Incorporating posterior model discrepancy into a hierarchical framework to facilitate out-of-the-box MCMC sampling for geothermal inverse problems and uncertainty quantification

Abstract

We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective. Our goal is to make standard, 'out-of-the-box' Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulation models. To do this, we first show how to pose the inverse and prediction problems in a hierarchical Bayesian framework. We then show how to incorporate so-called posterior model approximation error into this hierarchical framework, using a modified form of the Bayesian approximation error (BAE) approach. This enables the use of a 'coarse', approximate model in place of a finer, more expensive model, while also accounting for the additional uncertainty and potential bias that this can introduce. Our method requires only simple probability modelling and only modifies the target posterior - the same standard MCMC sampling algorithm can be used to sample the new target posterior. We show that our approach can achieve significant computational speed-ups on a geothermal test problem. A problem which would take around a year to carry out full MCMC sampling for, now only takes around a day or so using our approach. We also demonstrate the dangers of naively using coarse, approximate models in place of finer models, without accounting for model discrepancies. The naive approach tends to give overly confident and biased posteriors, while incorporating BAE into our hierarchical framework corrects for this while maintaining computational efficiency and ease-of-use.