Uncertainty in hydrogeophysics: electrical resistivity tomography with variational inference

Electrical resistivity tomography (ERT) is a widely used and effective tool for hydrogeological investigations. Conventional ERT inversion approaches are based on gradient-based algorithms, which typically provide deterministic optimal solutions, which are subject to uncertainty. Such uncertainty could have significant impact on hydrogeological interpretation using ERT. Model appraisal is a critical step after inversion, however, conventional appraisal methods are qualitative and thus subjective. To address these limitations, this study introduces a probabilistic variational inference (VI) method, referred to as Stein variational gradient descent (SVGD), to quantify both resistivity distributions and associated uncertainties in ERT inversions. Synthetic examples are conducted to investigate the effects of configurations and noise, and to compare the performance of SVGD with conventional inversion and model appraisal techniques. A field case study and its model validation are also presented to demonstrate the practical advantages of uncertainty quantification in field. The results indicate that SVGD can effectively reduce artifacts introduced by regularization and provide more comprehensive quantitative insights into subsurface structures compared to conventional approaches. The study also reveals limitations in the interpretation of basic statistics of uncertainty estimates, highlighting the need to examine the entire posterior distributions of parameter values. Additionally, this study demonstrates that the final uncertainty arises from a trade-off among multiple factors, such as geometry of subsurface structures, measurement techniques and data noise levels. Finally, we also discuss some comparisons with other probabilistic frameworks in hydrogeophysics, highlighting its potential to improve uncertainty and probability quantification in ERT, and possible future developments in hydrogeophysical coupled inversion.