Singularity points in multi-layered anisotropic medium

The slowness surfaces for P, S1 and S2 waves in anisotropic medium are defined by solving the Christoffel equation. The regular point on the slowness surface can be mapped on corresponding group velocity surface. The irregular (singularity) point on the slowness surface results in the plane curve in the group velocity domain (Stovas et al., 2024).

The characteristic equations for double and triple singularity points define the tangent cone of second and third order, respectively. If the plane wave passes through singularity points in some layers of multilayered model, the effective characteristic equation has order given by product of orders of characteristic equations from individual layers. Therefore, the order of effective characteristic equation can be computed as N=2^K3^L, where K and L are the number of layers with double and triple singularity points, respectively. The effective characteristic polynomial F_N(Δp₁,Δp₂,Δp₃)  (the N-th order tangential cone) for multi-layered model can be computed by resultant of individual characteristic polynomials,where Δp_j,j=1,2,3, are the increments in slowness projections.The number of individual branches is given by J=Floor[(N+1)/2]. The dual curve Φ_M (V₁,V₂,V₃)=0 is the group velocity umage of Nth-order singularity point, where M=N(N-1)-2n-3c (Quine, 1982), with n and c being respectively the number of nodes and cusps for curve F_N=0. It is shown that the tangential cone does not have cusps (c=0) but can have nodes if N≥6. The inflection points and bitangents for curve F_N=0 respectively result in cusps and nodes for dual curve Φ_M=0. The cusps affect the Gaussian curvature computed in vicinity of singularity point (Stovas et al., 2025). The irregularities in phase and group domain are illustrated in Figure by dots for two-layer model with double singularity points in both layers (N=4, J=2, n=2 and M=8).

Figure. Two-layer model with double singularity points. a) Curve F₄=0 in affine plane (phase domain). Four inflection points on converted wave branch are shown by black dots. Two bitangents are shown by dotted lines limited by gray dots. b) Group velocity image (Φ₈=0) of quartic singularity point (dual curve). Four cusps are shown by black dots, and two nodes are shown by gray dots. Solid and dashed lines stand for pure wave modes (S1S1 and S2S2) and converted (S1S2 and S2S1) waves, respectively.

References

Stovas, A., Roganov, Yu., and V. Roganov, 2024, Singularity points and their degeneracies in anisotropic media, Geophysical Journal International 238 (2), 881- 901.

Stovas, A., Roganov, Yu., and V. Roganov, 2025, Gaussian curvature of the slowness surface in vicinity of singularity point in anisotropic media, Geophysical Journal International 240 (3), 1917-1934.                                                                    

Quine, J.R., 1982, A Plücker equation for curves in real projective space, Proceedings of the American Mathematical Society, 85, no.1, 103-107.