• Convolutional Neural Network
• eletrical resistivity tomography
• inversion
• Residual Neural Network

This thesis assesses the applicability of a machine learning approach to solve the Electrical Resistivity Tomography (ERT). The ERT is a non-linear and ill-conditioned problem usually solved through gradient-based methods. These approaches ensure a rapid convergence toward a best-fitting model but suffer from the local linearization, and hence are prone to get trapped into local minima of the error function. Here we implement and apply an alternative approach in which a convolutional neural network is trained to learn the non-linear mapping between the pseudo resistivity (data) domain and the resistivity space. The computational cost of the training procedure is highly dependent on the dimensionality of the input and output of the network and for this reason, we also assess the applicability of the Discrete Cosine Transform (DCT) (an orthogonal transformation) to compress the number of data points and model parameters. The aim of the DCT is twofold: not only it reduces the dimensionality of the input and output of the network, but it also acts as a regularization operator in the model space that mitigates the ill-conditioning of the ERT inversion problem, while preserving the assumed spatial continuity pattern in the recovered solution. In addition to standard CNNs, we also apply a Residual Neural Network (RNN), a special kind of CNN that uses skip connections to avoid the so-called vanishing gradient problem that arises when training a deep network.
The implemented machine-learning ERT inversion includes four steps:
1) Generation phase: define an ensemble of 2D resistivity models and compute the associated pseudosections, which constitute the network output and input responses, respectively. The models are created according to a previously defined prior distribution and spatial variability pattern, while a finite-elements code constitutes the forward modeling engine.
2) Network Design: define a network architecture to approximate the non-linear mapping between the data and the model spaces.
3) Training phase: train the network by minimizing the differences between the predicted and desired output.
4) Prediction phase: once trained, use the network to project an observed pseudosection onto the associated resistivity values.
For what concerns the network design we analyze how different hyperparameter settings (i.e., number of layers, size of the convolutional filters, number of filters) affect the network performances that are evaluated on the training, validation, and test sets. We also assess the robustness and stability of the network predictions in case of erroneous assumptions about the noise and prior model statistics assumed during the learning stage. In this regard, we also demonstrate that transfer learning avoids retraining the network from scratch when the trained convolutional neural network is applied to a different scenario (i.e., a test model with different statistical properties). This technique is routinely employed in machine learning applications and consists of an additional training process with a small portion of target data, thereby allowing for a quick transfer of the learned features to a new task.
The implemented machine learning approaches for ERT inversion are first tested on synthetic data and then applied to real data measured along a river embankment. Our tests confirm the suitability of the proposed approach, opening the possibility to estimate the subsurface resistivity values in near real-time.