APPROXIMATE PHASE VELOCITIES

Probably the most common way to write the linearized equations for the phase velocities in VTI media (Thomsen, 1986; 2002; Rüger, 2002) is

$$\begin{eqnarray} V_p(\theta) \simeq V_{p0}\left[1 + \delta\sin^2\theta\cos^2\theta + \epsilon\sin^4\theta\right], \end{eqnarray}$$ (64)

$$\begin{eqnarray} V_{sv}(\theta) \simeq V_{s0}\left[1 + {{V_{p0}^2}\over{V_{s0}^2}}(\epsilon-\delta) \sin^2\theta\cos^2\theta\right], \end{eqnarray}$$ (65)

and

$$\begin{eqnarray} V_{sh}(\theta) \simeq V_{s0}\left[1 + \gamma\sin^2\theta\right]. \end{eqnarray}$$ (66)

The approximations in all three cases are made based on assumed smallness of the parameters $\delta$, $\epsilon$, $\gamma$, which in fact may or may not hold for any particular layered medium. However, the work in this paper demands better approximations than these, because the assumptions of weak anisotropy are always violated in the cases of most interest, i.e., when the SV-wave velocity actually does depend in a significant way on fluid content. Thomsen (2002) (in his Appendix III) quotes another form of the dispersion relation for $V_p(\theta)$ that is more useful for our purposes (modified here to correct an obvious typo in the leading term):

$$\begin{eqnarray} V_p^2(\theta) \simeq V_{p0}^2 + 2V_{nmo}^2(\delta\sin^2\theta + \eta\sin^4\theta), \end{eqnarray}$$ (67)

where $V_{p0}^2 = c/\rho$, $V_{nmo} \simeq V_{p0}(1+\delta)$, and $\eta = (\epsilon-\delta)/(1+2\delta)$ is the combination of parameters introduced by Alkhalifah and Tsvankin (1996). Although (67) is still an approximation, it is much closer in form to the dispersion relation quoted here in (34). The form (67) still assumes smallness of the anisotropy parameters, but the usual square root approximation has not been made yet, so the correspondence with (34) is easier to scan. If we neglect higher order contributions to $\Delta$ and thereby make the approximation that

$$\begin{eqnarray} \Delta \simeq 2k^2c(\epsilon-\delta)\sin^2\theta\cos^2\theta, \end{eqnarray}$$ (68)

then (34) becomes

$$\begin{eqnarray} V_p^2(\theta) \simeq V_{p0}^2 + 2V_{p0}^2\delta\sin^2\theta + 2V_{p0}^2(1+2\delta)\eta\sin^4\theta. \end{eqnarray}$$ (69)

If in addition we also make the small anisotropy approximations $V_{p0}^2\delta \simeq V_{nmo}^2\delta$and $V_{p0}^2(1+2\delta) \simeq V_{nmo}^2$ in (69), then the result recovers (67).

Our main goal in this Appendix is to make a direct comparison between the exact formulas (34) and (35), the approximate formulas resulting from (34) and (35) when $\Delta$ is replaced by its first approximation (37), and either standard equations (64) and (65) or approximation (67) and some yet to be determined companion equation for quasi-SV waves. The easiest and most consistent way to arrive at an appropriate approximate form for V_sv² is to use the exact relations (34) and (35) to determine what effective value of $\Delta_{eff}$ has been used in (67) and then use it again in (35). We find

$$\begin{eqnarray} \Delta_{eff}(\theta) \simeq 2k^2\rho V_{po}^2(\epsilon-\delta)\... ...ta\cos^2\theta -2k^2\rho V_{p0}^2\delta^2(2+\delta)\sin^2\theta. \end{eqnarray}$$ (70)

However, this formula has the undesirable characteristic that it does not vanish as it should for $\theta = 90^\circ$. The offending terms are second order in $\delta$ and therefore are usually neglected for weak anisotropy. But the weak anisotropy assumptions implicit in (67) are not valid in the present context, so this is nevertheless a problem for us here. Making proper allowance for this, we can arrive at a corrected $\Delta$ that has the desired behavior and still agrees with the prior results under weak anisotropy conditions:

$$\begin{eqnarray} \Delta_{corr}(\theta) \simeq 2k^2\rho V_{nmo}^2\eta\sin^2\theta\cos^2\theta, \end{eqnarray}$$ (71)

and, therefore, that a good choice for V_sv² to the same level of approximation is

$$\begin{eqnarray} V_{sv}^2(\theta) = V_{so}^2 + \Delta_{corr}(\theta)/k^2\rho. \end{eqnarray}$$ (72)

But these modifications have led us back to the approximation (34) and therefore provide nothing new. So instead of comparisons to (67) and (72), we will choose to make our comparisons to (64) and (65). In particular, these two equations amount to using

$$\begin{eqnarray} \Delta_{eff} = 2k^2\rho V_{p0}^2(\epsilon-\delta)\sin^2\theta\cos^2\theta, \end{eqnarray}$$ (73)

except for some higher order corrections in $\Delta_{eff}$ which would always be small if the anisotropy were really always weak. Although Eqs. (71) and (73) are apparently the same, this fact is a bit misleading since $\Delta$ and its corrections arise in the final results in different ways because of differing square root approximations and different assumptions about the presence or absence of strong anisotropy.

 

compvels
Figure 4 Compressional wave velocities as computed exactly from the dispersion relation (34), by (34) using approximation (37) for $\Delta$, and by the linear approximation (64). This layered model is the same as in Figures 2 and 3 for the case B = 1.

 

shearvels
Figure 5 Shear wave velocities as computed exactly from the dispersion relation (35), by (35) using approximation (37) for $\Delta$, and by the linear approximation (66). This layered model is the same as in Figures 2 and 3 for the case B = 1.

Numerical comparisons of these three sets of results for quasi-P and quasi-SV waves are summarized in Figures 4 and 5 for one strong anisotropy example. The comparison is obviously not a fair one for the weak anisotropy equations since they are being used beyond their acknowledged (and expected) range of validity. The main point of the exercise is to see that the approximations made here give reasonable approximations to the exact results for the full range of possible incidence angles for strong anisotropy conditions, while the standard results do not fair as well. All the methods agree quite well for compressional waves in this model. The evaluation of (35) using (37) to approximate $\Delta$ gives a clear improvement over (65) for the quasi-SV wave velocity in a range of intermediate angles. Overall, the weak anisotropy formulas (64) and (65) give better results for strong anisotropy in this case than might have been expected.