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不同先验分布下的后验分布确定土力学参数

魏德永 阮永芬 闫明 郭宇航 丁海涛

魏德永,阮永芬,闫明,郭宇航,丁海涛. 不同先验分布下的后验分布确定土力学参数 [J]. 应用数学和力学,2021,42(11):1136-1149 doi: 10.21656/1000-0887.410385
引用本文: 魏德永,阮永芬,闫明,郭宇航,丁海涛. 不同先验分布下的后验分布确定土力学参数 [J]. 应用数学和力学,2021,42(11):1136-1149 doi: 10.21656/1000-0887.410385
WEI Deyong, RUAN Yongfen, YAN Ming, GUO Yuhang, DING Haitao. Determination of Soil Mechanical Parameters From Posterior Distributions Under Different Prior Distributions[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1136-1149. doi: 10.21656/1000-0887.410385
Citation: WEI Deyong, RUAN Yongfen, YAN Ming, GUO Yuhang, DING Haitao. Determination of Soil Mechanical Parameters From Posterior Distributions Under Different Prior Distributions[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1136-1149. doi: 10.21656/1000-0887.410385

不同先验分布下的后验分布确定土力学参数

doi: 10.21656/1000-0887.410385
基金项目: 云南省重点研发计划(社会发展领域)(2018BC008)
详细信息
    作者简介:

    魏德永(1993—),男,硕士(E-mail:396431183@qq.com)

    阮永芬(1964—),女,教授,硕士生导师(通讯作者. E-mail:ryy64@163.com)

  • 中图分类号: TU443

Determination of Soil Mechanical Parameters From Posterior Distributions Under Different Prior Distributions

  • 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μσ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μσ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。
  • 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 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  4  未知参数的联合分布图

    Figure  4.  Joint distribution diagrams of unknown parameters

    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]
    fiEFG···H
    下载: 导出CSV

    表  2  土体c$ \varphi $原始试验样本

    Table  2.   The c and $ \varphi $ original samples of soil

    soil namesample size Nfak/ kPaindex nameindex value
    4-1 peaty soil20050c/kPa16.7, 50.6, 12.10, ···, 32.9, 28.7, 30.5, 46.4, 42.7, 3.3
    200$ \varphi /(^\circ ) $6.1, 3.8, 5.2, ···, 3.6, 6.9, 6.6, 6.1, 9.8, 9.8, 2.3
    4-2 silty soil19080c/kPa18.5, 20.1, 20.3, 20.6, ···, 31.5, 31.8, 32.7, 34.8
    190$ \varphi /{(}^\circ ) $15, 15, 14, 12.3, ···, 18, 18.1, 18.4, 18.6, 18.7, 19.3
    4-2 clay21090c/kPa34.44, 30.16, 28.53, 38.30,···, 29.66, 31.54, 6.64
    210$ \varphi /{(}^\circ ) $5.91, 5.69, 7.24, ···, 6.71, 5.10, 7.21, 5.21
    下载: 导出CSV

    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

    indexnumber in fig. 1CCVabsolute value of Sktype of density function
    μc,ps(a)0.232 070.015 89normal distribution
    μϕ, ps(b)0.216 970.023 64normal distribution
    μc, fs(c)0.163 660.016 38normal distribution
    μϕ, fs(d)0.088 320.022 41normal distribution
    μc, c(e)0.10 1950.027 14lognormal distribution
    μϕ, c(f)0.099 910.024 58normal distribution
    下载: 导出CSV

    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 namemechanical parameter$ \hat \alpha $$ \hat \beta $sample distribution $ f(\left. x \right|\mu ) \propto $posterior distribution $ \pi (\left. \mu \right|x) \propto $
    4-1 peaty soil$ {\mu _{\text{c}}} $29.612.8$ \exp \left\{ - \dfrac{{{{(x - 28.8)}^2}}}{{2 \times 10.37}}\right\} $$ \exp \left\{ -\left (\dfrac{{{{(x - 28.8)}^2}}}{{20.74}}{\text{ + }}\dfrac{{{{(x - 29.6)}^2}}}{{25.6}}\right)\right\} $
    $ {\mu _\varphi } $5.62.14$ \exp \left\{ - \dfrac{{{{(x - 5.95)}^2}}}{{2 \times 1.54}}\right\} $$ \exp \left\{ -\left (\dfrac{{{{(x - 5.6)}^2}}}{{4.28}}{\text{ + }}\dfrac{{{{(x - 5.95)}^2}}}{{3.08}}\right)\right\} $
    4-2 silty soil$ {\mu _{\text{c}}} $26.37.57$ \exp \left\{ - \dfrac{{{{(x - 26.75)}^2}}}{{2 \times 5.4}}\right\} $$ \exp \left\{ -\left (\dfrac{{{{(x - 26.75)}^2}}}{{10.8}}{\text{ + }}\dfrac{{{{(x - 26.3)}^2}}}{{15.54}}\right)\right\} $
    $ {\mu _\varphi } $16.832.59$ \exp \left\{ - \dfrac{{{{(x - 16.44)}^2}}}{{2 \times 2.4}}\right\} $${\rm{exp}}\left\{ {{\rm{ - }}\left( {\dfrac{{{{(x{\rm{ - 16}}.{\rm{44}})}^{\rm{2}}}}}{{{\rm{5}}.{\rm{18}}}}{\rm{ + }}\dfrac{{{{(x{\rm{ - 15}}.{\rm{9}})}^{\rm{2}}}}}{{{\rm{4}}.{\rm{8}}}}} \right)} \right\}$
    4-2 clay$ {\mu _{\text{c}}} $29.865.20$ \exp \left\{ - \dfrac{{{{(x - 29.4)}^2}}}{{2 \times 4.5}}\right\} $$ \exp \left\{ -\left (\dfrac{{{{(x - 29.86)}^2}}}{{10.4}}{\text{ + }}\dfrac{{{{(x - 29.4)}^2}}}{9}\right)\right\} $
    $ {\mu _\varphi } $5.591.1$ \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\} $
    下载: 导出CSV

    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 nameindexn$ \bar x $μS2rζkkn$ \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
    下载: 导出CSV

    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 typeprobability density functionnormal distribution μnormal distribution σ
    normal distribution$f(x,\mu ,\sigma ) = \dfrac{1}{ {\sqrt {2{\text{π}} } \sigma } }\exp \left\{ { - \dfrac{ { { {(x - \mu )}^2} } }{ {2{\sigma ^2} } } } \right\}$μσ
    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)}}$
    下载: 导出CSV
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  • 收稿日期:  2020-12-15
  • 修回日期:  2021-09-08
  • 网络出版日期:  2021-12-07
  • 刊出日期:  2021-11-30

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