The time-processing operators in VTI media, as mentioned earlier, depend primarily on two parameters,
and . This dependency becomes exact when the shear wave velocity (*V*_{S0}) is
set to zero (Alkhalifah, 1997c). Setting *V*_{S0}=0, although not practical for TI media, yields
remarkably accurate kinematic representations. Errors due to this approximation,
for practical VTI models,
are kinematically, in a worse-case scenario, less than 0.5 percent, which is far
within the limits of seismic accuracy.
This acoustic approximation yields simplified equations, including a simplified dispersion
relation. Because the vertical *P*-wave velocity does not have any significance in time-related
processing in VTI media (Alkhalifah and Tsvankin, 1995), I will refer to as the velocity and
denote it by the symbol *v* to simplify comparisons with isotropic media.
The zero-offset time-migration dispersion relation for VTI media, when *V*_{S0}=0, is given by

(33) |

(34) |

Figure 20

Following the same approach used in the previous sections for isotropic media, we set *p*_{x}=0, and
as a result, equation (B-10) reduces to

*a y ^{2}* +

c=X.
^{2} |
(35) |

(36) |

Expanding equation (B-10) using Taylors series,
around *p*_{h}=0 (),
and ignoring terms beyond the linear, yields

(37) |

(38) |

(39) |

Figure 21

Figure B-5 shows *p*_{h} given by equation (B-15)
and compares it with the accurate result
calculated numerically from equation (B-10) for three sets of .
Despite the large non-hyperbolic moveout associated with horizontal events in
such VTI media (Alkhalifah, 1997b), the approximation at *p*_{x}=0, which corresponds to a horizontal
event, is rather good. Combine this with the fact that the phase [equation (B-9)]
changes slowly as a function of (or rather insensitive to) *p*_{h} around the exact solution, the
result is a good approximation of the phase for all *p*_{x} (slopes).

The accuracy of equation (B-15) can be further enhanced, as in the isotropic case,
by fitting it to the exact
solution for *p*_{x}=*p*_{h}; the angular correspondence of this equality depends
on the offset-to-depth ratio. Following the same steps used to obtain the *p*_{h} for *p*_{x}=0, including
dropping terms beyond the quadratic in a similar quartic equation to that of equation (B-13),
the stationary point is given by

(40) |

*c*=*X ^{2}*.

Again, inserting equation (B-16) into equation (B-15) requires solving two equations and, similar to the case of isotropic media, the solution is given by

(41) |

Figure 22

Figure B-6 shows the numerically-driven curves (solid ones) along with the analytical curves calculated using the 3-point fitting of equation B-17. The 3-point equation provides the best estimation to the exact solution for .

11/11/1997