岩土强度参数正态–逆伽马分布的最大后验估计
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摘要
岩土强度参数的概率分布特征参数是确定强度参数标准值和可靠度分析及风险评估的基础,目前采用现场数据进行估计的方法存在小样本信息量不足的问题,为此,基于贝叶斯统计理论提出服从正态分布的岩土强度参数的概率特征参数服从于一个二维联合先验分布,并根据贝叶斯公式推导相应的共轭后验分布函数,以及岩土强度参数概率特征参数的最大后验估计量计算公式。以重庆万州区域内若干工程的泥岩和砂岩的黏聚力、内摩擦角的历史数据为例,建立先验分布函数,先统计单个工程的现场样本均值和方差,然后将若干工程的均值和方差组成新的样本,以此样本为基础采用参数估计得到理论推导确定的先验分布中的超参数,从而确定该区域泥岩和砂岩黏聚力及内摩擦角的先验分布函数,结合该区域内一个工程算例的现场数据,得到该工程在现场样本情况下泥岩和砂岩黏聚力及内摩擦角的概率分布特征参数的后验分布函数和最大后验估计值,并计算相应的黏聚力及内摩擦角的标准值,与传统方法确定的标准值进行比较,表明提出的贝叶斯方法综合了历史数据和现场数据的信息,更为科学合理。
The probability distribution characteristic parameters of geotechnical strength indices are necessary for determining the standard values of geotechnical strengths,analyzing reliability and evaluating risk. Existing method using in-situ data has the problem of insufficient information due to small samples. Based on Bayesian theory,in this paper,it is proposed that the probability characteristic parameters of geotechnical strengths obey a2 D Joint Prior Distribution,and a posterior distribution function and an equation for calculating the maximum posteriori estimator were deduced. Taking the cohesion and internal friction angle of mudstone and sandstone from Wanzhou for example,the hyper-parameter of the Prior Distribution Function was obtained based on the sample of means and variance of geotechnical strengths from different projects and consequently,the Prior Distribution Function of the cohesion and internal friction angle of mudstone and sandstone in Wanzhou was determined. By using the field data of an actual project,the posterior distribution and the maximum posteriori estimator of the probability characteristic parameters of mudstone and sandstone strengths were determined,and the standard values of the cohesion and the internal friction angle were calculated. Comparison with conventional methods shows that the new Bayesian method containing the information of historical and field data is more scientific and reasonable.
The probability distribution characteristic parameters of geotechnical strength indices are necessary for determining the standard values of geotechnical strengths,analyzing reliability and evaluating risk. Existing method using in-situ data has the problem of insufficient information due to small samples. Based on Bayesian theory,in this paper,it is proposed that the probability characteristic parameters of geotechnical strengths obey a2 D Joint Prior Distribution,and a posterior distribution function and an equation for calculating the maximum posteriori estimator were deduced. Taking the cohesion and internal friction angle of mudstone and sandstone from Wanzhou for example,the hyper-parameter of the Prior Distribution Function was obtained based on the sample of means and variance of geotechnical strengths from different projects and consequently,the Prior Distribution Function of the cohesion and internal friction angle of mudstone and sandstone in Wanzhou was determined. By using the field data of an actual project,the posterior distribution and the maximum posteriori estimator of the probability characteristic parameters of mudstone and sandstone strengths were determined,and the standard values of the cohesion and the internal friction angle were calculated. Comparison with conventional methods shows that the new Bayesian method containing the information of historical and field data is more scientific and reasonable.
引文
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[16]宫凤强,李夕兵,邓建.基于多源信息融合法的岩土力学参数概率分布推断[J].岩土力学,2007,28(3):599-602.(GONGFengqiang,LI Xibing,DENG Jian.Inferring probabilistic distribution of rock and soil mechanical parameters by using multiple sources information fusion method[J].Rock and Soil Mechanics,2007,28(3):599-602.(in Chinese))
[17]谢桂华,张家生.基于K-L信息距离的多源信息融合法[J].岩土力学,2010,31(9):2 983-2 986.(XIE Guihua,ZHANG Jiasheng.Multi-source information fusion based on K-L information distance[J].2010,31(9):2 983-2 986.(in Chinese))
[18]陈斌,刘宁,卓家寿.岩土工程反分析的扩展贝叶斯法[J].岩石力学与工程学报,2004,23(4):555-560.(CHEN Bin,LIU Ning,ZHUO Jiashou.Extended Bayesian method of inverse analysis in geo-engineering[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(4):555-560.(in Chinese))
[19]GELMAN A,CARLIN J B,HAL S,et al.Bayesian data analysis[M].3rd ed.London:CRC Press,2013:6-19.
[2]曾晟,徐华清,张家男,等.岩土工程参数Bayes法推断中先验与后验样本大小的探讨[J].南华大学学报:自然科学版,2015,29(3):30-33.(ZENG Sheng,XU Huaqing,ZHANG Jianan,et al.Discussion of priori and posteriori sample′s number of the geotechnical parameters with the Bayes inference method[J].Journal of University of South China:Science and Technology,2015,29(3):30-33.(in Chinese))
[3]张蕾,唐小松,李典庆,等.基于Copula函数的岩土结构物系统可靠度分析[J].岩土力学,2016,37(1):193-202.(ZHANG Lei,TANG Xiaosong,LI Dianqing,et al.System reliability analysis of geotechnical structures based on the Copula function[J].Rock and Soil Mechanics,2016,37(1):193-202.(in Chinese))
[4]KELLY R,HUANG J.Bayesian updating for one dimensional consolidation measurements[J].Canadian Geotechnical Journal,2015,52(9):1 318-1 330.
[5]中华人民共和国国家标准编写组.GB 50153-2008工程结构可靠性设计统一标准[S].北京:中国建筑工业出版社,2008.(The National Standards Compilation Group of People′s Republic of China.GB 50153-2008 Unified standard for reliability design of engineering structures[S].Beijing:China Architecture and Building Press,2008.(in Chinese))
[6]王新,黄磊群.随机加权法辅助小样本岩土参数概率分布的边坡稳定性分析[J].公路工程,2017,42(3):131-136.(WANG Xing,HUANG Leiqun.Analysis of slope stability based on probility distribution of small sample geotechnical parameters by random[J].Highway Engineering,2017,42(3):131-136.(in Chinese))
[7]蒋水华,魏博文,黄劲松.考虑参数空间变异性的失稳边坡参数概率反分析[J].岩土工程学报,2017,39(3):475-485.(JIANGShuihua,WEI Bowen,HUANG Jinsong.Probabilistic back analysis of slope failure considering spatial variability of soil properties[J].Chinese Journal of Geotechnical Engineering,2017,39(3):475-485.(in Chinese))
[8]肖欣,章杨松.基于贝叶斯网的岩体抗剪强度区间概率分析[J].勘察科学技术,2016,(2):5-26.(XIAO Xin,ZHANG Yangsong.Interval probability analysis of rock mass shear strength based on Bayesian network[J].Survey Science and Technology,2016,(2):5-26.(in Chinese))
[9]MARAL G,ALI N,AHMADREZA M A,et al.A worldwide SPT-based soil liquefaction triggering analysis utilizing gene expression programming and Bayesian probabilistic method[J].Journal of Rock Mechanics and Geotechnical Engineering,2017,(9):683-693.
[10]LUIS F C,EDWIN T,BMWN B M R.Bayesian data analysis to quantify the uncertain W of intact rock strength[J].Journal of Rock Mechanics and Geotechnical Engineering,2018,(10):11-31.
[11]赵宇飞,汪小刚,杨健.基于二维正态分布的岩土参数Bayes方法优化[J].中国水利水电科学研究院学报,2006,4(2):10-15.(ZHAO Yufei,WANG Xiaogang,YANG Jian.Bayes method of optimizing geotechnical parameters based no two-dimension normal distribution[J].Journal of China Institute of Water Resources and Hydropower Research,2006,4(2):10-15.(in Chinese))
[12]李红英,谭跃虎,赵辉.某滑坡体岩土参数概率分布统计分析方法研究[J].地下空间与工程学报,2012,8(3):659-665.(LIHongying,TAN Yuehu,ZHAO Hui.The statistical analysis technique research on probability distribution of geotechnical parameters of one landslide[J].Chinese Journal of Underground Space and Engineering,2012,8(3):659-665.(in Chinese))
[13]胡记磊,唐小微,裘江南.基于贝叶斯网络的地震液化概率预测分析[J].岩土力学,2016,37(6):1 745-1 752.(HU Jilei,TANGXiaowei,QIU Jiangnan.Prediction of probability of seismic-induced liquefaction based on Bayesian network[J].Rock and Soil Mechanics,2016,37(6):1 745-1 752.(in Chinese))
[14]程圣国,方坤河,罗先启,等.三峡库区新生型滑坡滑带土抗剪强度确定概率方法[J].岩石力学与工程学报,2007,26(4):840-845.(CHENG Shengguo,FANG Kunhe,LUO Xianqi,et al.Probability method to determine shear strength of slide zone clay for neogenic landslide in three gorgeservoir Area[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(4):840-845.(in Chinese))
[15]宫凤强,李夕兵,邓建.基于AHP先验分布融合法的岩土参数概率分布推断[J].岩土工程学报,2006,28(10):1 313-1 318.(GONGFengqiang,LI Xibing,DENG Jian.Probabilistic distribution of geotechnical parameters by using AHP prior distribution fusion method[J].Chinese Journal of Geotechnical Engineering,2006,28(10):1 313-1 318.(in Chinese))
[16]宫凤强,李夕兵,邓建.基于多源信息融合法的岩土力学参数概率分布推断[J].岩土力学,2007,28(3):599-602.(GONGFengqiang,LI Xibing,DENG Jian.Inferring probabilistic distribution of rock and soil mechanical parameters by using multiple sources information fusion method[J].Rock and Soil Mechanics,2007,28(3):599-602.(in Chinese))
[17]谢桂华,张家生.基于K-L信息距离的多源信息融合法[J].岩土力学,2010,31(9):2 983-2 986.(XIE Guihua,ZHANG Jiasheng.Multi-source information fusion based on K-L information distance[J].2010,31(9):2 983-2 986.(in Chinese))
[18]陈斌,刘宁,卓家寿.岩土工程反分析的扩展贝叶斯法[J].岩石力学与工程学报,2004,23(4):555-560.(CHEN Bin,LIU Ning,ZHUO Jiashou.Extended Bayesian method of inverse analysis in geo-engineering[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(4):555-560.(in Chinese))
[19]GELMAN A,CARLIN J B,HAL S,et al.Bayesian data analysis[M].3rd ed.London:CRC Press,2013:6-19.