## Gaussian beams in inhomogeneous anisotropic layered structures

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

Gaussian beams, approximate solutions of
elastodynamic equation concentrated close to rays
of high-frequency seismic body waves,
propagating in inhomogeneous anisotropic layered
structures, are studied. They have Gaussian
amplitude distribution along any straightline
profile intersecting the ray. At any point of the
ray, the Gaussian distribution of amplitudes is
controlled by the 2×2 complex-valued symmetric
matrix **M** of the second derivatives of the
traveltime field with respect to ray-centred
coordinates. Matrix **M** can be simply determined
along the ray if the ray propagator matrix is
known and if the value of **M** is specified at
a selected point of the ray. The ray propagator
matrix can be calculated along the ray by
solving the dynamic ray tracing system twice:
once for the real-valued initial plane-wave
conditions and once for the real-valued initial
point-source conditions. Alternatively, matrix
**M** can be determined along the ray by solving
the dynamic ray tracing system only once, but
for complex-valued initial conditions.
The dynamic ray tracing can be performed in various
coordinate systems (ray-centred, Cartesian, etc.).
Here we use the ray-centred coordinate
system, but propose a simple local transformation
to Cartesian coordinates. This simplifies
the computation of the Gaussian beams at the
observation points situated in the vicinity of the
central ray. The paper is self-contained and
presents all the equations needed in computing the
Gaussian beam. The proposed expressions for
Gaussian beams are applicable to general 3-D
inhomogeneous layered structures of arbitrary
anisotropy (specified by up to 21 independent
position-dependent elastic moduli).
Possible simplifications are outlined.

### Keywords

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

### Whole paper

The
image of the paper in GIF 150dpi (3174 kB !)
is designed for an instant screen preview.

The reprint is available in
PDF (275 kB).

*Geophys. J. int.*, **180** (2010), 798-812.

SW3D
- main page of consortium ** Seismic Waves in Complex 3-D Structures **.