ABSTRACTPrestack phaseshift migration is implemented by evaluating the offsetwavenumber (k_{h}) integral using the stationaryphase method. Thus, the stationary point along k_{h} must be calculated prior to applying the phase shift. This type of implementation allows for migration of separate offsets, as opposed to migration of the whole prestack data when using the original formulas. For zerooffset data, the stationary point (k_{h}=0) is known in advance, and, therefore, the phaseshift migration can be implemented directly. For nonzerooffset data, we first evaluate k_{h} that corresponds to the stationary point solution either numerically or through analytical approximations. The insensitivity of the phase to k_{h} around the stationary point solution (its maximum) implies that even an imperfect k_{h} obtained analytically can go a long way to getting an accurate image. In transversely isotropic (TI) media, the analytical solutions of the stationary point (k_{h}) include more approximations than those corresponding to isotropic media (i.e., approximations corresponding to weaker anisotropy). Nevertheless, the resultant equations, obtained using Shanks transforms, produce accurate migration signatures for strong anisotropy (0.3) and even large offsettodepth ratios (>2). The analytical solutions are particularly accurate in predicting the nonhyperbolic moveout behavior associated with anisotropic media, a key ingredient to performing an accurate nonhyperbolic moveout inversion for strongly anisotropic media. Although the prestack correction achieved using the phaseshift method can also be obtained using a cascade of normalmoveout correction, dipmoveout (DMO) correction, and zerooffset time migration, the prestack approach can handle sharper velocity models more efficiently. In addition, the resulting operator is cleaner than that obtained from the DMO method. Synthetic and field data applications of the proposed prestack migration demonstrate its usefulness.
