BayLUP: A Bayesian framework for conditional random field simulation of the liquefaction-induced settlement considering statistical uncertainty and model error

Assessing the spatial variability of the liquefaction-induced settlement at a site often involves spatial interpolation based on some stochastic models (e.g., random fields). Accuracy of the spatial interpolation results highly depends on the number of testing data and statistical model parameters. Statistical uncertainty in model parameters is inevitable due to a lack of sufficient testing data. Moreover, the liquefaction-induced settlement at testing locations can be estimated from site investigation data using semi-empirical models derived from previous testing data and observations, for which the model error is unavoidable. This paper develops a novel Bayesian framework, called BayLUP, based on Kriging-based conditional random field (CRF), which, simultaneously, accounts for spatial variability, statistical uncertainty, and model error into probabilistic characterization of the liquefaction-induced settlement. The proposed approach is comprised of three steps, i.e., learning step (L-step), updating step (U-step), and predicting step (P-step), which are formulated from a Bayesian perspective and are sequentially implemented using the ancestor sampling method. It is illustrated and validated using real and simulated data. Results show that the BayLUP framework provides reasonable spatial interpolation results of the liquefaction-induced settlement based on a limited number of testing data and prior knowledge. Under the BayLUP framework, the spatial variability, model error, and statistical uncertainty are taken into account in a quantifiable and rational way without compromising the computational efficiency of CRF simulation. Ignoring the statistical uncertainty and model error might lead to underestimation of the prediction uncertainty.