基于联立方程组的人工樟子松枝下高模型构建

李想; 董利虎; 李凤日

doi:10.13332/j.1000-1522.20170428

基于联立方程组的人工樟子松枝下高模型构建

东北林业大学林学院，黑龙江哈尔滨 150040

基金项目:

国家自然科学基金项目 31570626

详细信息

作者简介:
李想。主要研究方向：林分生长模型。Email：lx0326999@163.com 地址：150040 黑龙江省哈尔滨市香坊区和兴路26号东北林业大学林学院

责任作者:
李凤日，教授。主要研究方向：林分生长模型。Email：fengrili@126.com 地址：同上

中图分类号: S758.5
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出版历程
- 收稿日期: 2017-11-28
- 修回日期: 2018-04-16
- 发布日期: 2018-05-31

Building height to crown base models for Mongolian pine plantation based on simultaneous equations in Heilongjiang Province of northeastern China

School of Forestry, Northeast Forestry University, Harbin 150040, Heilongjiang, China

摘要

摘要:
目的基于黑龙江省帽儿山实验林场、横头山林场、孟家岗林场的61块樟子松人工林固定样地的5211株样木调查数据，构建了树高模型与枝下高模型的联立方程组。
方法首先，从8种常用的标准树高曲线，选出拟合效果较好的2个模型作为树高曲线的备选模型。再以5个枝下高预估模型作为基础模型，通过引入林木及林分变量(林木大小，竞争因子，立地条件)采用最优子集回归法筛选出3个变量少且拟合效果较好的模型作为枝下高备选模型。将树高曲线备选模型与枝下高备选模型分别两两联立，建立树高与枝下高联立方程组模型，采用似乎不相关回归(SUR)对模型参数进行求解。最后，对联立方程组进行评价。
结果树高(H)和枝下高(HCB)与林分断面积(G)和优势木平均高(H₀)呈正相关。最优的联立方程组预估树高时调整后相关系数(R_a²)为0.9520，均方根误差(RMSE)为1.17m；预估枝下高时的R_a²为0.9066，RMSE为1.36m，并且模型的各项检验指标数值较小。
结论整体来看，联立方程组的拟合效果较好，预估精度较高，同时联立方程组解决了树高与枝下高的内在相关性问题。本文所建立的含林分因子的树高模型与枝下高模型联立方程组可以很好地预估不同林分条件下樟子松人工林的树高和枝下高，为进一步研究樟子松树冠结构和动态提供了基础。
- 樟子松人工林 /
- 树高曲线 /
- 枝下高 /
- 似乎不相关回归 /
- 联立方程组
Abstract:
ObjectiveBased on the data of 5211 sample trees in 61 permanent sample plots in Mongolian pine plantations from Maoershan Experimental Forest Farm, Hengtoushan Forest Farm, Mengjiagang Forest Farm in Heilongjiang Province of northeastern China, the simultaneous equations for tree height model and height to crown base model were developed.
MethodAt first, 2 alternative height-diameter models had been selected by comparing the goodness of fit for 8 height-diameter models. From 5 basic height to crown base(HCB)models, 3 best HCB models including tree and stand variables (tree size, competition index, site condition) were selected as alternative models using the method of all subset regression. Based on the seeming unrelated regression (SUR), the parameters of the simultaneous equations model of height and HCB were estimated considering each kind of combinations for 2 alternative height-diameter models and 3 alternative HCB models, respectively. Finally, we evaluated the fitting effect of the simultaneous equation model.
ResultThe results showed that H and HCB were positively correlated with basal area (G) and average height of dominant tree (H₀). For the best simultaneous equations, the coefficient determination (R_a²) was 0.9520 and the root-mean-square error (RMSE) was 1.17m by fitting height (H), the R_a² was 0.9066, and RMSE was 1.36m by fitting HCB. The validation values of the best simultaneous equations were smaller.
ConclusionOn the whole, the simultaneous equations developed performed well in predicting the tree H and HCB simultaneously with the least predicting errors, and the model could handle correlations between tree H and HCB. The simultaneous equations considering stand variables developed in this paper could be suitable for predicting H and HCB for Mongolian pine plantations with different stand conditions and it will provide basis for future research on the crown structure and dynamics.
- Mongolian pine (Pinus sylvestris var. mogolica) plantation /
- tree height curve /
- height to crown base /
- seeming unrelated regression (SUR) /
- simultaneous equations

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图 1 联立方程组的树高预估值及枝下高预估值的残差分布

Figure 1. Residual distribution of predicted H and HCB of simultaneous equation model

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图 2 其他条件相同时(D₀=22.6cm, H₀=16m)不同竞争条件下树高和枝下高分别与胸径的关系曲线

Figure 2. Relationship between H and HCB with DBH considering D₀=22.6cm and H₀=16m for differentG

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图 3 其他条件相同时(D₀=22.6cm, G=33m²)不同立地条件下树高和枝下高分别与胸径的关系曲线

Figure 3. Relationship between H and HCB with DBH considering D₀=22.6cm and G=33m² for different H₀

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图 4 模型(7)和模型(25)拟合不同径阶树高的平均误差

Figure 4. ME obtained for diameter classes in the fitting H of model (7) and model (25)

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图 5 模型(22)和模型(25)拟合不同径阶树的枝下高的平均误差

Figure 5. ME obtained for diameter classes in the fitting HCB of model (22) and model (25)

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表 1 樟子松人工林样木及林分因子特征表

Table 1 Summary statistics of stand sample tree variables for Mongolian pine plantation

变量 Variable	平均值 Mean	最大值 Max.	最小值 Min.	标准差 S.D.
胸径Diameter at breast height (DBH)/cm	15.3	39.2	3.3	6.2
树高Tree height (H)/m	13.3	25.4	3.0	5.3
枝下高Height to crown base (HCB)/m	8.5	19.8	0.1	4.4
林分平均胸径Average stand DBH (D_g)/cm	18.3	33.4	6.3	6.5
林分平均高Mean stand height (H_m)/m	15.6	25.2	4.3	6.0
林分断面积/(m²·hm^-2) Stand basal area (G)/(m²·ha^-1)	33.1	48.0	9.6	8.2
大于对象木的断面积和/(m²·hm^-2) Basal area sum larger than subject tree (BAL)/(m ²·ha ^-1)	20.3	47.7	0.0	10.8
优势木平均胸径Mean DBH of dominant tree (D₀)/cm	23.0	10.0	9.4	6.0
优势木平均高Mean tree height of dominant tree (H₀)/m	16.0	24.7	5.2	5.2
年龄Stand age (A)/a	33	48	12	11.3
林分密度/(株·hm^-2) Stand density (SD)/(tree·ha^-1)	1 921	4 800	385	1 020.6

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表 2 8种树高曲线模型

Table 2 8 tree height-diameter functions

模型编号 Model No.	表达式 Expression	参考文献 Reference
(1)	$H=1.3+a_{0} G^{a_{1}}\left(1-e^{-a_{2} \times D B H}\right)$	[21]
(2)	$H=1.3+\left[a_{0}\left(\frac{1}{\mathrm{DBH}}-\frac{1}{D_{0}}\right)+\left(\frac{1}{H_{0}-1.3}\right)^{\frac{1}{3}}\right]^{-3}$	[22]
(3)	$H=1.3+\left(H_{0}-1.3\right)\left(\frac{\mathrm{DBH}}{D_{0}}\right)^{a_{0}}$	[23]
(4)	$H=1.3+\left(H_{0}-1.3\right) \frac{1-\mathrm{e}^{a_{0} \times \mathrm{DBH}}}{1-\mathrm{e}^{a_{0} D_{0}}}$	[23]
(5)	$H=1.3+\left(H_{0}-1.3\right) \mathrm{e}^{a_{0}\left(1-\frac{D_{\mathrm{g}}}{\mathrm{DBH}}\right.} )+a_{1}\left(\frac{1}{D_{\mathrm{g}}}-\frac{1}{\mathrm{DBH}}\right)$	[23]
(6)	$H=1.3+\left(H_{\mathrm{m}}-1.3\right) \mathrm{e}^{a_{0}\left(1-\frac{\mathrm{DBH}}{D_{\mathrm{g}}}\right.} )+a_{1}\left(\frac{\mathrm{DBH}}{D_{\mathrm{g}}}-\frac{1}{\mathrm{DBH}}\right)$	[23]
(7)	$H=1.3+\left(a_{0}+a_{1} H_{0}-a_{2} D_{\mathrm{g}}+a_{3} G\right) \mathrm{e}^{-a_{4} / \sqrt{\mathrm{DBH}}}$	[24]
(8)	$H=1.3+\left(a_{0}+a_{1} \times \mathrm{BAL}\right) \exp \left(-a_{2} / \mathrm{DBH}\right)$	[25]
注：a₀、a₁、a₂、a₃、a₄为模型参数。Notes:a₀, a₁, a₂, a₃, a₄ refer to model parameters.

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表 3 枝下高预估模型的基础形式

Table 3 Basic forms of predicting models for height to crown base

模型编号 Model No.	表达式 Expression	模型形式 Model form	因变量的范围 Range of dependent variable	参考文献 Reference
(9)	HCB=H(1.0－expX)	指数形式Exponential form	(-∞, H)	[10]
(10)	HCB=H(1.0－expX²)	指数形式Exponential form	(-∞, H)	[11]
(11)	HCB=H(1.0－c×expX^w)	指数形式Exponential form	(-∞, H)	[17]
(12)	HCB=H/(1.0+expX)	Logistic形式Logistic form	(0, H)	[11]
(13)	HCB=H/(1.0+c×exp X)^1/m	Logistic形式Logistic form	(0, H)	[17]
注：c、m、w为模型参数；X为关于林木大小、竞争因子、立地条件的函数。Notes:c, m and w refer to model parameters；X is the function about tree size competition index and site condition.

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表 4 树高曲线模型的参数估计值、拟合优度及检验结果

Table 4 Results of parameter estimates, goodness of fit and validation result of height-diameter models

模型编号 Model No.	参数 Parameter	参数估计值 Parameter estimate	标准误 Standard error	拟合优度 Goodness of fitting		检验结果 Validation result
模型编号 Model No.	参数 Parameter	参数估计值 Parameter estimate	标准误 Standard error	R_a²	RMSE/m	ME/m	MAE/m	TRE/%	MPE/%
(2)	a₀	1.0622	0.00955	0.9438	1.26	0.32	0.99	-2.19	6.42
(3)	a₀	0.4441	0.00370	0.9440	1.26	0.27	1.00	-1.94	6.39
(4)	a₀	-0.0769	0.000765	0.9441	1.26	0.20	0.99	-1.58	6.43
(6)	a₀	-0.0162	0.00656	0.9453	1.24	-0.08	0.95	-1.26	6.18
(6)	a₁	-0.1337	0.00552	0.9453	1.24	-0.08	0.95	-1.26	6.18
(7)	a₀	3.0443	1.3120	0.9520	1.17	-0.00	0.89	-0.43	5.90
	a₁	1.3103	0.0138
	a₂	2.6453	0.8350
	a₃	0.1984	0.0431
	a₄	2.3579	0.0493

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表 5 枝下高基础模型的参数估计值、拟合优度及检验结果

Table 5 Results of parameter estimates, goodness of fit and validations of basic height to crown base models

模型编号 Model No.	参数 Parameter	参数估计值 Parameter estimate	标准误 Standard error	拟合优度 Goodness of fitting		检验结果 Validation result
模型编号 Model No.	参数 Parameter	参数估计值 Parameter estimate	标准误 Standard error	R_a²	RMSE/m	ME/m	MAE/m	TRE/%	MPE/%
(9)	b₀	-0.877 7	0.018 3	0.8579	1.68	-0.28	1.45	-3.37	14.59
(9)	b₁	-0.0105	0.00865	0.8579	1.68	-0.28	1.45	-3.37	14.59
(10)	b₀	-0.978 8	0.010 8	0.856 8	1.69	-0.31	1.45	-3.72	14.52
(10)	b₁	-0.000273	0.000126	0.856 8	1.69	-0.31	1.45	-3.72	14.52
(11)	b₀	-0.978 8	0.011 4	0.856 8	1.69	-0.31	1.45	-3.72	14.52
(11)	b₁	-0.000 336	0.000 135	0.856 8	1.69	-0.31	1.45	-3.72	14.52
(12)	b₀	-0.3623	0.0287	0.8578	1.68	-0.28	1.45	-3.36	14.58
(12)	b₁	-0.015 6	0.001 51	0.8578	1.68	-0.28	1.45	-3.36	14.58
(13)	b₀	3.061 5	0.066 2	0.857 6	1.68	-0.28	1.45	-3.36	14.58
(13)	b₁	-0.0337	0.00332	0.857 6	1.68	-0.28	1.45	-3.36	14.58

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表 6 枝下高备选模型的拟合优度及检验结果

Table 6 Results of goodness of fitting and validations of height to crown base candidate models

模型编号 Model No.	表达式 Expression	拟合优度 Goodness of fitting		检验结果 Validation result
模型编号 Model No.	表达式 Expression	R_a²	RMSE/m	ME/m	MAE/m	TRE/%	MPE/%
(22)	HCB=H/(1+exp(b+c×DBH+dG+eH₀))	0.906 7	1.32	-0.01	1.01	-0.15	11.07
(23)	HCB=H/(1+exp(b+c×DBH+d×BAL+eH₀))	0.905 3	1.33	-0.01	1.02	-0.09	11.17
(24)	HCB=H/(1+exp(b+c×DBH+d×HDR+eH₀))	0.904 1	1.34	-0.01	1.03	-0.10	11.22

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表 7 联立方程组(25)的参数估计结果

Table 7 Results of parameter estimates for simultaneous equations model (25)

参数 Parameter	参数估计值 Parameter estimate	树高 Tree height (H)						枝下高 Height to crown base (HCB)
		拟合优度 Goodness of fitting		检验结果 Validation result				拟合优度 Goodness of fitting		检验结果 Validation result
		R_a²	RMSE/m	ME/m	MAE/m	TRE/%	MPE/%	R_a²	RMSE/m	ME/m	MAE/m	TRE/%	MPE/%
a₀	2.1629	0.9520	1.17	-0.00	0.89	1.67	6.34	0.906 6	1.36	0.02	1.01	-0.26	11.07
a₁	1.310 7
a₂	2.111 6
a₃	0.171 3
a₄	2.348 9
b	0.554 0
c	0.038 7
d	-0.015 2
e	-0.070 6

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