Bayesian approach for calibrating transformation model from spatially varied CPT data to regular geotechnical parameter

Cone Penetration Test (CPT) is widely utilized to gain regular geotechnical parameters such as compression modulus, cohesion coefficient and internal friction angle by transformation model in the site investigation. However, it is challenging to obtain simultaneously the unknown coefficients and error of a transformation model, given the intrinsic uncertainty (i.e., spatial variability) of geomaterial and the epistemic uncertainty of geotechnical investigation. A Bayesian approach is therefore proposed calibrating the transformation model based on spatial random field theory. The approach consists of three key elements: (1) three-dimensional anisotropic spatial random field theory; (2) classifications of measurement and error, and the uncertainty propagation diagram of geotechnical investigation; and (3) the unknown coefficients and error calibration of the transformation model given Bayesian inverse modeling method. The massive penetration resistance data from CPT, which is denoted as a spatial random field variable to account for the spatial variability of soil, are classified as type A data. Meanwhile, a few laboratory test data such as the compression modulus are defined as type B data. Based on the above two types of data, the unknown coefficients and error of the transformation model are inversely calibrated with consideration of intrinsic uncertainty of geomaterial, epistemic uncertainties such as measurement errors, prior knowledge uncertainty of transformation model itself, and computing uncertainties of statistical parameters as well as Bayesian method. Baseline studying indicates the proposed approach is applicable to calibrate the transformation model between CPT data and regular geotechnical parameter within spatial random field theory. Next, the calibrated transformation model was compared with classical linear regression in cross-validation, and then it was implemented at three-dimensional site characterization of the background project.     

Settlement prediction by spatial-temporal random process using Asaoka's method

A methodology was presented for observation-based settlement prediction with consideration of the spatial correlation structure of soil. The spatial correlation is introduced among the settlement model parameters and the settlements at various points are spatially correlated through these geotechnical parameters, which naturally describe the phenomenon. The method is based on Bayesian estimation by considering both prior information, including spatial correlation and observed settlement, to search for the best estimates of the parameters at any arbitrary points on the ground. Within the Bayesian framework, the optimised selection of auto-correlation distance by Akaike's Bayesian Information Criterion (ABIC) is also proposed. The application of the proposed approach in consolidation settlement prediction using Asaoka's method is presented in this paper. Several case studies were carried out using simulated settlement data to investigate the performance the proposed approach. It is concluded that the accuracy of the settlement prediction can be improved by taking into account the spatial correlation structure and the proposed approach gives the rational prediction of the settlement at any location at any time with quantified uncertainty.     

Reliability updating in geotechnical engineering including spatial variability of soil

At geotechnical sites, deformation measurements are routinely performed during the construction process. In this paper, it is shown how information from such measurements can be utilized to update the reliability estimate of the geotechnical site at future construction stages. A recently proposed method for Bayesian updating of the reliability is successfully applied in conjunction with a stochastic nonlinear geotechnical finite element model. Therein, uncertainty in the soil material properties is modelled by non-Gaussian random fields. The structural reliability evaluations required for the Bayesian updating are carried out by means of subset simulation, an efficient adaptive Monte Carlo method. The approach is demonstrated through an application to a sheet pile wall at a deformation-sensitive geotechnical construction site.     

Probabilistic outlier detection for sparse multivariate geotechnical site investigation data using Bayesian learning

Various uncertainties arising during acquisition process of geoscience data may result in anomalous data instances(i.e.,outliers)that do not conform with the expected pattern of regular data instances.With sparse multivariate data obtained from geotechnical site investigation,it is impossible to identify outliers with certainty due to the distortion of statistics of geotechnical parameters caused by outliers and their associated statistical uncertainty resulted from data sparsity.This paper develops a probabilistic outlier detection method for sparse multivariate data obtained from geotechnical site investigation.The proposed approach quantifies the outlying probability of each data instance based on Mahalanobis distance and determines outliers as those data instances with outlying probabilities greater than 0.5.It tackles the distortion issue of statistics estimated from the dataset with outliers by a re-sampling technique and accounts,rationally,for the statistical uncertainty by Bayesian machine learning.Moreover,the proposed approach also suggests an exclusive method to determine outlying components of each outlier.The proposed approach is illustrated and verified using simulated and real-life dataset.It showed that the proposed approach properly identifies outliers among sparse multivariate data and their corresponding outlying components in a probabilistic manner.It can significantly reduce the masking effect(i.e.,missing some actual outliers due to the distortion of statistics by the outliers and statistical uncertainty).It also found that outliers among sparse multivariate data instances affect significantly the construction of multivariate distribution of geotechnical parameters for uncertainty quantification.This emphasizes the necessity of data cleaning process(e.g.,outlier detection)for uncertainty quantification based on geoscience data.     

Geotechnical parameter estimation in tunnelling using relative convergence measurement

Accurate estimation of geotechnical parameters is an important and difficult task in tunnel design and construction. Optimum evaluation of the geotechnical parameters have been carried out by the back‐analysis method based on estimated absolute convergence data. In this study, a back‐analysis technique using measured relative convergence in tunnelling is proposed. The extended Bayesian method (EBM), which combines the prior information with the field measurement data, is adopted and combined with the 3‐dimensional finite element analysis to predict ground motion. By directly using the relative convergence as observation data in the EBM, we can exclude errors that arise in the estimation of absolute displacement from measured convergence, and can evaluate the geotechnical parameters with sufficient reliability. The proposed back‐analysis technique is applied and validated by using the measured data from two tunnel sites in Korea. Copyright © 2005 John Wiley & Sons, Ltd.     

The use of integral transforms in solving three dimensional problems in geomechanics

A method is presented which may be used to obtain solutions to geotechnical problems for the case where the soil profile is such that it is similar in one coordinate direction. If this is so, then Fourier transforms may be applied to the field quantities, thus reducing a problem which is essentially three dimensional to one involving only two spatial dimensions. This means great savings in computer storage and data preparation time as well as solution time. Solutions for the field quantities (i.e. displacements and stresses) at any point within the soil mass are found by applying inverse transforms and the required integration is carried out using numerical means. Examples of the application of the technique to common geotechnical problems are given to illustrate the use of the method.     

Bayesian updating revisited

Bayesian updating methods provide an alternate philosophy to the characterization of the input variables of a stochastic mathematical model. Here, a priori values of statistical parameters are assumed on subjective grounds or by analysis of a data base from a geologically similar area. As measurements become available during site investigations, updated estimates of parameters characterizing spatial variability are generated. However, in solving the traditional updating equations, an updated covariance matrix may be generated that is not positive-definite, particularly when observed data errors are small. In addition, measurements may indicate that initial estimates of the statistical parameters are poor. The traditional procedure does not have a facility to revise the parameter estimates before the update is carried out. alternatively, Bayesian updating can be viewed as a linear inverse problem that minimizes a weighted combination of solution simplicity and data misfit. Depending on the weight given to the a priori information, a different solution is generated. A Bayesian updating procedure for log-conductivity interpolation that uses a singular value decomposition (SVD) is presented. An efficient and stable algorithm is outlined that computes the updated log-conductivity field and the a posteriori covariance of the estimated values (estimation errors). In addition, an information density matrix is constructed that indicates how well predicted data match observations. Analysis of this matrix indicates the relative importance of the observed data. The SVD updating procedure is used to interpolate the log-conductivity fields of a series of hypothetical aquifers to demonstrate pitfalls and possibilities of the method.     

Bayesian updating revisited

Bayesian updating methods provide an alternate philosophy to the characterization of the input variables of a stochastic mathematical model. Here, a priori values of statistical parameters are assumed on subjective grounds or by analysis of a data base from a geologically similar area. As measurements become available during site investigations, updated estimates of parameters characterizing spatial variability are generated. However, in solving the traditional updating equations, an updated covariance matrix may be generated that is not positive-definite, particularly when observed data errors are small. In addition, measurements may indicate that initial estimates of the statistical parameters are poor. The traditional procedure does not have a facility to revise the parameter estimates before the update is carried out. alternatively, Bayesian updating can be viewed as a linear inverse problem that minimizes a weighted combination of solution simplicity and data misfit. Depending on the weight given to the a priori information, a different solution is generated. A Bayesian updating procedure for log-conductivity interpolation that uses a singular value decomposition (SVD) is presented. An efficient and stable algorithm is outlined that computes the updated log-conductivity field and the a posteriori covariance of the estimated values (estimation errors). In addition, an information density matrix is constructed that indicates how well predicted data match observations. Analysis of this matrix indicates the relative importance of the observed data. The SVD updating procedure is used to interpolate the log-conductivity fields of a series of hypothetical aquifers to demonstrate pitfalls and possibilities of the method.     

基于高性能并行计算的隧道开挖数值模拟

随着岩土工程规模的不断扩大、复杂性的增加以及计算参数的多样化和计算精度的提高, 人们对于计算机计算能力的要求越来越高, 然而单处理器无法满足这类大规模计算.从数据输入、区域分解、线性方程组的迭代求解、后处理等方面详细阐述高性能计算平台上并行有限元求解大规模岩土工程的关键问题.提出了利用MPI2的新特性进行海量数据的分段并行读入, 采用ParMetis软件并行地进行区域分解, 实现了前处理过程的完全并行化; 采用基于Jacobi预处理技术的预处理共轭梯度法(PCG)进行线性方程组的并行迭代求解; 采用Paraview软件实现了后处理的并行可视化.在深腾7000系统上对某隧道工程的三维开挖过程进行了数值模拟, 对其并行性能进行了分析和评价, 验证了采用的区域分解算法和系统方程组的求解方法的可行性, 并且具有较高的加速比和并行效率.      

基于遗传算法的岩土力学参数反演及其在 ABAQUS中的实现

岩土工程优化反分析是一个典型的复杂非线性函数优化问题,采用全局优化算法是解决这个问题的理想途径。结合ABAQUS有限元软件,提出遗传算法与有限元联合反演法,将有限元程序作为一个单独模块嵌入到遗传算法程序中,以测点的实测值与计算值建立误差函数,编制了遗传算法反演分析程序。并给出应用实例验证了该法的有效性,表明该方法可应用于岩土工程中的反演分析工作。     

Random field characterization of uniaxial compressive strength and elastic modulus for intact rocks

Rock properties exhibit spatial variabilities due to complex geological processes such as sedimentation,metamorphism, weathering, and tectogenesis. Although recognized as an important factor controlling the safety of geotechnical structures in rock engineering, the spatial variability of rock properties is rarely quantified. Hence, this study characterizes the autocorrelation structures and scales of fluctuation of two important parameters of intact rocks, i.e. uniaxial compressive strength(UCS) and elastic modulus(EM).UCS and EM data for sedimentary and igneous rocks are collected. The autocorrelation structures are selected using a Bayesian model class selection approach and the scales of fluctuation for these two parameters are estimated using a Bayesian updating method. The results show that the autocorrelation structures for UCS and EM could be best described by a single exponential autocorrelation function. The scales of fluctuation for UCS and EM respectively range from 0.3 m to 8.0 m and from 0.3 m to 8.4 m.These results serve as guidelines for selecting proper autocorrelation functions and autocorrelation distances for rock properties in reliability analyses and could also be used as prior information for quantifying the spatial variability of rock properties in a Bayesian framework.     

先验概率分布及似然函数模型的选择对边坡可靠度评价影响的定量评估

受工程勘察成本及试验场地限制,可获得的试验数据通常有限,基于有限的试验数据难以准确估计岩土参数统计特征和边坡可靠度。贝叶斯方法可以融合有限的场地信息降低对岩土参数不确定性的估计进而提高边坡可靠度水平。但是,目前的贝叶斯更新研究大多假定参数先验概率分布为正态、对数正态和均匀分布,似然函数为多维正态分布,这种做法的合理性有待进一步验证。总结了岩土工程贝叶斯分析常用的参数先验概率分布及似然函数模型,以一个不排水黏土边坡为例,采用自适应贝叶斯更新方法系统探讨了参数先验概率分布和似然函数对空间变异边坡参数后验概率分布推断及可靠度更新的影响。计算结果表明：参数先验概率分布对空间变异边坡参数后验概率分布推断及可靠度更新均有一定的影响,选用对数正态和极值I型分布作为先验概率分布推断的参数后验概率分布离散性较小。选用Beta分布和极值I型分布获得的边坡可靠度计算结果分别偏于保守和危险,选用对数正态分布获得的边坡可靠度计算结果居中。相比之下,似然函数的影响更加显著。与其他类型似然函数相比,由多维联合正态分布构建的似然函数可在降低对岩土参数不确定性估计的同时,获得与场地信息更为吻合的计算结果。另外,构建似然函数时不同位置处测量误差之间的自相关性对边坡后验失效概率也具有一定的影响。     

Probabilistic analysis of the inverse analysis of an excavation problem

This study presents the probabilistic analysis of the inverse analysis of an excavation problem. Two techniques are used during two successive stages. First, a genetic algorithm inverse analysis is conducted to identify soil parameters from in situ measurements (i.e. first stage of the construction project). For a given tolerable error between the measurement and the response of the numerical model the genetic algorithm is able to generate a statistical set of soil parameters, which may then serve as input data to a stochastic finite element method. The second analysis allows predicting a confidence interval for the final behaviour of the geotechnical structure (i.e. second stage of the project). The tools employed in this study have already been presented in previous papers, but the originality herein consists of coupling them. To illustrate this method, a synthetic excavation problem with a very simple geometry is used.     

Coupled boundary element/finite element analysis in geomechanics including body forces

A coupling scheme for boundary and finite elements using joint elements is proposed which includes the consideration of body forces. In this scheme the boundary and joint elements are formulated in a similar way as finite elements (i.e., the equivalent FE procedure). These joint elements are efficiently used to combine different BE regions. For the evaluation of a body forces, two methods are compared on computational efficiency and it is found that the method using Galerkin tensor is more efficient than the method dividing the problem domain into several internal cells. Two main geotechnical problems considering self weight are numerically examined using this coupling procedure.     

Comparative study of the effects of three data-interpretation methodologies on the performance of geotechnical back analysis

Back analysis can provide engineers with important information for better decision-making. Over the years, research on back analysis has focused mainly on optimisation techniques, while comparative studies of data-interpretation methodologies have seldom been reported. This paper examines the use of three data-interpretation methodologies on the performance of geotechnical back analysis. In general, there are two types of approaches for interpreting model predictions using field measurements, deterministic versus population-based, both of which are considered in this study. The methodologies that are compared are (a) error-domain model falsification (EDMF), (b) Bayesian model updating and (c) residual minimisation. Back analyses of an excavation case history in Singapore using the three methodologies indicate that each has strengths and limitations. Residual minimisation, though easy to implement, shows limited capabilities of interpreting measurement data with large uncertainty errors. EDMF provides robustness against incomplete information of the correlation structure. This is achieved at the expense of precision, as EDMF yields wider confidence intervals of the identified parameter values and predicted quantities compared with Bayesian model updating. In this regard, a modified EDMF implementation is proposed, which can improve upon the limitations of the traditional EDMF method, thus enhancing the quality of the identification outcomes.     

Segmentation of well logs by maximum-likelihood estimation

A maximum-likelihood procedure for segmenting digital well-log data is presented. The method is based on a univariate state variable model in which an observed log is treated as a time-series consisting of two terms: a Gauss-Markov signal remaining constant over a segment, and an additive Gaussian, but not necessarily stationary, noise. The signal jumps by a random amount at a segment boundary. The inverse problem of log segmentation consists of detecting the segment boundaries from a given log. The problem is solved using a Bayesian approach in which the unknown parameters, the locations of segment boundaries and the jumps in the signal value, are estimated by maximizing the likelihood function for the observed data. An algorithm based on Kalman smoothing and single most likelihood replacement (SMLR) procedure is proposed. The performance of the method is illustrated with a case study comprising of multisuite log data from an exploratory well. The method is found to be rapid and robust. The resulting segments are found to be geologically consistent.     

Statistical inverse analysis based on genetic algorithm and principal component analysis: Method and developments using synthetic data

This study concerns the identification of parameters of soil constitutive models from geotechnical measurements by inverse analysis. To deal with the non‐uniqueness of the solution, the inverse analysis is based on a genetic algorithm (GA) optimization process. For a given uncertainty on the measurements, the GA identifies a set of solutions. A statistical method based on a principal component analysis (PCA) is, then, proposed to evaluate the representativeness of this set. It is shown that this representativeness is controlled by the GA population size for which an optimal value can be defined. The PCA also gives a first‐order approximation of the solution set of the inverse problem as an ellipsoid. These developments are first made on a synthetic excavation problem and on a pressuremeter test. Some experimental applications are, then, studied in a companion paper, to show the reliability of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

Data assimilation using the particle filter for identifying the elasto‐plastic material properties of geomaterials

A computational method, incorporating the finite element model (FEM) into data assimilation using the particle filter, is presented for identifying elasto‐plastic material properties based on sequential measurements under the known changing traction boundary conditions to overcome some difficulties in identifying the parameters for elasto‐plastic problems from which the existing inverse analysis strategies have suffered. A soil–water coupled problem, which uses the elasto‐plastic constitutive model, is dealt with as the geotechnical application. Measured data on the settlement and the pore pressure are obtained from a synthetic FEM computation as the forward problem under the known parameters to be identified for both the element tests and the ground behavior during the embankment construction sequence. Parameter identification for elasto‐plastic problems, such as soil behavior, should be made by considering the measurements of deformation and/or pore pressure step by step from the initial stage of construction and throughout the deformation history under the changing traction boundary conditions because of the embankment or the excavation because the ground behavior is highly dependent on the loading history. Thus, it appears that sequential data assimilation techniques, such as the particle filter, are the preferable tools that can provide estimates of the state variables, that is, deformation, pore pressure, and unknown parameters, for the constitutive model in geotechnical practice. The present paper discusses the priority of the particle filter in its application to initial/boundary value problems for elasto‐plastic materials and demonstrates a couple of numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

基于最小二乘支持向量机的黄土湿陷性预测挖掘

运用数据挖掘技术进行了黄土湿陷性的预测挖掘,挖掘模型采用最小二乘支持向量机。建模过程中用主成份分析法进行数据的预处理,以剔除指标间的相关性,消除多指标信息冗余对挖掘模型的影响,并引入粒子群优化算法进行模型反演分析,确定最优参数。针对实际工程数据进行的预测挖掘表明：黄土的电阻率、剪切波速与土的结构特性、含水率、密度等指标密切相关,可较为全面地反映影响黄土湿陷性的因素;以电阻率、剪切波速及土层埋深作为模型的预测变量就可定量预测黄土的湿陷性;用所建模型和预测变量来预测黄土的湿陷性是可行的。     

An intelligent multi-objective EPR technique with multi-step model selection for correlations of soil properties

Current multi-objective evolutionary polynomial regression (EPR) methodology has difficulties on decision-making of optimal EPR model. This paper proposes an intelligent multi-objective optimization-based EPR technique with multi-step automatic model selection procedure. A newly developed multi-objective differential evolution algorithm (MODE) is adopted to improve the optimization performance. The proposed EPR process is composed of two stages: (1) intelligent roughing model selection and (2) model delicacy identification. In the first stage, besides two objectives (model accuracy and model complexity), the model robustness measured by robustness ratio is considered as an additional objective in the multi-objective optimization. In the second stage, a new indicator named selection index is proposed and incorporated to find the optimal model. After intelligent roughing selection and delicacy identification, the optimal EPR model is obtained considering the combined effects of correlation coefficient, size of polynomial terms, number of involved variables, robustness ratio and monotonicity. To show the practicality of the proposed EPR technique, three illustrative cases helpful for geotechnical design are presented: (a) modelling of compressibility, (b) modelling of undrained shear strength and (c) modelling of hydraulic conductivity. For each case, a practical formula with better performance in comparison with various existing empirical equations is finally provided. All results demonstrate that the proposed intelligent MODE-based EPR technique is efficient and effective.