Elastic wave-mode separation for TTI media |

The notations in this equation have the same definitions as in equation 2. For TTI media, the matrix has the elements

(14) | |||

(15) | |||

(16) | |||

(17) | |||

(18) | |||

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:

Here, the magnitude of the P-wave polarization vectors for a certain wavenumber is a constant:

(21) |

This ensures that for a certain wavenumber, P-waves obtained by projecting the elastic wavefields onto the polarization vectors are uniformly scaled. For comparison, the magnitudes of all three modes are respectively

(22) | |||

(23) | |||

(24) |

where is the polar angle of the propagating plane wave, i.e., the angle between vectors and . Figure 7 shows the polarization vectors of P-, SH-, and SV-modes computed using equations 13, 20, and 21, respectively. The P-wave polarization vectors in Figure 7(a) all have the same magnitude, but the SV and SH polarization vectors in Figures 7(c) and fig:polar3dS2 vary in magnitude. In the symmetry axis direction, they become zero. The zero amplitude of the shear modes in the symmetry axis direction is not an abrupt but a continuous change over nearby propagation angles. Using separators represented by solutions to equation 13 and expressions 20 and 21 to filter the wavefields, I obtain separated shear modes that are scaled differently than the P-mode. For a certain wavenumber, the shear modes are scaled by , with being the polar angle, which increases from zero in the symmetry axis to unity in the orthogonal propagation directions. Therefore, the separated SV- and SH-waves have zero amplitude in the symmetry axis direction, and the amplitudes of the shear modes are just kinematically correct.

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.

polar3d
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
represents the
symmetry axis, which is orthogonal to the isotropy plane. The vector
is the propagation direction of a plane wave. The wave-modes P, SV,
and SH are polarized in the direction
,
,
and
, respectively. The three modes are polarized orthogonal
to each other.
Figure 6. |
---|

polar3dP,polar3dS2,polar3dS1
The wave-mode polarization for P-, SH-, and SV-mode for a VTI medium
with parameters
km/s,
km/s,
, and
. The P-mode polarization is computed
using the 3D Christoffel equation, and SV and SH polarizations are
computed using Equations 21 and 20. Note that the SV- and
SH-wave polarization vectors have zero amplitude in the vertical
direction.
Figure 7. |
---|

filters3d
The separation
filters
,
, and
for the P, SV, and SH modes for a VTI
medium. The corresponding wavenumber-domain polarization vectors are
shown in Figure 7. Note that the filter
for the SH mode is blank because the
component of the
polarization vector is zero. The zoomed views show
samples out
of the original
samples around the center of the filters.
Figure 8. |
---|

Elastic wave-mode separation for TTI media |

2013-08-29