HOPFIELD NEURAL NETWORKS, AND MEAN-FIELD ANNEALING FOR SEISMIC DECONVOLUTION AND MULTIPLE ATTENUATION

We describe a global optimization method called mean field annealing ( MFA) and its application to two basic problems in seismic data process ing: Seismic deconvolution and surface related multiple attenuation. M FA replaces the stochastic nature of the simulated annealing method wi th a set of deterministic update rules that act on the average value o f the variables rather than on the variables themselves, based on the mean field approximation. As the temperature is lowered. the MFA rules update the variables in terms of their values at a previous temperatu re. By minimizing averages it is possible to converge to an equilibriu m state considerably faster than a standard simulated annealing method . The update rules are dependent on the form of the cost function and are obtained easily when the cost function resembles the energy functi on of a Hopfield network. The mapping of a problem onto a Hopfield net work is not a precondition for using MFA, but it makes analytic calcul ations simpler. The seismic deconvolution problem can be mapped onto a Hopfield network by parameterizing the source and the reflectivity in terms of binary neurons. In this context, the solution of the problem is obtained when the neurons of the network reach their stable states . By minimizing the cost function of the network with MFA and using an appropriate cooling schedule, it is possible to escape local minima. A similar idea can also be applied to design an operator that attenuat es surface related multiple reflections from plane-wave transformed se ismograms assuming a 1-D earth. The cost function for the multiple eli mination problem is based on the criterion of minim um energy of the m ultiple suppressed data.