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| Simulating propagation of separated wave modes in general anisotropic media,
Part II: qS-wave propagators | |
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For the qSV-wave, we should essentially build a projection from the elastic wavefields
to a pseudo-pure-mode wavefield
.
Naturally, we may introduce the following similarity transformation to the Christoffel
matrix, i.e.,
|
(19) |
with the intuitive projection matrix
defined by the reference polarization direction
:
|
(20) |
or its normalized form.
However, the resulting pseudo-pure-mode wave equation is very complicated and contains mixed
derivatives of time and space. To keep them simple, an intermediate
wavefield
is defined by
|
(21) |
So we project the vector displacement wavefields using:
|
(22) |
with an intermediate projection matrix:
|
(23) |
Accordingly, we apply the similarity transformation using
to
equation 3 and finally get an equivalent Christoffel equation:
|
(24) |
with
.
For a locally smooth medium, applying an inverse Fourier transformation to
equation 24, we obtain another coupled
forth-order linear system:
|
(25) |
or in its extended form:
|
(26) |
where
is an intermediate wavefield in the time-space domain, and
represents the corresponding Christoffel differential-operator matrix after the similarity
transformation.
The intermediate wavefield has the same horizontal components but a different vertical
component of the pseudo-pure-mode wavefield
.
Equation 21 indicates that the vertical component satisfies:
|
(27) |
Due to the symmetry property of a VTI material, we may sum the horizontal components and replace the vertical component
with the relation given in equation 27, and finally obtain a simpler second-order system that honors the kinematics of both qP- and qSV-waves:
|
(28) |
with
.
Note that pure SH-waves always polarize in the planes perpendicular to the symmetry axis with the polarization direction indicated by
,
which implies
, i.e.,
, for the SH-wave.
Therefore, the partial summation (after the similarity transformation) automatically removes the SH component from the transformed wavefields.
As a result, there are no terms related to
any more in equation 28.
In order to produce a pseudo-pure-mode scalar qSV-wave field, we sum all components of the transformed wavefields, namely
|
(29) |
For a 2-D VTI medium, equation 28 reduces to the following form:
|
(30) |
In fact, we can derive the same pseudo-pure-mode wave equation for a 2-D qSV-wave by
projecting the 2-D Christoffel matrix onto a reference vector
.
Similarly, a 2D pseudo-pure-mode scalar qSV-wave field is given by the summation:
.
If we apply the isotropic assumption by setting
and
,
and sum the two equations in equation 28, we get the scalar wave equation:
|
(31) |
with
representing a shear wave field, and
with
representing the propagation velocity of the isotropic shear wave.
The derived pseudo-pure-mode qSV-wave equations have some interesting and valuable features.
First, the projection using matrix
yields wave-mode separation to some
extent, because the chosen projection direction,
, represents the polarization direction of
the SV-wave in an isotropic medium.
As investigated by Tsvankin and Chesnokov (1990) and Psencik and Gajewski (1998), and also demonstrated in Figure 1b,
the difference in polarization directions between isotropic and VTI media is
generally quite small in most propagation directions for SV-waves.
Therefore, considering the mode separator (namely equation 9) and the small projection deviation,
summing all the pseudo-pure-mode wavefield components in equation 28 or 30
partially achieves wave-mode separation and produces a scalar wavefield dominated by
the energy of qSV-waves. This will be demonstrated in the numerical examples.
Second, the pseudo-pure-mode wave equations are easier to calculate
than the original elastic wave equation because they have no terms of mixed partial
derivatives. More importantly, the summation of the horizontal components further simplifies the
wave equations and reduces the number of parameters needed for scalar qSV-wave extrapolation.
These features are undoubtedly useful for performing multicomponent seismic imaging and
inversion that mainly use wavefield kinematics when it is necessary to include anisotropy.
|
|
|
| Simulating propagation of separated wave modes in general anisotropic media,
Part II: qS-wave propagators | |
|
Next: Removing of the residual
Up: Pseudo-pure-mode qSV-wave equation
Previous: Pseudo-pure-mode qSV-wave equation
2016-10-14