Gaussian beams in inhomogeneous anisotropic layered structures using dynamic ray tracing in Cartesian coordinates

Vlastislav Cerveny and 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 straight line profile intersecting the ray. At any point of the ray, the Gaussian distribution of amplitudes is controlled by the 2 x 2 complex-valued symmetric matrix M(y) of the second derivatives of the travel-time field with respect to wavefront orthonormal coordinates y1, y2, local Cartesian coordinates in a plane tangential to the wavefront with its origin at the central ray. Matrix M(y) can be simply determined along the ray if the real-valued propagator matrix of the dynamic ray tracing equations (ray propagator matrix) is known and if the value of M(y) is specified at an initial point of the ray. The ray propagator matrix can be calculated along the ray by solving twice the dynamic ray tracing system: once for the real-valued initial plane-wave conditions and once for the real-valued initial point-source conditions. Alternatively, matrix M(y) 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 (global Cartesian xi, local Cartesian yi, ray-centred qi, etc.).

Here we use three alternative variants of dynamic ray tracing in Cartesian coordinates: the global Cartesian system xi, the local Cartesian (wavefront orthonormal) coordinate system yi, and the simplified version of the DRT system in global Cartesian coordinates xi. In all these variants, the 2 x 2 matrix M(y) may be used to specify suitably the initial conditions for the dynamic ray tracing. We also present a simple local transformation of 2 x 2 matrix M(y) to 3 x 3 matrix of second order derivatives of travel times M(x) in global Cartesian coordinates. This 3 x 3 matrix simplifies considerably the computation of Gaussian beams at any paraxial observation point. The paper is self-contained and presents all the equations needed in computing Gaussian beams. The proposed expressions for Gaussian beams are applicable to general 3-D inhomogeneous layered structures of arbitrary anisotropy (specified by upto 21 independent position-dependent elastic moduli). Possible simplifications are outlined.

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In: Seismic Waves in Complex 3-D Structures, Report 20, pp. 127-168, Dep. Geophys., Charles Univ., Prague, 2010.