Merging parallel tempering with sequential geostatistical resampling for improved posterior exploration of high-dimensional subsurface categorical fields

The sequential geostatistical resampling (SGR) algorithm is a Markov chain Monte Carlo (MCMC) scheme for sampling from possibly non-Gaussian, complex spatially-distributed prior models such as geologic facies or categorical fields. In this work, we highlight the limits of standard SGR for posterior inference of high-dimensional categorical fields with realistically complex likelihood landscapes and benchmark a parallel tempering implementation (PT-SGR). Our proposed PT-SGR approach is demonstrated using synthetic (error corrupted) data from steady-state flow and transport experiments in categorical 7575- and 10,000-dimensional 2D conductivity fields. In both case studies, every SGR trial gets trapped in a local optima while PT-SGR maintains an higher diversity in the sampled model states. The advantage of PT-SGR is most apparent in an inverse transport problem where the posterior distribution is made bimodal by construction. PT-SGR then converges towards the appropriate data misfit much faster than SGR and partly recovers the two modes. In contrast, for the same computational resources SGR does not fit the data to the appropriate error level and hardly produces a locally optimal solution that looks visually similar to one of the two reference modes. Although PT-SGR clearly surpasses SGR in performance, our results also indicate that using a small number (16-24) of temperatures (and thus parallel cores) may not permit complete sampling of the posterior distribution by PT-SGR within a reasonable computational time (less than 1-2 weeks).