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

Vlastislav Cerveny & Ivan Psencik


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.


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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