Calculation of Gaussian curvature of the slowness surface in monoclinic media

Abstract

In this paper, we obtain formulas that determine the type of the slowness surface in the vicinity of a given point (regular or singular) on the vertical axis in a monoclinic medium with a horizontal plane of symmetry. The study uses a method based on a cylindrical coordinate system with the origin at a selected point. In this coordinate system, at a fixed azimuth, the polynomial defining the equation of the slowness surface is expanded into a Taylor series with respect to the distance from the given point. Then, vertical projections of the slowness for different types of waves are found in the form of Taylor series with respect to the distance from the given point. The leading terms of these series determine the shape of the slowness surface in the vicinity of the given point. Gaussians, mean, and principal curvatures are also presented as Taylor series. In this paper, we investigate the leading terms of the Taylor series of Gaussians, mean, and principal curvatures, for regular, double, and triple singular points. It is shown that a double singular point is always a point of the tangential type, i.e., at the double singular point, the slowness surface has a horizontal tangent plane. However, at the singular point, the Gaussian curvature does not exist, and in the vicinity of this point, it depends on the azimuth. Cases with a double singular point, when the leading term of the Gaussian curvature is locally independent of the azimuth, are investigated. The cases are also investigated for which in the vicinity of the singular point or at certain azimuths, the slowness surfaces of S1 and S2 waves are located close to each other. The presented results are demonstrated on two examples of monoclinic media. The analysis can be used to identify the amplitude anomalies in the modelled wavefield in anisotropic media with singularity points, as well as in ray tracing, and solving inverse seismic problems for monoclinic media.