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Next: Extrapolation operator in laterally Up: Shan and Biondi: Anisotropic Previous: Introduction

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):  
 \begin{displaymath}
\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} },\end{displaymath} (1)
where $\theta$ is the phase angle of the propagating wave, and f=1-(VS0/VP0)2. VP0 and VS0 are the qP- and qSV- wave velocities in the vertical direction, respectively. $\varepsilon$ and $\delta$ are anisotropy parameters defined by Thomsen (1986):

\begin{displaymath}
\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})},\end{displaymath}

where Cij 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:  
 \begin{displaymath}
\frac{V^2(\theta,\varphi)}{V^2_{P0}}=1+\varepsilon\sin^2(\th...
 ...t)^2-
\frac{2(\varepsilon-\delta)\sin^22(\theta-\varphi)}{f} }.\end{displaymath} (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. VP0 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 kx and kz by:  
 \begin{displaymath}
\sin \theta=\frac{V(\theta,\varphi)k_x}{\omega},\ \ \ \ \ \ \ \cos \theta=\frac{V(\theta,\varphi)k_z}{\omega},\end{displaymath} (3)
where $\omega$ is the temporal frequency. Squaring equation (2) and substituting (3) into (2), we can obtain a dispersion relation equation:

 
d4kz4+d3kz3+d2kz2+d1kz+d0=0, (4)

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

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

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 VP0 and anisotropy parameters $\varepsilon$ and $\delta$, the wavefield can be extrapolated by the phase-shift method Gazdag (1978):  
 \begin{displaymath}
P(z+\Delta z)=P(z)e^{-ik_z^a\Delta z},\end{displaymath} (5)
where kza is one of the roots of equation (4).
next up previous print clean
Next: Extrapolation operator in laterally Up: Shan and Biondi: Anisotropic Previous: Introduction
Stanford Exploration Project
10/23/2004