Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media |
Generally, the two-step time-marching pseudo-spectral solution is limited to a small time-step, as larger time-steps lead to numerical dispersion and stability issues. At more computational costs, high-order finite-difference (Dablain, 1986) can be applied to address this difficulty. As an alternative to second-order temporal differencing, a time integration technique based on rapid expansion method (REM) can provide higher accuracy with less computational efforts (Kosloff et al., 1989). As Du et al. (2014) demonstrated, one-step time marching schemes (Zhang and Zhang, 2009; Sun and Fomel, 2013), especially using optimized polynomial expansion, usually give more accurate approximations to heterogeneous extrapolators for larger time-steps. In this section, we discuss a strategy to extend the time-step for the previous two-step time-marching pseudo-spectral scheme according to the eigenvalue decomposition of the Christoffel matrix.
Since the Christoffel matrix is symmetric positive definite, it has a unique eigen-decomposition of the form:
The eigenvalues represent the frequencies and satisfy the condition given by,
According to above eigen-decomposition, we apply the -space adjustment to our pseudo-spectral scheme by modifying the eigenvalues of Christoffel matrix for the anisotropic elastic wave equation (see Appendix C), i.e.,
Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media |