S-wave singularities in tilted orthorhombic media

Quasi S-wave propagation in low-symmetry anisotropic media is complicated due to the existence of point singularities (conical points) — points in the phase space at which slowness sheets of the split S-waves touch each other. At these points, two eigenvalues of the Christoffel tensor (associated with the quasi S-waves) degenerate into one and polarization directions of the S-waves, which lay in the plane orthogonal to the polarization of the quasi longitudinal wave, are not uniquely defined. In the vicinity of these points, slowness sheets of the S-waves have complicated shapes, leading to rapid variations in polarization directions, multipathing, and cusps and discontinuities of the shear wavefronts. In a tilted orthorhombic medium, the point singularities can occur close to the vertical, distorting the traveltime parameters that are defined at the zero offset. We have analyzed the influence of the singularities on these parameters by examining the derivatives of the slowness surface up to the fourth order. Using two orthorhombic numerical models of different shear anisotropy strength and with different number of singularity points, we evaluate the complexity of the slowness sheets in the vicinity of the conical points and analyze how the traveltime parameters are affected by the singularities. In particular, we observe that the hyperbolic region associated with the singularity points in a model with moderate to strong shear anisotropy spans over a big portion of the slowness surfaces and the traveltime parameters are strongly affected outside the hyperbolic region. In general, the fast shear mode is less affected by the singularities; however, the effect is still very pronounced. Moreover, the hyperbolic region associated with the singularity points on the slow S-wave affects the slowness surface of the fast mode extensively. In addition, we evaluate a relation between the slowness surface Gaussian curvature and the relative geometric spreading, which has anomalous behavior due to the singularities.

Moderate smoothness of most Alexandrov surfaces

We show that, in the sense of Baire categories, a typical Alexandrov surface with curvature bounded below by κ has no conical points. We use this result to prove that, on such a surface (unless it is flat), at a typical point, the lower and the upper Gaussian curvatures are equal to κ and ∞, respectively.