## Ray propagator matrices in 3-D anisotropic inhomogeneous layered media

Vlastislav Cerveny & Tijmen Jan Moser

### Summary

Ray propagator matrices contain the complete solutions to the system of dynamic ray tracing equations connected with a given reference ray. They play an important role in studying the properties of complete four-parameteric systems of paraxial rays and offer many applications in both numerical modeling and practical interpretational problems of seismic ray fields in the high-frequency asymptotic approximation. Traditionally, ray propagator matrices have been expressed either in Cartesian coordinates or in ray-centered coordinates, connected with a reference ray. Both coordinate systems have certain advantages. For ray-centered coordinates, the dimensions of ray propagator matrices can be easily reduced from 6 x 6 to 4 x 4 (in a 3-D medium), thereby reflecting the strictly four-parameteric nature of a general paraxial ray field. On the other hand, in Cartesian coordinates, the computations are conceptually simpler and generally valid in isotropic and anisotropic media. In a Cartesian coordinate system, the dynamic ray tracing system and ray propagator matrices are well-known both for isotropic and anisotropic layered media. In ray-centered coordinates, the dynamic ray tracing system and ray propagator matrices are known for isotropic layered structures and for anisotropic smooth media, but not for anisotropic media with structural interfaces. We propose a simple and invertible transformation between ray-propagator matrices in both coordinate systems. It allows to perform conventional dynamic ray tracing in Cartesian coordinates, and to transform the resulting ray-propagator matrix to ray-centered coordinates at any point of the ray where we need it. This avoids dynamic ray tracing in ray-centered coordinates altogether. Vice versa, we can compute, at any point of the reference ray, the ray propagator matrix in Cartesian coordinates by dynamic ray tracing in ray-centered coordinates. We propose several alternative versions of the transformation, each of them equally valid in isotropic and anisotropic media.

### Keywords

Seismic ray theory, seismic anisotropy, lateral heterogeneities, layered media.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 16, pp. 87-114, Dep. Geophys., Charles Univ., Prague, 2006.
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