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

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## Adjustment to the pseudo-spectral solution

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:

 (9)

where 's are the eigenvalues and 's are the eigenvectors of , with . The three eigenvalues correspond to phase velocities of the three wave modes with (in which , and represents the phase velocity) representing the circular frequency, and the corresponding eigenvector represents the normalized polarization vector for the given mode. An alternative form of the above decomposition yields:

 (10)

with

 (11)

 (12)

Note is an orthogonal matrix, i.e., .

The eigenvalues represent the frequencies and satisfy the condition given by,

 (13)

in which is the maximum frequency of the source. Therefore, we suggest to filter out the high-wavenumber components in the wavefields beyond ( is the minimum phase velocity in the computational model) caused by the numerical errors to enhance numerical stability.

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.,

 (14)

Thus this adjustment inserts a modified Christoffel matrix, i.e., , into the original pseudo-spectral formula on the basis of equations 6 and 8. Note that the -space adjustment to the pseudo-spectral solution has been widely used in acoustic and ultrasound (Tabei et al., 2002; Bojarski, 1982) and elastic isotropic wavefield simulation (Firouzi et al., 2012; Liu, 1995).

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

Next: Propagating decoupled elastic wavefields Up: Propagating coupled elastic wavefields Previous: Pseudo-spectral solution of the

2016-11-21