Local Bayesian inversion: Theoretical developments

We derive a new Bayesian formulation for the discrete geophysical inverse p roblem that can significantly reduce the cost of the computations. The Baye sian approach focuses on obtaining a probability distribution (the posterio r distribution), assimilating three kinds of information: physical theories (data modelling), observations (data measurements) and prior information o n models. Once this goal is achieved, all inferences can be obtained from t he posterior by computing statistics relative to individual parameters (e.g . marginal distributions), a daunting computational problem in high dimensi ons. Our formulation is developed from the working hypothesis that the local (su bsurface) prior information on model parameters supercedes any additional i nformation from other parts of the model. Based on this hypothesis, we prop ose an approximation that permits a reduction of the dimensionality involve d in the calculations via marginalization of the probability distributions. The marginalization facilitates the tasks of incorporating diverse prior i nformation and conducting inferences on individual parameters, because the final result is a collection of 1-D posterior distributions. Parameters are considered individually, one at a time. The approximation involves throwin g away, at each step, cross-moment information of order higher than two, wh ile preserving all marginal information about the parameter being estimated . The main advantage of the method is allowing for systematic integration o f prior information while maintaining practical feasibility. This is achiev ed by combining (1) probability density estimation methods to derive margin al prior distributions from available local information, and (2) the use of multidimensional Gaussian distributions, which can be marginalized in clos ed form. Using a six-parameter problem, we illustrate how the proposed methodology w orks. In the example, the marginal prior distributions are derived from the application of the principle of maximum entropy, which allows one to solve the entire problem analytically. Both random and modelling errors are cons idered. The uncertainty measure for estimated parameters is provided by 95 per cent probability intervals calculated from the marginal posterior distr ibutions.