Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation

TTI Lowrank Finite Differences

The LFD approach is not limited to the isotropic case. In the case of transversely isotropic (TTI) media, the term $v(\mathbf{x})\,\vert\mathbf{k}\vert$ on the right-hand side of equation 6, can be replaced with the acoustic approximation (Fomel, 2004; Alkhalifah, 1998,2000),

$$\begin{array}{*{20}l} f(\mathbf{v},\mathbf{\hat{k}},\eta)=\;... ...{1+2\eta}v_1^2v_2^2\,\hat{k}_1^2\,\hat{k_2^2}}}\;, \end{array}$$ (17)

where $v_1$ is the P-wave phase velocity in the symmetry plane, $v_2$ is the P-wave phase velocity in the direction normal to the symmetry plane, $\eta$ is the anisotropic elastic parameter (Alkhalifah and Tsvankin, 1995) related to Thomsen's elastic parameters $\epsilon$ and $\delta$ (Thomsen, 1986) by

$$\frac{1+2\delta}{1+2\epsilon}=\frac{1}{1+2\eta}\,;$$ (18)

and $\hat{k}_1$ and $\hat{k}_2$ stand for the wavenumbers evaluated in a rotated coordinate system aligned with the symmetry axis:

$$\begin{array}{*{20}c} \hat{k}_1=k_1\cos{\theta}+k_2\sin{\the... ... \\ \hat{k}_2=-k_1\sin{\theta}+k_2\cos{\theta}\; \end{array}$$ (19)

where $\theta$ is the tilt angle measured with respect to vertical. Using these definitions, we develop a version of the lowrank finite-difference scheme for 2D TTI media.

Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation