Full-wave-equation depth extrapolation for migration

Abstract

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.

Sonic Imaging

“…These methods are effective only if the true speed of sound deviates from the background only a little, typically by not more than 10 %, as is the case in mammography. An improved version that is able to handle much greater variations of the velocity, as they occur, e.g., in seismics, can be found in [59].…”

Preliminary Investigation of Wavefield Depth Extrapolation by Two-Way Wave Equations

“…Seismic imaging based on the two-way depth migration, when high velocity contrast exists, the migrated image will contain low-frequency artifacts similar to the RTM-migrated data. Techniques like smoothing the migration velocity model [2,4] or applying a Laplacian filter to the final stacked image can effectively dampen the artifacts.…”

“…For a background with depth-dependent velocity and zero offset source-receiver configuration, the full wave partial differential equation is changed to a second-order ordinary differential equation, and they solved the equation by the fourth-order Runge-Kutta method. Sandberg and Beylkin [4] extended this method to laterally varying media and migrated the prestack seismic data in the source-receiver surveysinking style. Sandberg et al [2] modified the algorithm in Sandberg and Beylkin [4] to take a proper account of secondary reflections recorded at the surface for imaging.…”

“…Sandberg and Beylkin [4] extended this method to laterally varying media and migrated the prestack seismic data in the source-receiver surveysinking style. Sandberg et al [2] modified the algorithm in Sandberg and Beylkin [4] to take a proper account of secondary reflections recorded at the surface for imaging.…”

“…However, such a strategy also discarded certain propagating waves generated by the steeply dipping events, resulting in poor imaging of these structures. Sandberg and Beylkin [4] introduced spectral projectors to remove the evanescent wave modes and leave all propagating modes intact in a variable background.…”