YET ANOTHER VTI WAVE EQUATION

For $\eta$=0, the VTI acoustic wave equation reduces to the simple elliptically anisotropic wave equation (9), which is second order in t, and therefore has only two solutions. Assuming that $\eta$ is small, let us insert, when possible, the 2-D elliptical wave equation

   $$\begin{eqnarray} \frac{\partial^2 F}{\partial t^2} = v^2 \frac{\partial^2 F}{\partial x^2} +v_v^2 \frac{\partial^2 F}{\partial z^2}.\end{eqnarray}$$

in place of one $\frac{\partial^2 F}{\partial t^2}$ in each term of the 2-D version of equation (9),

   $$\begin{eqnarray} \frac{\partial^4 F}{\partial t^4} = (1+2 \eta) v^2 \frac{\parti... ...- 2 \eta v^2 v_v^2 \frac{\partial^4 F}{\partial x^2 \partial z^2}.\end{eqnarray}$$

The resultant formula after some manipulation is

   $$\begin{eqnarray} \frac{\partial^2}{\partial t^2}\left(v^2 \frac{\partial^2 F}{\p... ... z^4}+ 2 v^2 v_v^2 \frac{\partial^4 F}{\partial x^2 \partial z^2}.\end{eqnarray}$$

This equation is second order in time, and therefore has two solutions. To arrive at this form we had to ignore the difference between P and F, which is valid since our main objective is to treat the kinematics of wave propagation.

To understand this equation kinematically, we Fourier transform equation (39) in x, z, and t, and obtain the following dispersion relation:

$$\begin{eqnarray} {v^2}\,{{\omega }^2}\,{{{k_x}}^2} - {v^4}\,\left( 1 + 2\,\eta... ...k_x}}^2}\,{{{k_z}}^2}\,{{{v_v}}^2} - {{{k_z}}^4}\,{{{v_v}}^4}=0,\end{eqnarray}$$

with k_z satisfying  

$$k_z^2 = {\frac{{{\omega }^2} - 2\,{v^2}\,{{{k_x}}^2} \pm {... ...mega }^4} - 8\,{v^4}\,\eta \,{{{k_x}}^4}}}} {2\,{{{v_v}}^2}}}.$$ (41)

Figure 10 compares k_z given by this dispersion relation with that given by equation (4), which is the exact eikonal equation for acoustic VTI media. The difference is small for $\eta=0.2$, which implies that equation (39) can serve as a valid substitute, kinematically, for equation (11).

 

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Figure 10 The vertical component of slowness as a function of the horizontal component in a VTI medium with $\eta=0.2$ and v=2 km/s. The solid curve corresponds to using approximation (41), while the dashed curve corresponds to the exact acoustic eikonal equation for VTI media.