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Challenge for anisotropy with lower symmetry

Unlike the well-behaved qP-wave mode, the qS-wave modes do not consistently polarize as a function of propagation direction, and thus cannot be designated as SV- and SH-waves, except in isotropic and TI media (Winterstein, 1990). To demonstrate the difficulties of extending the methodology in this paper to anisotropic media with symmetry lower than TI, we first compare the polarization features of qS-waves in typical TI and orthorhombic anisotropic rocks. Figure 9 shows polarizations of qS1- and qS2-waves in a VTI material - Mesaverde shale (Thomsen, 1986), which has the parameters $ v_{p0}=3.749$ km/s, $ v_{s0}=2.621$ km/s, $ \epsilon=0.225$ , $ \delta=0.078$ , and $ \gamma=0.100$ . The polarization directions are either horizontal or vertical (in the symmetry plane), so that we can definitely designate qS-waves as qSV- and SH-wave modes, except at the kiss singularity. Figure 10 shows polarizations of qS1- and qS2-waves in a ``standard" orthorhombic anisotropic material - vertically fractured shale (Schoenberg and Helbig, 1997), which has the parameters $ v_{p0}=2.437$ , $ v_{s0}=1.265$ km/s, $ \epsilon_1=0.329$ , $ \epsilon_2=0.258$ , $ \delta_1=0.083$ , $ \delta_2=-0.078$ , $ \delta_3=-0.106$ , $ \gamma_1=0.182$ and $ \gamma_2=0.0455$ . The qS-waves polarize in a very complicated way and have point singularities in many propagation directions.

Shear-wave modeling is complicated by the presence of the shear-wave singularities. As investigated by Crampin and Yedlin (1981), a TI material only has line and kiss singularities, while other anisotropic materials excluding those with triclinic symmetry (e.g., orthorhombic and monoclinic anisotropic materials) have point singularities in many propagation directions. Line singularities occur only at a fixed angle from the symmetry axis and cause no distortion of phase velocity surfaces or polarization phenomena. For kiss singularities (along the direction of symmetry axis), qS-wave polarizations vary rapidly in their vicinity but are well-behaved because there is no distortion in phase-velocity surfaces. These features facilitate the derivations of pseudo-pure-mode qSV-wave and pure-mode SH-wave equations for TI media. For directions near point singularities, however, the polarization of plane qS-waves changes very rapidly, and amplitudes and polarizations of qS-waves with curved wavefronts behave quite anomalously. Therefore, although pseudo-pure-mode qP-wave equations exist for general anisotropic media, it may be more confusing than helpful to extend the proposed pseudo-pure-mode qS-wave equations to symmetry systems lower than TI.

polar3dvtiS1 polar3dvtiS2
polar3dvtiS1,polar3dvtiS2
Figure 8.
Polarization vectors of 3D qS-waves in a VTI material (Mesaverde shale): (a) qS1-wave; (b) qS2-wave.
[pdf] [pdf] [png] [png] [matlab] [matlab]

polar3dortS1 polar3dortS2
polar3dortS1,polar3dortS2
Figure 9.
Polarization vectors of 3D qS-waves in an orthorhombic anisotropic material: (a) qS1-wave; (b) qS2-wave.
[pdf] [pdf] [png] [png] [matlab] [matlab]


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Next: Conclusions Up: Discussion Previous: Kinematic and dynamic accuracy

2016-10-14