### ASA 125th Meeting Ottawa 1993 May

## 5pUW1. Finding eigenrays by optimization with application to tomography
and overcoming chaos.

**W. A. Kuperman
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Michael D. Collins
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*Naval Res. Lab., Washington, DC 20375
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Eigenrays may be determined with the initial-value approach of shooting.
The eikonal equation is solved repeatedly, with the initial conditions being
the position and direction of the ray at one of the end points, until the
correct initial conditions are found (i.e., until the ray nearly intersects the
other end point). It has been demonstrated that this approach is relatively
inefficient [B. R. Julian and D. Gubbins, J. Geophys. 43, 95--113 (1977)] and
prone to chaos [Smith et al., J. Acoust. Soc. Am. 91, 1939--1949 (1992)]. When
formulated in terms of Fermat's principle, finding eigenrays is a
boundary-value problem that should be free of the ill effects of chaos. If the
index of refraction is perturbed, the perturbed eigenrays may be determined
efficiently from the unperturbed eigenrays. This fact is exploited to perform
tomography efficiently with an optimization procedure that alternates between
attempting to satisfy Fermat's principle and attempting to match travel time.
After a sequence of iterations, both the index of refraction and the eigenrays
are obtained.