3D dispersion relation in tilted Ti media

In 3D VTI media, the phase velocity of P- and SV-waves in Thomsen's notation can be expressed as follows 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 P- and SV- wave velocities in the vertical direction, respectively. The anisotropy parameters $\varepsilon$ and $\delta$ are 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), $V(\theta)$ is the P-wave phase-velocity when the sign in front of the square root is positive, and the SV-wave phase velocity for a negative sign. Let $k_x^\prime$, $k_y^{\prime}$ and $k_z^{\prime}$ be the wavenumbers for VTI media in Cartesian coordinates. For plane-wave propagation, the phase angle $\theta$ is related to the wavenumbers $k_x^\prime$, $k_y^{\prime}$ and $k_z^{\prime}$ by the following relations:  

$$\sin \theta=\frac{V(\theta)k_r^{\prime}}{\omega},\ \ \ \ \ \ \ \cos \theta=\frac{V(\theta)k_z^{\prime}}{\omega},$$ (2)

where $\omega$ is the temporal frequency, and $k_r^{\prime}=\sqrt{(k_x^{\prime})^2+(k_y^{\prime})^2}$.From equations (1) and (2), we can derive the dispersion relation for 3D VTI media as follows:  

$$b_6(k_z^{\prime})^4+b_5(k_r^{\prime})^4+b_4(k_z^{\prime})^2(k_r^{\prime})^2+b_3(k_z^{\prime})^2+b_2(k_r^{\prime})^2+b_1=0,$$ (3)

where

$$\begin{array} {lll} b_6&=&f-1,\\ b_5&=& (f-1)(1+2\varepsil... ...ight),\\ b_1&=&\left(\frac{\omega}{v_p}\right)^4. \end{array}$$ (4)

For tilted TI media, the symmetry axis deviates from the vertical direction. We need two angles to describe the tilting direction, the tilting angle $\varphi$ and the azimuth of the tilting direction $\psi$. We first assume $\psi=0$, that is the symmetry axis is in the plane y=0. Then we generalize the dispersion relation to the case that $\psi\neq 0$ by coordinate rotation.

For a tilted TI medium, if we rotate the coordinates so that the symmetry axis is the axis $z^{\prime}$, it becomes a VTI medium in the new coordinates. Let k_x, k_y, and k_z be the wavenumbers for a tilted TI medium in Cartesian coordinates. $k_x^\prime$, $k_y^{\prime}$ and $k_z^{\prime}$, which are the wavenumbers for VTI media in Cartesian coordinates, can also be considered as the wavenumbers for tilted TI media in the rotated coordinates. For the case that $\psi=0$, the dispersion relation can be obtained from equation (3) by rotating the coordinates as follows:

$$\left( \begin{array} {l} k_x^\prime\\ k_z^{\prime} \end{... ...t) \left( \begin{array} {l} k_x\\ k_z \end{array} \right).$$ (5)

We can re-organize the the dispersion relation and obtain the equation for the wavenumber k_z as follows:

 

a₄k_z⁴+a₃k_z³+a₂k_z²+a₁k_z+a₀=0, (6)

where

$$\begin{array} {lll} a_4&=&(f-1)+2\varepsilon(f-1)\sin^2\var... ...c{\omega}{v_{p0}} \right)^2(2+\varepsilon-f)k_y^2. \end{array}$$ (7)

Equation (6) is a quartic equation in k_z. Given k_x, k_z, the velocity v_p0, the anisotropy parameters $\varepsilon$ and $\delta$, and the tilting angle $\varphi$, we can calculate all the coefficients of equation (6), and it can be solved analytically Abramowitz and Stegun (1972). Usually there are four solutions for equation (6). Two of them are related to the up- and down-going P-wave, and the other two are related to the up- and down-going SV-wave, respectively.

Let $k_x^{\prime\prime}$, $k_y^{\prime\prime}$ and $k_z^{\prime\prime}$ be the wavenumbers for tilted TI media with a general $\psi$ in the original coordinate system. For general $\psi$, after solving equation (6), we can get the wavenumber k_z by rotating coordinates (k_x,k_y) as follows:

$$\left( \begin{array} {l} k_x^{\prime\prime}\\ k_y^{\prime... ...t) \left( \begin{array} {l} k_x\\ k_y \end{array} \right).$$ (8)

Figure shows k_z as a function of k_x and k_y in a constant tilted TI medium. In this medium, the velocity is 2000 m/s, $\varepsilon$ is 0.4, $\delta$ is 0.2, $\varphi$ is $\frac{\pi}{6}$ and $\psi$ is 0. The frequency used in Figure 57 Hz.

 

dispersion
Figure 1 Dispersion relation of 3D tilted TI media.