Geophysical Journal International

Electrical resistivity tomography (ERT) is widely used to image the Earth’s subsurface and has proven to be an extremely useful tool in application to hydrological problems. Conventional smoothness-constrained inversion of ERT data is efficient and robust, and consequently very popular. However, it does not resolve well sharp interfaces of a resistivity field and tends to reduce and smooth resistivity variations. These issues can be problematic in a range of hydrological or near-surface studies, for example mapping regolith-bedrock interfaces. While fully Bayesian approaches, such as those using Markov chain Monte Carlo sampling, can address the above issues, their very high computation cost makes them impractical for many applications. Ensemble Kalman inversion (EKI) offers a computationally efficient alternative by approximating the Bayesian posterior distribution in a derivative-free manner, which means only a relatively small number of ‘black-box’ model runs are required. Although common limitations for ensemble Kalman filter-type methods apply to EKI, it is both efficient and generally captures uncertainty patterns correctly. We propose the use of a new EKI-based framework for ERT which estimates a resistivity model and its uncertainty at a modest computational cost. Our EKI framework uses a level-set parametrization of the unknown resistivity to allow efficient estimation of discontinuous resistivity fields. Instead of estimating level-set parameters directly, we introduce a second step to characterize the spatial variability of the resistivity field and infer length scale hyperparameters directly. We demonstrate these features by applying the method to a series of synthetic and field examples. We also benchmark our results by comparing them to those obtained from standard smoothness-constrained inversion. Resultant resistivity images from EKI successfully capture arbitrarily shaped interfaces between resistivity zones and the inverted resistivities are close to the true values in synthetic cases. We highlight its readiness and applicability to similar problems in geophysics.

Bayesian methods are extensively used to analyse geophysical data sets. A critical and somewhat overlooked component of high-dimensional Bayesian inversion is the definition of the prior probability density function that describes the joint probability of model parameters before considering available data sets. If insufficient prior information is available about model parameter correlations, then it is tempting to assume that model parameters are uncorrelated. When working with a spatially gridded model representation, this overparametrization leads to posterior realizations with far too much variability to be deemed realistic from a geological perspective. In this study, we introduce a new approach for structure-based prior sampling with Markov chain Monte Carlo that is suitable when only limited prior information is available. We evaluate our method using model structure measures related to standard roughness and damping metrics for l1- and l2-norms. We show that our structure-based prior approach is able to adequately sample the chosen prior distribution of model structure. The usefulness and applicability of the methodology is demonstrated on synthetic and field-based crosshole ground penetrating radar data. We find that our method provides posterior model realizations and statistics that are significantly more satisfactory than those based on underlying assumptions of uncorrelated model parameters or on explicit penalties on model structure within an empirical Bayes framework.