Ray propagator matrices in three-dimensional anisotropic inhomogeneous layered media

Vlastislav Cerveny & Tijmen Jan Moser


Ray propagator matrices contain the complete solutions to the system of dynamic ray-tracing (DRT) 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 modelling and practical interpretational problems of seismic ray fields in the high-frequency asymptotic approximation.

Traditionally, ray propagator matrices have been expressed either in Cartesian or in raycentred coordinates, connected with a reference ray. Both coordinate systems have certain advantages. For ray-centred coordinates, the dimensions of ray propagator matrices can be easily reduced from 6×6 to 4×4 (in a 3-D medium), thereby reflecting the strictly fourparameteric 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 DRT system and ray propagator matrices are well known both for isotropic and anisotropic layered media. In ray-centred coordinates, the DRT 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 DRT in Cartesian coordinates, and to transform the resulting ray propagator matrix to ray-centred coordinates at any point of the ray where we need it. This avoids DRT in ray-centred coordinates altogether. Vice versa, we can compute, at any point of the reference ray, the ray propagator matrix in Cartesian coordinates by DRT in ray-centred coordinates. We propose several alternative versions of the transformation, each of them equally valid in isotropic and anisotropic media.

For rays hitting a structural interface, the ray propagator matrix has to be transformed across the interface. The relevant transformation matrix is usually referred to as the interface propagator matrix. In Cartesian coordinates, the 6×6 interface propagator matrix has been published before, but in ray-centred coordinates only for the case of isotropic media. Based on the transformation from Cartesian to ray-centred coordinates, we present the 6×6 and 4×4 interface propagator matrices in ray-centred coordinates, valid for general isotropic and anisotropic media. The 4×4 interface propagator matrix in ray-centred coordinates can be factorized and used to derive the 4×4 surface-to-surface paraxial matrices. These matrices allow to relate the paraxial ray properties at different surfaces crossed by the reference ray and offer many important applications.


Lateral heterogeneities, layered media, seismic anisotropy, seismic ray theory.

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Geophys. J. int., 168 (2007), 593-604.
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