Discrete Cosine Transform Reparameterization for Bayesian Time-Lapse ERT Inversion

Time-Lapse electrical resistivity tomography (TL-ERT) is used to monitor dynamic processes through mapping the resistivity variations in the subsurface. Inversion of TL-ERT data is a highly non-linear and ill-conditioned problem characterized by non-unique solutions. For this reason, an accurate uncertainty appraisal is essential to quantify the ambiguity affecting the estimated resistivity model. We present a probabilistic TL-ERT inversion in which the Differential Evolution Markov Chain (DEMC) algorithm samples the posterior probability density function, while the Discrete Cosine Transform (DCT) is used to compress the model space. The model compression aims at mitigating both the ill-conditioned nature of the inversion problem and the curse of dimensionality issue. On the other hand, the DEMC combines principles coming from metaheuristic optimisation methods and Markov Chain Monte Carlo algorithms to speed up the probabilistic sampling. To draw essential conclusions about the reliability and applicability of the implemented algorithm, we focus on synthetic inversion experiments in which we simulate two data acquisitions at different time instants (t0 and t1) and we jointly estimate the resistivity model at t0 along with the resistivity changes at t1. The results demonstrate that the implemented method provides accurate model predictions and uncertainty estimations with an affordable computational cost.