The Levenberg‐Marquardt (LM) method is commonly used for inverting models used to describe geothermal, groundwater, or oil and gas reservoirs. In previous studies, LM parameter updates have been made tractable for highly parameterized inverse problems with large data sets by applying matrix factorization methods or iterative linear solvers to approximately solve the update equations. Some studies have shown that basing model updates on the truncated singular value decomposition (TSVD) of a dimensionless sensitivity matrix achieved using Lanczos iteration can speed up the inversion of reservoir models. Lanczos iterations only require the sensitivity matrix times a vector and its transpose times a vector, which are found efficiently using adjoint and direct simulations without the expense of forming a large sensitivity matrix. Nevertheless, Lanczos iteration has the drawback of being a serial process, requiring a separate adjoint solve and direct solve every Lanczos iteration. Randomized methods, developed for low‐rank matrix approximation of large matrices, are more efficient alternatives to the standard Lanczos method. Here we develop LM variants which use randomized methods to find a TSVD of a dimensionless sensitivity matrix when updating parameters. The randomized approach offers improved efficiency by enabling simultaneous solution of all adjoint and direct problems for a parameter update.

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 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 modeling, 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 speedups 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 Bayesian approximation error into our hierarchical framework corrects for this while maintaining computational efficiency and ease of use. Key Points: • We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective • We present a simple method for incorporating posterior-informed approximation errors into a hierarchical Bayesian framework • Our method makes standard out-of-the-box MCMC sampling feasible for more complex models while correcting for bias and overconfidence Supporting Information: • Supporting Information S1 , M. J. (2020). Incorporating posterior-informed approximation errors into a hierarchical framework to facilitate out-of-the-box MCMC sampling for geothermal inverse problems and uncertainty quantification. Water Resources Research, 56, e2018WR024240. https://

The Krafla area in north Iceland hosts a high‐temperature geothermal system within a volcanic caldera. Temperature measurements from boreholes drilled for power generation reveal enigmatic contrasts throughout the drilled area. While wells in the western part of the production field indicate a 0.5–1 km thick near‐isothermal (∼210°C) liquid‐dominated reservoir underlain by a deeper boiling reservoir, wells in the east indicate boiling conditions extending from the surface to the maximum depth of drilled wells (∼2 km). Understanding these systematic temperature contrasts in terms of the subsurface permeability structure has remained challenging. Here, we present a new numerical model of the natural, pre‐exploitation state of the Krafla system, incorporating a new geologic/conceptual model and a version of TOUGH2 extending to supercritical conditions. The model shows how the characteristic temperature distribution results from structural partitioning of the system by a rift‐parallel eruptive fissure and an aquitard at the transition between deeper basement intrusions and high‐permeability extrusive volcanic rocks. As model calibration is performed using a Bayesian framework, the posterior results reveal significant uncertainty in the inferred permeability values for the different rock types, often exceeding two orders of magnitude. While the model shows how zones of single‐phase supercritical vapor develop above the deep intrusive heat source, more data from deep wells is needed to better constrain the extent and temperature of the deep supercritical zones. However, the model suggests the presence of a significant untapped resource at Krafla.

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