Determination of Soil Mechanical Parameters From Posterior Distributions Under Different Prior Distributions
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摘要: 岩土工程中各土层参数的取值是根据现场及室内试验数据，采用经典统计学方法进行确定的，但这往往忽略了先验信息的作用。与经典统计学方法不同的是，Bayes法能从考虑先验分布的角度结合样本分布去推导后验分布，为岩土参数的取值提供一种新的分析方法。岩土工程勘察可视为对总体地层的随机抽样，当抽样完成时，样本分布密度函数是确定的，故Bayes法中的后验分布取决于先验分布，因此推导出两套不同的先验分布：利用先验信息确定先验分布及共轭先验分布。通过对先验及后验分布中超参数的计算，当样本总体符合N(μ,σ2)正态分布时，对所要研究的未知参数μ和σ展开分析，综合对比不同先验分布下后验分布的区间长度，给出岩土参数Bayes推断中最佳后验分布所要选择的先验分布。结果表明：共轭情况下的后验分布总是比无信息情况下的后验区间短，概率密度函数分布更集中，取值更方便。在正态总体情形下，根据未知参数μ和σ的联合后验分布求极值方法，确定样本总体中最大概率均值μmax和方差σmax作为工程设计采用值，为岩土参数取值方法提供了一条新的路径，有较好的工程意义。Abstract: The values of soil layer parameters in geotechnical engineering were determined according to field and laboratory test data with classical statistical methods, without use of the prior information. Unlike classical statistical methods, the Bayes method combines samples from the perspective of prior distribution to deduce the posterior distribution, providing a new analytical method for the evaluation of geotechnical parameters. The geotechnical engineering survey makes a random sampling of the overall strata. The density function of the sample distribution is determined when the sampling is completed. Therefore, the posterior distribution in the Bayes method depends on the prior distribution, and 2 different sets of prior distributions were derived: the prior distribution and the conjugate prior distribution were determined with the prior information. Through calculation of the parameters in the posterior distribution, with the sample generally conforming to the normal distribution of N(μ,σ2), unknown parameters μ and σ were analyzed, the interval lengths of the posterior distribution under different prior distributions were comprehensively compared, and the prior distribution selected for the optimal posterior distribution in the Bayes inference of the geotechnical parameter was given. The results show that, the posterior distribution in the conjugate case is always shorter than that in the absence of information, and the probability density function distribution is more centralized and the value determination is more convenient. Under the overall normal situation, the extreme value method obtained based on the joint posterior distribution of unknown parameters μ and σ to determine maximum probability mean μmax and variance σmax in the sample as the adopted values in the engineering design, provides a way for the value determination of geotechnical parameters, and has engineering significance.
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图 1 岩土力学参数平均值μ的样本概率分布图
Figure 1. Sample probability distribution diagrams of average value μ of rock and soil mechanics parameters
图 2 岩土力学参数均值μ的1/σ的Gamma密度函数分布图
Figure 2. Gamma density function distribution diagrams of 1/σ for mean value μ of rock and soil mechanical parameters
图 3 泥炭质土c，ϕ的均值μ的后验分布概率密度曲线
Figure 3. The posterior distribution probability density curves of mean value μ of peat c and ϕ
图 5 岩土力学参数最优后验分布密度曲线
Figure 5. The optimal posterior distribution density curves of rock and soil mechanical parameters
表 1 样本矩确定先验超参数表
Table 1. Sample moment determination of prior hyper parameters
$ \theta $ [x1+xi] [xi+1+xk] [xk+1+xj] ··· [xs+1+xn] fi E F G ··· H 
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表 2 土体c和
$ \varphi $ 原始试验样本Table 2.
The c and $ \varphi $ original samples of soilsoil name sample size N fak/ kPa index name index value 4-1 peaty soil 200 50 c/kPa 16.7, 50.6, 12.10, ···, 32.9, 28.7, 30.5, 46.4, 42.7, 3.3 200 $ \varphi /(^\circ ) $ 6.1, 3.8, 5.2, ···, 3.6, 6.9, 6.6, 6.1, 9.8, 9.8, 2.3 4-2 silty soil 190 80 c/kPa 18.5, 20.1, 20.3, 20.6, ···, 31.5, 31.8, 32.7, 34.8 190 $ \varphi /{(}^\circ ) $ 15, 15, 14, 12.3, ···, 18, 18.1, 18.4, 18.6, 18.7, 19.3 4-2 clay 210 90 c/kPa 34.44, 30.16, 28.53, 38.30,···, 29.66, 31.54, 6.64 210 $ \varphi /{(}^\circ ) $ 5.91, 5.69, 7.24, ···, 6.71, 5.10, 7.21, 5.21
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表 3 土体力学指标分布模型选择依据
Table 3. Basis for selection of soil mechanics index distribution model
index number in fig. 1 CCV absolute value of Sk type of density function μc,ps (a) 0.232 07 0.015 89 normal distribution μϕ, ps (b) 0.216 97 0.023 64 normal distribution μc, fs (c) 0.163 66 0.016 38 normal distribution μϕ, fs (d) 0.088 32 0.022 41 normal distribution μc, c (e) 0.10 195 0.027 14 lognormal distribution μϕ, c (f) 0.099 91 0.024 58 normal distribution
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表 4 利用先验信息计算先验及后验分布表
Table 4. Calculated prior and posterior distributions with prior information
soil name mechanical parameter $ \hat \alpha $ $ \hat \beta $ sample distribution $ f(\left. x \right|\mu ) \propto $ posterior distribution $ \pi (\left. \mu \right|x) \propto $ 4-1 peaty soil $ {\mu _{\text{c}}} $ 29.6 12.8 $ \exp \left\{ - \dfrac{{{{(x - 28.8)}^2}}}{{2 \times 10.37}}\right\} $ $ \exp \left\{ -\left (\dfrac{{{{(x - 28.8)}^2}}}{{20.74}}{\text{ + }}\dfrac{{{{(x - 29.6)}^2}}}{{25.6}}\right)\right\} $ $ {\mu _\varphi } $ 5.6 2.14 $ \exp \left\{ - \dfrac{{{{(x - 5.95)}^2}}}{{2 \times 1.54}}\right\} $ $ \exp \left\{ -\left (\dfrac{{{{(x - 5.6)}^2}}}{{4.28}}{\text{ + }}\dfrac{{{{(x - 5.95)}^2}}}{{3.08}}\right)\right\} $ 4-2 silty soil $ {\mu _{\text{c}}} $ 26.3 7.57 $ \exp \left\{ - \dfrac{{{{(x - 26.75)}^2}}}{{2 \times 5.4}}\right\} $ $ \exp \left\{ -\left (\dfrac{{{{(x - 26.75)}^2}}}{{10.8}}{\text{ + }}\dfrac{{{{(x - 26.3)}^2}}}{{15.54}}\right)\right\} $ $ {\mu _\varphi } $ 16.83 2.59 $ \exp \left\{ - \dfrac{{{{(x - 16.44)}^2}}}{{2 \times 2.4}}\right\} $ ${\rm{exp}}\left\{ {{\rm{ - }}\left( {\dfrac{{{{(x{\rm{ - 16}}.{\rm{44}})}^{\rm{2}}}}}{{{\rm{5}}.{\rm{18}}}}{\rm{ + }}\dfrac{{{{(x{\rm{ - 15}}.{\rm{9}})}^{\rm{2}}}}}{{{\rm{4}}.{\rm{8}}}}} \right)} \right\}$ 4-2 clay $ {\mu _{\text{c}}} $ 29.86 5.20 $ \exp \left\{ - \dfrac{{{{(x - 29.4)}^2}}}{{2 \times 4.5}}\right\} $ $ \exp \left\{ -\left (\dfrac{{{{(x - 29.86)}^2}}}{{10.4}}{\text{ + }}\dfrac{{{{(x - 29.4)}^2}}}{9}\right)\right\} $ $ {\mu _\varphi } $ 5.59 1.1 $ \exp \left\{ - \dfrac{{{{(x - 5.47)}^2}}}{{2 \times 0.9}}\right\} $ $ \exp \left\{ -\left (\dfrac{{{{(x - 5.59)}^2}}}{{2.2}}{\text{ + }}\dfrac{{{{(x - 5.47)}^2}}}{{1.8}}\right)\right\} $
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表 5 共轭先验、后验分布超参数计算表
Table 5. The hyper parameter calculation table of conjugate prior and posterion distributions
soil name index n $ \bar x $ μ S2 r ζ k kn $ \mu (\bar x) $ vn $ {\sigma _n} $ 4-1 peaty soil c 40 27.87 kPa 28.8 kPa 12.1 5.62 1.02 1.23 40.86 27.89 kPa 45.62 10.37 ϕ 40 5.44° 5.95° 2.14 7.52 4.02 0.72 40.72 5.45° 47.52 1.86 4-2 silty soil c 38 26.33 kPa 26.75 kPa 7.57 4.44 1.58 0.71 39.58 26.3 kPa 42.44 6.64 ϕ 38 16.84° 15.9° 2.58 3.86 6.26 0.93 41.86 16.8° 41.86 2.32 4-2 clay c 42 29.81 kPa 31.35 kPa 5.20 12.58 0.1 0.86 42.86 29.8 kPa 54.58 4.003 ϕ 42 5.68° 5.47° 1.1 14.84 0.54 0.82 42 .82 5.6° 56.84 0.803
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表 6 岩土工程参数概率分布函数转化表
Table 6. The transformation table of probability distribution functions of geotechnical engineering parameters
distribution type probability density function normal distribution μ normal distribution σ normal distribution $f(x,\mu ,\sigma ) = \dfrac{1}{ {\sqrt {2{\text{π}} } \sigma } }\exp \left\{ { - \dfrac{ { { {(x - \mu )}^2} } }{ {2{\sigma ^2} } } } \right\}$ μ σ lognormal distribution $f(x,\alpha ,\gamma ) = \dfrac{1}{ {\gamma x\sqrt {2{{\text{π}} } } } }{ {\rm{exp} }{\left\{ { - { {\left( {\dfrac{ {\ln x - \alpha } }{ {2\gamma } } } \right)}^2} } \right\} } }$ $\mu = { \rm{e}^{\alpha + { {\gamma ^2} }/2 } }$ $\sigma = {\rm{e}^{2\alpha + {\gamma ^2} } }( { {{\rm{e}}^{ {r^2} } } - 1})$ Gamma distribution $f(x;r,\lambda ) = \dfrac{ { {\lambda ^r} } }{ {\Gamma (r)} }{x^{r - 1} }{\rm{e}^{ - \lambda x} }$ $\mu = \dfrac{\gamma }{\lambda }$ $\sigma = \dfrac{\gamma }{{{\lambda ^2}}}$ exponential distribution $f(x;\lambda ) = \lambda {\rm{e}^{ - \lambda x} }$ $\mu = \dfrac{1}{\lambda }$ $\sigma = \dfrac{1}{{{\lambda ^2}}}$ Beta distribution $f(x;\alpha ,\beta ) = \dfrac{{\Gamma (\alpha + \beta )}}{{\Gamma (\alpha )\Gamma (\beta )}}{x^{\alpha - 1}}{(1 - x)^{\beta - 1}}$ $\mu = \dfrac{\alpha }{{\alpha + \beta }}$ $\sigma = \dfrac{{\alpha \beta }}{{{{(\alpha + \beta )}^2}(\beta + \alpha + 1)}}$
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