Direct optimization methods, ray propagation, and chaos. I. Continuous media

In classical ray tracing, eigenrays between a source and receiver are determined by an initial value or “shooting” approach. The launch angles of rays from a source point are varied until the rays intersect the receiver endpoint. In nonseparable range-dependent environments, the ray paths can be chaotic, putting a fundamental limit on tracing rays by shooting methods. In the present paper, an alternative approach based on Fermat’s principle of minimum propagation time is discussed. Rather than minimizing the travel time integral indirectly, by deriving the Euler–Lagrange equations for the ray paths, so-called “direct methods” of minimization, taken from the calculus of variations, are employed. Previous authors have demonstrated that simulated annealing, a direct method, can be used to find eigenrays in a particularly chaotic ray tracing problem. In the present paper, two direct methods are applied to the calculation of eigenrays in continuous media: the Rayleigh–Ritz technique, a classical direct method, and simulated annealing, a Monte Carlo direct method. Direct methods are compared to shooting techniques, and some of the advantages and drawbacks of both methods are shown using both nonchaotic and chaotic examples.