Generalized allometric volume and biomass equations for some tree species in Europe

Abstract

Since biomass is one of the key variables in ecosystem studies, widespread effort has aimed to facilitating its estimation. Numerous stand-specific volume and biomass equations are available, but these cannot be used for scaling up biomass to the regional level where several age-classes and structural types of stands coexist. Therefore simplified generalized volume and biomass equations are needed. In the present study, generalized biomass and volume regression equations were developed for the main tree species in Europe. These equations were based on data compiled from several published studies and are syntheses of the published equations. The results show that these generalized equations explain 64–99% of the variation in values predicted by the original published equations, with higher values for stem than for crown components.

Introduction

Rapid, easily implemented methods are needed for the assessment of standing biomass to estimate the degree of carbon sequestration by forest ecosystems. The method most often used for determining individual tree biomass and volume is the use of allometric relationships (Whittaker and Woodwell 1968). Normally, the volume or biomass of a tree is predicted as an equation of some easily measured variable, such as diameter-at-breast height (dbh) or height (h).

Whenever there is need for estimating the biomass of individual trees, the abundance of currently available predictive equations provides an alternative to the destructive sampling of trees for developing local equations. Comprehensive collections of stand-specific biomass equations are available in the literature (for North America Ter-Mikaelian and Korzukhin 1997, Jenkins et al. 2004; for Australia Eamus et al. 2000, Keith et al. 2000, Snowdon et al. 2000 and for Europe Zianis et al. 2005). Most published biomass equations were developed using trees sampled from specific study sites or from sites that represent small regions only. As a result, use of existing volume or biomass equations with forest inventory data at large spatial scales¹ is unreliable because the equations of previous studies may be site-specific, often disorganized and sometimes inconsistent (Pastor et al. 1983/1984; Jenkins et al. 2003; Wirth et al. 2004). Furthermore, unless an equation was developed exclusively for the species and study region of interest under conditions typical for the study site, it is impossible to know which equations to choose for a particular species and site. Reliable models of tree-level biomass to be used at large spatial scales are available mainly for Scandinavia and were compiled by Marklund (1987, 1988).

There may be additional value in deriving generalized equations for comparing different regions, since it is important to know that inconsistencies between regions are not due solely to the application of different regression equations, which yield contrasting values even for the same dataset but do not significantly vary among themselves (Pastor et al. 1983/1984). Despite these inconsistencies, or perhaps because of them, the need is clear for a consistent method and for generalized equations to estimate forest biomass at large scales (Jenkins et al. 2003).

To assess tree biomass on a large spatial scale one could (1) sample several trees of different sizes from a representative sample of species, regions and sites across the area of interest or (2) find an already existing equation for the geographically closest site (Ter-Mikaelian and Korzukhin 1997). Additionally, one could (3) use several available equations to estimate the range of biomass (Ter-Mikaelian and Korzukhin 1997) or (4) attempt as much as possible to collect sample data for reanalysis from all available sources of tree mensurational data (Wirth et al. 2004). Furthermore, one could (5) produce a generalized equation based on those reported in the literature (Schmitt and Grigal 1981; Pastor et al. 1983/1984; Zianis and Mencuccini 2003).

The first approach would ensure an unbiased sample of trees but would also be very expensive and time-consuming. The fourth approach is also difficult, since most scientists have not published the raw data from which their volume or biomass equations were developed. However, several authors reported that generalized regressions developed from field data can reasonably predict the biomass of trees from other sites (Schmitt and Grigal 1981; Wirth et al. 2004).

The aim here is to provide new generalized allometric volume and biomass equations using dbh for the most common tree species in Europe. The equations presented here should provide a consistent and unbiased basis for evaluating forest biomass across regional boundaries in analysis of the carbon budget of forests.

Methods

Material

Generalized allometric biomass (dry weight) and volume equations were developed for the most common tree species in Europe. Equations relating volume of the stem or biomass of the tree component (stem, branches, foliage and total aboveground) to dbh or dbh and h were compiled (Table 1) from the comprehensive study of Zianis et al. (2005). The equations were used to generate simplified specieswise and compartmentwise generalized regression equations.

Table 1 Equations relating volume of stem or biomass of tree component (stem, branches, foliage and total aboveground) to diameter-at-breast height or diameter and height used in this study. The equations were compiled from the comprehensive study of Zianis et al. (2005) and ID numbers here correspond to the reference ID in Zianis et al. (2005)

When several equations based on independent tree samples from different sites were reported, all were included in this study. In the present study we analysed the following tree species in Europe: Norway spruce [Picea abies (L.) Karst.], Scots pine (Pinus sylvestris L.), birch (Betula spp. L.), oak (Quercus spp. L.) and beech (Fagus spp. L.).

Meta-analysis

The meta-analysis was used to formulate a generalized regression equation and to summarize studies on the same topic by different contributors to obtain a combined overall mean among studies (Iyengar 1991). The four stages of the meta-analysis included (1) identification of a study problem, (2) retrieval of relevant studies, (3) extraction of appropriate data and (4) formulation of a statistical model for combining data. Incorporating datasets produced by different authors into a single comprehensive analysis introduces some inhomogeneity that particularly affects assessment of the accuracy of the resulting predictions. Estimates for the uncertainty of predicted values rely heavily on the assumption of independence of residuals from the fitted model, an assumption hardly met if data from different authors are combined.

There are two ways to produce generalized regression equations for volume and biomass: formal and modified meta-analytic techniques (Jenkins et al. 2003). The formal meta-analytic technique combines regression coefficients (Peña 1997) and all equations used in such meta-analyses must have identical forms and identical variable transformations. Application of formal meta-analytic techniques for combining regression coefficients is not applicable to the present study, with its aim of developing generalized regression equations based on as many unpublished and published previous equations as possible. Therefore, a modified version was used of a type of meta-analysis that generates volume and biomass data using various published equations and that fits an equation to the generated data, thus summarizing previous equations (Schmitt and Grigal 1981; Pastor et al. 1983/1984).

Assessment of the variability of the original equations was performed in the following manner. Equally spaced points at a 1-cm interval for dbh were generated from each of the compiled equations to form specieswise and compartmentwise pseudoobservations (Fig. 1). In Fig. 1, the pseudoobservations represent the actual situation. Separate pseudoobservations for the Temperate and Boreal Zones were produced (for definitions of vegetation zones see Ahti et al. 1968). These points covered the range of diameter values specific to each equation and were used to generate generalized equations for each species and compartment, thus ensuring more weight to the equations with a wider diameter range. If the data range of the sample trees behind the original equation was not reported, the equation was assumed to cover the range from dbh of 10 cm up to the species-specific maximum limit. For P. sylvestris, P. abies and Betula spp., the maximum limit for dbh was set to 40 cm and for Fagus spp. and Quercus spp. to 50 cm. Since not all studies reported the number of sample trees used in developing the original regression, the equations compiled do not have different weights based on sample size.

Fig. 1

Equally spaced points at a 1-cm interval for diameter-at-breast height were generated from each of the compiled equations to form specieswise and compartmentwise pseudoobservations that were used to estimate generalized equations

In some cases, the original equation has two explanatory variables: dbh and h. For these equations, the dbh–h relationships were utilized. For Fagus spp., Quercus spp., P. abies and P. sylvestris in the Temperate zone, the relationship between dbh and h was estimated based on data of forest management plans from the Czech Republic (Cienciala 2004, personal communication). For P. abies, P. sylvestris and Betula spp. in the Boreal Zone, tree measurements of permanent sample plots from the 8th National Forest Inventory of Finland were used. For both these datasets, the formula describing the dbh–h relationship developed by Näslund (1937) was fitted:

$$ h = 1.3 + \frac{{dbh^{2} }} {{{\left( {\beta _{0} + \beta _{1} \cdot dbh} \right)}^{2} }} $$

(1)

(see Table 2).

Table 2 Relationship between tree height and diameter-at-breast height

Generalizing

Finally, the pseudoobservations generated with the published equations were used to predict the relationships between dbh and either stem volume or biomass of the aboveground tree compartment (total aboveground, stem, foliage, branches). In this respect, the regressions were syntheses of the published equations (Fig. 1). The commonly used mathematical model for biomass studies takes the form of the power equation

$$ y_{i} = \beta _{{0}} \cdot dbh^{{\beta _{{1}} }} , $$

(2)

where β ₀ and β ₁ are the scaling coefficients, y is either the volume of the stem or the biomass of tree component i (Zianis and Mencuccini 2004). Marklund (1987, 1988) developed an alternative

$$ y_{i} = \exp {\left( {\beta _{{0}} + \beta _{{1}} \cdot \frac{{dbh}} {{dbh + \beta _{{2}} }}} \right)}. $$

(3)

The units used for dbh, volume and biomass are cm, m³ and kg, respectively. Both these equations were applied.

The dbh was used as the only predictive variable because it is the most common and the easiest variable to measure in the field (Pastor et al. 1983/1984). Although some previous biomass studies have also used h together with dbh, the advantage of such regressions over those using dbh alone is probably not practical with regard to the accuracy of h measurements and the increased fieldwork involved.

Model validation

Several procedures can be used to check the validity of a regression model: (1) comparison of the model predictions (ŷ) and coefficients with physical theory (see the global allocation rules below), (2) comparison of results with theoretical models and simulated data, (3) collection of new test data to check model predictions and (4) reservation of a portion of the available data to obtain an independent measure of the model prediction accuracy (Snee 1977).

When the validity of the generalized equations was checked, we first compared the model coefficients (β ₁) of Table 4 with global allocation rules (West et al. 1999; Enquist and Niklas 2001). These authors concluded that the extending allometric theory predicts that y is proportional to the 8/3 (∼2.667) power of the stem diameter dbh of any size class $({\text{i}}{\text{.e}}{\text{., }}y \propto dbh^{{8/3}} ).$ They also suggested that this allometric theory is almost universally applied in biology and that it originated in the common geometric and hydrometric principles that govern the transport of essential materials to support cellular metabolism. The accuracies of the generalized equations were tested by comparing the model predictions with independent test data compiled from reported volume and biomass values (Burger 1937, 1953; Dietrich 1968; Vinš and Šika 1981; Vyskot 1990; Do-Hyung 2001).

Results and discussion

When the form of the equation reported by Marklund (1987, 1988) was used, the generalized equations for stem volume fitted well (R ² = 96–100%)² with the pseudoobservations generated with the original allometric equations (Table 3). The high values for R ² are more or less self-evident, since the major parts of the variation had been removed before the pseudoobservations. Generalization accounted for 96–99% of the variation in values predicted by the original published equations for total aboveground biomass and stem biomass. Branch biomass was estimated accurately using generalized equations for P. sylvestris and Fagus spp. in the Temperate zone as well as for P. abies, P. sylvestris and Betula spp. under boreal conditions (R ² = 92–99%). For P. abies under temperate conditions, generalization adequately fitted the pseudoobservations (R ² = 85%). The gneralized equations for foliage biomass of P. abies and P. sylvestris in the Temperate Zone failed to predict the pseudoobservations accurately (R ² = 71–85%). Nonetheless, the foliage biomass of Fagus spp. under temperate conditions and P. abies, P. sylvestris and Betula spp. under boreal conditions were estimated accurately (R ² = 92–99%).

Table 3 Generalized volume and biomass equations (biomass kg dry matter)

The power equation, commonly used in biomass studies, predicted 92–99% of the total variation in pseudoobservations of stem volume (Table 4). Generalization explains 88–99% of the variation in values predicted by the original published equations for total aboveground biomass and branch biomass. Stem biomass was estimated accurately for P. abies and Fagus spp. in the Temperate zone and for P. abies and Betula spp. in the Boreal zone (R ² = 97–99%). For P. sylvestris under both temperal and boreal conditions, the generalizations fitted the pseudoobservations passably (81–90%). The generalized equations for the foliage biomass of P. abies and P. sylvestris under temperate conditions and P. sylvestris under boreal conditions failed to explain the variation in pseudodata (64–70%). Nonetheless, the generalized equations for foliage biomass of Fagus spp. in the Temperate Zone and P. abies and Betula spp. in the Boreal Zone fitted the pseudoobservations well (89–99%).

Table 4 Generalized volume and biomass equations (biomass kg dry matter)

The results show that the coefficients of determination ranged from 64 to 99%, with higher values for stem volume, stem biomass, branch biomass and total aboveground biomass than for foliage biomass. The higher R ² values for the generalizations indicate that the original equations used in the generalization are more similar or that only a few were used for the generalization. The equation form developed by Marklund (1987, 1988) (Eq. 3) appeared to fit better than power equation 2 (the most commonly used), especially in the lower diameters, where the power equation is rather stiff. Marklund’s simplified equations scale the relationship according to the dbh of the stump diameter. Equations 2 and 3 predicted similar types of biomass or volume values for the higher dbh values.

The coefficients β ₁ of Eq. 2 estimated in this study varied from 1.37 to 3.19, while the mean (∼2.32) was quite similar to the empirical power values $(1.16 - 3.32;\,\;\ifmmode\expandafter\bar\else\expandafter\=\fi{x}2.368)$ reported by Zianis and Mencuccini (2004). The theoretical universal power value (∼2.667) is slightly higher (West et al. 1999; Enquist and Niklas 2001), but Zianis and Mencuccini (2004) concluded that the mean of empirically determined coefficient β ₁ gave more aacurate predictions than the theoretical universal power value.

When the generalizations were compared against the reported volume and biomass values, the generalized equations very closely predicted the total aboveground, branch and foliage biomasses for temperate P. abies, and there was virtually no difference between the generalized estimations and test data (Figs. 2b, d, e). In contrast, the generalized equations did not accurately predict the stem volume and stem biomass (Figs. 2a, c).

Fig. 2

Comparison of generalized biomass equations (Eq. 3, see Table 3) against reported volume and biomass values of Temperate zone Picea abies (Burger 1937, 1953; Dietrich 1968; Vinš and Šika 1981; Vyskot 1990; Do-Hyung 2001): a stem volume, b total aboveground biomass, c stem biomass, d branch biomass and e foliage biomass. For mPRESS, RMSECV and RMECV, see Eqs. 4, 5 and 6 in the Appendix 1

In general, inaccurate estimations may clearly be obtained from generalized equations when applied to any particular stand (Zianis and Mencuccini 2003). However, over- and underestimations from generalized predictions may cancel out when these are applied to large geographical areas. The results were assumed to be applicable to nationwide studies for European countries within the Temperate and Boreal zones (for definitions of vegetation zones see Ahti et al. 1968). The equations produced separately for the Temperate and Boreal zones differed in a statistically significant manner (Table 5).

Table 5 Dissimilarity of parameters β ₁ (Table 3) between Temperate and Boreal zones. For calculation scheme, see Appendix 2

The results of this study are applicable to P. abies, P. sylvestris and Betula spp. with dbh values of 10–40 cm. For Fagus spp. and Quercus spp. the suitable range is from 10 to 50 cm. Any meta-analysis must face the diversity of methods used in different studies. These differed with respect to the set of biomass components considered and the strategy involved in sample tree selection.

Appendices

Appendix 1

The modelled relationships between dbh and h were tested, reserving a portion of the available data to obtain an independent measure of the model prediction accuracy. When no test set is available for model validation, a cross-validation criterion can be used (Stone 1974; Snee 1977). Model validation was accomplished with the leave-one-out (LOO) cross-validation. The dataset is split into a training set, on which a model is estimated, and a test set on which the model is evaluated. In this case the response value ŷ _(i) is predicted on a model that was estimated for the dataset minus the ith observation, while the test set contains only one observation (Stone 1974). The splitting procedure is repeated until all observations have once and only once been in the test set. Thus there are n models built, each using n−1 observations for model construction and the remaining observations for model validation. The LOO cross-validation criterion mPRESS (mean of the predictive error sum of squares) is most often used (Stone 1974; Snee 1977):

$$ {\text{mPRESS}} = \frac{1} {n}{\sum\limits_{i = 1}^n {{\left( {y_{i} - \hat{y}_{{(i)}} } \right)}^{2} } }, $$

(4)

in which n is the number of observations in the test set and y _i and ŷ _(i) are, respectively, the experimental and predicted response values. Taking the square root of this, we can derive the root-mean-square error of cross-validation:

$$ {\text{RMSECV}} = {\sqrt {\frac{1} {n}{\sum\limits_{i = 1}^n {{\left( {y_{i} - \hat{y}_{{(i)}} } \right)}^{2} } }} }. $$

(5)

The relative mean error of cross-validation was also calculated:

$$ {\text{RMECV}} = \frac{1} {n}{\sum\limits_{i = 1}^n {{\left| {\frac{{y_{i} - \hat{y}_{{(i)}} }} {{y_{i} }}} \right|.}} } $$

(6)

Appendix 2

The dissimilarity ${\left( {H_{0} :\beta ^{{{\text{BOR}}}}_{1} = \beta ^{{{\text{TEM}}}}_{1} ;H_{1} :\beta ^{{{\text{BOR}}}}_{1} \ne \beta ^{{{\text{TEM}}}}_{1} } \right)}$ of parameters $\beta ^{{{\text{BOR}}}}_{1}$ and $\beta ^{{{\text{TEM}}}}_{1}$ (Table 3) was tested using the test statistic t (Ranta et al. 1999):

$$ t = \frac{{\beta ^{{{\text{BOR}}}}_{1} - \beta ^{{{\text{TEM}}}}_{1} }} {{s_{{\beta ^{{{\text{BOR}}}}_{1} - \beta ^{{{\text{TEM}}}}_{1} }} }}, $$

(7)

where the standard error $s_{{\beta ^{{{\text{BOR}}}}_{1} - \beta ^{{{\text{TEM}}}}_{1} }}$is given by

$$ s_{{\beta ^{{{\text{BOR}}}}_{1} - \beta ^{{{\text{TEM}}}}_{1} }} = {\sqrt {\frac{{{\left( {s_{{Y \cdot X}} ^{2} } \right)}_{p} }} {{{\left( {n^{{{\text{BOR}}}} - 1} \right)}s_{{X^{{{\text{BOR}}}} }} ^{2} }} + \frac{{{\left( {s_{{Y \cdot X}} ^{2} } \right)}_{p} }} {{{\left( {n^{{{\text{TEM}}}} - 1} \right)}s_{{X^{{{\text{TEM}}}} }} ^{2} }}} }, $$

(8)

where the n _BOR and the n _TEM are the sample sizes. The error variance was assumed to be of equal size in both populations. The ${\left( {s_{{Y \cdot X}} ^{2} } \right)}_{p}$ is the so-called pooled variance estimator and is given by

$$ {\left( {s_{{Y \cdot X}} ^{2} } \right)}_{p} = \frac{{SS_{{{\text{RESIDUAL}}}} ^{{{\text{BOR}}}} + SS_{{{\text{RESIDUAL}}}} ^{{{\text{TEM}}}} }} {{{\left( {n^{{{\text{BOR}}}} - 2} \right)} + {\left( {n^{{{\text{TEM}}}} - 2} \right)}}} $$

(9)

The $SS_{{{\text{RESIDUAL}}}} ^{j}$follows:

$$ SS_{{{\text{RESIDUAL}}}} ^{j} = {\left[ {{\sum {{\left( {n^{j} - 1} \right)}s_{X} ^{2} {\left( j \right)}} }} \right]} - \frac{{{\left[ {{\sum {{\left( {n^{j} - 1} \right)}s_{{XY}} {\left( j \right)}} }} \right]}^{2} }} {{{\sum {{\left( {n^{j} - 1} \right)}s_{Y} ^{2} {\left( j \right)}} }}}. $$

(10)

The ${\left( {n^{j} - 1} \right)}s_{X} ^{2} {\left( j \right)}$, ${\left( {n^{j} - 1} \right)}s_{Y} ^{2} {\left( j \right)}$ and ${\left( {n^{j} - 1} \right)}s_{{XY}} {\left( j \right)}$ were calculated from

$$ {\left( {n^{j} - 1} \right)}s_{X} ^{2} {\left( j \right)} = {\sum\limits_{i = 1}^n {x_{i} ^{2} } } - \frac{{{\left( {{\sum\nolimits_{i = 1}^n {x_{i} } }} \right)}^{2} }} {n}, $$

(11)

$$ {\left( {n^{j} - 1} \right)}s_{Y} ^{2} {\left( j \right)} = {\sum\limits_{i = 1}^n {y_{i} ^{2} } } - \frac{{{\left( {{\sum\nolimits_{i = 1}^n {y_{i} } }} \right)}^{2} }} {n} $$

(12)

and

$$ {\left( {n^{j} - 1} \right)}s_{{XY}} {\left( j \right)} = {\sum\limits_{i = 1}^n {x_{i} y_{i} } } - \frac{{{\left( {{\sum\nolimits_{i = 1}^n {x_{i} } }} \right)}{\left( {{\sum\nolimits_{i = 1}^n {y_{i} } }} \right)}}} {n}, $$

(13)

respectively.

If the error variance is normally distributed and the errors of different values of the explanatory variable are independent, the test statistic follows the t-distribution with $n_{{{\text{BOR}}}} + n_{{{\text{TEM}}}} - 4$ degrees of freedom.