Most of the traditional approaches to migration by downwardextrapolation suffer from inaccuracies caused by usingone-way propagation, both in the construction of such propagatorsin a variable background and the suppression of propagatingwaves generated by, e.g., steep reflectors.We present anew mathematical formulation and an algorithm for downwardextrapolation that suppress only the evanescent waves.We show that evanescent wave modes are associated with thepositive eigenvalues of the spatial operator and introducespectral projectors to remove these modes, leaving all propagatingmodes corresponding to nonpositive eigenvalues intact.This approach suppresses evanescent modes in an arbitrarylaterally varying background. If the background velocityis only depth dependent, then the spectral projector may beapplied by using the fast Fourier transform and a filter in theFourier domain. In computing spectral projectors, we use aniteration that avoids the explicit construction of the eigensystem.Moreover,we use a representation of matrices leading tofast matrix-matrix multiplication and, as a result, a fast algorithmnecessary for practical implementation of spectral projectors.The overall structure of the migration algorithm issimilar to survey sinking with an important distinction of usinga new method for downward continuation. Using ablurred version of the true velocity as a background, steep reflectorscan be imaged in a 2Dslice of theSEG-EAGEmodel.
展开▼
