Various methods for modeling anisotropic seismic waves
have been proposed
and developed, especially for vertical transversely isotropic (VTI)
and tilted transversely isotropic (TTI) media.
In general, we can divide those methods into two broad categories:
methods that suffered from shear-wave artifacts, usually known as coupled equations where P and S-wave are coupled together
(Fowler et al., 2010; Duveneck and Bakker, 2011; Alkhalifah, 2000; Zhou et al., 2006a; Fletcher et al., 2009; Zhou et al., 2006b);
and decoupled (or pure) P-wave equations which are free of shear-wave artifacts
(Zhan et al., 2012; Chu et al., 2011; Liu et al., 2009; Etgen and Brandsberg-Dahl, 2009; Pestana et al., 2012).
Each has its pros and cons as summarized in Table 4.1.
| Category |
# of eqn. to solve |
method of solution |
S-wave artifacts |
numerical stability with variable angle |
cost |
|
FD |
w/ |
not stable |
efficient | ||
| Coupled Eqn. |
2 |
PS |
w/ |
not stable |
intensive |
| Decoupled Eqn. |
1 |
PS |
w/o |
more stable |
intensive |
The pure P-wave equation in the system of decoupled equations of Zhan et al. (2012) is in the wavenumber domain, and at each time step it requires eight fast Fourier transforms (FFTs) for 2D, and twenty-two FFTs for 3D. This imposes an unrealistic demand for practical migration of large-scale 3D field seismic data sets. In the work presented below, I follow the same derivations of Pestana et al. (2012) and Zhan et al. (2012), but reorganize and rewrite the wavenumber domain equation in a compact way for efficient computation. After some algebraic manipulations, it still requires eight FFTs per time step for 2D computation in the new formulation, but the number of FFTs needed per time step for 3D is reduced from twenty-two to fourteen.
To further reduce the computational cost introduced by numerous FFTs, I propose a new hybrid pseudospectral and finite-difference scheme to evaluate the equation by using the relation between the spatial derivative and the operator in the wavenumber domain. Both 2D and 3D reverse-time migration (RTM) examples with the new hybrid algorithm are tested and demonstrated to validate the uplift in efficiency.