地质统计学变异函数贝叶斯模型选择方法与比较

张一凡; 张璐璐; 徐加宝

doi:10.19509/j.cnki.dzkq.tb20240202

地质统计学变异函数贝叶斯模型选择方法与比较

doi: 10.19509/j.cnki.dzkq.tb20240202

上海交通大学海洋工程国家重点实验室，上海 200240

基金项目: 中国长江三峡集团有限公司科研项目（WWKY-2020-0741）；国家自然科学基金项目（52025094；51979158）

详细信息

作者简介:
张一凡：E-mail：zhangyifan_stu@126.com

通讯作者:
E-mail：lulu_zhang@sjtu.edu.cn

中图分类号: TU447
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- 文章访问数: 25
- PDF下载量: 0
- 被引次数: 0
出版历程
- 收稿日期: 2023-04-28
- 录用日期: 2023-07-01
- 修回日期: 2023-06-27
- 网络出版日期: 2025-03-21

Bayesian methods for geostatistical variogram model selection and comparative study

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

More Information

Author Bio:
E-mail：zhangyifan_stu@126.com

Corresponding author: E-mail：lulu_zhang@sjtu.edu.cn

摘要

摘要:
变异函数量化了空间2点地质属性的变异性，对地质统计分析至关重要。当地质数据随空间坐标呈现趋势变化时，正确选择和估计变异函数十分困难。为实现变异函数的模型选择和参数估计，提出了基于贝叶斯理论的变异函数选择方法，采用拉普拉斯近似方法将后验概率分布近似为高斯分布。首先计算出参数的后验概率分布，随后分别计算每个备选变异函数的贝叶斯模型证据，以确定最优模型。探讨了2种模型选择方法在变异函数选择中的适用性，包括贝叶斯模型证据（BME）、Akaike information criterion（AIC）识别准则和Bayesian information criterion（BIC）识别准则。通过实测静力触探试验的锥端阻力数据，说明了该方法，并从模型拟合度和复杂度罚值2个方面比较3种方法在变异函数模型选择中的差异性。研究表明，给定试验数据条件下，BME能够合理地考虑变异函数的拟合度和复杂性；而AIC和BIC识别准则在模型参数个数相同时，仅能反映不同变异函数的拟合度差异，因此，在这种情况下推荐采用BME选择变异函数。本研究方法能够在考虑趋势项参数条件下合理地选择地质统计学变异函数，所选最优变异函数与试验变异函数较一致，为地质统计学分析提供了有效的参考。
- 静力触探试验 /
- 贝叶斯理论 /
- 拉普拉斯近似 /
- 变异函数 /
- 模型选择 /
- BME /
- 地质统计学
Abstract: Objective
The variogram quantifies the variability of geological attributes between two spatial points and is of crucial significance for geostatistical analysis. When geological data exhibit a trend variation along spatial coordinates, the accurate selection and estimation of the variogram become exceptionally difficult.
Methods
To realize the model selection and parameter estimation of the variogram, this paper presents a variogram selection approach based on Bayesian theory, employing the Laplace approximation method to approximate the posterior probability distribution as a Gaussian one. Firstly, the posterior probability distribution of the parameters is computed, and subsequently, the Bayesian model evidence (BME) of each alternative variogram is calculated respectively to determine the optimal model. This study investigates the applicability of two model selection methods in the selection of variograms, encompassing Bayesian model evidence (BME), Akaike information criterion (AIC), and Bayesian information criterion (BIC).
Results
The proposed method is elucidated through the measured cone tip resistance data from static cone penetration tests, and the disparities among the three methods in the selection of variogram models are compared from the perspectives of model fitting and complexity penalty.
Conclusion
The research reveals that, under the given experimental data conditions, BME can rationally take into account the fitting degree and complexity of the variogram; while the AIC and BIC identification criteria can merely reflect the fitting degree differences of different variograms when the number of model parameters is the same. Consequently, in such circumstances, BME is recommended for the selection of variograms. The method proposed in this study is capable of reasonably selecting the geostatistical variogram considering the trend term parameters, and the selected optimal variogram is relatively consistent with the experimental variogram, providing an effective reference for geostatistical analysis.
- cone penetration test /
- Bayesian theory /
- Laplace approximation /
- variogram /
- model selection /
- BME /
- geostatistic

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图 1 试验变异函数和常用的理论变异函数模型

Figure 1. Experimental variation function and commonly used theoretical variation function model

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图 2 25套圆锥静力触探试验(CPT)测试的空间分布

Figure 2. Spatial distribution of 25 sets of CPT tests

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图 3 25套CPT测试的锥尖阻力q_c

Figure 3. Cone tip resistance for 25 CPT tests

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图 4 不同变异函数条件下模型参数的先验分布和后验分布

Figure 4. Prior distribution and posterior distribution of model parameters under different variogram conditions

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图 5 P1-6点位试验变异函数与理论变异函数对比

Figure 5. Comparison of experimental variance function with theoretical variance function

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图 6 25套CPT数据3种方法选择的最优模型

Figure 6. Optimal model selected by three methods for 25 CPT tests

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图 7 25套CPT数据不同模型选择方法拟合度和罚值比例对比

Figure 7. Comparison of fit degree and penalty ratio of different model selection methods for 25 CPT tests

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表 1 常用理论变异函数模型及其协方差函数公式

Table 1. Commonly used theoretical variance function model and its covariance function formula

模型	变异函数$ \gamma \left( h \right) $	协方差函数$C(h)$
指数模型	$ \left\{ {\begin{aligned} &{{c_0} + {c_1}(1 - \exp ( - h/a)){\text{ }},h > 0} \\ &{0 \qquad\qquad\qquad\qquad\;\; ,h = 0} \end{aligned}} \right. $	$\left\{ {\begin{aligned} &{{c_1}\exp ( - h/a){\text{ }},h > 0} \\ &{{c_0} + {c_1}\quad\quad\;\;,h = 0} \end{aligned}} \right.$
高斯模型	$\left\{ {\begin{aligned} &{{c_0} + {c_1}(1 - \exp ( - {h^2}/{a^2})){\text{ }},h > 0} \\ &{0 \qquad\qquad\qquad\qquad\quad\;,h = 0} \end{aligned}} \right.$	$\left\{ {\begin{aligned} &{{c_1}\exp ( - {h^2}/{a^2}){\text{ }},h > 0} \\ &{{c_0} + {c_1}\quad\quad\quad\;\;,h = 0} \end{aligned}} \right.$
球状模型	$ \left\{ \begin{aligned} &{c_0} + {c_1}\qquad\qquad\qquad\;{\text{ , }}h > a \\ &{{c_0} + {c_1}\left( {1.5\frac{h}{a} - 0.5\frac{{{h^3}}}{{{a^3}}}} \right){\text{ }},0 < h \leqslant a} \\ &{0\qquad\qquad\qquad\qquad\;{\text{ , }}h = 0} \\ \end{aligned} \right. $	$\left\{ {\begin{aligned} &{0 \qquad\qquad\qquad\qquad,h > a} \\ &{{c_1}\left(1 - 1.5\frac{h}{a} + 0.5\frac{{{h^3}}}{{{a^3}}}\right){\text{ }},h \leqslant a} \\ &{{c_0} + {c_1}\qquad\qquad\qquad,h = 0} \end{aligned}} \right.$
注：$ {c_0} $为块金值；$ {c_1} $为偏基台值；$ {c_0} + {c_1}{\text{ }} $为基台值；$a$为变差距离或相关长度；$h$为空间2点的距离；下同

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表 2 3种模型比选准则

Table 2. Three model selection criteria

模型比选	拟合度	罚值
BME	$ \ln \left( {{P} \left( {{\cambriabifont\text{z}}\|{{\boldsymbol{\theta }}^*},{C_i}} \right)} \right) $	$ - \ln \left( {{\mathrm{P}}_{\mathrm{r}} \left( {{{\mathbf{\theta }}^*}\|{C_i}} \right)} \right) - \dfrac{{{d}}}{2}\ln \left( {2\pi } \right) + \dfrac{1}{2}\ln \left( {\left\| \boldsymbol{H} \right\|} \right) $
AIC	$ 2\ln {P} {\left( {{\cambriabifont\text{z}}\|{\boldsymbol{\theta }},{C_i}} \right)_{\max }} $	$ 2d $
BIC	$ 2\ln {P} {\left( {{\cambriabifont\text{z}}\|{\boldsymbol{\theta }},{C_i}} \right)_{\max }} $	$ d\ln \left( {{N_m}} \right) $
注：表中代号含义详见正文

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表 3 模型参数的先验分布

Table 3. Prior distribution of model parameters

模型参数	${\beta _0}$	${\beta _1}$	${c_0}( \geqslant 0)$	${c_1}\left( { \geqslant 0} \right)$	$a\left( { \geqslant 0} \right)$
平均值	0	0	5	30	5
标准差	10	10	100	200	10
注：β₀. 常数项参数；β₁.一次项参数；下同

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表 4 P1-6点位CPT测试数据下3种模型的计算结果

Table 4. Calculation results of three models under P1-6 data

模型	后验均值					后验标准差					模型证据	模型选择概率
模型	${\beta _0}$	${\beta _1}$	${c_0}$	${c_1}$	$a$	${\beta _0}$	${\beta _1}$	${c_0}$	${c_1}$	$a$	模型证据	模型选择概率
球状模型	6.39	−1.31	0.95	8.85	2.08	0.99	0.41	0.34	9.39	0.06	−122.43	0.6814
高斯模型	6.57	−0.98	1.33	8.38	1.18	0.99	0.51	0.87	15.42	0.84	−150.81	0
指数模型	6.45	−1.19	0.52	9.23	1.22	1.00	0.46	0.44	13.85	0.44	−123.19	0.3186

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表 5 基于P1-6点值测试数据变异函数模型选择结果对比

Table 5. Comparison of selection results based on variation function model of P1-6 data

模型	BME			AIC识别准则			BIC识别准则
模型	BME值	拟合度	罚值	AIC值	拟合度	罚值	BIC值	拟合度	罚值
球状	−122.43	−110.01	−12.42	230.02	220.02	10	251.09	220.02	31.07
高斯	−150.81	−137.80	−13.01	285.60	275.60	10	306.67	275.60	31.07
指数	−123.19	−109.69	−13.50	229.38	219.38	10	250.45	219.38	31.07

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