天然栎类阔叶混交林林分平均高与平均胸径关系模型

娄明华; 张会儒; 雷相东; 白超; 杨同辉

doi:10.12171/j.1000-1522.20190463

天然栎类阔叶混交林林分平均高与平均胸径关系模型

1.
宁波市农业科学研究院，浙江宁波 315040
2.
中国林业科学研究院资源信息研究所，北京 100091
3.
宁波市测绘和遥感技术研究院，浙江宁波 315042

基金项目: 国家自然科学基金项目（31800539、31700563），宁波市科学技术局公益性计划项目（2019C10084）

详细信息

作者简介:
娄明华，博士，助理研究员。主要研究方向：森林可持续经营。Email：mhlou1987@163.com　地址：315040 浙江省宁波市鄞州区德厚街19号宁波市农业科学研究院

责任作者:
张会儒，研究员，博士生导师。研究方向：森林可持续经营。Email：huiru@ifrit.ac.cn　地址：100091 北京市海淀区香山路东小府1号中国林业科学研究院资源信息研究所

中图分类号: S758.5
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出版历程
- 收稿日期: 2019-12-10
- 修回日期: 2020-01-01
- 网络出版日期: 2020-09-16
- 发布日期: 2020-09-29

Relationship model between stand mean height and mean DBH for natural Quercus spp. broadleaved mixed stands

1.
Ningbo Academy of Agricultural Sciences, Ningbo 315040, Zhejiang, China
2.
Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China
3.
Ningbo Institute of Surveying and Mapping & Remote Sensing, Ningbo 315042, Zhejiang, China

摘要

摘要:
目的考虑天然混交林的林分密度、直径结构和树种结构，基于代数差分方程构建最适宜的林分平均高与平均胸径关系模型，为天然混交林的立地生产力估计与可持续经营提供理论依据。
方法以吉林省天然栎类阔叶混交林为研究对象，利用4期连续调查固定样地数据，基于Richards方程构建4种数据结构类型即typeC、typeD、typeE和typeF的基础代数差分方程，比较分析得出最优数据结构类型；基于最优数据结构类型，以5个林分密度指标即林木株数（N）、林分断面积（BA）、林分密度指数（SDIr）、可加林分密度指数（SDIa）和郁闭度（CD），5个直径多样性指数即Shannon均匀度指数（ShaI）、Simpson指数（SimI）、McIntosh均匀度指数（MceI）、Gini系数（GinI）和Berger-Parker指数（BerI），4个树种多样性指数即ShaI、SimI、MceI和BerI，构建并比较分析不同多样性代数差分方程的差异，得出最佳方程为最适宜林分平均高与平均胸径关系模型。
结果不同数据结构类型的建模效果由好到差排序：typeD > typeC > typeF > typeE。除了typeC，其他3个数据结构类型的模型参数b和r均显著不为零（P < 0.01），说明typeD拟合的模型参数检验效果最佳。林分密度指标SDIr的建模效果最好。无论使用哪个林分密度指标，其模型参数b₀、r和cSD均显著（P < 0.01），说明5个林分密度指标的模型参数检验效果均比较理想。直径多样性指数ShaI的建模效果最好。除了GinI，其他4个直径多样性指数的模型参数b₀、r、cSDIr和cDI均显著（P < 0.01），表明ShaI、SimI、MceI和BerI均为较理想的直径多样性指数。4个树种多样性指数的建模拟合效果和检验数据效果差别不大。BerI的模型参数b₀、r、cSDIr、cShaI和cSP均显著（P < 0.01），说明BerI是较理想的树种多样性指数。ShaI、SimI和MceI的模型参数b₀、r、cSDIr、cShaI和cSP均不能同时达到0.05显著水平，说明ShaI、SimI和MceI是不理想的树种多样性指数。
结论 typeD是最优的数据结构类型，林分密度、直径多样性和树种多样性对模型均有影响。其中，林分密度指标SDIr、直径多样性指数ShaI和树种多样性指数BerI建立的多样性代数差分方程拟合效果最佳，为最适宜的天然栎类阔叶混交林林分平均高与平均胸径关系模型。
- 林分平均高 /
- 数据结构类型 /
- 林分密度 /
- 直径多样性 /
- 树种多样性 /
- 代数差分方程
Abstract:
Objective Considering stand density, diameter structure and tree species structure, the optimal model for stand mean height and mean DBH relationship was constructed using algebraic difference approach. It may provide a theoretical basis for site productivity estimation and sustainable management of natural mixed forests.
Method Base algebraic difference approaches were modeled with 4 different data structure types, i.e. typeC, typeD, typeE and typeF based on Richards model using 4 inventory data of permanent sample plots in natural Quercus spp. broadleaved mixed stands. The 4 different base algebraic difference approaches were comparatively analyzed to get the optimal data structure type. Algebraic difference approach of diversity indices was constructed based on the optimal data structure type using 5 different stand density indices, including tree number (N), stand basal area (BA), stand density index (SDIr), additive stand density index (SDIa) and canopy density (CD), and the 5 different diameter diversity indices including Shannon evenness index (ShaI), Simpson index (SimI), McIntosh evenness index (MceI), Gini coefficient (GinI) and Berger-Parker index (BerI), and the 4 different species diversity indices including ShaI, SimI, MceI and BerI. The algebraic difference approach of diversity indices was comparatively analyzed to obtain the optimize algebraic difference approaches, i.e. the optimize stand mean height and mean DBH relationship.
Result Model fitting effects of calibration data in different data structure types were sorted from best to worst, and the ranking was: typeD > typeC > typeF > typeE. Except for typeC, model coefficients b and r of the other three data structure types were significant (P < 0.01), indicating that the model fitting effects of typeD were the best. Model fitting effects of SDIr were the best. Model coefficients b₀, r and cSD were significant (P < 0.01), regardless of which stand density index was used, indicating that model fitting effects of the 5 different stand density indices were reasonable. Model fitting effect of ShaI was the best. Except for GinI, model coefficients b₀, r, cSDIr and cDI of the other 4 diameter diversity indices were significant (P < 0.01), indicating that model fitting effects of ShaI, SimI, MceI and BerI were reasonable. Model fitting and validation effects had little difference among the 4 species diversity indices. Model coefficients b₀, r, cSDIr, cShaI and cSP of BerI were significant (P < 0.01), indicating that BerI was reasonable. However, model coefficients b₀, r, cSDIr, cShaI and cSP of ShaI, SimI and MceI were not significant at the level of 0.05, indicating that ShaI, SimI and MceI were not reasonable.
Conclusion TypeD is the best data structure type, stand density, diameter diversity and species diversity were significant for algebraic difference approach. Moreover, the model fitting effects of algebraic difference approach within SDIr, ShaI and BerI are the best, which is served as the optimize stand mean height and mean DBH relationship in natural Quercus spp. broadleaved mixed stands.
- stand mean height /
- data structure type /
- stand density /
- diameter diversity /
- tree species diversity /
- algebraic difference approach

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图 1 多次调查固定样地位置图

Figure 1. Location diagram of permanent sample plots for repeated surveying

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图 2 林分平均高与林分平均胸径的散点图

Figure 2. Scatter diagram of stand mean height and stand mean DBH

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图 3 多样性代数差分方程建模流程图

ModeO代表基础代数差分方程，即公式（3）。Sig. test表示参数显著性检验，Sig.表示参数检验显著，Not Sig.表示参数检验不显著，None表示模型中不加入f_SD、f_DD和f_SPD。 ModeO represents the basic algebraic difference equation, i.e. formula (3). Sig. test denotes significance test of parameters, Sig. denotes parameter test is significant, Not Sig. denotes parameter test is not significant, None denotes that f_SD, f_DD and f_SPD are not added into the model.

Figure 3. Modeling flowchart of diversity algebraic differential equations

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图 4 不同数据结构类型的残差图

Figure 4. Residual plots for different data structure types

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表 1 229个连续调查固定样地统计结果

Table 1 Summary statistics for 229 permanent sample plots inventory

特征值 Characteristic value	建模数据 Calibration data	检验数据 Validation data
连续调查2次 Continuously investigating twice	35	18
连续调查3次 Continuously investigating three times	57	23
连续调查4次 Continuously investigating four times	55	41
总样地个数 Total number of sample plots	147	82
林分年龄 Stand age	55 ± 31 (10, 152)	57 ± 28 (6, 149)
林分平均高 Stand mean height/m	13.0 ± 4.7 (2.0, 24.0)	13.8 ± 3.9 (5.0, 22.2)
林分平均直径 Stand mean diameter/cm	14.8 ± 4.9 (6.1, 28.9)	15.6 ± 4.2 (6.0, 25.9)
林分优势直径 Stand dominant diameter/cm	28.6 ± 11.3 (6.9, 57.7)	30.6 ± 9.3 (6.8, 56.4)
林木株数(N)/(株·hm⁻²) Tree number (N) /(plant·ha⁻¹)	1 133 ± 483 (200, 2 717)	1 100 ± 433 (250, 2267)
林分断面积(BA)/(m²·hm⁻²) Stand basal area (BA)/(m²·ha⁻¹)	20.100 ± 11.700 (0.833, 53.833)	20.800 ± 10.350 (1.150, 57.133)
注：小括号内的值表示范围，下同。连续调查是指前后两次调查的时间间隔为5年。Notes: values in parentheses denote scope, the same below. Continuously investigating refers to the time interval of 5 years between two surveys.

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表 2 林分密度指标

Table 2 Stand density indices

林分密度指标 Stand density index	林分密度指数 Stand density index (SDIr)	可加林分密度指数 Additive stand density index (SDIa)
公式 Formula	${\rm{SDIr} } = n{\left( {\dfrac{ { {D_{\rm{g} } } }}{ { {D_0} } } } \right)^{1.605} }, \; {D_{\rm{g} } } = \sqrt {\dfrac{1}{n}\displaystyle\sum\limits_{i = 1}^nd_i^2}$	${\rm{SDIa}} = \displaystyle\sum\limits_{i = 1}^n{\left( {\dfrac{{{d_i}}}{{{D_0}}}} \right)^{1.605}}$
注：n表示林木株数，d_i表示第i株林木的胸径，D_g表示林分平均胸径，D₀表示标准直径。Notes: n is the number of trees, d_i is the DBH of tree i, D_g is stand mean DBH, D₀ is standard diameter.

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表 3 229个连续调查固定样地的林分密度指标

Table 3 Stand density indices for 229 permanent sample plots inventory

林分密度指标 Stand density index	建模数据 Calibration data	检验数据 Validation data
林木株数(N)/(株·hm⁻²) Tree number(N)/(stem·ha⁻¹)	1133 ± 483 (200, 2717)	1100 ± 433 (250, 2267)
林分断面积(BA)/(m²·hm⁻²) Stand basal area(BA)/(m²·ha⁻¹)	20.100 ± 11.700 (0.833, 53.833)	20.800 ± 10.350 (1.150, 57.133)
SDIa	636.050 ± 315.833 (42.450, 1382.650)	651.817 ± 279.083 (57.117, 1522.033)
SDIr	696.250 ± 355.717 (42.650, 1540.833)	717.633 ± 318.400 (58.650, 1797.983)
郁闭度 Canopy density (CD)	0.8 ± 0.2 (0.2, 1.0)	0.8 ± 0.1 (0.3, 1.0)

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表 4 5个多样性指数

Table 4 Five diversity indices

多样性指数 Diversity index	公式 Formula	适用类型 Applicable type
均匀度指数 Shannon evenness index (ShaI)	${\rm{ShaI} } = \dfrac{ { - \displaystyle\sum\limits_{i = 1}^S{p_i}{\rm ln} { {p_i} } } }{ { {\rm ln} S } }$	直径多样性，树种多样性 Diameter diversity , tree species diversity
Simpson指数 Simpson index (SimI)	${\rm{SimI}} = 1 - \displaystyle\sum\limits_{i = 1}^S {p_i^2}$	直径多样性，树种多样性 Diameter diversity, tree species diversity
McIntosh均匀度指数 McIntosh evenness index (MceI)	${\rm{MceI} } = \dfrac{ { {\rm{BA} } - \sqrt {\displaystyle\sum\limits_{i = 1}^S {\rm{b} } } {\rm{a} }_i^2} }{ { {\rm{BA} } - \dfrac{ { {\rm{BA} } } }{ {\sqrt S } } } }$	直径多样性，树种多样性 Diameter diversity, tree species diversity
Berger-Parker指数 Berger-Parker index (BerI)	${\rm{BerI} } = 1 - \dfrac{ { {\rm{b} }{ {\rm{a} }_{\rm max} } } }{ { {\rm{BA} } } }$	直径多样性，树种多样性 Diameter diversity, tree species diversity
Gini系数 Gini coefficient (GinI)	${\rm{GinI} } = \dfrac{ {\displaystyle\sum\limits_{j = 1}^n {\left( {2j - n - 1} \right)} {\rm{b} }{ {\rm{a} }_{{j} } } } }{ {\displaystyle\sum\limits_{j = 1}^n {\rm{b} } { {\rm{a} }_j}\left( {n - 1} \right)} }$	直径多样性 Diameter diversity
注：p_i在直径多样性指数中表示第i个径阶的断面积比例，在树种多样性指数中表示第i个树种的株数比例；S在直径多样性指数中表示径阶数，在树种多样性指数中表示树种数；BA表示林分总断面积；ba_i表示第i个径阶的断面积；ba_j表示第j株林木的断面积；ba_max表示断面积最大所在径阶的断面积；n表示林木总株数。Notes: p_i is the proportion of basal area in diameter class i within diameter diversity index, or p_i is the proportion of tree number in tree species i within species diversity index; S is the number of diameter classes within diameter diversity index, or S is the number of tree species within species diversity index; BA is stand basal area; ba_i is the basal area in the diameter class i; ba_j is the basal area of the tree; ba_max is the basal area in the diameter class with the largest basal area; n is the total number of trees.

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表 5 229个连续调查固定样地的多样性指数

Table 5 Diversity indices for 229 permanent sample plots inventory

类型 Type	多样性指数 Diversity index	建模数据 Calibration data	检验数据 Validation data
	ShaI	0.900 ± 0.069 (0.490, 0.979)	0.904 ± 0.077 (0.421, 0.979)
	SimI	0.861 ± 0.097 (0.383, 0.937)	0.874 ± 0.105 (0.265, 0.937)
直径多样性 Diameter diversity	McI	0.903 ± 0.085 (0.394, 0.980)	0.909 ± 0.095 (0.323, 0.981)
	BerI	0.779 ± 0.122 (0.239, 0.901)	0.798 ± 0.125 (0.157, 0.906)
	GinI	0.543 ± 0.132 (0.142, 0.791)	0.573 ± 0.118 (0.105, 0.734)
树种多样性 Tree species diversity	ShaI	0.947 ± 0.068 (0.588, 0.998)	0.957 ± 0.062 (0.617, 1.000)
	SimI	0.961 ± 0.051 (0.611, 0.988)	0.965 ± 0.045 (0.675, 0.986)
	McI	0.949 ± 0.081 (0.488, 0.998)	0.959 ± 0.076 (0.537, 1.000)
	BerI	0.903 ± 0.102 (0.385, 0.975)	0.917 ± 0.095 (0.440, 0.973)

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表 6 参数b的组合模式

Table 6 Composition model of parameter b

表达式 Expression	说明 Description
${f}_{x}=1+{c}_{x}\dfrac{x-\bar{x} }{\bar{x} }$	参数标准化 Standard parameter
$b={b}_{0} {f}_{\rm{SD} }$	含SD的参数b Parameter b including SD
$b={b}_{0} {f}_{\rm{SD} } {f}_{\rm{DD} }$	含SD和DD的参数b Parameter b including SD and DD
$b={b}_{0} {f}_{\rm{DD} }$	含DD的参数b Parameter b including DD
$b={b}_{0} {f}_{\rm{SD} } {f}_{\rm{DD} } {f}_{\rm{SPD} }$	含SD、DD和SPD的参数b Parameter b including SD, DD and SPD
$b={b}_{0} {f}_{\rm{SD} } {f}_{\rm{SPD} }$	含SD和SPD的参数b Parameter b including SD and SPD
$b={b}_{0} {f}_{\rm{DD} } {f}_{\rm{SPD} }$	含DD和SPD的参数b Parameter b including DD and SPD
$b={b}_{0} {f}_{\rm{SPD} }$	含SPD的参数b Parameter b including SPD
注：b₀和c_x表示参数，SD表示林分密度指标，DD表示直径多样性指数，SPD表示树种多样性指数，f_x表示标准化函数，f_SD表示标准化的林分密度指标，f_DD表示标准化的直径多样性指数，f_SPD表示标准化的树种多样性指数。Notes: b₀ and c_x denote parameters, SD denotes stand density index, DD denotes diameter diversity index, SPD denotes tree species diversity index, f_x denotes standardize function, f_SD denotes standardize stand density index, f_DD denotes standardize diameter diversity index, f_SPD denotes standardize species diversity index.

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表 7 多样性代数差分方程

Table 7 Algebraic differential equations of diversity

模型 Model	方程 Equation	说明 Description
ModeA	${H}_{ {\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-{b}_{0}{f}_{\rm{SD} }{D}_{ {\rm{g} },ik})}{1-\exp (-{b}_{0}{f}_{\rm{SD} }{D}_{ {\rm{g} },ij})}\right)}^{r{b}_{0}{f}_{\rm{SD} } }+{\varepsilon }_{ik}$	含SD Including SD
ModeB	${H}_{{\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-{b}_{0}{f}_{\rm{SD} }{f}_{\rm{DD} }{D}_{ {\rm{g} },ik})}{1-\exp (-{b}_{0}{f}_{\rm{SD} }{f}_{\rm{DD} }{D}_{ {\rm{g} },ij})}\right)}^{r{b}_{0}{f}_{\rm{SD} }{f}_{\rm{DD} } }+{\varepsilon }_{ik}$	含SD和DD Including SD and DD
ModeC	${H}_{{\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-{b}_{0}{f}_{\rm{DD} }{D}_{ {\rm{g} },ik})}{1-\exp (-{b}_{0}{f}_{\rm{DD} }{D}_{ {\rm{g} },ij})}\right)}^{r{b}_{0}{f}_{\rm{DD} } }+{\varepsilon }_{ik}$	含DD Including DD
ModeD	${H}_{{\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-{b}_{0}{f}_{\rm{SD} }{f}_{\rm{DD} }{f}_{\rm{SPD} }{D}_{ {\rm{g} },ik})}{1-\exp (-{b}_{0}{f}_{\rm{SD} }{f}_{\rm{DD} }{f}_{\rm{SPD} }{D}_{ {\rm{g} },ij})}\right)}^{r{b}_{0}{f}_{\rm{SD} }{f}_{\rm{DD} }{f}_{\rm{SPD} } }+{\varepsilon }_{ik}$	含SD、DD和SPD Including SD, DD and SPD
ModeE	${H}_{{\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-{b}_{0}{f}_{\rm{SD} }{f}_{\rm{SPD} }{D}_{ {\rm{g} },ik})}{1-\exp (-{b}_{0}{f}_{\rm{SD} }{f}_{\rm{SPD} }{D}_{ {\rm{g} },ij})}\right)}^{r{b}_{0}{f}_{\rm{SD} }{f}_{\rm{SPD} } }+{\varepsilon }_{ik}$	含SD和SPD Including SD and SPD
ModeF	${H}_{{\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-{b}_{0}{f}_{\rm{DD} }{f}_{\rm{SPD} }{D}_{ {\rm{g} },ik})}{1-\exp (-{b}_{0}{f}_{\rm{DD} }{f}_{\rm{SPD} }{D}_{ {\rm{g} },ij})}\right)}^{r{b}_{0}{f}_{\rm{DD} }{f}_{\rm{SPD} } }+{\varepsilon }_{ik}$	含DD和SPD Including DD and SPD
ModeG	${H}_{{\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-{b}_{0}{f}_{\rm{SPD} }{D}_{ {\rm{g} },ik})}{1-\exp (-{b}_{0}{f}_{\rm{SPD} }{D}_{ {\rm{g} },ij})}\right)}^{r{b}_{0}{f}_{\rm{SPD} } }+{\varepsilon }_{ik}$	含SPD Including SPD

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表 8 具有4次观测数据的6种数据结构类型

Table 8 Six different types of data structure with four measurements

typeA	typeB	typeC	typeD	typeE	typeF
(H_m,i1, D_g,i1), (H_m,i4, D_g,i4)	(H_m,i1, D_g,i1), (H_m,i4, D_g,i4)	(H_m,i1, D_g,i1), (H_m,i2, D_g,i2)	(H_m,i1, D_g,i1), (H_m,i2, D_g,i2)	(H_m,i1, D_g,i1), (H_m,i2, D_g,i2)	(H_m,i1, D_g,i1), (H_m,i2, D_g,i2)
	(H_m,i4, D_g,i4), (H_m,i1 , D_g,i1)	(H_m,i2, D_g,i2), (H_m,i3, D_g,i3)	(H_m,i2, D_g,i2), (H_m,i1, D_g,i1)	(H_m,i1, D_g,i1), (H_m,i3, D_g,i3)	(H_m,i1, D_g,i1), (H_m,i3, D_g,i3)
		(H_m,i3, D_g,i3), (H_m,i4, D_g,i4)	(H_m,i2, D_g,i2), (H_m,i3, D_g,i3)	(H_m,i1, D_g,i1), (H_m,i4, D_g,i4)	(H_m,i1, D_g,i1), (H_m,i4, D_g,i4)
			(H_m,i3, D_g,i3), (H_m,i2, D_g,i2)	(H_m,i2, D_g,i2), (H_m,i3, D_g,i3)	(H_m,i2, D_g,i2), (H_m,i1, D_g,i1)
			(H_m,i3, D_g,i3), (H_m,i4, D_g,i4)	(H_m,i2, D_g,i2), (H_m,i4, D_g,i4)	(H_m,i2, D_g,i2), (H_m,i3, D_g,i3)
			(H_m,i4, D_g,i4), (H_m,i3, D_g,i3)	(H_m,i3, D_g,i3), (H_m,i4, D_g,i4)	(H_m,i2, D_g,i2), (H_m,i4, D_g,i4)
					(H_m,i3, D_g,i3), (H_m,i1, D_g,i1)
					(H_m,i3, D_g,i3), (H_m,i2, D_g,i2)
					(H_m,i3, D_g,i3), (H_m,i4, D_g,i4)
					(H_m,i4, D_g,i4), (H_m,i1, D_g,i1)
					(H_m,i4, D_g,i4), (H_m,i2, D_g,i2)
					(H_m,i4, D_g,i4), (H_m,i3, D_g,i3)
注：typeA、typeB、typeC、typeD、typeE、typeF分别为非下降最长组合、最长组合、非重叠非下降组合、非重叠组合、非下降所有可能组合、所有可能组合，下同。H_m,i1、H_m,i2、H_m,i3、H_m,i4分别表示第 i 个样地的第1、2、3、4次林分平均高的观测数据；D_g,i1、D_g,i2、D_g,i3、D_g,i4分别表示第 i 个样地的第1、2、3、4次林分平均胸径的观测数据。Notes: typeA, typeB, typeC, typeD, typeE, typeF are the longest nondescending combination, the longest combination, the nonoverlapping and nondescending combination, the nonoverlapping combination, all possible nondescending combination, all possible combinations，the same below. H_m,i1、H_m,i2、H_m,i3、H_m,i4represent the first, second, third and fourth observation data of stand mean height in the i-th sample plot; D_g,i1、D_g,i2、D_g,i3、D_g,i4 represent the observation data of the first, second, third and fourth of stand mean DBH in the i-th sample plot respectively.

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表 9 5个模型评价指标

Table 9 Five model evaluating indices

模型评价指标 Model evaluating index	公式 Formula
调整决定系数 Adjusted coefficient of determination ( $R_{\rm{a}}^{2}$ )	$R_{\rm{a}}^{2}=1-(1-{R}^{2}) \dfrac{n-1}{n-k-1}\begin{array}{ccc}& & {R}^{2}=1-\dfrac{\displaystyle\sum_{i=1}^{n}{\left({y}_{i}-{\hat{y}}_{i}\right)}^{2}}{\displaystyle\sum_{i=1}^{n}{\left({y}_{i}-\bar{y}\right)}^{2}}\end{array}$
均方根误差 Root mean square error (RMSE)	${\rm{RMSE}}=\sqrt{\dfrac{\displaystyle\sum_{i=1}^{n}{\left({y}_{i}-{\hat{y}}_{i}\right)}^{2}}{n-k-1}}$
平均绝对误差 Mean absolute error (MAE)	${\rm{MAE}} = \dfrac{1}{n} \displaystyle\sum\limits_{i = 1}^n {\left\| {{y_i} - {{\hat y}_i}} \right\|}$
相对平均绝对误差 Relative mean absolute error (RMAE)	${\rm{RMAE}} = \dfrac{1}{n} \displaystyle\sum\limits_{i = 1}^n {\dfrac{{\left\| {{y_i} - {{\hat y}_i}} \right\|}}{{{{\hat y}_i}}}}$
Akaike信息准则 Akaike information criterion (AIC)	$\mathrm{A}\mathrm{I}\mathrm{C}=-\log L+2k$
注：y_i表示观测值， ${\hat{y}}_{i}$ 表示估计值， $\bar{y}$ 表示平均观测值，n表示观测样本数，k表示模型参数个数，L表示似然函数值。Notes: y_i is observed value, ${\hat{y}}_{i}$ is estimated value, $\bar{y}$ is mean observed value, n is observed sample quantity, k is the number of model parameters, L is the likelihood function value.

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表 10 建模数据的不同数据结构类型拟合效果

Table 10 Fitting performance for different data structure types of calibration data

数据结构类型 Data structure type	R_a ²	RMSE	MAE	RMAE
typeC	0.720	2.019	1.336	0.121
typeD	0.767	1.980	1.315	0.111
typeE	0.562	2.373	1.628	0.149
typeF	0.681	2.280	1.572	0.133

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表 11 检验数据的不同数据结构类型拟合效果

Table 11 Fitting performance for different data structure types of validation data

数据结构类型 Data structure type	RMSE	MAE	RMAE
typeC	1.850	1.189	0.087
typeD	1.809	1.165	0.084
typeE	2.162	1.479	0.106
typeF	2.087	1.436	0.102

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表 12 不同数据结构类型的模型参数估计

Table 12 Model parameter estimates for different data structure types

数据结构类型 Data structure type	参数 Parameter	估计值 Estimate	标准差 Std. error	t值 t value	P值 P value
typeC	b	0.161 5	0.076 8	2.102 1	0.036 3
typeC	r	11.983 2	3.162 3	3.789 4	0.000 2
typeD	b	0.165 7	0.046 2	3.586 2	0.000 4
typeD	r	14.420 4	2.350 7	6.134 4	0.000 0
typeE	b	0.147 0	0.036 7	4.008 8	0.000 1
typeE	r	15.180 9	1.663 7	9.125 0	0.000 0
typeF	b	0.156 6	0.023 2	6.736 6	0.000 0
typeF	r	17.442 6	1.301 6	13.400 9	0.000 0

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表 13 建模数据的不同林分密度指标拟合效果

Table 13 Fitting performance for different stand density indices of calibration data

林分密度指标 Stand density index	R_a ²	RMSE	MAE	RMAE	AIC
N	0.768	1.976	1.312	0.111	2 641.729
BA	0.769	1.973	1.316	0.113	2 639.754
SDIa	0.769	1.973	1.317	0.113	2 639.648
SDIr	0.769	1.973	1.316	0.113	2 639.383
CD	0.768	1.978	1.319	0.113	2 643.066

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表 14 检验数据的不同林分密度指标拟合效果

Table 14 Fitting performance for different stand density indices of validation data

林分密度指标 Stand density index	RMSE	MAE	RMAE
N	1.828	1.193	0.087
BA	1.829	1.178	0.086
SDIa	1.829	1.180	0.086
SDIr	1.825	1.178	0.086
CD	1.808	1.160	0.084

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表 15 不同林分密度指标的模型参数估计

Table 15 Model parameter estimates for different stand density indices

林分密度指标 Stand density index	参数 Parameter	估计值 Estimate	标准差 Std. error	t值 t value	P值 P value
N	b₀	0.163 2	0.031 4	5.192 6	0.000 0
	r	16.757 4	2.543 3	6.588 9	0.000 0
	cSD	1.387 3	0.167 0	8.307 4	0.000 0
BA	b₀	0.174 9	0.032 3	5.407 8	0.000 0
	r	18.152 5	2.623 7	6.918 7	0.000 0
	cSD	1.081 8	0.180 9	5.979 0	0.000 0
SDIa	b₀	0.165 7	0.029 8	5.555 6	0.000 0
	r	17.725 9	2.438 1	7.270 3	0.000 0
	cSD	1.225 8	0.182 6	6.712 8	0.000 0
SDIr	b₀	0.168 5	0.030 8	5.474 9	0.000 0
	r	17.667 2	2.453 8	7.200 0	0.000 0
	cSD	1.172 1	0.192 2	6.099 6	0.000 0
CD	b₀	0.172 4	0.036 3	4.756 2	0.000 0
	r	16.061 2	2.440 0	6.582 3	0.000 0
	cSD	1.506 9	0.322 4	4.673 5	0.000 0
注：cSD是标准化后的5个林分密度指标参数。Note: cSD is the five stand density index parameters after standardization.

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表 16 建模数据的不同直径多样性指数拟合效果

Table 16 Fitting performance for different diameter diversity indices of calibration data

直径多样性指数 Diameter diversity index	R_a ²	RMSE	MAE	RMAE	AIC
ShaI	0.772	1.959	1.308	0.111	2 631.775
SimI	0.771	1.964	1.306	0.111	2 634.669
MceI	0.772	1.963	1.308	0.111	2 634.009
BerI	0.772	1.961	1.304	0.111	2 633.108
GinI	0.769	1.972	1.312	0.113	2 639.897

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表 17 检验数据的不同直径多样性指数拟合效果

Table 17 Fitting performance for different diameter diversity indices of validation data

直径多样性指数 Diameter diversity index	R_a ²	RMSE	MAE	RMAE	AIC
ShaI	0.772	1.959	1.308	0.111	2 631.775
SimI	0.771	1.964	1.306	0.111	2 634.669
MceI	0.772	1.963	1.308	0.111	2 634.009
BerI	0.772	1.961	1.304	0.111	2 633.108
GinI	0.769	1.972	1.312	0.113	2 639.897

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表 18 不同直径多样性指数的模型参数估计

Table 18 Model parameter estimates for different diameter diversity indices

直径多样性指数 Diameter diversity index	参数 Parameter	估计值 Estimate	标准差 Std. error	t值 t value	P值 P value
ShaI	b₀	0.186 1	0.027 4	6.792 0	0.000 0
	r	21.405 3	2.686 2	7.968 5	0.000 0
	cSDIr	1.279 6	0.120 2	10.646 4	0.000 0
	cDI	8.707 1	1.225 0	7.108 1	0.000 0
SimI	b₀	0.190 5	0.028 6	6.653 2	0.000 0
	r	22.628 8	2.900 7	7.801 2	0.000 0
	cSDIr	1.370 8	0.146 5	9.354 5	0.000 0
	cDI	6.965 8	1.879 7	3.705 7	0.000 2
MceI	b₀	0.180 1	0.027 5	6.544 5	0.000 0
	r	20.721 2	2.610 3	7.938 1	0.000 0
	cSDIr	1.299 1	0.132 2	9.823 5	0.000 0
	cDI	7.711 0	1.270 1	6.071 2	0.000 0
BerI	b₀	0.184 5	0.027 3	6.758 7	0.000 0
	r	22.645 4	2.798 0	8.093 4	0.000 0
	cSDIr	1.397 6	0.144 2	9.692 4	0.000 0
	cDI	5.400 2	0.775 3	6.965 5	0.000 0
GinI	b₀	0.183 7	0.034 6	5.311 0	0.000 0
	r	18.978 5	2.675 5	7.093 4	0.000 0
	cSDIr	1.228 1	0.175 9	6.981 0	0.000 0
	cDI	0.930 0	0.798 0	1.165 5	0.244 3
注：cSDIr是标准化后的林分密度指标SDIr参数，下同。cDI是标准化后的5个直径多样性指数参数。Note: cSDIr is the SDIr parameter of stand density index after standardization, the same below. cDI is the five diameter diversity index parameters after standardization.

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表 19 建模数据的不同树种多样性指数拟合效果

Table 19 Fitting performance for different tree species diversity index of calibration data

树种多样性指数 Tree species diversity index	R_a ²	RMSE	MAE	RMAE	AIC
ShaI	0.774	1.952	1.293	0.110	2 628.473
SimI	0.774	1.954	1.301	0.111	2 629.331
MceI	0.774	1.951	1.292	0.110	2 627.545
BerI	0.773	1.957	1.307	0.111	2 631.183

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表 20 检验数据的不同树种多样性指数拟合效果

Table 20 Fitting performance for different tree species diversity index of validation data

树种多样性指数 Tree species diversity index	RMSE	MAE	RMAE
ShaI	1.817	1.176	0.087
SimI	1.859	1.219	0.089
MceI	1.817	1.186	0.088
BerI	2.151	1.286	0.091

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表 21 不同树种多样性指数的模型参数估计

Table 21 Model parameter estimates for different tree species diversity index

树种多样性指数 Tree species diversity index	参数 Parameter	估计值 Estimate	标准差 Std. error	t值 t value	P值 P value
ShaI	b₀	0.317 2	0.064 0	4.957 9	0.000 0
	r	21.774 6	3.520 6	6.184 8	0.000 0
	cSDIr	0.594 5	0.296 4	2.005 5	0.045 3
	cShaI	−2.736 2	3.044 4	−0.898 8	0.369 1
	cSP	−24.003 0	5.118 3	−4.689 6	0.000 0
SimI	b₀	−0.037 3	0.019 9	−1.874 9	0.061 3
	r	−9.609 4	6.686 6	−1.437 1	0.151 2
	cSDIr	−1.993 7	0.482 0	−4.136 6	0.000 0
	cShaI	−1.692 0	2.139 2	−0.791 0	0.429 3
	cSP	50.617 9	15.604 2	3.243 9	0.001 2
MceI	b₀	0.420 9	0.095 5	4.407 6	0.000 0
	r	22.790 2	3.568 5	6.386 5	0.000 0
	cSDIr	0.720 0	0.279 4	2.577 3	0.010 2
	cShaI	−2.313 5	3.029 4	−0.763 7	0.445 3
	cSP	−30.040 1	4.594 6	−6.538 2	0.000 0
BerI	b₀	0.178 9	0.026 7	6.690 3	0.000 0
	r	21.947 4	2.934 8	7.478 3	0.000 0
	cSDIr	1.033 2	0.211 3	4.890 9	0.000 0
	cShaI	8.149 1	1.606 6	5.072 4	0.000 0
	cSP	3.606 4	0.762 0	4.733 0	0.000 0
注：cShaI表示标准化后的直径多样性指数ShaI参数，cSP表示标准化后的4个树种多样性指数参数。Notes: cShaI refers to the diameter diversity index ShaI parameters after standardization, and cSP refers to the four tree species diversity index parameters after standardization.

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天然栎类阔叶混交林林分平均高与平均胸径关系模型

作者简介: 娄明华，博士，助理研究员。主要研究方向：森林可持续经营。Email：mhlou1987@163.com 地址：315040 浙江省宁波市鄞州区德厚街19号宁波市农业科学研究院

责任作者: 张会儒，研究员，博士生导师。研究方向：森林可持续经营。Email：huiru@ifrit.ac.cn 地址：100091 北京市海淀区香山路东小府1号中国林业科学研究院资源信息研究所

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Relationship model between stand mean height and mean DBH for natural Quercus spp. broadleaved mixed stands

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作者简介:
娄明华，博士，助理研究员。主要研究方向：森林可持续经营。Email：mhlou1987@163.com　地址：315040 浙江省宁波市鄞州区德厚街19号宁波市农业科学研究院

责任作者:
张会儒，研究员，博士生导师。研究方向：森林可持续经营。Email：huiru@ifrit.ac.cn　地址：100091 北京市海淀区香山路东小府1号中国林业科学研究院资源信息研究所