W. A. Kuperman
Michael D. Collins
Naval Res. Lab., Washington, DC 20375
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.