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立地生产力为某一立地上植物生产潜力的量化估计[1]。在估计立地生产力的指标方面,同龄纯林的立地生产力指标已较为成熟,一般以3个基本原则即立地指数假说、Eichhorn’s原则和疏伐响应假说,及其原则的修改假说即Assmann’s收获水平理论和Assmann’s自然、最适宜和临界断面积理论为基础[2]。其中,立地指数是最为常用的立地生产力指标[3],该指标为特定树种特定年龄下的林分优势高[4]。目前,天然异龄混交林的估计立地生产力指标尚不成熟。由于天然异龄混交林的树种和年龄结构多样性,立地指数已不适用于天然异龄混交林[5-6]。为将立地指数应用于天然异龄混交林,学者们考虑了林分的树种和年龄结构多样性,发展了立地指数。考虑异龄混交林的树种多样性因素,发展了立地指数转移方程(site index conversion equation)[7-8];考虑年龄结构多样性,建立了含直径多样性的立地指数[9-11]。Ouzennou等[10]认为林分直径分布在异龄或不规则林分中与同龄林中的区别较大,同龄林中的林分直径分布一般呈正态分布;在异龄或不规则的林分中,林分直径分布在幼龄林时呈截尾倒J形(truncated reverse-J shape)分布,在近熟林时呈钟形或喇叭状(bell shape)分布,在成过熟林时呈现双峰(bi-modal)分布[12]。可见,一个林分不同演替阶段的直径分布变化情况与该林分不同演替阶段的立地质量和林分密度变化情况比较相似,且直径分布变化的速率更快[13],说明直径分布即直径多样性与立地质量存在着相关性[10]。事实上,在异龄混交林中的林分年龄很难确定[5, 14]。Huang等[5]认为直径−树高关系与立地质量密切相关,建立了基于直径−树高关系模型的立地型指数(site form index)[5, 15-16]。然而,立地型指数未考虑异龄混交林的直径结构和树种结构多样性。该立地型指数的直径−树高关系模型主要指林分优势胸径−优势高关系模型,在连续调查固定样地数据通常缺乏林分优势高数据,只有林分平均高数据[17-18],针对这一情形,有必要扩展立地型指数,研究林分平均高与平均胸径关系模型。与此同时,学者们研究认为林分平均高受林分密度的影响较大[19-21],故应将林分密度因子考虑到该模型中。
栎类分布于我国各省区,是我国天然林中的重要树种,也是组成森林的重要树种,在我国森林资源中占有十分重要地位[22-25]。栎类的经济价值与生态价值重要性也在逐步认识[26-27],栎类质量好坏直接关乎我国整体森林资源质量的提高[25]。因此,本文以连续调查固定样地数据的天然栎类(Quercus spp.)阔叶混交林为研究对象,借鉴Ouzennou等[10]的建模方法,基于代数差分方程构建林分平均高与平均胸径关系模型,并将林分密度、直径多样性和树种多样性引入该模型中,预期可提高模型拟合效果,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。
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数据来源于吉林省域尺度上的连续调查固定样地数据,固定样地大小均为600 m2,且每5年进行一次复查。本文收集了4期研究数据,即1994、1999、2004和2009年的数据,共有3 694个固定样地。设置3个筛选条件:(1)树种组成为栎类阔叶混交林,林分中每个树种的断面积占林分总断面积的比例均小于65%,所有阔叶树种的总断面积占林分总断面积65%以上;(2)未经过采伐的样地;(3)样地至少经过连续调查一次以上(这里的连续调查是指前后两次调查的时间间隔为5年)。依据以上3个筛选条件,共筛选得到229个天然栎类阔叶混交林的固定样地,覆盖了124°22′15″ ~ 131°16′15″E,40°55′10″ ~ 44°28′29″N的地区(图1)。其中:栎类断面积占林分总断面积的平均比例为42.1%,比例范围为14% ~ 64%;其他优势树种有紫椴(Tilia amurensis)、榆树(Ulmus pumila)、胡桃楸(Juglans mandshurica)、白桦(Betula platyphylla)、水曲柳(Fraxinus mandschurica)、大青杨(Populus ussuriensis)、落叶松(Larix gmelinii)、红松(Pinus koraiensis)、云杉(Picea asperata)等。
为建模需要,将229个样地随机分成两类,包括147个样地的建模数据和82个样地的检验数据。229个样地(包括建模数据和检验数据)的连续调查次数及林分特征因子见表1。由表1可知,根据样地个数及每个样地的连续调查次数,得到建模数据共461个,检验数据共269个,总样本数为730。
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. -
绘制461个样地的建模数据和269个样地的检验数据的林分平均高与林分平均胸径的散点图(图2)。由图2可见,本文选择以Richards方程为基础模型。
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本文选用常用的5个林分密度指标,包括林木株数(tree number,N)、林分断面积(stand basal area,BA)、林分密度指数(stand density index,SDIr)、可加林分密度指数(additive stand density index,SDIa)和郁闭度(canopy density,CD)。其中,SDIr是Reineke[28]提出的林分密度指数,SDIa是Stage[29]在Reineke[28]林分密度指数基础上扩展的可加林分密度[30],公式见表2。表2中,D0表示标准直径,在我国一般取20 cm[31],因此本文取D0 = 20。229个连续调查固定样地(建模数据和检验数据)的林分密度指标见表3。
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,di表示第i株林木的胸径,Dg表示林分平均胸径,D0表示标准直径。Notes: n is the number of trees, di is the DBH of tree i, Dg is stand mean DBH, D0 is standard diameter. 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) -
本文选择常用的5个多样性指数包括Shannon均匀度指数(Shannon evenness index,ShaI)、Simpson指数(Simpson index,SimI)、McIntosh均匀度指数(McIntosh evenness index,MceI)、Gini系数(Gini coefficient,GinI)和Berger-Parker指数(Berger-Parker index,BerI)。其中,ShaI、SimI、MceI和BerI指数均可以衡量直径多样性和树种多样性。在ShaI、MceI和BerI这3个多样性指数中,当该值为最小值0时,在直径多样性中表示林分中所有的林木均处在同一个径阶,在树种多样性中表示林分中所有的林木均为同一个树种;当该值为最大值1时,在直径多样性中表示林分中所有径阶的断面积呈现均匀分布即每个径阶具有相同的断面积,在树种多样性中表示林分中所有树种的株数呈均匀分布即每个树种具有相同的株数。在Simpson指数中,当该值为最小值0时,表明直径多样性或树种多样性的程度最小;当该值为最大值1时,表明直径多样性或树种多样性的程度最大。Gini系数是衡量直径异质性程度的大小,应用该指数时每个林木需按其断面积的大小进行从小到大排序。当Gini系数为最小值0时,表明每个林木均具有相同的直径;当该系数为最大值1时,表明林分中有一个林木的直径无穷大[32]。5个多样性指数的公式见表4,229个连续调查固定样地(建模数据和检验数据)的多样性指数(包括直径多样性指数和树种多样性指数)见表5。
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 diversitySimpson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 diversityMcIntosh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 diversityBerger-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 diversityGini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:pi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;bai表示第i个径阶的断面积;baj表示第j株林木的断面积;bamax表示断面积最大所在径阶的断面积;n表示林木总株数。Notes: pi is the proportion of basal area in diameter class i within diameter diversity index, or pi 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; bai is the basal area in the diameter class i; baj is the basal area of the tree; bamax is the basal area in the diameter class with the largest basal area; n is the total number of trees. 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) -
以Richards方程为基础模型(式(1))。将式(1)中的模型参数a视为随立地变化的参数,可得代数差分方程(式(2)):
$${H_{\rm m}} = 1.3 + a{(1 - {\rm{exp}}( - b{D_{\rm{g}}}))^c}$$ (1) $$\begin{aligned} \;\\ {H_{{\rm m},ik}} = 1.3 + \left( {{H_{{\rm m},ij}} - 1.3} \right) {\left( {\frac{{1 - {\rm{exp}}( - b{D_{{\rm{g}},ik}})}}{{1 - {\rm{exp}}( - b{D_{{\rm{g}},ij}})}}} \right)^c} + {\varepsilon _{ik}} \end{aligned}$$ (2) 式中:Hm,ij表示第i个样地的第j次调查的林分平均高,Hm,ik表示第i个样地的第k次调查的林分平均高,Dg,ij表示第i个样地的第j次调查的林分平均胸径,Dg,ik表示第i个样地的第k次调查的林分平均胸径,b和c表示模型参数,εik表示第i个样地的第k次调查的林分平均高观测误差。
Raulier[9]研究发现,参数b和c之间存在较强的线性相关。可见,参数c可写成关于参数b的线性函数,即c = rb,其中r表示斜率参数。因此,式(2)变成了如下公式:
$${H_{{\rm m},ik}} = 1.3 + \left( {{H_{{\rm m},ij}} - 1.3} \right) {\left( {\frac{{1 - {\rm{exp}}( - b{D_{{\rm{g}},ik}})}}{{1 - {\rm{exp}}( - b{D_{{\rm{g}},ij}})}}} \right)^{rb}} + {\varepsilon _{ik}}$$ (3) 本文将式(3)作为基础代数差分方程,以下简称ModeO。
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以N、BA、SDIa、SDIr和CD共5个指数计算林分密度指标,以ShaI、SimI、MceI、BerI和GinI共5个指数计算直径多样性指数,以ShaI、SimI、MceI和BerI共4个指数计算树种多样性指数。Raulier[9]和Ouzennou等[10]的研究认为将影响因子进行标准化后,再加入到模型参数中,效果较好。因此,本文先将林分密度指标、直径多样性指数和树种多样性指数进行标准化,根据模型参数检验是否显著,将标准化后的林分密度指标、直径多样性指数和树种多样性指数依次加入式(3)的参数b,得到对应的多样性差分方程,详见表6、表7和图3。
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cx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,fx表示标准化函数,fSD表示标准化的林分密度指标,fDD表示标准化的直径多样性指数,fSPD表示标准化的树种多样性指数。Notes: b0 and cx denote parameters, SD denotes stand density index, DD denotes diameter diversity index, SPD denotes tree species diversity index, fx denotes standardize function, fSD denotes standardize stand density index, fDD denotes standardize diameter diversity index, fSPD denotes standardize species diversity index. 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 -
代数差分方程建模应当考虑不同的数据结构类型,因为不同的数据结构类型会产生不同的模型拟合效果[33-34]。4次观测数据一般有6种不同的数据结构类型[33],包括非下降最长(longest nondescending)组合(typeA)、最长(longest)组合(typeB)、非重叠非下降(nonoverlapping and nondescending)组合(typeC)、非重叠(nonoverlapping)组合(typeD)、非下降所有可能(all possible nondescending)组合(typeE)和所有可能(all possible)组合(typeF),详见表8。由表8可知,typeA和typeB未充分利用观测数据,其他4个数据结构类型均充分利用了观测数据。因此,本文采用typeC、typeD、typeE和typeF依次构建代数差分方程。
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 (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i4, Dg,i4), (Hm,i1 , Dg,i1) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i2, Dg,i2), (Hm,i1, Dg,i1) (Hm,i1, Dg,i1), (Hm,i3, Dg,i3) (Hm,i1, Dg,i1), (Hm,i3, Dg,i3) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i3, Dg,i3), (Hm,i2, Dg,i2) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i2, Dg,i2), (Hm,i1, Dg,i1) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i4, Dg,i4), (Hm,i3, Dg,i3) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i4, Dg,i4) (Hm,i3, Dg,i3), (Hm,i1, Dg,i1) (Hm,i3, Dg,i3), (Hm,i2, Dg,i2) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i4, Dg,i4), (Hm,i1, Dg,i1) (Hm,i4, Dg,i4), (Hm,i2, Dg,i2) (Hm,i4, Dg,i4), (Hm,i3, Dg,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。Hm,i1、Hm,i2、Hm,i3、Hm,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;Dg,i1、Dg,i2、Dg,i3、Dg,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. Hm,i1、Hm,i2、Hm,i3、Hm,i4 represent the first, second, third and fourth observation data of stand mean height in the i-th sample plot; Dg,i1、Dg,i2、Dg,i3、Dg,i4 represent the observation data of the first, second, third and fourth of stand mean DBH in the i-th sample plot respectively. -
本研究选用了5个指标,分别为调整决定系数(adjusted coefficient of determination,Ra 2)、均方根误差(root mean square error,RMSE)、平均绝对误差(mean absolute error,MAE)、相对平均绝对误差(relative mean absolute error,RMAE)和Akaike信息准则(Akaike information criterion,AIC),公式见表9。
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:yi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: yi 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. -
利用基础代数差分方程ModeO分别拟合4种数据结构类型即typeC、typeD、typeE和typeF的林分平均高和平均胸径关系模型,采用Ra 2、RMSE、MAE和RMAE共4个指标评价模型的拟合效果。注意此处不选用AIC指标,AIC指标与样本数有关[35],而不同的数据结构类型的样本数相差较大,因此不宜采用AIC指标进行比较。比较结果见表10和表11。不同数据结构类型的模型参数检验结果见表12。不同数据结构类型的模型残差图见图4。
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 typeRa 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 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 typeRMSE 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 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 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 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 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 r 17.442 6 1.301 6 13.400 9 0.000 0 由表10可知,数据结构类型typeD的建模效果最好,因typeD具有最大Ra 2、最小RMSE、最小MAE和最小RMAE。其次为typeC,第三为typeF,建模效果最差的是typeE。从表11可见,依据RMSE、MAE和RMAE的3个评价指标,4个数据结构类型在检验数据中的效果从优至劣排序为:typeD > typeC > typeF > typeE。从表12的模型参数检验可知,利用typeC拟合的模型参数b(P= 0.036 3)在0.05水平下显著,而在0.01水平下不显著,其模型参数r在0.01水平下显著。除了typeC,其他3个数据结构类型拟合的模型参数b和r均显著不为零(P < 0.01),说明typeC拟合的模型参数检验效果较其他3个数据结构类型即typeD、typeE和typeF要差。
综合考虑不同数据结构类型模型拟合效果和参数检验效果可知,利用数据结构类型typeD模型拟合效果最理想。因此,选用typeD作为基础代数差分方程的数据结构类型,并用该类型进一步建立含有林分密度指数的多样性代数差分方程。
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以typeD为数据结构类型,将N、BA、SDIa、SDIr和CD共5个林分密度指标,引入表7中只含林分密度指标的多样性代数差分方程即ModeA建立林分平均高和平均胸径关系模型。利用Ra 2、RMSE、MAE、RMAE和AIC指标评价模型拟合效果,结果见表13和表14。不同林分密度指标的模型参数检验结果见表15。表15中,参数cSD表示标准化后的5个林分密度指标参数。
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 Ra 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 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 indexRMSE 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 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. 由表13可知,5个林分密度指标的建模效果均较接近。相对而言,SDIr是最好的林分密度指标,因其有最大Ra 2、最小RMSE和最小AIC,其MAE和RMAE略大于林分密度指标N。其他4个林分密度指标中,BA和SDIr是较好的林分密度指标,较差的是N和CD。从表14可知,依据RMSE、MAE和RMAE的3个评价指标,5个林分密度指标在检验数据中的效果为CD最好,其次SDIr,其余3个林分密度指标即N、BA和SDIa的检验效果比较相似。从表15的模型参数检验可知,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。
综合考虑5个林分密度指标的模型拟合效果和参数检验效果,可知SDIr是最佳的林分密度指标。因此,以SDIr为林分密度指标加入到代数差分方程中,并用该方程进一步建立含有林分密度指标和直径多样性指数的多样性代数差分方程。
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以SDIr为林分密度指标的多样性代数差分方程基础上,将ShaI、SimI、MceI、BerI和GinI引入表7中的ModeB建立林分平均高和平均胸径关系模型。采用Ra 2、RMSE、MAE、RMAE和AIC,共5个指标评价模型的拟合效果,结果见表16和表17。不同直径多样性指数的模型参数检验结果见表18。表18中,参数cSDIr表示标准化后的林分密度指标SDIr参数,参数cDI表示标准化后的5个直径多样性指数参数。
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 indexRa 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 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 indexRa 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 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. errort值
t valueP值
P valueShaI 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. 由表16可知,5个直径多样性指数的评价指标均较接近。相对而言,直径多样性指数ShaI的建模效果最好,因ShaI具有最大Ra 2、最小RMSE、最小RMAE和最小AIC,但MAE略高于BerI和MceI。其他4个直径多样性指数中,BerI最好,其次为MceI,第三为SimI,最差为GinI。由表17可知,从RMSE、MAE和RMAE的3个评价指标可看出,5个直径多样性指数在检验数据中的效果表现为GinI最好,其次为SimI,第三为ShaI,最差为MceI和BerI。从表18的模型参数检验可知,利用GinI的模型参数b0、r和cSDIr均显著(P < 0.01),但其cDI不显著(P > 0.1),说明GinI是不理想的直径多样性指数。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),表明其他4个即ShaI、SimI、MceI和BerI均为较理想的直径多样性指数。
综合考虑5个直径多样性指数的模型拟合效果和参数检验效果可知,ShaI是最佳的直径多样性指数。因此,选用ShaI作为直径多样性指数加入到多样性代数差分方程中。然后,进行下一步分析,即在加入林分密度指标SDIr和直径多样性指数ShaI的基础上,继续加入树种多样性指数。
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以SDIr为林分密度指标和ShaI为直径多样性指数的多样性代数差分方程基础上,分别加入4个树种多样性指数ShaI、SimI、MceI和BerI,即利用表7中的ModeD建立林分平均高和平均胸径关系模型。采用Ra 2、RMSE、MAE、RMAE和AIC的5个评价指标进行模型拟合效果评价,结果见表19和表20。不同树种多样性指数的模型参数检验结果见表21。表21中,参数cSDIr表示标准化后的林分密度指标SDIr参数,参数cShaI表示标准化后的直径多样性指数ShaI参数,参数cSP表示标准化后的4个树种多样性指数参数。
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 indexRa 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 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 indexRMSE 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 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. errort值
t valueP值
P valueShaI 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. 由表19可知,4个树种多样性指数的建模拟合效果差别不大。相对而言,以MceI为最好,其次为ShaI,第三为SimI,最差为BerI。从表20可知,依据RMSE、MAE和RMAE的3个评价指标,4个树种多样性指数在检验数据中的效果均相似,其中ShaI是最好的树种多样性指数,其次为MceI,第三为SimI,最差为BerI。从表21的模型参数检验可知,利用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。其他3个树种多样性指数即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。
从4个树种多样性指数模型拟合和参数检验效果来看,BerI是最佳的树种多样性指数,ShaI、SimI和MceI是较差的树种多样性指数。因此,选用BerI为树种多样性指数加入到多样性代数差分方程中。
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Ouzennou等[10]考虑直径多样性指数构建黑云杉(Picea mariana)异龄林的立地指数,结果表明含直径多样性的立地指数改善了树高预测效果。本文将直径多样性指数引入天然栎类阔叶混交林林分平均高与平均胸径关系模型中,提高模型拟合效果,这与Ouzennou等[10]的研究结果一致,说明直径多样性指数不仅适用于纯林异龄林,而且适用于天然栎类阔叶混交林。与此同时,本文将林分密度指标和树种多样性指数引入天然栎类阔叶混交林,进一步改善了模型拟合效果。然而,含林分密度指标、直径多样性指数和树种多样性指数的林分平均高与平均胸径关系模型是否适用于其他混交林,如亚热带常绿阔叶林、热带雨林等,这是下一步值得研究的问题。另外,将林分密度指标、直径多样性和树种多样性成功引入到天然栎类阔叶混交林林分平均高与平均胸径关系模型中,为研究天然林的其他各类模型提供了参考。
Wang等[33]研究火柜松(Pinus taeda)人工林的代数差分方程时表明typeC是最佳的数据结构类型。本文利用不同数据结构类型构建的代数差分方程,认为typeD为最佳数据结构类型,这与Wang等[33]的研究结果不一致,说明数据结构类型的适用性依赖于研究数据的特定属性,可能的特定属性如林分类型(如天然混交林或人工纯林)。本文的研究数据特定属性为天然混交林,Wang等[33]的研究数据特定属性为人工纯林,或者为其他特定属性如研究数据的定期观测次数等属性。此外,本文与Wang等[33]均选择以Richards模型为基础模型,模型类型的不同是否也影响着数据结构类型的适用性。因此,可根据研究数据的不同特定属性(或同时增加考虑基础模型类型)进一步探讨代数差分方程的数据结构类型的适用性。
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以Richards方程为基础模型,利用不同的数据结构类型,基于代数差分方程构建含多样性(包括林分密度指标、直径多样性指数和树种多样性指数)的天然栎类阔叶混交林林分平均高与平均胸径关系模型。可得出以下结论:(1)在基础代数差分方程中,typeD是最好的数据结构类型;(2)林分密度、直径多样性和树种多样性均对模型有影响;(3)以SDIr为林分密度指标、ShaI为直径多样性指数和BerI为树种多样性指数建立的多样性代数差分方程最佳,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。
模型具体形式:将表21中树种多样性指数BerI的5个参数b0 = 0.178 9、r = 21.947 4、cSDIr = 1.033 2、cShaI = 8.149 1和cSP = 3.606 4代入表7中含林分密度指标、直径多样性和树种多样性的多样性代数差分方程即ModeD,公式如下:
$$ {\widehat{H}}_{{\rm m},ik}=1.3+\left({H}_{{\rm m},ij}-1.3\right) {\left(\dfrac{1-\exp (-0.178\;9{f}_{\rm{SD}}{f}_{\rm{DD}}{f}_{\rm{SPD}}{D}_{{\rm{g}},ik})}{1-\exp (-0.178\;9{f}_{\rm{SD}}{f}_{\rm{DD}}{f}_{\rm{SPD}}{D}_{{\rm{g}},ij})}\right)}^{21.947\;4\times 0.178\; 9{f}_{\rm{SD}}{f}_{\rm{DD}}{f}_{\rm{SPD}}}$$ 其中,
$${f_{{\rm{SD}}}} = 1 + 1.033\;2 \times \frac{{{\rm{SDIr}} - \overline {{\rm{SDIr}}} }}{{\overline {{\rm{SDIr}}} }},\;{f_{{\rm{DD}}}} = 1 + 8.149\;1 \times \frac{{{\rm{ShaI}} - \overline {{\rm{ShaI}}} }}{{\overline {{\rm{ShaI}}} }},\;{f_{{\rm{SPD}}}} = 1 + 3.606\;4 \times \frac{{{\rm{BerI}} - \overline {{\rm{BerI}}} }}{{\overline {{\rm{BerI}}} }}$$ (4) 式中:
$ {\widehat{H}}_{m,ik} $ 表示第i个样地的第k次调查的林分平均高估计值,$\overline {{\rm{{\rm SDI}r}}} $ 表示林分密度指标SDIr的平均值,$\overline {{\rm{{\rm Sha}I}}} $ 表示直径多样性指数ShaI的平均值,$\overline {{\rm{BerI}}} $ 表示树种多样性指数BerI的平均值。在typeD的数据结构类型中,$\overline {{\rm SDI}r} $ = 48.936 6,$\overline {{\rm Sha}I} $ = 0.913 0,$\overline {{\rm Ber}I}$ = 0.932 0。
Relationship model between stand mean height and mean DBH for natural Quercus spp. broadleaved mixed stands
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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。 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. -
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fSD、fDD和fSPD。 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 fSD, fDD and fSPD are not added into the model.
Figure 3. Modeling flowchart of diversity algebraic differential equations
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. 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,di表示第i株林木的胸径,Dg表示林分平均胸径,D0表示标准直径。Notes: n is the number of trees, di is the DBH of tree i, Dg is stand mean DBH, D0 is standard diameter. 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) 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 diversitySimpson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 diversityMcIntosh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 diversityBerger-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 diversityGini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:pi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;bai表示第i个径阶的断面积;baj表示第j株林木的断面积;bamax表示断面积最大所在径阶的断面积;n表示林木总株数。Notes: pi is the proportion of basal area in diameter class i within diameter diversity index, or pi 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; bai is the basal area in the diameter class i; baj is the basal area of the tree; bamax is the basal area in the diameter class with the largest basal area; n is the total number of trees. 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) 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cx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,fx表示标准化函数,fSD表示标准化的林分密度指标,fDD表示标准化的直径多样性指数,fSPD表示标准化的树种多样性指数。Notes: b0 and cx denote parameters, SD denotes stand density index, DD denotes diameter diversity index, SPD denotes tree species diversity index, fx denotes standardize function, fSD denotes standardize stand density index, fDD denotes standardize diameter diversity index, fSPD denotes standardize species diversity index. 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 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 (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i1, Dg,i1), (Hm,i2, Dg,i2) (Hm,i4, Dg,i4), (Hm,i1 , Dg,i1) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i2, Dg,i2), (Hm,i1, Dg,i1) (Hm,i1, Dg,i1), (Hm,i3, Dg,i3) (Hm,i1, Dg,i1), (Hm,i3, Dg,i3) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i1, Dg,i1), (Hm,i4, Dg,i4) (Hm,i3, Dg,i3), (Hm,i2, Dg,i2) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i2, Dg,i2), (Hm,i1, Dg,i1) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i3, Dg,i3) (Hm,i4, Dg,i4), (Hm,i3, Dg,i3) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i2, Dg,i2), (Hm,i4, Dg,i4) (Hm,i3, Dg,i3), (Hm,i1, Dg,i1) (Hm,i3, Dg,i3), (Hm,i2, Dg,i2) (Hm,i3, Dg,i3), (Hm,i4, Dg,i4) (Hm,i4, Dg,i4), (Hm,i1, Dg,i1) (Hm,i4, Dg,i4), (Hm,i2, Dg,i2) (Hm,i4, Dg,i4), (Hm,i3, Dg,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。Hm,i1、Hm,i2、Hm,i3、Hm,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;Dg,i1、Dg,i2、Dg,i3、Dg,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. Hm,i1、Hm,i2、Hm,i3、Hm,i4 represent the first, second, third and fourth observation data of stand mean height in the i-th sample plot; Dg,i1、Dg,i2、Dg,i3、Dg,i4 represent the observation data of the first, second, third and fourth of stand mean DBH in the i-th sample plot respectively. 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:yi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: yi 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.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 typeRa 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 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 typeRMSE 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 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 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 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 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 r 17.442 6 1.301 6 13.400 9 0.000 0 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 Ra 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 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 indexRMSE 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 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. 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 indexRa 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 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 indexRa 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 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. errort值
t valueP值
P valueShaI 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. 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 indexRa 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 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 indexRMSE 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 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. errort值
t valueP值
P valueShaI 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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