Least-squares imaging of multiples

Applied to a common-midpoint gather, equation () produces an approximate unstacked zero-offset image of pseudo-primaries from water bottom multiple reflections. In this section, I introduce a least squares scheme to compute self-consistent images of primaries and pseudo-primaries which are in turn consistent with the data. First I define some terms: With these definitions in hand, we can now write the forward modeling operator for joint NMO of primaries and multiples of order 1 to p.  

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In words, equation () takes a collection of psuedo-primary panels, divides each by the appropriate reflection coefficient, applies inverse (adjoint) NMO to each, and then sums them together to create something that should resemble ``data''. We define the data residual as the difference between the input data and the forward-modeled data:  

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Viewed as a standard least-squares inversion problem, minimization of L₂ norm of the data residual by solution of the normal equations is underdetermined. Additional regularization terms, defined in later sections, force the problem to be overdetermined.