Hydrogeophysical parameter estimation using iterative ensemble smoothing and approximate forward solvers

In iterative ensemble smoother approaches and ensemble methods in general, the ensemble size governs the accuracy of the parameter estimates obtained. However, employing large ensembles may be computationally infeasible in applications with expensive forward solvers. Here, we reduce the computational cost of using large ensembles in iterative ensemble smoothing through the use of a proxy solver. To correct the proxy response for the corresponding model error, the latter of which can bias posterior parameter estimates if left untreated, we propose a local basis approach. With this approach, the discrepancy between the detailed and proxy solvers is learned for a subset of the ensemble and collected in a dictionary that grows with each iteration. For each ensemble member, the K-nearest neighbors in the dictionary are employed to build an orthonormal basis which is used to identify the model-error component of the residual by projection. The proposed methodology reduces the effects of overfitting the data with the proxy solver, but may lead to underfitting of the data in the absence of a sufficient number of dictionary entries, meaning that the number of ensemble members relative to the number of detailed-solver runs cannot be inflated arbitrarily. We present our approach in the context of the ensemble smoother with multiple data assimilations (ES-MDA) algorithm, and show its successful application to a high-dimensional synthetic example that involves crosshole ground-penetrating radar (GPR) travel-time tomography.