## Notes on summation of Gaussian beams and packets

**Ludek Klimes**
### Summary

A Gaussian beam is a high-frequency asymptotic time-harmonic
solution of the wave equation, with an approximately Gaussian profile
perpendicularly to the central ray.
A Gaussian packet is a high-frequency asymptotic space-time
solution of the wave equation.
The envelope of a Gaussian packet
at any given time is nearly Gaussian function.
The accuracy of a Gaussian beam or of a Gaussian packet depends
on its shape. The shape of a Gaussian beam or packet
should thus be optimized for the propagation between
a given source and a given receiver.

Gaussian beams and packets may serve as building blocks of
a wavefield.
The summation of Gaussian beams
and packets overcomes the problems of the standard ray theory with caustics.
The summation of Gaussian beams and packets is considerably
comprehensive and flexible.
It may be formulated in many ways and
depends primarily on the specification of the wavefield.
The summation of Gaussian beams and packets includes
two-parametric summation of Gaussian beams forming
a time-harmonic wavefield specified asymptotically in terms
of the amplitude and travel time,
three-parametric summation of Gaussian packets forming
a time-harmonic wavefield also specified asymptotically in terms
of the amplitude and travel time,
the Maslov method and its various generalizations,
coherent-state transform methods,
four-parametric decomposition of a general
time-harmonic wavefield into Gaussian beams,
six-parametric decomposition of a general
time-dependent wavefield into Gaussian packets
(used, e.g., for Gaussian-packet prestack depth migrations),
and the system of Gaussian packets scattered
from Gabor functions forming medium perturbations.

The integral superpositions of Gaussian beams and packets
are discretized into summations.
The discretization step should carefully be controlled in order
to preserve both accuracy and efficiency of the summation.
### Keywords

Gaussian beams, Gaussian packets, coherent states,
wavefield representation, summation methods, superposition integrals,
Maslov method, linear canonical transform, Gabor transform,
prestack depth migration.

### Whole paper

The
image of the paper in GIF 150dpi (920 kB)
is designed for an instant screen preview.

The paper is available in
PostScript (190 kB)
and GZIPped PostScript (59 kB).

In: Seismic Waves in Complex 3-D Structures, Report 14,
pp. 55-70, Dep. Geophys., Charles Univ., Prague, 2004.

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