Analytic study of the effective parameters for determination of the NMO velocity function in transversely isotropic media

In their studies of transversely isotropic media with a vertical symmetry axis (VTI media), Alkhalifah and Tsvankin observed that, to a high numerical accuracy, the normal moveout (NMO) velocity for dipping reflectors as a function of ray parameter p depends mainly on just two parameters, each of which can be determined from surface P-wave observations. They substantiated this result by using the weak-anisotropy approximation and exploited it to develop a time-domain processing sequence that takes into account vertical transverse isotropy. In this study, the two-parameter Alkhalifah-Tsvankin result was further examined analytically. It was found that although there is (as these authors already observed) some dependence on the remaining parameters of the problem, this dependence is weak, especially in the practically important regimes of weak to moderately strong transverse isotropy and small ray parameter. In each of these regimes, an analytic solution is derived for the anisotropy parameter η required for time-domain P-wave imaging in VTI media. In the case of elliptical anisotropy (η = 0), NMO velocity expressed through p is fully controlled just by the zero-dip NMO velocity—one of the AlkhalifahTsvankin parameters. The two-parameter representation of NMO velocity also was shown to be exact in another limit—that of the zero shear-wave vertical velocity. The analytic results derived here are based on new representations for both the P-wave phase velocity and normal moveout velocity in terms of the ray parameter, with explicit expressions given for the cases of vanishing onaxis shear speed, weak to moderate transverse isotropy, and small to moderate ray parameter. Using these formulas, I have rederived and, in some cases, extended in a uniform manner various results of Tsvankin, Alkhalifah, and others. Examples include second-order expansions in the anisotropy parameters for both the P-wave phasevelocity function and NMO-velocity function, as well as expansions in powers of the ray parameter for both of these functions. I have checked these expansions against the corresponding exact functions for several choices of the anisotropy parameters.