Inverse problems with non-trivial priors: efficient solution through sequential Gibbs sampling

Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis algorithm can be used to sample solutions to non-linear inverse problems. In principle, these methods allow incorporation of prior information of arbitrary complexity. If an analytical closed form description of the prior is available, which is the case when the prior can be described by a multidimensional Gaussian distribution, such prior information can easily be considered. In reality, prior information is often more complex than can be described by the Gaussian model, and no closed form expression of the prior can be given. We propose an algorithm, called sequential Gibbs sampling, allowing the Metropolis algorithm to efficiently incorporate complex priors into the solution of an inverse problem, also for the case where no closed form description of the prior exists. First, we lay out the theoretical background for applying the sequential Gibbs sampler and illustrate how it works. Through two case studies, we demonstrate the application of the method to a linear image restoration problem and to a non-linear cross-borehole inversion problem. We demonstrate how prior information can reduce the complexity of an inverse problem and that a prior with little information leads to a hard inverse problem, practically unsolvable except when the number of model parameters is very small. Considering more complex and realistic prior information thus not only makes realizations from the posterior look more realistic but it can also reduce the computation time for the inversion dramatically. The method works for any statistical model for which sequential simulation can be used to generate realizations. This applies to most algorithms developed in the geostatistical community.