Zero-offset phase-shift method of migration (Gazdag, 1978) is a robust and efficient way of seismic imaging. Gazdag migration ranks second only to Stolt (1978) in efficiency, and it can exactly (within the limit of ray theory) handle the kinematics of migrating data from v(z) media, whereas Stolt migration resorts to approximations to handle v(z) variations of velocity. Gazdag migration, with some modification (Gazdag and Sguazzero, 1984; Stoffa and Fokkema, 1990) can approximately handle smooth lateral velocity variation.
Extending phase-shift migration to treat prestack data is accomplished by using the double-square-root (DSR) equation of Yilmaz (1979). Its implementation in practice, however, is rare, if non existent; using the DSR equation in its original form requires that all offsets and common-midpoint (CMP) data be downward continued simultaneously. The output of such a prestack migration process is a stacked section that does not hold any explicit velocity information, or otherwise, some notable artifacts will occur during integration of the offset-wavenumber axis due to the sparseness in its sampling. Popovici (1995) showed that such artifacts can be avoided by sub-sampling the offset-wavenumber axis so that the integration range does not include evanescent waves. He also developed a technique to reserve the full complement of offsets free of artifacts after migration; he devised a method to downward continue separate offsets by evaluating the offset-wavenumber integral with the stationary-phase method. Determining the stationary point required solving a six-order polynomial which Popovici elected to avoid doing analytically. His numerical solutions of the stationary points, according to his assessment, were relatively expensive, because they required an additional integration.
In this paper, I discuss the stationary point solution associated with evaluating the offset-wavenumber integral, and show efficient numerical and approximate analytical solutions. The numerical implementation of the separate-offset phase-shift migration is based on tabulating the stationary point solutions. It is efficient enough to use in practice, especially for migrating transversely isotropic media. I also show analytical approximations of the stationary-point solution in both isotropic and anisotropic homogeneous media. Applications to synthetic and real data follow.