Reducing complexity of inverse problems using geostatistical priors

In a probabilistic formulation of inverse problems the solution can be given as a sample of the posterior probability distribution. All realizations retained in the posterior sample are consistent with both an assumed prior model and observed data. Some inverse problems are unsolvable, in that one can practically never hope to generate a posterior sample, others are just ’difficult’ and require special methods to become tractable, while others again are easily solved. We discuss how difficult nonlinear inverse problems can be handled such that their complexity, i.e. the time taken to obtain a posterior sample, can be reduced significantly using informed priors based on geostatistical models. We discuss two approaches to include such geostatistically based prior information. One is based on a parametric description of the prior likelihood that applies to 2-point based statistical models, and another approach makes use of conditional re-simulation to sample the prior that works for both 2-point and multiple point random models. The latter approach is shown to be superior in terms of computational efficiency. We quantify the information content given by a specific choice of prior model. This enables us to obtain a lower limit of, for example, the size of a grid cell in a grid-parametrized parameter space. The resulting decrease in effective dimension of the parameter space provides a much more efficient sampling of the posterior with orders of magnitude increase in computational efficiency.