2-D Bayesian deconvolution

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Abstract
Inverse problems can be solved in different ways. One way is to define natural criteria of good recovery and build an objective function to be minimized. If, instead, we prefer a Bayesian approach, inversion can be formulated as an estimation problem where a priori information is introduced and the a posteriori distribution of the unobserved variables is maximized. When this distribution is a Gibbs distribution, these two methods are equivalent. Furthermore, global optimization of the objective function can be performed with a Monte Carlo technique, in spite of the presence of numerous local minima. Application to multitrace deconvolution is proposed. In traditional 1-D deconvolution, a set of uni‐dimensional processes models the seismic data, while a Markov random field is used for 2-D deconvolution. In fact, the introduction of a neighborhood system permits one to model the layer structure that exists in the earth and to obtain solutions that present lateral coherency. Moreover, optimization of an appropriated objective function by simulated annealing allows one to control the fit with the input data as well as the spatial distribution of the reflectors. Extension to 3-D deconvolution is straightforward.

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