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 (VS0) is set to zero (Alkhalifah, 1997c). Setting VS0=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 VS0=0, is given by
(Alkhalifah, 1997c). Based on this relation, the normalized DSR equation, for VTI media, has the form
Following the same approach used in the previous sections for isotropic media, we set px=0, and as a result, equation (B-10) reduces to
Squaring both sides and expanding in ph, we get
where y=ph2 v2. This is a quartic equation in y, which, although,it has analytical solutions, is best solved numerically. Considering either or y (), or both, to be small, we can drop terms of y beyond the quadratic (perturbation theory; Bender and Orszag, 1978) , and as a result, the quartic equation reduces to
a y2 +b y +c =0,where
(Press et al., 1988). Thus,
which corresponds to large offset-to-depth ratio. For practical values of (<0.3), this condition is far within the typical mute zone. For example, if v=2 km/s, , and X=3 km, =1.14 s.
Expanding equation (B-10) using Taylors series, around ph=0 (), and ignoring terms beyond the linear, yields
Figure B-5 shows ph 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 px=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) ph around the exact solution, the result is a good approximation of the phase for all px (slopes).
The accuracy of equation (B-15) can be further enhanced, as in the isotropic case, by fitting it to the exact solution for px=ph; the angular correspondence of this equality depends on the offset-to-depth ratio. Following the same steps used to obtain the ph for px=0, including dropping terms beyond the quadratic in a similar quartic equation to that of equation (B-13), the stationary point is given by
c=X2.Equatio (B-16)n has a stricter condition for its validity (that is avoiding imaginary roots) than that of equation (B-11). Specifically,
which correspond to double the time for the horizontal reflector solution fitting. For v=2 km/s, , and X=3 km, =2.28 s.
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
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 .