Implementation and evaluation of Bayesian inversion algorithms for time-lapse ERT data

Time-Lapse Electrical Resistivity Tomography is a method to reconstruct the variation of the subsurface resistivity, widely used in near-surface geophysics. Different gradient- based strategies are currently available to solve the Time-Lapse inverse problem, with the standard outcome expressed by a single subsurface model without any measurement of the uncertainty. An alternative strategy is represented by the Bayesian algorithms, which express the solution as a posterior probability density function (pdf ). This thesis investigates two main approaches, the Markov Chain Monte Carlo (MCMC) and Data Assimilation (DA). Since the MCMC approaches require a considerable computational workload, we implement the Differential Evolution Markov Chain (DEMC) combined with the Discrete Cosine Transform (DCT) reparametrization, obtaining the DCT- DEMC algorithm. The multi-chain strategy of the DEMC allows for tackling the local minima issue, whereas the DCT reparametrization attenuates the ill-conditioning of the problem. The algorithm has been firstly applied to 2D data and then to the Time-Lapse. The strategy for the Time-Lapse DCT-DEMC is to invert two datasets simultaneously, obtaining the resistivity model and the resistivity variation. The results suggest the potential of the DCT-DEMC in uncertainty quantification but also highlight the limitation in terms of computational cost. The Data Assimilation methods guarantee the Bayesian framework of the inverse problem while reducing the computational workload. However, the drawback of these algorithms relies on the inaccurate uncertainty estimation in case of highly non-linear problems. The Ensemble Smoother Multiple Data Assimilation (ES-MDA) represents the most widespread DA algorithm. In this thesis, it has been implemented and validated by inverting both 2D and Time-Lapse data. Since it has been reported that the ES-MDA can be affected by overshooting issues, the adaptive ES-MDA restricted step (ES-MDA-RS) has been implemented. The results of both algorithms reveal the potential for posterior pdf assessment in a feasible computational burden. In particular, the uncertainties associated with the resistivity changes represent a valuable tool to reduce the over-interpretation of the results. Finally, the constrained Time-Lapse ES-MDA has been implemented, which follows a different strategy for reconstructing the resistivity variations. Specifically, the algorithm employs a reference model to converge faster than the simultaneous approach. Its application to field Time-Lapse datasets inversion suggests the consistency of the assessed pdf with the simultaneous approach while reducing the computational cost.