On the value of incorporating spatial statistics in large-scale geophysical inversions: the SABRe case

Electrical Resistance Tomography (ERT) is a popular method for investigating subsurface heterogeneity. The method relies on measuring electrical potential differences and obtaining, through inverse modeling, the underlying electrical conductivity field, which can be related to hydraulic conductivities. The quality of site characterization strongly depends on the utilized inversion technique. Standard ERT inversion methods, though highly computationally efficient, do not consider spatial correlation of soil properties; as a result, they often underestimate the spatial variability observed in earth materials, thereby producing unrealistic subsurface models. Also, these methods do not quantify the uncertainty of the estimated properties, thus limiting their use in subsequent investigations. Geostatistical inverse methods can be used to overcome both these limitations; however, they are computationally expensive, which has hindered their wide use in practice. In this work, we compare a standard Gauss-Newton smoothness constrained least squares inversion method against the quasi-linear geostatistical approach using the three-dimensional ERT dataset of the SABRe (Source Area Bioremediation) project. The two methods are evaluated for their ability to: a) produce physically realistic electrical conductivity fields that agree with the wide range of data available for the SABRe site while being computationally efficient, and b) provide information on the spatial statistics of other parameters of interest, such as hydraulic conductivity. To explore the trade-off between inversion quality and computational efficiency, we also employ a 2.5-D forward model with corrections for boundary conditions and source singularities. The 2.5-D model accelerates the 3-D geostatistical inversion method. New adjoint equations are developed for the 2.5-D forward model for the efficient calculation of sensitivities. Our work shows that spatial statistics can be incorporated in large-scale ERT inversions to improve the inversion results without making them computationally prohibitive.