Dynamic inversion for hydrological process monitoring with electrical resistance tomography under model uncertainties

Abstract

We propose an approach for imaging the dynamics of complex hydrological processes. The evolution of electrically conductive fluids in porous media is imaged using time‐lapse electrical resistance tomography. The related dynamic inversion problem is solved using Bayesian filtering techniques; that is, it is formulated as a sequential state estimation problem in which the target is an evolving posterior probability density of the system state. The dynamical inversion framework is based on the state space representation of the system which involves the construction of a stochastic evolution model and an observation model. The observation model that we use in this paper consists of the complete electrode model for ERT, with Archie's law relating saturations to electrical conductivity. The evolution model is an approximate model for simulating flow through partially saturated porous media. Unavoidable modeling and approximation errors in both the observation and evolution models are considered by computing approximate statistics for these errors. These models are then included in the construction of the posterior probability density of the estimated system state. This approximation error method allows the use of approximate, and therefore computationally efficient, observation and evolution models in the Bayesian filtering. We conside7r a synthetic example and show that the incorporation of an explicit model for the model uncertainties in the state space representation can yield better estimates than the frame‐by‐frame imaging approach.