VTI decoupled equations

Pestana et al. (2011) started with the exact phase velocity expression for VTI media (Tsvankin, 1996)

$$\frac{V^2(\bar{\theta})}{V^2_{p_0}} = 1+\epsilon \sin^2\bar{\thet... ...-\delta)\sin^22\bar{\theta}}{f\;(1+ \frac{2\epsilon\sin^2\bar{\theta}}{f})^2}},$$ (31)

where $\bar{\theta}$ is the phase angle measured to the symmetry axis and $f=1-V^2_{s_0}/V^2_{p_0}$ . Here, the plus sign corresponds to the P wave and the minus sign corresponds to the SV wave.

By revisiting equation 3.1 and expanding the square root to first order (i.e., $\sqrt{1-X}=1-X/2$ ), Pestana et al. (2011) got the following approximations for the P wave and SV wave phase velocities

$$\left\{ \begin{array}{ll} \displaystyle\frac{V^2(\bar{\theta})}{V... ...\theta}}{f})}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{(SV wave)} \end{array} \right. .$$ (32)

Equations 3.2 are good approximations for the P wave and SV wave dispersion relations when

$$\bigg\vert \frac{2(\epsilon-\delta)\sin^22\bar{\theta}}{f\;(1+ \frac{2\epsilon\sin^2\bar{\theta}}{f})^2} \bigg\vert \ll 1.$$ (33)

Since we have the following relations $\sin\bar{\theta}=V(\bar{\theta})k_r/\omega$ and $\cos\bar{\theta}=V(\bar{\theta})k_z/\omega$ with $k^2_r=k^2_x+k^2_y$ , where $\omega$ is the angular frequency, $k_x$ , $k_y$ and $k_z$ are spatial wavenumbers. So I get $V^2(\bar{\theta})=\omega^2/(k^2_r+k^2_z)$ . Plugging them into equations 3.2 gives

$$\left\{ \begin{array}{ll} \omega^2=V^2_{p_0} \bigg[ (1+2\epsilon)... ...r k^2_z}{F k^2_r+k^2_z} \bigg] \;\;\;\;\;\mbox{(SV wave)} \end{array} \right. ,$$ (34)

where

$$F=1+\frac{2\epsilon}{f}=\frac{(1+2\epsilon)V^2_{p_0}-V^2_{s_0}}{V^2_{p_0}-V^2_{s_0}}.$$ (35)

Equations 3.4 are equivalent to the P8 and SV8 approximations in the review paper of Fowler (2003). Setting $V_{s_0}=0$ (i.e., $F=1+2\epsilon$ ), equations 3.4 reduce to the dispersion relations used by Liu et al. (2009).