Trees (2004) 18: 467–479 DOI 10.1007/s00468-004-0333-z
ORIGINA L ARTI CLE
Huiquan Bi . John Turner . Marcia J. Lambert
Additive biomass equations for native eucalypt forest trees of temperate Australia
Received: 26 June 2002 / Accepted: 25 March 2004 / Published online: 18 May 2004 # Springer-Verlag 2004
Abstract Biomass additivity is a desirable characteristic of a system of equations for predicting component as well as total tree biomass since it eliminates the inconsistency between the sum of predicted values for components such as stem, bark, branch and leaf and the prediction for the total tree. Besides logical consistency, a system of additive biomass equations when estimated by taking into account the inherent correlation among the biomass components has greater statistical efficiency than separately estimated equations for individual components. Using mostly small sample data from both published and unpublished sources, a system of non-linear additive biomass equations was developed for 15 native eucalypt forest tree species of temperate Australia. Diameter at breast height was used as the independent variable for all 15 species, while the combined variable of diameter and tree height was used for 14 species with height data. The system of additive equations provided more accurate biomass estimates than the common approach of separately fitting total tree and component biomass equations using log transformed data through least squares regression. Residual error variances were collectively estimated for each species by pooling small sample data across species and using indicator variables to represent the scale factor for each species in a residual variance function. This method overcame a common problem in estimating heteroscedastic error variance in non-linear biomass equations with additive error terms for small samples. From the estimated residual H. Bi (*) . J. Turner . M. J. Lambert Research and Development Division, State Forests of New South Wales, Cooperative Research Centre for Greenhouse Accounting, P.O. Box 100 Beecroft, NSW, 2119, Australia e-mail:
[email protected] Tel.: +61-2-98720168 Fax: +61-2-98716941 Present address: J. Turner . M. J. Lambert ForSci Pty. Ltd, Unit 10, 124 Rowe Street, Eastwood, NSW, 2122, Australia
variance functions, approximate confidence bands containing about 95% of the observed data about the mean curve of predicted biomass were derived for all biomass components of each species. This system of additive biomass equations will prove to be useful for biomass estimation of native eucalypt forests of temperate Australia. Keywords Additive biomass equations . Residual error variance . Acacia . Angophora . Eucalyptus
Introduction Biomass equations for individual trees have appeared frequently in the ecological and forestry literature over the last 50 years because biomass estimation is a prerequisite for studies on forest productivity, nutrient cycling and for calculating biomass energy, carbon storage and sequestration of forests. The most common method of developing biomass equations is to relate total tree biomass and its components such as stemwood, bark, branches and foliage to diameter or both diameter and height using log transformed data through least squares regression (e.g. Satoo and Madgwick 1982; Overman et al. 1994; TerMikaelian and Korzukhin 1997; Burrows et al. 2000; O’Grady et al. 2000; Ketterings et al. 2001). The equation for each component is usually estimated separately without taking into account (1) the inherent correlation among the biomass components measured on the same sample trees and (2) the logical constraint between the sum of predicted biomass for tree components and the prediction for the total tree. As a result, many reported biomass equations are not estimated most efficiently and lack additivity among the component equations (Parresol 1999). The lack of additivity means inconsistency in logic in the sense that the predicted values from the biomass equations of tree components do not add up to the predicted value from the equation for the total tree biomass (Kozak 1970).
468
To eliminate this inconsistency, several model specifications and estimation methods have been suggested for forcing additivity on a system of biomass equations, both linear and non-linear. In the linear case, component biomass has been specified as a quadratic function of diameter (Kozak 1970; Chiyenda and Kozak 1984; Cunia and Briggs 1984, 1985) or as a function of diameter, height and live crown length (Parresol 1999). The estimation methods have included simple least squares (Kozak 1970; Cunia and Briggs 1984), restricted least squares (Chiyenda and Kozak 1984), weighted least squares and seemingly unrelated regressions (SUR) with cross-equation constraints (Cunia and Briggs 1984, 1985; Parresol 1999). In the non-linear case, power functions of diameter and height have been employed (Reed and Green 1985; Bi et al. 2001; Parresol 2001). The estimation methods have involved minimising a loss function which is the sum of the standardised sums of squared error from each of the component equations (Reed and Green 1985), non-linear seemingly unrelated regressions (NSUR) and maximum likelihood method (Bi et al. 2001; Parresol 2001; Carvalho and Parresol 2003). A review of most procedures for forcing additivity can be found in Parresol (1999, 2001).
Besides logical consistency, a system of additive biomass equations when estimated by taking into account the inherent correlation among the biomass components has greater statistical efficiency as demonstrated by Parresol (1999, 2001). Despite these merits, this approach has not been widely adopted in developing biomass equations, perhaps because it requires statistical analysis and software beyond ordinary least squares regression. There are several hundred native tree species in Australia (Boland et al. 1992), and many biomass equations have been reviewed by Eamus et al. (2000) and Keith et al. (2000). However, none of these equations have taken into account the inherent correlation among the biomass components, and none are additive, although data on component biomass are available. To overcome these weaknesses, this paper presents a system of additive biomass equations for 15 species of native eucalypt forest trees of temperate Australia using data from both published and unpublished sources. Furthermore, it demonstrates how the difficulty of evaluating the accuracy of biomass estimation for small samples may be overcome through collectively estimating the residual variance function using data pooled across species.
Table 1 Number of samples (n), the sample location and source of data for the 15 tree species analysed Species Acacia dealbata
n
23 4 Angophora costata 6 E. agglomerata 10 E. cypellocarpa 6 E. dalrympleana 7 E. diversicolor 6 6 E. fastigata 15 12 E. mannifera 7 E. muellerana 6 11 E. obliqua 5 3 5 3 11 E. pilularis 17 41 E. radiata 6 6 5 E. regnans 22 E. rossii 7 E. sieberi 19 10
State
Site
Age
Source
NSW NSW NSW Victoria NSW NSW WA WA NSW NSW NSW NSW Victoria Tasmania Tasmania Tasmania Victoria Victoria Queensland NSW NSW NSW Victoria Victoria NSW NSW Victoria
Glenbog Coffs Harbor Cumbland Genoa Yambulla Wee Jasper Pemberton Pemberton Glenbog Tallaganda Lidsdale Yambulla Genoa Emu Ground Retreat Turquoise Bluff Maroondah Strezlecki Ranges Fraser Island Coffs Harbour Lidsdale Wee Jasper Maroondah Warburton Lidsdale Yambulla Genoa
Regrowth 9 years Unknown Mixed age Mixed age Mature 4-11 years Unknown Regrowth Mature Regrowth 30-45 years Mixed age Mixed age 100 years 80-100 years 100 years Mature 70-90 years Regrowth and mature 3-42 years Regrowth 30-45 years Mature Mature 4-7 years Regrowth 30-45 years Mixed age Mixed age
Bi et al. (2001) Birk and Turner (1992) J. Turner (pers. comm.) Stewart et al. (1979) Turner et al. (1992) and Turner and Lambert (1986) Turner and Kelly (1985) Groves and Malajczuk (1985) Hinston et al. (1979) Bi et al. (2001) J. Turner (pers. comm.) Lambert (1979) Turner et al. (1992) and Turner and Lambert (1986) Stewart et al. (1979) Adams and Attiwill (1988) Adams and Attiwill (1988) Adams and Attiwill (1988) Feller (1980) Baker and Attiwill (1985) Applegate (1982) Turner and Lambert (2004) Lambert (1979) Turner and Kelly (1985) Feller (1980) Attiwill (1992) Lambert (1979) Turner et al. (1992) and Turner and Lambert (1986) Stewart et al. (1979)
469
Data Biomass data from about 300 individuals of 15 Australian tree species were collated over the past 20 years largely by the second author of this paper from published sources scattered across literature and unpublished internal reports (Table 1). The sample trees came from mostly native eucalypt forests, apart from Eucalyptus pilularis which included plantation trees. These native forests belong to the “open-forest” formation dominated by evergreen eucalypts which occur only in relatively narrow coastal zones where rainfall is relatively high and fairly reliable, that is, along the eastern and south-eastern coasts (including Tasmania), and in the far south-west of Western Australia (Specht 1970). Different from the “closed-forest” formation represented by rain forests (usually noneucalypt, moist, broad leaf forests), leafy crowns of the trees in the upper canopy of “open-forests” dominated by eucalypts do not touch or intermingle and the projected foliage cover is between 30 and 70% (Specht 1970; Florence 1996). The “open-forest” formation can be subdivided according to stand dominant height into three subformations: tall-open (>30 m), open (10-30 m) and low-open (5-10 m) forests (Specht 1970). In ecological and forestry literature, the tall-open forest has commonly been referred to as “wet sclerophyll forest”, and the open
and low-open forest as “dry sclerophyll forest” (Florence 1996). The sample trees in the data represented both wet and dry sclerophyll forests in five states of Australia (Table 1). The sample trees consisted of both young trees in regrowth stands and mature trees in mixed age stands (Table 1). For each sample tree, the data contained total tree biomass and four biomass components: stemwood, bark, branches and foliage. For most large tree species of wet sclerophyll forests, the sample trees covered a reasonable size range with the largest tree equal to or greater than 80 cm in diameter at breast height overbark (DBH). However, for E. regnans, only data from small trees were available (Fig. 1). Whilst DBH data were present for all trees, tree height data were not always complete. Tree height was not available at all for some species or was available only for some of the sample trees (Fig. 1). For each species represented in the biomass data, the biological characteristics, the forest type to which it belongs, the geographical distribution, climate and soil conditions are described in detail by Boland et al. (1992). Acacia dealbata is a medium-sized tree attaining 30 m in height and 40 cm in diameter and grows mainly within open-forest or tall-open forest. Angophora costata is usually a medium-sized tree attaining 15-25 m in height
Fig. 1 Height and diameter at breast height overbark (DBH) of the sample trees. For samples without tree height measurements, data points were plotted on the horizontal line at height equal to zero
470
and 50-100 cm in diameter. On hard sandstone sites near Sydney where the samples came from, trees are irregularly branched having rather open crowns and very short boles (Boland et al. 1992). Apart from these two species, all other species belong to the genus of Eucalyptus and most are tall forest trees which can attain height of more than 30 m. Among the Eucalyptus species, E. cypellocarpa, E. dalrympleana, E. diversicolor and E. mannifera belong to the subgenus Symphyomyrtus, all other Eucalyptus species belong to the subgenus Eucalyptus (or Monocalyptus). Differences in the distribution and ecological characteristics of the subgenera are given in detail by Florence (1996).
⋯,5), σij represents the covariance between the error term of the ith and the jth equation (j=1,⋯,5), ⊗ denotes the Kronecker product, T denotes the number of observations and IT is an identity matrix of order T. This model specification is based on that of Bi et al. (2001) and Parresol (2001). For parameter estimation, logarithmic transformation was taken on both sides of equations of model 1 such that y1 y2 y3 y4 y5
Model specification and estimation Two model specifications are compared in order to select the most appropriate specification for the 15 species. The first is a system of five equations with multiplicative error terms, cross-equation constraints on the structural parameters and cross-equation error correlation for four tree biomass components and total tree biomass with additivity: Y1 Y2 Y3 Y4 Y5
10
where ln denotes natural logarithm and yI ¼ ln YI . This model was fitted to the data of each species using NSUR in the PROC MODEL procedure of SAS (SAS 1988). The second model specification has the same deterministic component and cross-equation error correlation but with an additive error structure: Y1 Y2 Y3 Y4 Y5
11 "1
¼e D e ¼ e20 D21 e"2 ¼ e30 D31 e"3 ¼ e40 D41 e"4 ¼ ðe10 D11 þ e20 D21 þ e30 D31 þ e40 D41 Þe"5
¼ 10 þ 11 ln D þ "1 ¼ 20 þ 21 ln D þ "2 ¼ 30 þ 31 ln D þ "3 ¼ 40 þ 41 ln D þ "4 ¼ lnðe10 D11 þ e20 D21 þ e30 D31 þ e40 D41 Þ þ "5 (5)
¼ e10 D11 ¼ e20 D21 ¼ e30 D31 ¼ e40 D41 ¼ e10 D11
þ "1 þ "2 þ "3 þ "4 þ e20 D21 þ e30 D31 þ e40 D41 þ "5 (6)
(1) where Y1 to Y5 represent stem wood, stem bark, branch, leaf and total tree biomass in kg, respectively, D represents overbark diameter at breast height (1.3 m) in cm, βij are coefficients. The error terms can be expressed in the matrix algebra notation as follows: " ¼ ½"1 ; "2 ; "3 ; "4 ; "5 0
(2)
The properties of ε are E ð"Þ ¼ 0
(3)
and Covð"Þ ¼ E ð""0 Þ 2 11 12 6 21 22 6 ¼6 6 31 32 4 41 42 51 52
13 23 33 43 53
14 24 34 44 54
3 15 25 7 X 7 35 7 IT 7 IT ¼ 45 5 55
(4)
where E(ε) and Cov(ε) denote the expectation and covariance of ε, σii represents the variance of εi (i=1,
This model specification was first reported by Reed and Green (1985) when demonstrating the use of a loss function in forcing additivity among biomass equations. For relatively large samples, this model could be fitted to the data using weighted NSUR as demonstrated by Parresol (2001) to take into account the cross-equation error correlation in parameter estimation. However, with small samples it would be difficult to obtain weighting functions that can accurately portray the underlying pattern of variance of the error terms. So model 6 was fitted to the data of each species using the generalised method of moments (GMM) that produces efficient parameter estimates under heteroscedastic conditions without any specification of the nature of the heteroscedasticity (Greene 1999). For easier convergence in parameter estimation, model 6 was coded as Y1 Y2 Y3 Y4 Y5
¼ expð10 þ 11 ln DÞ þ "1 ¼ expð20 þ 21 ln DÞ þ "2 ¼ expð30 þ 31 ln DÞ þ "3 ¼ expð40 þ 41 ln DÞ þ "4 4 P ¼ expði0 þ i1 ln DÞ þ "5
(7)
i¼1
while fitting using the PROC MODEL procedure of SAS (SAS 1988).
471
Another problem in estimating model 6 was that the error term of the system equation for total tree biomass was a linear combination of other error terms, which led to a singular across equation variance and covariance matrix. For model 6 to be estimated as a system of seemingly unrelated equations, the matrix needs to be a positive definite matrix to preclude the possibility of any linear dependencies among the error terms in the system of equations (Srivastava and Giles 1987). The problem is overcome in SAS by computing a generalised inverse of the variance and covariance matrix through setting the part of the matrix for the total tree biomass to zero. The parameter estimates were equivalent to fitting the first four system equations while leaving that for total tree biomass out of the system. To select the model specification which best fitted the data, the difference between the observed and fitted values, ε, was calculated for each system equation of each species following parameter estimation. The fitted values from model 5, the specification with a multiplicative error structure, were back-transformed from logarithm for this purpose. Then mean squared error (MSE) was calculated for all component equations of models 5 and 6, respectively, since it is a measure of accuracy of estimation: MSE ¼ E "2 ¼ E ½" E ð"Þ þ E ð"Þ2 ¼ Varð"Þ þ ½E ð"Þ2
(8)
where Var(ε) and E(ε) are the error variance and expectation, indicating the precision and bias of estimation, respectively (Wackerly et al. 1996). To see whether model 6 is superior to model 5, a MSE ratio was calculated for each species and each system equation: Ri ¼ MSE5i =MSE6i , where MSE5i and MSE6i represent the MSE for the ith (i=1–5) system equation of models 5 and 6, respectively. In addition, fit index, a statistic analogous to R2, was calculated for each system equation of both models to indicate the goodness of fit as follows: 2 T P Yi Y^ i R2 ¼ 1 i¼1 T P
Yi Y
2
(9)
i¼1
where Yi and Y^ i are the observed and predicted values and Y is the mean observed value of biomass. A ratio of fit index of model 5 to that of model 6 was also calculated for each system equation of each species. Values of the MSE ratio were generally greater than 1 with the average R1, R2, R3, R4 and R5 being 1.92, 1.85, 2.78, 1.57 and 2.49, respectively. The fit index ratio was generally less than 1. Therefore, model 6 was more accurate in biomass estimation and it was chosen as the model specification for all species.
For a number of species, biomass data consisted of sample trees from two or more sites (Table 1). During analysis appreciable differences in the relations of component biomass to diameter were detected between the New South Wales and Queensland data of E. pilularis. Log transformed data of each biomass component was regressed against log diameter using a dummy variable representing site in the intercept and/or the slope term. Parameter estimates associated with the dummy variable for all biomass components apart from total tree biomass was significantly different from zero at α=0.05. Such within species variability was not found for other species. So the data of E. pilularis were analysed separately to derive two sets of system equations for the same species at two locations. The differences in biomass estimates between the two sets of system equations were evaluated relatively. For each biomass component at a given diameter, the absolute difference between the predicted value for New South Wales and that for Q ueensland was calculated at first. Then the absolute difference was divided by the predicted value for New South Wales and multiplied by 100 to obtain a percentage difference. For species with tree height data, the combined variable of diameter and tree height, D2H, was used in place of D in Eq. 7 to develop another set of system equations so that tree height data can be used together with diameter in biomass estimation for these species. In comparison with the results of using D as the independent variable, the combined variable generally gave smaller values of mean square error and larger values of fit index for stem and bark components, but not so for branch and leaf components. The comparison was made for those species with diameter and height measurement for all trees, i.e. species with the same number of samples when either D or D2H was used in Eq. 7. The following model specification was finally adopted for species with tree height data: Y1 Y2 Y3 Y4
¼ expð10 þ 11 ln D2 H Þ þ "1 ¼ expð20 þ 21 ln D2 H Þ þ "2 ¼ expð30 þ 31 ln DÞ þ "3 ¼ expð40 þ 41 ln DÞ þ "4 2 P ¼ expði0 þ i1 ln D2 H Þ
(10)
i¼1
Y5 þ
4 P
expði0 þ i1 ln DÞ þ "5
i¼3
Evaluating prediction accuracy Since the system of equations for many species are based on small samples, it is difficult to obtain any reliable statistics on their predictive accuracy in the way shown by Bi and Hamilton (1998), and Bi (1999,2000) for volume and taper functions using large samples. However, to evaluate their prediction accuracy at least indicatively, the system of equations of model 6 with D as the independent variable was compared with the common approach of separately fitting total tree and component biomass
472
equations using log transformed data through least squares regression. A MSE ratio and a ratio of fit index were calculated in the same way as described above for this purpose. In addition, approximate confidence bands containing about 95% of the observed data about the mean curve of predicted biomass were derived for all biomass components of each species through modelling residual variance as a function of diameter. For a biomass component of a species, the residual variance, Var(ε), is assumed to be some power of the estimated mean, Y^ , multiplied by a scale factor σ2 such that Varð"Þ 2 Y^ b
(11)
The theoretical background and applications of such methods are discussed in detail in the context of Estimated Generalised Least Squares in econometric texts (Amemiya 1985; Greene 1999). Parresol (2001) demonstrated a particular example of the use of such methods in estimating weighting functions for weighted NSUR for biomass estimation. Since reliable estimates of a variance function were particularly difficult to obtain for species with small samples, data were pooled across species for analyses to overcome this difficulty. As in modelling heteroscedasticity in the context of Estimated Generalised Least Squares (Amemiya 1985; Greene 1999), the squares of residuals were used as representative of their variances and as the dependent variable in the following regression model: ln "2 ¼ ln 12 þ
16 X
ln s2 Is þ b ln Y^
(12)
s¼2
where ln stands for natural logarithm, ε represents the residual of a system equation, σ12 to σ162 are the scale factors associated with each of the 16 species groups, and Is is an indicator variable equalling 1 for the species it represents and 0 otherwise. Graphical examination of residuals against the predicted values showed that Acacia dealbata and Angophora costata were within the magnitude of residual variations of the Eucalyptus species for all biomass components, so they were analysed together with the eucalypts in modelling residual variance. Since the scale factor in Eq. 12 is species-specific and Y^ is a species-specific function of either D or D2H, the collective analysis results in species-specific residual variance functions of either D or D2H. Once the residual variance function is obtained, approximate confidence bands containing about 95% of the observed data about the mean curve of predicted biomass were derived for all biomass components of each species by assuming conditional normal distribution of residuals at any given value of the predictor variable. A detailed example can be found in Bi and Hamilton (1998).
Results The estimated coefficients of the system of additive biomass equations for the 15 species with diameter as the only independent variable showed different degrees of variation among the four biomass components (Table 2). As expected, the estimated exponent for stem biomass, ^11 , was the most consistent among the four biomass components. It ranged from 2.00 to 2.70 with an average
Table 2 Estimated coefficients of the system equations (Eq. 6) and fit indices (R2) for the 15 tree species where D is the only independent variable in the equations Species
Acacia dealbata Angophora costata E. agglomerata E. cypellocarpa E. dalrympleana E. diversicolor E. fastigata E. mannifera E. muellerana E. obliqua E. pilularis (NSW) E. pilularis (QLD) E. radiata E. regnans E. rossii E. sieberi
N
27 6 16 8 10 12 27 8 21 29 41 17 17 22 9 29
Stem
Bark
β10
β11
R
−2.544 −1.104 −2.572 −1.725 −2.751 −3.035 −3.022 −2.864 −2.312 −1.870 −3.031 −2.122 −1.738 −2.685 −1.131 −2.035
2.569 2.049 2.421 2.258 2.512 2.701 2.595 2.488 2.368 2.218 2.500 2.340 2.260 2.352 2.000 2.311
0.963 0.952 0.959 0.978 0.964 0.978 0.977 0.993 0.975 0.887 0.982 0.998 0.992 0.993 0.949 0.992
2
Branch
β20
β21
R
−2.497 −3.917 −1.851 −2.067 −2.220 −4.913 −4.233 −1.033 −1.056 −1.360 −3.482 −2.320 −0.756 −4.474 −1.537 −2.541
2.051 2.453 1.884 1.919 1.852 2.663 2.539 1.528 1.640 1.598 2.377 1.907 1.546 2.200 1.667 2.187
0.868 0.987 0.850 0.996 0.985 0.991 0.979 0.987 0.819 0.756 0.960 0.993 0.976 0.993 0.820 0.969
2
Foliage
β30
β31
R
−2.626 −8.123 −1.985 −3.868 −4.405 −6.797 −6.062 −4.202 −2.915 −9.399 −3.934 −8.914 −5.470 −5.143 −1.404 −4.407
2.149 3.784 1.990 2.398 2.453 3.190 2.924 2.326 2.198 3.629 2.650 3.702 2.715 2.612 1.501 2.663
0.872 0.985 0.867 0.982 0.971 0.815 0.714 0.979 0.934 0.883 0.924 1.000 0.997 0.981 0.813 0.961
2
Tree
β40
β41
R
−2.393 −5.650 −2.729 −4.733 −3.618 −3.250 −5.440 −3.402 −2.401 −5.760 −2.229 −2.097 −3.744 −4.427 −2.541 −2.584
1.303 1.917 1.629 2.060 1.635 1.763 2.205 1.623 1.490 2.326 1.550 1.386 1.691 2.235 1.383 1.546
0.618 0.968 0.837 0.996 0.982 0.886 0.845 0.981 0.809 0.836 0.994 0.858 0.984 0.971 0.893 0.951
2
R2 0.974 0.976 0.976 0.984 0.970 0.995 0.981 0.994 0.981 0.956 0.982 0.999 0.994 0.997 0.960 0.991
473
of 2.37. The estimated exponent for branch biomass, ^31 , was the most variable, ranging from 1.50 to 3.78 with an average of 2.68. There were appreciable differences in parameter estimates between the New South Wales and Queensland data for E. pilularis, particularly for the bark and foliage biomass (Table 2). The predicted values from the system of equations for New South Wales were smaller than those for Queensland for stem biomass, but for other components they were generally greater (Fig. 2). The differences in predicted component biomass changed with diameter, and for a large part of the diameter range the relative difference was well over 20% of the predictions of the system equations for New South Wales (Fig. 3). In comparison, differences in the predicted total tree biomass were smaller, but over 10% for diameter ranging from 27 to 86 cm (Fig. 3). The system of additive equations in model 6 with D as the independent variable was more accurate in estimating biomass than the common approach of separately fitting total tree and component biomass equations using log transformed data through least squares regression. Values of the ratio between the MSE of the common approach and that of model 6 were generally greater than 1 for all biomass components (Table 3). For E. pilularis from Queensland the ratios were particularly high for the stem, bark and branch components. Values of the ratio between the fit index of the common approach and that of model 6 were generally less than 1. So the system of additive equations in model 6 resulted in smaller residual variance and better fit to the data. Table 3 Mean squared error ratios and fit index ratios for comparing prediction accuracy of the system of additive equations in Eq. 6 and the common approach of separately fitting total tree and component biomass equations using log transformed data through least squares regression (see text for details)
Species
Acacia dealbata Angophora costata E. agglomerata E. cypellocarpa E. dalrympleana E. diversicolor E. fastigata E. mannifera E. muellerana E. obliqua E. pilularis (NSW) E. pilularis (QLD) E. radiata E. regnans E. rossii E. sieberi Minimum Mean Median Maximum
The estimated exponent of the residual variance function, i.e. parameter b in Eq. 12 was shown in the heading row of Table 4. It varied within a relatively narrow range between 1.41 and 1.48. The values were similar for stem and bark components and so for branch and leaf components. For total tree biomass, the estimated exponent was 1.41, the smallest. In contrast, the estimated scale factor showed much greater variation among the 16 species groups. For stem biomass, the scale factors, σ12 to σ162 in Eq. 12, varied between 0.01 for E. regnans and 1.26 for E. dalrympleana. For bark biomass it ranged between 0.003 for E. regnans and 0.91 for E. muellerana. The largest estimated scale factor for branch biomass was 2.47 for E. diversicolor, and that for leaf biomass was 0.38 for E. agglomerata and 0.36 for E. pilularis from Queensland. For total tree biomass, E. dalrympleana had the largest estimated scale factor of 1.32. Using the estimated variance functions in Table 4, approximate confidence bands within which about 95% of the prediction error would fall were drawn for all biomass components of each species (Fig. 2). The estimated coefficients of the system equations with both the combined variable and diameter as the independent variables were given in Table 5. The estimated exponent for stem biomass, ^11 , varied between 0.93 and 1.02 for all species apart from Angophora costata and E. rosii for which ^11 was 0.81 and 0.85. The estimated exponent for bark biomass, ^21 , varied between 0.61 and 0.94 among the 14 species. For branch and bark components, the estimated coefficients were similar to that when D was the only independent variable in the
Mean square error ratio
Fit index ratio
Stem
Bark
Branch
Leaf
Tree
Stem
Bark
Branch Leaf
Tree
1.437 2.696 1.203 2.220 1.191 1.574 1.284 3.542 1.390 1.028 1.506 69.989 1.071 2.006 2.117 2.334 1.028 6.037 1.540 69.989
1.021 1.002 1.088 2.320 0.985 5.711 1.643 4.968 1.102 1.027 1.449 11.951 1.844 1.821 1.147 2.000 0.985 2.567 1.546 11.951
1.039 0.998 2.210 1.898 1.019 1.908 1.004 0.941 2.262 3.051 1.019 433.708 9.559 3.833 1.057 1.696 0.941 29.200 1.797 433.708
1.105 1.670 2.191 1.026 1.096 1.141 1.006 1.109 1.425 1.222 1.420 1.251 3.682 5.489 1.061 1.338 1.006 1.702 1.237 5.489
1.375 2.282 2.016 1.549 1.053 5.175 1.366 1.875 1.799 1.158 1.279 2.872 0.993 1.351 1.905 1.580 0.993 1.852 1.565 5.175
0.999 0.979 0.999 0.991 0.997 1.002 0.998 0.998 1.000 0.999 0.997 0.997 1.000 0.998 0.978 0.998 0.978 0.996 0.998 1.002
1.001 1.000 0.993 0.999 1.000 0.988 0.998 0.997 0.989 0.997 0.995 0.994 0.995 0.999 0.992 0.995 0.988 0.996 0.996 1.001
1.000 1.000 0.935 1.002 1.000 0.899 1.003 1.002 0.976 0.949 0.999 0.999 1.000 0.986 0.993 0.998 0.899 0.984 0.999 1.003
1.000 0.995 0.995 0.997 0.999 0.998 0.999 0.998 0.997 1.000 0.999 1.000 1.000 1.000 0.986 0.999 0.986 0.998 0.999 1.000
0.998 0.993 0.928 1.000 1.000 0.995 0.999 1.002 0.964 0.995 0.999 0.989 0.990 0.985 0.999 0.996 0.928 0.989 0.996 1.002
474
Fig. 2 Observed values of biomass components and total tree biomass plotted against diameter for the 15 species together with the fitted curves and approximate confidence bands containing about 95% of the observed data about the mean curve of predicted biomass
475 Table 4 Estimated scale factors ^i2 (I=1, 2, 3, 4, 5) and exponents (in the heading row) of the residual variance function for stem (1), bark (2), branch (3), leaf (4) and total tree biomass (5) of each species where D is the only independent variable in the system of additive equations
Species
^12 Y^ 11:439
^ 22 Y^ 21:422
^32 Y^ 31:482
^42 Y^ 41:478
^ 52 Y^ 51:408
Acacia dealbata Angophora costata E. agglomerata E. cypellocarpa E. dalrympleana E. diversicolor E. fastigata E. mannifera E. muellerana E. obliqua E. pilularis (NSW) E. pilularis (QLD) E. radiata E. regnans E. rossii E. sieberi R2
0.292 0.085 0.658 0.592 1.261 0.157 0.357 0.289 0.526 0.956 0.194 0.367 0.993 0.011 0.430 0.215 0.75
0.228 0.011 0.365 0.044 0.111 0.142 0.155 0.104 0.906 0.358 0.207 0.240 0.179 0.003 0.160 0.622 0.78
0.142 0.054 0.912 0.320 0.141 2.468 0.271 0.104 0.416 0.940 0.898 0.678 0.066 0.018 0.089 0.548 0.81
0.184 0.015 0.378 0.006 0.015 0.082 0.045 0.032 0.223 0.244 0.017 0.361 0.059 0.023 0.036 0.014 0.61
0.250 0.189 1.045 0.474 1.322 0.080 0.419 0.122 1.006 0.762 0.542 0.349 0.831 0.005 0.487 0.512 0.81
system equations. The estimated residual variance function was given in Table 6.
Discussion Biomass additivity has long been recognised as a desirable characteristic of a system of equations for predicting component as well as total tree biomass since it eliminates the logical inconsistency between the sum of predicted values for the tree components and the prediction for the total tree (Kozak 1970). Besides logical consistency, a system of additive biomass equations when estimated by
taking into account the inherent correlation among the biomass components has greater statistical efficiency as demonstrated by Parresol (1999, 2001). Furthermore, our results in Table 3 showed that the system of additive equations with D as the independent variable was more accurate in estimating biomass than the common approach of separately fitting total tree and component biomass equations using log transformed data through least squares regression. The deterministic part of the model specification as described in Eq. 1 of this paper is the same as that reported by Reed and Green (1985) for demonstrating the use of a loss function in the estimation of the system of equations
Fig. 3 Percentage differences in component and total tree biomass predictions between the two sets of system equations for E. pilularis in New South Wales and Queensland
476 Table 5 Estimated coefficients of the system equations (Eq. 10) and fit indices (R2) for the 14 tree species where both D and D2H are independent variables in the equations Species
Acacia dealbata Angophora costata E. agglomerata E. cypellocarpa E. dalrympleana E. fastigata E. mannifera E. muellerana E. obliqua E. pilularis E. radiata E. regnans E. rossii E. sieberi
N
27 6 9 8 10 27 8 13 26 41 12 22 9 29
Stem
Bark
β10
β11
R
−3.569 −1.894 −4.293 −3.372 −3.630 −3.902 −3.989 −3.772 −4.176 −4.438 −3.035 −4.223 −2.269 −3.955
0.970 0.807 1.004 0.939 0.952 0.975 1.016 0.960 0.984 0.987 0.933 0.970 0.850 0.973
0.983 0.954 0.969 0.977 0.978 0.964 0.991 0.965 0.908 0.995 0.998 0.997 0.992 0.989
2
Branch β21
R
β30
β31
R
−3.300 −4.621 −1.381 −3.411 −3.322 −4.971 −1.511 −1.188 −3.102 −4.784 −1.319 −5.787 −2.860 −4.589
0.770 0.939 0.617 0.793 0.740 0.945 0.607 0.585 0.716 0.935 0.611 0.890 0.749 0.942
0.897 0.992 0.802 0.997 0.977 0.962 0.988 0.958 0.773 0.963 0.981 0.991 0.870 0.962
−2.645 −7.685 −1.829 −3.868 −4.174 −5.544 −4.001 −2.727 −12.350 −3.861 −5.725 −4.670 −0.761 −4.414
2.158 3.648 1.956 2.398 2.393 2.789 2.284 2.142 4.283 2.635 2.773 2.424 1.320 2.663
0.872 0.987 0.951 0.982 0.972 0.718 0.978 0.958 0.902 0.924 0.997 0.986 0.823 0.961
for predicting biomass from diameter with additivity. However the specification of the error structure in this paper explicitly incorporates the cross-equation error correlation that naturally exists among biomass components of the same tree. Taking this correlation into account should result in more efficient estimation of the system of equations than non-linear ordinary least squares, and the gain in efficiency can be expected to increase with sample size according to statistical theory (Zellner 1962; Srivastava and Giles 1987; Judge et al. 1988). A common problem encountered in estimating nonlinear biomass equations with additive error terms is the estimation of heteroscedastic error variance with small samples. Because of the time-consuming nature of destructive biomass measurements, samples for developing biomass equations are usually small, often not more than 20 trees, especially when large trees are involved Table 6 Estimated scale factors ^i2 (I=1, 2, 3, 4, 5) and exponents (in the heading row) of the residual variance function for stem (1), bark (2), branch (3), leaf (4) and total tree biomass (5) of each species where both D and D2H are independent variables in the equations
Foliage
β20
2
2
Tree
β40
β41
R
R2
−2.318 −5.679 −2.238 −4.733 −3.459 −5.595 −3.397 −0.735 −6.943 −2.311 −3.192 −4.329 0.377 −2.562
1.283 1.927 1.499 2.060 1.595 2.245 1.633 1.063 2.542 1.569 1.563 2.206 0.547 1.540
0.617 0.967 0.905 0.996 0.982 0.845 0.981 0.899 0.836 0.994 0.987 0.971 0.857 0.951
0.991 0.979 0.969 0.983 0.982 0.959 0.993 0.985 0.936 0.987 0.999 0.998 0.993 0.991
2
(Applegate 1982; Ter-Mikaelian and Korzukhin 1997; Burrows et al. 2000). In some cases, sample size is less than ten trees (e.g. Feller 1980; Ter-Mikaelian and Korzukhin 1997). With small samples, it has been particularly difficult to obtain functions that accurately portray the underlying pattern of residual variance of biomass components as one would with large samples or with log transformed data. This may explain why many opt for log transformation instead of resorting to weighting functions to overcome heteroscedasticity. For a system of additive biomass equations with heteroscedastic additive error terms, one may use the GMM to obtain efficient parameter estimates without any specification of the nature of the heteroscedasticity (Greene 1999), and thus avoid the difficulties of estimating heteroscedastic error variance with small samples. Alternatively, the method introduced in this paper to collectively estimate residual variance
Species
^ 12 Y^ 11:346
^ 22 Y^ 21:767
^32 Y^ 31:466
^ 42 Y^ 41:166
^ 52 Y^ 51:566
Acacia dealbata Angophora costata E. agglomerata E. cypellocarpa E. dalrympleana E. fastigata E. mannifera E. muellerana E. obliqua E. pilularis E. radiata E. regnans E. rossii E. sieberi R2
0.126 0.080 1.294 0.455 2.181 0.810 0.486 1.565 0.513 0.380 0.402 0.011 0.059 0.195 0.79
0.028 0.009 0.143 0.014 0.023 0.032 0.011 0.019 0.026 0.026 0.032 0.005 0.087 0.077 0.78
0.158 0.030 0.631 0.339 0.144 0.319 0.185 0.710 3.342 0.877 0.050 0.025 0.131 0.619 0.82
0.300 0.014 1.084 0.011 0.048 0.108 0.034 0.386 0.921 0.051 0.034 0.015 0.145 0.035 0.55
0.026 0.010 0.421 0.112 0.398 0.131 0.049 0.232 0.293 0.104 0.070 0.003 0.048 0.122 0.77
477
Fig. 4 Component and total tree biomass curves for ten Eucalyptus species with a comparable range of DBH (see text)
functions for individual species by pooling small sample data across species and using indicator variables to represent the scale factor for each species in Eq. 12 may prove to be useful in overcoming this difficulty. The estimated residual error variance functions in Tables 4 and 6 can be used to compare and evaluate the prediction accuracy of the system of additive biomass equations across biomass components and species. For example, the values of the scale factors for stem and bark for E. regnans were relatively small, possibly reflecting the fact that the sample contained only small juvenile trees from dense stands regenerated after intense forest fires (Attiwill 1992). The approximate confidence bands containing about 95% of the observed data about the mean curve of predicted biomass appeared to be narrower for stem and total tree biomass, but wider for other components with foliage being the least precisely estimated component (Fig. 2). The generally wider confidence bands for branch and foliage biomass reflect their inherent variability. Branch and foliage biomass are naturally more variable than stem and bark since they are influenced to a greater extent by factors internal to stand growth such as stand density and competition from neighbouring trees, and by external factors such as altitude, position on slope, climate variations, seasonal changes, wind damage and atomospheric pollution (Satoo and Madgwick 1982). For the same species of E. pilularis from two different locations, the approximate confidence bands for the New South Wales data were narrower than those for the Queensland samples for all biomass
components apart from the total tree biomass (Fig. 2). Such approximate confidence bands would not be possible for small samples without the method introduced in this paper to collectively estimate residual variance functions for individual species. Among the species with data from two or more sites (Table 1), only E. pilularis exhibited site differences in biomass estimates that warranted the development of sitespecific equation systems. The New South Wales samples were from several stands on residual soils, while the Queensland trees were from moist climate growing on siliceous sands adjacent to rainforest. There were recognised differences in tree form between the populations. The within species variability of E. pilularis in biomass estimates is a reflection of the variation in tree form among its populations. E. pilularis occurs over a wide latitudinal and edaphic range (Boland et al. 1992). It occupies a far greater habitat diversity than other eucalypt species in this study, and furthermore maintains an exceptional vigour and dominance throughout its range of distribution (Florence 1961). After examining four populations of E. pilularis, one on Fraser Island, Queensland and three in Northern New South Wales, Florence (1961) revealed variations in crown form, bark, branch, leaf and other morphological characteristics among these populations. In comparison to populations in New South Wales, E. pilularis on Fraser Island have quite distinctive morphological characteristics. Such variations are an indication of both a broad regional influence and a strong edaphic influence on tree characteristics and may also be correlated
Fig. 5 Coefficient of variation in biomass estimates for the biomass curves in Fig. 4
478
with changes in associated inter-breeding species (Florence 1961). The variability in biomass estimates among species is much greater than the variability within E. pilularis. This is evident when curves of estimated biomass shown for individual species in Fig. 2 are plotted together. Apart from E. regnans, E. rosii and E. diversicolor, the remaining ten Eucalyptus species had a comparable range of diameter in the data (Fig. 1). Therefore the curves of estimated biomass for these ten species were plotted together to serve as an example for discussion on the variability in biomass estimates among species (Fig. 4). Visual examination of these curves suggests that it will not be possible to form species groups for developing generic equation within the framework of system of additive biomass equations. The reason is that two species may have similar total tree biomass at a given diameter, but the distribution of total tree biomass over the components can be very different as in the case of the two E. pilularis populations (Fig. 3). Hence generic equations for species groups may be possible only if biomass components are considered individually. Coefficient of variation of estimated biomass at a given diameter was also calculated for all biomass components of these species to see the magnitude of variation in relation to diameter. It was generally greater than 13% for stem, 23% for bark, 52% for branch, 30% for foliage and 13% for total tree. The coefficient of variation followed a concave curve with diameter for all biomass components; it decreased initially towards a minimum and then increased thereafter (Fig. 5). As described by Jacobs (1955), eucalypts have distinctive growth stages including sapling, pole, mature and overmature in terms of tree morphologies. The patterns of change with diameter in the coefficient of variation indicate that the variability in biomass estimates among species may be associated with the growth stage of eucalypts. Most biomass equations reported so far in the literature are non-additive and estimated using least squares regression and log transformed data. The merits of SUR will make it the method of choice for developing additive biomass equations in the future. This study together with other recent works on additive biomass equations (Parresol 1999, 2001; Bi et al. 2001; Carvalho and Parresol 2003) are useful examples for understanding the basic principles of this method and the analytical procedures involved. For a comprehensive understanding of the statistical theory behind this approach, one has to refer to the econometrics literature (e.g. Zellner 1962; Amemiya 1985; Gallant 1975; Srivastava and Giles 1987; Judge et al. 1988; Greene 1999). The method is easy to implement with a proper software such as SAS/ETS (SAS 1988) or other econometrics software. If access to commercial software is difficult, some free econometrics software for regression analysis could be used. One such example is EasyReg International (Bierens 2002), which does most of the econometrics tasks available in competing commercial software (Sephton 1998) and can be downloaded and
distributed freely from http://www.econ.la.psu.edu/~hbierens/EASYREG.HTM. Acknowledgements We would like to thank Shimin Cai for technical assistance and our colleagues Drs. Yushan Long, Craig Barton, Sandra Roberts and Mr. Jack Simpson for helpful comments on the manuscript.
References Adams MA, Attiwill PM (1988) Nutrient cycling in forests of northeast Tasmania. Research Report No. 1. Tasmanian Forest Research Council, Hobart Amemiya T (1985) Advanced econometrics. Harvard University Press, Cambridge Applegate GB (1982) Biomass of blackbutt (Eucalyptus pilularis Sm.) forests on Fraser Island. MSc Thesis. University of New England, Armidale Attiwill PM (1992) Productivity of Eucalyptus regnans forest regenerating after bushfire. S Afr For J 160:1-6 Baker TG, Attiwill PM (1985) Above-ground nutrient distribution and cycling in Pinus radiata (D. Don) and Eucalyptus obliqua (L’Herit) forests in south-eastern Australia. For Ecol Manage 13:41-52 Bi H (1999) Predicting stem volume to any height limit for native tree species in southern New South Wales and Victoria. N Z J For Sci 29:318-331 Bi H (2000) Trigonometric variable-form taper equations for Australian eucalypts. For Sci 46:397-409 Bi H, Hamilton F (1998) Stem volume equations for native tree species in southern New South Wales and Victoria. Aust For 61:275-286 Bi H, Birk E, Turner J, Lambert M, Jurskis V (2001) Converting stem volume to biomass with additivity, bias correction, and confidence bands for two Australian tree species. N Z J For Sci 31:298-319 Bierens HJ (2002) EasyReg International. Department of Economics, Pennsylvania State University, University Park, Pa. Birk EM, Turner J (1992) Response of flooded gum (E. grandis) to intensive cultural treatments: biomass and nutrient content of eucalypt plantations and native forests. For Ecol Manage 47:128 Boland DJ, Brooker MIH, Chippendal GM, Hall N, Hyland BPM, Johnson RD, Kleinig DA, Turner JD (1992) Forest trees of Australia. CSIRO, Australia Burrows WH, Hoffmann MB, Compton JF, Back PV, Tait LJ (2000) Allometric relationships and community biomass estimates for some dominant eucalypts in Central Queensland woodlands. Aust J Bot 48:707-714 Carvalho JP, Parresol BR (2003) Additivity in tree biomass components of Pyrenean oak (Quercus pyrenaica Willd.). For Ecol Manage 179:269-276 Chiyenda SS, Kozak A (1984) Additivity of component biomass regression equations when the underlying model is linear. Can J For Res 14:441-446 Cunia T, Briggs RD (1984) Forcing additivity of biomass tables: some empirical results. Can J For Res 14:376-384 Cunia T, Briggs RD (1985) Forcing additivity of biomass tables: use of the generalised least squares method. Can J For Res 15:2328 Eamus D, McGuinness K, Burrows W (2000) Review of allometric relationships for estimating woody biomass for Queensland, the Northern Territory and Western Australia. National Carbon Accounting System, Technical Report No. 5a. Australian Greenhouse Office, Canberra Feller MC (1980) Biomass and nutrient distribution in two eucalypt forest ecosystems. Aust J Ecol 5:309-333 Florence RG (1961) Ecology of blackbutt. PhD thesis. Sydney University, Sydney
479 Florence RG (1996) Ecology and silviculture of eucalypt forests. CSIRO, Collingwood, Australia Gallant AR (1975) Seemingly unrelated non-linear regressions. J Econometrics 3:35-50 Greene WH (1999) Econometric analysis, 4th edn. Prentice Hall, Upper Saddle River, N.J. Groves TS, Malajczuk N (1985) Biomass production in trees and understory shrubs in an age-series of Eucalyptus diversicolor F. Muell. stands. For Ecol Manage 11:59-74 Hinston FJ, Turton AG, Dimmock GM (1979) Nutrient distribution in karri (Eucalyptus diversicolor F. Muell.) ecosystems in southwest Western Australia. For Ecol Manage 2:133-158 Jacobs MR (1955) Growth habits of the eucalypts. Commonwealth Government Printer, Canberra Judge GG, Hill RC, Griffiths WE, Lutkepohl H, Lee TC (1988) Introduction to the theory and practice of econometrics, 2nd edn. Wiley, New York Keith H, Barrett D, Keenan R (2000) Review of allometric relationships for estimating woody biomass for New South Wales, the Australian Capital Territory, Victoria, Tasmania and South Australia. National Carbon Accounting System, Technical Report No. 5b. Australian Greenhouse Office, Canberra Ketterings QM, Coe R, Noordwijk M van, Ambagau Y, Palm CA (2001) Reducing uncertainty in the use of allometric biomass equations for predicting above-ground tree biomass in mixed secondary forests. For Ecol Manage 146:199-209 Kozak A (1970) Methods for ensuring additivity of biomass components by regression analysis. For Chron 46:402-404 Lambert MJ (1979) Sulphur relationships of native and exotic tree species. MSc (Hons) Thesis. Macquarie University, Australia O’Grady AP, Chen X, Eamus D, Hutley LB (2000) Composition, leaf area index and standing biomass of eucalypt open forests near Darwin in the Northern Territory, Australia. Aust J Bot 48:629-638 Overman JPM, Witte HJL, Saldarriaga JG (1994) Evaluation of regression models for above-ground biomass determination in Amazon rainforest. J Trop Ecol 10:207-218
Parresol BR (1999) Assessing tree and stand biomass: a review with examples and critical comparison. For Sci 45:573-593 Parresol BR (2001) Additivity of nonlinear biomass equations. Can J For Res 31:865-878 Reed DD, Green EJ (1985) A method of forcing additivity of biomass tables when using nonlinear models. Can J For Res 15:1184-1187 SAS (1988) SAS/ETS user’s guide, version 6, 2nd edn. SAS Institute, Cary Satoo T, Madgwick HAI (1982) Forest biomass. Nyhoff-Junk, The Hague Sephton PS (1998) EasyReg: version 1.12. J Appl Econometrics 13:203-207 Specht RL (1970) Vegetation. In: Leeper GW (ed) The Australian environment, 4th edn. The Dominion, North Blackburn, Victoria, Australia, pp 44-67 Srivastava VK, Giles DA (1987) Seemingly unrelated regression equations models: estimation and inference. Dekker, New York Stewart HTL, Flinn DW, Aeberli BC (1979) Above-ground biomass of mixed eucalypt forest in Eastern Victoria. Aust J Bot 27:725740 Ter-Mikaelian MT, Korzukhin MD (1997) Biomass equations for sixty-five North American tree species. For Ecol Manage 97:124 Turner J, Kelly J (1985) Effect of radiata pine on soil chemical characteristics. For Ecol Manage 11:257-270 Turner J, Lambert MJ (1986) Effects of forest harvesting nutrient removals on soil nutrient reserves. Oecologia 70:140-148 Turner J, Lambert MJ (2004) Nutrient cycling in two Eucalyptus plantation species. New For (in press) Turner J, Lambert MJ, Holmes G (1992) Nutrient cycling in forested catchments in southeastern New South Wales. I. Biomass accumulation. For Ecol Manage 55:135-148 Wackerly DD, Mendenhall W, Scheaffer RL (1996) Mathematical statistics with applications. Duxbury, Belmont Zellner A (1962) An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J Am Stat Assoc 57:348-368
