基于贝叶斯理论的抗剪强度参数最优Copula函数识别
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摘要
提出基于贝叶斯理论的抗剪强度参数最优Copula函数识别方法,首先简要介绍了基于Copula函数的岩土体抗剪强度参数相关结构表征方法,给出常用的识别最优Copula函数的最小平方欧氏距离法和AIC(akaike information criterion)准则。其次,采用蒙特卡洛模拟方法验证了贝叶斯理论识别最优Copula函数的有效性,比较了3种方法的最优Copula函数识别能力,并分析了影响贝叶斯理论识别精度的主要因素。最后,收集了实际工程共23组抗剪强度参数试验数据,研究了贝叶斯理论在抗剪强度参数最优Copula函数识别中的应用。结果表明,贝叶斯理论能够有效地识别表征抗剪强度参数间相关结构的最优Copula函数,且能有效考虑先验信息对识别结果的影响;与传统的最小平方欧氏距离法和AIC准则相比,贝叶斯理论的识别能力和识别精度都更高;抗剪强度参数的样本数目、相关性大小、真实Copula函数类型以及先验信息都对贝叶斯理论的识别精度具有重要的影响。此外,常用的Gaussian Copula函数并不总是表征抗剪强度参数间相关结构的最优Copula函数。
This paper proposes a Bayesian Copula identification method for shear strength parameters of soils and rocks. First, the characterization of dependence structure between shear strength parameters using Copulas is presented. Two commonly-used methods, namely least square method of Euclidean distance and akaike information criterion(AIC), for identifying the best-fit Copula, are given. Then, Monte Carlo simulations are conducted to validate the Bayesian Copula identification method. Moreover, the identification accuracy in the three methods is compared, and the main factors affecting the accuracy in the Bayesian Copula identification are identified. Finally, a total of twenty-three sets of shear strength data are compiled to demonstrate the application of Bayesian theory Copula model identification. The results indicate that with limited project-specific data and prior information, the Bayesian Copula identification method can successfully identify the best-fit Copula from a set of alternative Copulas for shear strength parameters. In comparison with the least square method of Euclidean distance and AIC, the Bayesian Copula identification method produces more accurate results for identifying the best-fit Copula. The sample size, correlation, the type of the true Copula and prior information of shear strength parameters has a significant impact on the accuracy of the Bayesian Copula selection method. Furthermore, the commonly adopted Gaussian copula for characterizing the dependence structure between shear strength parameters does not always provide the best fit to the shear strength data.
This paper proposes a Bayesian Copula identification method for shear strength parameters of soils and rocks. First, the characterization of dependence structure between shear strength parameters using Copulas is presented. Two commonly-used methods, namely least square method of Euclidean distance and akaike information criterion(AIC), for identifying the best-fit Copula, are given. Then, Monte Carlo simulations are conducted to validate the Bayesian Copula identification method. Moreover, the identification accuracy in the three methods is compared, and the main factors affecting the accuracy in the Bayesian Copula identification are identified. Finally, a total of twenty-three sets of shear strength data are compiled to demonstrate the application of Bayesian theory Copula model identification. The results indicate that with limited project-specific data and prior information, the Bayesian Copula identification method can successfully identify the best-fit Copula from a set of alternative Copulas for shear strength parameters. In comparison with the least square method of Euclidean distance and AIC, the Bayesian Copula identification method produces more accurate results for identifying the best-fit Copula. The sample size, correlation, the type of the true Copula and prior information of shear strength parameters has a significant impact on the accuracy of the Bayesian Copula selection method. Furthermore, the commonly adopted Gaussian copula for characterizing the dependence structure between shear strength parameters does not always provide the best fit to the shear strength data.
引文
[1]TANG X S,LI D Q,RONG G,et al.Impact of copula selection on geotechnical reliability under incomplete probability information[J].Computers and Geotechnics,2013,49:264-278.
[2]CHERUBINI C.Reliability evaluation of shallow foundation bearing capacity on c?,??soils[J].Canadian Geotechnical Journal,2000,37(1):264-269.
[3]LOW B K.Reliability-based design applied to retaining walls[J].Geotechnique,2005,55(1):63-75.
[4]范明桥,盛金保.土强度指标?,c的互相关性[J].岩土工程学报,1997,19(4):100-104.FAN Ming-qiao,SHENG Jin-bao.Cross correlation of soil strength indexes?and c[J].Chinese Journal of Geotechnical Engineering,1997,19(4):100-104.
[5]赵宇飞,杨钧.岩土强度参数二维正态分布拟合研究[J].华北水利水电学院学报,2006,27(2):79-81.ZHAO Yu-fei,YANG Jun.The fitting reaesrch of two-dimension normal distribution of geotechnical strength parameters[J].Journal of North China Institute of Water Conservancy and Hydroelectric Power,2006,27(2):79-81.
[6]李典庆,唐小松,周创兵.含相关非正态变量边坡可靠度分析的认知聚类分区方法[J].岩土工程学报,2011,33(6):875-882.LI Dian-qing,TANG Xiao-song,ZHOU Chuang-bing.Reliability analysis of slope stability involving correlated non-normal variables using knowledge-based clustered partitioning method[J].Chinese Journal of Geotechnical Engineering,2011,33(6):875-882.
[7]NELSEN R B.An introduction to Copulas[M].New York:Springer,2006.
[8]唐小松,李典庆,周创兵,等.基于Copula函数的基桩荷载-位移双曲线概率分析[J].岩土力学,2012,33(1):171-178.TANG Xiao-song,LI Dian-qing,ZHOU Chuang-bing,et al.Probabilistic analysis of load-displacement hyperbolic curves of single pile using Copula[J].Rock and Soil Mechanics,2012,33(1):171-178.
[9]唐小松,李典庆,周创兵,等.基于Copula函数的抗剪强度参数间相关性模拟及边坡可靠度分析[J].岩土工程学报,2012,34(12):2284-2291.TANG Xiao-song,LI Dian-qing,ZHOU Chuang-bing,et al.Modeling dependence between shear strength parameters using Copulas and its effect on slope reliability[J].Chinese Journal of Geotechnical Engineering,2012,34(12):2284-2291.
[10]杨超,黄达,张永兴.基于Copula理论岩体质量Q值及波速与变形模量多变量相关性研究[J].岩石力学与工程学报,2014,33(3):507-513.YANG Chao,HUANG Da,ZHANG Yong-xing.Multivariable correlation of rock mass quality index Q system value,wave velocity and deformation modulus based on Copula theory[J].Chinese Journal of Rock Mechanics and Engineering,2014,33(3):507-513.
[11]LI D Q,ZHANG L,TANG X S,et al.Bivariate distribution of shear strength parameters using Copulas and its impact on geotechnical system reliability[J].Computers and Geotechnics,2015,68:184-195.
[12]HUFFMAN J C,STRAHLER A W,STUEDLEIN A W.Reliability-based serviceability limit state design for immediate settlement of spread footings on clay[J].Soils and Foundations,2015,55(4):798-812.
[13]MOTAMEDI M,LIANG R Y.Probabilistic landslide hazard assessment using Copula modeling technique[J].Landslides,2014,11(4):565-573.
[14]WU X Z.Development of fragility functions for slope instability analysis[J].Landslides,2014,12(1):165-175.
[15]DURRLEMAN V,NIKEGHBALI A,RONCALLI T.Which copula is the right one[J].Social Science Research Network Electronic Journal,2000.
[16]AKAIKE H.A new look at the statistical model identification[J].IEEE Transactions on Automatic Control,1974,19(6):716-723.
[17]PANCHENKO V.Goodness-of-fit test for Copulas[J].Statistical Mechanics and its Applications(Physica A),2005,355(1):176-182.
[18]唐小松,李典庆,周创兵,等.基于Bootstrap方法的岩土体参数联合分布模型识别[J].岩土力学,2015,36(4):913-922.TANG Xiao-song,LI Dian-qing,ZHOU Chuang-bing,et al.Bootstrap method for joint probability distribution identification of correlated geotechnical parameters[J].Rock and Soil Mechanics,2015,36(4):913-922.
[19]WANG Y,AU S K,CAO Z J.Bayesian approach for probabilistic characterization of sand friction angles[J].Engineering Geology,2010,114(3):354-363.
[20]WANG Y,CAO Z J.Probabilistic characterization of Young’s modulus of soil using equivalent samples[J].Engineering Geology,2013,159:106-118.
[21]CAO Z J,WANG Y.Bayesian approach for probabilistic site characterization using cone penetration tests[J].Journal of Geotechnical and Geoenvironmental Engineering,2013,139(2):267-276.
[22]CAO Z J,WANG Y.Bayesian model comparison and characterization of undrained shear strength[J].Journal of Geotechnical and Geoenvironmental Engineering,2014,140(6):04014018.
[23]CAO Z J,WANG Y.Bayesian model comparison and selection of spatial correlation functions for soil parameters[J].Structural Safety,2014,49:10-17.
[24]于清波.土石坝坝坡稳定可靠度研究及其工程应用[硕士学位论文D].南京:河海大学,2006.YU Qing-bo.Reliability analysis of dam slope stability and its application to engineering[Master thesis D].Nanjing:Hohai University,2006.
[25]汪小刚,董育坚.岩基抗剪强度参数[M].北京:中国水利水电出版社,2010.WANG Xiao-gang,DONG Yu-jian.The shear strength of rock mass[M].Beijing:China Water Power Press,2010.
[26]FORREST W S,ORR T L L.Reliability of shallow foundations designed to Eurocode 7[J].Georisk,2010,4(4):186-207.
[27]HATA Y,ICHII K,TSUCHIDA T,et al.A practical method for identifying parameters in the seismic design of embankments[J].Georisk,2008,2(1):28-40.
[28]KADAR I.Some characteristic values of the stability analysis of MAL dams[C]//Proceedings of Second Conference of Junior Researchers in Civil Engineering.Budapest,Magyarország:[s.n.],2013:100-104.
[29]LUMP P.Safety factors and the probability distribution of soil strength[J].Canadian Geotechnical Journal,1970,7(3):225-242.
[30]MATSUO M,KURODA K.Probabilistic approach to design of embankments[J].Soils and Foundations,1974,14(2):1-17.
[31]SCHULTZE E.Frequency distributions and correlations of soil properties[C]//Proceedings of the First International Conference on Applications of Statistics and Probability in Soil and Structural Engineering(ICASP).Hong Kong:Hong Kong University Press,1971:372-387.
[32]SPEEDIE M G.Selection of design value from shear test results[J].New Zealand Engineering,1955,10(11):377-378.
[33]RAHARDJO H,SATYANAGA A,LEONG E C,et al.Variability of residual soil properties[J].Engineering Geology,2012,141:124-140.
[34]GAY G,SCHAD H.Landslides and rockfall in keuper[J].Otto-Graf-Journal,2001,12:201-214.
[35]YOUNG D S.Probabilistic analysis of blasting impact on open pit stability[C]//Proceedings of the 25th US Symposium on Rock Mechanics(USRMS).Evanston:American Rock Mechanics Association,1984:1031-1041.
[2]CHERUBINI C.Reliability evaluation of shallow foundation bearing capacity on c?,??soils[J].Canadian Geotechnical Journal,2000,37(1):264-269.
[3]LOW B K.Reliability-based design applied to retaining walls[J].Geotechnique,2005,55(1):63-75.
[4]范明桥,盛金保.土强度指标?,c的互相关性[J].岩土工程学报,1997,19(4):100-104.FAN Ming-qiao,SHENG Jin-bao.Cross correlation of soil strength indexes?and c[J].Chinese Journal of Geotechnical Engineering,1997,19(4):100-104.
[5]赵宇飞,杨钧.岩土强度参数二维正态分布拟合研究[J].华北水利水电学院学报,2006,27(2):79-81.ZHAO Yu-fei,YANG Jun.The fitting reaesrch of two-dimension normal distribution of geotechnical strength parameters[J].Journal of North China Institute of Water Conservancy and Hydroelectric Power,2006,27(2):79-81.
[6]李典庆,唐小松,周创兵.含相关非正态变量边坡可靠度分析的认知聚类分区方法[J].岩土工程学报,2011,33(6):875-882.LI Dian-qing,TANG Xiao-song,ZHOU Chuang-bing.Reliability analysis of slope stability involving correlated non-normal variables using knowledge-based clustered partitioning method[J].Chinese Journal of Geotechnical Engineering,2011,33(6):875-882.
[7]NELSEN R B.An introduction to Copulas[M].New York:Springer,2006.
[8]唐小松,李典庆,周创兵,等.基于Copula函数的基桩荷载-位移双曲线概率分析[J].岩土力学,2012,33(1):171-178.TANG Xiao-song,LI Dian-qing,ZHOU Chuang-bing,et al.Probabilistic analysis of load-displacement hyperbolic curves of single pile using Copula[J].Rock and Soil Mechanics,2012,33(1):171-178.
[9]唐小松,李典庆,周创兵,等.基于Copula函数的抗剪强度参数间相关性模拟及边坡可靠度分析[J].岩土工程学报,2012,34(12):2284-2291.TANG Xiao-song,LI Dian-qing,ZHOU Chuang-bing,et al.Modeling dependence between shear strength parameters using Copulas and its effect on slope reliability[J].Chinese Journal of Geotechnical Engineering,2012,34(12):2284-2291.
[10]杨超,黄达,张永兴.基于Copula理论岩体质量Q值及波速与变形模量多变量相关性研究[J].岩石力学与工程学报,2014,33(3):507-513.YANG Chao,HUANG Da,ZHANG Yong-xing.Multivariable correlation of rock mass quality index Q system value,wave velocity and deformation modulus based on Copula theory[J].Chinese Journal of Rock Mechanics and Engineering,2014,33(3):507-513.
[11]LI D Q,ZHANG L,TANG X S,et al.Bivariate distribution of shear strength parameters using Copulas and its impact on geotechnical system reliability[J].Computers and Geotechnics,2015,68:184-195.
[12]HUFFMAN J C,STRAHLER A W,STUEDLEIN A W.Reliability-based serviceability limit state design for immediate settlement of spread footings on clay[J].Soils and Foundations,2015,55(4):798-812.
[13]MOTAMEDI M,LIANG R Y.Probabilistic landslide hazard assessment using Copula modeling technique[J].Landslides,2014,11(4):565-573.
[14]WU X Z.Development of fragility functions for slope instability analysis[J].Landslides,2014,12(1):165-175.
[15]DURRLEMAN V,NIKEGHBALI A,RONCALLI T.Which copula is the right one[J].Social Science Research Network Electronic Journal,2000.
[16]AKAIKE H.A new look at the statistical model identification[J].IEEE Transactions on Automatic Control,1974,19(6):716-723.
[17]PANCHENKO V.Goodness-of-fit test for Copulas[J].Statistical Mechanics and its Applications(Physica A),2005,355(1):176-182.
[18]唐小松,李典庆,周创兵,等.基于Bootstrap方法的岩土体参数联合分布模型识别[J].岩土力学,2015,36(4):913-922.TANG Xiao-song,LI Dian-qing,ZHOU Chuang-bing,et al.Bootstrap method for joint probability distribution identification of correlated geotechnical parameters[J].Rock and Soil Mechanics,2015,36(4):913-922.
[19]WANG Y,AU S K,CAO Z J.Bayesian approach for probabilistic characterization of sand friction angles[J].Engineering Geology,2010,114(3):354-363.
[20]WANG Y,CAO Z J.Probabilistic characterization of Young’s modulus of soil using equivalent samples[J].Engineering Geology,2013,159:106-118.
[21]CAO Z J,WANG Y.Bayesian approach for probabilistic site characterization using cone penetration tests[J].Journal of Geotechnical and Geoenvironmental Engineering,2013,139(2):267-276.
[22]CAO Z J,WANG Y.Bayesian model comparison and characterization of undrained shear strength[J].Journal of Geotechnical and Geoenvironmental Engineering,2014,140(6):04014018.
[23]CAO Z J,WANG Y.Bayesian model comparison and selection of spatial correlation functions for soil parameters[J].Structural Safety,2014,49:10-17.
[24]于清波.土石坝坝坡稳定可靠度研究及其工程应用[硕士学位论文D].南京:河海大学,2006.YU Qing-bo.Reliability analysis of dam slope stability and its application to engineering[Master thesis D].Nanjing:Hohai University,2006.
[25]汪小刚,董育坚.岩基抗剪强度参数[M].北京:中国水利水电出版社,2010.WANG Xiao-gang,DONG Yu-jian.The shear strength of rock mass[M].Beijing:China Water Power Press,2010.
[26]FORREST W S,ORR T L L.Reliability of shallow foundations designed to Eurocode 7[J].Georisk,2010,4(4):186-207.
[27]HATA Y,ICHII K,TSUCHIDA T,et al.A practical method for identifying parameters in the seismic design of embankments[J].Georisk,2008,2(1):28-40.
[28]KADAR I.Some characteristic values of the stability analysis of MAL dams[C]//Proceedings of Second Conference of Junior Researchers in Civil Engineering.Budapest,Magyarország:[s.n.],2013:100-104.
[29]LUMP P.Safety factors and the probability distribution of soil strength[J].Canadian Geotechnical Journal,1970,7(3):225-242.
[30]MATSUO M,KURODA K.Probabilistic approach to design of embankments[J].Soils and Foundations,1974,14(2):1-17.
[31]SCHULTZE E.Frequency distributions and correlations of soil properties[C]//Proceedings of the First International Conference on Applications of Statistics and Probability in Soil and Structural Engineering(ICASP).Hong Kong:Hong Kong University Press,1971:372-387.
[32]SPEEDIE M G.Selection of design value from shear test results[J].New Zealand Engineering,1955,10(11):377-378.
[33]RAHARDJO H,SATYANAGA A,LEONG E C,et al.Variability of residual soil properties[J].Engineering Geology,2012,141:124-140.
[34]GAY G,SCHAD H.Landslides and rockfall in keuper[J].Otto-Graf-Journal,2001,12:201-214.
[35]YOUNG D S.Probabilistic analysis of blasting impact on open pit stability[C]//Proceedings of the 25th US Symposium on Rock Mechanics(USRMS).Evanston:American Rock Mechanics Association,1984:1031-1041.