A Three-Level Model System of Biomass and Carbon Storage for All Forest Types in China

Abstract

Forest biomass and carbon storage models are crucial for inventorying, monitoring, and assessing forest resources. This study develops models specific to China’s diverse forests, offering a methodological foundation for national carbon storage estimation and a quantitative basis for national, regional, and global carbon sequestration projections. Utilizing data from 52,700 permanent plots obtained during China’s 9th national forest inventory, we calculated biomass and carbon storage per hectare for 35 tree species groups using respective individual tree biomass models and carbon factors. We then constructed a three-level volume-based model system for forest biomass and carbon storage, applying weighted regression, dummy variable modeling, and simultaneous equations with error-in-variables. This system encompasses one population of forests, three forest categories (level I), 20 forest types (level II), and 74 forest sub-types (level III). Finally, the assessment of these models was carried out with six evaluation indices, and comparative analyses with previously established biomass models of three major forest types were conducted. Determination coefficients (R²) for the population average model, and three dummy models on levels I, II, and III, exceed 0.78, 0.85, 0.92, and 0.95, respectively, with corresponding mean prediction errors (MPEs) of 0.42%, 0.34%, 0.24%, and 0.19%, and mean percent standard errors (MPSEs) of approximately 22%, 21%, 15%, and 12%. Models for 20 forest types and 74 sub-types yield R² values above 0.87 and 0.85, with MPE values below 3% and 5%, respectively. Notably, the estimates of previous biomass models of three major forest types demonstrated considerable uncertainty, with TRE ranging from −20% to 74%. However, accuracy has improved with larger sample sizes. In total biomass and carbon storage estimations, the R² values of dummy models for levels I, II, and III progressively increase and MPSE and MPE values decrease, whereas TRE approximates zero. The tiered model system of simultaneous equations developed herein offers a quantitative framework for precise evaluations of biomass and carbon storage on different scales. For enhanced accuracy in such estimations, applying level III models is recommended whenever feasible, especially for national estimation.

1. Introduction

Forest biomass and carbon storage, akin to forest volume, are vital for monitoring forest resources and are integral indicators of forest ecosystem function and productivity [1,2,3]. As global climate concerns rise, the examination of forest carbon storage and its sequestration potential is increasingly prioritized [4,5]. Forest biomass can be determined through individual tree biomass models [6,7] or by formulating stand-level biomass models and conversion factors [2,6]. Forest carbon storage estimates are derived by multiplying biomass by the forest’s average carbon factor [2].

Reviewing work by Luo et al. [8], it is evident that from 1978 to 2013, Chinese researchers developed 5924 tree biomass models covering approximately 200 species. Since 2014, the State Forestry Administration of China has developed tree biomass models and related carbon accounting parameters for the primary tree species or groups, resulting in a series of official standards [9,10,11,12,13,14,15,16,17,18,19,20,21]. Yet, comparatively, stand-level biomass models [7,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] are less prevalent than individual tree models [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,26,27,42,43,44,45,46]. Given that stand-level models are more commonly utilized on large scales, such as national, regional, and global levels, the development of these models for all forest types in China is of substantial practical significance.

Among the stand-level biomass models for China’s forest types, Fang et al. [29,30] stand out with volume-derived biomass models for 21 forest types from 418 sample plots, which have seen broad application [35,38]. Wang et al. [32] created hyperbolic relationships between biomass and volume for 16 forest types using 1266 sample plots from forest inventory, and Zhang et al. [41] developed power function models for 10 types with 1828 sample plots from forest inventory, along with 21 biomass models for various regions and forest categories.

Evaluating the performance of these models reveals three main deficiencies: First, the modeling sample sizes are often insufficient, with Fang et al. [30] basing 18 models on fewer than 30 sample plots, among the models for 21 forest types, and only three models for larch (Larix spp.), Chinese fir (Cunninghamia lanceolata), and Chinese pine (Pinus tabulaeformis) were based on more than 30 plots. Similarly, Wang et al. [32] used fewer than 50 plots for 10 models, and Zhang et al. [41] used fewer than 50 plots for two models. Second, the modeling methods tend to be oversimplified, commonly using ordinary least square (OLS) regression, which does not account for the heteroscedasticity of the biomass and volume data. Third, the evaluation indices are typically limited, with only R² [30,41] or R [32] provided, lacking other error-related indices, making it difficult to gauge their uncertainty in the application.

Moreover, as the discourse on climate change intensifies and strategies for ‘carbon peak and carbon neutrality’ are implemented, the precise estimation of forest carbon storage has garnered escalating interest. Researchers often rely on existing biomass models to gauge carbon storage trends, typically using a universal carbon factor of 0.5 or 0.47 for all forest types [37,38,39,40]. Nevertheless, carbon factors vary among different forest types, a fact underscored by Zeng et al. [31] who formulated simultaneous models for volume, biomass, and carbon storage across 10 major forest types in northeast China using 2000 plots.

The primary aims of this study are twofold: (1) To develop a tiered system of models for forest biomass and carbon storage based on the measured data of 52,700 sample plots from the ninth national forest inventory (NFI) of China, which includes one population, three forest categories (level I), 20 forest types (level II), and 74 forest sub-types (level III). This system employs weighted regression, dummy variable modeling, and simultaneous equations with error-in-variables [31,47,48,49,50]. (2) To assess the evaluation indices of the three-level models, providing a basis for understanding the uncertainty of the models when applied on different scales.

2. Materials and Methods

2.1. Data Description

The dataset for this research is derived from the permanent sample plots in the ninth National Forest Inventory (NFI) of China. Nationwide, 52,700 effective sample plots were analyzed, each with a non-zero stand volume. In-depth analysis of these plots included calculating forest volume and biomass (accounting for above- and below-ground biomass, but not understory vegetation) as well as carbon storage per hectare. These calculations were based on official standards on tree biomass models and carbon factors for major tree species, which were developed using the measurement data of 7534 sample trees from destructive sampling [9,10,11,12,13,14,15,16,17,18,19,20,21,43]. On the tree level, 35 species/groups were classified according to the national standard [43], and the estimates of volume, biomass, and carbon storage at the plot level were obtained by summing the estimates of individual trees, which were based on one-variable models. Even though tree height is the second most important variable to estimate biomass, the contribution to above- or below-ground biomass estimation is not highly great [43]. These plots are organized by dominant tree species into three categories and 20 forest types, categorized based on area and stand volume, encompassing a diverse range of species such as fir (Abies), spruce (Picea), and others, including broadleaved and coniferous types. To ensure robust model development and subsequent validation, the sample plots were divided, with two-thirds utilized for model construction and one-third reserved for validation.

Table 1. The basic data pertaining to the modeling samples and validation samples for 20 forest types.

Forest Type		Number of Plots	Modeling Samples			Validation Samples
			Number of Plots	Max Volume (m3/ha)	Max Biomass (t/ha)	Number of Plots	Max Volume (m3/ha)	Max Biomass (t/ha)
Coniferous	Abies spp.	534	355	1439	764	179	1364	792
	Picea spp.	1353	900	941	564	453	800	487
	Larix spp.	2495	1665	485	381	830	522	379
	Cunninghamia lanceolata	3152	2100	456	257	1052	454	262
	Cupressus spp.	1328	885	511	768	443	388	622
	Pinus massoniana	2607	1740	319	325	867	333	290
	P. tabulaeformis	1186	790	306	377	396	275	343
	P. yunnanensis	766	510	340	250	256	355	242
	Other coniferous	1681	1125	471	323	556	414	327
Mixed	Conifer mixed	1898	1265	881	600	633	789	487
	Conifer–broadleaf mixed	4364	2910	650	606	1454	502	436
	Broadleaf mixed	13,073	8715	732	870	4358	706	723
Broadleaves	Quercus spp.	4474	2980	706	875	1494	402	538
	Betula spp.	2201	1465	472	359	736	420	360
	Populus spp.	3924	2615	396	367	1309	380	298
	Robinia pseudoacacia	830	550	188	248	280	193	245
	Eucalyptus spp.	1036	690	261	296	346	195	248
	Hevea brasiliensis	701	465	298	241	236	370	296
	Other hardwood	3368	2245	488	549	1123	443	509
	Other softwood	1729	1150	381	366	579	422	321

provides a detailed breakdown of the data used for both modeling and validation across the different forest categories and types.

2.2. Modeling Methods

2.2.1. Forest Classification

To enhance the practicability of the modeling results, based on the data available, 15 forest types among the 20 were subdivided according to dominant tree species (group) [51] or geographical region [52], excluding Chinese pine, Yunnan pine, Robinia, eucalyptus, and rubber-woods. One type (other coniferous) was divided into 8 sub-types by dominant tree species (group), and two types (other hardwood broadleaved and other softwood broadleaved) were divided into 12 and 6 sub-types, respectively, by both dominant tree species (group) and geographical region, and the remaining 12 types were divided by geographical region from 2 to 6 sub-types. The whole country is categorized into six geographical regions, that is, North China (NC), Northeast (NE), East China (EC), Central South (CS), Southwest (SW), and Northwest (NW). These regions can also be combined as North (N = NE + NC + NW), South (S = EC + CS + SW), West (W = NW + SW), and other regions as necessary. Consequently, 20 forest types were reclassified into 74 forest sub-types (see

Table 2. A three-level system of forest classification.

Forest Category	Forest Type	Forest Sub-Type/Level III
Level I	Level II	Number	Name
Coniferous	Abies spp.	2	Abies spp. I (N); Abies spp. II (SW)
	Picea spp.	2	Picea spp. I (N); Picea spp. II (SW)
	Larix spp.	3	Larix spp. I (NE); Larix spp. II (NC); Larix spp. III (W)
	Cunninghamia lanceolata	3	C. lanceolata I (EC); C. lanceolata II (CS); C. lanceolata III (SW)
	Cupressus spp.	3	Cupressus spp. I (NE + NC); Cupressus spp. II (NW); Cupressus spp. III (S)
	Pinus massoniana	3	P. massoniana I (EC); P. massoniana II (CS); P. massoniana III (SW)
	P. tabulaeformis	1	P. tabulaeformis
	P. yunnanensis	1	P. yunnanensis
	Other coniferous	8	P. sylvestris; P. armandii; P. densata; P. K.T.D.; Foreign pine; Other pine; Cryptomeria spp.; Other coniferous
Mixed	Conifer mixed	4	Conifer mixed I (N); Conifer mixed II (EC); Conifer mixed III (CS); Conifer mixed IV (SW)
	Conifer–broadleaf mixed	5	Conifer–broadleaf I (NE); Conifer–broadleaf II (NC + NW); Conifer–broadleaf III (EC); Conifer–broadleaf IV (CS); Conifer–broadleaf V (SW)
	Broadleaf mixed	6	Broadleaf mixed I (NE); Broadleaf mixed II (NC); Broadleaf mixed III (NW); Broadleaf mixed IV (EC); Broadleaf mixed V (CS); Broadleaf mixed VI (SW)
Broadleaves	Quercus spp.	5	Quercus spp. I (NE); Quercus spp. II (NC); Quercus spp. III (NW); Quercus spp. IV (SE); Quercus spp. V (SW)
	Betula spp.	3	Betula spp. I (NE); Betula spp. II (NC); Betula spp. III (W)
	Populus spp.	4	Populus spp. I (NE); Populus spp. II (NC); Populus spp. III (W); Populus spp. IV (SE)
	Robinia pseudoacacia	1	R. pseudoacacia
	Eucalyptus spp.	1	Eucalyptus spp.
	Hevea brasiliensis	1	H. brasiliensis
	Other hardwood	12	F.J.P.; C.S.P.; Ulmus spp.; Schima superba; Juglans regia; Castanea mollissima; Quercus variabilis; Other non-wood; Other hardwood I (NE + NC); Other hardwood II (NW); Other hardwood III (EC); Other hardwood IV (CS + SW)
	Other softwood	6	Tilia tuan; Salix spp.; Other softwood I (NE + NC); Other softwood II (NW); Other softwood III (SE); Other softwood IV (SW)

Note: (i) China is divided into six geographic regions, that is, North China (NC), Northeast (NE), East China (EC), Central South (CS), Southwest (SW), and Northwest (NW). NC includes Beijing, Tianjin, Hebei, Shanxi, and Inter Mongolia; NE includes Liaoning, Jilin, and Heilongjiang; EC includes Shanghai, Jiangsu, Zhejiang, Anhui, Fujian, Jiangxi, and Shandong; CS includes Henan, Hubei, Hunan, Guangdong, Guangxi, and Hainan; SW includes Chongqing, Sichuan, Guizhou, Yunnan, and Tibet; NW includes Shannxi, Gansu, Qinghai, Ningxia, and Xinjiang. NE, NC, and NW are merged to North (N); Northwest and Southwest are merged to West (W); East China and Central South are merged to Southeast (SE); and Southeast and Southwest are merged to South. The above forest classification and geographical division do not include Taiwan, Hong Kong, and Macau. (ii) Abbreviations in parentheses indicate geographic regions. (iii) P. K.T.D denotes P. koraiensis, P. thunbergii, and P. densiflora; F.J.P. means Fraxinus mandshurica, Juglans mandshurica, and Phellodendron amurense; C.S.P. represents Cinnamomum spp., Sassafras spp., and Phoebe spp.; and other non-wood includes the non-wood forests composed of other tree species except Hevea brasiliensis, Juglans regia, Castanea mollissima, and Quercus variabilis.

).

2.2.2. Model Development

Forest biomass is closely related to its volume, and the volume-derived biomass models have been extensively applied in previous studies [27,28,32,35,38]. According to Fang et al. [30], there is a linear correlation between forest biomass and volume for 21 forest types in China. This linearity was confirmed by a scatterplot depicting the relationship between forest biomass and volume data per hectare from the 52,700 sample plots. Moreover, linear models are advantageous for application across different scales and can avoid the scaling-up errors associated with overestimates of nonlinear models [37].

Additionally, both above- and below-ground biomass require consideration. For instance, reporting both kinds of biomass data in the Global Forest Resources Assessment by the FAO [3] is mandatory. The ratio of below-ground to above-ground biomass, known as the root-to-shoot ratio (RSR), varies across forest types [2]. Accurate estimation of below-ground biomass for various forest types depends on the reliable values of RSR, but these data are not available. After estimating total forest biomass, forest carbon storage is calculated by multiplying the biomass by the average carbon factor, usually 0.5 or 0.47 in previous studies [2,29,30]. Nonetheless, it should be recognized that carbon factors may differ among forest types [31].

Given the recursive nature of the relationship between total biomass and both above-ground biomass (or below-ground biomass) and carbon storage, this study employed simultaneous equations with error-in-variables and dummy variables [47,48,49,50] to develop a model system at three levels. The equations are as follows:

$$\left\{\begin{array}{l}{B}_T=\Sigma a_i{S}_i+(\Sigma b_i{S}_i)V+{\epsilon }_1\\ B_A=(\Sigma c_i{S}_i){\widehat{B}}_T+{\epsilon }_2\\ C=(\Sigma d_i{S}_i){\widehat{B}}_T+{\epsilon }_3\end{array}\right.$$

(1)

Here, B_T represents observed total biomass per hectare (t/ha), B_A indicates above-ground biomass (t/ha), and C refers to carbon storage (t/ha). V is the forest volume (m³/ha) regarded as error-free-variable, ${\widehat{B}}_T$ is estimated total biomass per hectare (t/ha) regarded as an error-in-variable in the second and third equations, and a_i, b_i, c_i, and d_i are parameters where i denotes the ith category, type, or sub-type. S_i is a dummy variable equal to 1 when data belong to the ith category, type, or sub-type, and 0 otherwise. ε₁, ε₂, and ε₃ are error items, assumed to follow a normal distribution with a mean of zero.

By dividing the first equation in model system (1) by V, we obtain a biomass conversion factor (BCF) model:

$$BCF=B_T/V=b_i+a_i/V$$

(2)

In this model, BCF combines basic wood density (WD), biomass expansion factor (BEF), and the ratio of shoot-to-root (RSR) as outlined in the IPCC Guidelines for national greenhouse gas inventories [2], where BCF = WD·BEF·(1 + RSR). The d_i parameter in the third equation in model system (1) aligns with the carbon factor (CF). From the c_i parameter in the second equation in model system (1), one can deduce the RSR:

$$RSR=B_B/B_A={(1-c}_i)/c_i$$

(3)

Like forest volume data, both forest biomass and carbon storage data display heteroscedasticity. This research advocates the use of the weighted regression method [47], with a weight function defined as w = 1/V^0.5, which was determined on compromise consideration of total relative error and average systematic error of the models [48]. Additionally, considering the recursive relationship between total biomass, above-ground biomass, and carbon storage, it is crucial to employ simultaneous equations with error-in-variables to fit accurately the model system (1) [50].

2.2.3. Model Evaluation

Six indices were employed to evaluate the models: coefficient of determination (R²), standard error of the estimate (SEE), total relative error (TRE), average systematic error (ASE), mean prediction error (MPE), and mean percentage standard error (MPSE) [53,54], which were commonly used to evaluate the integrated performance of models [55,56]. TRE, ASE, MPE, and MPSE are defined as follows:

$$\mathrm{TRE}=\sum (y_i-{\widehat{y}}_i)/\sum {\widehat{y}}_i\times 100$$

(4)

$$\mathrm{ASE}=\sum (y_i-{\widehat{y}}_i)/{\widehat{y}}_i/n\times 100$$

(5)

$$\mathrm{MPE}=t_{\alpha }\cdot (SEE/\overline{y})/\sqrt{n}\times 100$$

(6)

$$\mathrm{MPSE}=\sum \left|(y_i-{\widehat{y}}_i)/{\widehat{y}}_i\right|/n\times 100$$

(7)

In these equations, y_i represents observed values, ${\widehat{y}}_i$ represents estimated values, $\overline{y}$ is the mean of observed values, n is the number of plots, and t_α is the t-value at the confidence level α. For the models developed, values of the six indices were computed and used for evaluation.

Practically, models are expected to have MPE values less than 3% or 5%, MPSE values less than 15% or 20%, TRE values within ±3%, and ASE values within ±5%. For extensive applicability assessments, TRE and ASE must also be independently verified using validation samples, with TRE ideally within ±3% or below the MPE value, and ASE within ±5%. If ASE exceeds ±5% but is within ±10%, the model has a moderate systematic error; if it exceeds ±10%, the systematic error is considered high.

3. Results

For this study, data from 35,120 plots were utilized to calibrate model system (1), applying simultaneous equations with error-in-variables and dummy variables. This encompassed one population, three forest categories, 20 forest types, and 74 forest sub-types. The parameter estimates and associated evaluation indices for the first equation in model system (1) are presented in

Table 3. The parameter estimates of three-level simultaneous models of forest biomass and carbon storage and evaluation indices of biomass model in China.

Level	Type	Parameter Estimate					Evaluation Indices of Biomass Model
		ai	bi	ci	RSR	di (CF)	R2	SEE/t	TRE/%	ASE/%	MPE/%	MPSE/%
Population	Forest	1.4864	0.9792	0.7973	0.2542	0.4891	0.781	34.62	0.00	5.30	0.42	22.38
Level I	Coniferous	1.6355	0.8129	0.8069	0.2393	0.4994	0.787	35.69	−0.01	8.52	0.80	24.60
	Mixed	1.7434	1.0463	0.7956	0.2569	0.4868	0.884	25.39	−0.04	3.45	0.45	16.81
	Broadleaved	0.2986	1.1133	0.7959	0.2564	0.4838	0.887	23.06	−0.01	3.93	0.55	21.68
Level II	Abies spp.	5.5779	0.5937	0.8180	0.2225	0.4951	0.873	47.30	−0.06	2.20	2.21	17.30
	Picea spp.	1.6095	0.7157	0.7991	0.2514	0.4901	0.914	29.89	−0.01	5.50	1.23	15.65
	Larix spp.	1.0427	0.8228	0.7794	0.2830	0.4888	0.936	14.99	0.00	2.70	0.82	13.28
	Cunninghamia lanceolata	0.5904	0.7118	0.8109	0.2332	0.4969	0.936	12.06	0.00	4.36	0.87	14.04
	Cupressus spp.	0.4836	1.5336	0.8007	0.2489	0.5013	0.947	22.57	0.00	2.88	1.65	16.47
	Pinus massoniana	1.7401	0.9726	0.8297	0.2053	0.5162	0.954	11.20	0.02	0.91	0.67	10.01
	P. tabulaeformis	0.0901	1.1255	0.8098	0.2349	0.5130	0.966	16.18	0.00	0.05	1.66	15.60
	P. yunnanensis	0.9361	0.6499	0.8453	0.1830	0.5047	0.973	11.89	0.00	3.94	1.71	13.79
	Other coniferous	0.9884	0.8313	0.8041	0.2436	0.5009	0.874	20.90	−0.03	6.63	1.64	20.41
	Conifer mixed	0.5828	1.2964	0.8088	0.2364	0.5014	0.921	19.64	0.06	1.69	1.22	14.21
	Conifer–broadleaf mixed	0.5018	1.0149	0.7961	0.2561	0.4920	0.894	20.58	−0.01	3.37	0.89	15.80
	Broadleaf mixed	0.0077	0.9022	0.7938	0.2598	0.4835	0.916	22.67	−0.03	3.19	0.47	15.44
	Quercus spp.	0.6954	1.5388	0.7930	0.2610	0.4810	0.917	26.78	−0.01	3.27	0.85	17.44
	Betula spp.	0.3333	1.1740	0.7821	0.2786	0.4867	0.898	17.39	0.00	1.77	1.03	14.36
	Populus spp.	0.8226	0.8607	0.8246	0.2127	0.4725	0.933	12.21	0.00	1.86	0.78	12.83
	Robinia pseudoacacia	0.3439	1.2130	0.7832	0.2768	0.4838	0.934	10.75	−0.01	1.94	2.00	14.68
	Eucalyptus spp.	0.5703	0.9935	0.7793	0.2832	0.5238	0.962	8.98	0.00	1.32	1.18	8.88
	Hevea brasiliensis	4.0799	0.8272	0.7981	0.2530	0.4956	0.991	4.10	−0.01	2.48	0.53	5.96
	Other hardwood	2.6170	0.9515	0.7880	0.2690	0.4856	0.933	14.37	−0.01	2.95	1.17	17.67
	Other softwood	1.5064	1.1073	0.7954	0.2572	0.4873	0.896	18.66	−0.02	5.29	1.79	19.75
Level III	Abies spp. I	1.9730	0.7767	0.8017	0.2473	0.4955	0.989	13.10	0.00	0.56	1.02	5.09
	Abies spp. II	3.6239	0.5359	0.8267	0.2096	0.4950	0.971	23.16	0.01	1.52	1.36	8.97
	Picea spp. I	1.4994	0.7351	0.7918	0.2629	0.4900	0.899	31.11	0.00	5.75	1.42	16.25
	Picea spp. II	3.4253	0.6405	0.8282	0.2074	0.4903	0.978	17.05	−0.01	1.63	1.63	8.27
	Larix spp. I	0.6576	0.9237	0.7452	0.3419	0.4885	0.967	9.24	0.00	1.35	0.87	9.95
	Larix spp. II	0.4562	0.7999	0.7836	0.2762	0.4888	0.939	12.49	0.00	0.81	1.12	10.26
	Larix spp. III	2.3320	0.7342	0.8285	0.2070	0.4895	0.973	14.02	0.01	2.47	1.37	11.69
	Cunninghamia lanceolata I	0.7850	0.6990	0.8100	0.2346	0.4965	0.920	14.32	0.00	4.60	1.40	16.21
	Cunninghamia lanceolata II	0.4758	0.7229	0.8120	0.2315	0.4974	0.954	8.85	0.00	4.52	1.29	11.56
	Cunninghamia lanceolata III	0.4392	0.7268	0.8115	0.2323	0.4973	0.953	10.36	0.00	2.99	1.56	12.57
	Cupressus spp. I	0.0735	1.7647	0.7997	0.2505	0.5020	0.947	9.14	0.00	0.89	2.92	11.70
	Cupressus spp. II	2.0093	1.6802	0.8006	0.2491	0.5030	0.956	28.59	0.00	2.14	2.54	12.57
	Cupressus spp. III	1.0093	1.3584	0.8012	0.2481	0.4995	0.968	13.95	0.00	2.07	1.55	10.56
	Pinus massoniana I	1.3985	0.9519	0.8303	0.2044	0.5157	0.947	11.84	0.02	1.10	1.29	11.00
	Pinus massoniana II	1.8000	1.0408	0.8278	0.2080	0.5166	0.953	10.28	0.02	0.23	1.12	10.08
	Pinus massoniana III	1.6417	0.9385	0.8309	0.2035	0.5163	0.979	8.03	0.01	0.38	0.75	7.13
	P. tabulaeformis	0.0940	1.1255	0.8098	0.2349	0.5130	0.916	16.18	0.00	0.03	1.66	15.60
	P. yunnanensis	0.9394	0.6499	0.8453	0.1830	0.5047	0.909	11.89	0.00	3.94	1.71	13.79
	P. sylvestris var. mongolica	0.5574	0.8918	0.7950	0.2579	0.5163	0.873	21.25	0.00	4.26	4.68	17.69
	P. K.T.D.	0.8327	0.8267	0.8028	0.2456	0.5065	0.864	23.09	0.05	9.86	5.22	24.67
	P. armandii	0.5115	0.9231	0.8021	0.2467	0.5175	0.981	7.78	0.00	1.02	1.84	8.62
Level III	P. densata	3.1127	0.6639	0.8267	0.2096	0.4904	0.970	10.63	0.00	0.97	1.86	8.42
	Foreign pine	0.7032	0.9988	0.7793	0.2832	0.4763	0.976	6.55	0.00	1.67	1.48	8.85
	Other pine	3.9524	0.8538	0.8331	0.2003	0.5070	0.869	21.56	0.05	1.61	3.99	22.33
	Cryptomeria spp.	0.9519	0.7296	0.7809	0.2806	0.5052	0.968	11.52	−0.01	4.84	2.89	10.82
	Other coniferous	0.8331	0.7273	0.8079	0.2378	0.4993	0.855	22.33	0.00	1.00	6.42	24.38
	Conifer mixed I	5.2106	0.7982	0.7963	0.2558	0.4944	0.929	23.92	0.06	4.28	2.41	17.35
	Conifer mixed II	2.4575	0.8529	0.8116	0.2321	0.5033	0.915	14.73	−0.02	1.76	1.76	13.81
	Conifer mixed III	1.2449	0.9207	0.8132	0.2297	0.5047	0.959	9.48	−0.01	0.82	1.73	10.35
	Conifer mixed IV	4.4595	0.8099	0.8159	0.2256	0.5044	0.906	24.82	0.05	1.71	2.97	13.66
	Conifer–broadleaf mixed I	1.8584	0.8430	0.7829	0.2773	0.4870	0.930	17.73	0.00	2.52	1.37	10.22
	Conifer–broadleaf mixed II	3.8893	1.1026	0.7938	0.2598	0.4934	0.877	21.66	0.01	0.89	2.35	16.44
	Conifer–broadleaf mixed III	1.7448	1.0257	0.7965	0.2555	0.4918	0.935	16.07	−0.01	2.12	1.42	14.12
	Conifer–broadleaf mixed IV	1.1497	1.0444	0.7974	0.2541	0.4950	0.940	10.91	−0.01	2.41	1.31	12.54
	Conifer-broadleaf mixed V	2.2794	0.8772	0.8121	0.2314	0.4945	0.923	19.06	−0.04	2.89	1.88	14.32
	Broadleaf mixed I	0.4864	0.9751	0.7927	0.2615	0.4798	0.909	19.82	0.00	1.37	0.69	11.40
	Broadleaf mixed II	−0.0928	1.3711	0.7894	0.2668	0.4838	0.927	16.58	0.00	0.75	1.85	14.87
	Broadleaf mixed III	−0.0875	1.3001	0.7965	0.2555	0.4849	0.954	16.91	0.00	1.28	1.02	11.90
	Broadleaf mixed IV	1.3414	1.2308	0.7905	0.2650	0.4833	0.952	17.96	−0.01	3.33	0.80	13.05
	Broadleaf mixed V	1.8432	1.1634	0.7911	0.2641	0.4865	0.960	15.28	−0.02	1.40	0.79	11.35
	Broadleaf mixed VI	0.9649	1.0453	0.8035	0.2446	0.4863	0.956	19.22	−0.03	3.09	1.06	12.15
	Quercus spp. I	−1.0068	1.1471	0.7818	0.2791	0.4795	0.970	13.96	0.00	1.13	0.80	9.89
	Quercus spp. II	−0.0049	1.5550	0.7821	0.2786	0.4808	0.946	15.61	0.00	−0.06	1.14	11.37
	Quercus spp. III	−0.2365	1.4851	0.7968	0.2550	0.4815	0.961	19.38	0.00	0.87	1.34	9.94
	Quercus spp. IV	1.0500	1.4135	0.7912	0.2639	0.4823	0.957	18.15	−0.02	2.81	1.77	14.07
	Quercus spp. V	−0.1926	1.1182	0.8217	0.2170	0.4820	0.969	23.36	0.01	4.84	1.54	11.61
	Betula spp. I	0.1938	0.9346	0.7613	0.3135	0.4862	0.984	6.21	0.00	−0.73	0.88	6.91
	Betula spp. II	1.5641	1.0444	0.7850	0.2739	0.4868	0.858	18.71	0.00	2.50	1.40	15.79
	Betula spp. III	0.7607	1.0042	0.7970	0.2547	0.4870	0.885	22.14	−0.01	3.63	2.82	12.73
	Populus spp. I	0.1542	0.7329	0.7968	0.2550	0.4731	0.982	7.14	0.00	1.33	1.17	11.25
	Populus spp. II	−0.0628	0.9791	0.8170	0.2240	0.4724	0.973	7.49	0.00	1.56	0.72	9.17
	Populus spp. III	−0.3020	0.8946	0.8195	0.2203	0.4729	0.956	11.94	0.01	0.81	1.97	13.26
	Populus spp. IV	−0.3962	0.9141	0.8601	0.1627	0.4721	0.907	12.13	0.01	1.31	1.50	14.09
	Robinia pseudoacacia	0.7447	1.5370	0.7832	0.2768	0.4838	0.934	10.74	0.00	1.75	2.01	14.68
	Eucalyptus spp.	0.3307	1.1740	0.7793	0.2832	0.5238	0.962	8.98	0.00	1.33	1.18	8.88
	Hevea brasiliensis	0.8497	0.8604	0.7981	0.2530	0.4956	0.991	4.09	−0.01	2.43	0.53	5.93
	F.J.P.	0.4219	0.9889	0.7929	0.2612	0.4822	0.908	14.30	0.00	4.38	3.49	14.22
	C.S.P.	1.0519	1.1024	0.7934	0.2604	0.4894	0.975	8.00	−0.01	1.08	1.96	7.44
	Ulmus spp.	0.7656	1.0968	0.7844	0.2749	0.4839	0.938	10.41	0.00	4.89	2.73	16.44
	Schima superba	0.4144	1.2017	0.7728	0.2940	0.4746	0.942	15.88	0.00	1.67	3.81	14.17
	Juglans regia	0.3120	1.0375	0.7907	0.2647	0.4952	0.968	2.65	−0.02	1.59	2.76	11.32
	Castanea mollissima	0.2104	1.1672	0.7958	0.2566	0.4943	0.943	8.21	−0.01	1.72	3.46	13.91
	Quercus variabilis	−0.2029	1.5204	0.7823	0.2783	0.4945	0.963	10.46	0.00	−0.95	3.77	12.58
	Other non-wood	0.4858	1.0736	0.7882	0.2687	0.4944	0.964	5.32	−0.01	2.30	1.86	12.61
	Other hardwood I	0.4491	1.4058	0.7852	0.2736	0.4834	0.919	11.75	−0.01	3.80	4.08	15.92
	Other hardwood II	−0.1049	1.4276	0.7986	0.2522	0.4842	0.980	11.53	0.00	1.87	2.25	13.11
	Other hardwood III	1.0589	1.3343	0.7842	0.2752	0.4829	0.958	14.14	−0.01	5.09	2.74	15.51
	Other hardwood IV	0.4022	1.2749	0.7886	0.2681	0.4827	0.950	18.16	−0.01	4.25	3.67	15.88
	Tilia tuan	1.7288	0.8643	0.7949	0.2580	0.4617	0.879	19.92	−0.04	3.26	4.21	15.76
	Salix spp.	0.3393	0.9938	0.7948	0.2582	0.4929	0.932	9.15	−0.01	2.21	4.00	16.19
	Other softwood I	0.4991	0.9423	0.7900	0.2658	0.4888	0.933	14.43	−0.01	6.15	3.11	21.26
	Other softwood II	0.8547	1.3636	0.8010	0.2484	0.4938	0.898	19.83	−0.02	4.27	3.89	19.19
	Other softwood III	0.6294	0.9392	0.7895	0.2666	0.4937	0.963	9.45	−0.01	4.70	2.82	16.84
	Other softwood IV	1.3184	0.9434	0.8002	0.2497	0.4891	0.953	13.90	−0.04	3.00	2.53	10.55

Note: Only the parameter estimates in italics are not statistically significant, all others are highly significant (p < 0.01); the RSR values are calculated from Equation (3).

. Notably, when assessing the SEE values for the second and third equations in model system (1) relative to the first equation, significant discrepancies emerged. However, for the other five evaluation indices, only minor differences were noted. Due to their negligible variance, the indices for the second and third equations in model system (1) have been excluded from Table 3 for succinctness.

A thorough analysis of the six evaluation indices in Table 3 revealed several insights into the model’s performance. The TRE values were nearly zero, which suggests a strong agreement between the observed and estimated values across the models. The average values of the other five evaluation indices demonstrated a positive trend moving from one population to three forest categories and further to 20 forest types and 74 forest sub-types, except for a slight uptick in the average MPE values from 0.42% to 0.60%, 1.22%, and 2.11%, attributable to sample size variations. The average R² values increased progressively from 0.781 for the population average model to 0.853, 0.928, and 0.941 for levels I, II, and III models, respectively, while the average values of SEE, ASE, and MPSE decreased. For instance, the average MPSE values at the three levels fell from 22.38% for the population average model to 21.03%, 14.68%, and 12.96%, a reduction of approximately 6%, 34%, and 42% respectively.

Regarding the models for the three forest categories, the R² values exceeded 0.78, MPE values were below 1%, ASE values fell within ±10% (with two categories within ±5%), and MPSE values were around 20%. The coniferous biomass model did not perform so well due to the large variation in forest volume stock. Subsequently, data from 17,580 plots, classified as validation samples in Table 1, underwent independent validation. The outcomes affirmed that TRE values stayed within ±0.3% and ASE values within ±5%.

For the 20 forest-type models, the R² values were above 0.87, and MPE values were below 3% (with eight types under 1%). ASE values were within ±10% (18 types within ±5%), and MPSE values were under 20% (with the exception of the ‘other coniferous’ type and two types under 10%). Independent cross-validation confirmed that TRE values for most types were within ±3%, barring the third equation in model system (1) for the ‘other coniferous’ type at −3.16%; and ASE values were within ±5%, except for two types. The R² values and other evaluation indices, which were different for each forest type, depended on the size and structure of the modeling samples.

For the 74 forest sub-type models, the R² values surpassed 0.85, and MPE values remained under 5% (except for two sub-types), with 60 sub-types below 3%. ASE values were within ±10% (70 sub-types within ±5%), and MPSE values were under 20% (with four sub-types being the exception, and 55 sub-types under 15%). Independent cross-validation indicated that TRE values for most sub-types were within ±3%, with only six sub-types deviating, and ASE values for the majority were within ±5%, except for eight sub-types, and only two sub-types exceeding ±10%.

The model evaluations described above pertain especially to forest category, forest type, and forest sub-type, respectively, across three levels. If the evaluation indices target the entire population of forests, a comparable trend in the average values is observed, as seen in the three-level models. The distinction lies in the uniform sample size of the three dummy-variable simultaneous models, which results in a consistent decrease in MPE values from the categories to types and sub-types, all falling below 0.5% (refer to

Table 4. The evaluation indices of three-level simultaneous models for the population.

Model	Level	R2	SEE	TRE/%	ASE/%	MPE/%	MPSE/%
(1) Total biomass	Population	0.781	34.62	0.00	5.30	0.42	22.38
	Level I	0.857	28.04	−0.02	5.07	0.34	20.73
	Level II	0.928	19.88	−0.01	3.01	0.24	15.11
	Level III	0.955	15.80	0.00	2.15	0.19	12.42
(2) Above-ground biomass	Population	0.796	27.07	0.22	4.74	0.41	21.69
	Level I	0.864	22.07	−0.01	4.35	0.34	20.10
	Level II	0.933	15.55	0.00	2.31	0.24	14.69
	Level III	0.957	12.51	0.00	1.52	0.19	12.11
(3) Carbon storage	Population	0.786	16.72	0.05	5.48	0.42	22.36
	Level I	0.851	13.94	−0.07	5.32	0.35	21.11
	Level II	0.927	9.79	−0.05	3.06	0.24	15.15
	Level III	0.954	7.76	−0.04	2.14	0.19	12.43

).

As indicated by Table 4, the evaluation indices for forest biomass, above-ground biomass, and carbon storage models at each respective level exhibit minimal difference, barring the dimensionally variable SEE values, which, from a practical standpoint, can be disregarded. However, when comparing the indices across the three levels, it is evident that the level III models outperform others, followed by level II, with level I models lagging behind. Using forest carbon storage models as an example, the R² value improved from 0.851 at level I to 0.954 at level III, the MPE value diminished from 0.35% to 0.19%, and the MPSE value dropped from 21.11% to 12.43%, respectively, signifying a significant enhancement in model fit (as shown in Figure 1). The three-level model system has provided a quantitative framework for accurate evaluations of biomass and carbon storage on national, regional, and global scales. To diminish the uncertainty of forest biomass and carbon storage estimates, employing the level III models is highly recommended wherever feasible in application, especially for national estimation.

4. Discussion

4.1. Comparison with Related Models

In the introduction, we noted that, of the 21 biomass models developed by Fang et al. [30], only three models for larch (Larix spp.), Chinese fir (Cunninghamia lanceolata), and Chinese pine (Pinus tabulaeformis) were based on sample sizes larger than 30 plots. To provide a comparative analysis, we assessed the biomass models established by Fang et al. [30], Wang et al. [32], and Zhang et al. [41], using both validation samples and all samples from these three forest types, as shown in

Table 5. Comparison of estimation results of biomass models between this study and other three sources.

Forest Type	Validation Samples		All Samples		Biomass Models	Source	Sample Size
	TRE/%	ASE/%	TRE/%	ASE/%
Larix spp.	−10.13	−19.82	−10.41	−19.79	B = 33.806 + 0.6096V	Fang et al. [30]	34
	13.40	12.01	13.11	12.00	B = V/(1.1111 + 0.0016V)	Wang et al. [32]	39
	8.97	6.33	8.88	6.44	B = 1.4091V0.8752	Zhang et al. [41]	241
	−0.29	3.13	−0.10	3.32	B = 0.6986 + 0.8262V	This study	1665
Cunninghamia lanceolata	6.10	−9.55	6.68	−8.75	B = 22.5410 + 0.3999V	Fang et al. [30]	56
	14.01	13.12	14.37	13.45	B = V/(1.2917 + 0.0022V)	Wang et al. [32]	70
	17.08	10.84	17.40	11.39	B = 1.2877V0.8427	Zhang et al. [41]	88
	−0.28	3.75	−0.09	4.21	B = 0.5743 + 0.7120V	This study	2100
Pinus tabulaeformis	33.26	20.89	33.88	20.99	B = 5.0928 + 0.7554V	Fang et al. [30]	82
	38.56	33.33	39.52	33.28	B = V/(1.0529 + 0.0020V)	Wang et al. [32]	147
	24.27	14.11	24.99	14.14	B = 1.7969V0.8416	Zhang et al. [41]	699
	−0.60	−0.47	−0.20	−0.58	B = 0.2112 + 1.1235V	This study	790

.

Table 5 reveals that the TRE values of three model sets exceed ±3%, with those for two models for Chinese pine being even higher than 30%; and the ASE values also surpass ± 5%, with the model by Wang et al. [32] for Chinese pine also higher than 30%. It is important to note that the performance of the three model sets generally improves with increased sample size. In general, Zhang et al.’s models [41] exhibit better performance than those Wang et al. [32] and Fang et al.’s models [30].

Examining the parameters from the models in our study shows that Fang et al.’s models [30] possess larger intercept parameters but smaller slope parameters (see Table 5). The first potential reason for these differences in parameter estimates might be the use of an improper estimation method. Using ordinary regression instead of weighted regression, especially in the presence of heteroscedasticity, may result in such distorted outcomes. Secondly, the sample size is also a significant factor. This is evident from the test results of the three forest types with more than 30 sample plots for modeling as indicated in Table 5, and their TRE values range from −10.41% to 33.88%. The modeling sample sizes for the other 18 forest types are all less than 30, and two types have even fewer than 10. As a result, the errors in forest biomass estimates are likely to be greater. Since the classification of forest types by Fang et al. [30] does not correspond exactly to that in this study, direct comparison is not possible. Eight forest types from the remaining 18 were selected for further validation—these include spruce and fir, cypress, Pinus massonina, P. yunnanensis, P. sylvestris var. mongolica, P. armandii, birch, poplar, and eucalyptus—which correspond to the forest type, sub-type, or their combinations in this study. The results showed that the TRE values ranged from −20.02% to 74.01%, and the ASE values from −19.32% to 77.62%, indicative of significant uncertainty.

To illustrate the differences between the three model sets more clearly, the validation results of the forest biomass model in this study and the three biomass models for Pinus tabulaeformis using all 1186 plots are compared in Figure 2. The models display substantial deviations, and the regression trend line (solid black) clearly diverges from the y = x line (dotted red), resulting in a non-random distribution of residuals and high TRE and ASE values. The validation results for the biomass models of the other two forest types, Larix spp. and Cunninghamia lanceolata, are similarly conclusive and are omitted here for brevity.

4.2. Correction of Negative Intercept Parameters

For the biomass models at level III in Table 3, the intercept parameters of 11 out of 74 forest sub-types are negative, leading to negative biomass estimates for young stands with a small volume or close to zero. Hence, the model parameters for these 11 sub-types required re-estimation, using simultaneous equations with error-in-variables and dummy variables for estimating two sub-types of broadleaf mixed forests, three sub-types of poplar forests, and four sub-types of oak forests, while the same approach without dummy variables was applied for the remaining two sub-types. If the intercept parameter remained negative, the weight function was modified to ensure a positive intercept. The updated parameter estimates and evaluation indices for the biomass model for the 11 forest sub-types are presented in

Table 6. The parameter estimates of simultaneous forest biomass and carbon storage models and evaluation indices of biomass model for 11 forest sub-types.

Forest Sub-Type	Parameter Estimates					Evaluation Indices of Biomass Model
	ai	bi	ci	RSR	Di(CF)	R2	SEE/t	TRE/%	ASE/%	MPE/%	MPSE/%
Broadleaf mixed II	0.3940	1.3619	0.7893	0.2669	0.4838	0.927	16.60	−0.01	−0.82	1.86	14.74
Broadleaf mixed III	0.6511	1.2911	0.7964	0.2557	0.4849	0.955	16.90	0.01	−0.31	1.02	11.38
Quercus spp. I	0.2090	1.1363	0.7817	0.2793	0.4795	0.970	14.16	0.01	−2.87	0.81	8.46
Quercus spp. II	0.3647	1.5479	0.7820	0.2788	0.4808	0.946	15.61	0.01	−1.68	1.14	11.84
Quercus spp. III	0.3070	1.4795	0.7968	0.2550	0.4815	0.961	19.33	0.00	−0.67	1.34	10.01
Quercus spp. V	0.7385	1.1107	0.8216	0.2171	0.4820	0.969	23.29	0.00	1.98	1.53	11.01
Populus spp. II	0.3080	0.9727	0.8170	0.2240	0.4724	0.973	7.48	0.01	−0.92	0.72	10.08
Populus spp. III	0.0941	0.8891	0.8194	0.2204	0.4729	0.956	11.96	0.00	−1.23	1.97	12.92
Populus spp. IV	0.4810	0.9021	0.8600	0.1628	0.4721	0.908	12.10	0.01	−0.82	1.50	13.76
Quercus variabilis	0.0492	1.5131	0.7823	0.2783	0.4945	0.963	10.52	0.00	−4.88	3.79	12.73
Other hardwood II	0.1409	1.4235	0.7985	0.2523	0.4842	0.980	11.49	0.00	0.04	2.24	13.33

Note: Only the parameter estimates in italics are not statistically significant, all others are highly significant (p < 0.01); the RSR values are calculated from Equation (3).

.

In comparison with the original models in Table 3, the evaluation indices of the updated models for the 11 sub-types show minimal overall changes. Independent cross-validation confirmed that the TRE values fell within ±3% except for one sub-type of Populus spp. IV; the ASE values were within ±5% except for two sub-types, broadleaf mixed II and Quercus variabilis.

Thus, it is both straightforward and practical to develop simultaneous models of forest biomass and carbon storage for various types at once using simultaneous equations with error-in-variables and dummy variables. Nevertheless, if some model parameters are found to be atypical, separate re-estimation is necessary for adjustment and refinement.

5. Conclusions

Based on the measured data of 52,700 permanent plots from the ninth national forest inventory of China, a tiered volume-based forest biomass and carbon storage model system has been developed in this study according to one population, three forest categories (level I), 20 forest types (level II), and 74 forest sub-types (level III) through the use of weighted regression, dummy variable modeling, and simultaneous equations with error-in-variables. Additionally, the evaluation indices of the three-level models were examined. From the study results, the following conclusions can be drawn.

The performance of models generally improved as the modeling sample size increased. This study used almost all permanent plots in forests from the national forest inventory to develop a forest biomass and carbon storage model system, which is the first time to obtain the largest and most representative modeling samples.

The three-level models developed in this study will serve as a solid foundation for accurately assessing the status and changes of forest biomass and carbon storage at national, regional, and global scales; and the evaluation indices of the models would provide a scientific base for calculating the uncertainty in the estimates.

This study has provided several alternatives to select the models at different scales. To minimize uncertainty in forest biomass and carbon storage estimates, the use of level III models is recommended wherever feasible, especially for national or sub-national estimation.