Seismic-waveform effects of conical points in gradually varying anisotropic media

Abstract

SUMMARY We describe the effects of anisotropic slowness-surface conical points (acoustic axes) on quasi-shear wavefronts and waveforms in variable elastic media. Conical points have quite complicated geometrical consequences even for a point source in or a wave refracting into a homogeneous anisotropic medium. A hole develops in the fast quasi-shear wavefront and the swallowtail catastrophe plays an important role in the geometry of the slow quasi-shear front, which becomes folded with numerous self-intersections. The two fronts are joined along the rim of the hole. This geometry influences the waveforms, which show Hilbert-transform and diffraction effects. Therefore, standard ray theory is inapplicable even for a uniform medium and the Maslov method is needed to describe waveforms. The introduction of elastic gradients further complicates the geometry of the problem, because rays bend sharply as their slowness approaches that of the axis. An initially smooth, single-valued slow quasi-shear front will evolve in the gradient region into a front which is folded and multivalued and once again the swallowtail is important. However, in contrast to a homogeneous medium, no ‘hole’ develops in the fast quasi-shear front and the slow and fast fronts separate completely. While such geometrical factors are included in the Maslov method, waveforms are also affected by coupling of the fast and slow waves on nearing the axis, where the rays and polarizations rotate most rapidly and their slownesses differ by very little. Numerical examples are presented for a cubic and an orthorhombic material. The differences between these two examples show that the fine structure of ‘continuously varying internal conical refraction’ can vary considerably from material to material, though its basic principles are clearly defined. Waveforms are presented for a point source in a uniform medium and for fast and slow shear waves in a gradient, with and without coupling. Overall, we conclude that the wavefront-folding effects cause the most drastic waveform distortions. Coupling becomes most important when signals merge, as at cuspidal edges or at lower frequencies, since the net waveform could be altered significantly if its components are varied. Conical refraction complicates and yet could be decisive for the identification of seismic anisotropy and rock properties.