## 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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