


 Simulating propagation of separated wave modes in general anisotropic media, Part I: qPwave propagators  

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We have proposed an alternative approach
to simulate propagation of separated wave modes in general anisotropic media.
The key idea is splitting the onestep wave mode separation into two cascaded steps based on the following
observations: First, the Christoffel equation derived from the original elastic wave equation
accurately represents the kinematics of all wave modes; Second, various coupled
secondorder wave equations can be derived from the Christoffel equation
through similarity transformations. Third, wave mode separation can be achieved by projecting the original
elastic wavefields onto the given mode's polarization directions, which are usually calculated
based on the local material parameters using the Christoffel equation.
Accordingly, we have derived the pseudopuremode qPwave equation by applying a similarity transformation
aiming to project the elastic wavefield onto the wave vector, which
is the isotropic references of qPwave polarization for an anisotropic medium.
The derived pseudopuremode equations not only describe propagation of all wave modes
but also implicitly achieve partial mode separation once the wavefield components are summed.
As shown in the examples,
the scalar pseudopuremode qPwave fields
are dominated by qPwaves while the residual
qSwaves are weaker in energy, because the projection deviations of qPwaves are generally far less than those of the qSVwaves.
Synthetic example of Hess VTI model demonstrates successful application of the pseudopuremode qPwave
equation to RTM for conventional seismic exploration.
To completely remove the residual qSwaves, a filtering step has been proposed
to correct the projection
deviations resulting from the difference between polarization direction and its isotropic reference.
In homogeneous media, it can be efficiently implemented by applying wavenumber domain filtering to each
wavefield component. In heterogeneous media, nonstationary spatial filtering using
pseudoderivative operators are applied to finish the second step for wave mode separation.
In a word, pseudopuremode wave equations plus corrections of projection deviations provide us an efficient
and flexible tool to simulate propagation of separated wave modes in anisotropic media.
In spite of the amplitude properties, this approach has some advantages over the classical solution
combining elastic wavefield extrapolation and wave mode separation:
First, the pseudopuremode wave equations could be directly used for migration of seismic data recorded
with singlecomponent geophones without computationally expensive wave mode separation (as shown in the last example).
Second, because partial wave mode separation is automatically achieved during wavefield extrapolation and the correction step
to remove the residual qSwaves is optional depending on the strength of anisotropy,
our approach provides better flexibilty for seismic modeling,
migration and parameter inversion in practice;
Third, computational cost is reduced at least one third for the 2D cases
if the finite difference algorithms are used thanks to the simpler structure of pseudopuremode wave equations
(i.e., having no mixed derivative terms for VTI and vertically orthorhombic media).
For the 3D TI media, computational cost is further reduced about one third because two instead of three
equations are used to simulate wave propagation.
Unlike the pseudoacoustic wave equations, pseudopuremode wave equations have no approximation in
kinematics and allow for
provided that the stiffness tensor is positivedefinite.
Moreover, they provide a possibility to extract artifactfree separated wave mode
during wavefield extrapolation.
Although we focus on propagation of separated qPwaves using the pseudopuremode qPwave equation,
our approach also works for qSwaves in TI media. This will be demonstrated in the second paper of this series.



 Simulating propagation of separated wave modes in general anisotropic media, Part I: qPwave propagators  

Next: ACKNOWLEDGMENTS
Up: Cheng & Kang: Propagation
Previous: Discussion
20140624