Group velocity image of singularities for converted reflected and transmitted waves in orthorhombic anisotropic media

Abstract

We define double singularity points, lines and surfaces and their group velocity images for converted reflected and transmitted PS1 and PS2 waves in layered orthorhombic media. This is performed by considering the dominant term in Taylor series for Christoffel equation defined for converted waves. We show that the group velocity image for converted wave has the same functional form as the one for pure wave modes (ellipse in 3D). The position and area of the ellipse in the group velocity domain and offset domain are also analysed. We also expand this method for two-layer model with S1S2 wave singularity in both layers. It results in quartic (S1S1, S2S2, S1S2 and S2S1 waves) characteristic equation. In the group velocity domain, it gives two close curves: quasi-elliptical (for pure S1S1 and S2S2 waves) and very complex curve with deflection points (for converted S1S2 and S2S1 waves).