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(33) |
(34) |
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 toa y2 +b y +c =0,
where andc=X2. | (35) |
(36) |
Expanding equation (B-10) using Taylors series, around ph=0 (), and ignoring terms beyond the linear, yields
(37) |
(38) |
(39) |
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
(40) |
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
(41) |
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 .