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![]() | Elastic wave-mode separation for TTI media | ![]() |
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When constructing shear mode separators, one faces an additional complication: SV- and SH-waves have the same velocity along the symmetry axis of a 3D TI medium, and this singularity prevents one from obtaining polarization vectors for shear modes in this particular direction by solving the Christoffel equation (Tsvankin, 2005). In 3D TI media, the polarization of the shear modes around the singular directions are non-linear and cannot be characterized by a plane-wave solution. Consequently, constructing 3D global separators for fast and slow shear modes is difficult.
To mitigate the effects of the shear wave-mode singularity, I use
the mutual orthogonality among the P, SV, and SH modes depicted
in Figure 6. In this figure, vector
represents the
symmetry axis of a TTI medium, with
and
being the tilt
and azimuth of the symmetry axis, respectively. The wave vector
characterizes the propagation direction of a plane wave. Vectors
,
, and
symbolize the
compressional, and fast and slow shear polarization directions, respectively.
For TI media, plane waves propagate in symmetry planes, and the symmetry axis
and any wave vector
form a symmetry plane. For a plane
wave propagating in the direction
, the P-wave is polarized in
this symmetry plane and deviates from the vector
; the SV- and
SH-waves are polarized perpendicular to the P-mode, in and out of the
symmetry plane, respectively.
Using this mutual orthogonality among all three modes, I first
obtain the SH-wave polarization vector
by cross multiplying
vectors
and
, which ensures that the SH mode is
polarized orthogonal to symmetry planes:
Then I calculate the SV polarization vector
by
cross multiplying polarization vectors P and SH modes, which ensures
the orthogonality between SV and P modes and SV and SH modes:
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The components of the polarization vectors for P-, SV-, and SH-waves can be transformed back to the space domain to construct spatial filters for 3D heterogeneous TI media. For example, Figure 8 illustrates nine spatial filters transformed from the Cartesian components of the polarization vectors shown in Figure 7. All these filters can be spatially varying when the medium is heterogeneous. Therefore, in principle, wave-mode separation in 3D would perform well even for models that have complex structures and arbitrary tilts and azimuths of TI symmetry.
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polar3d
Figure 6. A schematic showing the elastic wave-modes polarization in a 3D TI medium. The three parallel planes represent the isotropy planes of the medium. The vector ![]() ![]() ![]() ![]() ![]() |
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polar3dP,polar3dS2,polar3dS1
Figure 7. The wave-mode polarization for P-, SH-, and SV-mode for a VTI medium with parameters ![]() ![]() ![]() ![]() |
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filters3d
Figure 8. The separation filters ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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![]() | Elastic wave-mode separation for TTI media | ![]() |
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