ull-wave-equation depth extrapolation for migration

Most of the traditional approaches to migration by downward extrapolation suffer from inaccuracies caused by using one-way propagation, both in the construction of such propagators in a variable background and the suppression of propagating waves generated by, e.g., steep reflectors. We present a new mathematical formulation and an algorithm for downward extrapolation that suppress only the evanescent waves. We show that evanescent wave modes are associated with the positive eigenvalues of the spatial operator and introduce spectral projectors to remove these modes, leaving all propagating modes corresponding to nonpositive eigenvalues intact. This approach suppresses evanescent modes in an arbitrary laterally varying background. If the background velocity is only depth dependent, then the spectral projector may be applied by using the fast Fourier transform and a filter in the Fourier domain. In computing spectral projectors, we use an iteration that avoids the explicit construction of the eigensystem. Moreover, we use a representation of matrices leading to fast matrix-matrix multiplication and, as a result, a fast algorithm necessary for practical implementation of spectral projectors. The overall structure of the migration algorithm is similar to survey sinking with an important distinction of using a new method for downward continuation. Using a blurred version of the true velocity as a background, steep reflectors can be imaged in a 2D slice of the SEG-EAGE model.