• ill-conditioning
• inverse problems
• machine learning
• deep learning
• Dix inversion

This thesis was realized through a collaboration between Eni, the University of Pisa, and the Montan University of Leoben, to exploit supervised-learning techniques (artificial neural networks) in the context of the Dix inversion, to retrieve the interval from the root mean squared velocities, while evaluating the robustness of the trained machine-learning models.
Due to the lack of available labelled real data, an algorithm was implemented to generate synthetic data to train and test different various artificial neural networks. Eventually, fully connected artificial neural networks trained only with synthetic data were tested to invert a real marine seismic dataset, the line 12 of the 2D Viking-graben marine dataset.
To better understand how supervised-learning techniques might be exploited to solve an inverse problem, a top-down approach was followed.
First the linearized Dix inversion was solved with stochastic gradient descent (SGD) techniques, implementing two different architectures considering the knowledge of the linear inverse operator to be approximated. The architectures were characterized by N and 2N-1 degrees of freedom, where N is the number of interval velocity values to be inverted. The first architecture converged toward the correct solution of the linear inverse problem, while the second architecture provided un-physical results due to the under-determination of this formulation. Finally, the first architecture was re-trained introducing a physics-informed regularizer in the cost function to be minimized, to simulate a physics-informed neural network (PINN). Then, progressively deeper fully connected artificial neural networks (FC-ANNs) were exploited to solve the non-linearized Dix inversion. Specifically, FC-ANNs composed by 1 to 3 hidden layers (with N neurons per layer) were implemented, testing multiple combination of activation functions, to evaluate their capability to learn a mapping between 1D regularly sampled rms and interval velocity profiles. To test the capability of FC-ANNs to learn a mapping attenuating the effect of noise, a dataset composed by noise contaminated inputs and noise free labels was built. The robustness of the implemented FC-ANNs was assessed simulating incorrect assumptions about the noise statistics. The noise sensitivity and robustness tests allowed to evaluate the capability of FC-ANNs to tackle the ill-conditioning of the Dix inversion.
Comparing the performance of the investigated FC-ANNs, the deepest architecture was designated to be tested also on real marine seismic data. The predictions of the FC-ANNs were compared with the outcome of the analytic Dix inversion and that of a standard model-driven technique. Using the ProMAX software, the Pre-stack depth migration algorithm was used to image the same dataset with different interval velocity models. The flatness of the common reflection point gathers allowed to fairly compare the results obtained with model-driven and data-driven inversion techniques.
In the last part of this thesis, two possible extensions of this work are briefly discussed: the possibility to learn a regularizer from a training dataset and the need of methods aimed to decipher the “black-box” nature of machine learning models.