Simulating propagation of separated wave modes in general anisotropic media, Part II: qS-wave propagators

Removing of the residual qP-waves

The pseudo-pure-mode qSV-wave equations are derived by using a similarity transformation that projects the vector displacement wavefield onto the isotropic reference of the qSV-wave's polarization direction. As demonstrated in Figure 1b, even for a very strong VTI medium, the difference bewteen the two directions is generally quite small in most propagation directions. However, this difference does result in some qP-wave energy remaining in the pseudo-pure-mode scalar qSV-wave fields. To remove the residual qP-waves, we have to correct the projection deviations before summing the pseudo-pure-mode wavefield components. For heterogeneous VTI media, this can be implemented through nonstationary spatial filtering defined by the projection deviations (Cheng and Kang, 2014).

The filters can be constructed once the qSV-wave polarization directions are determined by solving the Christoffel equation based on local medium properties for every grid point. However, this operation is computationally expensive, especially in 3D heterogeneous TI media. We may further reduce the computational cost using a mixed-domain integral algorithm using a low-rank approximation (Cheng and Fomel, 2014). We shall observe in the examples that the residual qP-waves in the pseudo-pure-mode qSV-wave fields are quite weak, even if the anisotropy becomes strong. As explained in Cheng and Kang (2014), it is not necessary to apply the filtering at every time step for many applications, such as RTM.

In the case of transverse isotropy with a tilted symmetry axis (TTI), the elastic tensor loses its simple form and the terminology ``in-plane polarization" and ``cross-plane polarization" is to be preferred for qSV- and qSH-waves. The generalization of pseudo-pure-mode wave equation to a TTI medium involves no additional physics but greatly complicates the algebra. One strategy for deriving the wave equations is to locally rotate the coordinate system so that its third axis coincides with the symmetry axis; and to make use of the simple form of the wave equation in VTI media (see Cheng and Kang (2014)). Alternatively, we may use some new strategies to derive more numerically stable pseudo-pure-mode wave equations for TTI media with strong variations of parameters (Bube et al., 2012; Zhang et al., 2011). Moreover, the filter to correct the projection deviation can also be constructed with the coordinate rotation.

Simulating propagation of separated wave modes in general anisotropic media, Part II: qS-wave propagators