A convolutional neural network approach to electrical resistivity tomography

Electrical resistivity tomography (ERT) is an ill-posed and non-linear inverse problem commonly solved through deterministic gradient-based methods. These algorithms guarantee fast convergence toward the final solution but hinder accurate uncertainty assessments. On the contrary, numerical Markov Chain Monte Carlo algorithms provide accurate uncertainty appraisals but at the expense of a considerable computational effort. In this work, we develop a novel approach to ERT that guarantees an extremely fast inversion process and reliable uncertainty appraisals. The implemented method combines a Discrete Cosine Transform (DCT) reparameterization of data and model spaces with a Convolutional Neural Network. The CNN is employed to learn the inverse non-linear mapping between the DCT-compressed data and the DCT-compressed 2-D resistivity model. The DCT is an orthogonal transformation that here acts as an additional feature extraction technique that reduces the dimensionality of the input and output of the network. The DCT also acts as a regularization operator in the model space that significantly reduces the number of unknown parameters and the ill-conditioning of the inversion procedure, thereby preserving the spatial continuity of the resistivity values in the recovered solution. The estimation of model uncertainties is a key step of geophysical inverse problems and hence we implement a Monte Carlo simulation framework that propagates onto the estimated model the uncertainties related to both noise contamination and network approximation (the so-called modeling error). We first apply the approach to synthetic data to investigate its robustness in case of erroneous assumptions on the noise and model statistics used to generate the training set. Then, we demonstrate the applicability of the method through inverting real data measured along a river embankment. We also demonstrate that transfer learning avoids retraining the network from scratch when the statistical properties of training and target sets are different. Our tests confirm the suitability of the proposed approach, opening the possibility to estimate the subsurface resistivity values and the associated uncertainties in near real-time.