## 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.

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