Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media |

in which , and represent the three components of the elastic wavefields in wavenumber-domain at the time of .

For a VTI or orthorhombic medium, we express the stiffness tensor as a Voigt matrix:

(27) |

in which there are only five independent coefficient with , , and , for a VTI medium. Therefore, the propagation matrix has the following extended formulation,

If the principal axes of the medium are not aligned with the Cartesian axes, e.g., for the tilted TI and orthorhombic media, we should apply the Bond transformation (Carcione, 2007; Winterstein, 1990) to get the stiffness matrix under the Cartesian system. This will introduce more mixed partial derivative terms in the wave equation, which demands lots of computational effort if a finite-difference algorithm is used to extrapolate the wavefields. Fortunately, for the pseudo-spectral solution, it only introduces negligible computation to prepare the propagation matrix and no extra computation for the wavefield extrapolation.

Similarly, we can write the propagation matrix (in equation 20) for the decoupled elastic waves in its extended formulation:

in which with represents the scalar qP-, qS - and qS -wave fields. So these wavefields satisfy the same scalar wave equation

The standard leapfrog scheme for this equation can be expressed as

It is well known that this solution is limited to small time-steps for stable wave propagation.

Fortunately, there is an exact time-steping solution for the second-order time derivatives allowing for any size of time-steps for homogeneous medium (Cox et al., 2007; Etgen and Brandsberg-Dahl, 2009), namely:

Comparing equations C-3 and C-4 shows that, it is possible to extend the length of time-step without reducing the accuracy by replacing with . This opens up a possibility by replacing with as a -space adjustment to the spatial derivatives, which may convert the time-stepping pseudo-spectral solution into an exact one for homogeneous media, and stable for larger time-steps (for a given level of accuracy) in heterogeneous media (Bojarski, 1982).

Nowadays, the -space scheme is widely used to improve the approximation of the temporal derivative in acoustic and ultrasound (Fang et al., 2014; Cox et al., 2007; Tabei et al., 2002). As far as we know, Liu (1995) was the first to apply -space ideas to elastic wave problems. He derived a -space form of the dyadic Green's function for the second-order wave equation and used it to calculate the scattered field iteratively in a Born series. Firouzi et al. (2012) proposed a -space scheme on the base of the first-order elastic wave equation for isotropic media.

Accordingly, we apply the -space adjustment to improve the performance of our two-step time-marching pseudo-spectral solution of the anisotropic elastic wave equation. To propagate the elastic waves on the base of equations 6 and 8, we need modify the eigenvalues of Christoffel matrix as in Equation 14.

Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media |

2016-11-21