## Elementary Green function as an integral superposition
of Gaussian beams in inhomogeneous anisotropic layered
structures in Cartesian coordinates

**Vlastislav Cerveny** **&**
**Ivan Psencik**
### Summary

Integral superposition of Gaussian 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
raytracing. 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 the DRT in Cartesian
coordinates just for the point-source initial
conditions. Moreover, instead of seeking 3×3
paraxial matrices in Cartesian coordinates, it is
sufficient to seek just 3×2
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).

### Keywords

Body waves, seismic anisotropy, theoretical seismology, wave propagation.

### Whole paper

The reprint is available in
PDF (273 kB).

*Geophys. J. int.*, **210** (2017), 561-569.