VTI media in tilted coordinates

A VTI medium is a medium that is transversely isotropic and has a vertical axis of symmetry. The phase-velocity $V(\theta)$ of P-waves in VTI media can be expressed in Thomsen notation 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, and $\varepsilon$ 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), $V(\theta)$ 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.

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

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

where $\omega$ is the temporal frequency. From Cartesian coordinates to tilted coordinates, we do a transformation as follows:

$$\left( \begin{array} {l} x^{\prime}\\ z^{\prime} \end{ar... ...right) \left( \begin{array} {l} x\\ z \end{array} \right).$$ (3)

where $\varphi$ is the rotation angle.

Figure illustrates this change of symmetry axis during the coordinate transformation. The layered structure has a symmetry axis in the vertical direction in Cartesian coordinates (x,z) in the left panel. When we rotate the coordinates from (x,z) to $(x^\prime,z^\prime)$, the symmetry axis in the new coordinates $(x^\prime,z^\prime)$ deviates from the $z^\prime$ direction.

In tilted coordinates $(x^\prime,z^\prime)$, we have the following relation between wavenumber $k_{x^{\prime}},k_{z^{\prime}}$ and phase angle $\theta^{\prime}$: 

$$\sin \theta^{\prime}=\frac{V(\theta^{\prime})k_{x^{\prime}}}... ...theta^{\prime}=\frac{V(\theta^{\prime})k_{z^{\prime}}}{\omega}.$$ (4)

The symmetry axis is not vertical in tilted coordinates, so the angle between the direction of wave propagation and the symmetry axis is not the phase angle $\theta^{\prime}$, but $\theta^{\prime}-\varphi$, where $\varphi$ is the tilting angle of the tilted coordinates.. Therefore, VTI media in Cartesian coordinates become tilted TI media. The P-wave phase velocity in tilted coordinates can be expressed as  

$$\frac{V^2(\theta^{\prime},\varphi)}{V^2_{P0}}=1+\varepsilon\... ...{2(\varepsilon-\delta)\sin^2(2(\theta^{\prime}-\varphi))}{f} }.$$ (5)

 

vtitilt
Figure 1 VTI media in tilted coordinates. Left panel is a layer structure in Cartesian coordinates. Right panel is the layer structure in tilted coordinates.

Thus for a VTI medium in tilted coordinates we need a wave-extrapolation operator that can downward extrapolate the wavefield in tilted TI media. From equations (4) and (5), we can solve $k_{z^{\prime}}$ as a function of $\varphi, \varepsilon, \delta$, and $\omega/V_{P0}$. The wavefield can be extrapolated in two steps. First, the wavefield is extrapolated by an isotropic operator as follows:  

$$\bar{P}(z+\Delta z)=P(z)e^{ik_z^{iso}\Delta z},$$ (6)

where $k_z^{iso}=\sqrt{(\omega/V_{P0})^2-(k_x^{\prime})^2}$ is the isotropic vertical spatial wavenumber. This isotropic operator can be implemented by the split-step method Stoffa et al. (1990), the general screen propagator Huang and Wu (1996), or Fourier finite difference (FFD) Ristow and Ruhl (1994). Next, the wavefield is corrected by an explicit correction operator. This correction operator is designed in the Fourier domain, and is implemented in the space domain. The correction operator in the Fourier domain is a phase-shift operator:

$$P(z+\Delta z)=\bar{P}(z+\Delta z)e^{i(k_z^{\prime}-k_z^{iso})\Delta z}.$$ (7)

In the space domain, this correction operator is a convolution filter. The coefficients of the convolution filter depend on $\varphi, \varepsilon, \delta$, and $\omega/V_{P0}$, and they can be estimated by the weighted least-squares method Shan and Biondi (2004b).