Properties of S waves near a kiss singularity: a comparison of exact and ray solutions

We have studied the properties of S waves generated by a point source in a homogeneous, transversely isotropic, elastic medium, propagating in directions close to that of a kiss singularity, which coincides with the symmetry axis of the medium. We have proved analytically as well as numerically that the ray solution can describe the S waves correctly far from the source in all directions, including that of the kiss singularity. We have found that, in contrast to the far-field P wave, which can be reproduced satisfactorily by the zeroth-order ray approximation in all directions from the source, the far-field S waves can be reproduced satisfactorily by the zeroth-order ray approximation only in directions far from the kiss singularity. In directions near the kiss singularity, the first-order ray approximation must also be considered, because the zeroth- order ray approximation yields distorted results. The first-order ray approximation can be of high frequency and can be detected in the far field.     

Simple analytical Green's functions for ray perturbations in layered media

The total Green's function for two-point boundary-value problems can be related to the propagator for initial-value problems. A very simple expression for the Green's function is obtained when the unperturbed medium may be described by material with a constant gradient in quadratic slowness. The derivation requires a correct understanding of assumptions made in the propagator solution. Expressions are also obtained for Green's function in multilayered media.     

First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.