GNGTS 2023 - Atti del 41° Convegno Nazionale

Session 3.3 ______ ___ GNGTS 2023 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 statistical 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 modelling 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. Method NNs are computing models made of computational units (neurons) able to perform mathematical operations on a multidimensional array (Haykin, 2009; Moseley et al. , 2020). In our work, the problem consists of a function approximation, so we need to perform supervised learning: each example provided to the network is a pair (input and target output, also known as ground truth) and the NN is trained to produce outcomes as close as possible to the desired output. The learning process is performed through a continuous update of network parameters (weights associated with connections and biases associated with neurons) that brings the prediction as close as possible to the ground truth. Therefore, the training corresponds to an optimization problem in which a cost function (often called loss) is minimized through an iterative algorithm. The network used for our purpose is a Convolutional Neural Network (CNN). Unlike regular NNs, CNNs treat inputs as images and their layers have neurons arranged in 3D (width, height, depth). Different typologies of layers perform different operations on the neurons they act on (convolution, downsampling, normalization, etc … ) allowing the network to learn and combine features or local information from the input images. In this work, we want to insert a NN in the process of ERT inversion in order to substitute the forward modelling and the subsequent objective function value (i.e., data misfit) calculation. In Fig. 1 it is sketched a workflow showing the implemented process to build the dataset of resistivity model and data misfit pairs needed to train the CNN. Indeed, to implement the ML-based inversion in a DCT compressed domain, we have to provide the network with the DCT coefficients retained to approximate a resistivity model as input and the corresponding data misfit value as output. The first step is the definition of a prior distribution from which to extract plausible resistivity models. We assume that the resistivity values follow a stationary (i.e., spatial invariant) log-Gaussian prior distribution with mean and covariance calculated directly from the synthetic true model and with a spatial variability pattern defined by a 2D Gaussian variogram model. After the generation of