Compatibility of height-diameter curve equations for <i>Cunninghamia lanceolata</i> stands in Hunan Province

LIU Weiyi; DENG Huafeng; RAN Qixiang; HUANG Guosheng; WANG Xuejun

doi:10.11833/j.issn.2095-0756.2017.06.012

Volume 34 Issue 6

Nov. 2017

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LIU Weiyi, DENG Huafeng, RAN Qixiang, HUANG Guosheng, WANG Xuejun. Compatibility of height-diameter curve equations for Cunninghamia lanceolata stands in Hunan Province[J]. Journal of Zhejiang A&F University, 2017, 34(6): 1051-1058. doi: 10.11833/j.issn.2095-0756.2017.06.012

Citation:

LIU Weiyi, DENG Huafeng, RAN Qixiang, HUANG Guosheng, WANG Xuejun. Compatibility of height-diameter curve equations for Cunninghamia lanceolata stands in Hunan Province[J]. Journal of Zhejiang A&F University, 2017, 34(6): 1051-1058. doi: 10.11833/j.issn.2095-0756.2017.06.012

Compatibility of height-diameter curve equations for Cunninghamia lanceolata stands in Hunan Province

doi: 10.11833/j.issn.2095-0756.2017.06.012

1.
College of Foresty, Beijing Forestry University, Beijing 100083, China
2.
Academy of Forest Inventory and Planning, State Forestry Administration, Beijing 100714, China

Received Date: 2016-11-17
Rev Recd Date: 2017-03-19
Publish Date: 2017-12-20

Abstract

To improve the forest production mode of Cunninghamia lanceolata artificial and natural forests in Hunan Province, their height-diameter curves were studied. Based on periodic inventory data of C. lanceolata in Hunan, six common empirical models were used to fit height-growth and diameter-growth curves, and selected the best one from them. On the basis of the optimal model, differences in stand origin were considered and compatible height-diameter curve equations for a group of stands with different stand origins were built by introducing dummy variables and using the measurement error model method. These growth status for stand in different origins were compared and analyzed by drawing scatter plots. Results showed that the fitted coefficient of determination for the C. lanceolata height-growth model with the dummy variables was better than this without dummy variables. The fitted coefficients for the tree diameter-growth curve was 0.820 5 with prediction accuracy of 98.30%, and for tree height was 0.758 0 with prediction accuracy of 98.01%. Also, D_MAD and E_RMSE indicators were superior to the traditional methods. The growth progress of C. lanceolata artificial and natural forests were the same; however, in the young stage, tree height and diameter growth rate of the artificial forest were higher than the natural forest. Moreover, tree height and diameter in the artificial forest reached a maximum value earlier than in natural forests. In conclusion, the model of tree height growth was established using the model with dummy variables, which not only improved the accuracy of the model but also solved the problem of incompatibility between models.
- forest mensuration,
- Cunninghamia lanceolata,
- model with measurement error,
- dummy variable,
- height-diameter curve

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沈阳化工大学材料科学与工程学院沈阳 110142

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树高和胸径是反映林木生长状况的2个基本且重要的数量指标，然而在外业调查中，树高测量困难，耗时费力且测量结果误差较大。目前，获取树高数据的方法主要有利用测高器直接测量和通过树高曲线模型间接推算2种，但利用测高器进行测量时对该点的地势、立地等有一定的限制要求，且测量结果受人为操作影响较大^[1]，因此，目前使用较多的还是树高曲线模型推算法。与树高获取相比，胸径数据具有获取简单、准确等优点，且胸径和树高间存在一定的生物学关系，利用这种密不可分的联系建立合理的树高曲线模型则能有效地推算林分的树高。国内外学者对树高生长模型已经进行了广泛研究，如关于落叶松Larix gmelinii^[2-4]，马尾松Pinus massoniana^[5]和红松Pinus koraiensis^[6]等的树高曲线的报道已有很多，但这些研究多集中在单木水平^[7-10]和径阶水平层面，而像2015年高东启等^[11]考虑树高、胸径与年龄三者之间的相关关系从而建立蒙古栎Quercus mongolica林分树高曲线这样从林分水平对树高曲线进行的研究较少。杉木Cunninghamia lanceolata是中国特有的针叶商品材树种，在中国南方地区栽培广泛，具有生长快、材质好、经济价值高等优点。众多学者对杉木的胸径、树高、材积生长规律等方面进行了大量深入研究^[12-13]。2012年魏晓慧等^[14]针对福建杉木人工林建立了Richards树高生长模型，模型的预估能力大幅度提升。2016年姜鹏等^[15]先按生长势对杉木成熟林进行分级再对其树高曲线进行研究，对指导杉木的营林生产起到了积极作用。然而，对不同林分起源的杉木树高曲线研究较少。笔者在前人研究的基础上，以湖南省杉木为例，引入以林分起源为特征的哑变量，运用度量误差模型方法，将天然林和人工林合并起来建立林分树高曲线方程组，使得树高、胸径、年龄3个因子互相预测时保持相容性和一致性，并将天然林和人工林不同林龄段的生长状况进行对比分析，以期为湖南省杉木林分的生长和收获预估提供可靠理论依据。

1. 研究区概况

湖南省地处长江中游，25°46′~29°38′N，108°47′~114°15′E，全省地貌以山地、丘陵为主。气候类型为大陆性亚热带季风气候，全年日照充足，热量丰富，雨水充沛。湖南省植物资源丰富，种类繁多，主要树种有马尾松，水杉Metasequoia glyptostroboides，樟树Cinnamomum camphora等，此外有银杏Ginkgo biloba，红豆杉Taxus chinensis，珙桐Davidia involucrata等60多种珍贵树种。杉木在湖南全省均有分布，其西南地区是全国著名产地之一。

2. 数据与方法

2.1. 数据来源

本次研究所用数据为湖南省第6，7，8次3期一类连续清查数据。从数据库中挑选杉木样地，所选样地均为固定复测样地，面积为0.066 7 hm²^[16]。剔去记载不详、数据离散异常的样地，最后选出157块样地，其中天然林62块，人工林95块。样地的基本情况如表 1所示。从157块样地中随机抽取119块建模，其中天然林47块，人工林72块；剩余38块样地用于检验，其中天然林15块，人工林23块。

样地类别	样地数/块	平均年龄/a			平均胸径/cm			平均树高/m
样地类别	样地数/块	最小	最大	平均	最小	最大	平均	最小	最大	平均
天然林	62	8	47	21	7.1	17.9	11.5	5.3	12.4	8.6
人工林	95	6	50	22	6.5	18.9	12.1	4.5	14.3	9.1
建模样地	119	6	50	23	6.5	18.9	12.0	4.5	13.9	9.0
检验样地	38	6	40	18	7.2	17.5	11.3	4.8	14.3	8.6

Table 1. Basic situation of plots

2.2. 树高生长曲线与胸径生长曲线方程

用来拟合林分生长过程曲线的方程很多，有Schumacher模型、对数型模型、幂函数模型等经验方程和Logistic模型、Mitscherlich模型、Korf模型等理论方程。本研究选取6种常用经验模型对杉木树高生长曲线和胸径生长曲线进行初步拟合，具体模型如表 2所示，同时考虑到林分生长到胸高位置需要一定的年限，故将经验模型进行一定变换作为最终拟合模型。现以Richards方程为例，林分树高曲线方程和胸径生长过程曲线方程的表达式如下：

模型序号及名称	模型形式
(1) Logistic模型	y=a/[1+bexp(-ct)]
(2) Richards模型	y=a [1-exp(-ct)]^b
(3) Weibull模型	y=a[1-exp(-bt)]
(4) Korf模型	y=a exp(-bt^c)
(5) Mitscherlich模型	y=a [1-bexp(-ct)]
(6) Schumacher模型	y=aexp(-b/t)]

Table 2. Common experience models of stand growth process

式（1）和式（2）中：D为林分平均胸径（cm）；H为林分平均树高（m）；T为林分平均年龄（a）；T_1.3为林木生长到胸高时的年龄（由林分平均树高生长曲线推算而得）；a，b，c为待估参数。

2.3. 非线性度量误差联立方程组

一般情况下，回归模型都是在自变量的观测值不含误差而因变量的观测值有误差的条件下成立的。导致误差的因素有多种，如抽样方法和测量仪器等，这类误差统称为度量误差。如果自变量和因变量的观测值中均含有误差，则不能再用普通的回归模型估算方法，而必须采用度量误差模型方法^[17]。通常在研究林分树高曲线时，假设不含误差的自变量林分胸径的观测值是可能存在抽样或观测等度量误差的，因为林分胸径一般由单木胸径推算而来，此时应使用度量误差方法对其进行研究。以往对于树高曲线和胸径生长过程曲线都是单独建模的，很少考虑它们之间的相关性，事实上林分的平均胸径、平均树高和年龄三者不是相互孤立而是相互关联的，以林分平均胸径D作为桥梁，就可以把树高曲线、胸径生长过程曲线和树高生长过程曲线三者紧密地联系到一起，从而达到保证三者相容性的目的，为此可采用误差变量联立方程组^[18-20]。

多元非线性误差变量联立方程组，也称非线性度量误差模型的向量形式为^[21]:

式（3）中：x_i是q维无误差变量的观测数据，y_i是p维误差变量的观测数据，f是m维向量函数，Y_i是y_i的未知真值，c是参数，e_i是度量误差，E（e_i）是e_i的期望，cov（e_i）=σ²φ渍为误差的协方差矩阵，其中φ渍是e_i的误差结构矩阵，σ²为估计误差。

本研究中将式（1）和式（2）联立构成式（4）树高曲线联立方程组。式（4）中的林分胸径（D）是由单木胸径推算而来的，林分树高（H）是估测的，因此将D和H作为误差变量，T作为无误差变量。

2.4. 哑变量的引入

哑变量又称虚拟变量，是自变量经量化后所得的，一般取值0或者1^[21-22]。在本研究中引入哑变量，可以将人工林和天然林2种类型的林分用定性代码0或1来表示。将其中一种起源的林分样地定义为S_i，将定性数据S_i转化为（0，1）形式。

式（5）中：i=1, 2；S₁，S₂分别表示人工林分和天然林分的定性代码。

以Richards方程为例，通过试验知在第1个参数上加入哑变量时模型收敛且效果最好，因此，最终加入哑变量的联立方程组形式如下：

式（6）中：D为胸径；H为树高；T为林龄；T_1.3为林木生长到胸高时的年龄；S₁和S₂分别为天然林和人工林的定性代码；b₁~b₈为待估参数。

2.5. 参数估计与模型检验

先运用Excel和SPSS对数据进行初步处理，再利用Forstat中的非线性度量误差联立方程组方法进行参数估计。本研究采用并计算平均绝对偏差（D_MAD）, 均方根误差（E_RMSE）, 决定系数（R²）和预估精度（P）等几个指标对不同方法的模型进行检验评价分析。

式（7）~（10）中：y_i为林分胸径（D）或林分树高（H）的实测值，ŷ_i为林分胸径（D）或林分树高（H）的模型预估值，$ \hat{\bar{y}}$为林分胸径（D）或林分树高（H）的模型预估值的平均值，n为样本数，p为模型参数个数，t_0.05为置信水平α=0.05时t分布值。

4. 结论与讨论

本研究在分析比较初步拟合结果后确定：Richards方程为最优基础模型，引入代表林分起源的哑变量，采用度量误差模型方法，建立了湖南省杉木林分的树高曲线方程组。通过对比分析得出了哑变量方法优于传统方法的结论，所建立的含哑变量的方程组中树高曲线和胸径生长过程曲线的拟合决定系数为0.758 0和0.820 5，且对杉木林分的平均树高和平均胸径的预估精度均达到了98.00%以上。利用特征为林分起源的哑变量将天然林人工林合并建模，不仅减少了建模的工作量还解决了模型间不相容的问题，而且所建立的树高曲线联立方程组保证了树高、胸径和林龄三者在生长模型预测中的相容性。综上所述，采用含哑变量的度量误差模型方法建立的树高曲线联立方程组在考虑树高、胸径的度量误差的基础上解决了多个相容性的问题，使模型具有更高的拟合精度和更好的预估能力，从而为从林分水平预测林木生长情况和统筹管理杉木林分提供了参考依据。

本研究拟合得到的胸径生长过程曲线和树高生长过程曲线的总体走势与前人研究结果一致^{[14, 23-25]}，都呈现出生长速率先增加然后逐渐趋于平稳的倒“J”型，但由于建模数据有限，天然林和成、过熟龄阶段林分的数据样本较少，引入哑变量后方程的拟合系数没有出现大幅度的提升。在下一步研究中，应增加建模的数据量以提高拟合系数和保证树高曲线的完整性，同时还应考虑时间效应和样地效应对模型的影响以期建立更加合理可信的模型，为杉木天然林和人工林的经营提供理论基础。

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模型序号及名称	R²	D_MAD	E_RMSE	P/%
(1) Logistic模型	0.778 6	0.795 9	0.912 3	96.79
(2) Richards模型	0.789 9	0.700 3	0.856 7	97.25
(3) Weibull模型	0.788 5	0.705 7	0.859 6	97.23
(4) Korf模型	0.785 1	0.725 6	0.889 3	97.15
(5) Mitscherlich模型	0.787 6	0.708 9	0.862 9	97.21
(6) Schumacher模型	0.782 7	0.706 9	0.871 3	97.20

模型序号及名称	R²	D_MAD	E_RMSE	P/%
(1) Logistic模型	0.812 3	0.925 4	1.154 6	98.25
(2) Richards模型	0.824 2	0.898 0	1.076 6	98.35
(3) Weibull模型	0.753 4	1.217 1	1.511 7	97.67
(4) Korf模型	0.814 7	0.908 7	1.111 9	98.30
(5) Mitscherlich模型	0.811 5	0.933 2	1.165 4	98.21
(6) Schumacher模型	0.778 2	1.155 6	1.456 7	97.87

模型公式	参数
H=1.3+a₁(1-exp (-a₂D))^a₃	a₁=37.212 866	a₂=0.029 574	a₃=1.303 845
D=a₄[1-exp(-a₅(T-T_1.3))]^a₆	a₄=98.224 975	a₅=0.000 099	a₆=0.331 043
H=1.3+(b₁S₁+b₂S₂)(1-exp(-b₃D))^b₄	b₁=34.500 583	b₂=35.032 262	b₃=0.032 024	b₄=1.320 568
D=(b₅S₁+b₆S₂) [1-exp(-b₇(T-T_1.3))]^b₈	b₅=85.512 397	b₆=87.866 852	b₇=0.000 138	b₈=0.329 027

方法	模型	建模数据				检验数据
方法	模型	R²	D_MAD	E_RMSE	P/%	R²	D_MAD	E_RMSE	P/%
传统方法	树高曲线	0.740 3	0.765 2	0.952 5	96.54	0.700 2	0.8107	1.121 7	95.07
	胸径生长曲线	0.814 6	0.909 5	1.100 9	97.20	0.826 3	0.848 6	0.821 6	97.32
哑变量方法	树高曲线	0.758 0	0.702 4	0.898 2	98.01	0.724 6	0.775 2	0.967 4	95.74
	胸径生长曲线	0.820 5	0.810 3	0.982 1	98.30	0.834 9	0.747 5	0.9143	97.97

Compatibility of height-diameter curve equations for Cunninghamia lanceolata stands in Hunan Province

doi: 10.11833/j.issn.2095-0756.2017.06.012