None of the leading approaches to the migration of seismic sections—the Kirchhoff‐summation method, the finite‐difference method, or the frequency‐domain method—readily migrates seismic reflections to their proper positions when overburden velocities vary laterally. For inhomogeneous media, the diffraction curve for a localized, buried scatterer is no longer hyperbolic and its apex is displaced laterally from the position directly above the scatterer. Hubral observed that the Kirchhoff‐summation method images seismic data at emergent “image ray” locations rather than at the desired positions vertically above scatterers. In addition, distortions in diffraction shapes lead to incorrect imaging (i.e., incomplete diffraction collapse) and, hence, to further displacement errors for dipping reflections. The finite‐difference method has been believed to continue waves downward correctly through inhomogeneous media. In conventional implementations, however, both the finite‐difference method and frequency‐domain approach commit the same error that the Kirchhoff method does. Synthetic examples demonstrate how conventional migration fails to image events completely. Hubral’s solution to this migration problem is two‐ (or three‐) dimensional mapping of imaged time sections into depth. This mapping, “depth migration,” replaces simple vertical conversion from time to depth. Such depth migration can be postponed until after efficient image‐ray modeling has been performed to (1) support the final choice of velocity model, and (2) determine whether depth migration is necessary. Comparisons between depth‐migrated and conventionally depth‐converted sections of both synthetic and field data properly show that significant lateral displacement is often required to position reflectors properly. Monte Carlo studies show that the lateral corrections can be important not only in absolute terms but also in relation to errors expected from an inaccurate velocity model.
Computation of wavefields in vertically inhomogeneous media by a frequency domain finite-difference method and application to wave propagation in earth models with random velocity and density perturbations
