Elementary Green function as an integral superposition of Gaussian beams in inhomogeneous anisotropic layered structures in Cartesian coordinates
Abstract
Integral superposition ofGaussian beams is a useful generalization of the standard ray theory. It removes some of the deficiencies of the ray theory like its failure to describe properly behaviour of waves in caustic regions.
It also leads to a more efficient computation of seismic wavefields since it does not require the time-consuming two-point ray tracing.We present the formula for a high-frequency elementary Green function expressed in terms of the integral superposition of Gaussian beams for inhomogeneous, isotropic or anisotropic, layered structures, based on the dynamic ray tracing (DRT) in Cartesian coordinates. For the evaluation of the superposition formula, it is sufficient to solve theDRT inCartesian coordinates just for the point-source initial conditions.Moreover, instead of seeking 3 x 3 paraxial matrices in Cartesian coordinates, it is sufficient to seek just 3 x 2 parts of these matrices.
The presented formulae can be used for the computation of the elementary Green function corresponding to an arbitrary direct, multiply reflected/transmitted, unconverted or converted, independently propagating elementary wave of any of the three modes, P, S1 and S2. Receivers distributed along or in a vicinity of a target surface may be situated at an arbitrary part of the medium, including ray-theory shadow regions.
The elementary Green function formula can be used as a basis for the computation of wavefields generated by various types of point sources (explosive, moment tensor).