Gaussian beams in inhomogeneous anisotropic layered structures

Vlastislav Cerveny & Ivan Psencik


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.


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

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