Fast algorithms for elastic-wave-mode separation and vector decomposition using low-rank approximation for anisotropic media

A 2D two-layer TI model

We first test our approach on a two-layer TI model with the size of $N_{x}=401\times401$ .The first layer is a VTI medium with $v_{p0}=2500 m/s$ , $v_{s0}=1200 m/s$ , $\epsilon=0.25$ , and $\delta=-0.25$ , and the second layer is a TTI medium with $v_{p0}=3600 m/s$ , $v_{s0}=1800 m/s$ , $\epsilon=0.2$ , $\delta=0.1$ , and the tilt angle $\theta=30^{\circ}$ . A point-source is placed at the center of this model. To aim for the relative accuracy, rank $N=M=2$ is required for both mode separation and vector decomposition. Thanks to small approximation errors in low-rank decompositions (see Figures 1 and Figure 3), we obtain good mode separation and vector decomposition for the synthesized elastic wavefields (see Figures 2 and 4). It took CPU time of 7.5 seconds to construct the separated forms (as equation 15 expressed) of the mode separation matrixes for qP- and qSV-waves. For one time step, it took CPU time of 0.12, 0.21, and 0.33 seconds to extrapolate, separate and decompose the elastic wavefields, repectively.

Fast algorithms for elastic-wave-mode separation and vector decomposition using low-rank approximation for anisotropic media