Unlike zero-offset phase-shift migration, the prestack version, obtained by evaluating the offset-wavenumber integral with the stationary phase method requires solving for the stationary point of the horizontal component of offset slowness (ph) prior to applying the phase shift. Exact analytical solutions of ph are hard to obtain, especially for anisotropic media. Numerical solutions of ph, however, are efficiently obtainable through the use of numerous precomputed tables at different stages of the migration. Given that the phase of the kx integrand is rather insensitive to ph around its maximum (around the stationary point solution), ph can also be estimated using analytical approximations. These analytical equations are obtained by fitting them to the exact solution at different slopes. Errors on the order of 1 in ph result in less than 0.1 error in the phase because of its relatively low sensitivity to ph.
For anisotropic media, the analytical solutions include additional approximations based on weak anisotropy. The resultant equations, nevertheless, produce accurate migration signatures for relatively strong anisotropy (i.e., ) and even for large offsets (i.e., offset-to-depth ratios larger than 2). The accuracy of the anisotropic result at zero reflector dip is particularly amazing. Analytically, the accuracy far exceeds anything previously derived using Taylor series expansions (Tsvankin and Thomsen, 1994, Alkhalifah, 1997b).
The numerical implementation of the proposed prestack migration is fast. The cost of prestack migration of a single common-offset section is about 10 higher than the cost of zero-offset migration of the same size section. The additional cost (spent calculating ph) is reduced even further (percentage wise) when migrating large volumes of data, where the cost of this overhead becomes more and more insignificant.