Inverse theory; Hydrogeophysics; Sub-resolution effects; Hydrogeology; Approximate Bayesian computation; Conceptual models; Lithological tomography; Markov chain Monte Carlo; Model selection
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Geophysical data are often affected by heterogeneities at scales smaller than those resolved by traditional geophysical inversion. These sub-resolution effects may provide insights about pore or fine-scale properties, but can also severely degrade Markov chain Monte Carlo (MCMC) inversion results. Similar effects occur when disregarding modeling errors. Statistical inference from sampled realizations can thus be misleading when unaccounted sub-resolution effects and modeling errors are present and when the prior probability density function (pdf) is imperfect. Even in the absence of such limitations, it might happen that model realizations similar to the model used to generate noise-contaminated data are not part of the sampled MCMC realizations. This happens when other parameter combinations (possibly with low geological realism, but still in agreement with the prior pdf) exist with significantly lower data misfits. These problems are increasing when working with large data sets having small errors and model parameterizations involving many degrees of freedom. There is a need for new approaches to account for these effects.Economic and logistic constraints often lead to situations where high-resolution near-surface geophysical data only cover a small fraction of study areas of interest (e.g., a catchment, the hyporheic zone along a river, or an aquifer). Geophysical data are typically used to derive local subsurface information by inverting site-specific data, but the resulting images may provide few insights about expected subsurface characteristics at other locations. An alternative approach is based on formal model selection, in which multiple conceptual subsurface models are compared against the data and the prior pdf. The focus is then placed on inferring general system properties (e.g., a statistical model based on two-point statistics or representative lithological units) that may be used in subsequent probabilistic simulations across the study area. General algorithms for model selection are very time-consuming. An alternative based on summary statistics used in approximate Bayesian computation (ABC) will be investigated. Instead of relying on likelihood functions based on pair-wise comparisons of the observed and simulated data, ABC uses lower-order likelihood-free representations. These so-called summary statistics may, for example, include the spread of the data. In ABC, realizations from the prior pdf are kept as realizations from the posterior pdf if the corresponding summary statistics are within pre-defined distances from those of the observed data. We postulate that working with summary statistics and corresponding thresholds can help to decrease the effects of sub-resolution heterogeneity and modeling errors compared with classical MCMC. We also expect that summary statistics are better suited to target data characteristics that are representative of a given conceptual model. We will work with and adapt state-of-the-art MCMC models to ABC and explore the use of sequential Monte Carlo for model selection with summary statistics. These developments will improve the efficiency of probabilistic inversions and model selection procedures, and provide less biased estimates. We will test and develop algorithms against numerical and field-based test cases with competing conceptual subsurface models and important sub-resolution effects. To focus on common methods with contrasting physics, we will primarily consider the use of crosshole ground penetrating radar (GPR) and surface-based electrical resistivity tomography (ERT) in different hydrological applications.