non-linear inverse problem

Geophysical inverse problems seek to provide quantitative information about geophysical characteristics of the Earth’s subsurface for indirectly related data and measurements. It is generally formulated as an ill-posed non-linear optimization problem commonly solved through deterministic gradient-based approaches. Using these methods, despite fast convergence properties, may lead to local minima as well as impend accurate uncertainty analysis. On the contrary, formulating a geophysical inverse problem in a probabilistic framework and solving it by constructing the multi-dimensional posterior probability density (PPD) allow for complete sampling of the parameter space and the uncertainty quantification. The PPD is numerically characterized using Markov Chain Monte Carlo (MCMC) approaches. However, the convergence of the MCMC algorithm (i.e. sampling efficiency) toward the target stationary distribution highly depends upon the choice of the proposal distribution. In this paper, we develop an efficient proposal distribution based on perturbing the model parameters through an eigenvalue decomposition of the model covariance matrix in a principal component space. The covariance matrix is retrieved from an initial burn-in sampling, which is itself initiated using a linearized covariance estimate. The proposed strategy is first illustrated for inversion of hydrogeological parameters and then applied to synthetic and real geo-electrical data sets. The numerical experiments demonstrate that the presented proposal distribution takes advantage of the benefits from an accelerated convergence and mixing rate compared to the conventional Gaussian proposal distribution.