Bayesian approach for probabilistic characterization of sand friction angles

Site characterization is a unique problem of geotechnical engineering that utilizes both prior information (including engineering judgment) and project-specific information from test borings, in-situ testing, and/or laboratory testing. The problem is further complicated by inherent spatial variability of geo-materials and the fact that only a small portion of geo-materials are examined during site characterization. This paper describes a Bayesian approach to integrate prior information and project-specific test results for probabilistic characterization of soil properties from a limited number of tests. The Bayesian framework is developed in conjunction with cone penetration tests to estimate the sand effective friction angle and with random field theory to model the inherent spatial variability. Posterior distributions of uncertain parameters are derived. An approximate method is used to by-pass multi-dimensional integration involved in obtaining the marginal distributions, while removing the need of using traditional conjugate prior distributions. Using conditional variance formula, it is shown analytically that the posterior variance of the friction angle arises from three sources, namely, the spatial variability (aleatory) and its uncertainty (epistemic) as well as the uncertainty in the mean value. This provides a means to determine whether the amount of project-specific information (e.g., in-situ and/or laboratory tests) is sufficient in site characterization. Analytical solutions are also derived for two asymptotic cases of posterior mean, which can be used as reference cases for checking the results from the Bayesian approach. The Bayesian approach is illustrated through a set of real results of cone penetration tests at a National Geotechnical Experimental Site in the USA.