AN EVALUATION OF NUMERICAL INTEGRATION OF TAPER FUNCTIONS FOR VOLUME ESTIMATION IN EUCALYPTUS SALIGNA STANDS
J. N. Fonweban
Department of Forestry, University of Dschang, P.O. Box I8O Dschang, Cameroon
Received July 1996_____________________________________________________
FONWEBAN, J. N. 1999. An evaluation of numerical integration of taper functions for volume estimation in Eucalyptus saligna stands. Results obtained in this study show that numerical integration of variable-form taper functions constructed for Eucalyptus saligna can give volume estimates of comparable accuracy to those provided by volume-ratio equations for the same species. However, caution should be excercised when using taper functions to estimate tree volumes as some may be biased. A numerical integration routine, based on Romberg integration, was written in QuickBasic for numerical integration of the taper functions. The program can be modified and adapted to other non-integrable taper functions for other species. The routine should be useful to those who do not have easy access to standard numerical analysis packages.
Key words: Numerical integration - taper functions - volume estimation - accuracy - Eucalyptus saligna - Cameroon
FONWEBAN, J. N. 1999. Penilaian penyatuan angka bagi fungsi tirus untuk mengira isipadu di dirian Eucalyptus saligna. Keputusan yang diperoleh dalam kajian ini menunjukkan penyatuan angka bagi bentuk pemboleh ubah fungsi tirus yang dibina untuk Eucalyptus saligna dapat memberikan anggaran isipadu ketepatan bandingan kepada persamaan nisbah-isipadu untuk spesies yang sama. Bagaimanapun, apabila menggunakan fungsi tirus untuk menganggarkan isipadu pokok, kita mestilah berhati-hati kerana sesetengahnya mungkin berat sebelah. Rutin penyatuan angka, berasaskan penyatuan Romberg, disediakan dalam QuickBasic untuk penyatuan angka bagi fungsi tirus. Program tersebut dapat diubah dan disesuaikan dengan fungsi tirus tidak bergabung yang lain bagi spesies yang lain. Rutin tersebut sepatutnya berguna kepada mereka yang tidak mempunyai akses yang senang kepada pakej analisis angka yang piawai.
Introduction
Precise and accurate methods for estimating tree and log volumes are vital for efficient forest inventory and sound forest management. Even though advance- ment has taken place in this field in temperate regions, much still has to be done in the tropics. Volume equations have been the most widely used technique for estimating tree volumes. However, with multiple product demands and changing utilisations standards of forest products there has been a need for volume estimates
410
to different merchantability limits. This could be accomplished by fitting separate volume equations to different merchantable limits; however, such an approach will not only be expensive but could lead to illogical estimates (Cao et al. 1980).
Volume-ratio equations have been proposed as a more flexible and consistent method of volume estimation to any merchantable limit (Bukhart 1977, Cao &
Bukhart 1980). Still, taper equations have an added flexibility because they are capable of describing stem profiles as well as predicting volume to various merchantability limits when integrated (Demaerschalk 1971). Taper functions have also been shown to be useful in silvicultural research, especially in the evaluation of fertilisation or thinning impacts on tree form (Lowell 1986, Bouillet 1993).
While several taper functions of varying complexities have been constructed to take advantage of the aforementioned qualities, the general tendency is to use volume or volume-ratio equations for volume estimation whenever they are available. The reluctance to use taper functions may be ascribed to two problems:
the fact that some taper functions with complex expressions (examples in Hilt 1980, Cao et al. 1980, Martin 1981, 1984) do not have implicit integrals, hence require lengthy numerical integration techniques and; secondly, the fact that some functions, when integrated, provide volume estimates with some inherent bias (e.g. Cao et al. 1980, Martin 1981, Gordon 1983). Even though the problem of numerical integration can be easy to overcome, especially with the advent of modern computing, standard computer packages for numerical analysis are still not readily available for this purpose, especially in less developed tropical coun- tries.
Most recently, Kozak (1988) and Newnham (1988) proposed the use of variable- form or variable-exponent taper functions, which have been shown to describe stem profiles relatively better than existing taper functions, but require numerical integration to estimate tree volume (Kozak 1988, Newnham 1990, Perez et al. 1990).
Fonweban and Houllier (1997) developed variable-form taper functions for Euca- lyptus saligna stands in Cameroon. The objective of this study was to evaluate the accuracy of volume estimates based on numerical integration of these variable-form taper functions.
Materials and methods Numerical methods
Numerical integration methods are needed when a function lacks an anti-derivative (Swokoski 1986). The history of numerical integration goes back to the invention of calculus: the fact that integrals of some elementary functions could not be computed analytically brought about the search for numerical methods during the whole of the 18th and 19th centuries (Press et al. 1992).
Numerical methods commonly used include the Trapezium, Simpson, Romberg and Monte Carlo methods. The Trapezium rule decomposes a surface into several trapezia; the sum of areas (or volumes) of these trapezia provides an
estimate of the area or volume of the surface or solid of revolution. Simpson's rule uses the sum of quadratic pieces (second order polynomials) to approximate areas or volumes. Romberg quadrature is a refinement of the Trapezium and Simpson's rule through successive elimination of truncation error terms. Technical details of these methods are treated in Swokowski (1986), Yokowitz and Szidarovsky (1986), and Press et al. (1986, 1992).
The Romberg method has been shown to be a more generalised and better choice of the three (Goulding 1971, 1979, Press et al. 1992). In this study, the Romberg method required only 5 iterations to converge to results that required 20 to 40 iterations or more using the other two methods. Results obtained using the Trapezium and Simpson methods showed only slight differences compared to those using the Romberg method. Based on these facts and earlier findings (e.g.
Goulding 1971, 1979), we have presented results only for the Romberg method.
Monte Carlo integration serves as an alternative to numerical integration techniques (Yang & Robinson 1986), especially when we are faced with complicated multi-dimensional problems (Karlos & Whitlock 1986). In Monte Carlo integra- tion random points are generated and the fraction of points that fall on or below a given curve is counted. This fraction multiplied by the area of rectangle that completely encloses the curve or a portion of the curve of interest gives the integral or area enclosed under that portion of the function. Details of this method have been treated in Yang and Robinson (1986), Press et al. (1986) and Barrodale and Roberts (1971). The Monte Carlo method has not been considered in this study because preliminary work with it did not give promising results.
Equations used
Following a comparative study of eight volume equations, Fonweban and Houllier (1997) obtained the best results with the Schumacher and Hall (1933) equation. This equation was used to construct volume-ratio equations for Eucalyp- tus saligna in six forest reserves (Baleng, Baham, Melap, Bana, Bafut-Ngemba and Bali-Ngemba Forest Reserves) in Cameroon. The volume-ratio estimation system, denoted VRATIO (cf. Table 1), is composed of two equations: a total volume equation [1] and a volume-ratio equation [2]. The combined form [3] can be written as:
Vd = PflBH1*2 //' [1 - p4 (dt/DBH) P'J where Vd = Volume to a top diameter limit (dt)
The data set, which consisted of 636 Eucalyptus saligna trees measured in the six forest reserves, has been described in detail in Fonweban and Houllier (1997).
The same data set was used to construct variable-form taper functions as described by Newnham (1988) (cf. Fonweban & Houllier 1997). However, because these functions lack explicit integrals, numerical integration is needed to estimate tree volumes from them. A numerical integration routine was written
in QuickBasic following the Romberg method (cf. Yakowitz & Szidarovsky 1986) and used to integrate the taper functions to obtain volume estimates. The aim was to determine whether volume estimates obtained by this method are of comparable accuracy to those provided by the volume-ratio estimation system
(VRATIO)
Table 1. Equation forms used in this study (cf. Fonweban & Houllier 1997)
Model Designation
[1]
[2]
[3]
[4]
Forest reserve
Baleng Baham Melap
Bana Bafut
"
Total volume equation (Schumacher & Hall 1933) 1 - /J, (dl/DUH) ' Volume ratio equation
V(/J; Merchantable volume equation l)[(Hl-k)/(Hl-l30)]"«'> Variable taper function (see below)
Taper function
d=DBH [
d = DBH [(Ht-h)/(Hl-130)] l/ eXP (".«3(i37-().472fils.W(;)BH/HO-o.«()480fi(»A)) d=DBH [(Ht-h)/(
d=DHH [(#/-/,)/ (W<-1. 30)] 1 / e xP «>.««» 17-O..W4355.W-0.05(I35HH A) >
d = DliH [ ( Hl-h) / ( Ht-l .30) ] ' d=DBH [(Ht^h)/(Ht-l .30)] l
where x = [(Hi- h) (Ht-1.30)]
Accuracy testing
The criteria used for evaluating the accuracy of the methods was based on the residuals of differences (D.) between observed volumes (V0) and predicted volumes
By letting Di=Vo-Vp, we defined the evaluation criteria as follows:
Average residual or bias:
B = £ D./n
where n = the number of trees in the sample.
Variance of residuals or precision:
Var (B) = £ (D.-B)* / ( n - 1 )
Mean square error (MSE), which combines bias and precision, was defined as:
MSE = B2 + Var (Cochran 1977)
and root mean square error as
RMSE = MSE = root mean square error.
The later composite criteria (RMSE) was used in evaluating the overall accuracy of the estimations.
In addition to these expected errors, error margins or confidence intervals provided by each estimation method are also important. The interval estimation approach proposed by Reynolds (1984) and implemented in BASIC ATEST and SASATEST by Rauscher (1986) and Gribko and Wiant (1992) respectively was adopted. In this approach, a normality test is conducted as described by Filliben (1975) to determine whether errors are normally distributed or not. In case of non-normality, a 10% trimmed mean and jack-knifed standard deviation are calculated. Rauscher's ATEST source code (Rauscher 1986) was used to write a program in QuickBasic.
Results and discussion
Results for bias (B), 95% confidence intervals (CI), standard deviation (S.D.) and root mean square error (RMSE) are presented in Tables 2 to 5. Results have been given for each forest reserve (Baleng, Baham, Melap, Bana, Bafut and Bali-Ngemba reserves), with the suffix T to denote estimates obtained by numerical integration of the taper function (Romberg method) and the suffix V to denote estimates obtained using the volume-ratio equation (VRATIO).
Values for bias (Tables 2 and 3) are generally low except for Bali forest.
Estimates using VRATIO are generally more precise (lower S.D.), more accurate (lower RMSE) and have narrower confidence intervals than those obtained using numerical integration. This may be due to the fact that the volume estimation system was obtained by directly fitting volume data whereas the taper system was obtained by fitting diameter data. In order to obtain tree volumes from the taper function, numerical integration was needed. Since this process also required an iterative estimation of tree heights to given diameter limits, some errors may have been incurred leading to less accurate estimates of tree volumes.
Munro and Demaerschalk (1974) have shown that a totally unbiased taper equation does not exist and that very few taper-based systems provide reasonably satisfactory results for both taper and volume estimation. Cao et al. (1980) observed that this assertion was true during an evaluation study of 12 volume-ratio and taper equations, in which they found that no single equation (even compatible taper functions) consistently performed best in predicting tree diameters, tree volumes and tree heights to various top merchantability limits. However, Munro and Demaerschalk (1974) acknowledged the fact that an unbiased taper function can
produce reasonably accurate volume estimates when integrated. In this study, volume estimates obtained from numerical integration are generally unbiased and compare fairly well with those provided by the volume estimation system. Other encouraging conclusions have also been reached for other species using variable- form taper functions (e.g. Kozak 1988, Newnham 1990).
Table 2. Values for bias, 95% confidence interval (Cl), standard deviation (S.D.) and root mean square error (RMSE) for total volume estimation of Eucalyptus saligna in Baleng, Baham, Melap, Bana, Bafut and Bali Forest Reserves
Reserve Baleng, Balengv BahamT, Bahamv MelapT MelapV
BanaV
BanaV
BafutT
BafutV
BaliT Baliv
N 126 126 91 91 140 140 100 100 96 96 83 83
Bias (m-3) - 0.0083 - 0.0034 0.0043 0.0003 0.0013 - 0.0005 - 0.0009 -0.0016 -0.0179 -0.0008 - 0.0273*
0.0005
[±]CI(m3) 0.0123 0.0128 0.0175 0.0147 0.0129 0.0104 0.0138 0.0092 0.0353 0.0233 0.0202 0.0183
S.D.(m3) 0.0695 0.0723 0.0841 0.0705 0.0772 0.0623 0.0694 0.0464 0.1741 0.1149 0.0925 0.0838
RMSE(m3) 0.0699 0.0724 0.0842 0.0705 0.0772 0.0623 0.0694 0.0464 0.1750 0.1149 0.0964 0.0838
*Bias is significant at 5%. Bias is significant if the 95% confidence interval (CI) does not contain zero.
Table 3. Values for bias, 95% confidence interval (CI), standard deviation (S.D.) and root mean square error (RMSE) for merchantable volume estimation of Eucalyptus saligna in Baham, Bana, Bafut and Bali Forest Reserves
Reserve N Bias(m3) S.D.(m3) RMSE m()3
Baham, Baliamv BanaT BanaV
BafutT
BafutV
BaliT
Baliv
91 91 100 100 96 96 83 83
0.0066 0.0115 - 0.0003 - 0.0074 -0.0161 0.0109 0.0213*
0.0071
0.0177 0.0147 0.0138 0.0092 0.0355 0.0235 0.0203 0.0185
0.0848 0.0706 0.0696 0.0466 0.1753 0.1159 0.0929 0.0847
0.0850 0.0715 0.0696 0.0472 0.1760 0.1165 0.0953 0.0850
*Bias is significant at 5%. Bias is significant if the 95% confidence interval (CI) does not contain zero.
Graphical displays for the biased model for Bali-Ngemba Forest Reserve were made to assess the general performance of the model over the full range of data used (Figures 1 & 2)1. Predicted versus observed total volumes indicate a close
1S i m i l a r graphs were made for the other forest reseves. Because similar trends were obtained, they have been
left out to reduce redundancy in the presentation.
relationship between volumes obtained by the numerical method and those from the volume-ratio equation system (Figure 1), with some slight dispersion for large trees. Residual plots (observed - predicted volumes) indicate some tendency towards underestimation for large trees using both methods (Figure 2). This may be an indication of non-randomness in the error structure and hence, a violation of the homogeneity of variance assumption in the models. This problem is common in volume and biomass studies in forestry and can be dealt with via some transformations or weighting (Cunia 1964) so as to stabilise the variance.
2.b
ST 2
CD
|1.5
O
* Volume equation
• Numerical method
T3 3 1
0.5
0.5 1 1.5 Observed volume (m3)
2.5
Figure 1. Predicted versus observed total volume for Eucalyptus saligna in Bali-Ngemba Forest Reserve
_ 0.3
•§• 0.2 o>
I 0.1
=5 •O-1 0)
Q. -0.2 I
"g -0.3
.a
° -0.4 -0.5
» Volume equation • Numerical method
Observed volume (m3)
Figure 2. Residuals versus observed total volume for Eucalyptus saligna in Bali-Ngemba Forest Reserve
Table 4 gives merchantable height (height to 7 cm top diameter) estimates obtained by numerical iteration of the taper function. Results indicate that the estimates are biased with an overestimation of close to 0.65 m (65 cm). However, these figures may be misleading given that actual merchantable heights did not exist, but were obtained through linear interpolation from the field data. The results should therefore be interpreted with some caution. However, despite this bias in merchantable height estimation, merchantable volumes obtained by nu- merical integration (Table 3) are not biased.
Table 4. Values for bias, 95% confidence interval (CI), standard deviation (S.D.) and root mean square error (RMSE) for merchantable height estimation (using the taper function) of Eucalyptus saligna in Baham, Bana and Bafut Forest Reserve
Reserve N Bias(m) [±]CI(m) S.D. (m) RMSE(m)
Baham Bana Bafut Bafut
91 100 96 83
- 0.668*
-0.579*
-0.697*
- 0.378*
0.255 0.156 0.337 0.348
1.225 0.784 1.665 1.592
1.395 0.975 1.712 1.636
* = bias is significant at 5%. Bias is significant if the 95% confidence interval (CI) does not contain zero in this interval.
Results in Table 5 are based on a limited independent data set of 57 trees obtained from Bana Forest Reserves. Usually, for purposes of model validation, it is important to use an independent data set so that results obtained mimic reality.
Again, these results do not show any apparent bias for total and merchantable volume estimations.
Table 5. Values for bias, 95% confidence interval (CI), standard deviation (S.D.) and root mean square error (RMSE) for total and merchantable volume and merchantable height estimations of Eucalyptus saligna in the Bana Forest Reserve
Method Bias ( m3) S.D.(m3) RMSE(m3)
VRATIO Romberg
VRATIO Romberg
- 0.009 -0.005
- 0.00002 -0.004
Total volume 0.011 0.019 Merchantable volume
0.011 0.019
0.042 0.071
0.042 0.072
0.043 0.071
0.042 0.072
Summary and application
Based on the evaluation carried out in this study, we can conclude that:
• Numerical integration of the variable-form taper equations constructed for Eucalyptus can provide volume estimates with comparable accuracy to those obtained from corresponding volume-ratio equations for the species in the various forest reserves. However, caution should always be exercised when using taper functions to estimate tree volumes because some of them could be biased.
• Numerical integration that constitutes a major drawback to the use of some taper functions has been specifically considered in this study by writing a simple computer routine in QuickBasic, a common programming language available in the later versions of the disk operating system (DOS). The routine can be easily adapted to other non-integrable taper functions con- structed for other species.
• The program will allow the user to compute total bole volume and volumes to various merchantability limits, heights to various top diameter limits, diameters to various height limits. Inputs required include tree dbh (in cm) and total height (in m). These inputs can be entered iteratively on the keyboard, but it is preferable to access data through a previously created file, especially when we need to estimate volumes for many trees.
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