## Lyapunov exponents for 2-D ray tracing without interfaces

**Ludek Klimes**
### Summary

The Lyapunov exponents asymptotically quantify the exponential
divergence of rays.
The "Lyapunov exponent" for a finite 2-D ray and
the average "Lyapunov exponents"
for a set of finite 2-D rays or lines and for a 2-D velocity model
are introduced.
The equations for the estimation of the average
"Lyapunov exponents" in a given smooth 2-D velocity model
without interfaces are proposed and illustrated by a numerical example.
The equations allow to estimate the average exponential
divergence of rays and exponential growth of the number
of travel-time branches in the velocity model, prior to ray tracing.

### Keywords

Velocity models, travel times, ray tracing, paraxial rays,
deterministic chaos.

### Whole paper

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### Revised version

Klimes, L.:
Lyapunov exponents for 2-D ray tracing without interfaces.
*Pure and appl. Geophys.*, **159** (2002), 1465-1485.

### Expanded abstract

Klimes, L.:
Lyapunov exponents for 2-D ray tracing without interfaces.
Expanded Abstracts of 70th Annual Meeting (Calgary),
pp. 2293-2296, Soc. Explor. Geophysicists, Tulsa, 2000.

In: Seismic Waves in Complex 3-D Structures, Report 8,
pp. 83-96, Dep. Geophys., Charles Univ., Prague, 1999.

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