Stronger lateral velocity variations?

The numerical scheme described by equation (7) can probably handle some amount of lateral velocity variations, but it would be inaccurate for more complex velocity functions. In the derivation presented in the previous section there is no assumption of mild lateral velocity variations up to equation (3). The problem with equation (3) is that it is fourth order in time. In addition to the desired solution it has another solution that can generate artifacts and cause instability. Therefore, a direct solution by finite-differences would encounter problems with the spurious solution. Alkhalifah (1998) describes a similar problem when solving an acoustic wave equation for anisotropic media.

However, it is fairly straightforward to derive an approximation to equation (3) that is more accurate than equation (6). Equation (3) can be easily solved for $\omega^2$because it contains only the even powers of $\omega$.We can then approximate the square root that appears in the formal solution for $\omega^2$ as  

$$\sqrt{1 - \frac {\Delta_s \left[k_z^4 + k_z^2\left(k_{x_m}^... ...ht]} {2\left(\Delta_s k_{x_m}k_{x_h}- \Sigma_s k_z^2\right)^2}.$$ (10)

The useful solution of equation (3) can then be approximated as  

$$\omega^2=\frac{1}{2\left(\Sigma_s - \Delta_s k_{x_m}k_{x_h}\... ..._{x_m}^2 + k_{x_h}^2 + \frac{k_{x_m}^2k_{x_h}^2}{k_z^2}\right).$$ (11)

Equation (11) degenerates to equation (6) when $\Delta_s$ is zero. It is more accurate than Equation (6) for $\Delta_s$ different than zero, but it is likely to break down for strong lateral velocity variations.

Equation (11) shares with equation (6) the fundamental problem of instability for horizontally propagating waves. Therefore, I have not implemented a numerical scheme to solve equation (11) yet. However, it is possible to define a mixed implicit-explicit method to solve equation (11), similar to the one proposed by Klíe and Toro (2001) to solve Alkhalifah's acoustic wave equation for anisotropic media.