As far as ray diagrams and propagation losses computed according to clas- 

 sical ray theory are concerned, past experience indicates that cubic splines 

 produce the best representations for analj^tic type sound speeds. However, as 

 the input data become more irregular, the curve fitting procedure becomes more 

 difficult to automate.^ A second disadvantage of cubic splines is that the corre- 

 sponding ray tracing equations cannot be integrated in closed form, which is a 

 process that can be accomplished with piecewise linear and quadratic fits. 



Many of the statements made above are also true when the sound speed 

 varies with one or more horizontal coordinates as well as depth. If, for example, 

 the input data are fitted with triangular planes, the ray tracing equations may 

 be integrated in closed form, but anomalies due to discontinuous gradients are 

 possible. 



ASYMPTOTIC TREATMENT OF CAUSTICS 



In the lastfewyears, significant advances in practical ray tracing techniques 

 involved the treatment of caustics rather than improvements in curve fitting 

 algoritltms. The problem may be illustrated when the sound speed decreases 

 inversely as the square root of depth, as shown in figure 4. We see that the 

 ray diagram, figure 5, forms a well defined caustic. 



VELOCITY (kyd/sec) 

 1 2 



GRADIENT (1/sec) 

 -2.0 -1.2 -0.4 0. 



Figure 4. Sound Speed and Gradient Studied by 

 Pedersen and Gordon 



101 



