Prestack time migration for anisotropic media 

Tariq Alkhalifah

tariq@sep.stanford.edu

ABSTRACT

Prestack phase-shift migration is implemented by evaluating the offset-wavenumber (k_h) integral using the stationary-phase 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 zero-offset data, the stationary point (k_h=0) is known in advance, and, therefore, the phase-shift migration can be implemented directly. For non-zero-offset 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 ($\eta \approx$0.3) and even large offset-to-depth ratios (>2). The analytical solutions are particularly accurate in predicting the non-hyperbolic moveout behavior associated with anisotropic media, a key ingredient to performing an accurate non-hyperbolic moveout inversion for strongly anisotropic media. Although the prestack correction achieved using the phase-shift method can also be obtained using a cascade of normal-moveout correction, dip-moveout (DMO) correction, and zero-offset 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.