Multivariate Distribution And Probabilistic Transformation Models For Multiple Geotechnical Parameters Using Copulas

Water resources and hydropower engineering is an important part of national strategy for clean energy priority development and securing the decisive victory in poverty-alleviation,with a pressing risk of inducing geologic hazards.It is of great significance to make full use of geotechnical investigation data and adequately grasp the information for key design parameters,in order to evaluate the probability of potential geological hazard and to formulate engineering measures and emergency plans.Due to the limitation in technical and economic conditions,the direct investigation data of key design parameters,such as hydraulic parameters(permeability coefficient etc.),deformation parameters(elasticity modulus etc.),strength parameters(undrained shear strength etc.),in engineering practice is very limited,which brings great uncertainty for geotechnical reliability analysis.Therefore,with a transformation model based on the investigation data of existing similar projects,converting the inexpensively and frequently measured parameters(such as CPT test indices and SPT test index)to design parameters,is a practical way for reducing the uncertainty of design parameters and improving the effectiveness of reliability analysis.However,the geological conditions between the existing engineering projects and investigated site may not be completely consistent,and there exists uncertainties in site investigation data of a single existing project caused by geologic origin,composition,structure and so on.The existence of a variety of uncertainty factors usually leads to a gap between the predicted values obtained by transformation model and the actual values of design parameters.With a careful consideration of the influence of the above uncertainties,it is imperative to construct a probabilistic transformation model which can reasonably represent the probability information of actual design parameters.Moreover,distinct related mechanisms exist between different pairs of measured parameters and design parameters,while some of the pairwise correlations among geotechnical parameters may be tail-dependent and asymmetric.How to characterize the tail dependence,asymmetry and even diversity in correlations among multiple geotechnical parameters is a key problem in geotechnical investigation.Additionally,geotechnical investigation data is often incomplete,with various sources and characteristics of data missing.The classification of the data missing mechanisms and characteristics,and the accurate construction of probabilistic transformation model of geotechnical design parameters based on incomplete investigation data,are of great significance for geotechnical investigation.To address the above three key issues,this thesis aims to introduce the copula theory into the transformation model construction of geotechnical design parameters and to explore the joint multivariate distribution and probabilistic transformation model construction of geotechnical parameters based on complete and incomplete investigation data.Specifically,the problems studied in this thesis include:(1)the applicability assessment of multiple construction methods for multivariate Gaussian Copula in multivariate distribution modeling of multiple geotechnical parameters;(2)the probabilistic transformation model construction method for design parameters based on multivariate Gaussian Copula and the validation method of transformation models;(3)the probabilistic transformation model construction method for design parameters based on multivariate t Copula and the application on global investigation database;(4)the construction method of multivariate distributions and transformation models for multiple geotechnical parameters based on vine copula;(5)the classification of data missing mechanisms and characteristics for incomplete geotechnical investigation data with the construction method of multivariate distributions and transformation models for multiple geotechnical parameters considering the different types of investigation data missing.The implementation details and some conclusions are listed as follows:(1)This thesis first presents the background and significance of multivariate distribution and transformation model construction of multiple geotechnical parameters using copulas.The existing transformation models for geotechnical design parameters are classified as theoretical transformation models,empirical transformation models and probabilistic transformations models,with the advantages and disadvantages of each kinds of transformation models studied.The existing methods for uncertainty modeling for geotechnical parameters are outlined,and a detailed review of the applications of copula theory in geotechnical engineering is presented.The key issues for the uncertainty modeling of geotechnical parameters under incomplete investigation data are analyzed and identified.Thus the research ideas and main content are established.(2)The multivariate copula theory is introduced.With the definition and basic properties of copulas,the commonly used bivariate copulas are presented.The main multivariate copulas in statistics,multivariate Gaussian copula,multivariate t Copula and multivariate Archimedean copulas,are introduced from definition and simulation aspects,while multivariate generalized t copulas and nested Archimedean copulas are outlined.The estimation of multivariate copula parameters using maximum likelihood method and three commonly used indexes to identify best-fitted copulas are presented.(3)The multivariate correlations underlying the probabilistic transformation models for geotechnical design parameters has not been studied extensively.Therefore,a Gaussian-copula-based method is proposed to model the joint probability distributions of multiple geotechnical parameters.The applicability of four construction methods of multivariate Gaussian copulas is theoretically and practically evaluated from four aspects: calculating costs,universality,goodness of fit and simulation errors.Commonly used Pearson correlation coefficient method only applies to joint distribution modeling of lowly variable geotechnical parameters for regional investigation database.Correlation matrix may be not positively definite for highly correlated geotechnical parameters with Kendall correlation coefficient method applied.Spearman correlation coefficient is recommended with little amplified effect and high applicability.Maximum likelihood method is suggested as a reference method with high precision and expensive calculating costs.(4)A method for probabilistic transformation model of geotechnical design parameters is proposed based on multivariate Gaussian copula,with calculating formulas for transformation model parameters derived.The proposed method is validated by leave-one-out design exercises based on the multivariate investigation databases,and then the probabilistic transformation models for common geotechnical design parameters are deduced.The effect of construction methods for multivariate Gaussian copula on effectiveness of probabilistic transformation models is studied.The proposed Gaussian-copula-based method can consider the effect of site investigation information on uncertainty and correlation of design parameters,and the deduced probabilistic transformation models based on global investigation database are simple to use and of favorable practical value.(5)For the potential tail correlations among geotechnical parameters,a method to construct probabilistic transformation model of geotechnical design parameters is proposed based on multivariate t copula,with the formulas for univariate and multivariate transformation model parameters developed.The tail correlations among geotechnical parameters are studied by comparison between multivariate t copula and multivariate Gaussian copula.The study results based on global investigation databases indicate that different levels of tail dependence exist among multiple geotechnical parameters,especially between strength parameters and CPTU test indices.With the tail dependence taken into consideration,the similarity in investigation data between the investigated site and existing sites can be considered by the probabilistic transformation models based on multivariate t copula.Uncertainty of design parameters obtained by the proposed probabilistic transformation model decreases with the similarity increases.The proposed t-copula-based method provides an efficient and sensible way to estimate uncertainty of design parameters under insufficient investigation data of existing projects.(6)Construction of probabilistic transformation model considering diversity in the pairwise correlations among geotechnical parameters is a challenging problem.A vinecopula-based approach is proposed to model joint multivariate distributions and probabilistic transformation models for geotechnical parameters,and validated by the existing multivariate investigation databases.The results indicate that the pairwise correlations among multiple geotechnical parameters are diversiform.With full consideration of diversity in pairwise correlations,the vine-copula-based joint distributions have better fitting effect than the joint distributions based on multivariate Gaussian copula and multivariate t copula.The proposed method provide an effective tool for adequate construction of multivariate distribution and probabilistic transformation model by fully considering diversity in pairwise correlations among geotechnical parameters.(7)For the main issues in uncertainty modeling of geotechnical parameters under incomplete investigation data,a new mothed to construct joint distribution and probabilistic transformation models of geotechnical parameters based on investigation data-missing types is proposed.The phenomena of investigation data missing are classified as discrete data missing and continuous data missing by data missing causes and characteristics,with both types of data missing statistically belonging to completely random missing.On that basis,likelihood function for the joint distribution of geotechnical parameters under incomplete investigation data is deduced,which provides a criterion of goodness for uncertainty modeling of multiple geotechnical parameters under incomplete investigation data.A method based on ECM algorithm is proposed to construct the joint distributions and probabilistic transformation models with consideration of different types of investigation data missing.Compared with discrete data missing,continuous data missing brings more uncertainty into the constructed multivariate distributions,while the uncertainty of probabilistic transformation models increases with the number of measure parameters.The proposed method possesses better robustness than Spearman correlation coefficient method under both data-missing types and all rates of data missing,which provides a robust tool for joint distribution modeling and probabilistic transformation model construction of geotechnical parameters under incomplete investigation data.