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## VTI homogeneous media

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)
Clearly, for =0, the above equations reduce to there isotropic counterparts. The stationary point (or points) satisfy the following relation
 (34)

ph1eta2m
Figure 20
Left: Values of ph as a function px calculated numerically (solid curves), and calculated analytically (dashed curves) using equation (B-1). Right: The absolute difference between the two curves on the right. The medium is homogeneous and transversely isotropic with v=2.0 km/s and =0.2.. The black curve corresponds to =1.0 (km/s), the dark-gray curve corresponds to =2.0 (km/s), and the light gray curve corresponds to =3.0 (km/s).

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

and

 c=X2. (35)

Because can be small, as small as zero for isotropic media, I prefer to use the following form of solution of the quadratic equation:

(Press et al., 1988). Thus,
 (36)
where the positive-sign root is used. The negative-sign root results in an imaginary ph solution which is clearly not the solution we are looking for. Another, yet better, approximation is described in Appendix C using perturbation series as well as Shanks transformation. For =0 (isotropic media), a=0, and this equation reduces to equation (B-3), which is its isotropic counterpart. This is amazing considering we have dropped some terms in solving the quartic equation. Equation (B-11) is not the exact solution for horizontal reflectors in VTI media. However, as we will see bellow, it is a good approximation; better than similar approximations based on Taylors series expansion of traveltimes around the zero-offset point. Also, to insure we do not get imaginary roots for ph, b2-4ac must be greater or equal zero. This results in the following condition that must be satisfied:

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
 (37)
As a result, the stationary point is given by
 (38)
Again, I insert equation (B-11) in place of , and obtain
 (39)
where a, b, and c are given by equation (B-11). Equation (B-15) is exact for (ph=0). However, unlike the isotropic medium case, equation (B-15) is an approximation at px=0. [Recall that we dropped terms beyond the quadratic in equation (B-13).] Despite the approximation used at px=0, I will refer to this equation as the 2-point solution for VTI media.

ph2eta2m
Figure 21
Same as in Figure B-1, but with the analytical solution evaluated using the 2-point fitting of equation (B-15).

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)
where

and

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)
where setting px=0 gives b=1 and setting px=phs results in

ph3eta2m
Figure 22
Same as in Figure B-1, but with the analytical solution evaluated using the 3-point fitting of equation (B-17).

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 .

Next: Better stationary-phase approximations Up: Analytical Approximations of the Previous: Isotropic homogeneous media
Stanford Exploration Project
11/11/1997