conclusion

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., $\eta=0.4$) 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.