By applying two different similarity transformations to the original Christoffel equation, which aim to project the vector displacement
wavefields onto the isotropic SV- and SH-waves' polarization directions, we have derived the pseudo-pure-mode qSV-wave equation
and the pure-mode SH-wave equation for 2D and 3D heterogeneous TI media, respectively.
These equations are simpler than the original elastic wave equation and involve less material parameters, which reduces computational
cost at least by half if the finite-difference scheme is used in practice.
The theoretical analysis and numerical examples have demonstrated that, the pseudo-pure-mode
qS-wave propagators for TI media have the following features:
First, the qSV-wave equations honor the kinematics for both qP and qSV modes, while the pure-mode SH-wave equation guarantees
the kinematics for the scalar SH-wave.
Second, although qP-waves still remain in the pseudo-pure-mode qSV-wave fields, their horizontal and vertical components have almost
opposite polarities in most propagation directions. As a result, the summation of all components
produces a pseudo-pure-mode scalar qSV-wave field with very weak qP-wave energy.
Third, the non-SH parts in the pseudo-pure-mode vector SH-wave field have completely opposite polarities,
and thus are thoroughly removed from the scalar SH-wave field once all components are summed.
In addition,
a filtering step taking into account the polarization deviation can be used to thoroughly remove the residual qP-waves
for pseudo-pure-mode scalar qSV-wave extrapolation.
These features indicate the potential of the proposed qS-wave propagators for developing promising seismic imaging and
inversion algorithms in heterogeneous TI media.
Like the pseudo-acoustic or pseudo-pure-mode qP-wave equations,
the proposed pseudo-pure-mode qS-wave equations take into account ``scalar anisotropy" and may distort the dynamic elasticity
of the real anisotropic media.
Simulating propagation of separated wave modes in general anisotropic media,
Part II: qS-wave propagators