Accounting for Petrophysical Prediction Uncertainty in Hydrogeophysical Inversion Using the Correlated Pseudo-Marginal Method

Abstract:

Hydrogeophysical investigations aim at obtaining information about hydrogeological properties or processes from geophysical data. We consider an inverse problem in which hydrogeological parameters are derived from geophysical data and the intermediate geophysical properties are treated as latent variables. The petrophysical relationship linking the hydrogeological and geophysical properties is generally non-linear and includes significant scatter. Instead of solving the inverse problem in a two-step approach by first inferring the geophysical properties and afterwards the hydrogeological parameters, we use a Metropolis-Hastings scheme to infer directly the hydrogeological parameters from the geophysical data. In doing so, we need to estimate the intractable likelihood of observing the geophysical data given the hydrogeological parameters. The Pseudo-Marginal method relies on an unbiased approximation of this likelihood based on Monte-Carlo averaging over samples from the petrophysical relationship, thereby ensuring that the scattered nature of the petrophysical relationship is taken into account. To increase the efficiency of the resulting Metropolis-Hastings scheme, we ensure low-variance approximations of the likelihood ratio by correlating the samples used in the proposed and current steps of the Markov chain. We assess the performance of this Correlated Pseudo-Marginal method with a synthetic example in which we invert for porosity, using crosshole ground-penetrating radar (GPR) travel times as geophysical data. For comparison purposes, we also consider an approach in which the petrophysical uncertainty is accounted for by using only one brute-force Monte Carlo sample at each Metropolis step (so-called lithological tomography). In data rich geophysical settings with low noise levels, it is essential to rely on importance sampling distributions that well represent the posterior random field of petrophysical scatter around the current hydrogeological property field. We demonstrate how this can be achieved when the petrophysical scatter is conceptualized as a Gaussian random field. We find that the Correlated Pseudo-Marginal method outperforms both the lithological tomography and the Pseudo-Marginal method by enhancing the efficiency and reducing the computational cost.