Machine Learning to approximate the objective function in a global inversion of ERT data

An inverse problem aims to determine the parameters of a model from the measured data. During the inversion, the parameters are iteratively updated and the associated data are calculated from the resulting models through the forward modeling operation. This last step is usually the most time consuming of the process. In this work we aim to reduce this computational effort by replacing the forward modeling and the data misfit calculation with a properly trained Neural Network (NN) model that approximates the objective function. NNs, the basis of Machine Learning (ML) algorithms, consist of computing models that can learn the underlying relationship between their inputs and their outputs. We decide to apply the ML-based inversion to Electrical Resistivity Tomography (ERT), a nonlinear and ill-posed inverse problem, usually solved through deterministic gradient-based methods that linearize the problem around an initial solution (Aleardi et al., 2021). We tackle the problem with a global optimization method, Genetic Algorithms (GA): this allows a good exploration of the solution space treating the models as individuals of populations and performing on them operations inspired by the natural selection processes. In this way, and compared to local approaches, GAs reduce the risk of entrapment in a local minimum of the objective function. Besides the problem of a forward model computationally expensive, there is another issue that makes a global optimization ERT inversion computationally challenging: the large dimensional parameter space and the associated curse of dimensionality problem. To this end we employ the Discrete Cosine Transform (DCT) to reduce the number of unknowns to invert for. This technique is similar to the Fourier Transform, but it uses only cosines as bases function to signal reconstruction so that the computed coefficients are real numbers. Therefore, the DCT of a signal (expressing, in this case, the resistivity model) indicates the signal’s energy distribution in the frequency domain. Since most of the signal’s energy is usually expressed by low-order DCT coefficients, retaining only these ones we can use this technique for model compression. In the inverse problem, the selected coefficients become the unknown parameters to retrieve (Vinciguerra et al., 2021). By reducing the dimensionality of the problem, the computational effort required to deal with it also decreases. The first part of the work (not shown here for the lack of space) consisted of a test phase in which different types of NNs were applied to approximate analytical objective functions (De Jong, Rastrigin, Schwefel, Styblinski-Tang and Ackley) that mimic some characteristics of objective function usually encountered in geophysical data inversion. Then, the method has been applied to the ERT inversion. To draw essential conclusions about the feasibility of the proposed approach we limit our attention to synthetic data inversion, in which the statistic characteristics of the subsurface resistivity are assumed to be perfectly known when training the NN model. In this preliminary study, we verified that the combination of NN, DCT and GAs makes the ERT inversion converge toward plausible predictions comparable to that achieved when an accurate, but computationally expensive, Finite Elements code is used for the forward modeling phase during the GA optimization. The idea is to build a methodology that allows inverting ERT data in a reasonable time, in order to have preliminary results in just a few minutes after data acquisition. This provides an almost real-time inversion algorithm, whose results can be used as a starting point for a subsequent and more accurate inversion.