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.

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.