A Maslov-propagator seismogram for weakly anisotropic media

Abstract

SUMMARY We introduce a formalism to calculate shear-wave seismograms for weakly-anisotropic and inhomogeneous media. The method is based on the combination of the forward-propagator method, which accounts for shear-wave interaction along a single reference ray, and the Maslov ray-summation, which incorporates amplitude and phase information from neighbouring rays to account for waveform and diffraction effects at caustics and in shadow regions. The approach is based on the assumption that the multiply split shear waves, on the way to a given receiver, travel along a common ray path that can by obtained from ray tracing in an isotropic reference medium (i.e. the common-ray approximation). The forward propagator and the Maslov amplitude are expressed with respect to radial and transverse coordinates (perpendicular to the ray propagation direction) that are defined uniquely by the initial conditions. Local polarizations and slownesses of the fast and slow shear-waves in the direction of propagation are obtained from the eikonal equation. The Maslov-propagator phase is given by the average shear-wave traveltime along the reference ray. Phase advances and delays of individual shear wave components are accounted for by the propagator. The geometrical-spreading information required for the Maslov integration is supplied by dynamic ray tracing in the isotropic reference medium. In the high-frequency limit effective phase functions are defined to assess the validity of the Maslov propagator phase information. For a homogeneous isotropic reference medium, we find good agreement with exact Maslov phase functions for anisotropic perturbations of up to 20 per cent. As a numerical application we consider effects of inhomogeneous anisotropy in a shear-wave cross-hole survey. The variations of the transversely-isotropic medium require 2-D slowness integrals. The method can handle discontinuities of the fast polarization along the ray path and also for neighbouring rays which is important for the slowness integration. Smooth transitions between isotropic and anisotropic regions along the ray path can be accounted for without the need to switch between numerical formulations.