## 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 6×6 propagator
matrix in Cartesian coordinates into
a 6×6 surface-to-surface paraxial matrix.
This matrix defines the transformation
of paraxial ray quantities from one surface
to another. The redundant non-eikonal
and ray-tangent solutions of the dynamic
ray-tracing system in Cartesian coordinates
can be easily eliminated from the 6×6
surface-to-surface paraxial matrix,
and it can be reduced to 4×4 form. Both the 6×6
and 4×4 surface-to-surface paraxial matrices
satisfy useful properties, particularly
the symplecticity. In their 4×4 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
surface-to-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.

### Whole paper

The reprint is available in
PostScript (1174 kB),
GZIPped PostScript (419 kB),
and PDF (218 kB).

*Geophys. Prospecting*, **55** (2007), 21-37.

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