TTI decoupled equations

The dispersion relations 3.4 hold for VTI media. The same dispersion relations for TTI media can be deduced from equations 3.4 through variable exchanges (here I choose $F=1$ , this simplification only affects the resulting amplitudes in weak anisotropic media, i.e., $\epsilon$ is small.)

$$\left\{ \begin{array}{ll} \omega^2=V^2_{p_0} \bigg[ (1+2\epsilon)... ...hat{k}^2_r+\hat{k}^2_z} \bigg] \;\;\;\;\;\mbox{(SV wave)} \end{array} \right. ,$$ (36)

where $\hat{k}^2_r=\hat{k}^2_x+\hat{k}^2_y$ , $\hat{k}_x$ , $\hat{k}_y$ and $\hat{k}_z$ are spatial wavenumbers in the rotated coordinate system

$$\left(\begin{matrix}\hat{k}_x \cr \hat{k}_y \cr \hat{k}_z \end{ma... ...end{matrix}\right) \left(\begin{matrix}k_x \cr k_y \cr k_z \end{matrix}\right).$$ (37)

Using a square root approximation, Chu et al. (2011) obtained an approximate dispersion relation for pure P wave in acoustic TTI media which shares the same expression as the P wave dispersion relation in equations 3.6, but his derivation is based on Liu's (2009) VTI decoupled wave equations.

From equation 3.7, we have

$$\left\{ \begin{array}{ll} \hat{k}^2_r = k^2_r - \sin^2\theta (\co... ...~~~~~~ - \sin2\theta(\cos\phi\;k_xk_z + \sin\phi\;k_yk_z) \end{array} \right. .$$ (38)

We also have

$$\hat{k}^2_r + \hat{k}^2_z = k^2_r + k^2_z.$$ (39)

Substituting equations 3.8 and 3.9 into equations 3.6, and after some algebraic manipulations, one can formulate the approximated P wave and SV wave dispersion relations for TTI media. Below is the 2D version

$$\left\{ \begin{array}{ll} \omega^2 = V^2_{p_0}\bigg\{\;k^2_x+k^2... ...^2_z}{k^2_x+k^2_z} \bigg] \;\bigg\}\;\; \mbox{(SV wave)} \end{array} \right. .$$ (310)

Multiplying both sides of equations 3.10 by the wavefield function $p(\omega,k_x,k_z)$ in the Fourier domain, followed by an inverse Fourier transform to both sides and then using the relation $i\omega\leftrightarrow \partial / \partial t$ , I finally derive the decoupled P wave and SV wave equations in the time-wavenumber domain for 2D TTI media

$$\left\{ \begin{array}{ll} \displaystyle\frac{1}{V^2_{p_0}} \disp... ...{k^2_x+k^2_z} \bigg] \;\bigg\} P_{\scriptscriptstyle SV} \end{array} \right. .$$ (311)

Even though my derivation started from a different point based on VTI decoupled equations, the resulting P wave equation is as same as equations derived by Du et al. (2005) and Zhang et al. (2005). However, because I used a rotation matrix, I can easily extend the results to the 3D case for both the decoupled P and SV wave modes (see Appendix A). In addition, the form is general enough to contain the special cases of VTI, HTI (transversely isotropic with a horizontal symmetry axis) and tilted elliptical anisotropy. In Appendix A, I show how the general form reduces to the special cases of HTI and tilted elliptical anisotropy. Because of the appearance of the wavenumbers in the denominators of the terms in all of these cases, they are best implemented with the pseudospectral and REM methods (see Appendix B) described below.