Integrating geological prior information into the inverse gravimetric problem: the Bayesian approach

It is well known that the inverse gravimetric problem is generally ill-posed and therefore its solution requires some restrictive hypotheses and strong numerical regularization. However, if these initial assumptions are improperly used, the final results could be theoretically and physically admissible but far from the actual mass density distribution. In this work, a Bayesian approach to estimate the mass density distribution from gravity data coupled with a-priori geological information is presented. It requires to model the masses in voxels, each of them characterized by two random variables: one is a discrete label defining the type of material (or the geological unit), the other is a continuous variable defining the mass density (considered constant inside the single voxel). The a-priori geological information is translated in terms of this model, providing for each class of material the mean density and the corresponding variability and for each voxel the a-priori most probable label. Basically the method consists in a simulated annealing aided by a Gibbs sampler with the aim to find the MAP (maximum a posteriori) of the posterior probability distribution of labels and densities given the observations and the a-priori geological model. Some proximity constrains between labels of adjacent voxels are also introduced into the solution. The proposed Bayesian method is here tested on two simulated scenarios. In particular the first is an example of bathymetry recovering, while the second a salt dome shape estimation. These experiments show the capability of the method to correct the possible inconsistencies between the a-priori geological model and the gravity observations: 86% and 60% of wrong voxels have been corrected in the first and second test respectively.