TTI pure P-wave equation

A similar expression for TTI media can be deduced from equation 4.4 through variable exchanges (Zhan et al., 2012)

$$-L^2 = v_v^2 k_{\tz }^2 + v_h^2 k_{\tr }^2 + \bigg( v_n^2 - v_h^2 \bigg) \frac{k_{\tr }^2 k_{\tz }^2}{k_{\rho}^2},$$ (45)

where $k_{\tr }^2=k_{\tx }^2 + k_{\ty }^2$ ; $k_{\tx }$ , $k_{\ty }$ and $k_{\tz }$ are spatial wavenumbers in the rotated coordinate system

$$\begin{bmatrix} k_{\tx } \\ k_{\ty } \\ k_{\tz } \end{bmatr... ...ta \end{bmatrix} \begin{bmatrix} k_{x} \\ k_{y} \\ k_{z} \end{bmatrix}.$$ (46)

Here $\theta$ and $\phi$ are dip and azimuth, and the following relation hold

$$k_{\tx }^2+k_{\ty }^2+k_{\tz }^2=k_x^2 + k_y^2 + k_z^2.$$ (47)

In the case of elliptical anisotropy where $\varepsilon = \delta$ (i.e., $v_h=v_n$ ), the last term of equation 4.5 with wavenumbers in the denominator disappears. Therefore, the first two terms in equation 4.5 represent the properties of elliptical anisotropy, while the last term compensates for anelliptical anisotropic effects due to rotation of the symmetry axis.

According to the rotation matrix 4.6, and denoting $\sin\theta \cos\phi=\Gamma_x$ , $\sin\theta \sin\phi=\Gamma_y$ , $-\cos\theta =\Gamma_z$ , we can rewrite $k_{\tz }$ in the rotated system in terms of $k_x$ , $k_y$ and $k_z$

$$k_{\tz }=-\bigg( k_x \sin\theta \cos\phi + k_y \sin\theta \sin\phi - k_z \cos\theta \bigg) =-\bigg( k_x \Gamma_x+k_y \Gamma_y+k_z \Gamma_z \bigg).$$ (48)

Hence the three wavenumber terms in equation 4.5 can be computed in the following order

where $\Gamma_{ij}=\Gamma_i\Gamma_j$ ; $V$ , $H$ and $T$ are differential operators in the wavenumber domain that operate along the symmetry axis direction, the symmetry plane perpendicular to the symmetry axis, and the tilted direction, respectively.