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

Discussion

For the general anisotropic media, qP- and qS-wave modes are intrinsically coupled. The elastic wave equation must be solved at once to get correct kinematics and amplitudes for all modes. The scalar wavefields, however, are widely used with the help of wave mode separation or by using approximate equations derived from the elastic wave equation for many applications such as seismic imaging. As demonstrated in the theoretical parts, the pseudo-pure-mode wave equation is derived from the elastic wave equation through a similarity transformation to the Christoffel equation in the wavenumber domain. The components of the transformed wavefield essentially represent the spatial derivatives of the displacement wavefield components. This transformation preserves the kinematics of wave propagation but inevitablely changes the phases and amplitudes for qP- and qS-waves as the elastic wave mode separation procedure using divergence-like and curl-like operations (Yan and Sava, 2009; Zhang and McMechan, 2010; Dellinger and Etgen, 1990). The filtering step to correct the projection deviation is indispensable for complete removing the residual qS-waves from the extrapolated pseudo-pure-mode qP-wave fields. This procedure does not change the phases and amplitudes of the scalar qP-waves because the deviation operator is computed using the normalized wave and polarization vectors.

In fact, it is not even clear what the correct amplitudes should be for "scalar anisotropy". Like the anisotropic pseudo-analytic methods (Zhan et al., 2012; Song and Alkhalifah, 2013; Fomel et al., 2013; Etgen and Brandsberg-Dahl, 2009), the pseudo-pure-mode wave equation may distort the reflection, transmission and conversion coefficients of the elastic wavefields when there are high-frequency perturbations in the velocity model. Therefore, the converted qP-waves remaining in the separated qP-wave fileds only have reliable kinematics. What happens to the qP-wave's amplitudes and how to make use of the converted qP-waves (for seismic imaging) on the base of the pseudo-pure-mode qP-wave equation need further investigation in our future research.

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