Simulating propagation of separated wave modes in general anisotropic media, Part II: qS-wave propagators |
Finally, we demonstrate pseudo-pure-mode qSV-wave propagation in the 2D BP TTI model (see Figure 6). The space grid size is 12.5 m and the time step is 1 ms for high-order finite-difference operators. Here the vertical velocities for the qSV-wave are set to half of the qP-wave velocities. Figure 7 displays snapshots of wavefield components at the time of 1.4s synthesized by using the original elastic wave equation and the pseudo-pure-mode qSV-wave equation. In the elastic wavefields, we observe strong scattered and mode-converted energy in the region with a rapidly varying anisotropic symmetry axis direction. For comparison, in Figures 7c and 7d, we also show the separated qP- and qSV-wave scalar fields obtained using the approach proposed by Cheng and Fomel (2014). Not that in the pseudo-pure-mode qSV-wave fields (see Figure 7g), the incident qP-waves as well as scattered and converted qP-waves are effectively suppressed. The spatial filtering appears to remove residual qP-waves and accurately separates qSV-wave data (including the converted qP-qSV waves) from the pseudo-pure-mode wavefields in this complex model (Figure 7h). Vertical slices through the scalar fields (Figure 8) provide further proof to evaluate the performance of the proposed qS-wave propagators. As we observed, in heterogeneous rough zones with strong variations in tilt angle, there are differences between the elastic and pseudo-pure-mode qSV-wave fields. Fortunately, the pseudo-pure-mode qSV-wave equation still captures the shear wave kinematics to a great extent. For a single time-step, it respectively takes CPU times of 2.71 and 1.22 seconds to extrapolate the elastic and pseudo-pure-mode qSV-wave fields, and about 7.50 seconds to separate the qSV-wave fields from both wavefields using low-rank approximate mixed-domain integral operations based on the qSV-wave's polarization directions (Cheng and Fomel, 2014).
vp0,epsi,del,the
Figure 6. Partial region of the 2D BP TTI model: (a) vertical qP-wave velocity, Thomsen coefficients (b) and (c) , and (d) the tilt angle . |
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Elasticx,Elasticz,ElasticSepP,ElasticSepSV,PseudoPureSVx,PseudoPureSVz,PseudoPureSV,PseudoPureSepSV
Figure 7. Synthesized wavefield snapshots on BP 2007 TTI model using original elastic wave equation and pseudo-pure-mode qSV-wave equation respectively: (a) x- and (b) z-components synthesized by elastic wave equation; (c) scalar qP- and (d) scalar qSV-wave fields separated from the elastic wavefield; (e) x- and (f) z-components synthesized by pseudo-pure-mode qSV-wave equation; (g) pseudo-pure-mode scalar qSV-wave field and (h) pure-mode scalar qSV-wave field separated from the pseudo-pure-mode qSV-wave field. |
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Simulating propagation of separated wave modes in general anisotropic media, Part II: qS-wave propagators |