Uncertainty Quantification in Inverse Scattering Problems.

Inverse scattering problems seek anomalies in a medium given data measured after the interaction with emitted waves. Due to noise, predictions about the nature of these inclusions should be complemented with uncertainty estimates. To this end, we propose a progressive framework for inverse scattering from low- to high-dimensional Bayesian formulations depending on the prior information and the problem complexity. We aim to reduce computational costs by exploiting educated prior information. When we look for a few well-separated inclusions in a known medium with information about their number, we resort to low-dimensional parameterizations in terms of a few random variables representing their shape and material constants. We test this approach detecting anomalies in tissues and deposits in stratified subsoils. In more complex situations where the anomalies may overlap, we propose high-dimensional parameterizations obtained from Karhunen-Loève (KL) or Fourier expansions of the density and velocity fields. We employ these methods to characterize oil and gas reservoirs in a salt dome configuration, where the screening effect of the dome cap prevents the obtention of adequate prior information. We characterize the posterior probability by means of affine invariant ensemble and functional ensemble MCMC samplers depending on dimensionality. This provides information on configurations with the highest a posteriori probability and the uncertainty around them, identifying factors that could reduce the uncertainty. In high-dimensional setups, techniques based on KL developments are more effective and stable. A recurring issue is the choice of the a priori covariance (which strongly affects the results) and the choice of its hyperparameters. Here, we use educated choices. Formulations that include them as additional parameters could be a next step at a higher cost.