Here, I consider the simplest and probably most practical anisotropic model,
that is, a transversely isotropic (TI)
medium with a vertical symmetry axis. Such a medium is regarded in the anisotropic world
with the same importance that *v*(*z*) velocity variation has in the inhomogeneous world.
Although more complicated kinds of anisotropies can exist (i.e., orthrohombic anisotropy), the large
amount of shales
present in the subsurface implies that the TI model has the most influence on P-wave data (Banik, 1984).

In homogeneous transversely isotropic media with a vertical symmetry axis (VTI media),
*P*- and *SV*-waves
^{} can be described by the vertical velocities
*V*_{P0} and *V*_{S0} of *P*- and *S*-waves, respectively, and two
dimensionless parameters and (Thomsen, 1986).

Alkhalifah and Tsvankin (1995) further demonstrated that a new representation in terms of
just *two* parameters is sufficient for performing all time-related processing,
such as normal moveout correction (including non-hyperbolic moveout correction, if necessary),
dip-moveout correction, and prestack and post-stack time
migration. These two parameters are the normal-moveout velocity for a horizontal reflector

(1) |

(2) |

Moreover, Alkhalifah and Tsvankin (1995) further show that these two parameters,
and , are obtainable solely from surface seismic
*P*-wave data: specifically, from estimates of stacking
velocity for reflections from interfaces having two distinct dips. These two parameters can
also be resolved by examining the behavior of moveout at far offsets (Alkhalifah, 1997b).
The third parameter, *V*_{P0}, is needed for time-to-depth conversion only.
The two-parameter representation and inversion also holds in *v*(*z*) media (Alkhalifah, 1997a).
For that situation, these two parameters are expressed in terms of the vertical time .

Because the main assumption in the new parameterization is that the data remain in the time rather
than depth domain, the post-stack migration considered here is primarily a time one^{}.

11/11/1997