Dimensionality reduction for efficient Bayesian estimation of groundwater flow in strongly heterogeneous aquifers

We focus on the Bayesian estimation of strongly heterogeneous transmissivity fields conditional on data sampled at a set of locations in an aquifer. Log-transmissivity, Y, is modeled as a stochastic Gaussian process, parameterized through a truncated Karhunen–Loève (KL) expansion. We consider Y fields characterized by a short correlation scale as compared to the size of the observed domain. These systems are associated with a KL decomposition which still requires a high number of parameters, thus hampering the efficiency of the Bayesian estimation of the underlying stochastic field. The distinctive aim of this work is to present an efficient approach for the stochastic inverse modeling of fully saturated groundwater flow in these types of strongly heterogeneous domains. The methodology is grounded on the construction of an optimal sparse KL decomposition which is achieved by retaining only a limited set of modes in the expansion. Mode selection is driven by model selection criteria and is conditional on available data of hydraulic heads and (optionally) Y. Bayesian inversion of the optimal sparse KLE is then inferred using Markov Chain Monte Carlo (MCMC) samplers. As a test bed, we illustrate our approach by way of a suite of computational examples where noisy head and Y values are sampled from a given randomly generated system. Our findings suggest that the proposed methodology yields a globally satisfactory inversion of the stochastic head and Y fields. Comparison of reference values against the corresponding MCMC predictive distributions suggests that observed values are well reproduced in a probabilistic sense. In a few cases, reference values at some unsampled locations (typically far from measurements) are not captured by the posterior probability distributions. In these cases, the quality of the estimation could be improved, e.g., by increasing the number of measurements and/or the threshold for the selection of KL modes.