Anisotropic phase-shift in tilted TI media

In a VTI media, the phase velocity of qP- and qSV-waves in Thomsen's notation can be expressed as Tsvankin (1996):  

$$\frac{V^2(\theta)}{V^2_{P0}}=1+\varepsilon\sin^2(\theta)-\fr... ...{f}\right)^2- \frac{2(\varepsilon-\delta)\sin^2(2\theta)}{f} },$$ (1)

where $\theta$ is the phase angle of the propagating wave, and f=1-(V_S0/V_P0)². V_P0 and V_S0 are the qP- and qSV- wave velocities in the vertical direction, respectively. $\varepsilon$ and $\delta$ are anisotropy parameters defined by Thomsen (1986):

$$\varepsilon=\frac{C_{11}-C_{33}}{2C_{33}}, \delta=\frac{(C_{11}+C_{44})^2-(C_{33}-C_{44})^2}{2C_{33}(C_{33}-C_{44})},$$

where C_ij are elastic moduli. In equation (1), $V(\theta)$ the is qP-wave phase-velocity when the sign in front of the square root is positive, and the qSV-wave phase velocity for a negative sign.

If we rotate the symmetry axis from vertical to a tilted angle $\varphi$, we obtain the phase velocity of a tilted TI medium whose symmetry axis forms an angle $\varphi$ with the vertical direction:  

$$\frac{V^2(\theta,\varphi)}{V^2_{P0}}=1+\varepsilon\sin^2(\th... ...t)^2- \frac{2(\varepsilon-\delta)\sin^22(\theta-\varphi)}{f} }.$$ (2)

Here, in contrast to equation (1), $\varepsilon$ and $\delta$ are now defined in a direction tilted by the angle $\varphi$ from the vertical direction. V_P0 is the qP-wave velocity in the direction parallel to the symmetry axis.

For plane-wave propagation, the phase angle $\theta$ is related to the wavenumbers k_x and k_z by:  

$$\sin \theta=\frac{V(\theta,\varphi)k_x}{\omega},\ \ \ \ \ \ \ \cos \theta=\frac{V(\theta,\varphi)k_z}{\omega},$$ (3)

where $\omega$ is the temporal frequency. Squaring equation (2) and substituting (3) into (2), we can obtain a dispersion relation equation:

 

d₄k_z⁴+d₃k_z³+d₂k_z²+d₁k_z+d₀=0, (4)

where the coefficients d₀,d₁,d₂,d₃, and d₄ are as follows:

$$\begin{array} {l} d_0=(2+2\varepsilon\cos^2\varphi-f)\left( ... ...phi-\frac{f}{2}(\varepsilon-\delta)\sin^22\varphi. \end{array}$$

The dispersion relation equation (4) is a quartic equation. It can be solved analytically Abramowitz and Stegun (1972) or numerically by Newton's Method Stoer and Bulirsch (1992). Equation (4) has four roots, which are related to up-going and down-going qP- and qSV- waves, respectively. For a medium without lateral change in the velocity V_P0 and anisotropy parameters $\varepsilon$ and $\delta$, the wavefield can be extrapolated by the phase-shift method Gazdag (1978):  

$$P(z+\Delta z)=P(z)e^{-ik_z^a\Delta z},$$ (5)

where k_z^a is one of the roots of equation (4).