Local Bayesian inversion: theoretical developments

Abstract

Summary We derive a new Bayesian formulation for the discrete geophysical inverse problem that can significantly reduce the cost of the computations. The Bayesian approach focuses on obtaining a probability distribution (the posterior distribution), assimilating three kinds of information: physical theories (data modelling), observations (data measurements) and prior information on models. Once this goal is achieved, all inferences can be obtained from the posterior by computing statistics relative to individual parameters (e.g. marginal distributions), a daunting computational problem in high dimensions. Our formulation is developed from the working hypothesis that the local (subsurface) prior information on model parameters supercedes any additional information from other parts of the model. Based on this hypothesis, we propose an approximation that permits a reduction of the dimensionality involved in the calculations via marginalization of the probability distributions. The marginalization facilitates the tasks of incorporating diverse prior information 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 throwing away, at each step, cross-moment information of order higher than two, while preserving all marginal information about the parameter being estimated. The main advantage of the method is allowing for systematic integration of prior information while maintaining practical feasibility. This is achieved by combining (1) probability density estimation methods to derive marginal prior distributions from available local information, and (2) the use of multidimensional Gaussian distributions, which can be marginalized in closed form. Using a six-parameter problem, we illustrate how the proposed methodology works. 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 considered. The uncertainty measure for estimated parameters is provided by 95 per cent probability intervals calculated from the marginal posterior distributions.