An efficient PDE-constrained stochastic inverse algorithm for probabilistic geotechnical site characterization using geophysical measurements

Abstract This paper develops an efficient, PDE-constrained stochastic inverse analysis methodology to probabilistically estimate site-specific elastic parameters of soil from sparse geophysical test measurements by accounting for the uncertain spatial variability of soil deposits and any measurement uncertainty associated with the geophysical experiment. Hypothesizing the soil parameters at any site to be three-dimensional, heterogeneous, anisotropic random fields, the methodology first probabilistically simulates the geophysical experiment using the finite element method in conjunction with a stochastic collocation approach to compute statistical measures of a quantity of interest such as the soil displacement or acceleration throughout the soil domain. To this end, the random fields are discretized into finite number of random variables by utilizing a Gaussian mixture model that allows for mimicking the soil formation process. The parameters of the random fields are initially assumed based on the generic data available in the literature for the geological soil type. The stochastic collocation approach utilizes a recently developed non-product quadrature method, conjugate unscented transformation, to accurately estimate the statistical moments corresponding to the model response variables in a computationally efficient manner. The methodology, then, employs a minimum variance framework to fuse the finite element model output with sparse real measurements to update the initially assumed soil statistical parameters. The methodology is illustrated through numerical geophysical experiments at a fictitious geotechnical site and is verified with three very different true profiles of the soil modulus. Moreover, a probabilistic sensitivity analysis is carried out by varying the number and locations of sensors. It is observed that by judiciously selecting the sensor locations, following a set of information maps, obtained by exploiting the equations of the minimum variance scheme, more information may be extracted from any geophysical experiments, leading to less uncertain estimates of the soil parameters.