基于贝叶斯理论及MCMC-MH算法推演地基土材料阻尼比的概率分布模型

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振动与冲击 ›› 2021, Vol. 40 ›› Issue (8) : 216-222.

论文

曹艳梅，李东伟，张玉玉，杨林

作者信息 +

Inversion of a probability distribution model of soil damping ratio based on the Bayesian theory and the MCMC-MH algorithm

CAO Yanmei， LI Dongwei， ZHANG Yuyu， YANG Lin

Author information +

文章历史 +

摘要

浅层土壤的材料阻尼比参数分布可以通过多通道表面波(MASW)分析法提取的表面波衰减曲线来反演识别，但是衰减曲线对于较大深度和小空间尺度土壤性质的变化不敏感，反演的土壤材料阻尼比分布是非唯一的和不确定的。基于该研究建立了土体材料阻尼比随深度变化的先验概率分布模型，利用Nataf变换和Karhunen-Loeve将其分解为标准高斯变量与特征值及特征向量的乘积和；随后依据贝叶斯理论，以TLM-PML模型结合频率波数域-半功率带宽法对衰减曲线进行正演，并与试验数据联合构建似然函数，使用蒙特卡洛马尔科夫链(MCMC)-Metropolis(MH)算法得到土体材料阻尼比的后验概率分布模型；对马尔科夫链的收敛性和独立性进行检验获得了多组相互独立的后验样本数据；用独立的后验样本计算出自由场振动响应，利用核密度估计得到具有一定置信度的置信区间，并与试验数据进行比较，验证了该研究提出的土体阻尼比非确定性概率模型的合理性和可靠性。

Abstract

The parameter distribution of the material damping ratio of shallow soil can be identified by inversion through the surface wave attenuation curve extracted by the multi-channel surface wave analysis method (MASW). However, the attenuation curves of surface waves are insensitive to the property changes of soil with larger depths and small spatial scales, and the damping ratio distribution of subsoil material is non-unique and uncertain. Therefore, in this paper, the variation of the damping ratio with depth is expressed by a non-Gaussian prior probability distribution model, and then is decomposed into the sum of standard Gaussian variables and also eigenvalues and eigenvectors by using the Nataf transformation and the Karhunen-Loeve decomposition. Moreover, on the basis of the Bayesian theory, the TLM-PML theory was combined with a half-power bandwidth method in the frequency-wavenumber domain to simulate the attenuation curves, which is compared with the experimental data to construct the likelihood function. The Monte Carlo Markov chain (MCMC)-Metropolis (MH) algorithm was used to obtain the posterior probability distribution model of soil damping ratio from the posterior sample data, and the convergence and independence of the Markov chain were validated so that multiple sets of independent posterior sample data were obtained. Finally, the independent posterior samples were used to calculate the vibration responses of the free field. The confidence interval with a certain degree of confidence was obtained using the kernel density estimation and thus was compared with the experimental data, which validates the rationality and reliability of the non-deterministic probability model of soil damping ratio proposed in this paper.

导出引用

曹艳梅，李东伟，张玉玉，杨林. 基于贝叶斯理论及MCMC-MH算法推演地基土材料阻尼比的概率分布模型[J]. 振动与冲击, 2021, 40(8): 216-222

CAO Yanmei， LI Dongwei， ZHANG Yuyu， YANG Lin. Inversion of a probability distribution model of soil damping ratio based on the Bayesian theory and the MCMC-MH algorithm[J]. Journal of Vibration and Shock, 2021, 40(8): 216-222

参考文献

［1］林署炯, 曾平, 林鹏威，等. 高速列车引起的地基土振动研究综述［J］. 石家庄铁道大学学报（自然科学版）, 2016, 29(4):40-47.
LIN Shujiong, ZENG Ping, LIN Pengwei, et al. A review on ground vibration induced by high-speed train［J］. Journal of Shijiazhuang Railway University (Natural Science Edition), 2016, 29(4): 40-47.
［2］曹艳梅, 夏禾. 基于Betti-Rayleigh动力互易定理求解移动荷载引起的地基土振动［J］. 岩石力学与工程学报, 2009, 28(7): 1467-1476.
CAO Yanmei, XIA He. Solution of foundation soil vibrations induced by moving loads based on Betti-Rayleigh dynamic reciprocity theorem［J］. Chinese Journal of Rock Mechanics and Engineering, 2009, 28(7): 1467-1476.
［3］SCHEVENELS M. The impact of uncertain dynamic soil characteristics on the prediction of ground vibration［D］. Leuven: Katholieke Universiteit Leuven, 2007.
［4］AUERSCH L. Technically induced surface wave fields, Part I: measured attenuation and theoretical amplitude-distance laws［J］. Bulletin of the Seismological Society of America, 2010, 100(4): 1528-1539.
［5］KARL L, HAEGEMAN W, DEGRANDE G. Determination of the material damping ratio and the shear wave velocity with the seismic cone penetration test［J］. Soil Dynamics and Earthquake Engineering, 2006, 26(12):1111-1126.
［6］BADSAR S A, SCHEVENELS M, HAEGEMAN W, et al. Determination of the material damping ratio in the soil from SASW tests using the half-power bandwidth method［J］. Geophysical Journal International, 2010, 182: 1493-1508.
［7］王薇. 移动荷载作用下地基土振动的TLM-PML理论分析方法及振动特性研究［D］. 北京: 北京交通大学, 2016.
［8］LAI C. Simultaneous inversion of Rayleigh phase velocity and attenuation for near-surface site characterization［D］. Atlanta:Georgia Institute of Technology,1998.
［9］LAI C, RIX G, FOTI S. Simultaneous measurement and inversion of surface wave dispersion and attenuation curves［J］. Soil Dynamics and Earthquake Engineering, 2002, 22(9/10/11/12): 923-930.
［10］RIX G J, LAI C G, WESLEY SPANG Jr A. In situ measurement of damping ratio using surface waves［J］. Journal of Geotechnical and Geoenvironmental Engineering, 2000, 126(5): 472-480.
［11］SCHEVENELS M, LOMBAERT G, DEGRANDE G, et al. A probabilistic assessment of resolution in the SASW test and its impact on the prediction of ground vibrations［J］. Geophysical Journal International, 2008, 172: 262-275.
［12］肖张波. 地震数据约束下的贝叶斯随机反演方法研究［D］. 北京: 中国石油大学, 2013.
［13］高山, 郑向远, 黄一. 非高斯随机过程的短期极值估计：复合Hermite模型［J］. 工程力学, 2019, 36(1): 23-31.
GAO Shan, ZHENG Xiangyuan, HUANG Yi. Hybrid Hermite models for short term extrema estimation of non-Gaussian processes［J］. Engineering Mechanics, 2019, 36(1): 23-31.
［14］李锦华, 陈水生. 非高斯随机过程模拟与预测的研究进展［J］. 华东交通大学学报, 2011, 28(6): 1-6.
LI Jinhua, CHEN Shuisheng. Advances in simulation and prediction of non-Gaussian stochastic processes［J］. Journal of East China Jiaotong University, 2011, 28(6): 1-6.
［15］MINASNY B. The Matérn function as a general model for soil variograms［D］. Sydney: The University of Sydney, 2005.
［16］范文亮, 韩杨, 周擎宇. 概率信息不完全系统的统计矩估计方法［J］. 工程力学, 2017, 34(2): 34-41.
FAN Wenliang, HAN Yang, ZHOU Qingyu. Point estimation for statistical moments of systems with incomplete probability information［J］. Engineering Mechanics, 2017,34(2):34-41.
［17］XIAO Q. Evaluating correlation coefficient for Nataf transformation［J］. Probabilistic Engineering Mechanics, 2014, 37: 1-6.
［18］陈绵书, 王园园, 桑爱军, 等. 基于多维矢量矩阵理论的KL变换［J］. 吉林大学学报, 2016, 46(2): 627-631.
CHEN Mianshu, WANG Yuanyuan, SANG Aijun, et al. KL transformation based on the theory of multidimensional vector matrix［J］. Journal of Jilin University, 2016, 46(2): 627-631.
［19］蒋伟, 刘纲. 基于多链差分进化的贝叶斯有限元模型修正方法［J］. 工程力学, 2019, 36(6): 101-108.
JIANG Wei, LIU Gang. Complicated finite element model updating based on Bayesian method［J］. Engineering Mechanics, 2019, 36(6): 101-108.［20］王佐仁, 杨琳. 贝叶斯统计推断及其主要进展［J］. 统计与信息论坛, 2012, 27(12):3-8.
WANG Zuoren, YANG Lin. Bayesian inference and main progress［J］. Journal of Statistics and Information, 2012, 27(12): 3-8.
［21］刘纲, 罗钧, 秦阳, 等. 基于改进MCMC方法的有限元模型修正研究［J］. 工程力学, 2016, 33(6): 138-145.
LIU Gang, LUO Jun, QIN Yang, et al. A finite element model updating method based on improved MCMC method［J］. Engineering Mechanics, 2016, 33(6): 138-145.
［22］王禹玺, 李典庆, 唐小松. 边坡系统可靠度区间估计方法［J］. 武汉大学学报（工学版），2019, 52(11): 941-950.
WANG Yuxi, LI Dianqing,TANG Xiaosong. An interval estimation method of slope system reliability［J］. Engineering Journal of Wuhan University, 2019, 52(11): 941-950.