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We consider geothermal inverse problems and uncertainty quantification from aBayesian perspective. Our main goal is to make standard, `out-of-the-box'Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulationmodels by using suitable approximations. To do this, we first show how to poseboth the inverse and prediction problems in a hierarchical Bayesian framework.We then show how to incorporate so-called posterior-informed modelapproximation error into this hierarchical framework, using a modified form ofthe Bayesian approximation error (BAE) approach. This enables the use of a`coarse', approximate model in place of a finer, more expensive model, whileaccounting for the additional uncertainty and potential bias that this canintroduce. Our method requires only simple probability modelling, a relativelysmall number of fine model simulations, and only modifies the target posterior-- any standard MCMC sampling algorithm can be used to sample the newposterior. These corrections can also be used in methods that are not based onMCMC sampling. We show that our approach can achieve significant computationalspeed-ups on two geothermal test problems. We also demonstrate the dangers ofnaively using coarse, approximate models in place of finer models, withoutaccounting for the induced approximation errors. The naive approach tends togive overly confident and biased posteriors while incorporating BAE into ourhierarchical framework corrects for this while maintaining computationalefficiency and ease-of-use.

Incorporating Posterior-Informed Approximation Errors into a  Hierarchical Framework to Facilitate Out-of-the-Box MCMC Sampling for  Geothermal Inverse Problems and Uncertainty Quantification