Bayesian inversion with total variation prior for discrete geologic structure identification

The characterization of geologic heterogeneity that affects flow and transport phenomena in the subsurface is essential for cost‐effective and reliable decision‐making in applications such as groundwater supply and contaminant cleanup. In the last decades, geostatistical inversion approaches have been widely used to tackle subsurface characterization problems and quantify their corresponding uncertainty. Some well‐established geostatistical methods use models that assume gradually varying parameters. However, in many cases, the subsurface can often be better represented as consisting of a few relatively uniform geologic facies or zones with abrupt changes at their boundaries. We advance a Bayesian inversion approach with the gradient represented not through a Gaussian but a Laplace prior, also known as total variation prior, for the case that there are reasons to believe that discrete geologic structures with relatively homogeneous properties predominate in the subsurface but their number, locations, and shapes are unknown a priori. Structural parameters (or hyperparameters of the inversion scheme) are determined in a Bayesian framework by maximizing the marginal distribution of these parameters using an expectation‐maximization approach; this allows proper weighting of prior versus data information and produces results with realistic uncertainty quantification. We present here three applications of the method: a time‐varying extraction rate estimation at a well, a linear cross‐well seismic tomography, and a nonlinear hydraulic tomography. These results are compared with those achieved in the classical geostatistical method and it is shown that the Bayesian inversion approach with total variation prior can be a useful tool to identify discrete geologic structures.