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SUMMARY In general, the inverse problem of electrical resistivity tomography (ERT) is treated using a deterministic algorithm to find a model of subsurface resistivity that can numerically match the apparent resistivity data acquired at the ground surface and has a smooth distribution that has been introduced as prior information. In this paper, we propose a new deep learning algorithm for processing the 3-D reconstruction of ERT. This approach relies on the approximation of the inverse operator considered as a nonlinear function linking the section of apparent resistivity as input and the underground distribution of electrical resistivity as output. This approximation is performed with a large amount of known data to obtain an accurate generalization of the inverse operator by identifying during the learning process a set of parameters assigned to the neural networks. To train the network, the subsurface resistivity models are theoretically generated by a geostatistical anisotropic Gaussian generator, and their corresponding apparent resistivity by solving numerically 3-D Poisson's equation. These data are formed in a way to have the same size and trained on the convolutional neural networks with SegNet architecture containing a three-level encoder and decoder network ending with a regression layer. The encoders including the convolutional, max-pooling and nonlinear activation operations are sequentially performed to extract the main features of input data in lower resolution maps. On the other side, the decoders are dedicated to upsampling operations in concatenating with feature maps transferred from encoders to compensate the loss of resolution. The tool has been successfully validated on different synthetic cases and with particular attention to how data quality in terms of resolution and noise affects the effectiveness of the approach.

SUMMARY We present an inversion algorithm to reconstruct the spatial distribution of the electrical conductivity from the analysis of magnetometric resistivity (MMR) data acquired at the ground surface. We first review the theoretical background of MMR connecting the generation of a magnetic field in response to the injection of a low-frequency current source and sink in the ground given a known distribution of electrical conductivity in the subsurface of the Earth. The forward modelling is based on sequentially solving the Poisson equation for the electrical potential distribution and the magnetostatic (Biot and Savart) equation for the magnetic field. Then, we introduce a Gauss–Newton inversion algorithm in which the logarithm of the electrical conductivity field is parametrized by using the chaos polynomial expansion in order to reduce the number of model parameters. To illustrate how the method works, the algorithm is successfully applied on four synthetic models with 3-D heterogeneous distribution of the electrical conductivity. Finally, we apply our algorithm to a field case study in which seepage was known to be occurring along an embankment of a headrace channel to a power station.

We propose a cross-hole imaging approach based on seismoelectric conversions (SC) associated with the transmission of seismic waves from seismic sources located in a borehole to receivers (electrodes) located in a second borehole. The seismoelectric (seismic-to-electric) problem is solved using Biot theory coupled with a generalized Ohm's law with an electrokinetic streaming current contribution. The components of the displacement of the solid phase, the fluid pressure, and the electrical potential are solved using a finite element approach with Perfect Match Layer (PML) boundary conditions for the seismic waves and boundary conditions mimicking an infinite material for th electrostatic problem. We develop an inversion algorithm using the electrical disturbances recorded in the second borehole to localize the position of the heterogeneities responsible for the SC. Because of the ill-posed nature of the inverse problem (inherent to all potential-field problems), regularization is used to constrain the solution at each time in the SC-time window comprised between the time of the seismic shot and the time of the first arrival of the seismic waves in the second borehole. All the inverted volumetric current source densities are aggregated together to produce an image of the position of the heterogeneities between the two boreholes. Two simple synthetic case studies are presented to test this concept. The first case study corresponds to a vertical discontinuity between two homogeneous sub-domains. The second case study corresponds to a poroelastic inclusion (partially saturated by oil) embedded into an homogenous poroelastic formation. In both cases, the position of the heterogeneity is recovered using only the electrical disturbances associated with the SC. That said, a joint inversion of the seismic and seismoelectric data could improve these results.