1. Introduction
Plantation forests typically have high growth rates and thereby absorb large amounts of carbon dioxide and help mitigate global climate change. Larch (
Larix olgensis Henry) is an important tree species for afforestation and acquiring commercial timber in Northeast China. This species is the fourth most important in China, and it is used to establish fast growing and high-yielding plantation forests, covering a wide geographical range from the northeast to northern subalpine areas of China [
1]. 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]. In addition, tree height (
) can also be used as a predictor. 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]. 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-in-variable models (TSEM), and nonlinear seemingly unrelated regression (NSUR) [
6,
7,
21,
22]. Nonadditive models separately fit the aboveground and component biomass data, and they ignore the inherent correlation among the aboveground and component biomass of the same sample tree. Thus, the biomass models established fall short of statistical efficiency in parameter estimation and fail to consider the additivity among the aboveground and component biomass [
7,
23]. For nonadditive models, the base model (BM) fitted by nonlinear ordinary least squares (OLS) is believed to be the most widely used technique on parameter estimation, and it is appropriate for datasets with general structures containing random and independent observations [
7,
24,
25,
26]. For hierarchically structured data (e.g., trees within plots), mixed effects models (MEM) are likely to be a better choice; they are characterized by fixed parameters corresponding to population and random parameters corresponding to each subject. Some studies have applied MEM to establish component biomass models [
27,
28,
29].
Thus far, two model structures have been used to achieve additivity of the aboveground and component biomass models. Parresol [
30] proposed an aggregation model system (referred to as AMS1), in which a nonlinear model is specified for each component, and these component models for stems, branches, and foliage are aggregated to the aboveground biomass. Other researchers proposed that the aboveground biomass model may not occur in Parresol’s model system, i.e., that Parresol’s model system may only fit the component biomass models, rather than the aboveground and component biomass models (referred to as AMS2) [
23,
31]. The NSUR methods are often used to simultaneously compute the aboveground and component biomass for the aggregation model system [
6,
7,
23,
24,
25,
26,
27]. The aggregation model system has become a standard for developing new biomass models because it can easily ensure additivity among the aboveground and component biomass predictions [
16,
20,
23,
31]. Tang et al. [
32] proposed a disaggregation model system (referred to as DMS1), in which the aboveground biomass model is first developed, and then, the estimated aboveground biomass is disaggregated into different tree components (e.g., stems, branches, and foliage) based on their proportions in the aboveground biomass. Furthermore, some researchers have developed an extended disaggregation model system (referred to as DMS2), in which the aboveground and component biomass models can be fitted simultaneously [
21,
22]. The TSEM and NSUR methods are often used to estimate the parameters of the aboveground and component biomass models jointly, and to guarantee the additivity of the aboveground and component biomass [
7,
21,
22]. Several studies have shown that the prediction accuracies of the two fitting methods, TSEM and NSUR, were virtually identical for each component and aboveground biomass. However, the advantage of the NSUR method over the TSEM method lies in the fact that it can be readily implemented by PROC MODEL procedure in SAS version 9.3 and nlsystemfit procedure in R version 3.5.1 [
7,
21,
22,
33,
34].
We aimed to explore the difference in biomass predictions of the BM, MEM, AMS1, AMS2, DMS1, and DMS2. Therefore, the objectives of our study were: (1) to construct two nonadditive biomass models (i.e., BM and MEM) of each component based on only or both and ; (2) to construct four additive biomass models (i.e., AMS1, AMS2, DMS1, and DMS2) based on only or both and with weighted NSUR; (3) to verify the performance of the biomass models with jackknife resampling; and (4) to compare the accuracy of model fitting and performance of the different nonadditive and additive biomass models.
4. Discussions
For the selection of biomass model variables,
is an indispensable predictor of biomass models. In practice, tree biomass models constructed with only
require basic forest inventory data in their application [
10,
20,
38]. Our results showed that
was the primary explanatory variable in the component biomass models. This may originate from the intimate correlations between components and tree diameter [
7,
10,
20,
39]. However, within the given
, there is usually some variation among the aboveground and component biomass values, which highlight that it is insufficient to predict the aboveground and component biomass by the biomass model constructed with only
. Thus, to improve the prediction accuracy of the aboveground and component biomass models, another variable should be added into the biomass model [
20]. An increasing number of scholars have often considered
as another commonly used and vital predictor variable to reduce biased estimates of biomass models because tree height usually reflects the site factors [
15,
18]. In our study, to simulate the biomass allometric relationships for larch trees,
and
were selected to construct basic equations. Six biomass modeling approaches were constructed and validated with the jackknifing technique. Our findings were consistent with previous studies, i.e., that a combination of
and
significantly improved the prediction accuracy of aboveground and component biomass [
6,
7,
9,
40].
Tree biomass models are categorized as nonadditive or additive models. Nonadditive models cannot synchronously consider the aboveground and component biomass data, leading to unequal aboveground biomass. The additive biomass models comprise a desirable characteristic for a system of equations used for the tree biomass prediction, which explicate the instinctive correlations among component biomass of the same sample, and thus, they have a great statistical efficiency [
23,
30,
41]. In this study, we applied six biomass modeling approaches to develop the biomass models. The BM and MEM have separately fitted aboveground and component biomass models, and they did not hold the additivity property for aboveground biomass. Therefore, the sums of the predictions of tree component models were usually larger or smaller than the predictions of the aboveground biomass models, although the differences were small. In contrast, AMS1, AMS2, DMS1, and DMS2 successfully accounted for the correlations among component biomass by a covariance matrix, in which the aboveground biomass prediction was aggregated from the predictions of the tree component models or disaggregated into tree component biomass. Thus, the AMS1, AMS2, DMS1, and DMS2 held the additivity property for the aboveground biomass.
The AMS1 and AMS2 was fitted with independent nonlinear biomass models, in which there is no random effect in each model. The AMS2 contains no constraint, while AMS1 contains one constraint that guarantees that aboveground predictions will be exactly equal to the sum of the biomass prediction of the stem, branch, and foliage component. Our results indicated that both AMS1 and AMS2 fitted the data and performed well in terms of the average prediction errors for aboveground and component biomass predictions. In this study, we demonstrated the differences between AMS1 and AMS2 because of the constraints imposed on the model system. Furthermore, the AMS1 and AMS2 in our study had smaller standard errors of parameters compared to BM, although AMS1, AMS2, and BM possess similar
, RMSE%, MPE, MPE%, MAE, and MAE%, which was consistent with results from Parresol [
30]. In comparison to the BM, the AMS1 and AMS2 accounted for correlations between component biomass and focus on additivity. Therefore, we recommend AMS1 and AMS2 over the BM. In addition, AMS2 was indeed better for predicting the aboveground and component biomass than AMS1, even though AMS1 actually uses aboveground biomass model as a dependent equation. These data are in accordance with those from Zhao et al. [
23]. Thus, the AMS2 is more suitable to construct the biomass models for aggregated model systems.
Both DMS1 and DMS2 maintained the properties of additivity for the aboveground and component biomass. In our study, the prediction accuracy of the aboveground and each component biomass model using DMS1 were higher than those from using DMS2. This is likely because disaggregated model systems depend on the aboveground biomass model, which is commonly thought to be the most accurate among the aboveground and component biomass models [
7]. Although DMS1 was slightly superior to AMS2, the advantage of AMS2 over DMS1 lies in the fact that it has been successfully implemented for individual biomass estimation, and that it is more maneuverable in practical applications.
The results in
Table 5,
Table 6, and
Figure 4 show that the biomass models were obviously improved on the fit and validation statistics after including the sample plot as a random effect into MEM. Consequently, the MEM in the above six approaches selected is probably more suitable for this study. Many studies have shown that biotic factors (e.g.,
,
) and abiotic factors (e.g., origin, site, and climate) affect the biomass prediction accuracy [
7,
9,
21,
23,
27]. The random effect “plot” added into the MEM could relate to climatic factors or site factors. In other words, BM, AMS1, AMS2, DMS1, and DMS2 consider the influence of biotic factors only, the MEM also takes abiotic factors into account, making it the most efficient among the six biomass modeling approaches. In fact, the fixed effects parameters in MEM has larger standard errors among all six approaches (
Table 2). Thus, when a subsample of biomass is available to predict the random effects, the MEM is more efficient than the other five biomass modeling approaches. However, if the subsample is available, the MEM without the random effects would obtain less efficient estimates. At this point, we would recommend the use of the AMS2 or DMS1.
5. Conclusions
We developed six biomass modeling approaches based on only and a combination of , for larch trees occupying a relatively large geographical area in Northeast China. The BM, MEM, AMS1, AMS2, DMS1, and DMS2 separately fitted the component biomass. We compared the six biomass modeling approaches based on , RMSE%, MPE, MPE%, MAE, and MAE%. Our results indicated the MEM with random effects had better , RMSE%, MPE, MPE%, MAE, and MAE% than the BM, AMS1, AMS2, DMS1, and DMS2, and thus, it was selected as the most suitable for this study. However, when no subsample is available to calculate the random effects, the aggregated and disaggregated model systems are recommended because these model systems had better fitting and smaller standard errors of the parameters than the BM did; furthermore, they also accounted for correlations among the aboveground and component biomass. Between the aggregated model systems, AMS2 was better for predicting the aboveground and component biomass than AMS1; DMS1 was better than DMS2. Furthermore, with regard to biomass estimation, there was no single model or system to predict biomass that was best for the aboveground and component biomass. For this study, the overall ranking based on the fit and validation statistics obeyed the following order: MEM > DMS1 > AMS2 > AMS1 > DMS2 > BM.
Our future work will aim to use conifer and deciduous tree species to verify the differences between the aggregated and disaggregated model systems, as well as compare the differences between AMS1 against AMS2, or DMS1 against DMS2. The biomass models described in our study are useful tools for the prediction of the aboveground and component biomass for larch trees in different locations and supply basic information to the Chinese National Forest Inventory.