Soil-Sensitive Weibull Distribution Models of Larix principis-rupprechtii Plantations across Northern China

Abstract

Tree diameter distribution models are important tools for forest management decision making. Soil variables affect tree growth and thus diameter distribution. However, few studies have been conducted on diameter distribution models describing the effects of soil. This study developed a soil-sensitive diameter distribution model based on 213 sample plots of Larix principis-rupprechtii plantations in northern China. The Weibull distribution model was modified by a compatible simultaneous system and the percentile method with the inclusion of soil variables. The most significant factors influencing the diameter distribution of L. principis-rupprechtii in terms of both scale and shape were stand characteristics and available K and alkali-hydrolysable N. The adjusted coefficient of determination for parameter γ significantly improved by 16.0%, while the root mean square error for parameter β decreased by 10.4%. The F test indicated a substantial difference between the models with and without soil variables. From the perspective of adjustable R² values, the Akaike information criterion, root mean square error, relative error index, and absolute error index, the inclusion of stand and soil factors in the tree diameter distribution model enhanced its performance compared to the model that did not consider soil factors. The soil-sensitive diameter distribution model is proven to be effective and accurate.

1. Introduction

The diameter structure of a stand is the most basic and significant component of the stand structure. Numerous studies have demonstrated a close relationship between changes in diameter structure, which describes the distribution status of trees of various sizes and diameters in forest stands according to their diameter order and variables such as tree height, stem shape, volume, species, crown, and biomass [1,2,3]. Therefore, research on the distribution of forest stand diameter structure can potentially serve as a foundation for methodologies and approaches related to forest management and tree measurement and tabulation. The distribution characteristics of stand diameter classes are described by a variety of probability density functions (PDFs) or models, including the β model, Johnson’s sb model, gamma model, normal model, lognormal model, and Weibull model [4,5]. Weibull’s three-parameter model, which includes a closed cumulative density formula and convenient parameter extraction, is a versatile tool for modeling the diameter distribution of Larix principis-rupprechtii plantations [6]. The fundamental variation in forest characteristics and ecological traits is likely to be captured by environmental variables such as soil conditions and climate, clarifying the biological linkages between the environment and tree growth. Recent studies have shown a significant impact of the environment on carbon and tree allotrophic growth, particularly the influence of soil factors [7,8]. Tree development and thus the distribution of tree diameter can be influenced by environmental variables. Several diameter distribution models, such as those by Sanquetta et al. (2014) and Guo et al. (2022) [6,9], describe the impacts of climatic characteristics. However, few studies have investigated how soil conditions influence the distribution of diameters. Larix principis-rupprechtii, as the primary afforestation tree in China, plays a significant role in wood production, carbon sequestration, and ecological services. According to the National Forestry and Grassland Administration in China (2019), arbor forests cover approximately 6.72% of the total area, with 29.19% of this area being planted. As indicated by Zang et al. (2016) and Lei et al. (2016) [10,11], the species under consideration is climate-sensitive. Due to this sensitivity, numerous studies have been conducted to develop growth and biomass models that are sensitive to climate factors, both at the individual tree level and at the stand level [12,13]. Nevertheless, there is a lack of understanding regarding the impact of soil variables on the diameter class distribution of Larix principis-rupprechtii plantations. The size structure of trees has been shown to be associated with the diversity of stand structure, which has been shown to have an impact on forest growth, biomass, productivity, and multifunctionality [13,14,15,16,17]. Forest managers employ silvicultural practices to manipulate stand density and preserve a desirable diameter distribution [18]. Additionally, the structure of stands plays a significant role in shaping the response of forests to climate and silviculture practices [15]. A lack of comprehensive understanding regarding the impact of changes in soil variables on the structure of larch forests can pose a significant challenge for forest managers in devising adaptive strategies to mitigate the potential consequences of these effects. Hence, the objectives of our study were twofold: (1) to establish a soil-sensitive diameter distribution model utilizing the Weibull distribution model by utilizing permanent sample plots from larch plantations located in the northern and northeastern regions of China and (2) to simulate the effects of soil on the diameter distribution. This study will help improve our understanding of the impact of soil on growth prediction.

2. Materials and Methods

2.1. Sample Plot Data

The data for the sample plots were obtained from the 6th National Forest Inventory of China, which was conducted between 1999 and 2003 in four provinces, namely, Beijing, Hebei, Inner Mongolia, and Shanxi Provinces (Figure 1). These plots are rectangular with an area from 0.06 ha to 0.08 ha in different provinces listed in

Table 1. Descriptive statistics of the stand variables.

Provinces	Plot Size	Statistic	SAA /Years	AH /m	QMD/cm	SI/m	BA/m2/ha	S/ha
Beijing	0.0667 ha	Mean	24.56	7.02	10.86	26.47	9.29	773
		Standard Deviation	9.43	1.32	3.64	3.93	10.04	509
		Min	13.00	5.58	7.16	14.01	0.79	195
		Max	43.00	9.76	18.70	46.35	28.84	1694
Hebei	0.06 ha	Mean	17.86	6.30	9.39	19.25	7.57	1040
		Standard Deviation	5.27	1.27	3.49	3.30	5.82	610
		Min	9.00	4.77	5.65	9.70	0.75	217
		Max	31.00	9.85	20.00	33.42	24.11	2733
Inner Mongolia	0.06 ha	Mean	21.33	7.97	11.68	23.00	10.64	939
		Standard Deviation	5.96	1.80	3.39	3.33	6.31	496
		Min	13.00	5.89	8.32	14.01	1.9	350
		Max	33.00	11.17	18.09	35.57	18.84	2133
Shanxi	0.0667 ha	Mean	22.16	6.64	9.60	23.89	8.58	1128
		Standard Deviation	8.84	1.58	2.67	2.66	5.86	522
		Min	14.00	4.82	6.20	15.09	0.54	180
		Max	50.00	12.24	18.11	53.90	24.72	1979
Total		Mean	19	7.30	9.32	8.84	7.81	1076
		Standard Deviation	8	2.01	3.21	2.56	5.86	180
		Min	5	4.52	5.59	4.68	0.48	653
		Max	50	17.80	23.69	19.20	28.84	3820

. Only Larix principis-rupprechtii plantations that had not been harvested were chosen for this study. There were 213 plots in the dataset, with a total of 9980 trees. The plots that were examined were rectangular in shape and had areas that varied between 0.06 and 0.08 hectares in different provinces. The primary variables under investigation in the plots were tree diameter at breast height (DBH) greater than 5 cm, average age of the stand, average height of the trees, canopy closure, slope position, aspect, slope, elevation, soil thickness, and soil type. Other stand variables, such as the stand basal area (BA), quadratic mean diameter (QMD), density stems (S), minimum diameter (MinD), maximum diameter (MaxD), stand average age (SAA), average height (AH), and stand site index (SI), were derived from tree list measurements. The site indices were computed based on the model proposed by Zang et al. (2016) [10] in the literature. The summary statistics of the variables pertaining to the stand are presented in Table 1.

2.2. Site and Soil Data

The soil data utilized in this study were derived from a comprehensive dataset of soil properties specific to China, which was collected and compiled by Shangguan and Dai (2014) [19]. This dataset served as the foundation for land surface modeling in the present research. The source data for this dataset were derived from the 1:1 million soil map of China and 8595 soil profiles. The dataset provided in raster format encompassed various parameters, such as pH value, soil organic matter, total N, total P, total K, alkali-hydrolysable N, available P, available K, cation exchange capacity, exchangeable Al3+, and others, totaling 16 indices (refer to

Table 2. Description of soil and site variables for model development.

Category	Variable Unit	Description	Statistic
			Maximum	Minimum	Mean
Soil chemical properties	TN (g/100 g)	Soil total nitrogen	0.48	0.04	0.19
	TK (g/100 g)	Soil total potassium	2.51	1.41	1.93
	TP (g/100 g)	Soil total phosphorus	0.13	0.04	0.06
	SOM (g/100 g)	Soil organic matter	8.90	0.55	3.69
	pH	pH value (H2O)	8.39	6.07	6.88
	AN (mg/kg)	Alkali-hydrolysable nitrogen	284.15	24.78	140.64
	AK (mg/kg)	Available potassium	254.44	49.22	140.25
	AP (mg/kg)	Available phosphorus	13.81	2.19	6.66
	AL (Me/100 g)	Exchangeable Al3+	3.21	0.03	0.37
	CEC (Me/100 g)	Cation exchange capacity (CEC)	27.90	6.39	16.84
Soil physical property	PDEP (cm)	Soil profile depth	147.85	43.86	85.62
	GRAV	Rock fragment	43.73	0.25	18.30
	BD (g/cm3)	Bulk density	1.39	0.83	1.26
	CL	Clay	22.03	9.53	15.73
	SI	Silt	55.34	22.22	41.97
	SA	Sand	68.25	29.59	42.31
Site	LA (m)	Latitude	2383.00	690.00	1471.55
	SLD (cm)	Soil layer depth	100.00	1.00	24.69

). Each soil physical and chemical property in the database included data for eight different soil layer depths: 0–0.045 m, 0.045–0.091 m, 0.091–0.166 m, 0.166–0.289 m, 0.289–0.493 m, 0.493–0.829 m, 0.829–1.383 m, and 1.383–2.296 m. The 8 soil grid data points were weighted averages and registered with the basic geographic data of Shanxi province, Hebei province, and the Inner Mongolia Autonomous Region and were used as the soil environmental variables. In addition, altitude and soil thickness were also added as environmental variables.

2.3. Weibull Distribution Model and Parameter Estimation Methods

The three-parameter Weibull distribution model was selected because of its extensive utilization and flexibility, as shown by previous studies [20,21]. The probability density function (PDF) f(x) for the three-parameter Weibull distribution can be expressed as follows:

$$f(x)=({\displaystyle \frac{\gamma }{\beta }})({\displaystyle \frac{x-\alpha }{\beta }}{)}^{\gamma -1}\mathit{exp}[-({\displaystyle \frac{x-\alpha }{\beta }}{)}^{\gamma }]$$

(1)

$$\alpha \le x\le \infty ,\beta >0,\gamma >0$$

where x represents the measured tree diameter at breast height (DBH) and α, β, and γ are the position, scale, and shape parameters of the Weibull distribution, respectively.

This study compared the percentile method (PM) and maximum likelihood estimation method (MLE) for estimating PDF parameters. Both methods involve predicting the percentiles of the diameter distribution, which are dependent on the distribution parameters. However, PM has been found to have a stronger correlation with stand characteristics than with PDF parameters [22]. The probability proportional to size is a statistical method that directly represents the PDF parameters as a function of stand variables, as demonstrated by Liu et al. (2004) [23]. In this article, the Kolmogorov–Smirnov (KS) test [24,25] was employed to compare the suitability of the three-parameter Weibull distribution in fitting sample plots between PM and the MLE. The three-parameter Weibull distribution was fitted using maximum likelihood estimation [26] based on the tree list for each plot measurement (observation) in the moment method. The concept of PM can be defined as Equations (2)–(4) [23].

$$\alpha ={\displaystyle \frac{n^{0.3333}{\mu }_0-{\mu }_{50}}{n^{0.3333}-1}}$$

(2)

$$\gamma ={\displaystyle \frac{2.343088}{\ln ({\mu }_{95}-\alpha )-\ln ({\mu }_{25}-\alpha )}}$$

(3)

$$\beta ={\displaystyle \frac{\alpha \Gamma (1+\frac{1}{\gamma })}{\Gamma (1+\frac{2}{\gamma })}}+sqrt\{{\displaystyle \frac{{\alpha }^2}{{\Gamma }^2(1+\frac{1}{\gamma })}}[{\mathsf{\Gamma }}^2(1+{\displaystyle \frac{1}{\gamma }})-\Gamma (1+{\displaystyle \frac{2}{\gamma }})]+{\displaystyle \frac{\overline{u_q^2}}{\Gamma (1+\frac{2}{\gamma })}}]\}$$

(4)

In the given context, n represents the number of trees present in a plot, where ${\mu }_0$, ${\mu }_{25}$, ${\mu }_{50}$, and ${\mu }_{95}$ correspond to the 0th, 25th, 50th, and 95th percentiles, respectively, in the distribution of trees. The gamma function Γ was utilized, and the mean square diameter $u_q$ was also considered. If the estimated value of α is less than 0, α is forced to be set to 0.

2.4. Diameter Distribution Model with Soil Factors

First, the estimation of the three parameters of the Weibull distribution for each plot was conducted using the equations provided. Equations (2)–(4) were derived from the PM. Additionally, separate relationships were established between the parameters and stand variables, as well as between the parameters, the stand, and additional soil variables. Stand and soil variables were chosen through the use of SUR regression, which is a linear model that is employed to estimate sparse parameters and is particularly well suited for reducing the number of parameters [27]. A commonly employed method in regression analysis to identify multicollinearity is the variance inflation factor (VIF) [28]. The severity of multicollinearity increases with increasing VIF. Generally, a VIF exceeding 10 indicates the presence of a significant collinearity issue within the model. Moreover, to further elucidate the potential internal relationships among the three parameters of the Weibull distributions, we employed simultaneous equations to impose a constraint. In this study, the VIF was set at a threshold of 10 to examine the presence of multicollinearity. The calculations were conducted by the glmnet and Mass package in R4.4.0 [29,30].

$$\left\{\begin{array}{c}\alpha =f1(stand)\\ \beta =f2(stand)\\ \gamma =f3(stand)\end{array}\right.$$

(5)

$$\left\{\begin{array}{c}\alpha =f4(stand,soil)\\ \beta =f5(stand,soil)\\ \gamma =f6(stand,soil)\end{array}\right.$$

(6)

In the mathematical expression 5 and 6, the variable “stand” in this study refers to a set of stand variables, namely, BA, S, MaxD, MinD, QMD, and SI, as listed in Table 1. On the other hand, the term “soil” represents a collection of soil variables listed in Table 2.

2.5. Model Test

2.5.1. Kolmogorov–Smirnov Test for the Weibull Distribution

The Kolmogorov–Smirnov (KS) test, as proposed by Smirnov (1948) and further discussed by Reynolds et al. (1988) [25], was employed at a 95% confidence level to assess the suitability of fitting the diameter distribution of a sample plot using the three-parameter Weibull distribution through cumulative frequency (Equation (7)).

$$D=\underset{-\infty <\theta <\infty }{limsup}\left|F^{*}(v)-F(v)\right|$$

(7)

where lim sup is the maximum value, $v$ represents the measured tree diameter at breast height (DBH), and D is the KS statistic. $F^{*}\left(v\right)$ and $F(v)$ are the theoretical and realized probabilities of the diameter distribution of the plots, respectively. The original hypothesis is H0: the distributions of the two variables ($F^{*}\left(v\right)$ and $F(v)$) are consistent, or the data conform to the theoretical distribution. When the observed value D exceeds the critical value D (n, alpha), the null hypothesis H0 is rejected; otherwise, the null hypothesis H0 is accepted based on the assumption, where n represents the number of trees in a single plot and alpha denotes the specified test level, which is set at 0.05.

2.5.2. Model Evaluation for Regression

Leave-one-out cross-validation (LOOCV) was employed to assess the performance of the Weibull distribution parameter models (mathematical expression 5 and 6) [31]. Root mean square error ($RMSE$) and adjusted coefficient of determination (${\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{R}}^2$) were calculated in Equations (10) and (11). The paper compared the differences between two models through the F-test in Equation (12). The Akaike information criterion (AIC) is a standard for measuring the goodness of fit of statistical models in Equation (13) [32]. E1 and E2 are the absolute error index and relative error index in Equations (14) and (15) respectively, that are specific to the ith plot measurement.

$$SSE={\sum }_{i=1}^N(y_i-{\widehat{y}}_i{)}^2$$

(8)

$$SSR={\sum }_{i=1}^N(y_i-{\overline{y}}_i{)}^2$$

(9)

$$RMSE=\sqrt{{\displaystyle \frac{SSE}{N}}}$$

(10)

$${adjustedR}^2=1-{\displaystyle \frac{SSE\left(N-1\right)}{\left(N-k-1\right)SSR}}$$

(11)

$$F={\displaystyle \frac{SSR/(d{f}_s-d{f}_c)}{SSE/d{f}_c}}$$

(12)

$$AIC=e^{({\displaystyle \frac{2p}{N}})}{\displaystyle \frac{SSE}{N}}$$

(13)

$$\mathrm{E}1={\sum }_{\mathrm{i}}^{\mathrm{N}}\left|{\mathrm{N}}_{\mathrm{i},\mathrm{j}}-\widehat{{\mathrm{N}}_{\mathrm{i},\mathrm{j}}}\right|$$

(14)

$$\mathrm{E}2={\sum }_{\mathrm{i}}^{\mathrm{N}}\left|{\displaystyle \frac{{\mathrm{N}}_{\mathrm{i},\mathrm{j}}}{\sum {\mathrm{N}}_{\mathrm{i},\mathrm{j}}}}-{\displaystyle \frac{\widehat{{\mathrm{N}}_{\mathrm{i},\mathrm{j}}}}{\sum \widehat{{\mathrm{N}}_{\mathrm{i},\mathrm{j}}}}}\right|$$

(15)

where SSEc and SSEs are the sum of square errors of the regression model (Equations (8) and (9); $y_i$ and ${\widehat{y}}_i$ are the observed and predicted numbers of trees for the ith plot, respectively; $N$ is the total number of plots; p is the number of independent variables; dfc and dfs are the degrees of freedom for Equations (5) and (6), respectively; and ${\mathrm{N}}_{\mathrm{i},\mathrm{j}}$ and $\widehat{{\mathrm{N}}_{\mathrm{i},\mathrm{j}}}$ are the observed and predicted numbers of trees per hectare within the jth diameter class (2 cm width), respectively.

3. Results

3.1. Parameter Estimation Methods for the Weibull Distribution Models

According to the Kolmogorov–Smirnov (KS) test, of a total of 213 samples, 208 samples were successfully fitted to parameters via the percentile method (PM), while 134 plots were successfully fitted to parameters via the maximum likelihood estimation method (MLE). Therefore, the use of PM is deemed more appropriate for modeling the diameter distribution of Larix principis-rupprechtii plantations. The parameter estimates obtained from PM exhibit the following ranges: 2.50 ≤ β ≤ 7.25, 3.33 ≤ β ≤ 16.25, and 1.28 ≤ γ ≤ 7.50 (

Table 3. Summary and statistics of diameter distribution parameters.

Variables	$\mathit{\alpha }$	β	γ
Mean	2.89	7.49	3.99
Standard Deviation	0.90	2.99	1.19
Max	7.25	16.25	7.50
Min	2.50	3.33	1.28

).

3.2. Comparison of Models for Estimating Diameter Distribution Parameters with and without Soil Variables

The β and γ parameters of the Weibull distribution, obtained through the PM method, were utilized as dependent variables. Using the SUR method, two sets of parameter prediction equations for the Weibull distribution were derived: the traditional model, which includes only stand variables (Equation (16)), and the soil-sensitive model, which incorporates both stand and soil variables (Equation (17)). Stand variables, such as the maximum diameter (MaxD), minimum diameter (MinD), diameter growth (QMD), and stand density (SD), were included in the analysis. The soil and site variables that exhibited statistical significance were bulk specific gravity (BSG), altitude, available nitrogen, and DPEP.

Table 4. Parameter estimates and fit statistics for Models (16) and (17).

Parameters	Type	Variables	Estimate	Standard Deviation	Error t Value	Pr	VIF
β	Stand	Intercept	2.37 × 10−3	7.17 × 10−2	0.033	0.97366	-
		MinD	−4.98 × 10−1	1.15 × 10−2	−43.309	<2.00 × 10−16	2.20
		QMD	1.05 × 100	6.34 × 10−3	165.833	<2.00 × 10−16	2.39
		LA	−1.37 × 10−4	4.44 × 10−5	−3.083	0.00233	1.48
		Density	1.07 × 10−4	2.58 × 10−5	4.138	5.10 × 10−5	1.12
		Soil depth	1.04 × 10−3	5.21 × 10−4	2.004	0.04641	1.22
	With soil	Intercept	−1.60 × 10−1	9.62 × 10−2	−1.661	0.09825	-
		MinD	−4.96 × 10−1	1.13 × 10−2	−43.969	<2.00 × 10−6	2.40
		QMD	1.05 × 100	6.42 × 10−3	162.829	<2.00 × 10−6	2.75
		LA	−1.01 × 10−4	4.54 × 10−5	−2.215	0.02788	1.75
		Density	9.93 × 10−5	2.53 × 10−5	3.921	0.00012	1.21
		SLD	1.52 × 10−3	5.18 × 10−4	2.923	0.003856	1.36
		AP	2.68 × 10−2	6.96 × 10−3	3.847	0.00016	2.86
		AL	6.58 × 10−2	1.97 × 10−2	3.333	0.001022	1.34
		AK	−1.55 × 10−3	4.32 × 10−4	−3.584	0.000424	2.77
		PDEP	1.93 × 10−3	6.89 × 10−4	2.802	0.005577	1.15
γ	Stand	(Intercept)	3.298216	0.271389	12.153	<2.00 × 10−16
		MaxD	−0.19526	0.013603	−14.354	<2.00 × 10−16	1.59
		MinD	0.446559	0.036948	12.086	<2.00 × 10−16	1.42
		LA	0.000775	0.000172	4.515	1.06 × 10−5	1.38
		SLD	−0.00848	0.002084	−4.069	6.70 × 10−5	1.22
	With soil	(Intercept)	2.163463	0.37838	5.718	3.82 × 10−8	1.59
		(Intercept)	−0.19918	0.013451	−14.808	<2.00 × 10−16	1.76
		MaxD	0.47691	0.035632	13.384	<2.00 × 10−16	1.49
		MinD	0.000609	0.000168	3.623	0.000368	1.49
		LA	−0.00889	0.002054	−4.329	2.35 × 10−5	1.33
		AL	0.208315	0.093252	2.234	0.026581	1.86
		AK	0.003616	0.001422	2.543	0.011721	1.87
		PDEP	0.005869	0.002973	1.974	0.049712	1.33
		AN	0.005541	0.001422	3.897	0.000132	2.84
		TN	−3.02752	1.124804	−2.692	0.007704	4.35

and

Table 5. Model valuation indices with leave-one-out cross-validation.

Parameters	Type	RMSE	Multiple R2	AIC
β	Stand	0.2005	0.9955	−68.23
	With Soil	0.1864	0.9961	−87.12
γ	Stand	0.806	0.5472	523.83
	With Soil	0.750	0.6080	503.09

(Note: root mean square error and adjusted coefficient of determination).

present the calculated values of the parameters β and γ obtained through Equations (16) and (17) using the seemingly unrelated regression (SUR) method. Compared to Equation (13), the adjusted R² value improved (16.0% for γ), while the RMSE decreased (3.5% for β and 10.4% for γ). Figure 2 and Figure 3 demonstrate a higher level of concordance between the predicted and observed values for the parameter and γ when soil factors were considered. Therefore, the inclusion of soil variables in the models resulted in an increase in model performance. The severity of multicollinearity increased as the variance inflation factor (VIF) increased. Generally, when the variance inflation factor (VIF) exceeds 5, it indicates the presence of a significant collinearity issue within the model. According to Table 4, the variance inflation factor (VIF) values for all of the models ranged from 1.12 to 2.86, which are below the threshold of 5. There was a notable disparity observed, indicating that the models incorporating soil variables outperformed those without soil variables. This conclusion is supported by the fact that the Akaike information criterion (AIC) of Equation (17) was smaller than that of Equation (15) for both variables, γ. The β results indicate a significant improvement in the models, as evidenced by the F test, with the value decreasing from 100.80 to 49.25. The AIC for parameter β increased from −87.12 for the model without soil factors to −68.23 for the model with soil factors. The AIC for parameter γ increased from 503.09 for the model without soil factors to 523.82 for the model with soil factors.

$$\left\{\begin{array}{c}\alpha =0.5\ast MinD\\ \beta =f2\left(s\mathit{tan}d\right)=a_0+a_{1}Dg+a_{2}MinD+a_{3}DE+a_{4}SLD\\ \gamma =f3(s\mathit{tan}d)=c_0+c_{1}MaxD+c_{2}MinD+c_{3}LA+a_{4}SLD\end{array}\right.$$

(16)

$$\left\{\begin{array}{c}\alpha =0.5\ast MinD\\ \beta =f4\left(s\mathit{tan}d,soil\right)=a_0+a_{1}Dg+a_{2}MinD+a_{3}DE+a_{4}SLD+b_{1}AP+b_{2}AL+b_{3}AK+b_{4}PDEP\\ \gamma =f5\left(s\mathit{tan}d,soil\right)=c_0+c_{1}MaxD+c_{2}MinD+c_{3}LA+a_{4}SLD+d_{1}AL+d_{2}AK+d_{3}DPEP+d_{4}DPE+d_{5}AN+d_{6}TN\end{array}\right.$$

(17)

The absolute error index (E1) ranged from 33.3 ≤ E1 (stand) ≤ 93.6 to 32.1 ≤ E1 (soil) ≤ 73.4 (

Table 6. Absolute error index E1 and relative error index E2 in different provinces.

	E1 (Stand)			E1 (Soil)			E2 (Stand) %			E2 (Soil) %
Provinces	Mean	Min	Max	Mean	Min	Max	Mean	Min	Max	Mean	Min	Max
Hebei	33.3	0	145	32.1	0	136	4.08	0.13	48.8	4.03	1.05	43.3
Beijing	93.6	2	477	73.4	0	347	4.31	0.23	14.1	4.26	0.05	13.8
Shanxi	74.9	0	400	68.9	2	364	5.38	0.31	35.2	4.96	0.23	31.3
Inner Mongolia	57.4	8	202	48.9	8	175	9.2	0	43.2	8.4	0	38.3

). The relative error (E) ranged from 4.08 ≤ E2 (stand) ≤ 9.26 to 4.03 ≤ E2 (soil) ≤ 8.4 (Table 6). Compared to Equation (15), the mean and maximum values of E1 and E2 decreased with the soil factors in the four provinces. In terms of the E1 index, the model that considered soil factors had a 3% to 21% decrease in the four provinces compared to the model that did not consider soil factors. Figure 2 and Figure 3 show the fitting graphs of parameter β and parameter γ considering only forest factors and both soil factors and stand factors for predicted and measured values, respectively. Figure 2 shows slightly improved models when considering both soil factors and stand factors for parameter β because both models are much more accurate. Figure 3 shows a larger improved effect for models considering both soil factors and stand factors for parameter γ from scatter plots.

4. Discussion

4.1. Attributes of the Diameter Distribution Model

Although many diameter class distribution models have been developed, soil variables are rarely introduced [9]. In this study, a soil-sensitive diameter distribution model was developed for Larix principis-rupprechtii plantations in northern China. When fitting with the PM method, 97.65% of the plots passed the KS test, while compared to the PPM method, only 62.91% of the plots passed the test. This result is the same as that of Guo et al. (2022) [6]. The PM method is more suitable for simulating the diameter distribution of L. principis-rupprechtii than the MLE method.

The distribution of tree diameter is influenced by intricate interactions among environmental, stand, and tree factors. Despite the existence of various diameter class distribution models [33], the incorporation of soil variables has been infrequently observed. In this research, a soil-sensitive diameter distribution model was developed using the SUR method. This method was employed to incorporate stand and soil variables into the model, aiming to improve the performance parameters for Larix principis-rupprechtii plantations in northern China. In our study, the model incorporating both stand and soil factors exhibited superior performance compared to the model that included only stand factors as predictors. Compared to Equation (13), the adjusted R² value improved (16.0% for γ), while the RMSE decreased (3.5% for β and 10.4% for γ). Compared with the model not considering soil factors, the model considering soil factors increased the AIC for both parameter β and parameter γ. The absolute error index (E) for both the mean value and maximum value decreased with increasing soil factors in the four provinces. From the perspective of adjustable R², AIC, RMSE, E1, and E2 values, the inclusion of stand and soil factors in the tree diameter distribution model enhanced its performance compared to that of the model that did not consider soil factors. Figure 4 shows the observed and fitting distribution with only stand factors and with the inclusion of stand and soil factors for four plots from Beijing, Heibei, Inner Mongolia, and Shanxi. We also can obtain the result that the fitting distribution with the inclusion of stand and soil factors is improved. The soil-sensitive diameter distribution model is proven to be effective and accurate.

The regression analysis examining the relationship between the scale parameter and the quadratic mean diameter in cork demonstrated a significant proportion of the variability in the β parameter (R² > 0.99). This was observed in both the models that included soil factors and those that did not. These findings are in line with previous studies that have demonstrated the positive impact of using the mean square diameter as an input variable [34,35]. Additionally, these studies reported a strong correlation between the mean square diameter and the observed outcomes [35,36].

A significant proportion (55%) of the plots exhibited γ values exceeding 3.6, with a mean value of 3.98. The distribution of diameters exhibited a mound-shaped pattern, with the majority of plots displaying negative asymmetry. These findings align with the typical attributes of the forests in the designated study area, which are primarily composed of stands of similar ages. Negative asymmetry is commonly observed in these forests due to low thinning and/or low competition, as noted by García Güemes et al. (2002) [36].

4.2. Influence of Stand Factors and Soil Factors on the Parameters of the Weibull Distribution Model

Independent variables play a crucial role in influencing the parameters β and γ. Prior studies have highlighted the significance of various stand variables. The scale parameter β was effectively represented by the quadratic mean diameter under cork, resulting in a high adjusted R² value of 0.99. The shape parameter γ was estimated by incorporating the maximum diameter, minimum diameter, and plot elevation (R² adj = 0.40) [37]. Stand density is also a significant factor. The frequency and cumulative percentage of the diameter order distribution of the tree count suggest that low-density stands exhibit faster growth in diameter, with a greater number of trees in the larger diameter class. In this study, the scale parameter β was modeled based on the quadratic mean diameter and stand density, while the shape parameter γ was modeled using the maximum diameter, minimum diameter, and altitude. The results obtained from this analysis are in line with the findings reported in previous studies.

Tree growth has been found to be influenced by various soil factors, including topography, soil texture, soil moisture [38,39,40,41,42,43], and soil nutrient availability [44]. Soil, which serves as a crucial medium for tree growth, is subject to the influence of various soil-forming factors, such as topography, climatic variations, parent material, and living organisms (among others). The impact of nutrient variability on plant growth has been extensively investigated in grasslands, agricultural land systems, and certain forest ecosystems [44,45]. Numerous articles have indicated that soil properties exhibit a strong correlation with aboveground biomass, belowground biomass, and total basal area, suggesting a positive relationship between soil fertility and tree growth [46]. Although there is existing documentation on the impact of soil factors on tree growth, particularly in terms of tree diameter growth, there is limited knowledge regarding the effects of soil factors on tree diameter distribution. According to the Pr values of the soil factors, the minimum value was observed for AN. The results revealed the order of significance for the soil factors and β as AK, AP, AL, and PDEP. The results of the study revealed the order of significance for soil factors and γ as follows: AN, AK, AL, TN, and PDEP. The parameter γ exhibited significant correlations with AK, AL, and DPEP. The parameter β exhibited robust and statistically significant positive correlations with AP but strong and statistically significant negative correlations with AK. The parameter γ exhibited a robust and statistically significant positive correlation with AN. In general, the results of the study indicated that AK and AN are the most significant soil factors influencing the diameter distribution of L. principis-rupprechtii plantations.

4.3. Insufficient Research and Future Directions

Many techniques, including PM, MLE, the moment method (MOM), and hybrid methods, have been used in the field of dynamic prediction of stand diameter structure utilizing probability distributions to forecast these parameters in diverse tree species and forest types [47]. In regard to parameter estimation, using other techniques, such as MOM and the hybrid method, can produce very interesting outcomes.

Numerous studies have demonstrated that Weibull mixture distributions exhibit greater accuracy than the Weibull model. However, further research is required to develop robust analytical methods for Weibull mixture distributions. It is worth noting that this paper focuses solely on standard variables and soil factors, and it is possible that the prediction accuracy could be improved by incorporating climate factors.

5. Conclusions

PM is more suitable for predicting the parameters of the three-parameter Weibull diameter distribution function than the maximum likelihood estimation method (MLE) for the construction of Larix principis-rupprechtii plantations. Furthermore, compared to models that only consider stand factors, multiple regression models that consider the distribution parameters of both the stand and soil components are more accurate. The diameter distribution of L. principis-rupprechtii was found to be influenced primarily by stand density and soil composition, especially by AK and AN soil variables for both β and γ. One efficient way to estimate stand density, diameter distribution structure, and individual tree diameter is to use a three-parameter Weibull diameter distribution model that takes soil factors into account. As such, this model is a useful instrument for controlling the density and organization of Larix principis-rupprechtii plantations in China.