Ensemble-Based Time-Lapse ERT Inversion with Model and Data Space Compression Through Deep Variational Autoencoders

Time-lapse electrical resistivity tomography (TL-ERT) aims to image resistivity changes in the subsurface. This is an ill-posed and non-unique inverse problem and hence the estimation of the model uncertainties is of crucial importance. To reduce the computational cost of the probabilistic inversion, model and data can be re-parameterized into low-dimensional spaces where the inverse solution can be computed more efficiently. Among the many compression methods, deep learning algorithms based on deep generative models provide an efficient approach to reduce model and data spaces. Here, we propose a TL-ERT probabilistic inversion where the data and model spaces are compressed through deep variational autoencoders, while the optimization procedure is driven by the ensemble smoother with multiple data assimilation, an iterative ensemble-based algorithm that performs a Bayesian updating step at each iteration. This method provides multiple realizations for the quantification of the uncertainty by iteratively updating an initial ensemble of models that we generate according to previously defined prior model and spatial variability pattern. A finite-element code constitutes the forward operator. We test the method on synthetic data computed over a schematic subsurface model. Our tests demonstrate the applicability and the reliability of the proposed TL-ERT inversion.