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 (*p*_{h}) prior to applying the phase shift.
Exact analytical solutions of *p*_{h} are hard to obtain, especially for anisotropic media.
Numerical solutions of *p*_{h}, however, are efficiently obtainable through the use of numerous
precomputed tables at different stages of the migration.
Given that the phase of the *k*_{x} integrand is rather insensitive to *p*_{h} around its maximum
(around the stationary point solution), *p*_{h} 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 *p*_{h} result in less than
0.1 error in the phase because of its relatively low sensitivity to *p*_{h}.

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 *p*_{h}) is reduced even further
(percentage wise) when migrating large volumes of data,
where the cost of this overhead becomes more and more
insignificant.

11/11/1997