.94 0.98 0.98 0.96 0.94 0.98 0.98 0.96 0.94 0.98 0.98 0.96 0.94 - 114 -The t a b l e s were c o n s t r u c t e d f o r i n s i d e b a r k volumes e s t i m a t i o n but may a l s o be used f o r overbark volumes (assuming that bark -vo lume w i l l have con s t an t r e l a t i v e changes throughout the s tem) . - 115 -5. DISCUSSION ON THE RESULTS OF THE VARIOUS SECTIONS The results of t h i s study have been b r i e f l y discussed i n the various chapters. Where necessary, references were continually being made to. the other chapters addressing si m i l a r problems. I t i s necessary to point out , i n greater d e t a i l , a few of the important observations already made. In reference to the f i t t i n g of popular volume estimation models, together with some modifications of the same, i t has been shown that the logarithmic volume model (Schumacher and H a l l , 1933) and weighted models conditioned through the o r i g i n , generally give good estimates. Four models were found to be giving almost s i m i l a r estimates. These were models 1, 13, 14 and 15. Their s l i g h t differences might not be s i g n i f i c a n t i n practice. Models weighted by current approach were far much i n f e r i o r to these models. I t i s possible that the theory behind which current weighting i s done i s only r e a l i s t i c i f the variable used a weight i s not intended to be one of the variables i n the f i n a l model. An important c h a r a c t e r i s t i c observed while f i t t i n g a l l the various volume models was that by using d i f f e r e n t f i t t i n g techniques different variables could become s i g n i f i c a n t . For example, the Australian model 2 did not have D term being s i g n i f i c a n t when f i t t e d by the current weighting approach from P_. patula data. Also the model adopted for estimating the P_. patula volume, together with a corresponding model for C_. l u s i t a n i c a ( i . e . model 15), when f i t t e d by the current weighting - 116 -approach i n d i c a t e d t h a t some of the v a r i a b l e s used i n the models were not s i g n i f i c a n t . They were , however, a l l s i g n i f i c a n t when f i t t e d by the proposed w e i g h t i n g a t 5% s i g n i f i c a n c e l e v e l . I t was a l s o observed t h a t the assumption of s y s t e m a t i c b i a s of the l o g a r i t h m i c volume model was not a l l that s y s t e m a t i c . When the es t imates a t the v a r i o u s d iameter c l a s s e s were observed , i t was noted tha t i n most d iameter c l a s s e s (7 out of 12 f o r C_. l u s i t a n i c a ) the model was a c t u a l l y o v e r e s t i m a t i n g the vo lumes. A l though on the average ( o v e r a l l ) the model was u n d e r e s t i m a t i n g the vo lumes, such underest imates were so s m a l l tha t they d i d not seem i m p o r t a n t . The a p p l i c a t i o n of Meyer ' s c o r r e c t i o n f a c t o r o n l y s l i g h t l y improved the e s t i m a t e s . In f a c t , f o r some diameter c l a s s e s , the a p p l i c a t i o n of the f a c t o r made the volumes to be o v e r -es t imated even more. The a p p l i c a t i o n of the f a c t o r was cons idered to be unnecessary . The model cou ld a l s o have been f i t t e d u s i n g the n o n - l i n e a r r e g r e s s i o n approach . T h i s would have e l i m i n a t e d the n e c e s s i t y to c o r r e c t f o r the b i a s . T a b l e s XXIX (a) to (c) and XXX (a) t o (c) g i v e the s e c t i o n a l volume e s t i m a t e s and the 95% c o n f i d e n c e l i m i t s of the e s t i m a t e s g i v e n by models 1 5 , 22 and 35 f o r P_. p a t u l a , and models 1 , 2 1 , 32 and 34 f o r JC. l u s i t a n i c a . I t should be n'oted that taner model 32 was r e j e c t e d when i t was found to be s e r i o u s l y o v e r e s t i m a t i n g the d iameters i n the lower p a r t s of the t r e e and most of the upper p a r t s of the t r e e (see F i g u r e 10) T a b l e XXIX ( a ) . p . p a t u l a ; S e c t i o n a l Volume E s t i m a t e s and th*» C o r r e s p o n d i n g 95% C o n f i d e n c e L i m i t s f o r t h e E s t i m a t e s U s i n g M o d e l s 1 5 , 22 and 35 f o r a s m a l l t r e e T r e e S i z e : DBH « 12.4 cm R t 9 . 5 m Measurement P o i n t R e j e c t e d New T a p e r Mode l P r e d i c t e d Volume 95% C o n f i d e n c e L i m i t s . A c c e p t e d New V o l u m e - b a s e d Volume M o d e l Taper M o d e l Taper M o d e l ( O r d i n a r y )  95% 95% 95% P r e d i c t e d C o n f i d e n c e P r e d i c t e d C o n f i d e n c e P r e d i c t e d C o n f i d e n c e Volume L i m i t s Volume L i m i t s Volume L i m i t s S t a n d a r d o r A c t u a l Volume ( S m a l i a n ' s f o r m u l a used) 0 . 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 3 0 . 0 0 3 0 . 0 0 1 - 0 . 0 0 5 0 . 0 0 3 ; 0 . 0 0 1 - 0 . 0 0 5 0 . 0 0 4 0 . 6 0 . 0 0 6 0 . 0 0 4 - 0 . 0 0 8 0 . 0 0 5 ; 0 . 0 0 1 - 0 . 0 0 9 0 . 0 0 7 0 . 9 0 . 0 0 9 0 . 0 0 7 - 0 . 0 1 1 0 . 0 0 8 0 . 0 0 2 - 0 . 0 1 4 0 . 0 1 0 1 . 3 0 . 0 1 3 0 . 0 0 9 - 0 . 0 1 7 0 . 0 1 1 0 . 0 0 5 - 0 . 0 1 7 0 . 0 1 4 I 0 . 0 1 9 0 . 0 1 5 - 0 . 0 2 3 0 . 0 1 7 0 . 0 0 9 - 0 . 0 2 5 0 . 0 2 0 I I 0 . 0 2 5 0 . 0 2 1 - 0 . 0 2 9 0 . 0 2 2 0 . 0 1 4 - 0 . 0 3 0 0 . 0 2 5 I I I 0 . 0 3 1 0 . 0 2 3 - 0 . 0 3 9 0 . 0 2 7 0 . 0 2 1 - 0 . 0 3 3 0 . 0 3 0 IV 0 . 0 3 6 0 . 0 2 6 - 0 . 0 4 6 0 . 0 3 1 0 . 0 2 3 - 0 . 0 3 9 0 . 0 3 3 V 0 . 0 4 0 0 . 0 2 6 - 0 . 0 5 6 0 . 0 3 4 ' 0 . 0 2 6 - 0 . 0 4 2 0 . 0 3 6 V I 0 . 0 4 3 0 . 0 2 5 - 0 . 0 6 1 0 . 0 3 7 0 . 0 2 9 - 0 . 0 4 5 0 . 0 3 8 V I I 0 . 0 4 6 0 . 0 2 4 - 0 . 0 6 8 0 . 0 3 9 0 . 0 2 9 - 0 . 0 4 9 0 . 0 3 9 V I I I 0 . 0 4 8 0 . 0 2 4 - 0 . 0 7 2 0 . 0 4 0 0 . 0 3 0 - 0 . 0 5 0 0 . 0 4 0 I X 0 . 0 4 8 0 . 0 2 4 - 0 . 0 7 2 0 . 0 4 1 0 . 0 2 9 - 0 . 0 5 3 0 . 0 4 0 X 0 . 0 4 8 0 . 0 2 4 - 0 . 0 7 2 0 . 0 4 1 0 . 0 2 9 - 0 . 0 5 3 0 . 0 4 1 0 . 0 2 1 - 0 . 0 6 1 0 . 0 4 0 Table XXIX (b). f - ^ ^ r Sectional Volume Estimates and the Corresponding 95% Conifdence Limits for the Estimates Using Models 15, 22 and 35 for a medium-sized tree. e s Tree Size: D3H «• 27.6 c n Ht - 27.7 Measurement Foint Rejected New Taper Model 0.0 0.3 0.6 0.9 1.3 I II III IV V VI VII VIII IX X • S E T *S£~ ™ < * ™ « o J S 95% Accepted New Taper Model 95% if id Limits Volume-based Taper Model Volume 95% lfide Limits Volume Model (Ordinary) 95% Volume ience Limits 0.000 0.000 0.000 0.000 0.016 0.012-0.020 0.017 0.013-0.021 0.032 0.024-0.040 0.033 0.025-0.041 0.048 0.036-0.060 0.049 0.035-0.063 0.068 0.050-0.086 0.070 0.050-0.090 0.190 0.151-0.229 0.201 0.150-0.252 0.297 0.240-0.354 0.317 0.244-0.390 0.392 0.296-0.488 0.419 0.333-0.505 0.476 0.368-0.584 0.505 0.403-0.607 0.550 0.415-0.685 0.577 0.452-0.702 0.613 0.450-0.776 0.635 0.486-0.784 0.660 0.472-0.848 0.678 0.504-0.852 0.691 0.483-0.899 0.709 '' 0.515-0.903 0.705 0.485-0.920 0.726 0.524-0.928 0.708 0.483-0.933 0.731 0.527-0.935 Standard or Actual Volume (Smalian's formula used) 0.731 0.537-0.925 0.000 0.016 0.031 0.046 0.065 0.183 0.294 0.398 0.490 0.571 0.640 0.695 0.733 0.751 0.755 co i T r e e S i z e : DBH - 4 3 . 6 cm Ht - 3 1 . 0 m Measurement Point 0 . 0 0 . 3 0 . 6 0 . 9 1 . 3 I II III IV V VI VII VIII IX X Rejected New Taper Model v j Predicted Volume 95% Confidence Limits ' Accepted New Taper Model Volume-based Taper Model Volume Model (Ordinary) • , 95% 9 5 % g^-Predicted Confidence Predicted Confidence Predicted Confidence • V ° l u m e ' L l m l t.g Volume Limits V 0 l u m e Limits 0.000 0.000 0-.000 0.000 0.041 0.033-0.049 0.039 0.027-0.051 0.081 0.065-0.097 0.078 0.054-0.102 0.120 0.098-0.142 0.116 0.085-0.147 0.171 0.146-0.196 0.166 0.131-0.201 0.518 0.442-0.594 0.517 0.450-0.584 0.820 0.693-0.947 0.827 0.698-0.956 1.088 0.912-1.264 1.098 0.914-1.282 1.327 •1.098.-1.556 1.329 1.098-1.560 1'. 538 1.256KL.820 1.521 1.245^ -1.797 1.714 •1.379-2.049 1.675. 1.354-1.996 1.849 1.465-2.233 1.792 1.424-2.160 1.936 1.511-2.361 1.873 1.463-2.283 1.977 1.528-2.426 1.919 1.482-2.356 1.984 1.531-2.437 1.933 1.488-2.378 Standard or Actual Volune (Smalian's formula used) 1 . 9 3 3 1 . 4 8 8 - 2 . 3 7 8 0.000 0.042 0.083 0.122 0.171 0.499 0.791 1.056 1.290 1.493 1.663 1.791 1.875 1.912 1.920 Table XXX (a). C. lusitanica: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 1, 21 and 34 for a small tree. Tree Size: DBH - 11.3 cm Ht - 9.5 m Measurement Point Rejected New Taper Model Accepted New Taper Model Volume-based Taper Model Volume Model (Ordinary) Predicted Volume 95% Confidence Limits Predicted Volume 95% Confidence Limits Predicted ' Volume 95% Confidence Limits Predicted Volume 95% Confidence Limits Standard or Actual Volume (Smalian's formula used) 0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.3 0.004 0.002-0.006 0.004 0.002-0.006 0.004 0.004-0.004 0.005 0.6 0.007 0.005-0.009 0.007 0.005-0.009 0.008 0.006-0.010 0.008 0.9 0.010 0.008-0.012 0.010 0.008-0.012 0.012 0.008-0.016 0.011 1.3 0.013 0.009-0.017 0.013 0.009-0.017 0.016 0.010-6.022 0.015 I 0.019 0.013-0.025 0.019 0.013-0.025 0.024 0.014-0.034 0.023 II • 0.023 0.013-0.033 0.024 0.016-0.032 0.031 0.019-0.043 0.030 III 0.028 0.014-0.042 0.029 0.019-0.039 0.035 0.023-0.047 0.036 IV 0.032 0.016-0.048 0.033 0.021-0.045 0.039 0.029-0.049 0.041 V 0.035 0.019-0.051 0.036 0.022-0.050 0.Q42 0.034-0.050 0.045 VI 0.037 0.019-0.055 0.038 0.022-0.054 0.043 0.035-0.051 0.048 VII 0.039 0.021-0.057 0.040 0.024-0.056 0.044 0.036-0.052 0.049 VIII 0.039 0.021-0.057 0.041 0.024-0.057 0.045 0.037-0.053 0.050 IX 0.040 0.022-0.058 0.041 0.024-0.057 0.045 0.037-0.053 0.051 X 0.040 0.022-0.058 0.041 0.024-0.057 0.045 0.037-0.053 0.045 0.035-0.055 0.051 Table XXX (b). C_. lusitanica: Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 1, 21 and 34 for a medium-sized tree. Tree Size: DBH - 36.1 cm Ht - 27.4 m Measurement Point Rejected New Taper Model Accepted New Taper Model Volume-based Taper Model Volume Model (Ordinary) 95% 95% 95% 95% Predicted Confidence Predicted Confidence Predicted Confidence Predicted Confidence Volume Limits Volume Limits Volume Limits Volume Limits Standard or Actual Volume (Smalian's formula used) 0.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.3 0.042 0.020-0.064 0.038 0.009-0.067 0.037 0.008-0.066 0.049 0.6 0.081 0.061-0.101 0.074 0.043-0.105 0.074 0.043-0.105 0.085 0.9 0.117 0.097-0.137 0.107 0.076-0.138 0.110 0.085-0.135 0.118 1.3 0.161 0.139-0.183 0.149 0.122-0.176 6.156 0.186-0.176 0.157 I 0.377 0.346-0.408 0.365 0.332-0.398 0.426 0.301-0.551 0.375 II 0.533 0.466-0.600 0.531 0.462-0.600 0.640 0.426-0.854 0.553 III 0.668 0.548-0.788 0.674: 0.564-0.784 0.806 0.565-1.047 0.710 IV 0.791 0.630-0.952 0.799: " 0.652-0.946 0.930 0.705-1.155 0.846 V 0.897 0.711-1.083 0.902 . 0.724-1.080 1.018 0.828-1.208 0.958 VI 0.977 0.765-1.189 0.978 . 0.768-1.188 1.076 0.913-1.239 1.045 VII 1.029 0.786-1.272 1.026 0.900-1.152 1.111 0.948-1.274 1.1.06 VIII 1.056 0.786-1.326 1.051 0.772-1.329 1.129 0.949-1.309 1.142 IX 1.066 0.786-1.326 1.060 0.764-1.356 1.135 0.941-1.329 1.156 X 1.068 0.778-1.358 1.062 ' ' 0.762-1.362 1.136 0.938-1.334 1.134 0.950-1.318 1.159 Table XXX (c). C. lusitanica; Sectional Volume Estimates and the Corresponding 95% Confidence Limits for the Estimates Using Models 1, 21' and 34 for a large tree. Tree Size: DBH •» 66.4 cm Ht - 31.5 m Measurement Rejected New Accepted New Volume-based Volume Model Standard or Actual Volume Point. Predicted Volume .95% Confidence Limits Predicted Volume 95% Confidence Limits Predicted Volume 95% Confidence Limits Predicted Volume 95% Confidence Limits (Smalian's : 0.0 0.000 0.000 0.000 0.000 0.000 . 0.000 0.000 0.3 0.141 0.029-0.253 0.127 0.000-0.274 0.114 0.000-0.296 0.160 0.6 0.273 0.130-0.416 0.246 0.052-0.440 0.225 0.000-0.476. 0.287 0.9 0.396 . 0.239-0.553 0.359 0.155-0.563 0.335 0.068-0.602 0.398 1.3 0.548 0.372-0.724 0.501 0.303-0.699 0.477 0.224-0.730 0.533 I 1.404 1.194-1.614 1.351 1.130-1.572 1.433 1.172-1.694 1.388 II 2.011 1.692-2.330 1.991 1.644-2.338 2.195 1.593-2.797 2.049 III 2.534 2.026-3.042 2.540 ' 2.042-3.038 2.784 2.031-3.537 2.615 IV 3.007 2.382-3.632 3.018 ! 2.410-3.626 3.224 •, 2.073-3.495 3.103 V 3.413 2.729-4.097 3.411 2.725-4.097 3.537 2.563-3.885 3.497 VI 3.722 2.981-4.463 3.700 2.936-4.464 3.744 3.007-4.481 3.788 VII 3.921 3.113-4.729 3.883 3.042-4.724 3.868 2.996-4.740 3.970 VIII 4.024 3.136-4.912 3.976 3.055-4.897 3.931 2.939-4.923 4.065 IX 4.061 3.120-5.002 4.010 3.036-4.984 3.954 2.896-5.012 4.102 X 4.067 3.118-5.016 4.015 3.031-4.999 3.957 2.887-5.028 3.954 3.094-4.814 4.108 - 123 From these tables i t w i l l be seen that for p r a c t i c a l purposes,traditional volume models and volume-based taper models give i d e n t i c a l t o t a l volume estimates. However, the computed confidence l i m i t s do not t o t a l l y agree. This was because the standard errors of estimates given by each of the two approaches were d i f f e r e n t . By considering the width of the confidence i n t e r v a l s the taper model would be considered i n f e r i o r to the volume model from which i t was derived for the large trees of C_. l u s i t a n i c a . However, for P_. patula the estimates given the volume-based taper model are more precise. Another important observation from these tables i s that the model which was rejected as overestimating the sectional diameters of (3. l u s i t a n i c a trees seems to be giving s i m i l a r volume estimates to the accepted model although the accepted model was known to be giving much better sectional diameter estimates. For the large trees, i t was giving s l i g h t l y better estimates of the standard volumes. Even then, the t o t a l volume estimates given by the rejected model were s t i l l a l i t t l e less than the standard volumes. This depicts the effect of volume d i s t r i b u t i o n along the tree p r o f i l e s . I f a taper model overestimates only a short stem section and gives good estimates at most other parts, the volume overestimates r e s u l t i n g from such diameter overestimations w i l l be n e g l i g i b l e . Likewise, diameter underestimation that extends over longer stem sections, especially i n those parts of the tree where there i s more (percentage of the tot a l ) volume, would r e s u l t i n underestimation of the t o t a l volume. Considering the three sets of tables f o r each spcies, i t can be said that the pattern of volume over- or under- estimation shown by the adopted taper models i s almost random. There does not seem to be d i s t i n c t size class effect. Even the - 124 -taper model which was r e j e c t e d as o v e r e s t i m a t i n g s e c t i o n a l d iameters i n /J. l u s i t a n i c a t r e e s i s shown to be good enough f o r volume e s t i m a t i o n . Table X X I I I shows t h a t even the adopted taper model f o r C_. l u s i t a n i c a was s t i l l s l i g h t l y o v e r e s t i m a t i n g the s e c t i o n a l d iameters a t a number of p o i n t s and yet Table XXV i n d i c a t e s t h a t the i n t e g r a t i o n of the model was p r o d u c i n g s l i g h t l y l e s s volume than the s tandard a t most p o i n t s . Because the vo lume-based taper models seem to be g i v i n g i d e n t i c a l t o t a l volume e s t i m a t e s to the co r respond ing volume models ( i . e . those f i t t e d i n the t r a d i t i o n a l way) i t can be assumed t h a t the p r o f i l e s g i ven by the vo lume-based t a p e r models a re the t r u e p i c t u r e s of the p r o f i l e s assumed by t r a d i t i o n a l volume models . I f t h i s assumpt ion i s c o r r e c t then the forms o f t r e e s assumed must be f a r from t r u e . Because volume-based taper models have no i n f l e c t i o n p o i n t s , the forms assumed are j u s t not r e a l i s t i c . T rees do not have such l i n e a r p r o f i l e s . For C^ . l u s i t a n i c a , : t h e . t o t a l volume e s t i m a t e s .g iven by both the volume-based taper model (when i n t e g r a t e d ) and the t r a d i t i o n a l volume models are s t i l l a l i t t l e l e s s than the s tand ar d ( S m a l i a n ' s f o r m u l a ) . In taper s t u d i e s i t i s apparent t h a t g r e a t e r emphasis needs t o be g i v e n to d e s c r i b i n g the stem p r o f i l e than to comparing volume e s t i m a t e s e i t h e r w i t h o t h e r models o r . w i t h other s t a n d a r d s . Because the fo rmalae u s u a l l y used to compute the s tandard volumes a re not always e f f i c i e n t i n every c a s e , there may be cases where the t r e e forms are such that the s tandard fo rmula may be g i v i n g worse e s t i m a t e s than the taper mode l . T h i s would l i k e l y be so i f the s tandard volumes are computed from c r o s s - c u t l o g s . Some p a r t s of the main stem would have been l o s t d u r i n g c r o s s - c u t t i n g . I f a model i s o b t a i n e d that g i v e s good s e c t i o n a l d iameter e s t i m a t e s from the t r e e base to the t i p , i t should a l s o g i ve volume - 125 -e s t i m a t e s which a re ve ry c l o s e to the t rue vo lumes, when i n t e g r a t e d . The most important p a r t of the t r e e that a taper model should best d e s c r i b e i s the p a r t above dbh . R e l e n t l e s s e f f o r t to get models that p e r f e c t l y d e s c r i b e the base may make the models i n e f f i c i e n t i n d e s c r i b i n g the upper b o l e which c o n t a i n s more u t i l i z a b l e vo lumes. The percentage of t o t a l t r e e volume that i s c o n t a i n e d below dbh i s i s f a r much l e s s than i s c o n t a i n e d above the dbh, and can s a f e l y be forgone by g i v i n g b e t t e r d iameter e s t i m a t e s above dbh. Even f o r the poor models , one which s e r i o u s l y o v e r e s t i m a t e s the upper r e g i o n s of t r e e stem w i l l g i v e poorer volume e s t i m a t e s than another model tha t s e r i o u s l y underes t imates the b u t t . T h e t r u t h i s that t r e e s taper comparably s lower i n the upper p a r t s than c l o s e to the ground. Consequent l y , o v e r - o r u n d e r - e s t i m a t i o n s of d iameters i n the upper p a r t s vrould extend over a g r e a t e r l e n g t h of the stem than would s i m i l a r e s t i m a t i o n s near the b a s e . In p r a c t i c e , some of the f o r e g o i n g d i s c u s s i o n may not be i m p o r t a n t . Some of the taper models t e s t e d and r e j e c t e d have been used f o r a l o n g t i m e . The magnitude of the b i a s e s of the models may be more important i n d iameter e s t i m a t i o n than i n volume e s t i m a t i o n . S i n c e the f i n a l o b j e c t i v e of t r e e taper s t u d i e s i s to e s t i m a t e t r e e volumes more e f f e c t i v e l y , the b i a s of the d iameter e s t i m a t e s become l e s s i m p o r t a n t . The percentage volume b i a s i s a c t u a l l y too low to make models complete ly unacceptabe . The observed c h a r a c t e r i t i c s of the adopted taper models should be c o n s i d e r e d mere ly as improvements to the e s t i m a t e s -*- e s p e c i a l l y d iameter e s t i m a t e s . Taper model 25 would h a v e , p e r h a p s , g i v e n b e t t e r volume e s t i m a t e s than the o ther popular taper models t e s t e d i f i t had not been f i t t e d w i t h o u t an i n t e r c e p t . Because i t was g i v i n g f a i r l y good t r e e p r o f i l e d e s c r i p t i o n , i t s f a i l u r e to g i v e good volume e s t i m a t e s might have been because the i n t e r c e p t was s i g n i f i c a n t l y d i f f e r e n t from z e r o . T h i s would have s i g n i f i c a n t e f f e c t on the volume e s t i m a t e s upon i n t e g r a t i o n . - 126 -In general, t h i s study has shown that the changes i n diameter along the tree p r o f i l e s are products of distance from the ground and distance from the top to the di f f e r e n t points where diameters are to be estimated. I m p l i c i t i n t h i s i s the effect of t o t a l height which naturally becomes an inherent factor, to establish r e l a t i v e heights for trees of a l l sizes. I t has also been shown that the rate of p r o f i l e changes up the trees although d i f f e r e n t between species (especially for the two species studied) does not seem to depend much on the tree diameter. Even i n P_. patula where the size effect was noticed i t was f a i r l y small and only i n some diameter classes (less than 24 cm). Of course, we do not always have to use taper or volume models produced by regression studies, to get required diameters and volumes. This study has shown that some natural ch a r a c t e r i s t i c s when studied could give good indi c a t o r s . From such examination, Tables XX, XXI, XXVI, XXVII and XXVIII have been produced i n sections 4.3.2.1 and 4.4 which could be used to give quick estimates of both diameters and volumes. F i n a l l y , a word of caution: I t i s important to remember the c h a r a c t e r i s t i c s of the data used i n t h i s study. The s i z e , d i s t r i b u t i o n (country-wise) and range of the sample trees may impose a l i m i t a t i o n on the generalization of some of the results obtained i n t h i s study. This i s l i k e l y to be important with respect to the use of the recommended volume models developed i n the t r a d i t i o n a l approach. As mentioned e a r l i e r the sample size might not have been large enough for volume study although i t was considered s u f f i c i e n t for taper study. As such volume - 127 -models o b t a i n e d by i n t e g r a t i n g the taper model are more l i k e l y to g i v e e s t i m a t e s c l o s e r to the t r u e volumes than the models developed i n the t r a d i t i o n a l way. A l s o the f a c t tha t l o c a l i t y does not q u i t e a f f e c t the forms of the t r e e s , t a p e r - b a s e d volume models are more l i k e l y t o account f o r volumes of t r e e s o u t s i d e E lburgon (where data was c o l l e c t e d ) than would the o r d i n a r y volume models . For P_. p a t u l a where the s i z e range d i d not s a t i s f a c t o r i l y cover the expected range i n p r a c t i c e , i t has been shown i n Tab le XXI t h a t t r e e s l a r g e r than 24 cm overbark a t b r e a s t - h e i g h t have b a s i c a l l y s i m i l a r stem forms. As such t a p e r - b a s e d volume model based on the d a t a used i n t h i s study should g i v e good volume e s t i m a t e s even o u t s i d e the range of the data used i n t h i s s t u d y . Moreover , i t has been shown, by e x a m i n i n g . t h e v a r i o u s t a b l e s , where d iameter e s t i m a t e s of the v a r i o u s taper models were t e s t e d , tha t d iameter e s t i m a t e s a re not a f f e c t e d by the t o t a l h e i g h t of the v a r i o u s t r e e s . As such the f e a r , p r e v i o u s l y h e l d , that ' the samples should have been d i s t r i b u t e d w i t h r e s p e c t to age r a t h e r . t h a n . s i z e , c l a s s ( d b h ) - d o e s — -not seem to h o l d . However, i t should be r e a l i z e d tha t the accepted taper models e s t a b l i s h e d t h a t the most important r e l a t i o n s h i p i n d e s c r i b i n g t r e e p r o f i l e i s the r a t i o ~ — . To g i v e the form d e s c r i p t i o n s p e c i f i c to the d i f f e r e n t s p e c i e s i n t e r a c t i o n between d i s t a n c e from the ground and d i s t a n c e from the top i s a necessary added term. T h i s , t h e r e f o r e , e s t a b l i s h e s the r e l a t i v e forms of t r e e s of a l l s i z e s . Because not a l l p o i n t s , f rom where measurements were taken to e s t a b l i s h the r e l a t i o n s h i p s i i the mode ls , were r e l a t i v e , t h e r e may be a d i s c r e p a n c y i n some of the d iameter e s t i m a t e s . T h i s would be p a r t i c u l a r l y important i n Cypress - 128 -where the adopted model was s l i g h t l y o v e r e s t i m a t i n g the reg ions j u s t below dbh. Because dbh i s not a r e l a t i v e p o i n t but f i x e d on each t r e e , i f a t r e e which i s abnormal ly t a l l w i l l have i t s s e c t i o n a l d iameters es t imated u s i n g the m o d e l , there i s a h i g h p o s s i b i l i t y that the o v e r e s t i m a t i o n s g i v e n by the model , below dbh, w i l l extend to r e g i o n s a l i t t l e above dbh. As such the i n s i d e d iameter e s t i m a t e a t the dbh may be a l i t t l e g r e a t e r than the cor respond ing diameter o u t s i d e b a r k . The d iameter e s t i m a t e s a t the other p o i n t s above dbh may not be a f f e c t e d . The t o t a l he ight d i s t r i b u t i o n may not be important i n taper study but would most l i k e l y a f f e c t the volume e s t i m a t e s g i v e n by t r a d i t i o n a l volume models . I t shou ld be remembered t h a t a l l the t e s t s as c a r r i e d out i n t h i s study were based on the data which was used i n deve lop ing the same models . As such they need to be cons idered merely as i n i t i a l t e s t s . To g i v e the models b e t t e r and more r i g o r o u s t e s t i n g i t would have been necessary to use c o m p l e t e l y new s e t s of d a t a . That s o r t of data, was not a v a i l a b l e . - 129 -6. SUMMARY AND CONCLUSIONS A f t e r examin ing , d e v e l o p i n g , and t e s t i n g the v a r i o u s models f o r volume and taper e s t i m a t i o n , i t can be g e n e r a l l y concluded t h a t : (1) Of the popu la r techniques f o r f i t t i n g volume e q u a t i o n s , the weighted r e g r e s s i o n approach and the l o g a r i t h m i c t r a n s f o r m a t i o n s , g e n e r a l l y , g i v e good e s t i m a t i o n . However, when one of the v a r i a b l e s n o r m a l l y p resent i n an unweighted e q u a t i o n i s to be used as w e i g h t , b e t t e r e s t i m a t e s would be ob ta ined by f i t t i n g a model c o n d i t i o n e d through the o r i g i n . A techn ique has been suggested f o r do ing t h i s . (2) Volume-based taper models were found to be g i v i n g b i a s e d t r e e p r o f i l e s . They l a c k e d i n f l e c t i o n p o i n t s . G e n e r a l l y , they o v e r -e s t i m a t e the d iameters a t lower p a r t s of the t r e e and underest imate the d iameter i n the upper p a r t s . When i n t e g r a t e d f o r vo lumes, however, they seemed to be g i v i n g good t o t a l volume e s t i m a t e s . Of the models t r i e d , the vo lume-based taper models and t h e i r co r respond ing volume models ( i . e . volume models f rom which they were developed) seemed to g i v e t o t a l volume e s t i m a t e s which were c l o s e s t to the s tandard ( S m a l i a n ' s fo rmula) i n most d iameter c l a s s e s . The volume-based taper models a r e , however, not recommended f o r e s t i m a t i n g the volumes to the o ther p o i n t s a l o n g the t r e e p r o f i l e s because of t h e i r b i a s i n e s t i m a t i n g the s e c t i o n a l d i a m e t e r s . - 130 -(3) Most popu lar taper models ( i n c l u d i n g the volume-based taper models) a re j u s t not r e a l i s t i c i n t h e i r s e c t i o n a l d iameter e s t i m a t i o n . They l a c k the i n f l e c t i o n p o i n t s which a re necessary f o r them to g i v e proper t r e e p r o f i l e d e s c r i p t i o n s . As such they g i v e b i a s e d s e c t i o n a l d iameter e s t i m a t e s . I f too b i a s e d i n d iameter e s t i m a t e s , even the volume e s t i m a t e may be b i a s e d . However, s l i g h t d iameter b i a s e s may not s e r i o u s l y a f f e c t the volume e s t i m a t e s . T h i s study has shown that models which g i v e g r e a t e r d iameter b i a s below dbh and a l i t t l e b e t t e r es t imates i n the upper p a r t s , g e n e r a l l y , g i v e b e t t e r volume e s t i m a t e s than those model which tend t o g i v e b e t t e r d iameter e s t i m a t e s below dbh and worse es t imates i n the upper p a r t s . (4) The c u r r e n t b e l i e f tha t not a s i n g l e taper model can be developed tha t i s capab le of d e s c r i b i n g the t r e e p r o f i l e w e l l throughout i t s l e n g t h i s not c o m p l e t e l y r i g h t . A l though the two taper models developed i n t h i s s t u d y , which have been recommended . f o r g e n e r a l taper e s t i m a t i o n , - - -were g i v i n g s l i g h t b i a s e s a t a few p o i n t s a long the t r e e p r o f i l e s , i t i s thought tha t the s l i g h t weakness was not on the models but on the d a t a . The models d e s c r i b e the t r e e forms i n r e l a t i v e te rms. Hence, a l l the data needed to have been c o l l e c t e d on the same r e l a t i v e p o i n t s on each t r e e . The data c o l l e c t e d a t the dbh, and below i t were not c o l l e c t e d at s t r i c t l y r e l a t i v e p o i n t s . The p o i n t s were s i m i l a r on ly on a b s o l u t e terms but were d e f i n i t e l y d i f f e r e n t (on r e l a t i v e terms) depending on the t o t a l h e i g h t of the t r e e . Because more samples are needed at t h i s r e g i o n to account f o r the b u t t s w e l l , s t r i c t adherence to r e l a t i v e p o i n t s would have - 131 -resulted i n worse estimates. L u c k i l y , the s l i g h t diameter biases have no effect on volume estimates. We, therefore, have to accept to l i v e with the s l i g h t weakness. (5) For the construction of volume tables i n the t r a d i t i o n a l way, any of the four models recommended i n th i s study can be used. However, the tables so produced using the data i n th i s study cannot be guaranteed for accuracy. The data was not large enough for such a study. Moreover, i t was not properly distributed'(country-wise) to make the tables applicable at other Divisions. However, volume and taper tables produced from the recommended taper models are considered good enough for use anywhere, not only i n Kenya but also i n any other countries where the two species studied are also planted. The models describe tree forms i n r e l a t i v e terms. Since tree forms do not quite depend on l o c a l i t y , the tables are applicable anywhere. Volume tables produced from the recommended volume models (that is, overbark volumes) and taper models (that i s , underbark volumes) are given i n Appendices 10 to 13. Appendix 14 gives an example of the detailed underbark volumes and taper tables to be produced for use i n Kenya. The advantage of "this kind of table i s that i t can be used for t o t a l volumes and for merchantable volumes to any desired upper diameters (underbark). The tables w i l l be made available to the Forest Department, Kenya, for examination. In a general sense, taper models developed from taper data (not volume-based taper models) are recommended for the construction of both taper and volume tables i f the taper models are giving proper description of the tree p r o f i l e s . - 132 -LITERATURE CITED Adlard, P.G. and K.F. Richardson. 1975. 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Dept. Tech. Note No. 81. , J.S.G. McCulloch and E.S. Waweru. 1964. A volume table compilation for Pinus radiata in Kenya. E. Afr. Agric. and For. J. 29: 326-329. Ek, A.R. and M.C. Kaltenberg. 1975. Generalization of tree taper and volume models. Paper presented at Midwest Forest Mensuration Meeting. Oct. 1975. Kentucky, U.S.A. Evert, F. 1968. Form height and volume per square foot of basal area. J. For. 66(4): 358-359. . 1969. Use of form factor in tree volume estimation. J. For. 67: 126-128. . 1973. New form-class equations improve volume estimates. Can. J. For. Res. 3: 338-341. Forest Club. 1971. Forestry Handbook for British Columbia. Univ. of B.C., Vancouver. 815p. Freese, F. 1960. Testing Accuracy. For. Sci. 6: 139-145. Fry, G. 1963a. Temporary revision of Cypress volume table. Kenya For. Dept. Tech. Note No. 92. . 1963b. Mensuration studies on P. radiata and P. patula.. Kenya For. Dept. Tech. Note No. 94. Furnival, G.M. 1961. An index for comparing equations used in constructing volume tables. For. Sci. 7: 337-341. Gerrard, D.J. 1966. The construction of standard tree volume tables by weighted multiple regression. Fac. For.,. Univ. of Toronto, Tech. Report No. 6, 35p. - 135 -Golding, D.L. and O.F. Hall. 1961. Tests of precision of cubic-foot tree volume equations on aspen, Jack pine and White spruce. • For. Chron. 37: 123-132. Goulding, C.J. and J.C. Murray. 1976. Polynomial taper equations that are compatible with tree volume equation. N.Z. J. For. Sci. 5: 313-322. Gray, H.R. 1956. The form and taper of forest tree stems. Inst. Paper No. 32. Imperial For. Inst., Oxford. 79p. Gregory, R.A. and P.M. Haack. 1964. Equations and tables for estimating cubic-foot volume of interior Alaska tree species. USDA For. Serv. Res. Note NOR-6, 21p. Grosenbaugh, L.R. 1954. New tree measurements concepts: height accumulation, giant tree, taper and shape. USDA For. Serv. South. For. Exp. Sta., Occasional Paper No. 134. 32p. . 1966. Tree form definition, interpolation, extrapolation For. Chron. 42: 443-456. Haack, P.M. 1963. Volume tables for hemlock and sitka spruce on the Ehugach National Forest, Alaska. USDA For. Serv., Res. Note NOR-4. 4p. Hejjas, J. 1967. Comparison of absolute and relative standard errors and estimates of tree volumes. Univ. of B.C., Fac. For., MF Thesis. 58p. Honer, T.G. 1964. The use of height and squared diameter ratios for the estimation of merchantable cubic-foot volume. For. Chron. 40: 324-331. _. 1965a. A new total cubic-foot volume function. For. Chron. 41: 476-493. . 1965b. Volume distribution in individual trees. Woodlands Review, Pulp and Paper Magazine of Canada. Woodlands Section Index 2349 (F-2), WR: 499-508. . 1967. Standard volume tables arid merchantable conversion factors for the commercial tree species of Central and Eastern Canada. For. Mgmt. Res. and Serv. Inst. Ottawa. Information Report FMR-X-5. and L. Syn-Wittgenstein. 1963. Report of the committee on forest mensuration problems. J. For. 61: 663-667. Hummel, F.C. 1955. The volume/basal area line; a study in forest mensuration. U.K. For. Comm. Bull. No. 24. - 136 -Husch, B. , C.I. Miller and T.W. Beers. 1972. Forest Mensuration. Ronald Press Co., N.Y. 410p. Jensen, C.E. 1973. Marchacurve 3. Multiple-component and multi-dimensional models for natural resource studies. USDA For. Serv. Res. Pap. INT-146. . 1976. Marchacurve 4. Segmented mathematical descriptors for asymmetric curve forms. USDA For. Serv. Res. Pap. INT-182. Jensen, C.E. and Homeyer, J.W. 1970. Marchacurve 1. for Algebraic transformations to describe signoid or bell-shaped curves. USDA For. Serv. Intermount. For. Range Exp. Stn. 1971. Marchacurve 2. for Algebraic transformations to <J • — describe curves of the class Xn. USDA For. Serv. Res. Pap. INT-106. Kingston, B. 1972. Volume tables for Pinus patula. Uganda For. Dept. Tech. Note No. 192/72. Kozak, A. 1966. Multiple Correlation Coefficient tables up to 100 independent variables. Fac. For., Univ. of B.C., Res. Note No. 57. and J.H.G. Smith. 1966. C r i t i c a l analysis of multivariate techniques for estimating tree taper suggests that simpler methods are best. For. Chron.. 42: 458-463. ,. ., D.D. Munro and J.H.G. Smith. 1969. Taper functions and their application in forest inventory. For. Chron. 45: 278-283. Kullervo Knusela. 1965. A method for estimating the volume and taper of tree stems and for preparing volume functions and tables. Comm. Inst. For. Fenniae 20: 1-18. L i , Jerome, C.R. 1964. S t a t i s t i c a l Inference I and I I . Ann Arbor, Edwards Brothers. Loetsch, F. and K.E. Haller. 1964. Forest Inventory. Vol. 1, BLV Verlagsgesellchaft. Munchen. (trans, by E.F. Brunig).436p. , F. Zohrer and K.E. Haller. 1973. Forest Inventory, Vol. 2, BLV Verlagsgesellchaft. Munchen. (trans, by K.F. Panzer). 469p. Matte, L. 1949. The taper of coniferous species with special reference to lob l o l l y pine. For. Chron. 25: 21-31. - 137 -Max, T.A. and H.E. Burkhart. 1976. Segmented polynomial regression applied to taper equations. For. Sci. 22: 283-289. Meyer, H.A. 1938. The standard error of estimate of tree volume from the logarithmic volume equation. J. For. 36: 340-341. . 1944. A correction factor for a systematic error occurring in the application of the logarithmic volume equation. The Pennsylvania State Forest School, State College, Penn. Res. Paper No. 7 3p. . 1953. Forest Mensuration. Penns Valley Publishers, Inc., State College, Pennsylvania. 357p. Munro, D.D. 1964. Weighted least squares solutions improve precision of tree volume estimates. For. Chron. 40: 400-401. and J.P. Demaerschalk. 1974. Taper-based versus volume-based compatible estimating systems. For. Chron. 50(5): 197-199 Myers, CA. 1963. Taper table for pole size ponderosa pines in Arizona and New Mexico. USDA For. Serv. Res. Note RM-8. . 1964. Taper tables, Bark thickness and diameter relationships for lodgepole pines in Colorado and Wyoming. USDA For. Serv. Res. Note RM-31. • and C.B. Edminster. 1972. Volume tables and point sampling factors for Engelmann spruce i n Colorado and Wyoming. USDA For. Serv., Res. Paper RM-95, 23p. Newham, R.M. 1958. A study of form and taper of stems of Douglas-fir, western hemlock and western red cedar on the University research forest, Haney, B.C., Univ. of B.C., Fac. For., MF Thesis, 71p. ~ Ogaya, N. 1968. Kubierungsformeln und bestandesmassenformeln. Thesis, Univ. Freiburg i . Br. 85 pp. (See Loetsch et a l . , 1973). Osumi, S. 1959. Studies on the stem form of the forest trees (1). J. Jap. For. Soc. 41: 471-479 (in Japanese, abstract in English). Paterson, D.N. 1967. Volume tables to normal saw-timber merchantable limits for three important East African exotic softwoods. EAAFRO. Forest. Tech. Note. No. 20. Pearman, P.J. 1963. Preliminary volume tables for Pinus radiata and Pinus patula in Kenya. Kenya Forest Department. Res. Bulletin No. 25, Univ. of Oxford. - 138 -Pudden, H.H.C. 1958. New volume tables for Cypress. Kenya For. Dept. Tech. Note No. 56. Schumacher, F.X. and F.S. Hall. 1933. Logarithmic expression of timber tree volume. J. Agric. Res. 47: 719-734. Smalley, G.W. 1973. Weighting tree volume equations for young loblolly and short-leaf pines USDA Forest Service, Res. Note SO-61. 5p. and D.E. Beck. 1971. Cubic-foot volume table and point sampling factors for white pine plantations in the southern Appalachians. USDA For. Serv. Res. Note SO-118. 2p. Smith, J.H.G. and R.E. Breadon. 1964. Combined variable equations and volume-basal area ratios for total cubic-foot volumes of the commercial trees of B.C. For. Chron.' 40: 258-261. 2 and J.W. Ker. 1957. Timber volume depends on D H. British I 1 • ~ T 1 ~ Columbia Lumberman, September 1957. pp.28-30 J.W. Ker and J. Csizmazia. 1965. Economics of reforestation of Douglas-fir, western hemlock and western redcedar i n the Vancouver Forest D i s t r i c t . Fac. For. Bull. No. 3, Univ. of B.C. Spurr,S.H. 1952. Forest Inventory. Ronald Press Co., N.Y. 476p. Stoate, T.N. 1945. The use of a volume equation in pine stands. Australian forestry 9: 48-52. Walpole,R.E. 1968. Introduction to s t a t i s t i c s . MacMillan Publishing Co., Inc., N.Y. and Collier MacMillan Publishers, London. 340p. Wanene,A.G. 1975. A provisional yield table f or Pinus patula grown in Kenya. Kenya For. Dept. Tech. Note No. 143. and P. Wachiori. 1975. Variable density yield tables for Cupressus lusitanica group in Kenya. Kenya For. Dept. Tech. Note No. 144. Wright, W.G. 1927. Taper as a factor i n the measurement of standing timber. Forest Service, Canada. B u l l e t i n No. 79. (Possibly made accessible i n 1923) Append ix 1 . Unweighted M o d e l s . P i n u s p a t u l a SEE R 2 (2) VO = 0.072605A + 0.00003456A9D 2 H 0 .155176 0 . 9 7 2 2 V I = 0.047A513 + 0.0000312360D 2 H 0 .147616 0 . 9 6 9 3 (3&13) VO = - 0 . 0 6 2 6 3 2 7 8 4 + 0.01071502H - 0 .00028729702D 2 + 0 .000040971685D 2 H 0 .129350 0 . 9 8 1 0 VI = - 0 . 0 5 8 2 5 9 4 2 9 + 0.00842301H - 0 .000232270263D 2 + 0 .000036493541D 2 H 0 .131527 0 . 9 7 6 0 (4&14) VO - 0 . 0 3 A 2 4 0 4 - 0 .00024764264D 2 + 0 .000031761966D 2 H + 0.0005809075DH 0 . 1 2 1 1 0 0 . 9 8 3 4 V I =t - 0 . 0 3 7 0 0 1 7 - 0 .00020023495D 2 + 0 .000029132431D 2 H + 0.00046093495DH 0 . 1 2 6 3 0 0 . 9 7 7 9 (15) VO - 0 . 3 3 7 0 3 2 - 0.145974H + 0 .0008A6011D 2 + 0 . 0 0 3 4 5 1 6 7 H 2 + 0 . 6 8 5 0 6 4 1 o g H 2 0.128664 0 . 9 8 1 4 V I = - 0 . 2 7 7 7 9 4 - 0.131494H + 0 . 0 0 0 7 1 5 7 7 3 D 2 + 0 . 0 0 3 1 0 4 7 5 H 2 + 0 . 5 9 6 9 8 7 1 o g H 2 0.127868 0.9776 Cupressus l u s i t a n i c a (2) VO SC 0.140209 + 0 .0000290443D 2 H 0 .237143 0 . 9 8 3 1 V I = 0.118940 + 0 .0000270572D 2 H 0 .216100 0 . 9 8 3 8 (3&13) VO - 0 . 2 0 7 2 2 9 7 1 + 0.019031981H + 0 .00018918319D 2 + 0 .000021118841D 2 H 0 . 2 0 2 4 0 . 9 8 7 8 VI = - 0 . 1 7 6 4 6 1 2 5 + 0.01539572H + 0 .00023544957D 2 + 0 .000018125046D 2 H 0 . 1 8 8 1 0 . 9 8 7 9 (4&14) VO = - 0 . 0 8 9 6 6 9 4 4 4 + 0 .00018024185D 2 + 0 .00016355849D 2 H + 0.00058037224DH 0 . 1 9 3 0 0 . 9 8 8 9 V I at - 0 . 0 8 1 4 1 5 0 6 + 0 .00022822866D 2 + 0 .000014269865D 2 H + 0.00046963A67DH 0 .1814 0 . 9 8 8 7 (15) VO =x 3.02497 + 0.117082D + 0 . 0 0 1 0 0 8 0 8 H 2 + 1.A70A71ogD 2 H 0 .269538 0 . 9 7 8 4 V I 2 .78832 + 0.110A57D + 0 .0008239A5H 2 + 1 . 3 6 1 3 A l o g D 2 H 0 . 2 5 4 9 6 0.9777 Appendix 2(a). Plots of residuals of volume estimates for model 1 - specific for P. patula volume estimates. O . I O O D « » * * • a * •2 2* * 2» • * » * * * • •c.2;oc C.5C0C 0-33333 1.CCC0 1.C667 2.3333 3.CS00 * 3.66c7 °-° 0.66667 1.3333 2.0000 2.6667 3.3333 *.C000 Observed volume inside-bark (m3) Appendix 2(b). Plots of residuals of volume estimates for model 2 - specific for patula volume estimates. C.40C0 . 2 * » » « * » * » * * • * * * » . « • *• * * « « * 2 -.' * « * . * ** * * . • * 2 • »* * ft . •«?•« • * -0.2000 . »»*» * « • * • 5 ft * * * ft 0.33333 1.0000 I.t6c7 2.3333 - 3.C000 3.6667 0.0 0.66667 1.3333 2.0000 2.6667 3-3333 4.0C00 Observed volume inside-bark( m 3) Appendix 2(c). Plots of residuals of volume estimates for models 3 and 4 - specific for P_. patula volume estimates. a 3 •a •H m C . 7 J C 0 0 . 4 0 C 0 2 • ? 7« 2 * 24 . * ft • •* * c. i : c o . . » . « • * • • ' c 2* * « * « * w » 2 •»• » • • * « « CQ • # * « * * - ft * * ' ft g . » • * « * - > ft rH . • • 2 ft IS) - 0 . 5 C C C - 0 . 8 0 5 0 0 . 3 3 3 3 3 1.CC00 1 .66*7 2 . 3 3 3 3 3 . 0 0 0 0 3 - 6 6 6 7 1 . 3 3 3 3 2 . 0 0 0 0 2 Observed volume inside-bark(m ) 0 - 0 0 . 6 6 6 6 7 1 . 3 3 3 3 2 . 0 0 0 0 ^ 2 . 6 6 6 7 3 . 3 3 3 3 4 . 0 0 0 0 Appendix 2(d). f^ s of residuals of volume estimates for m o d e l s 1 3, 1 4 a n d 1 5 specific for P. patula volume estimates. 0.70CC C.4CCC 2 * a.ICCC n . « « « « , E • * • « • » • « ** ^ 3 « « 2 » » 2 « « « • CO . * » * . » a • * * * rH o > -3.?0CC rH <0 3 T3 •H CO CU C4 « * « * * * ** * • « « - lUii -Appendix 3: Sample p r o l a n s for estimating the free-parameter values i n volume-based taper noddy. H I S T •scuwc.e* 1 tR * F TN • 2 REAL LT 3 P=3 .0 4 OC 5 J M t3 5 CP TO f B t 9 . I C ) . J 6 8 C PH = 11 .3 7 H T « 9 . 5 P CO TO 7 9 9 CPH = 3 C 1 IC HT-27.4 11 GC TO 7 I? 10 Cew = 66 . 4 . 13 H 1=31. 5 14 7 CCNTINIIE 15 S H T M H T - l .31/10 16 DO 6 1=1.15 17 I F(1 .GE.12) CQ TO 35 18 LT=(1 -VI*SHT i«; H=MT-LT 20 GC TC 3 21 35 CCNTINUE 22 I F I I.F0.12) LT=LT*0.4C 23 1 F M.GT.I2> LT=LT*0.3D 24 H=H T-L T 25 3 CONTINUE 26 S0=0.662 93 9''.»P*Ceil**l .8066 306*HT«*( 1. 06 01C157 - P M L T * * ( p - l > 27 I Ft SO,LE .0) GC TC 2 ?8 C'SQKTISC) 29 7 K M T E ^ . I O J I CRH.HT .C .LT 30 100 Fr>P.MAT(6X,<.PlC.2) 33 6 Cf KT I.V.JO 32 5 CONTINUE 33 RFTIIKN 34 EMI END Of F K F Appendix 3a. t L I S l *Sf:UPCF* 1 1FIJS * FT'4 ? REM IT 3 P=3 4 nr i j = i , 4 0 5 T t l A S O . 6 N = 0 7 SSO=C 8 S S 1 1 = 0 9 15 PE.'.Ot ^,105.EKr> = 25) 00 ,H T ,D ,H , LT . 10 105 FORMAT(6X,5F10.2) 11 N=N*1 12 PP=C.6 62r>394*P*nR**l.eCt63C6*HT**( 1. C601 857-P ) *L T**( P - l 1 13 • IF(PC.EC.O) GC TC 12 14 PD^SCRTtPDI 15 12 C r \Tl»fl !E 16 P1AS=G-P0 17 TniAS=Tn|ASfBIAS 18 SG>= (D-PD l**2 19 sso=ssr:*sn 2C CC'TC 15 ' 21 25 CCNTINUE 22 SEE*SC<U(SSn/(N-4l • 23 KP.1TEI6.1C8) S E E.TMAStP 24 1C8 F C K M A T 1 6 X,2F10 . 4 . F10 . 2 I 25 I M J . L F . 2 C I r ! A l = J*C.C5 26 I F f J . G T . 2 C I CIAL=( J - 2 O I * C 0 5 27 lFI.J.t. E . 2 0 1 F = 3«CIAL 26 I F I J . G T . 201 P='.-ntAL 29 REWIND 5 3C 1 CCNTINUE 31 RETURN s 32 END END OF FILE Appendix 3b. - 145 -Appendix 4. P. patula: Taper tables for data used in taper studies - given by diameter classes. By row: 1 =distance from ground; 2=inside--bark diameter at that point. '. class 1 2 3 4 5 6 7 0.0 | 0 . 0 0 . 2 0 . 0 1 3 . 1 0 . 0 2 1 . 3 0 . 0 2 6 . 6 0 . 0 3 4 . 0 0 . 0 4 2 . 8 0 . 0 4 9 . 4 0.3 0 . 3 7 . 0 0 . 3 1 2 . 1 0 . 3 2 0 . 7 0 . 3 2 5 . 6 0 . 3 33 . 3 0 . 3 4 1 . 7 0 . 3 4 9 . 1 0.6 0 . 6 6 . 5 0 . 6 1 1 . 4 0 . 6 1 9 . 9 0 . 6 2 5 . 1 0 . 6 3 2 . 5 0 . 6 4 1 . 1 0 . 6 4 8 . 0 0.9 0 . 9 6 . 1 0 . 9 1 0 . 0 0 . 9 1 9 . 2 0 . 9 2 4 . 6 0 . 9 31 . 7 0 . 9 4 0 . 2 0 . 9 4 6 . 8 1.3 1 . 3 5 . 5 1 . 3 1 0 . 1 1 .3 1 0 . 3 1.3 2 4 . 1 1 . 3 3 0 . 9 1.3 3 8 . 7 1.3 4 5 . 3 I 1 .7 5 . 0 2 . 1 9 . 2 3 . 6 1 7 . 5 3 . 9 2 3 . 2 4 . 2 2 9 . 7 4 . 3 3 6 . 0 4 . 3 4 2 . 4 I I . 2 . 1 '•.6 2 . 9 8 . 5 5 . 9 1 6 . 8 6 . 6 2 2 . 7 7 . 1 2 8 . 4 7 . 2 3 4 . 5 7 . 4 4 1 . 1 H I . 2 . 5 4 . 0 3 . 8 7 . 8 8 . 2 1 5 . 6 9 . 2 2 1 . 6 1 0 . 0 2 7 . 1 10 .2 3 2 . 7 10 .4 3 9 . 3 IV . 2 . 9 3 . 6 4 . 6 6 . 7 1 0 . 5 1 5 . 0 11.8 2 0 . 3 1 3 . 0 2 5 . 3 13 .2 3 0 . 4 1 3 . 5 3 5 . 6 V, 3 . 3 3 . 2 5 . 4 5 . 8 1 2 . 8 1 3 . 6 1 4 . 5 18 .8 1 5 . 9 2 3 . 7 16.2 2 8 . 5 1 6 . 5 3 3 . 1 V I 3 . 7 2 . 6 6 . 2 4 . 9 1 5 . 1 1 2 . 2 17 .1 1 7 . 1 1 8 . 8 21 . 5 19 .1 2 5 . 2 1 9 . 5 3 1 . 4 V I I 4 . 1 2 . 1 7 . 0 3 . 8 1 7 . 4 1 0 . 3 1 9 . 8 1 5 . 2 2 1 . 7 1 8 . 5 2 2 . 1 2 1 . 5 2 2 . 6 2 6 . 4 V I I I - 4 . 5 1 .7 7 . 8 2 . 8 1 9 . 7 7 . 8 2 2 . 4 1 1 . 0 2 4 . 6 1 3 . 6 2 5 . 1 15 .7 2 5 . 6 2 0 . 1 IX 4 . 9 1 .2 8 . 7 2 . 0 2 2 . 0 4 . 4 2 5 . 0 6 . 1 2 7 . 5 7 . 8 2 0 . 1 7 . 0 2 8 . 7 10 .0 X 5 . 3 0 . 0 9 . 5 0 . 0 2 4 . 3 0 . 0 2 7 . 7 0 . 0 3 0 . 4 0 . 0 3 1 . 0 0 . 0 3 1 . 7 0 . 0 6-5 50X, of T&7.5 i l »5 1* If 3 Appendix 5. DBH class By 0.0 . 0 . 0 9 . 6 14.^1 0.3 O.i fi. t 0. 3 1 2 . 3 0.6 0. A 7 . 4 0 .6 1 1 . 4 0.9 . • 0 . 9 6 . 9 0.1 1 1 . 0 1.3 1.3 6 . 6 1.3 1 0 . 6 I 1.1 6 . 1 2 . 1 1 0 . 0 II 2 . 4 5 . 5 2 . 9 9 . 2 III 2 . 9 5 . 0 3 . 3 8 . 3 IV 3 . 5 4. 3 4 . 6 7 .4 V 4 . 0 3 . 6 5 .4 6 . 0 VI 4 . 6 2 . 9 6 . ? 4 . 9 VII 5 . 1 2 . 2 7 . 0 3 . 7 VIII 5 . 6 1 . 5 7 . a 2 . 5 IX 6 . 2 0 . 7 S . 7 1 .0 X 6 . 7 0 . 0 9 . 5 0 . 0 AVer ESM 6-9 /, 0 . 0 2 5 . 5 0. 0 3 6 . 9 0 . 3 7 1 . 0 0 .3 3 2 . 0 0 . 6 1 9 . 9 0 . 6 2 9 . 6 0 . 9 1 9 . 0 0 . 9 2 3 . 3 1 .3 1 3 . 5 1.3 2 7 . 4 2 . 7 1 7 . 5 3 . 2 2 4 . 7 4 . 1 1 6 . 7 5 . 2 2 3 . 1 5 . 5 1 5 . 4 6 . 9 1 3 . 3 8 . 3 1 2 . 0 9 . 7 9 . 9 U . l 7 . 2 1 2 . 5 4 . 5 1 3 . 9 2 .0 15. 3 0 . 0 N°-°F-RUts It IS 7 . 1 2 0 . R » . l 11 . 0 1 6 . 6 13 .0 1 3 . 4 14.9 9 . 9 16 .9 6 . 3 18. 8 2 . 9 2 0 . 7 0 . 0 f. 0 . 0 49.ft 0 . 0 • 5 7 . 6 0 . 3 4 0 . 6 0 . 3 4 9 . 6 0 . 6 3 7 . 3 0 . 6 4 6 . 1 0 . 9 3 6 . 3 0 . 9 4 3 . 7 1 .3 3 4 . 6 1.3 4 1 . 5 3 . 9 3 0 . 2 4 . 3 3 5 . 7 6 . 5 2 8 . 6 9 . 1 2 6 . 6 1 1 . 8 2 4 . 7 1 4 . 4 2 1 . 3 1 7 . 0 1 8 . 9 1 9 . 6 1 4 . 9 2 2 . 2 1 0 . 3 2 4 . 8 4 . 9 2 7 . 4 0 . 0 7 . 3 3 3 . 5 1 0 . 3 3 1 . 5 13. 2 2 3 . 3 1 6 . 2 2 5 . 7 1 9 . 2 2 2 . 3 2 2 . 2 1 6 . 6 2 5 . 2 11 .0 2 8 . 2 4 . 9 3 1 . 2 0 . 0 7 1 0 . 3 /. <1. 9 0 . 0 77. 7 0 . 3 5 9 . 7 0 . 3 6 6 . 9 0 .6 5 6 . 4 0 . 6 6 1 . 0 0 . 9 5 3 . 5 0 . 9 5 8 . 9 1.3 5 C . 9 1 . 3 5 6 . 6 4 . 3 4 3 . 6 4 . 4 4 7 . 4 • ' 4 7 . 5 IS 15 3 9 . 9 1 0 . 4 3 7 . 0 1 3 . 5 3 3 . 9 1 6 . 5 3 0 . 4 1 9 . 5 2 6 . 0 2 2 . 6 20 . 0 2 5 . 6 1 2 . 5 2 3 . 7 6 . 0 3 1 . 7 0 . 0 10. 5 4 0 . 1 1 3 . 6 3 6 . 7 1 6 . 7 3 2 . 4 1 9 . 3 2 7 . 8 2 2 . 9 2 1 . 0 2 5 . 9 1 2 . 9 2 9 . 0 5 . 5 3 2 . 1 0 . 0 0 . 0 fl.1.0 0 . 3 7 5 . 9 0 . 6 7 0 . 5 0 . 9 6 6 . 6 52-fc 2 2 . 5 2 3 . 1 2 5 . 5 1 5 . 8 2 8 . 5 6 . 8 31 .'5 0 . 0 fcfe-4r 10 0 . 0 9 i .:• O.i 3 2 . 4 0 . 6 7 9 . 1 0 . 9 7 6 . 2 1 . 3 2 6 . 9 2 2 . 7 3 0 . 5 1 4 . 2 3 4 . 1 8 . 3 3 7 . 8 3 . 0 7SS 0 . 0 ' 13 7 . 7 0 . 3 1 0 5 . 0 0 . 6 9 3 . 0 0 . 3 3 5 . 5 6 4 . 3 7 3 . 2 8 5 . 2 4 . 3 5 5 . 3 4 . 9 5 9 . 0 4 . 1 7 7 . 0 7 . 3 5 0 . 1 8 . 6 5 0 . 3 6 . 9 6 7 . 7 1 0 . 4 4 7 . 3 1 2 . 3 4 8 . 2 9 . e 6 3 . 5 13 .4 , 4 3 . 0 1 5 . 9 4 0 . 1 1 2 . 6 4 7 . 0 1 6 . 4 3 7 . 9 1 9 . 6 3 3 . 6 1 5 . 4 3 4 . 3 19. 4 3 1 . 4 2 3 . 2 2 9 . 1 1 8 . 2 2 5 . 0 2 1 . 0 2 0 . 5 2 3 . 9 1 9 . 0 2 6 . 7 1 1 . 7 2 9 . 5 0 . 0 I S - 147 -Appendix 6. In ?Sr«JJ^  d l S t r i b U t i ° n a l ° ^ t h e P - f ^ s of tree used By rows: l=volume from ground; 2=distance from ground. DBH class 1 0.0 0.0 0.0 0.3 0.6 0.9 1.3 I I I I I IV VI VII VI I I IX o.ooi 0. 30 0.00? 0. 60 0.003 C.90 0.005 1.30 0.005 1.70 0.006 2. 10 0.007 2.50 0.007 2.90 0.008 3.30 0. OCR 3.70 0.008 4.C9 0.008 '+.4 9 0. 0C8 4.89 0, 008 5.29 0.0 0.0 0.004 0.30 0.007 0.60 0.010 0.90 0.014 1.30 0.020 2.12 0.025 2.93 0.030 3.75 0.033 4.57 0. 036 5.38 0.038 6.20 0.039 7.02 0.040 7.83 0.040 8.65 0.04 0 9.47 3 0.0 0.0 0.011 0.30 0.020 0.60 0.029 0.90 0.041 1.30 C. 100 3.60 0.154 5. 91 0.203 8.21 0.246 10.52 0. 284 12.82 0.315 15. 13 0.339 17.43 0,355 19.74 0.3 63 22.04 0.365 24.35 0.0 0.0 0.016 0.30 0.0 31 0.60 0.045 0.90 0.065 1.30 0.193 3.94 0.294 6.57 0.398 9.21 0.490 11.85 0.571 14„49 0.640 17.12 0.695 19.76 0.733 22.40 0.751 25.04 0.75C 27.67 0.0 0.0 0.027 0.30 0.052 0.60 0.077 0.90 0. 103 1.30 0.321 4.21 0.517 7.13 0.697 10.04 0. 657 12.96 0.997 15. 87 1.116 18.79 1 .210 2L .70 1.2 72 24.6? 1.302 27.53 1.310 30.45 0.0 0.0 0.042 0.30 0.C83 0.60 0. 122 0.90 0.171 1.30 0.499 4.27 0.791 7.25 1.056 10.22 1-290 13.20 1 .493 16. 17 1.663 19.15 1.791 22.12 1.875 25. 10 1.912 20.07 1.920 31.05 0.0 0.0 0.057 0.30 0. 113 0.60 0.166 0.90 0.233 1.30 0.698 4.34 1.116 7.38 1.501 10.42 I .836 13.4C 2.119 16.50 2.368 19.54 2.569 22.58 2.701 25.62 2.762 28.66 2.774 31 .70 Appendix 7 . g.lusitanica: Volume distribution along the profiles of tree used in taper studies By rows: l=volume from ground; 2=distance from ground. studies. DBH class 1 2 3 A 5 6 7 S q 0 . 0 0 . 3 0 . 6 0 . 9 X 0.0 0.30 • 0.0 0. 0 0.0 c. 0 0.0 0.0 0.0 0.0 0.0 0.0 0.332 O.50 0.005 0.3 0 0.013 0.30 0.028 0.33 0. 049 0.30 0.069 0.30 0.003 o.io o.ooa 0.60 0.023 0.60 0.051 0.-6 0 0.C85 0 .60 0.123 0.60 C.C04 0.93 o.ou 0.90 0. 032 0.90 0.071 0.90 0.118 0.90 • 0.170 0. 90 0. 0C6 1.30 0.015 1.33 0.043 1.30 0.095 1.30 0.157 1.30 0.228 1.30 0.038 1.84 0.023 2. 12 0.079 2.70 0.201 3.24 0.375 3.91 0.5E2 4.29 1 3 9 1 0 1 1 - W • !:! !:! SS." SriS' a? 1:1" V.l? s-f" ?••» ••>« a - & ° 0.60 0.60 0.60 0.251 0.309 0.393 0.474 ,0. 781 0.90 0.90 0.93 0.90 0.93 . , 0 0C6 0 015 n . n u « - - 0.336 0.414 0.533 0.649 1.021 i ' J 1 .30 1.30 1.30 1.30 1.30 0 038 0.023 0.079 n -> - 0.876 1.072 1.3B8 1.916 2.4S2 I 4.34 4.38 4.32 4.95 4. 12 T T 0.009 0.030 0.112 0.290 0.553 0.864 1.297 1.574 2.049 2.785 3.646 I I 2.33 2.94 4.10 5.19 6.53 7.27 7.33 7.46 7.34 3.60 6.04 0.010 0.C36 0. 140 0.365 0.710 V.1 14 1.654 1.993 2.615 3.433 4.600 III 2.93 3.75 5.51 7.13 9.14 10.26 10.42 10.54 ' 10.36 12.2-5 9.76 C.OU 0.04-1 0.164 0.429 0.846 1.326 1.960 » -IV 3 . A 7 ^.57 <. 0 1 '.-sr l i :;r ±1? 2.211 16.50 ^ il;it8 .?.a65 7 V 0.012 0.045 o 133 n 4 a I 1 5 - " ° t ! " "« ^ ' .Sir i i r l J :jr .!:•? 0.012 4.55 0.048 6.21 C. 196 9.71 0.519 12.96 0.013 5.09 0.049 7.02 C.2C5 11.11 0.543 14.91 0.313 5.63 0. 050 - 7.84 0.209 12.52 0.555 16.85 0.013 6.13 0.051 8.66 0.210 13.92 0.559 18.30 0.013 6.72 0.051 9.43 0.210 15.32 0.560 20.74 VI VII VIII IX I'M1 .?•"«> 0-559 1 .156 19-43 23.20 13.*22' AS' J:J? £11' , , . « . .W »':,T *S» „•:!.« -AS' 1.804 2.631 3.n 5 « • « » . . r M:.v ^ " .ar .SJP AS? ,-«- > { : A. - 149 -Appendix 8. Derivation of volume models from the proposed taper models. Taper model 31 d = b n + b DL + b Dh2L  H H 3 0 0 o o o o o / o o d = b + b D L + bD h L + b b DL + b b DhJL + b b DL + R2 1 R6 H H 3 H (i) ( i i ) 2 2 2 2 2 2 2 b ^ D h L + bQb;Dh L + b b D h i 4 3 4 H H H ( i i i ) (iv) (v) Further Simplification of the parts numbered in the above equation: Replace h with H-L since H=h+L 9 9 9 9 9 9 9 9 9 9 9 (i) = bp (H-L) L ; = b^ D L (H -2HL+L ) (H -2HL+L ) = b^D2L2(H4-2H3L+H2L2-2H3L+4H2L2-2HL3+H2L2-2HL3+L4) 0 0 0 A 0 0 O / b2D L (H -4H L+6H L -4HL +L ) b 2D 2LV-4b 2D 2LV +6b 2DVL 4-4b 2D 2L\ +b 2D 2L 6 H 6 2 H 6 H 6 2"7" b 2A 2-4b 2W + 6 b 2 D V - 4b2D2jA b2pV H2 H 3 H4 H5 H6 ( i i ) = b Qb 2D(H-L) 2L ; = bQb2DL(H2-2HL+L2) H 3 H 3 - 150 -- V W - 2 b o b 2 ^ + b W H IT H = b 0b 2pj,- Zb^DL 2. + oQb2m? H H 2 H 3 ( i i i ) = b ] [b 2D 2L 2(H-L) 2 ; = b 1b 2D 2L 2(H 2-2HL+L 2) H 4 = b 1b 2D 2L 2H 2- 2b 1b 2D 2L 3H + b^D 2!, 4 4 A ' 4 4 H H H b ] [b 2D 2L 2 - 2b 1b 2D 2L 3 + b b D 2L 4 H 2 H 3 H 4 (iv) = b b DL(H-L) 2 ; = bQb2DL(H_2-2HL+L2) H 3 H 3 - b ^ D j V - 2b W H + b ^ D I ? 11 II H = b0b2DL. - 2b ( )b 2DL 2 r + b 0b 2DI? H H 2 H 3 (v) = b 1b 2]A 2_(H-L) 2 ; = b b D 2L 2(H 2-2HL+L 2) 4 4 H 11 b 1 b 2 l A j l 2 - 2b1b2D^L3_H + b ] [b 2D 2L 4 H H H 2 2 2 3 2 4 b.bjTL - 2b.b_pjV + b b D L 1 l Y 1 1 -\ 4 i r H H - 151 -By combining a l l the s i m p l i f i e d forms of the equations above the f o l l o w i n g equation i s obtained:-d 2 " "0 + < 2Vl + 2V2>£| + <b2 + b2 + 2blb,)DV - 4b„b,a! » • ' • ^ o U z „ H H Z 2b 0b 2DL^ - 4 ( b 2 + b l b 2 ) D V + (6b 2 + 2b 1b 2)pV_ - 4b^D 2L 5 H 3 H 3 H 4 H 5 H 6 3 t h i n t e g r a t e d w i t h respect to L. This he equation that i s i n t e g r a t e d f o r volume. I t i s f d L = V cr. " t h e r e f o r e , the volume i s computed from the f o l l o w i n g equation V = 0.000078540^ + (b b + b b W + ( b 2 + bl + 2 b 1 b j D 2 L 3 H 2 3H 4b ( )b 2DL 3_ - b 0 b 2 D L ^ - ( b 2 + b 1b 2)pV. + ( 6 b 2 + 2 b 1 b 2 ) D 2 L 5 3H 2 2H 3 H 3 5H 4 2 b 2 D 2 L 6 + b 2 D 2 L 7 ) 3H 5 7H 6 To compute volume between some s p e c i f i c p o i n t s the i n t e g r a t i o n i s done as shown below: - 152 -L2 V = J d L s h The volume equation then becomes: V g = 0.00007854(b2 + ( b ^ + b Qb 2)D(L 2 - L 2) + e.t.c.) H Where L 2 and are respectively, lower and upper distances from the top of the tree. Taper model 33 d = b + b DL + b DL 3h + b DL 4h H 4 5~ H H d2 = b2 + b 2D 2L 2 + b^D2L6h2 + b 2D 2L 8h 2 + b.b DL + b_b DjA_+ H 2 H 8 H 1 0 H 4 bQb3DL4h + b^ b^ DL + b^pV^h + b^  D2L5h + bQb2DL3h + H 5 H H 5 H 6 H 4 b b0D2L4h + b b j W + b b DL4h + b.b D2L5h + b b D_Vh2 H 5 H 9 H 5 H 6 H 9 By replacing h with H-L and following steps similar to those outlined, in detail for model 31 above, the following model is obtained:-d 2 = b 2 + 2b 0b ]DL + b 2D 2L 3 + 2b Qb 2DL 3 + (b Qb -2b 0b 2)DL^ + H H 2 H 3 H 4 2 4 2b.b D L 1 1 4 H 2b Qb 3DL 5 + (.2b ] [b 3-b 1b 2)D 2L 5 +(b 2-2b 1b 3)D 2L 6 H" II" - 153 -(2b 2b 3-2b 2)D 2L 7 + (b 2+b 2-4b 2b 3)D 2L 8 + (2b 2b 3-2b 2)p 2L 9 + H7 H 8 H9 b 2 D 2 L 1 0 H 1 0 When integrated for volume this equation becomes: V = 0.00007854(b2 + b^DL^ + b 2p 2L 3 + b ^ P L ^ + (2b b -2b b )DL^ + H 3H2 " 2H3 5H4 2 b l b 2 ^ " V34 + ( b l V b l b 2 ^ + (b 2- 2 b lb,)Pl7 + 5H 3H 3H 6 ( b 2 b 3 - b 2 ) P 2 L 8 + (b 2+b 2-4b 2b 3)p\ 9_ + ( b 2 b 3 - b 2 ) P 2 L 1 0 + 4H7 9H8 5H9 b 2 p 2 L n ) Appendix 9a. Pinus patula: Plot of diameter residuals for Model 31. 6 . 1 0 0 • * * * * • * * * * « • * *** * . * * 6 • • «*««*« * 2 *« « » „ . • «« * **« * J * »« * * . « « * * * «« • * * * -2.400 * » * 4.5CC0 13.500 22.5CC 31.500 * 40.500 * 4V.5O0 °-° 9 - o o c o 18.000 27.000 36.000 <,5.0C0 54.000 Observed diameter{cm) Append ix 9 b . Cupressus l u s i t a n i c a : P l o t o f d i a m e t e r r e s i d u a l s f o r Mode l 3 3 . 36.20 19.40 . * * , * * * * 2.6CC • * * * *** * * * »* * * **5*3*«237*** 4* 22 *2 ** » ft* ft « +24 **2 * * * 2 **** 2 ** * . * ft * *** ft ft * a * ** • * « ft ft -14.2C 11.500 34.500 57.500 80.500 133.50 126.50 23.COO 46.0C0 69.000 92.000 115.00 13.3.00 Observed D iameter (cm) I 7.0-1 Appendix 9c. Pinus patula: Plot of residuals of distances from top of trees for Model 1 0 . 6 0 2«« CJ • * * o * , C & • * » * * « t i t t 2 .'•««««. « . « * » » CO . * * « 2« •« * ••* • « • * * * - 2 . 2 0 0 . > , ,_, •a • • • ^ • H • . • , | CO • ' 0 0 3 , 1 6 4 7 A 9 , 5 0 0 0 I 5 # S 3 3 2 2 ' ' 1 6 7 ' 2 9 - 5 ° ° " 3 4 . 8 3 3 ° '° 6 ' 3 " 3 1 9 . 0 0 0 2 5 . 3 3 1 3 1 . 6 6 7 3 0 . 0 0 0 Observed Distance from top of tree(m) 1 7 . 0 0 Appendix 9(d).Cupressus lusitanica: Plot of residuals of distances from top of trees for Model 1 0 . 6 0 E cu o a rt o 3 T3 • H co cu Pi 4.200 * * »* *» * * * * t * * * * . t, . 2 »*"2 2* 3 *3 • 2* * • « » « 2 2 2 * * * » » 2 * * « • * « * -2.200 . 2 * ** . * « • * » * I M I -3.600 . 9 - 5 3 0 3 , , l 5 ' 8 3 3 22-l*T 28.500 * 34.833 6 ' 3 3 3 3 l 2 - 6 6 7 19.303 25.333 31.667 38.000 Observed Distance from top of tree(m) Appendix 10. P. patula: Total volume overbark. 4 7 -i 0 . 0 CO" s r.o-o 0.022 "2 C.017 0.040 16 0.06 2 23 32 36 4 ° 52 56 60 64 6 3 72 76 80 S'i 8? 92 96 I 00 109 10 13 16 19 22 25 23 31 34 37 40 0 .0 13 0. .034 .0 . U 4 5 o, .063 0. 035 0. 103 0. 100 0. 137 0. 1 74 0.2 12 0. "44 0. 199 0.2S4 0.3 C 9 0. 196 0 . 272 0.343 0 .424 0.500 0. 257 C. 357 0.157 0.557 0.657 C.757 0.85) 0.452 0.580 0.7CS 0. 835 C. 963 1.090 1.2 13 0. 559 0.717 0.876 1. 034 1.193 1.351 1.509 1.663 0.677 0.669 1 .062 1. 254 1.447 1.6 39 1. 832 2.024 0.806 1.036 1.266 1.495 1.725 1.9 55 2. 185 2.415 2.645 .1.217 1.4G7 1.753 2.023 2.299 2. 569 2.843 3. 110 3.281 1.412 1.726 2. 041 2.355' 2.670 2.984 3.259 3. 513 3.923 1.621 1.983 2.345 2.707 3.069 3. 430 3.792 4. 154 S.216 1-8*6 2.258 2.670 3.083 3.495 3.907 4.320 4.732 5. 144 2.CS4 2.550 3.017 3. 453 3.949 4.415 4.8 32 5.343 5.314 2.337 2.851 3. 384 3.907 4.431 4.954 5.473 6.001 6. 524 2.605 3.133 . 3.772 4.356 4. 940 5.524 6.103 6.692 7.275 3.5 3* 4.182 4.329 5.477 6. 125 6.772 7.420 3.C67 3.897 4.6 12 5.327 6. 041 6.756 7.471 8. 135 3.900 0.279 5.064 5. 849 6.634 7.419 3.204 8. 939 S.774 4.577 5. 536 6. 395 7. 253 3. 112 3.971 9.329 10.633 5.094 6. 030 6.965 7.901 8.836 9.772 IC.703 11. 6 3 5.523 6. 544 7.550 8.575 5.592 10.603 11.623 12.639 5.960 7. OfiO 8. 179 9.279 10.378 11.477 12. S77 13. 6 76 6.4 50 7. 636 8.823 10.009 11.195 12.331 13.563 14.754 Appendix 11. P patula: Total volume underbark ZPH «C«.) . . HEIGHT f 1 3 l & IS 22 25 •C.027 C.C62 0.076 C l l l 0.1 26 0. 162 0.174 0.2)4 0.254 0.251 0.3C9 0.3c6 C.424 7 .3 4 C.CC2 G.C04 0.0C5 S C. CC3 0.015 0.0? 1 12 c . c i ; 0.C33 0 . 043 16 C.OfiO o.rss 20 0. 134 24 0. 193 23 0.243 32 3 c 40 44 4 S 52 56 60 64 63 72 76 ec 84 38 32 96 100 104 23 31 34 37 4C C.342 0.421 0.5CC C.S73 C.658 0.736 0.447 0.654 0.757 C.C60 ' 0.963 1.066 0.567 C.6?7 0.828 C.959 1.089 1.220 1.351 • 1.4S2 C.7C5 0.861 1.C23 1.185 1.246 1.50S 1.66? 1.33; 0.S47. 1.043 1.239 1.434. 1.630 1.E25 2.C21 2.217 .1.242 1.4T5 1.7C3 l . S ' - l 2.174 2.406 2.639 1.45S 1.732 2.CC5 2.279' 2.552 2.825 3.099 1.692 2.CCS 2.326 2.644 2.961 3.278 3.595 1.943 2.3C7 2.672 3.026 3.4C0 3.765 4.129 2.211 2.626 3.041 3.455 3.870 4.2£4 4.699 2.4S7 2.*•«.<• 3.434 3.9C2 4.370 4.333 5.306 2.SCO 3.325 ' 3.850 4.375 4.9C0 5.425 5.950 3.706 4.291 4.876 5.461 6.046 6.631 4.1C7 4.755 5.404 6.C52 6.701 7.349 4.529 5.244 5.959 6.674 7.3E9 9.1C4 8.110 8.395 2.412 2.E72 3.105 3. 372 3.6*6 3.913 4.230 4.493 4.S57 5.114 5.52S 5.775 6.243 6.475 7.000 7.216 7.301 7.993 3.646 8. 319 9.534 9.680 1C.465 4.971 5.7'j6 £.541 7.3 2-b 5.434 6.292 7.150 3.CCS 8.866 9.724 10.532 11.440 5.917 6. 852 7.786 3.720 9.655 10.589 11.523 12.453 6.422 7.436 £.449 9.463 10.477 11.491 12.505 13.519 6.94c 3.043 ^.143 10.237 11.333 12.430 13.527 14.624 Appendix 12. £, l u s i t a n i c a ; Total volume overbark 7:3 2 2 35 i 3 c 3 72 75 SO 33 92 9 6 ICC 104 EPIC-H3 C.C03 C.C05 0.01". C.020 0.022 C.040 0, Tfi7 10 13 16 1 9 22 25 22 31 34 37 40 C.003 C.023 C .033 G.059 G.C77 0. 096 • 0.09S 0. 129 0. 1 51 0. 193 0.145 0. 192 0.239 0.287 0.201 0. 265 0.331 0.397 0.464 0. 255 0.3 50 0.436 0.523 0. 611 G.6 9s> 0.783 0.-44 0. 5 53 0.663 0.774 0.867 1.000 I.I 13 0.547 0. 6S2 0.813 C. 955 1.0 94 1. 23 7. 1.373 1.514 0.6 60 0.323 0.S87 1. 152 1.319 1.4S7 1.6 57 1.827 0.732 0. 975 1.159 1. 355 1.563 1.762 T.963 2.165 2.367 1. 1 33 1.365 1. 594 1.325 2.053 2.292 2.527 2.764 3. 0 02 1. 312 1.574 1. S 33 2. 105 2.373 • 2.643 2.914 3. 187 3.461 1.497 1.796 2.093 2.402 2. 70S 3.016 3. 325 3. 637 3.950 1.6 93 2.031 2. 372 2.715 3.062 3. 4 10 3.760 4. 112 4.466 1. 899 2. 278 2.661 3.046 3.424 3.8 25 4.2 13 4. 613 5.010 2.116 2.528 2.954 3. 3 93 3. 826 4.261 4.699 5. 139 5.531 2.342 2.810 3.2 81 2.7 57 4.236 4.7 18 5.202 5. 639 6. 179 3.094 3.613 1.1 37 4.6 64 5.194-' 5.728 6.264 6.803 3.390 3.959 4.532 5.110 5.691 6.276 6.863 7. 454 • 3.6 97 4. 3 18 4. 943 5.574 6.208 6.845 7.486 8. 130 4.C16 4. 691 5.370 S.055 6.744 7.436 8. 133 8.832 4.347 ' 5.077 • 5. 8 13 6.554 7.299 8.Q.49 3. 802 9.560 4.690 5.477 6.'270 7.069 7.874 8.682 9. 495 10.312 5.043 5.890 6.743 7.6C2 8.467 9.337 10.211 11.090 5.4CS 6.316 7.231 8. 152 9.080 10.012 10. 950 11.892 I—1 O S o Appendix 13. C. l u s i t a n i c a : Total volume underbark. C 6 u M 32 36 4C 44 56 63 64 co 7? 76 80 ?4 g-3 96 ICO 104 4 7 C.CC? 0.005 C C O " C.CI 5 C O l ^ 0.034 0.05 7 10 1 3 1 f 19 22 25 23 31 34 37 40 0.007 0.323 C.C30 o.c--? 0.06 3 0.0 77 0 .03? 0.107 0. 1 31 0. 156 0. 125 C. 162 0.2CC 0.237 0.1^7 0. 230 0. 2S3 0. 336 0. 3P3 0.2:17 3.320 0.3S0 0.451 0.522 C.593 0.664 0.399 0.491 C.5S3 0.675 C.767 0.3 59 0.951 0.5C1 0.6 17 0.732 0. 343 C.964 1.079 1.195 1.311 0.515 0.757 0.899 1.041 1.183 1.324 1.466 I. 60S 0.740 0.911 1.082 1.253 1.424 1 .595 1.765 1.936 2. 107 1. 050 1.2 82 1.485 1 .637 1 .890 2.092 2.294 2.497 2.699 1.263 1.499 1.736 1.973 2.210 2.446 2. 683 2.923 3.157 1.4 60 1.734 2.CC7 2.231 2.555 2.823 3.102 3.376 3.650 1.671 I .965 2.298 2.611 ' 2.925 3.238 3.552 3.S65 4.178 1.897 2.253 2.6C9 2.964 3.320 3.676 4.0 31 4.3 37 4.743 2. 137 2.538 2.939 3.339 3 .740 4.141 4.542 4. 943 5.343 2.392 2. 840 3.239 2.737 4.136 4.634 5.032 5. 53 1 5.979 3.159 3.6 53 4.157 4.656 5.154 •5;653 6.152 6.651 3.495 4. 047 4.599 5.151 5.703 6.255 6.307 7.353 3.R48 4.4 5.i 5.063 5.671 6.278 6.386 7.494 8. 102 4.2 IS 4.834 5.550 6.217 6.832 7.543 3.215 8.881 4.6C5 5.332 6.0C-0 6.737 7.5 14 3.241 3.963 9.695 5.CC9 5.300 I. 591 7.382 8.173 8.954 9.755 10.546 •5.430 6.2 37 7. 145 3.003 3.360 9.717 10.575 11.432 5.368 6. 7S5 7.721 8.643 9.574 10.531 11.427 12.354 Appendix IA. Extract pages of comprehensive volume-and-taper table. can- 4.c HT= 4.0 HI = 5 . 0 HT = 6 . 0 HT-= 7.0 HT= 8 . 0 H I - * 0 H 0 v c H 0 VC H 0 VC H 0 vc - H 0 VC H 0 VC Q.O 3 . 7 o . ' o o o 0 . 0 3. 7 0 . 0 0 0 0 . 0 3 . 7 0 . 0 0 0 0 . 0 3 . 7 0 . 0 0 0 0 . 0 3 . 7 0. C 0 3 0 . 0 3 . 7 0 . TOO 0.15 3 . 6 o . o o o 0 . 1 5 ' 3 . 6 0 . 0 0 0 0 . 1 5 3 . 6 c . o o c • 0 . 1 5 3 . 7 0 . 0 0 0 0 . 1 5 3 . 7 0. 0 0 0 0 . 1 5 3 . 7 0 . 0 3 0 0 . 3 3 3 . 5 . 0 . o o o 0 . 3 0 3 . 5 0 . 0 0 0 0 . 3 0 3 . 6 0 . 0 0 0 C . J O 3 . 6 0 . 0 0 0 0 . 3 0 3 . 6 C . CCO 0 . 3 0 2 . 6 0 . 0 0 3 0 . 6 0 3 . 3 0 . 0 0 1 o . t o 3.4 0 . 0 0 1 0 . 6 0 . 3 . 4 0 . 0 0 1 0 . 6 0 3 . 5 0 . 0 0 1 0 . 6 0 3 . 5 0 . 0 0 ! .0 . 6 0 3 . 5 0 . 0 0 1 0 . 9 0 3 . 1 0 . 0 0 1 0 . 9 0 3 . 2 0 . 0 0 1 0 . 9 0 3 . 3 0 . 0 0 1 C . 9 0 3 . 3 0 . 0 0 1 0 . 9 0 3 . 4 0 . C 0 1 0 . 9 0 3 . 4 O . O 0 1 1 . 3 0 2 . 9 0 . 0 0 1 1 . 3 0 3 . 1 0 . 0 0 1 1 . 3 0 3 . 1 0 . 0 0 1 I . 3 0 3 . 2 0 . 0 0 1 1 . 3 0 3 . 3 0 . C 0 1 . 1 . 3 0 3 . 3 0. 0 0 1 1 . 4 3 2 . 9 0 . 0 0 1 1 . 4 3 3 . 0 0 . 0 0 1 1 . 5 3 3 . 1 0 . 0 0 1 i . 58 3 . 1 0 . 0 0 1 1 . 6 3 3 . 2 0 . C 0 2 1 . 6 8 2.2 0 . 0 0 2 1 . 5 7 2 . S O . 0 C 1 1 . 6 7 2 . 9 0 . 0 0 1 1 . 7 7 . 3 . 0 0 . 0 0 2 1 - 6 7 3 . 0 C . 0 0 2 1 . 9 7 3 . 1 0 . CC2 2 . 0 7 i. 1 C . C 0 2 1 . 7 0 2 . 8 0 . 0 0 1 I . 8 5 2 . 9 0 . 0 0 2 2 . 0 0 2 . 9 0 . 0 0 2 2 . 1 5 3 . 0 0 . 0 0 2 2 . 3 0 2 . 0 0 . C 0 2 2 . 4 5 3 . C 0 . 0 5 2 1 . 3 4 2 . 7 0 . 0 0 1 2 . 0 4 2 . 8 0 . 0 0 2 2 . 2 4 2 . 9 - C . 0 C 2 2 . 4 4 2.9 0 . 0 0 2 2 . 6 4 2 . 9 . 0.CC2 2 . 8 4 3.0 0 . 0 0 2 1 . 9 7 2 . 6 0 . 0 0 2 2 . 2 2 2 . 7 0 . 0 0 2 2 . 4 7 2 . 8 0 . 0 0 2 2 . 7 2 2 . 8 0 . 0 0 2 2 . 5 7 2 - 9 0 . C C 2 3 . 2 2 2- 9 0 . 0 0 3 2 . U 2 . 5 0 . 0 0 2 2 . 4 1 2 . 6 0 . 0 0 2 2 . 7 1 2 . 7 0 . O C 2 . 2 . C 1 2 . 8 0 . 0 - 0 2 3 . 3 1 2 . 8 0 . 0 0 3 3 . 5 1 2 . e 0 . 0 0 3 2 . 2 4 2 . 5 O . 0 0 2 2 . 5 5 2.6 0 . 0 0 2 2 . 9 4 2 . 6 0 . 0 0 2 3 . 2 9 2 . 7 0 . 0 0 3 3 . 6 4 2 . 7 B . CC3 3 . 9 9 2 . 7 C . G 0 3 2 . 3 3 2.4 0 . 0 0 2 2 . 7 3 2 . 5 0 . 0 0 2 - 3 . 18 2 . 5 0 . 0 C 2 3 . 5 8 2 . 6 0 . 0 03 3 . 9 8 2 . 6 0 . C C 3 4 . 3 8 2 . t O . C 0 3 2 . 5 1 2 . 2 0 . 0 0 2 2 . 9 6 2 . 4 0 . 0 0 2 3 . 4 2 2 . 4 0 . 0 C 3 3 . 8 6 2 . 5 0 . 0 0 3 4 . 3 1 2 . 5 C . C C 3 4 . 7 6 '/.. 5 0 . 3 0 4 2 . 6 5 2 . 1 0 . 0 0 2 3 . 1 5 2 . 2 0 . 0 0 2 3 . 6 5 2 . 3 0 . 0 0 3 4 . 1 5 2 . 4 0 . 0 0 3 4 . 6 5 2 . 4 C . C C 3 5 . 1 5 2 . 4 C . 0 C 4 2 . 7 3 2 . 0 0 . 0 0 2 3 . 3 3 2 . 1 O . 0 0 2 3 . 88 2 . 2 0 . 0 0 3 4 . 4 3 2 . 2 0 . 0 03 4 . 9 8 2 . 3 0 . 0 0 4 5 . 5 3 2 . 3 0 . 0 0 4 2 . 9 3 l . B 0 . 0 0 2 3 . 5 2 2 . 0 0 . 0 0 2 4 . 1 2 2 . 0 0 . 0 0 3 4 . 7 2 2 . 1 0 . 0 0 3 5 . 3 2 2 . 1 0. CC4 5 . 9 2 2 . 1 0 . 0 0 4 3 . 0 5 1 . 7 0 . 0 0 2 3 . 7 0 1 . S 0 . 0 0 2 4 . 3 5 1 . 9 0 . 0 0 3 5 . 0 0 1 . 9 0 . 0 03 " 5 . 6 5 1 . 9 0 . C 0 4 6 . 3 0 2 . 0 0. 0 0 4 3 . 1 9 1 . 5 0 . J 0 2 3 . 8 9 1 . 6 0 . 0 0 2 4 . 5 9 1 . 7 0 . 0 0 3 5 . 2 9 1 . 7 0 . 0 O 3 5 . 5 9 1 . 7 0 . C 0 4 6 . 6 9 1 . 3 0 . 0 0 4 3 . 3 2 1 . 3 0 . 0 0 2 . 4 . 0 7 1 . 4 0 . 0 0 2 4 . 8 2 1 . 4 0 . 0 0 3 5 . 5 7 1 . 5 ~ 0 . 0 0 3 6 . 3 2 1 . 5 0 . C 0 4 7 . 0 7 i . L 0 . 0 0 ' . 3 . 4 6 1 . 1 0 . 0 0 2 4 . 2 6 1 . 1 0 . 0 0 2 5 . 06 1 . 2 0 . 0 C 3 5 . 8 6 1 . 2 0 . 0 03 6 - 6 6 1 . 3 0 . C 0 4 7 . 4 6 1 . 3 0 . 0 0 4 3 . 5 9 c a 0 . 0 0 2 4 . 4 4 0 . 9 0 . 0 0 3 5 . 2 9 0 . 9 0 . 0 0 3 6 . 14 1.0 0 . 0 0 4 6.5-rf 1.0 0. C 0 4 7 . S 4 " 1.0 0 . C C 5 3 . 7 3 0 . 5 0 . 0 0 2 4 . 6 3 0 . 6 0 . 0 0 3 5 . 5 3 0 . 6 0 . 0 0 3 6 . 4 3 C . 6 0 . 0 0 4 7 . 3 3 0 . 7 0. C 0 4 8 . 2 3 0 . 7 C . 0 0 5 3 . S 6 0 . 2 0 . 0 0 2 4 . 8 1 0 . 3 0 . 0 0 3 5 . 7 6 0 . 3 C . C C 3 6 . 7 1 0 . 3 0 . 0 0 4 7 . 6 6 0 . 3 0 . 0 0 4 S . 6 1 0 . 3 0 . 0 0 5 4.00 0.0 0 . 0 0 2 5 . 0 0 0.0 0 . 0 0 3 6.00 0.0 0 . 0 0 3 7.00 C O 0 . 0 0 4 ' 8.CO 0 . 3 S.CC4 9.00 S. 0 0 . 0 C 5 HT«13.0 HT =>20.C H 0 vc H 0 VC o . a 42.2 0.000 0.0 42.2 o . o c o 0. 15 41.3 0.021 0. 1 5 4 i . 9 0.021 0.30 41.5 0.041 - 0.30 41.5 0.04 1 0.60 40.8 0.031 0.60 41.0 0.061 0.90 40.2 0. 120 0.90 40.4 0.120 1.30 39.4 0. 170 1.30 39.7 0.171 2.13 36.0 0.260 2.23 3 8. 2 0.262 2.97 36.3 0. 359 3.17 36.9 0.386 3.60 35.7 0.445 4.10 35.8 0.433 4. 64 34.7 0.527 5.04 34.3 0.574 5.47 33.6 0.603 5.97 33.9 .0.661 6.31 32.9 0.676 6.91 33. 0 0. 743 7.14 31.'9 0.745 7.64 32.0 0.820 7.93 31.0 0. 3 !0 S.73 31.1 0.693 . S.31 29.9 0.B71 9.7! 30. 0 0.962 9.65 23.7 0.927 10.65 26.3 1.025 10.43 27.3 0.979 11.56 2 7.4 1.C84 11.32 25.7 1.02S 12.52 25.8 1.136 12.15 23.9 1.065 13.45 24.0 1.1B1 12.99 21.7 1. C M 14.39 21.6 1.220 13.32 19.2 1.127 15.32 19.3 1.251 14.66 16.2 1. 143 16.26 16.3 1.2(4 15.49 13.0 I . l o 2 17.19 13.0 1.290 . 16.33 9.1 1.170 13.13 9.2 1 .300 17.16 4.3 1. 173 19.06 4. 6 1.303 18.03 0.0 1.173 20.00 C O 1.304 DBH= 44.C HT*22.0 HT=24.0 a D VC H 0 VC 0.0 42.2 0.000 0.0 42.2 0.0 00 0.15 41.9 0.021 0.15 41.9 0.021 0.30 4 1.6 " 0.04 1 0.30 41 . 6 0.041 0.60 41.1 0.0B2 0.60 41.1 0.082 0.90 40, 5 0.121 0.90 40.7 0.121 1.30 39.9 0.1 72 1.30 40. 1 0. 1 72 2.33 30.4 0.296 2.43 36 .5 0.310 3.37 37. 1 0.412 3.57 37.2 0.437 4.40 35.9 0.520 4.70 36.0 0.557 5.44 34.9 6.622 5.64 35.0 C.669 6.47 33.9 0.713 6.S7 34.0 0.775 7.51 33.0 0.309 8.11 33.1 0.075 3.54 32. 1 0.89 5 9.24 32.2 0.9 70 9.53 31.2 0.977 10.38 11.2 1.060 10.61 30. 1 1.053 11.51 20.2 1.144 ! i.65 28.9 1.124 12.65 ::9.0 1.222 12.63 27.5 l . i & 9 13.73 27.6 1.2 93 13.72 25.9 1.247 14.92 26.0 1.353 14.75 24.0 1.296 16.05 24.1 1.414 15.79 21.9 1. 34 1 17.19 21.9 1.461 16.B2 19.3 1.375 13.32 19.4 1.499 17.36 16.4 1.401 19.46 16.5 1.523 IS.69 13.1 1.419 20.59 13.1 1.548. 'l9 . ' . '3 9.2 1.429 21.73 9.3 1.559 20.96 4.8 1.434 22*66 •V.9 1.564 22.CO 0.0 1.434 24. CO 0.0 1.565 H'f*26.0 KT • 28. C H 0 VC H D vc 0.0 42.2' 0.000 0.0 •^2.2 o.oo 'o 0. 15 41.9 C.021 0. 15 • 1 . 9 0.021 0.30 41.7 0.041 0.3G -.1.7 0.041 0.60 41.2 C. C 62 0.6C 41.3 0.0S2 0.90 40 .3 0.122 0.90 40.9 0.122 1. 30 40.2 0. 173 1.30 40.3 0.174 2.53 28.6 0. 324 2.63 : e . 7 0. 337 3.77 37.3 C.463 3.97 57.4 0.439 ' 5. CO 36.1 0. 554 . 5.30 36.2 0.631 6.24 35.0 0.716 6 .64 IS. 1 0.764 7.47 •34.1 0. 832 "7.97 34. 1 0.839 8.71 33.2 0.942 9.31 32. 2 1.008 9.94 32.2 1.046 10.64 22.3 1.12! 11. 18 31.3 1. 143 11.98 11.3 1.22 7 12.41 30.2 1.235 13.31 30. 3 1.326 13.65 29.0 1.320 14.65 2 9: 1 1.419 14.38 27.6 1.393 15. 93 21. 7 1.503 16. 12 26.0 1.463 17.32 i i . 1 1.579 17.35 24.2 i.E30 • 13.65 24.2 1 .644 111.59 22.0 1.532 19.99 22. 1 1.702 19.82 19.5 1.623 21.32 1 9 . i 1.743 21.06 16.5 1. 655 22.66 14.6 1.782 22.29 13.2 1.677 23.99 13.2 1.805 23.53 9.3 1.639 25.33 •3.3 1.E19 24.76 4.9 1.694 26.66 4. 9 1.325 26. CO 0.0 1.495 28 .90 0.0 1.825 H7-30.0 H 0 5-0 ' 42.2 0-15 42.0 0.30 41.7 0.60 41.3 0.90 41.0 1.30 40.5 2.73 33.S 4.17 -3?.4 5.60 36.2 7.04 35.2 e.47 34.2 9.91 32.2 11.34 32.3 12-78 31.* 14.21 30.3 15.65 29.1 17.08 27.7 13.52 26,1 19.95 24.3 21.39 22.1 22.82 19.6 24.26 16.6 25.65 13.2 27.13 9.3 23.56 4.9 30.00 0.0 VC 0. 000 0.021 0. 041 0.002 0.;22 0.17-5 C.35I 0.515 0. 668 c a n 0. 947 1.075 1.196 1. 310 1.417 1.517 1.60S 1.690 1.762 1.623 1. 6 72 1.909 l.?34 1.949 1.555 1.956 Hl*32.0 H D VC 0.0 42.2 0.000 0.13 .42.0 0.021 0.30 41.8 C.042 0.50 41.4 0.082 0.90 41.0 0 . 122 1.3C 40.6 0.175 2.83 38.5 0.365 4.37 37.5 0.541 5.50 36.3 C. 70S 7.44 35.2 0.658 8.97 34.2 1.G04 10.51 33.3 I . M l 12.04 32.4 1.271 13.58 31.4 1.353 15.11 30.3 1.503 16.65 29.1 1.615 18.18 27.3 i»713 19.72 26.2 1.80! 21.25 24.3 1.878 22.79 22.1 1.543 24.32 19.6 1.996 25.86 16.7 2.03o 27.39 13.3 2.063. 28.93 9.4 2.079 30.46 4.? 2.035 32.00 0.0 2.036 H H 0.0 0. 15 0.30 0.60 0.90 1.30 2.93 4.57 6. 20 7.04 9.47 11.11 12.74 14.38 16.01 17.65 19.23 20 .92 .22.55 24.19 25.82 27.46 29.09 30,73 32.36 34.00 BDK- 44.0 ••34.0 HT*36.0 0 VC H O 42.2 0.000 0.0 42.2 42.0 0.02.1 0. 15 42.0 41.8 0.042 0.30 41.e 41.4 0-062 0.60 41.5 41.1 0.122 C.50 41.1 40.6 0.175 1.30 40.7 35.0 0.370 3.C3 39.1 37,6 0.566 4.77 37.6 36.3 0.741 6.50 36.4 35.2 '0.906 8.24 35.3 34.3 1.061 9.57 34.3 33.3 1.207 U . 7 i 33.3 32.4 1.346 13.44 32.4 31.4 1.477 15.16 31.4 30.4 1.595 16.51 30.4 29.2 1.713 18.65 29.2 27.8 1.818 20.38 27.3 26.2 1.912 22.12 26.2 24.4 1.'.'94 23.85 24.4 22.2 2.064 25.59' 22.2 19.6 2.120 27.32 19.7 16.7 2. 163 29.06 16.7 13.3 2.152 3C.79 13.3 9.4 2.209 32.53 9.4 4.9 2.215 34.26 • 5.0 0.0 2.217 36.00 O.C VC 0.000 0.021 0.042 0.082 0.123 0.1 75 0.392 0.5 52 0.773 0.953 1.116 1.274 1.421 1.560 1.6 90 1.812 1.922 2.022 2.110 2 .1 84 2.244 2.2 90 2.321 2.333 2.346 2.347 Hr*3S.O H 0 0.0 42.2 0. 15 42 .0 0.30 41.3 0.60 41.5 0. 90 41.? 1. 30 40.8 3. 13 39.1 4. 57 37.7 6.80 36.4 6.64 35.3 10.47 34.3 .12.31 33.4 14. 14 32.4 15.93 31.5 17.81 30.4 19.65 29.2 21.48 27.9 23.32 26.3 25.15 24.4 26.99 22.2 28.82 19-7 30.66 16.7 32.45 13.3 34.33 9.4 36.16 5.0 33.CO 0.0 VC 0. OCO 0.021 0. C42 C.0S2 0. 123 0. 176 0.405 0.613 0. «1S I.OCO 1. 175 1.340 1.456 1.642 1.781 1. 510 2. C27 2. 133 2.226 2.305 2.36E 2.416 2.445 2 2.476 2.477 68 HT«-4C. C H • 0 VC 0.0 42.2 0.000 C.15 42.0 0.021 0.30 41.5 0.042 0.60 41.J C.033 0.90 41.2 0.123 1.30 40.9 0.176 3.23 39.2 0.419 5.17 37. 7 0.643 7.10 36. 4 0.852 9.04 35. 3 1.043 10.57 3<-3 1-232 12.91 33.4 1.406 -14.S4 32. 5 1.571 16.73 31. 5 1. 727 18.71 30.4 1.872 20.65 25.2 2.COS 22.58 27.5 2.132 24 .52 26.3 2.244 26.45 24. 4 2.342 23.39 12.3 2.425 30.32 2 9. 7 2.492 22.26. It.8 2.543 34.19 13.4 2.573 36.13 5.4 2.593 38.06 5.0 2.606 40.00 0.0 2.608 

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