Multiple-point Statistics and Bayesian Inverse Theory - Some Inspirations from Mean-field and Curie-weiss Theories

For petroleum geostatistics, modelling of rock facies is of leading order importance: they are the dominant predictor of flow, and the dominant control of remote sensed data. Stronger geological control is desirable, and this is effectively introduced via multipoint statistics (MPS). The obscure character of MPS algorithms has hitherto prevented their clean integration with Bayesian inverse theory. We show that expressing MPS priors in terms of Gibbs energies makes this possible, and meets the dual requirements of modelling low entropy images well, and allowing rapid probability recomputations under local perturbations. By their close relation to ``standard'' Markov random field models via mean-field theory, albeit with a complex graph, their parameter inference problems are rendered easier by some analogies with classical Curie-Weiss type models. The data assimilation problem leads to an NP-hard problem equivalent to constrained binary quadratic programming, even for simple priors. This gives access to newer discrete optimisation methods like semidefinite programming (SDP). These relaxations provide remarkably good lower bounds on the optimisation, and serve as helpful validation of direct heuristic methods like annealing. Some demonstration problems on seismic AVO inversion illustrate the Gibbsian MPS formulation and its successful optimisation via both SDP and annealing.