Elastic wave-mode separation for VTI media |
I then consider a 2D anisotropic model similar to the previous model shown in Figures 10(a)-fig:separate3-ro (with , representing the vertical P and S wave velocities), and additionally characterized by the parameters and shown in Figures 10(d) and fig:separate3-delta, respectively. The parameters and vary gradually from top to bottom and left to right, respectively. The upper left part of the medium is isotropic and the lower right part is highly anisotropic. Since the difference of and is great at the bottom part of the model, the qS waves in this region are severely triplicated due to this strong anisotropy.
Figure 12 illustrates the pseudo derivative operators obtained at different locations in the model defined by the intersections of coordinates 0.3, 0.6, 0.9 km and coordinates 0.3, 0.6, 0.9 km. Since the operators correspond to different combination of the parameters and , they have different forms. The isotropic operator at coordinates km and km, shown in Figure 12(a), is purely vertical and horizontal, while the anisotropic operators (Figure 12(b) to fig:separate3-aop22) have ``tails'' radiating from the center. The operators become larger at locations where the medium is more anisotropic, for example, at coordinates km and km.
Figure 13(a) shows the vertical and horizontal components of one snapshot of the simulated elastic anisotropic wavefield, Figure 13(b) shows the separation to qP and qS modes using conventional isotropic and operators, and Figure 13(c) shows the mode separation obtained using the pseudo operators constructed using the local medium parameters. A comparison of Figure 13(b) and 13(c) indicates that the spatially-varying derivative operators successfully separate the elastic wavefields into qP and qS modes, while the and operators only work in the isotropic region of the model.
aoppos,vs,ro,epsilon,delta
Figure 10. A km km model with parameters (a) km/s except for a low velocity Gaussian anomaly around km and km, (b) km/s except for a low velocity Gaussian anomaly around km and km, (c) g/cm in the top layer and g/cm in the bottom layer, (d) smoothly varying from 0 to from top to bottom, (e) smoothly varying from 0 to from left to right. A vertical point force source is located at km and km shown by the dot in panels (b), (c), (d), and (e). The dots in panel (a) correspond to the locations of the anisotropic operators shown in Figure 12 . |
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uI-f06,qI-f06,pI-f06
Figure 11. (a) One snapshot of the isotropic wavefield modeled with a vertical point force source at =0.6 km and =0.6 km for the model shown in Figure 10, (b) isotropic P and S wave modes separated using and , and (c) isotropic P and S wave modes separated using pseudo derivative operators. Both (b) and (c) show good separation results. |
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aop00,aop10,aop20,aop01,aop12,aop21,aop02,aop12,aop22
Figure 12. The order anisotropic pseudo derivative operators in the and directions at the intersections of =0.3, 0.6, 0.9 km and =0.3, 0.6, 0.9 km for the model shown in Figure 10. |
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uA-f06,qA-f06,pA-f06
Figure 13. (a) One snapshot of the anisotropic wavefield modeled with a vertical point force source at =0.6 km and =0.6 km for the model shown in Figure 10, (b) anisotropic qP and qS modes separated using and , and (c) anisotropic qP and qS modes separated using pseudo derivative operators. The separation of wavefields into qP and qS modes in (b) is not complete, which is obvious at places such as at coordinates km km. In contrast, the separation in (c) is much better, because the correct anisotropic derivative operators are used.. |
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Elastic wave-mode separation for VTI media |