Direct optimization methods, ray propagation and chaos 
         II. Propagation with discrete transitions 

         Martin A. Mazur and Kenneth E. Gilbert

         The Applied Research Laboratory and the Graduate Program   
		 in Acoustics 
         Pennsylvania State University 
         P. O. Box 30  
         State College, Pennsylvania 16804  
         Telephone: (814) 863-4782 (Mazur), (814) 863-8291 (Gilbert) 

Eigenrays between a source and receiver are classically determined by an
initial-value or `shooting' approach. In range-dependent environments the ray
paths can be chaotic, putting a fundamental limit on the accuracy of
classical ray tracing methods. An alternative approach can be based on
Fermat's principle of minimum propagation time. Using the so-called `direct
methods' of the calculus of variations, the travel time integral can be
minimized directly, rather than by means of the Euler-Lagrange differential
equations used in shooting. In the current paper, examples are studied where
the changes in a ray's trajectory occur at discrete points. Discrete
mappings, analogous to the Euler-Lagrange equations used in shooting
techniques in continuous problems, are introduced. It is demonstrated that
direct methods can be applied to the travel time summations in discrete
problems, just as they can be applied to the travel time integral in
continuous problems. It is shown that direct methods can be used to calculate
certain arrivals (e.g. `head waves') that cannot be produced by
shooting. Using discrete examples, direct methods are compared to the more
conventional discrete mapping (shooting) approach, without the complications
of numerical analysis and infinite-dimensionality found in continuous
problems. Two examples are studied that are associated with a standard
discrete mapping known to be chaotic. Direct methods are used to find
eigenrays for these chaotic examples. The advantages and limitations of
direct methods in discrete problems with chaos are discussed.