## Paraxial ray methods for anisotropic inhomogeneous media

**Tijmen Jan Moser**
&
**Vlastislav Cerveny**
### Summary

A new formalism of surface-to-surface paraxial matrices allows a very
general and flexible formulation of the paraxial ray theory, equally
valid in anisotropic and isotropic inhomogeneous layered media. The
formalism is based on conventional dynamic ray tracing in Cartesian
coordinates along a reference ray. At any user-selected pair of
points of the reference ray, a pair of surfaces may be defined. These
surfaces may be arbitrarily curved and oriented, and may represent
structural interfaces, data recording surfaces, or merely formal
surfaces. A newly obtained factorization of the interface propagator
matrix allows to transform the conventional 6 x 6 propagator matrix in
Cartesian coordinates into a 6 x 6 surface-to-surface paraxial matrix.
This matrix defines the transformation of paraxial ray quantities from
one surface to another. The redundant noneikonal and ray-tangent
solutions of the dynamic ray-tracing system in Cartesian coordinates
can be easily eliminated from the 6 x 6 surface-to-surface paraxial
matrix, and it can be reduced to 4 x 4 form. Both the 6 x 6 and 4 x 4
surface-to-surface paraxial matrices satisfy useful properties,
particularly the symplecticity. In their 4 x 4 reduced form, they can
be used to solve important boundary-value problems of a
four-parametric system of paraxial rays, connecting the two surfaces,
similarly as the well-known surfaceto- surface matrices in isotropic
media in ray-centred coordinates. Applications of such boundary-value
problems include the two-point eikonal, relative geometrical
spreading, Fresnel zones, the design of migration operators, and more.

### Keywords

Dynamic ray tracing, ray propagator matrices, two-point problems, seismic anisotropy,
inhomogeneity, structural interfaces.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 16,
pp. 115-144, Dep. Geophys., Charles Univ., Prague, 2006.

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