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Nonlinear simultaneous systems of tree biomass models have been developed for various tree species around the world. To deal with the additivity property between total tree biomass and component biomass predictions, Parresol's (Parresol 2001) aggregation approach, in which a nonlinear model is specified for each of the tree components and the total tree biomass is set up as the sum of these component models, has been widely adapted. The aggregation nature of the systems requires each tree component to be estimated to obtain the total or subtotal biomass. A relatively large prediction error in any component biomass model can affect the prediction accuracy of the total or subtotal tree biomass. In this study, we used a three-step proportional weighting (3SPW) system to deal with the additivity of nonlinear biomass models, which disaggregated the model prediction of the total tree biomass into subtotals (e.g., aboveground or crown), and then the estimated biomass of the subtotals was proportionally divided into tree components (e.g., stems, branches, or foliage). Our results indicated that the 3SPW system guaranteed the stepwise additivity of total, subtotals, and tree components, as well as provided a biomass model for each subtotal and tree component. The results of model fitting and validation revealed that the 3SPW system performed as well as Parresol's aggregation systems and offers a good alternative for ensuring the additivity property of nonlinear biomass model systems.

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... The disaggregative strategy is the "component proportion" approach in which total tree biomass is disaggregated into tree components based on their estimated proportions (Tang et al. 2000;Jenkins et al. 2003;Zhao et al. 2019). Tang et al. (2000) developed a disaggregation approach in which a total biomass model is first developed, and the biomass component models are defined and then the component proportions are derived from these biomass component models and total biomass model (also see Dong et al. 2015). The estimated value of a dependent (endogenous) variabletotal biomassis used as explanatory variables to solve the parameters of biomass component equations with two-stage nonlinear error-in-variable models (TSEM) (Tang et al. 2001). ...

... Thus, negative terms tend to "outweigh" positive terms in the computation of E% although it is possible to obtain positive values of E as reported in Tables 2 and 3. A model equation that is performing well would be expected to have a negative but small E% value and a small E value. Furthermore, there was no single system to predict biomass that was best for all components and total tree biomass, as demonstrated in the current study and others (Zhao et al. 2015(Zhao et al. , 2019Dong et al. 2015). So, the better practice for evaluating different systems of biomass equations is to compare their overall predictive performances based on an array of statistics in absolute units and percentages for each biomass component and total tree biomass, using a ranking system as we did in the current study. ...

... The heteroscedasticity problem that always exists in biomass model residuals could be addressed by having each equation with its own weighting function as we did in this study and others (Zhao et al. 2015(Zhao et al. , 2019Dong et al. 2015). In addition to this problem, mathematical relationships between biomass equation themselves and relationships between the error terms of biomass equations determine how to develop and estimate the system of biomass equations. ...

A system of nonlinear biomass component equations was developed for slash pine (Pinus elliottii var. elliottii) trees using an econometric approach in which endogenous right-hand-side variables were included in some equations. The system was fitted to component biomass data from 306 slash pine trees sampled in the southeastern US with weighted two-stage (2SLS) and three-stage (3SLS) least squares and full information maximum likelihood (FIML) estimation methods. The predictive performances of the system fitted with these three estimation methods were ranked based on an array of statistics, and the ranking follows the order of FIML > 3SLS > 2SLS. The new system performed as well or better than previously published biomass equation systems developed using the aggregation and disaggregation approaches and fitted to the same data. The results demonstrated that the econometric approaches such as FIML and 3SLS have potential to be useful for tree biomass modeling.

... Such a top-down approach of allocating component biomass was developed within the framework of error-in-variable models by Tang et al. (2000Tang et al. ( , 2001 and has been predominantly used by researchers in China for developing their so-called compatible biomass equations. But it has had only limited exposure in the English literature (Dong et al. 2015;Wang et al. 2017). ...

... In such cases, the system of allocative equations can also be estimated by WNSUR and GMM using the PROC MODEL Procedure of SAS/ETS without resorting to the specialist software ForStat 2.2 developed by the Chinese Academy of Forestry and documented in detail by Tang et al. (2008). As exemplified by Dong et al. (2015) and Wang et al. (2017), the SAS procedure and ForStat produced almost identical parameter estimates and validation statistics. ...

... The other is the top-down allocative approach, which was developed within the framework of error-in-variable models by Tang et al. (2000Tang et al. ( , 2001 and has been used mostly in China. Although the two approaches differ in model specification and also in the method of parameter estimation, their predictive performance is, by and large, comparable, being either equivalent or with one being slightly superior to the other (Dong et al. 2015;Fu et al. 2016). Even when the additive and allocative systems of nonlinear biomass equations were specified within a probabilistic framework and fitted using Gaussian maximum likelihood estimation, their predictive performances were hardly differentiable (Affleck and Diéguez-Aranda 2016). ...

Two systems of additive equations were developed to predict aboveground stand level biomass in log products and harvest residue from routinely measured or predicted stand variables for Pinus radiata plantations in New South Wales, Australia. These plantations were managed under three thinning regimes or stand types before clear-felling at rotation age by cut-to-length harvesters to produce sawlogs and pulpwood. The residue material following a clear-fell operation mainly consisted of stumps, branches and treetops, short off-cut and waste sections due to stem deformity, defects, damage and breakage. One system of equations did not include dummy variables for stand types in the model specification and was intended for more general use in plantations where stand density management regimes were not the same as the stand types in our study. The other system that incorporated dummy variables was for stand type-specific applications. Both systems of equations were estimated using 61 plot-based estimates of biomass in commercial logs and residue components that were derived from systems of equations developed in situ for predicting the product and residue biomass of individual trees. To cater for all practical applications, two sets of parameters were estimated for each system of equations for predicting component and total aboveground stand biomass in fresh and dry weight respectively. The two sets of parameters for the system of equations without dummy variables were jointly estimated to improve statistical efficiency in parameter estimation. The predictive performances of the two systems of equations were benchmarked through a leave-one-plot-out cross validation procedure. They were generally superior to the performance of an alternative two-stage approach that combined an additive system for major components with an allocative system for sub-components. As using forest harvest residue biomass for bioenergy has increasingly become an integrated part of forestry, reliable estimates of product and residue biomass will assist harvest and management planning for clear-fell operations that integrate cut-to-length log production with residue harvesting.

... The modern methods of modelling the biological productivity of trees and tree stands have been developed towards additivity of biomass components (Bi et al., 2010;Dong et al., 2015) and towards transition from "pseudo-generic" allometric models to really generic, involving regionalization of biomass model by introducing dummy variables (Fu et al., 2012), that usually fulfilled on local sets of actual biomass of trees and tree stands. We generated the database of forest stand biomass for the main forest species in Eurasia (Usoltsev, 2010;Usoltsev, 2013), that has enabled these modern methodologies to be implemented on the entirely different, higher level, namely to begin modelling additive biomass on transcontinental level. ...

... Analysis of biomass forest stands is made on the basis of allometric additive models. According to the structure of disaggregation three-step model (Tang et al., 2000;Dong et al., 2015), biomass value, estimated by the total biomass equation, exploded into components according to the scheme presented in Fig. 1. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of biomass of all the components -total, intermediate (steps 1 and 2) and initial (step 3a, b) (Dong et al., 2015). ...

... According to the structure of disaggregation three-step model (Tang et al., 2000;Dong et al., 2015), biomass value, estimated by the total biomass equation, exploded into components according to the scheme presented in Fig. 1. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of biomass of all the components -total, intermediate (steps 1 and 2) and initial (step 3a, b) (Dong et al., 2015). ...

Today, estimating of biological productivity or carbon-depositing ability of forests is going on the global level, and its increase is one of the major factors of climate stabilization. In recent years, two trends in the harmonization of allometric models of tree biomass have been developing. The first of them is related to ensuring the additivity of the biomass component composition, and the second one-to the search for the so-called generic model applicable to a wide range of environmental conditions. However, all "generic" models give significant biases in their application in local conditions. In our modeling, we adhere to the principle of biomass additivity, split "generic" model into regional variants by introducing dummy variables, and build the model at the transcontinental level for the first time. When using the unique in terms of the volume of database on the level of stand of the genus Populus sp. in a number of 212 sample plots, the trans-Eurasian additive allometric models of biomass of stands for Eurasian Populus forests are developed, and thereby the combined problem of model additivity and generality is solved. The additive model of forest biomass of Populus is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of forest stands not only on total, aboveground and underground biomass, but also on its component structure, i.e. it reflects the regional peculiarities of the component structure of biomass.

... However, when assessing the regional component composition of biomass, such models give biases (Usoltsev et al., 2017). Modern methods of modelling the biological productivity of single-trees and forest stands are developed in terms of biomass component additivity (Bi et al., 2010;Dong et al., 2015) and the transition from "pseudo-generic" allometric models to really generic, supposing regionalization of biomass model by introducing dummy variables (Fu et al., 2012). The database of single-tree biomass compiled for forest-forming species in Eurasia (Usoltsev, 2016) has enabled these modern methodologies to begin modelling additive tree biomass on transcontinental level. ...

... According to the structure of disaggregating three-step additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ccording to the structure of disaggregating three-step additive model system Dong et al., 2015), total biomass, estimated by the total equation, exploded into ents according to the scheme presented in Figure 2. The coefficients of the regression for all three steps are evaluated simultaneously, which ensures additivity of the all ents: total, intermediate and initial ones (Dong et al., 2015). ...

... According to the structure of disaggregating three-step additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ccording to the structure of disaggregating three-step additive model system Dong et al., 2015), total biomass, estimated by the total equation, exploded into ents according to the scheme presented in Figure 2. The coefficients of the regression for all three steps are evaluated simultaneously, which ensures additivity of the all ents: total, intermediate and initial ones (Dong et al., 2015). ...

Abstract. Today, estimating of biological productivity or carbon-depositing ability of forests is going on the global level, and its increase is one of the major factors of climate stabilization. In recent years, two trends in the harmonization of allometric models of tree biomass have been developing. The first of them is related to ensuring the additivity of the biomass component composition, and the second one – to the search for the so-called generic model applicable to a wide range of environmental conditions. However, all "pseudo-generic" models give significant biases in their application in local conditions. In our modeling, we adhere to the principle of biomass additivity, split "generic" model into regional variants by introducing dummy variables, and build the model at the transcontinental level for the first time. When using the unique in terms of the volume of database of trees of the genus Larix Mill. In a number of 420 sample trees, the trans-Eurasian additive allometric models of biomass of trees for Eurasian larch forests are developed, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Larix is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes on their biomass, i.e. it reflects the regional peculiarities of the component structure of tree biomass.

... However, when assessing the regional component composition of biomass, such models give biases (Usoltsev et al., 2017). Modern methods of modelling the biological productivity of single-trees and forest stands are developed in terms of biomass component additivity (Bi et al., 2010;Dong et al., 2015) and the transition from "pseudo-generic" allometric models to really generic, supposing regionalization of biomass model by introducing dummy variables (Fu et al., 2012). The database of single-tree biomass compiled for forest-forming species in Eurasia (Usoltsev, 2016) has enabled these modern methodologies to begin modelling additive tree biomass on transcontinental level. ...

... According to the structure of disaggregating three-step additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ccording to the structure of disaggregating three-step additive model system Dong et al., 2015), total biomass, estimated by the total equation, exploded into ents according to the scheme presented in Figure 2. The coefficients of the regression for all three steps are evaluated simultaneously, which ensures additivity of the all ents: total, intermediate and initial ones (Dong et al., 2015). ...

Today, estimating of biological productivity or carbon-depositing ability of forests is going on the global level, and its increase is one of the major factors of climate stabilization. In recent years, two trends in the harmonization of allometric models of tree biomass have been developing. The first of them is related to ensuring the additivity of the biomass component composition, and the second one – to the search for the so-called generic model applicable to a wide range of environmental conditions. However, all "pseudo-generic" models give significant biases in their application in local conditions. In our modeling, we adhere to the principle of biomass additivity, split "generic" model into regional variants by introducing dummy variables, and build the model at the transcontinental level for the first time. When using the unique in terms of the volume of database of trees of the genus Larix Mill. In a number of 420 sample trees, the trans-Eurasian additive allometric models of biomass of trees for Eurasian larch forests are developed, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Larix is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes on their biomass, i.e. it reflects the regional peculiarities of the component structure of tree biomass.

... As it has been known, directly measuring the stem volume and weights of the tree components are undeniably the most precise approach for estimating individual tree biomass [19,20]. Thus, biomass is often estimated by utilizing the biomass allometric equations or merely by multiplying the stem volume approximations with biomass conversion and expansion factors (BCEF s ) or basic wood density [13,[21][22][23][24][25]. The BCEF s would be more favorable in case the tree volume is the only existing data; thus, it is possible to convert the stem volume data into the tree components' (i.e., root, stem, branch, and foliage) dry biomass [26,27]. ...

... When establishing models to predict the total biomass and that of several primary tree tissues, two definite attributes are widely known as an outcome of employing several different approaches in the model's fitting process, namely: non-additive and additive models [22]. The non-additive model individually fits the data of each tree tissue and total biomass, neglecting the inherent correlation between the tissues measured on the same sample trees. ...

... The non-additive model individually fits the data of each tree tissue and total biomass, neglecting the inherent correlation between the tissues measured on the same sample trees. On the contrary, the additive models require the tree tissues and total biomass data to be simultaneously fitted, reckoning the intrinsic correlation of the biomass tissues sampled from the same individuals [22,37,41]. Consequently, the total tree biomass estimation will be equal to the sum of the prediction of biomass tissues. ...

Short-rotation forestry is of interest to provide biomass for bioenergy and act as a carbon sink to mitigate global warming. The Poplar tree (Populus × xiaohei) is a fast-growing and high-yielding tree species in Northeast China. In this study, a total of 128 Populus × xiaohei trees from the Songnen Plain, Heilongjiang Province, Northeastern China, were harvested. Several available independent variables, such as tree diameter at breast height (D), tree’s total height (H), crown width (CW), and crown length (CL), were differently combined to develop three additive biomass model systems and eight stem volume models for Populus × xiaohei tree. Variance explained within the three additive biomass model systems ranged from 83% to 98%, which was lowest for the foliage models, and highest for the stem biomass models. Similar findings were found in the stem volume models, in which the models explained more than 94% of the variance. The additional predictors, such as H, CL, or CW, evidently enhanced the model fitting and performance for the total and components biomass along with the stem volume models. Furthermore, the biomass conversion and expansion factors (BCEFs) of the root (118.2 kg/m3), stem (380.2 kg/m3), branch (90.7 kg/m3), and foliage (31.2 kg/m3) were also calculated. The carbon concentrations of Populus × xiaohei in root, stem, branch, and foliage components were 45.98%, 47.74%, 48.32%, and 48.46%, respectively. Overall, the newly established models in this study provided complete and comprehensive tools for quantifying the biomass and stem volume of Populus × xiaohei, which might be essential to be specifically utilized in the Chinese National Forest Inventory.

... However, when assessing the regional component composition of biomass, such models give biases (Usoltsev et al., 2017). Modern methods of modelling the biological productivity of single-trees and forest stands are developed in terms of biomass component additivity (Bi et al., 2010;Dong et al., 2015) and the transition from "pseudo-generic" allometric models to really generic, supposing regionalization of biomass model by introducing dummy variables (Fu et al., 2012). The database of single-tree biomass compiled for forest-forming species in Eurasia (Usoltsev, 2016) has enabled these modern methodologies to begin modelling additive tree biomass on transcontinental level. ...

... The distribution of sample plots, on which sample trees are taken in different ecoregions of Eurasia, is shown in Figure 1. According to the structure of disaggregating three-step additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ...

... The distribution of sample plots, on which sample trees are taken in different ecoregions of Eurasia, is shown in Figure 1. According to the structure of disaggregating three-step additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ...

The aim of current study was to develop a generic pseudo-allometric model of the biomass structure of larch (Larix spp.) trees growing in Eurasia, and to assess the impacts of temperature and precipitation. It was assumed that this model will create a prerequisite for predicting changes in the structure of the tree biomass of the genus Larix spp. under the influence of current climate shifts. According to the Trans-Eurasian hydrothermal gradients of Eurasia harvest biomass database was compiled from 510 sample trees. The data adequacy was determined by the level of variability and it accounted for 87–99% variability as the proposed by regression models. It was found that the increase in temperature by 1 °C at the constant level of precipitation causes decrease in the aboveground, stem, foliage and branches of equal-sized and equal-aged larch trees. The increase of precipitation by 100 mm at the constant level of temperature causes decrease in the aboveground and stem biomass and increase of foliage and branches.

... The modern methods of modelling the biological productivity of trees and tree stands have been developed towards additivity of biomass components (Bi et al 2010, Dong et al 2015 and towards transition from "pseudogeneric" allometric models to really generic, involving regionalization of biomass models by introducing dummy variables (Fu et al 2012), that usually fulfilled on local sets of actual biomass of trees and tree stands. The database of forest stand biomass for the main forest species in Eurasia (Usoltsev 2010(Usoltsev , 2013, that has enabled these modern methodologies to be implemented on the entirely different, higher level, namely to begin modelling additive biomass on transcontinental level. ...

... Analysis of biomass forest stands is made on the basis of allometric additive models. According to the structure of disaggregation three-step model (Tang et al 2000, biomass value, estimated by the total biomass equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additive biomass of all the components-total, intermediate and initial (Dong et al 2015). ...

... The equations (2) are modified according to the algorithm proposed by chinese researchers (Dong et al 2015) (Table 3), and the final transcontinental additive model of birch biomass component composition on the level of forest stand is given in the Table 4. The model is valid in the range of actual data of stand age, mean tree height, mean stem diameter and tree density, listed in the Table 1, and is characterized by a double harmonization: one of which provides the principle of biomass component additivity, and the second one relates to the introduction of dummy variables, localizing the model according to ecoregions of Eurasia. ...

When using the unique in terms of the volume of database on the level of stand of the genus sp., the trans-Eurasian additive Betula allometric models of biomass of stands for Eurasian birch forests are developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of forest biomass of is harmonized in two ways: it eliminated the internal Betula contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of forest stands not only on total, aboveground and underground biomass, but also on its component structure, i.e. it reflects the regional peculiarities of the component structure of biomass.

... The modern methods of modelling the biological productivity of trees and tree stands have been developed towards additivity of biomass components (Bi et al., 2010;Dong et al., 2015) and towards transition from "pseudo-generic" allometric models to really generic, involving regionalization of biomass model by introducing dummy variables (Fu et al., 2012), that usually fulfilled on local sets of actual biomass of trees and tree stands. Because different biomass components are characterized by different rates both their growth and mortality, they make a different contribution to matter cycling in the forest ecosystem and should be estimated not only in total but also separately. ...

... Analysis of biomass forest stands is made on the basis of allometric additive models. According to the struc-ture of disaggregation three-step model (Tang et al., 2000;Dong et al., 2015), biomass value, estimated by the total biomass equation, exploded into components according to the scheme presented in Figure 1. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of biomass of all the components -total, intermediate (steps 1 and 2) and initial (step 3a,b) (Dong et al., 2015). ...

... According to the struc-ture of disaggregation three-step model (Tang et al., 2000;Dong et al., 2015), biomass value, estimated by the total biomass equation, exploded into components according to the scheme presented in Figure 1. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of biomass of all the components -total, intermediate (steps 1 and 2) and initial (step 3a,b) (Dong et al., 2015). The distribution of sample plots, on which the oak forest biomass is measured in ecoregions of Eurasia, is shown in Figure 2. ...

... As a result, the subtotal or total biomass predictions will be equivalent to the summation of biomass component estimations [8,15,16]. Several parameter estimation techniques have been proposed by researchers to address the compatibility property for a system of biomass equations [6,[17][18][19][20]. Among these methods, seemingly unrelated regression (SUR) and nonlinear seemingly unrelated regression (NSUR) appear to be more popular than others since they are more general and flexible in application [17,19,[21][22][23][24]. ...

... Several parameter estimation techniques have been proposed by researchers to address the compatibility property for a system of biomass equations [6,[17][18][19][20]. Among these methods, seemingly unrelated regression (SUR) and nonlinear seemingly unrelated regression (NSUR) appear to be more popular than others since they are more general and flexible in application [17,19,[21][22][23][24]. ...

... Aggregated model systems with three parameter restrictions Referring to another of Parresol's [17] model structure, which has been applied in Dong et al. [19], the AMS3 ensures the additivity between the tree component and total biomass with three parameter restrictions, i.e., (1) tree crown biomass is equivalent to the summation of the branch and foliage biomass; (2) tree aboveground biomass is equivalent to the summation of the stem, branch, and foliage biomass; (3) tree total biomass is equivalent to the sum of the whole tree components biomass. The structural system of AMS3 are given as follows: ...

Three systems of additive biomass models were developed and the effects of tree components, tree sizes, and tree growing regions on the carbon concentration were analyzed for Mongolian oak (Quercus mongolica Fisch. ex Ledeb.) in the natural forests of Northeastern China. The nonlinear seemingly unrelated regression (NSUR) method was used to fit each of the three systems simultaneously; namely, aggregated model systems with no parameter restriction (AMS0), aggregated model systems with one parameter restriction (AMS1), and aggregated model systems with three parameter restrictions (AMS3). A unique weighting function for each biomass model was applied to address the heteroscedasticity issue. The systems assertively guarantee the additivity property, in which, the summation of the respective predicted tree components (i.e., root, stem, branch, and foliage) will match the prediction of subtotals (i.e., crown and aboveground) and total biomass. Using one-, two-, and three-predictor combinations (i.e., D (diameter at breast height), D and H (total height), and D, H, and CL (crown length)) as the general model underlying formats, three systems of additive biomass model were developed. Our results indicate that (1) all of the aggregated model systems performed well and the differences between the systems were relatively small; (2) the rank order of the three systems based on an array of statistics are as follows: AMS0 > AMS1 > AMS3; (3) the carbon concentration significantly varied depending on the types of tree tissues and growing regions; (4) the regional respective component carbon concentration and regional weighted mean carbon concentration multiplied by observed biomass value appeared to be the best approach to calculate carbon stock.

... Thus, to calculate plantation productivity and study forest health, fuel, nutrient cycling, accurate quantification of tree biomass for larch is critical and essential [2][3][4][5]. A biomass estimation model constructed by direct measurement data of tree biomass is undoubtedly the most appropriate and accurate method for practical applications [6][7][8][9]. To date, hundreds of biomass models have been developed worldwide, in which the diameter at breast height ( ) is a commonly used and most reliable predictor of aboveground and component biomass [8,[10][11][12][13][14][15][16]. ...

... Adding into biomass quantification can significantly improve model fitting and performance, and it can help explain the potential limitation of intra-species divergence. Many studies have shown that biomass models with both and parameters can more reliably predict tree biomass [5,7,9,[17][18][19][20]. At present, model specifications of developing biomass equations for aboveground and components have evolved from nonadditive models to additive models [7,9,20]. ...

... Many studies have shown that biomass models with both and parameters can more reliably predict tree biomass [5,7,9,[17][18][19][20]. At present, model specifications of developing biomass equations for aboveground and components have evolved from nonadditive models to additive models [7,9,20]. Meanwhile, various model estimation methods have been developed to ensure the additivity property for nonlinear biomass models such as the generalized method of moments (GMM), two-stage nonlinear error-invariable models (TSEM), and nonlinear seemingly unrelated regression (NSUR) [6,7,21,22]. ...

Accurate quantification of tree biomass is critical and essential for calculating carbon storage, as well as for studying climate change, forest health, forest productivity, nutrient cycling, etc. Tree biomass is typically estimated using statistical models. Although various biomass models have been developed thus far, most of them lack a detailed investigation of the additivity properties of biomass components and inherent correlations among the components and aboveground biomass. This study compared the nonadditive and additive biomass models for larch (Larix olgensis Henry) trees in Northeast China. For the nonadditive models, the base model (BM) and mixed effects model (MEM) separately fit the aboveground and component biomass, and they ignore the inherent correlation between the aboveground and component biomass of the same tree sample. For the additive models, two aggregated model systems with one (AMS1) and no constraints (AMS2) and two disaggregated model systems without (DMS1) and with an aboveground biomass model (DMS2) were fitted simultaneously by weighted nonlinear seemingly unrelated regression (NSUR) and applied to ensure additivity properties. Following this, the six biomass modeling approaches were compared to improve the prediction accuracy of these models. The results showed that the MEM with random effects had better model fitting and performance than the BM, AMS1, AMS2, DMS1, and DMS2; however, when no subsample was available to calculate random effects, AMS1, AMS2, DMS1, and DMS2 could be recommended. There was no single biomass modeling approach to predict biomass that was best for all aboveground and component biomass except for MEM. The overall ranking of models based on the fit and validation statistics obeyed the following order: MEM > DMS1 > AMS2 > AMS1> DMS2 > BM. This article emphasized more on the methodologies and it was expected that the methods could be applied by other researchers to develop similar systems of the biomass models for other species, and to verify the differences between the aggregated and disaggregated model systems. Overall, all biomass models in this study have the benefit of being able to predict aboveground and component biomass for larch trees and to be used to predict biomass of larch plantations in Northeast China.

... The tree height (H), another variable of the tree, can also be used as a predictor [9,25,26]. Adding H into biomass or carbon equations as another predictor notably improves the model performance by explaining the divergences and avoiding the potential limitations [25,[27][28][29]. Thus, the regression with D and H was more reliable for prediction of the carbon in a forest. ...

... The nonadditive equations cannot suit the total and tissue carbon data synchronously, leading to unequal total tissue carbon derived from the tissue and the total carbon model. The additive carbon equations fit the total and tissue carbon data simultaneously, which explicates the instinctive correlations among carbon tissues of the same sample [25,[27][28][29], where the sum of carbon predictions from the tissue carbon model and from the total carbon models are the same [30,31]. For the additive carbon equations, various parameter estimation methods and model specifications were used in linear and nonlinear models [30][31][32][33]. ...

... Among these, nonlinear seemingly unrelated regression (NSUR) and seemingly unrelated regression (SUR) are more widely used. An advantage of using SUR and NSUR is the low variance of the total tree carbon model because of their own predictor variables and weighting functions for heteroscedasticity, which make SUR and NSUR the popular methods of parameter estimation in nonlinear and linear carbon and biomass equations [10,24,25,29,33,34]. Although additivity is included in this property by most researchers, additivity is still ignored in most carbon equations. ...

... The tree height (H), another variable of the tree, can also be used as a predictor [9,25,26]. Adding H into biomass or carbon equations as another predictor notably improves the model performance by explaining the divergences and avoiding the potential limitations [25,[27][28][29]. Thus, the regression with D and H was more reliable for prediction of the carbon in a forest. ...

... The nonadditive equations cannot suit the total and tissue carbon data synchronously, leading to unequal total tissue carbon derived from the tissue and the total carbon model. The additive carbon equations fit the total and tissue carbon data simultaneously, which explicates the instinctive correlations among carbon tissues of the same sample [25,[27][28][29], where the sum of carbon predictions from the tissue carbon model and from the total carbon models are the same [30,31]. For the additive carbon equations, various parameter estimation methods and model specifications were used in linear and nonlinear models [30][31][32][33]. ...

... Among these, nonlinear seemingly unrelated regression (NSUR) and seemingly unrelated regression (SUR) are more widely used. An advantage of using SUR and NSUR is the low variance of the total tree carbon model because of their own predictor variables and weighting functions for heteroscedasticity, which make SUR and NSUR the popular methods of parameter estimation in nonlinear and linear carbon and biomass equations [10,24,25,29,33,34]. Although additivity is included in this property by most researchers, additivity is still ignored in most carbon equations. ...

In this study, the effects of tree species, tissue types, and tree size on the carbon concentration were studied, and the two additive systems, one with tree diameter (D), and the other with both D and tree height (H), were developed to estimate the stem, root, branch, and foliage carbon content of 10 broadleaf species in northeast China. The coefficients of the two systems were estimated with the nonlinear seemingly unrelated regression (NSUR), while the heteroscedasticity of the model residual was solved with the weight function. Our results showed that carbon concentrations varied along with tree species and size; the tissues and foliage contained higher carbon concentration than other observed tissues. The two additive carbon equation systems exhibited good predictive and fitting performance, with Ra 2 > 0.87, average prediction error of approximately 0, and small average absolute error and absolute error percentage. The carbon equation system constructed with D and H exhibited better fit and performance, particularly for the stem and total carbon. Thus, the additive carbon equation systems estimated the tree carbon of 10 broadleaf species more accurately. These carbon equations can be used to monitor the carbon pool sizes for natural forests in the Chinese National Forest Inventory.

... Therefore the trend of the development of generic allometric models is replaced gradually by the phasing out of them and moving on to the concepts of their harmonizing and compatibility. The mentioned concept includes at least two directions: (1) designing the systems of compatible regional allometric models based on dummy variables (Fu et al., 2012;Zeng, 2015) and (2) the development of so-called "compatible" models based on the principle of additivity of biomass component composition (Parresol, 2001;Dong et al., 2015;Stankova et al., 2016). The latter relate almost exclusively to allometric patterns performed at the level of sample trees. ...

... The modelling procedure involves several stages. At the first stage the independent allometric equations are calculated in the following order (Dong et al., 2015): first -for total biomass, then -for the aboveground (intermediate component) and underground biomass (Step 1), then -for intermediate components -tree crown and stem above bark (Step 2) and, finally, for the original (initial) components -needle and branches (Step 3a) and wood and bark of the stem (Step 3b) according to their adopted structure; ...

... All the regression coefficients for numerical variables in equations (2) are significant at the level of probability P 0.95 or higher, and the equations are adequate to harvest data. On the second stage the structure of an additive model is compiled when modifying (2) and using the methodology by Chinese scientists (Tang et al., 2000;Dong et al., 2015), and the final form of transcontinental additive model of biomass component composition of spruce and fir stands is listed in the Table 2. ...

... The tree height (H), another variable of the tree, can also be used as a predictor [9,25,26]. Adding H into biomass or carbon equations as another predictor notably improves the model performance by explaining the divergences and avoiding the potential limitations [25,[27][28][29]. Thus, the regression with D and H was more reliable for prediction of the carbon in a forest. ...

... The nonadditive equations cannot suit the total and tissue carbon data synchronously, leading to unequal total tissue carbon derived from the tissue and the total carbon model. The additive carbon equations fit the total and tissue carbon data simultaneously, which explicates the instinctive correlations among carbon tissues of the same sample [25,[27][28][29], where the sum of carbon predictions from the tissue carbon model and from the total carbon models are the same [30,31]. For the additive carbon equations, various parameter estimation methods and model specifications were used in linear and nonlinear models [30][31][32][33]. ...

... Among these, nonlinear seemingly unrelated regression (NSUR) and seemingly unrelated regression (SUR) are more widely used. An advantage of using SUR and NSUR is the low variance of the total tree carbon model because of their own predictor variables and weighting functions for heteroscedasticity, which make SUR and NSUR the popular methods of parameter estimation in nonlinear and linear carbon and biomass equations [10,24,25,29,33,34]. Although additivity is included in this property by most researchers, additivity is still ignored in most carbon equations. ...

In this study, the effects of tree species, tissue types, and tree size on the carbon concentration were studied, and the two additive systems, one with tree diameter (D), and the other with both D and tree height (H), were developed to estimate the stem, root, branch, and foliage carbon content of 10 broadleaf species in northeast China. The coefficients of the two systems were estimated with the nonlinear seemingly unrelated regression (NSUR), while the heteroscedasticity of the model residual was solved with the weight function. Our results showed that carbon concentrations varied along with tree species and size; the tissues and foliage contained higher carbon concentration than other observed tissues. The two additive carbon equation systems exhibited good predictive and fitting performance, with Ra2 > 0.87, average prediction error of approximately 0, and small average absolute error and absolute error percentage. The carbon equation system constructed with D and H exhibited better fit and performance, particularly for the stem and total carbon. Thus, the additive carbon equation systems estimated the tree carbon of 10 broadleaf species more accurately. These carbon equations can be used to monitor the carbon pool sizes for natural forests in the Chinese National Forest Inventory.

... To overcome the heteroscedasticity of the stand biomass model residuals, logarithmic transformation or a weighted regression should be performed before the construction of each carbon model [12,13]. To acquire an ideal result from logarithmic regression, a correction is necessary after the antilog transformation, i.e., the predicted values are multiplied by a correction factor [31,32]. However, when determining the total and component equations of stand biomass, after applying the correction factor to the logarithmic equations of the additive system, realizing additivity is difficult [32]. ...

... To acquire an ideal result from logarithmic regression, a correction is necessary after the antilog transformation, i.e., the predicted values are multiplied by a correction factor [31,32]. However, when determining the total and component equations of stand biomass, after applying the correction factor to the logarithmic equations of the additive system, realizing additivity is difficult [32]. Thus, the weighted regression successfully overcomes the heteroscedasticity of the total and component biomass model residuals in an additive system [12,13]. ...

... We previously developed species-specific tree biomass allometric equations with only tree D as the predictor for the tree total and component biomass (i.e., the stand total biomass ( ), the stand root biomass ( ), the stand stem biomass ( ), the stand branch biomass ( ), and the stand foliage biomass ( )) [12,32,33], and they were applied to each tree within the permanent sample plots. The stand biomass (Mg·ha −1 ) was determined on an area basis for each sample plot. ...

Currently, forest biomass estimation methods at the regional scale have attracted the
greatest attention from researchers, and the development of stand biomass models has become
popular a trend. In this study, a total of 5074 measurements on 1053 permanent sample plots were
obtained in the Eastern Da Xing’an Mountains, and three additive systems of stand biomass
equations were developed. The first additive system (M-1) used stand variables as the predictors
(i.e., stand basal area and average height), the second additive system (M-2) utilized stand volume
as the sole predictor, and the third additive system (M-3) included both stand volume and biomass
expansion and conversion factors (BCEFs) as the predictors. The coefficients of the three model
systems were estimated with nonlinear seemingly unrelated regression (NSUR), while the
heteroscedasticity of the model residuals was solved with the weight function. The jackknifing
technique was used on the residuals, and several statistics were used to assess the prediction
performance of each model. We comprehensively evaluated four stand biomass estimation methods
(i.e., M-1, M-2, M-3 and a constant BCEF (M-4)). Here, we showed that the (1) three additive systems
of stand biomass equations showed good model fitting and prediction performance, (2) M-3
significantly improved the model fitting and performance and provided the most accurate
predictions for most stand biomass components, and (3) the ranking of the four stand biomass
estimation methods followed the order of M-3 > M-2 > M-4 > M-1. Our results demonstrated these
additive stand biomass models could be used to estimate the stand aboveground and belowground
biomass for the major forest types in the Eastern Da Xing’an Mountains, although the most
appropriate method depends on the available data and forest type.

... In the proposed article we try to partially answer this question, namely, we undertake, in essence, the first attempt to model changes in the additive component composition of biomass of forest phytocenoses on Trans-Eurasian gradients of mean temperatures and precipitation on the example of the genus Larix sp. In the development of additive systems of equations, preference is given to the principle "from general to particular", in which the equation for the total biomass is "splitted" into additive equations for each of the constituent components by proportional weighing [9,10]. ...

... Since in the North of Eurasia the mean annual temperature in January has negative values, the corresponding independent variable is modified to the form (Tm+50). In contrast to the two-step disaggregation additive model for above-ground biomass [10] and to the three-step disaggregation additive model for above-ground and under-ground biomass [9], in our study, the total biomass of forest phytocenosis (tree stand and understory), estimated by the initial equation, is divided into components according to the four-step scheme of proportional weighing presented in Figure 1. ...

... The equations obtained are modified to additive form according to the above mentioned algorithm [9] in the sequence shown in the scheme (see Figure 1), and the final form of the transcontinental additive model of component composition of biomass of larch phytocenoses is shown in Table 3. When tabulating additive model (1), a problem arises, which consists in the fact that we can specify the indices of only the forest stand age, temperature and precipitation, but the values of the stem volume and tree density can be entered into the resulting table in the form of calculated values obtained by a system of auxiliary recursive equations. ...

The first attempt of modeling changes in additive component composition of biomass of Larix sp. communities on the Trans-Eurasian hydrothermal gradients based on regional peculiarities of age and morphology of the forests is attempted. The increase of all biomass components of the tree layer with increasing temperature at the constant precipitation and its decrease with increasing precipitation at the constant temperature is established. The positive relationship of the understory biomass with the temperature in the areas of insufficient moisture as the transition to moisture-rich areas is replaced by the opposite one. The development of such models for basic forest-forming species in Eurasia will give possibility to predict any changes in the biological productivity of forest cover of Eurasia in relation to climate change.

... In the development of trivial empirical models, the additivity of component composition is not provided, according to which the total biomass of components (stems, branches, needles, roots) obtained by component equations would be equal to the value of biomass obtained by the common equation (Dong et al., 2015). According to Sanquetta et al. (2015), independent (without additivity) fitting of coefficients for biomass components and total biomass is not satisfactory, but this is not observed when simultaneous fitting is used accounting the additivity principle, which results in more effective estimators. ...

... According to the structure of the disaggregation three-step additive biomass model (Tang et al., 2000;Dong et al., 2015), the total biomass estimated from the initial equation is divided into its components according to the scheme presented in Fig. 3 and Table 1. We used a schematic map of the contours of the mean January temperature (stage of deep winter dormancy in forest trees), rather than the mean annual temperature, as warming is most pronounced in the cold half of the year (Golubyatnikov, Denisenko, 2009;Laing, Binyamin, 2013;Felton et al., 2016). ...

... Biomass components of the crown respond ambiguously to changes in precipita- Table 1. The structure of the three-step additive model sold under proportional weighting supposed by Dong et al. (2015). Symbols here and further as per Fig. 3 and Eq. ...

Since ancient times, climate change has largely determined the fate of human civilisation, which was related mainly to changes in the structure and habitats of forest cover. In the context of current climate change, one must know the capabilities of forests to stabilise the climate by increasing biomass and carbon-depositing abilities. For this purpose, the authors compiled a database of harvest biomass (t/ha) in 900 spruce (Picea spp.) sample plots in the Eurasian area and used the methodology of multivariate regression analysis. The first attempt at modelling changes in the biomass additive component composition has been completed, according to the Trans-Eurasian hydrothermal gradients. It is found that the biomass of all components increases with the increase in the mean January temperature, regardless of mean annual precipitation. In warm zonal belts with increasing precipitation, the biomass of most of the components increases. In the process of transitioning from a warm zone to a cold one, the dependence of all biomass components upon precipitation is levelled, and at a mean January temperature of ˗30°C it becomes a weak negative trend. With an increase in temperature of 1°C in different ecoregions characterised by different values of temperature and precipitation, there is a general pattern of decrease in all biomass components. With an increase in precipitation of 100 mm in different ecoregions characterised by different values of temperature and precipitation, most of the components of biomass increase in warm zonal belts, and decrease in cold ones. The development of such models for the main forest-forming species of Eurasia will make it possible to predict changes in the productivity of the forest cover of Eurasia due to climate change.

... However, it is necessary that the sum of the best above-ground component biomass models equal the total in the AGB models. For this purpose, we suggest a one-step proportional weighting system for AGB based on a disaggregated model structure (namely, a two-step proportional weighting system, the TSEM method) proposed by Tang et al. [52]; this structure was successfully interpreted as a three-step proportional weighting system, the 3SPW method, and has been implemented in China [54] and Russia [55]. Using the disaggregation method, the total tree predicted AGB, Ŷ , stem (wood + bark) biomass, (X), branch biomass, (X), and foliage biomass, (X), are separately fitted, and the best models are selected. ...

... The sum of biomass predictions from separate tree component models may not equal the biomass prediction of the total tree biomass model [50,51]. To eliminate this inconsistency, several model specifications and estimation methods have been suggested for forcing additivity on a series of biomass equations, both linear and nonlinear [45,[51][52][53][54][55]. The property of additivity assures regression functions that are consistent with one other. ...

... However, it is necessary that the sum of the best above-ground component biomass models equal the total in the AGB models. For this purpose, we suggest a one-step proportional weighting system for AGB based on a disaggregated model structure (namely, a two-step proportional weighting system, the TSEM method) proposed by Tang et al. [52]; this structure was successfully interpreted as a three-step proportional weighting system, the 3SPW method, and has been implemented in China [54] and Russia [55]. Using the disaggregation method, the total tree predicted AGB,Ŷ a , stem (wood + bark) biomass, f s (X), branch biomass, f b (X), and foliage biomass, f f (X), are separately fitted, and the best models are selected. ...

Understanding the contribution of forest ecosystems to regulating greenhouse gas emissions and maintaining the atmospheric CO2 balance requires the accurate quantification of above-ground biomass (AGB) at the individual tree species level. The main objective of this study was to develop species-specific allometric equations for the total AGB and various biomass components, including stem, branch, and foliage biomass in Khangai region, northern Mongolia. We destructively sampled a total of 183 trees of five species (22–74 trees per species), including Siberian stone pine (Pinus sibirica Du Tour.), Asian white birch (Betula platyphylla Sukacz.), Mongolian poplar (Populus suaveolens Fisch.), Siberian spruce (Picea obovata Ldb.), and Siberian larch (Larix sibirica Ldb.), across this region. The results showed that for the five species, the average biomass proportion for the stems was 75%, followed by branches at 20% and foliage at 5%. The species-specific component and total AGB models for the Khangai region were developed using tree diameter at breast height (D) and D² and tree height (H) combined ( D 2 H ); and both D and H were used as independent variables. The best allometric model was lnŶ = lna + b × lnD + c × lnH for the various components and total AGB of B. platyphylla and L. sibirica, for the stems and total AGB of P. suaveolens, and for the stem and branch biomass of P. obovata. The equation lnŶ = lna + b × ln( D 2 × H ) was best for the various components and total AGB of P. sibirica, for the branch and foliage biomass of P. suaveolens, and for AGB of P. obovata. The equation lnŶ = lna + b × ln(D) was best only for the foliage biomass of P. obovata. Our results highlight that developing species-specific tree AGB models is very important for accurately estimating the biomass in the Khangai forest region of Mongolia. Our biomass models will be used at the tree level inventories with sample plots in the Khangai forest region.

... All above mentioned models are internally contradictory, they are not harmonized by the biomass structure, i.e. they do not provide the additivity of component composition, according to which the total biomass of components (stems, branches, needles, roots) obtained by "component" equations would be equal to the value of biomass obtained by the total biomass equation (Dong et al., 2015). The influence of climatic changes on the biomass of a particular tree species in the format of additive models according to transcontinental hydrothermal gradients has not been studied at all. ...

... According to the structure of the disaggregation model of a three-step additive equation system (Tang et al., 2000;Dong et al., 2015), the total biomass P t , estimated by the initial equation, is divided into component biomass estimated by corresponding equations according to the scheme presented in Figure 4 and Table 2. Figure 4. The pattern of disaggregating three-step proportional weighting additive model. ...

... The structure of the three-step additive model, sold under proportional weighting when using 122 sample trees of Larix gmelinii Rupr. (Dong et al., 2015). Symbols here and further as per equation (1) Step 1 ...

The analysis of the biomass of larch (genus Larix spp.) trees on the total component composition based on regression equations having the additive biomass structure. Two trends of changes in the tree biomass structure are revealed: due to the mean January temperature and due to the mean annual precipitation. It was shown for the first time that both trends are mutually determined: the intensity of biomass trend in relation to the temperature is changing when depending on the level of precipitation, and the intensity of biomass trend in relation to precipitation level is changing during to a transition from the cold zone to the warm one and vice versa.

... In recent years, the scientific branch associated with calculating allometric models of trees and stands in the aspect of their harmonization. Harmonization implies at least two directions: (1) designing of compatible regional models based on dummy variables (Zeng, 2015;Fu et al., 2017) and (2) designing of compatible models based on the principle of additivity of biomass component composition (Parresol, 2001;Dong et al., 2015). ...

... Analysis of biomass of tree biomass is made on the basis of allometric additive models. According to the structure of disaggregating three-step additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in: (Dong et al., 2015). The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ...

... Analysis of biomass of tree biomass is made on the basis of allometric additive models. According to the structure of disaggregating three-step additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in: (Dong et al., 2015). The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ...

– When using the unique in terms of the volume of database on the level of a single-tree of the genus Betula sp., the trans-Eurasian additive allometric model of biomass of trees for Eurasian birch forests is developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Betula sp. is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes not only on total, aboveground and underground biomass, but also on its component structure, i.e. it reflects the regional peculiarities of the component structure of tree biomass.

... According to the structure of the disaggregated three-step model (Dong, Zhang, & Li, 2015), its total biomass, estimated by the original equation, is exploded into component equations according to the scheme presented in Exhibit 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures the additivity of the biomass components: the total, intermediate, and initial ones (Dong et al., 2015). ...

... According to the structure of the disaggregated three-step model (Dong, Zhang, & Li, 2015), its total biomass, estimated by the original equation, is exploded into component equations according to the scheme presented in Exhibit 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures the additivity of the biomass components: the total, intermediate, and initial ones (Dong et al., 2015). ...

... The equations are given in an additive form in accordance with the above-mentioned algorithm (Dong et al., 2015) in However, when tabulating model (1), there is a problem, which is that we can know and give only the value of stand age, mean annual precipitation, and mean annual temperature. The remaining two variables N, tree density, and M, stem volume, can be entered into a table in the form of calculated values obtained by the system of auxiliary recursive equations (Usoltsev et al., 2017). ...

... In the present study, we make the first attempt to model changes in the additive component composition of biomass and NPP of forest ecosystems by Trans-Eurasian gradients of mean January temperatures and mean annual precipitation on the example of twoneedled pines (subgenus Pinus sp.). In the development of additive systems of equations, preference is given to the principle "from general to particular", in which the equation for the total biomass is "splitted" into additive equations for each component by the method of proportional weighing (Dong et al., 2015). ...

... Because mean January temperature in northern part of Eurasia has negative values, corresponding independent variable is modified to the form (Тm+40). Equations (1) and (2) form a recursive system in which the dependent variable of the first of them is included in the second equation as one of the independent variables. In contrast to the three-step structure of the disaggregation model of additive system of equations (Dong et al., 2015) shown in Table 1, in our study the total biomass estimated from the initial equation is subdivided into much more biomass component according to the four-step scheme of proportional weighting presented in Figure 1. Step 1 Step ...

... The structure of the three-step additive model, implemented according to the principle of proportional weighting according to the 122 trees of Larix gmelinii(Dong et al., 2015) ...

Modelling forest biomass sensitive to climate change is fulfilled at the levels as forest stands and single-trees, but mostly on a local or regional level, often without regard to the age, morphology of the forest stands and species composition. With this, it does not provide additive component composition, according to which the total of biomass components (stems, branches, needles, and roots), obtained by component equations, would be equal to the value of the biomass obtained by the general equation. The influence of climate change on the biomass of a tree species in the format of additive models for transcontinental hydrothermal gradients has not yet been studied. In the present study, the first attempt is made to model changes in the additive component composition of the stand biomass and NPP of two-needled pines along Trans-Eurasian hydrothermal gradients. In the process of modelling the database of pine stand biomass in a number of 2460 sample plots with the definitions of biomass and 760 plots with the definitions of biomass and annual NPP compiled by the authors, is used.

... Thus, the modern methods of modelling the biological productivity of trees and tree stands have been developed towards additivity of biomass components (Bi et al., 2010;Dong et al., 2015b) and towards transition from "pseudo-generic" allometric models to really genericl, involving regionalization of biomass models by introducing dummy variables (Fu et al., 2012), that usually fulfilled on local sets of actual biomass of trees and tree stands. We generated the database of forest stand biomass for the main forest species in Eurasia (Usoltsev, 2010(Usoltsev, , 2013, that has enabled these modern methodologies to be implemented on the entirely different, higher level, namely to begin modelling additive biomass on transcontinental level. ...

... The distribution of sample plots, on which the larch forest biomass is measured in ecoregions of Eurasia, is shown in Figure 1. According to the structure of disaggregation three-step model (Tang et al., 2000;Dong et al., 2015b), biomass value, estimated by the total biomass equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of biomass of all the components -total, intermediate and initial (Dong et al., 2015b). Figure 2. The pattern of disaggregating three-step proportional weighting additive model. ...

... The distribution of sample plots, on which the larch forest biomass is measured in ecoregions of Eurasia, is shown in Figure 1. According to the structure of disaggregation three-step model (Tang et al., 2000;Dong et al., 2015b), biomass value, estimated by the total biomass equation, exploded into components according to the scheme presented in Figure 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of biomass of all the components -total, intermediate and initial (Dong et al., 2015b). Figure 2. The pattern of disaggregating three-step proportional weighting additive model. ...

1
Modeling the additive allometric of stand biomass of Larix sp.
for Eurasia
Vladimir Аndreevich Usoltsev1, 2, Seyed Omid Reza Shobairi2*, Viktor Petrovich Chasovskikh2
1Botanical Garden, Russian Academy of Sciences, Ural Branch, 8 Marta 202a St, Yekaterinburg, 620144 Russian Federation
2Ural State Forest Engineering University, Sibirskii Trakt 37 St, Yekaterinburg,
620100 Russian Federation,
*e-mail: Omidshobeyri214@gmail.com
Received: 6 November 2018 / Accepted: 30 January 2019
Abstract. When using the unique in terms of the volumes of database on the level of a stand of the genus Larix Mill., the trans-Eurasian additive allometric models of biomass for Eurasian larch forests are developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of forest biomass of Larix is harmonized in two levels, one of which provides the principle of additivity of biomass components, and the second one is associated with the introduction of dummy independent variables localizing model for eco-regions of Eurasia. Comparative analysis of the biomass structure of larch stands of different ecoregions at the age of 100 years shows, that the greatest values of biomass (210-450 t/ha) correspond to the regions adjacent to the Atlantic and Pacific coasts, as well as to the regions, located at the southern limit of larch growing area and the lowest – to northern taiga regions of Siberia, where larch grows on permafrost. The biomass indices of different ecoregions differed not only in absolute value but also in biomass ratios of different components; for example, the proportion of needles in the aboveground biomass is maximum (5.0-7.3%) in the northern taiga of Central Siberia and the Far East on permafrost and is minimum (1.4-1.9%) in larch forests of upper productivity having biomass values 210-450 t/ha. The proposed model and corresponding tables for estimating stand biomass makes them possible to calculate larch stand biomass on Eurasian forests when using measuring taxation.
Key words: allometric models, biological productivity, biomass of forests; Larix Mill., sample plots.

... Allometric models of tree biomass are harmonized or by ensuring the additivity of component composition (Dong et al., 2015), either by their regionalization (localization) using dummy variables (Fu et al., 2012;Usoltsev et al., 2017a) or by coding (marking) several tree species in a single model by dummy variables (Zeng, 2017) that is typically fulfilled on local data sets of tree biomass. ...

... The distribution of sample plots, on which sample trees are taken in different ecoregions of Eurasia is shown in fig. 1. *Author for correspondence : E-mail: omidshobeyri214@gmail.com According to the structure of disaggregating threestep additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in fig. 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ...

... The distribution of sample plots, on which sample trees are taken in different ecoregions of Eurasia is shown in fig. 1. *Author for correspondence : E-mail: omidshobeyri214@gmail.com According to the structure of disaggregating threestep additive model system (Tang et al., 2000;Dong et al., 2015), total biomass, estimated by the total equation, exploded into components according to the scheme presented in fig. 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones (Dong et al., 2015). ...

When using the unique in terms of the volumes of database on the level of a tree of the subgenus Pinus spp., the trans-Eurasian additive allometric model of biomass of trees for Eurasian forests are developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Pinus is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes on total, aboveground and underground biomass. The proposed model and corresponding tables for estimating tree biomass makes them possible to calculate two-needled pine biomass (t/ ha) on Eurasian forests when using measuring taxation.

... The development of generic allometric biomass models [5,6,22,25,27,29,32,38,42] is replaced by the phasing out of them and moving on to the concept of their harmonizing. Harmonization implies at least two directions: (1) designing of compatible regional models based on dummy variables [13,14,15,16,18,19,31,33,36,37,39,40,41] and (2) designing of compatible models based on the principle of additivity of biomass component composition [2,3,4,9,10,11,12,21,28]. ...

... Analysis of biomass of tree biomass is made on the basis of allometric additive models. According to the structure of disaggregating three-step additive model system [10,30], total biomass, estimated by the total equation, exploded into components according to the scheme presented in Fig. 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones [10]. ...

... Analysis of biomass of tree biomass is made on the basis of allometric additive models. According to the structure of disaggregating three-step additive model system [10,30], total biomass, estimated by the total equation, exploded into components according to the scheme presented in Fig. 2. The coefficients of the regression models for all three steps are evaluated simultaneously, which ensures additivity of the all components: total, intermediate and initial ones [10]. ...

When using the unique in terms of the volume of database on the level of a single-tree of the genus Betula sp., the trans-Eurasian additive allometric model of biomass of trees for Eurasian birch forests is developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Betula sp. is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes not only on total, aboveground and underground biomass, but also on its component structure, i.e. it reflects the regional peculiarities of the component structure of tree biomass.

... Therefore the trend of the development of generic allometric models is replaced gradually by the phasing out of them and moving on to the concepts of their harmonizing and compatibility. The mentioned concept includes at least two directions: (1) designing the systems of compatible regional allometric models based on dummy variables (Fu et al., 2012;Zeng, 2015) and (2) the development of so-called "compatible" models based on the principle of additivity of biomass component composition (Parresol, 2001;Dong et al., 2015;Stankova et al., 2016). The latter relate almost exclusively to allometric patterns performed at the level of sample trees. ...

... The modelling procedure involves several stages. At the first stage the independent allometric equations are calculated in the following order (Dong et al., 2015): first -for total biomass, then -for the aboveground (intermediate component) and underground biomass (Step 1), then -for intermediate components -tree crown and stem above bark (Step 2) and, finally, for the original (initial) components -needle and branches (Step 3a) and wood and bark of the stem (Step 3b) according to their adopted structure; ...

... All the regression coefficients for numerical variables in equations (2) are significant at the level of probability P 0.95 or higher, and the equations are adequate to harvest data. On the second stage the structure of an additive model is compiled when modifying (2) and using the methodology by Chinese scientists (Tang et al., 2000;Dong et al., 2015), and the final form of transcontinental additive model of biomass component composition of spruce and fir stands is listed in the Table 2. ...

For the first time using the unique database of eight species of the genera Picea spp. and of eight species of the genera Abies spp. in a number of 670 and 255 sample plots correspondingly growing on the territory of Eurasia a comparative analysis of biomass (stem, branches, foliage, roots) structure of forest stands is fulfilled. The result represents the additive allometric model, harmonized on three levels: one of which provides the principle of additivity of biomass components, the second one relates to the involving dummy variables, localizing the model along to ecoregions of Eurasia, and the third one harmonizes the structure of a transcontinental model of spruce and fir stands by involving the binary variable into biomass model. It is shown that the model demonstrates differences between spruce and fir stand biomass not only for its absolute values for stem, needles, branches and roots (as is typical for trivial independent models that includes only dummy variables), but also for their ratios, i.e. the differences in the biomass structure. Negative correlation of calculated indices of spruce and fir biomass with continentality index by V. Tsenker is shown, characterizing by a correlation coefficient-0.92 to total phytomass and-0.89 for aboveground ones. The model can be developed further on the third level of harmonization, on which one can include a block of dummy variables coding distribution of actual biomass data between main forest-forming wood species in Eurasia.

... An alternative disaggregation approach is based on the development of component biomass fraction equations (Tang et al. 2000;Dong et al. 2015). In this approach, a total biomass model is first developed, and the estimated total tree biomass is disaggregated into tree components based on their proportions in the total. ...

... To verify the efficiency of parameter estimates obtained using the weighted NSUR, the primary functions were also estimated using weighted OLS estimation with the same weighting functions used in NSUR. Tang et al. (2000) initially developed a disaggregation strategy (also see Dong et al. 2015). In their strategy, a total biomass model y T = f T (X T ,  T ) needs to be first developed. ...

... Then the estimated total biomass ŷ T is disaggregated into the component biomass: y m ϭ p m × ŷ T + m (m = 1, 2, …, M), that is, ŷ m ϭ p m × ŷ T . The parameters associated with biomass component models are estimated through jointly fitting these component models with two-stage nonlinear errorin-variable models (TSEM) (Tang et al. 2001;Tang and Wang 2002) or NSUR (Dong et al. 2015). ...

Both aggregative and disaggregative strategies were used to develop additive nonlinear biomass equations for slash pine trees in the southeast US. In the aggregative approach, total tree biomass equation was specified by aggregating the expectations of component biomass models, and their parameters were estimated by jointly fitting all component and total biomass equations using weighted nonlinear seemingly unrelated regression (NSUR) (SUR1) or by jointly fitting component biomass equations using weighted NSUR (SUR2). In an alternative disaggregative approach (DRM), the biomass component proportions were modeled using Dirichlet regression, and the estimated total tree biomass was disaggregated into biomass components based on their estimated proportions. There was no single system to predict biomass that was best for all components and total tree biomass. The ranking of the three systems based on an array of fit statistics followed the order of SUR2 > SUR1 > DRM. All three systems provided more accurate biomass predictions than previously published equations.

... The monitoring of forest biomass and carbon stocks and the establishment of biomass models that are suitable for larger areas are therefore increasingly important. Over recent decades, researchers have developed more than 2600 biomass models for over 100 species of tree around the world [3], most of which are used to estimate aboveground biomass [4][5][6][7]. There are much fewer estimates of the belowground biomass of plants due to the difficulties of excavation and high consumption costs. ...

... Only the root biomass is estimated as a component in these systems, which is less accurate for root prognosis when using Wang's additive model for birch plantation development [18]. Moreover, in contrast to the aggregative method, the disaggregation strategy focuses first on the specification of the allometric equation for the total biomass and then constrains the form or parameters of the component equations by fitting the proportion of total biomass that is observed in each component to ensure additivity [3,19]. ...

Most of the forest biomass models that have been developed so far focus on the study of the aboveground biomass of forest trees and the prediction of belowground biomass remains obviously insufficient. Moreover, most of the existing studies on the estimation of the belowground biomass of trees have considered roots as a whole, ignoring the differences in composition and function of roots within different diameter classes. In this study, we measured the root biomass of birch plantation forests in northeastern China using extensive destructive sampling, in which we divided the root system into three parts: coarse, medium, and fine roots. We selected the best model base form from three common allometric biomass equations and determined the most appropriate error structure for the two sets of models using likelihood comparisons. The additive and disaggregated models were fitted using maximum likelihood with open-source software. We also added the site factor as a dummy variable into the two models. Finally, the competency of the two models was tested using ten-fold cross-validation. The results showed that both models could provide relatively accurate estimates of birch root biomass but that the disaggregated model performed slightly better than the additive model.

... To date, the most widely used parameter estimation method of additive model system is nonlinear seemingly unrelated regression (NSUR) method (Parresol, 2001). The advantage of NSUR is that each model can use its own independent variable(s) and weighting function for solving heteroscedasticity, resulting in a lower variance for the total tree biomass model (Dong et al., 2015). But the disadvantage of NSUR model system is that there is no separate total tree biomass model developed and there is no constraints for subtotal biomass (e.g. ...

... But the disadvantage of NSUR model system is that there is no separate total tree biomass model developed and there is no constraints for subtotal biomass (e.g. aboveground or crown biomass) (Parresol, 2001;Dong et al., 2015). For disaggregation model system, the total biomass model is first developed and the estimated total biomass is disaggregated into tree components based on their proportions (Tang et al., 2001). ...

It is important to guarantee the property of biological compatibility when estimating tree biomass of the total and components for carbon accounting under global climate change. The issue was successfully considered in traditional nonlinear regressions, but not for machine learning methods. A new method for approaching the compatibility of tree biomass estimation in ANN (Artificial Neural Network) was developed by using the multi-task loss function, which had the desire features of minimizing residuals and approaching biomass compatibility. The method was tested by two tree species biomass dataset and showed the desired feature. Leave-one-out validation results showed that comparing ANN model with simultaneously fitting 7 outputs (stem, bark, branch, leaf, crown, trunk, aboveground) and classical loss function, the RMSE of aboveground estimation (AGB) and the mean absolute relative difference between AGB and the sum of component biomass estimations from the model developed by our new method decreased from 166.864 (kg) to 154.860 (kg) and from 4.757% to 0.071%, respectively for Abies nephrolepis dataset, and from 49.18 (kg) to 33.060 (kg) and from 5.314% to 0.636%, respectively for Acer mono dataset. It provided a trade-off solution for the error accumulation and the compatibility among components and the total estimations when using ANN for tree biomass modelling, and was useful for carbon accounting using machine learning methods.

... Samples were then taken to the laboratory for oven drying and carbon content determination. The in-depth explanations regarding the field measurements and laboratory analysis have been entirely discussed in Dong et al. [14,22,40]. The descriptive statistics of the measured dry weight biomass and the independent variables (i.e., DBH and H) are presented in Table 1. ...

... For each biomass equation, a unique weighting function was determined using the power function to independently fit the predictors (D i ) and the error variance (ε i ) of the ith individuals. A more detailed theoretical explanation can be found in Harvey [44], while the practical application using the SAS/ETS PROC MODEL procedure [45] in Dong et al. [14] and Balboa-Murias et al. [46] ...

The population of natural Korean pine (Pinus koraiensis) in northeast China has sharply declined due to massive utilization for its high-quality timber, while this is vice versa for Korean pine plantations after various intensive afforestation schemes applied by China’s central authority. Hence, more comprehensive models are needed to appropriately understand the allometric relationship variations between the two origins. In this study, we destructively sampled Pinus koraiensis from several natural and plantation sites in northeast China to investigate the origin’s effect on biomass equations. Nonlinear seemingly unrelated regression with weighted functions was used to present the additivity property and homogenize the model residuals in our two newly developed origin-free (population average) and origin-based (dummy variable) biomass functions. Variations in biomass allocations, carbon content, and root-to-shoot ratio between the samples obtained from plantations and natural stands were also investigated. The results showed that (1) involving the origin’s effect in dummy variable models brought significant improvement in model performances compared to the population average models; (2) incorporating tree total height (H) as an additional predictor to diameter at breast height (D) consistently increase the models’ accuracy compared to using D only as of the sole predictors for both model systems; (3) stems accounted for the highest partitioning proportions and foliage had the highest carbon content among all biomass components; (4) the root-to-shoot ratio ranged from 0.18–0.35, with plantations (0.28 ± 0.04) had slightly higher average value (±SD) compared to natural forests (0.25 ± 0.03). Our origin-based models can deliver more accurate individual tree biomass estimations for Pinus koraiensis, particularly for the National Forest Inventory of China.

... Вторая процедура согласования уравнений имеет прямо противоположный алгоритм, согласно которому рассчитывается обобщенное уравнение, которое по специальной схеме расчленяется на исходные, и их суммарный итог равен итоговому значению обобщенного уравнения. Этот метод является альтернативой упомянутому и известен как трехшаговый метод пропорционального взвешивания по принципу «от общего -к частному» (Dong et al., 2015). При этом общая биомасса дерева делится на над-и подземную части в соответствии с их долями в общей, представленными соответствующими «фракционными» зависимостями (шаг 1), затем полученная надземная биомасса разделяется аналогичным образом на ствол в коре и крону дерева (шаг 2), крона расчленяется на хвою и ветви (шаг 3а), а ствол -на древесину и кору (шаг 3б). ...

... Исходные уравнения модели, числовые параметры которой показаны в табл. 1, путем процедуры пропорционального взвешивания (Dong et al., 2015) приведены к виду искомой модели (табл. 2). ...

Kozak, 1970). Среди них выделяются две группы: одна ориентирована на расчет уравнений отдельно для каждой фракции с последующей их моди-фикацией по специальным алгоритмам таким Поступила в редакцию 23.08.2018 г. Биомасса лесов является ключевой экосистемной составляющей и важнейшим компонентом глобального углеродного цикла. Разработка регрессионных моделей биомассы ведется сегодня, во-первых, в ограничен-ных экорегионах и, во-вторых, без согласования по фракционному составу. Согласованные по фракциям био-массы модели называются аддитивными. Среди них выделяются две группы: одна ориентирована на расчет уравнений отдельно для каждой фракции с последующей их модификацией по специальным алгоритмам таким образом, что суммарный результат исходных уравнений равняется результату обобщенного уравне-ния. Вторая процедура согласования уравнений имеет прямо противоположный алгоритм, согласно которому рассчитывается обобщенное уравнение, которое по специальной схеме расчленяется на исходные, и их сум-марный итог равен итоговому значению обобщенного уравнения. Сформированная авторами база данных о биомассе 1035 деревьев елей и пихт в их евразийских ареалах использована в качестве основы для выявления различий равновеликих деревьев двух древесных родов по структуре биомассы в пределах их ареалов при обеспечении принципа согласованности по второму из названных вариантов. Гармонизированная модель, позволяющая сравнивать структуру биомассы деревьев двух родов на континентальном уровне, предложена впервые. Установлено, что пихта при равных с елью высоте и диаметре ствола превышает ель по общей, над-и подземной биомассе деревьев соответственно на 13, 11 и 20 %. Но по соотношению над-и подземной биомассы в общей и по соотношению массы кроны и ствола в надземной биомассе различий между елями и пихтами не наблюдается, тогда как по соотношению масс хвои и ветвей в массе кроны и по соотношению дре-весины и коры различия существуют. Разработанная модель аддитивной структуры биомассы деревьев елей и пихт дает возможность определять согласованную по фракционному составу биомассу соответствующих древостоев на основе данных наземной таксации. Ключевые слова: ель и пихта в ареале, гармонизированная по биомассе модель, регрессионные уравнения, согласованность фракционного состава, трансевразийская закономерность.

... Harmonization implies at least two directions: (1) designing of compatible regional models based on dummy variables [40,36,38,47,10,22,20,50,49,15,33,19,52,16,18,17,51,42,43,44] and (2) designing of compatible models based on the principles of additivity of biomass component composition [23,5,3,2,9,31,13,12,14,31,42,46]. ...

... At the second stage of the study, the structure of the additive model and its calculating algorithm proposed by Chinese researchers [34,12] are modified in accordance with the specifics of our study, and the result obtained in the form of additive and regionally distributed model of triple harmonization is shown in the Table 1. The model is valid in the range of actual data of stand age, mean height, mean diameter and tree density and is characterized by the triple harmonization: one of which provides the principle of additivity of biomass components, the second one is related to the introduction of dummy variables that localizing model for eco-region of Eurasia and the third one conforms (harmonizes) the biomass structure of spruce and fir through the binary variable. ...

... Анализ фитомассы деревьев и древостоев выполнен на основе аллометрических аддитивных моделей. Согласно структуре «дисагрегированной» (расчленяемой) (disaggregation model) трехшаговой аддитивной системы моделей (Tang et al., 2000;Dong et al., 2015), общая фитомасса, оцененная по исходному уравнению, расчленяется на фракции согласно схеме, представленной на рис. 3. Коэффициенты регрессионных моделей всех трех шагов оцениваются одновременно, что обеспечивает аддитивность фитомассы всех фракций -общей, промежуточных и исходных (Dong et al., 2015). ...

... Анализ фитомассы деревьев и древостоев выполнен на основе аллометрических аддитивных моделей. Согласно структуре «дисагрегированной» (расчленяемой) (disaggregation model) трехшаговой аддитивной системы моделей (Tang et al., 2000;Dong et al., 2015), общая фитомасса, оцененная по исходному уравнению, расчленяется на фракции согласно схеме, представленной на рис. 3. Коэффициенты регрессионных моделей всех трех шагов оцениваются одновременно, что обеспечивает аддитивность фитомассы всех фракций -общей, промежуточных и исходных (Dong et al., 2015). ...

V.А. Usoltsev, I.S. Tsepordey, V.P. Chasovskikh, A.A. Osmirko
ADDITIVE REGIONAL MODELS OF TREE AND STAND BIOMASS FOR
EURASIA. MESSAGE 2: GENUS Betula sp.
Key words: genus Betula sp., equations additivity, biosphere role of forests, biomass of trees and forests, allometric models, sample plots, biological productivity, transcontinental tables of biomass.
When using the unique in terms of the volumes of databases on the levels of a tree and stand of the genus Betula sp., the trans-Eurasian additive allometric models of biomass of trees and forests for Eurasian birch forests are developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Betula is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes not only on total, aboveground and underground biomass, but also on its component structure, i.e. it reflects the regional peculiarities of the component structure of tree biomass. The additive model of forest biomass of Betula is harmonized in two levels too, one of which provides the principle of additivity of biomass components, and the second one is associated with the introduction of dummy independent variables localizing model for eco-regions of Eurasia.

... Таблица 2 Схема кодирования региональных массивов фактических данных фитомассы 384 древостоев лиственницы Согласно структуре «дисагрегированной» (расчленяемой) (disaggregation model) трехшаговой аддитивной системы моделей (Tang et al., 2000;Dong et al., 2015), общая фитомасса, оцененная по исходному уравнению, расчленяется на фракции согласно схеме, представленной на рис. 3. Коэффициенты регрессионных моделей всех трех шагов оцениваются одновременно, что обеспечивает аддитивность фитомассы всех фракций -общей, промежуточных и исходных (Dong et al., 2015). ...

... Таблица 2 Схема кодирования региональных массивов фактических данных фитомассы 384 древостоев лиственницы Согласно структуре «дисагрегированной» (расчленяемой) (disaggregation model) трехшаговой аддитивной системы моделей (Tang et al., 2000;Dong et al., 2015), общая фитомасса, оцененная по исходному уравнению, расчленяется на фракции согласно схеме, представленной на рис. 3. Коэффициенты регрессионных моделей всех трех шагов оцениваются одновременно, что обеспечивает аддитивность фитомассы всех фракций -общей, промежуточных и исходных (Dong et al., 2015). ...

V.А. Usoltsev, I.S. Tsepordey, V.P. Chasovskikh, A.A. Osmirko
ADDITIVE REGIONAL MODELS OF TREE AND STAND BIOMASS FOR
EURASIA. MESSAGE 1: GENUS Larix sp.
Key words: genus Larix Mill., equations additivity, biosphere role of forests, biomass of trees and forests, allometric models, sample plots, biological productivity, transcontinental tables of biomass.
When using the unique in terms of the volumes of databases on the levels of a tree and stand of the genus Larix Mill., the trans-Eurasian additive allometric models of biomass of trees and forests for Eurasian larch forests are developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Larix is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes not only on total, aboveground and underground biomass, but also on its component structure, i.e. it reflects the regional peculiarities of the component structure of tree biomass. The additive model of forest biomass of Larix is harmonized in two levels too, one of which provides the principle of additivity of biomass components, and the second one is associated with the introduction of dummy independent variables localizing model for eco-regions of Eurasia.
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... Affleck & Diéguez-Aranda (2016) showed that aggregated non-linear equations could be fit using the Gaussian maximum likelihood (ML), and that this approach can be useful for integrating logged tree data collected under different protocols. Dong et al. (2015) implemented a disaggregation method, which also ensures additivity by predicting the proportion of each component. In the same way way, Zhao et al. (2016) tested three proportion modelling approaches, highlighting the Dirichlet method as superior to the fractional multinomial logit and logratio approaches. ...

The biomass content and carbon captured by forest plantations is of interest, for example in the context of climate change and carbon budgets.The main objective of our study was to develop functions to estimate the total biomass and its components (stem, branches, bark and leaves) of Pinus radiata D. Don trees in Chile. The methodology proposed for the model fitting uses the maximum likelihood method in a multivariate equation system fitting simultaneously. The fit strategy incorporates additivity restrictions in the estimation functions and in the variance functions to incorporate the heteroskedasticity of biomass, and three structures of the variance–covariance matrix were evaluated to assess the dependence of the different components of tree biomass. Non-linear biomass functions that used the variable $D^2H$ performed best according to several indicators of goodness-of-fit (log-likelihood, Akaike Information Criterion and Bayesian Information Criterion) and estimation precision (root mean square error (RMSE), Bias and EI). The simple structure of both biomass and variance estimation functions was $\beta _1 (D^2H)^{\beta _2}$, and in the modelling system for total tree biomass RMSE between 54.1-54.4 kg (28-36%) were obtained. The three variance–covariance matrix structures evaluated did not generate clear differences in relation to the RMSE, bias and Error Index indicators. The structure of the variance–covariance matrix that incorporated explicitly in the system equations allowed modelling of the relationship between biomass components.

... When estimating a system of additive biomass equations, considering the inherent correlation among the biomass components results in greater statistical efficiency [16,17]. Some studies have shown that the two methods, TSEM and NSUR, have the same accuracy in predicting each component and total biomass, but the NSUR method is superior to the TSEM method in that it is easy to implement through the nlsystemit process in R version 3.5.1 [13,14,18,19]. Therefore, this study used the NSUR method to estimate the model parameters of stump and coarse root biomass and those for the biomass of each component and ensured the additivity of stump and coarse root biomass and component biomasses. ...

The stump and coarse root biomass remaining after tree harvesting are often overlooked by researchers, which may lead to underestimation of their role in carbon cycling, so we constructed two sets of additive models for larch (Larix olgensis Henry) plantations in Northeast China. Due to the absence of tree diameter at breast height data after harvesting, only the sole predictor variable stump disc diameter could be used to predict stump and coarse root biomass, and the results showed that stump disc diameter predicted stump biomass with higher accuracy than coarse root biomass predictions. In addition, to investigate the effect of the site class of complex stands on the predictive capability of the model, the generic model in this study was employed with all site class data and a specific model was developed and employed with all the site class data. We found that the generic model had different degrees of error compared to the predicted results for each site class, overestimating the total biomass by 15% and underestimating it by 10%, especially for site class IV. In conclusion, to obtain a biomass prediction model with reliable results, the impact of more complex site class effects should be considered.

... Affleck & Diéguez-Aranda (2016) showed that aggregated non-linear equations could be fit using the Gaussian maximum likelihood (ML), and that this approach can be useful for integrating logged tree data collected under different protocols. Dong et al. (2015) implemented a disaggregation method, which also ensures additivity by predicting the proportion of each component. In the same way way, Zhao et al. (2016) tested three proportion modelling approaches, highlighting the Dirichlet method as superior to the fractional multinomial logit and logratio approaches. ...

The biomass content and carbon captured by forest plantations is of interest, for example in the context of climate change and carbon budgets.The main objective of our study was to develop functions to estimate the total biomass and its components (stem, branches, bark and leaves) of Pinus radiata D. Don trees in Chile. The methodology proposed for the model fitting uses the maximum likelihood method in a multivariate equation system fitting simultaneously. The fit strategy incorporates additivity restrictions in the estimation functions and in the variance functions to incorporate the heteroskedasticity of biomass, and three structures of the variance-covariance matrix were evaluated to assess the dependence of the different components of tree biomass. Non-linear biomass functions that used the variable D 2 H performed best according to several indicators of goodness-of-fit (log-likelihood, Akaike Information Criterion and Bayesian Information Criterion) and estimation precision (root mean square error (RMSE), Bias and EI). The simple structure of both biomass and variance estimation functions was β 1 (D 2 H) β 2 , and in the modelling system for total tree biomass RMSE between 54.1-54.4 kg (28-36%) were obtained. The three variance-covariance matrix structures evaluated did not generate clear differences in relation to the RMSE, bias and Error Index indicators. The structure of the variance-covariance matrix that incorporated explicitly in the system equations allowed modelling of the relationship between biomass components.

... Theoretically, if some tree compartments were measured from the same individuals, it is reasonable to expect that the summation of each compartment's estimate not to surpass the prediction of the total tree. Thus, the additivity property in developing individual tree biomass models is fundamental to be included and has been addressed by several researchers (Affleck and Di eguez-Aranda, 2016;Bi et al., 2004;Dong et al., 2015a;Men endez-Migu elez et al., 2013;Zhao et al., 2019). Various alternatives of model specifications and coefficient estimations are available in the literature to deliver the compatibility property for the biomass equations system. ...

The area of larch (Larix gmelinii) plantations in northeast China has increased following the massive afforestation program implemented by China's central government to provide biomass for global carbon sink and the development of industrial sectors. Hence, a better comprehension of equations used to characterize the estimation of regional biomass is much needed. A total of 301 sample trees of both planted and natural larch from northeast China were used to construct the additive systems of general and dummy variable models using weighted nonlinear seemingly unrelated regression (NSUR) to investigate the effect of forest origin on tree compartments and total biomass equations. Three different combinations of predictors were used in the two additive systems, and each biomass equation has its own specific weighting function to achieve the homoscedasticity in model residuals. Biomass partitioning, root-to-shoot ratio (RSR), and carbon concentration between the two origins were also analyzed. The results indicated that (1) the additive system of origin-based dummy variable model clearly outperformed the general additive model according to their respective predictors' combinations and the difference between them were found to be significant; (2) based on the jackknife models' validation, the two additive systems delivered good biomass predictions, of which the best models' R² were 0.939, 0.987, 0.910, 0.878, and 0.989 for root, stem, branch, foliage, and total biomass equations, respectively; (3) biomass allocations were varied between the two origins, and the RSR was higher in natural forests (0.32 ± 0.16) compared to plantations (0.25 ± 0.06) both averagely (±SD) and entirely across all diameter classes; (4) the mean carbon concentrations of root, stem, branch, and foliage observed in naturally regenerated trees were 1.81%, 2.90%, 1.23%, and 2.30% higher than those in plantations, respectively. The newly developed additive systems of origin-based equations can provide a more precise estimation for single-tree biomass of Larix gmelinii in China's National Forest Inventory.

... The coefficients of the regression equations of all two steps are evaluated simultaneously, that ensures the additivity of the biomass of all components (Dong et al. 2015). Since the regression coefficients in the designed model have been calculated on the log-transformed data, a corresponding correction has been introduced in the equations to eliminate the displacements caused by ...

We used our database of tree biomass with a number of 433 sample trees of Larix from different ecoregions of Eurasia, involving 61 trees from Mongolia for developing an additive model of biomass tree components. Our approach solved the combined problem of additivity and regionality of the model. Our additive model of tree aboveground biomass was harmonized in two ways: first, it eliminated the internal contradictions of the component and of the total biomass equations, secondly, it took into account regional (and correspondingly species-specific) differences of trees in its component structure. A significant excess of larch biomass in the forest-tundra is found that may be explained by permafrost conditions, by tree growth in low-yielding stands with a high basic density of stem wood and relatively high developed tree crown in open stands. The aboveground biomass of larch trees in Mongolia does not stand out against the background of the most ecoregions of Eurasia. Based on our results, we conclude that the growing conditions of larch in Mongolia are not as tough as it was suggested earlier by other scientists. Biomass relations between regions may be explained by unknown and unaccounted factors and errors of measurements in all their phases (assessment of age, diameter, height of a tree, the selection of supposedly representative samples of component biomass, their drying, weighing, etc.). The question what explains the regional differences in the structure of biomass of trees with the same linear dimensions of their stems, remains open. Undoubtedly, the differences in tree age here play an important role. Also, important factor is the variation in the morphological structure of stands, which, in turn, is determined by both climatic and edaphic factors. The obtained models allow the determination of larch forest biomass in different ecoregions of Eurasia with the help of height and diameter data.

... There have been various forest biomass modeling studies carried out. The widely used model fitting methods are ordinary least square (OLS) regression [14,15], nonlinear seemingly unrelated regression (NSUR) [16,17], dummy variable approach [17][18][19], error-in-variable (EIV) approach [20,21], linear and nonlinear simultaneous estimation [17,22,23], and mixed-effect modeling [4,5,19,24,25]. Out of the studies, only a few have included climate variables into the models [3][4][5]. ...

Accurate estimate of tree biomass is essential for forest management. In recent years, several climate-sensitive allometric biomass models with diameter at breast height [Formula: see text] as a predictor have been proposed for various tree species and climate zones to estimate tree aboveground biomass (AGB). But the allometric models only account for the potential effects of climate on tree biomass and do not simultaneously explain the influence of climate on [Formula: see text] growth. In this study, based on the AGB data from 256 destructively sampled trees of three larch species randomly distributed across the five secondary climate zones in northeastern and northern China, we first developed a climate-sensitive AGB base model and a climate-sensitive [Formula: see text] growth base model using a nonlinear least square regression separately. A compatible simultaneous model system was then developed with the climate-sensitive AGB and [Formula: see text] growth models using a nonlinear seemingly unrelated regression. The potential effects of several temperature and precipitation variables on AGB and [Formula: see text] growth were evaluated. The fitting results of climatic sensitive base models were compared against those of their compatible simultaneous model system. It was found that a decreased isothermality ([mean of monthly (maximum temperature-minimum temperature)]/(Maximum temperature of the warmest month-Minimum temperature of the coldest month)) and total growing season precipitation, and increased annual precipitation significantly increased the values of AGB; an increase of temperature seasonality (a standard deviation of the mean monthly temperature) and precipitation seasonality (a standard deviation of the mean monthly precipitation) could lead to the increase of [Formula: see text]. The differences of the model fitting results between the compatible simultaneous system with the consideration of climate effects on both AGB and [Formula: see text] growth and its corresponding climate-sensitive AGB and [Formula: see text] growth base models were very small and insignificant [Formula: see text]. Compared to the base models, the inherent correlation of AGB with [Formula: see text] was taken into account effectively by the proposed compatible model system developed with the climate-sensitive AGB and [Formula: see text] growth models. In addition, the compatible properties of the estimated AGB and [Formula: see text] were also addressed substantially in the proposed model system.

... В качестве гармонизации моделей в терминах аддитивности нами использован алгоритм, альтернативный ранее часто применяемому (Parresol, 2001;Návar et al., 2004), а именно трехшаговая схема, разработанная в Китае (Tang et al., 2000;Dong et al., 2015) и показанная ранее (Усольцев и др., 2018). ...

TREE BIOMASS OF TWO-NEEDLED PINES IN EURASIA:
ADDITIVE MODELS IN CLIMATIC GRADIENTS
V. А. Usoltsev1, 2, I. S. Tsepordey1, V. P. Chasovskikh2
1 Botanical Garden, Russian Academy of Sciences, Ural Branch
8 Marta str., 202а, Yekaterinburg, 620144 Russian Federation
2 Ural State Forest Engineering University
Sibirskiy trakt, 37, Yekaterinburg, 620100 Russian Federation
E-mail: Usoltsev50@mail.ru; ivan.tsepordey@yandex.ru; u2007u@yandex.ru
The analysis of studies on the relations between tree and forest stand biomass and climatic conditions revealed a wide
variety of independent variables and their combinations involved as predictors. There are significant contradictions
and uncertainties found in modeling of dependences of tree and stand biomass upon temperature and precipitation
using both empirical and process-based models. The database on biomass of 2100 single-trees of two-needled
pines (subgenus Pinus L.) of Eurasia compiled by the authors, enables to design for the first time a trans-Eurasian
harmonized model on the tree biomass structure and to estimate quantitatively the influence of January temperatures
and annual precipitation on tree biomass. The harmonization is achieved with additivity of biomass component
composition, which means that the total of biomass components (stems, branches, foliage, roots) derived from
component equations is equal to the result obtained using a common biomass equation. It is stated, that in cold
climatic zones any increase in precipitation leads to a corresponding decrease in the biomass values, but in warm
zones – to its increase. In wet areas, the rise in temperature causes an increase of biomass values, but in arid areas
– their reductions. Geometric view of this model represented by a «propeller-shaped» surface is consistent with the
results formerly revealed by the other authors in Russia on local and regional levels. The proposed transcontinental
model of additive structure of tree biomass makes it possible to predict a change of biomass structure in relation to
simultaneous increase or decrease of January temperature and annual precipitation. The development of such models
for basic forest-forming species grown in Eurasia enables one to forecast any changes in the biological productivity
of forest cover of Eurasia in relation to climate change.
Keywords: tree biomass, additive biomass model, annual temperature, precipitation.
How to cite: Usoltsev V. А., Tsepordey I. S., Chasovskikh V. P. Tree biomass of two-needled pines in Eurasia: additive
models in climatic gradients // Sibirskij Lesnoj Zurnal (Sib. J. For. Sci.). 2019. N. 1: 44–56 (in Russian with English
abstract).
DOI: 10.15372/SJFS20190104

... A number of tree biomass models were reported with most studies focused on model construction. For instance, studies have addressed the types of variables that should be included in the equation (diameter at breast height (DBH), tree height, crown radius, or some combination), investigated whether the equation is consistent with the additivity principle [16], and compared models with different regression methods [17,18]. These aspects are important for the development of accurate biomass equations but neglect the influences of stand characteristics, such as stand age, on equation fitting. ...

We studied the effects of stand age on allocation and equation fitting of aboveground and below-ground biomass in four Quercus acutissima stands (14, 31, 46, and 63 years old) in the Central Loess Plateau of China. The stem wood, stem bark, branch, foliage, and belowground biomass of each of the 20 destructive harvesting trees were quantified. The mean total biomass of each tree was 28.8, 106.8, 380.6, and 603.4 kg/tree in the 14-, 31-, 46-, and 63-year-old stands, respectively. Aboveground biomass accounted for 72.22%, 73.06%, 75.98%, and 80.26% of the total tree biomass in the 14-, 31-, 46-, and 63-year-old stands, respectively, and stem wood was the major component of tree biomass. The proportion of stem (with bark) biomass to total tree biomass increased with stand age while the proportions of branch, foliage, and belowground biomass to total tree biomass decreased with stand age. The ratio of belowground biomass to aboveground biomass decreased from 0.39 in the 14-year-old stand to 0.37, 0.31, and 0.24 in the 31-, 46-, and 63-year-old stands, respectively. Age-specific biomass equations in each stand were developed for stem wood, stem bark, aboveground, and total tree. The inclusion of tree height as a second variable improved the total tree biomass equation fitting for middle-aged (31-year-old and 46-year-old) stands but not young (14 years old) and mature (63 years old) stands. Moreover, biomass conversion and expansion factors (BCEFs) varied with stand age, showing a decreasing trend with increasing stand age. These results indicate that stand age alters the biomass allocation of Q. acutissima and results in age-specific allometric biomass equations and BCEFs. Therefore, to obtain accurate estimates of Q. acutissima forest biomass and carbon stocks, age-specific changes need to be considered.

... Анализ фитомассы деревьев и древостоев выполнен на основе аллометрических аддитивных регрессионных уравнений, структурированных согласно трехшаговой аддитивной системе (Tang et al., 2000;Dong et al., 2015) (см. предыдущие статьи настоящего выпуска). ...

V.А. Usoltsev, I.S. Tsepordey, V.P. Chasovskikh, A.A. Osmirko.
ADDITIVE REGIONAL MODELS OF TREE AND STAND BIOMASS FOR
EURASIA. MESSAGE 4: GENUS Quercus sp.
Key words: genus Quercus sp., equations additivity, biosphere role of forests, bio-mass of trees and forests, allometric models, sample plots, biological productivity, transcon-tinental tables of biomass.
When using the unique in terms of the volumes of databases on the levels of a tree and stand of the genus Quercus sp., the trans-Eurasian additive allometric models of biomass of trees and forests for Eurasia are developed for the first time, and thereby the combined problem of model additivity and generality is solved. The additive model of tree biomass of Quer-cus is harmonized in two ways: it eliminated the internal contradictions of the component and the total biomass equations, and in addition, it takes into account regional differences of trees of equal sizes not only on total, aboveground and underground biomass, but also on its component structure, i.e. it reflects the regional peculiarities of the component structure of tree biomass. The additive model of forest biomass of Quercus is harmonized in two levels too, one of which provides the principle of additivity of biomass components, and the second one is associated with the introduction of dummy independent variables localizing model for ecoregions
of Eurasia.

In the context of current climate change, it is important to know the patterns characterising the response of forest trees to the dynamics of air temperature and precipitation. In this study, the first attempt to model changes of additive component composition of genera Larix spp. and Quercus spp. aboveground biomass according to Eurasian gradients of January's mean temperature and annual mean precipitation is made, taking into account regional particularities of tree age and morphology structure. In the process of modelling, the database of single-tree biomass for forest-forming species in Eurasia is used. According to our results, the factors limiting the biomass of trees differ not only between the two tree genera but also between different components of biomass within the genus. In larches, the reaction of the biomass of all components to an increase in precipitation in cold zones is directly opposite in comparison with oaks, i.e. it decreases as precipitation increases. But in warm areas, the reactions of the two genera to increased precipitation coincide, i.e. precipitation does not affect the biomass of all components, both in larches and oaks. In wet areas, larch biomass components react to temperature increases in the opposite way, i.e. the aboveground and stem biomass increases, but the biomass of foliage and branches decreases. In dry areas, the reaction to the temperature of all larch and oak biomass components is unambiguous and opposite, i.e. there is a decrease in the larch biomass of all components as temperatures rise, and in oak biomass vice versa. This situation is discussed in terms of limiting factors.

Allometric equations have been used as a main tool to quantify forest above-ground biomass. Therefore, the development of species-specific allometric equations is essential to accurately estimate forest biomass. The present study aims at developing species-specific equations for above-ground biomass (AGB) estimation for Pinus sylvestris L. in northern Mongolia. A total of 35 sample trees were harvested from natural Scots pine forests in the northeastern Khangai and western Khentii forest regions. Evaluating the statistical relationships of AGB against predictor variables, three allometric equations were formulated. Stem diameter at breast height (D) and height (H) were measured, and the biomass of stem, branches and needles were weighed, separately. AGB was regressed against stem D and H individually, and in combination. The best-fitting equations with a higher coefficient of determination and lower residual standard error were selected using model performance statistics. The most well performing biomass model was the logarithmic equation lnŶ = lna + b x lnD + c x lnH for the estimation of the total AGB, the stem, branch and needle masses. The biomass model is the important tool for accurate estimation of Scots pine forest biomass and carbon stocks in Mongolia.

Climate change, especially modified courses of temperature and precipitation, has a significant impact on forest functioning and productivity. Moreover, some alterations in tree biomass allocation (e.g. root to shoot ratio, foliage to wood parts) might be expected in these changing ecological conditions. Therefore, we attempted to model fir stand biomass (t ha −1) along the trans-Eurasian hydrothermal gradients using the data from 272 forest stands. The model outputs suggested that all biomass components, except for the crown mass, change in a common pattern, but in different ratios. Specifically, in the range of mean January temperature and precipitation of −30°C to +10°C and 300 to 900 mm, fir stand biomass increases with both increasing temperature and precipitation. Under an assumed increase of January temperature by 1°C, biomass of roots and of all components of the aboveground biomass of fir stands increased (under the assumption that the precipitation level did not change). Similarly, an assumed increase in precipitation by 100 mm resulted in the increased biomass of roots and of all aboveground components. We conclude that fir seems to be a perspective taxon from the point of its productive properties in the ongoing process of climate change.

Usoltsev V.А., Tsepordey I.S., Osmirko А.А., Kovyazin V.F., Chasovskikh V.P.,
Аzarenok V.А., Аzarenok М.V., Kuz’min N.I. Modeling of the additive biomass
structure of Pinus L. stands in climatic gradients of Eurasia. Izvestia Sankt-
Peterburgskoj Lesotehniceskoj Akademii, 2018, is. 225, pp. 28–46 (in Russian with
English summary). DOI: 10.21266/2079-4304.2018.225.28-46
Forest biomass is a key ecosystem part and an important component of the global
carbon cycle. Modelling of biomass, sensitive to climate change, is fulfiled up-to-date
at levels as forest stands and sample trees. However, all current studies of this matter
are carried out within limited ecoregions. The database on forest biomass of the
subgenus Pinus L. in Eurasia in a number of 2460 sample plots compiled by the
authors is the basis for revealing transcontinental regularities. The first attempt is made
to develop a biomass structure model harmonized by means of additive component
composition algorithm describing biomass change in trans-Eurasian hydrothermal
gradients, namely, mean annual precipitation and mean January air temperature.
Additivity of biomass component composition means that the total of biomass
components (stems, branches, foliage, roots) derived from component equations is
equal to the result obtained using the common biomass equation. It is stated that in
cold climatic zones any increase in precipitation leads to corresponding decrease in the
biomass values, but in warm zones – to its increase. In wet areas, the rise in
temperature causes an increase of biomass values, but in arid areas – their reductions.
Geometric view of this model represented by a «propeller-shaped» surface is
consistent with the results, formerly revealed by the other authors in Russia on local
and regional levels. The proposed transcontinental model of additive structure of forest
biomass gives a possibility to predict the change of biomass structure in relation to
simultaneous increase or decrease of January temperature and annual precipitation.
The development of such models for basic forest-forming species grown in Eurasia
enables to forecast any changes in the biological productivity of forest cover of
Eurasia in relation to climate change.
Ke ywo r d s : two-needled pines of Eurasia, forest biomass, additive biomass
model, mean January air temperature, mean annual precipitation.

Assessment of aboveground biomass stocks in the coniferous forests of the inland northwest USA is important for timber, bioenergy, and carbon inventories, as well as for wildfire risk determination. In this study, individual tree biomass equation systems are developed for 7 regionally important conifer species using data from 470 felled trees sampled across 84 stands and spanning a range of diameters at breast height (1.37 m; dbh) running from 5 cm to 105 cm. The equation systems permit estimation of crown biomass components (i.e., foliage, dead branches, and live branches by size class) and stem components (abovestump stemwood and stembark), as well as compatible estimates of (sub)totals. The systems draw on commonly collected inventory variables including dbh, tree height, and live crown length. All biomass components scaled approximately linearly with dbh on the logarithmic scale, but equation systems drawing on both dbh and height provided more accurate estimates for all species; systems drawing additionally on live crown length provided more accurate estimates still for all species but one. In line with previous work, incorporation of live crown length improved live crown component equations most, but also improved stem component equations for two species. Across species and systems, stem components and subtotals were most accurately estimated (mean absolute errors ∼10%) while dead branch biomass estimation proved least tractable (mean absolute errors >50%). Overall, the reported biomass equation systems draw on the largest felled tree samples collected from the region, and provide the most comprehensive basis developed to date for regional forest biomass assessments over the inland northwest.

Based on the data of tree biomass for Korean pine (Pinus koraiensis) plantations, the optimal model of tree biomass for Korean pine had been selected by using the method of non-linear simultaneous equations with measure error. Based on the total biomass and stem biomass as restrictions, the compatible tree biomass equations between the total biomass and each components of tree (stem, branches, foliages, and roots) were developed. Meanwhile, the weighted regression method was used to eliminate heteroscedasticity. The results showed that the coefficients of determination (R2) and the modeling efficiency (EF) were 0.80-0.99 and 0.82-0.98 respectively for the optimal base models and compatible biomass models. The most of the prediction accuracy was larger than 82% for tree biomass models of total and each component, where the effects of prediction accuracy were better for stem biomass and the worst for foliage. On the whole, the compatible tree biomass models developed on the total biomass as restriction were little better than stem biomass as restriction. There were no significant differences between statistics of goodness-of-fit and validation for the optimal base models and compatible models of tree biomass. The conclusion is that the two kinds of compatible biomass equations developed in the paper can effectively estimate the tree biomass for Korean pine plantations.

There are increasing requirements for forest management agencies to estimate not only wood volume for timber production, but also biomass accumulation and carbon sequestration rates of their forests for environmental purposes. The common methods of biomass estimation have been to develop allometric equations to predict the biomass of individual trees from diameter or both diameter and height. The biomass equations are usually based on small samples, especially for large trees, due to the time-consuming nature of destructive biomass sampling. Consequently, the predictive performance of biomass equations has been seldom evaluated. Most forest management agencies do, however, have reliable volume estimates that are based on large samples. Converting stem or stand volume estimates, that are already available in forest inventory and growth and yield systems, to biomass seems to be the most convenient and reliable way to estimate forest biomass over a large management area. Adopting this approach, we developed a system of additive equations for converting stem volume into four biomass components (stemwood, bark, branches, foliage) and total above-ground tree biomass using data for two Australian tree species as an example. To correct log transformation bias and at the same time maintain additivity among the component equations, we proposed a regression-based bias correction factor and simultaneously estimated the biomass correction factors for the component equations. The distributional properties of the error in stem volume prediction were incorporated in stochastic simulations of the system of equations to determine the confidence bands of the biomass conversions. Such results would provide a clear indication whether the required precision of biomass prediction is met for a particular objective of investigation and, if not, where improvements can be made.

Establishing biomass models is a major way of biomass estimate. To establish biomass model compatible with volume table is the base of realizing the combination of forest biomass survey and growing stock inventory. Based on this point, the progressive variable selection method is used to establish biomass model compatible with volume table for Larix olgensis and Tilia amurensis, and weighted least squares method is used to estimate parameters for clearing up phenomenon of non-homogeneous variance. In the meantime, the paper presented five indices for evaluation of models, they are coefficient of variation for parameters C(vi), total relative error E(r), average relative error E(m) average absolute value of relative error E(a) and prediction precision P. All the results show that biomass models established in this paper not only realize compatibility with volume table, but also get remarkable improvement in estimate effect and prediction precision comparing with previous model CAR. According to the results, some reference models are given.

Aboveground biomass was studied in Castanea sativa Mill. coppice stands in north-west Spain, and biomass equations were fitted at three levels (individual tree, stool and stand). Four systems of biomass estimation were developed. In two of the systems, the following individual tree variables were taken into account: standing tree variables and stump dimension variables. In the other two systems, biomass was estimated at stool and stand level, respectively.In order to represent the existing range of ages, stand densities and sites in the study area, samples of 120 trees (for the individual tree level), 45 stools (for the stool level) and 70 plots (for the stand level) were chosen for study. The trees were felled and destructively sampled to separate biomass into the following components: wood, bark, thick branches, medium branches, thin branches and leaves. Several equations for quantifying the biomass of the different biomass components were evaluated. Heterocedasticity was corrected for by weighted fitting. To guarantee the additivity of the different biomass components, the equations were fitted simultaneously by nonlinear seemingly unrelated regressions (NSURs).The different biomass levels considered accounted for between 60% and 90% of the total variability, depending on the level and component evaluated. Most of the equations developed in this study were evaluated with an independent dataset, which confirmed the good performance of the biomass equations for prediction purposes.

The paper presents a comprehensive review of the biomass equations for 65 North American tree species. All equations are of the form M = aDb, where M is the oven-dry weight of the biomass component of a tree (kg), D is diameter at breast height (DBH) (cm), and a and b are parameters. Equations for the following tree components were included in the review: total aboveground biomass, stem wood, stem bark, total stem (wood and bark), foliage, and branches (wood and bark). A total of 803 equations are presented with the range of DBH values of the sample, sample size, coefficient of determination R2, standard error of the estimate, fitting method used to estimate the parameters a and b, correction factor for a bias introduced by logarithmic transformation of the data, site index and geographic location of the sampled stand(s), and a reference to the paper in which the equation (or the data) was published. The review is a unique source of equations that can be used to estimate tree biomass and/or to study the variation of biomass components for a tree species.

Multiple-stemmed tree species are often used in agricultural settings, playing a significant role in natural resource conservation and carbon sequestration. Biomass estimation, whether for modeling growth under different climate scenarios, accounting for carbon sequestered, or inclusion in natural resource inventories, requires equations that can accurately describe biomass in these species. Russian-olive (Elaeagnus angustifolia) is a common tree species used in Great Plains shelterbelts and has a growth form typical to open-grown, multiple-stemmed tree species. Using shelterbelt-grown Russian-olive, we present a procedure of choosing predictors, formulating models, and determining equations by optimizing the accuracy in above-ground woody biomass estimates associated with labor costs for open-grown, multiple-stemmed tree species. Trunk (a primary stem) diameter at breast height and/or tree height were satisfactory for trunk biomass prediction but insufficient for determining branch (secondary stems and limbs) biomass, a major component of biomass in these trees. Incorporating the diameters of the three largest stems into the branch biomass equations improved the prediction satisfactorily. Two sets of equations, each of which includes two equations for trunk and branches, respectively, are presented. One set has the cost-saving-preferred (CSP) equations having lower precision but only requiring easily measured DBH variables of trunk and stems. The other set has the precision-preferred (PP) equations that have better precision but at the added cost required for taking an additional measurement of height and the inconvenient measurements of stem diameters at branch bark ridge. Both sets of equations were used to estimate the biomass of the same representative shelterbelts. The results indicated that the PP equations consistently gave better precision for trunk, branches, and whole tree than the CSP equations, but reduced the relative error in whole-tree biomass estimates by only 0.8–1.2%. Ultimately, the decision to use the CSP or the PP equations will depend on the desired precision level and/or available budget. The procedure we have presented, along with the chosen predictors and formulated models, provides a reference for estimating above-ground woody biomass of other open-grown, multiple-stemmed tree species in agricultural settings.

The estimation of aboveground biomass density (organic dry mass per unit area) is required for balancing Canadian national forest carbon budgets. Tree biomass equations are the basic tool for converting inventory plot data into biomass density. New sets of national tree biomass equations have therefore been produced from archival biomass data collected at the beginning of the 1980s through the ENergy from the FORest research program (ENFOR) of the Canadian Forest Service. Since the sampling plan was not standardized among provinces and territories, data had to be harmonized before any biomass equation could be considered at the national level. Two features characterize the new equations: estimated biomass of the compartments (foliage, branch, wood, and bark) are constrained to equal the total biomass, and dependence among error terms for the considered compartments of the same tree is taken into account in the estimates of both the model parameters and the variance prediction. The estimation method known to economists as oseemingly unrelated regressiono allowed the inclusion of dependencies among the error terms of the considered biomass compartments. Sets of equations based on diameter at breast height (dbh) and on dbh and height have been produced for 33 species, groups of hardwood and softwood, and for all species combined. Biomass predicted by the new equations was compared with that estimated from provincial equations to evaluate the loss of accuracy when scaling up from the regional to the national scale. Bias and error of prediction from the set of national equations based on dbh and height were generally more similar to those from provincial equations than to those of predictions from the set of equations based on dbh alone.

Detailed ground-based quantifications of total carbon stocks in tropical forests are few despite their importance in science and ecosystem management. Carbon stocks in live aboveground and belowground biomass, necromass, and soils were measured in a heterogeneous landscape composed of secondary and primary forest. A total of 110 permanent plots were used to estimate the size of these carbon pools. Local biomass equations were developed and used to estimate aboveground biomass and coarse root biomass for each plot. Herbaceous vegetation, fine roots, coarse and fine litter, and soil carbon to 4m depth were measured in subplots. In primary forests, mean total carbon stocks (TCS) were estimated as 383.7±55.5MgCha−1 (±S.E.). Of this amount, soil organic carbon to 4m depth represented 59%, total aboveground biomass 29%, total belowground biomass 10%, and necromass 2%. In secondary forests, TCS was 228.2±13.1MgCha−1, and soil organic carbon to 4m depth accounted for 84% of this amount. Total aboveground biomass represented only 9%, total belowground biomass 5%, and total necromass 1% of TCS in secondary forests. Monte Carlo methods were used to assess the uncertainty of the biomass measurements and spatial variation. Of the total uncertainty of the estimates of TCS, the variation associated with the spatial variation of C pools between plots was higher than measurement errors within plots. From this study it is concluded that estimates of aboveground biomass largely underestimate total carbon stocks in forest ecosystems. Additionally, it is suggested that heterogeneous landscapes impose additional challenges for their study such as sampling intensity.

The temperate forest in northeastern China accounts for more than one-third of Chinese forest resources (both area and stocking volume), and plays a key role in the national and global carbon budgets and climatic system. However, few allometric equations exist for accurately estimating biomass and carbon budgets of the forest. In this study, allometric equations were developed relating component biomass to diameter at breast height (DBH) and tree height (H) to DBH for 10 co-occurring tree species in the Chinese temperate mixed forest. The 10 species were Korean pine (Pinus koraiensis Sieb. et Zucc.), Dahurian larch (Larix gmelinii Rupr.), Mongolian oak (Quercus mongolica Fisch.), white birch (Betula platyphylla Suk.), Amur cork-tree (Phellodendron amurense Rupr.), Manchurian walnut (Juglans mandshurica Maxim.), Manchurian ash (Fraxinus mandshurica Rupr.), aspen (Populous davidiana Dode), maple (Acer mono Maxim.), and Amur linden (Tilia amurensis Rupr.). The biomass components included stem, current-year branch, older branch, current-year foliage, older foliage, stump, and coarse root (diameter >= 5 mm). Harvested tree DBH ranged from 2.4 to 57.1 cm. Generalized biomass allometric equations that ignored tree species, based on DBH and fitted on a log-log scale, explained more than 90% of variability in woody component biomass, but the species effect was significant (alpha = 0.05). Including tree height as the second independent variable in the allometric equations improved the accuracy of biomass estimates, especially for foliage biomass. The relative differences in biomass estimated from the generalized, DBH-only, and DBH-H combined equations varied from -50.6% to 43.9% depending upon model form, species, and biomass component. Foliage biomass was more variable than other component biomass both across and within tree species. Some potential sources of error in biomass estimation were also discussed.

In tree biomass estimations, it is important to consider the property of additivity, i.e., the total tree biomass should equal the sum of the components. This work presents functions that allow estimation of the stem and crown dry weight components of Pyrenean oak (Quercus pyrenaica Willd.) trees. A procedure that considers additivity of tree biomass components is presented, and applied to a particular case. The application of a simultaneous equations system estimation procedure that used parameter restrictions and considered residual contemporaneous correlations allowed more efficient estimates and consistent prediction intervals.

The Republic of Korea, henceforth referred to as Korea, has successfully implemented intensive programs of reforestation and forest management over the last 30 years to restore its once-rich forests. This nationwide effort has resulted in a massive accumulation of less than 30-year-old tree biomass, which now accounts for about 72% of the total forest biomass in Korea. Here we use a forest tree inventory data set for Korea to calculate the effectiveness of these planted trees in absorbing excess carbon dioxide from the atmosphere during the period 1954–2000. The forest carbon density in Korea has increased from 5–7 megagrams of carbon per hectare (Mg C ha−1, Mg = 106 grams) in the period 1955–1973 to more than 30 Mg C ha−1 in the late 1990s. The calculated carbon uptake has increased from a mean rate of 0.001 petagrams of carbon per year (Pg C yr−1, Pg = 1015 grams) in the period 1955–1973 to as high as 0.012 Pg C yr−1 in recent years, largely due to the 30-year implementation of reforestation and forest management projects. The contemporary rate of carbon uptake by the total Korean tree biomass is approximately one-half of the 1994–1998 mean rate of carbon uptake by the total Chinese forest biomass of 0.026 Pg C yr−1 [Fang et al., 2001]; the Chinese forest biomass has recently been found to be a significant carbon sink in northern temperate regions. The observed uptake rate for Korea is remarkably high, considering the fact that the total area of Korean forests is approximately 16 times smaller than that of Chinese forests. Our results show that long-term rates of carbon sequestration by nationwide forests can be increased substantially through reforestation and forest management.

A procedure is presented for estimating the coefficients of allometric models for predicting tree component biomass. The resulting equations force the sum of the component estimates to be equal to the estimate of total biomass. An illustration of the procedure is given using published biomass data and the relationship of this procedure to previously published procedures is discussed.

The objectives of this study were to evaluate visual and digital estimates of percent cover as source data and to develop cover-based allometric models for the prediction of aboveground biomass of Canada yew (Taxus canadensis Marsh.). Cover was determined from visual assessment and digital images captured over 25 plots (1 m2) at a model training site near Timmins, Ontario. Linear and power functions were fit to the cover–biomass data to develop models of foliage, stem, and total aboveground biomass. Both sources of cover data produced models that explained between 70% and 85% of the variance in the training data, with root mean square error estimates ranging from 27 g·m–2 (foliage) to 85 g·m–2 (total). Models based on visual cover data performed consistently better and were tested on independent data. Stem and total biomass were underestimated in the model testing data set; however, prediction statistics indicated that the linear and power forms of foliage biomass models were validated by the testing data. Final models of foliage biomass were developed from the entire data set, with mean absolute errors of 18.3 and 18.7 g·m–2 for the linear and power forms, respectively. Additional variables (e.g., plant height, age) may be required to provide general predictions of the woody biomass of Canada yew.