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# GENERAL FORMULATION OF THE METHOD

Multiple suppression in the velocity domain can be formulated as an optimization problem. The aim is to find some model" in the space-time domain, which, when added to the original CMP gather, minimizes the power within a pre-specified window in the velocity-time domain. The precise mathematical formulation follows.

Let represent the original CMP gather, its velocity transform, and N the velocity transform operator
 (1)
Then, we want to find the unknown model that minimizes the objective function
 (2)
where is a windowing operator that selects the region of the velocity spectrum in which the energy of the multiples is dominant. To find the solution that minimizes we need to solve

which is a system of linear equations to be solved for the unknown , where the data is the windowed velocity transform of the original data. The system can be over- or under determined, depending on how the operators and are defined, but it will usually be large enough, since is typically 105 points long. Of course is one of the solutions of equation 4. However the window operator transfer" the events that map outside the window (which are predominantly primaries) to the null space. The use of conjugate gradients (CG) seems to be a suitable choice for solving this problem especially because a few iterations would not only be enough but would also contribute to the stabilization of the process, when noise is present. We have chosen to use a very simple form for the velocity transform operator , the nearest neighbor normal moveout-stack. The advantages of choosing this operator are its computational speed and the simplicity of its transpose , which is needed in the conjugate algorithm (Claerbout, 1991).