Stochastic electrical resistivity tomography with ensemble smoother and deep convolutional autoencoders

To reduce both the computational cost of probabilistic inversions and the ill-posedness of geophysical problems, model and data spaces can be re-parameterized into low-dimensional domains 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 for model and data space reduction. We present a probabilistic electrical resistivity tomography inversion in which the data and model spaces are compressed through deep convolutional variational autoencoders, while the optimization procedure is driven by the ensemble smoother with multiple data assimilation, an iterative ensemble-based algorithm. This method iteratively updates an initial ensemble of models that are generated according to a previously defined prior model. The inversion outcome consists of the most likely solution and a set of realizations of the variables of interest from which the posterior uncertainties can be numerically evaluated. We test the method on synthetic data computed over a schematic subsurface model, and then we apply the inversion to field measurements. The model predictions and the uncertainty assessments provided by the presented approach are also compared with the results of an MCMC sampling working in the compressed domains, a gradient-based algorithm, and with the outcomes of an ensemble-based inversion running in the uncompressed spaces. A finite-element code constitutes the forward operator. Our experiments show that the implemented inversion provides most likely solutions and uncertainty quantifications comparable to those yielded by the ensemble-based inversion running in the full model and data spaces, and the MCMC sampling, but with a significant reduction of the computational cost.