  We consider geothermal inverse problems and uncertainty quantification from a
Bayesian perspective. Our main goal is to make standard, `out-of-the-box'
Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulation
models by using suitable approximations. To do this, we first show how to pose
both the inverse and prediction problems in a hierarchical Bayesian framework.
We then show how to incorporate so-called posterior-informed 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
accounting for the additional uncertainty and potential bias that this can
introduce. Our method requires only simple probability modelling, a relatively
small number of fine model simulations, and only modifies the target posterior
-- any standard MCMC sampling algorithm can be used to sample the new
posterior. These corrections can also be used in methods that are not based on
MCMC sampling. We show that our approach can achieve significant computational
speed-ups on two geothermal test problems. We also demonstrate the dangers of
naively using coarse, approximate models in place of finer models, without
accounting for the induced approximation errors. 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.
