## Gaussian beams and paraxial ray approximation
in three-dimensional elastic inhomogeneous media

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

The elastodynamic Gaussian beams in 3D elastic inhomogeneous media
are derived as asymptotic high-frequency one-way solutions of
the elastodynamic equation concentrated close to rays of P and S
waves. In this case, the elastodynamic equation is reduced to
a parabolic (Schrödinger) equation which further leads to a matrix
Riccati equation and the transport equation. Both these equations
can be simply solved along the ray, the first numerically and
the other analytically. The amplitude profile of the principal
displacement component of the elastodynamic Gaussian beams is
Gaussian in the plane perpendicular to the ray, with its maximum
at the ray. The Gaussian beams are regular along the whole ray,
even at caustics. As a limiting case of infinitely broad Gaussian
beams, the paraxial ray approximation is obtained. The properties
and possible applications of Gaussian beams and paraxial ray
approximations in the numerical modelling of seismic wave fields
in 3D inhomogeneous media are discussed.

### Keywords

Seismic waves, 3D elastodynamic equation, paraxial ray approximation,
Riccati equation, elastodynamic Gaussian beams.

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*J. Geophys.*, **53** (1983), 1-15.

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