Probabilistic inversions of electrical resistivity tomography data with a machine learning-based forward operator

Casting a geophysical inverse problem into a Bayesian setting is often discouraged by the computational workload needed to run many forward modelling evaluations. Here we present probabilistic inversions of electrical resistivity tomography data in which the forward operator is replaced by a trained residual neural network that learns the non-linear mapping between the resistivity model and the apparent resistivity values. The use of this specific architecture can provide some advantages over standard convolutional networks as it mitigates the vanishing gradient problem that might affect deep networks. The modelling error introduced by the network approximation is properly taken into account and propagated onto the estimated model uncertainties. One crucial aspect of any machine learning application is the definition of an appropriate training set. We draw the models forming the training and validation sets from previously defined prior distributions, while a finite element code provides the associated datasets. We apply the approach to two probabilistic inversion frameworks: A Markov chain Monte Carlo algorithm is applied to synthetic data, while an ensemble-based algorithm is employed for the field measurements. For both the synthetic and field tests, the outcomes of the proposed method are benchmarked against the predictions obtained when the finite element code constitutes the forward operator. Our experiments illustrate that the network can effectively approximate the forward mapping even when a relatively small training set is created. The proposed strategy provides a forward operator that is three orders of magnitude faster than the accurate but computationally expensive finite element code. Our approach also yields the most likely solutions and uncertainty quantifications comparable to those estimated when the finite element modelling is employed. The presented method allows solving the Bayesian electrical resistivity tomography with a reasonable computational cost and limited hardware resources.