A reason for using a quadratic (or a higher-order polynomial) surface 

 is that the second partial derivatives can be obtained. This fact was 

 used in the Griswold-Mehr program to obtain values for Beta (Munk and 

 Arthur, 1952), the coefficient of refraction, and H/Hq. (As mentioned, 

 however, their interpolation scheme rendered their results invalid.) It 

 is apparent that certain problems with the quadratic (or higher-order 

 polynomial) interpolation schemes must be resolved before such sophisticated 

 parameters as Beta or H/Hq can be estimated. It should be possible to 

 derive a quadratic surface by heavily weighting the velocity values at the 

 central four grid intersections while still using the surrounding eight 

 velocity values. This procedure may produce a good interpolation surface; 

 if so, it would be quite worth while to calculate estimates of the above 

 parameters. 



Ray Curvature Approximation 



An entire grid of velocity values is not a smooth and continuous 

 surface when observed as a set of planes in which each grid cell is 

 represented by a specific plane of best fit to its four bounding velocity 

 values. With this in mind, it is not surprising that all curvature approxi- 

 mations (see MOVE subroutine of MAIN 1620, Appendix D; or MOVE subroutine of 

 MAIN 7094, Appendix E) fail to converge to a single value when determining 

 a new point along a ray path. This is especially apparent when adjacent 

 planes present a large discontinuity in wave-velocity values at their 

 common edges. This fact is the reason for the variable MIT included in 

 MOVE subroutine. In case the curvature approximations oscillate among 

 three or more values after 20 iterations, the ray is terminated, as no 

 valid curvature approximation can be made. Although infrequent, sometimes 

 (not shown in OUTPUT, Appendix D) the curvature approximations oscillate 

 between two values. In this case, the message "CURVATURE APPROXIMATED" is 

 included with the output in order that the operator note that the curvature 

 used for the given point was an average of two values. The Griswold-Mehr 

 program did not have such a check on the curvature approximations; the 

 average of the last two approximations after 10 iterations was used as the 

 new curvature value, if the values did not converge. This is another reason 

 for the erratic ray behavior near the shore in Figure 9C. 



Grid Considerations 



For reasons outlined in a previous section ("Future Interpolation 

 Schemes") assume that a grid of depth values will be input and used for 

 derivation of the interpolation surfaces. It will be necessary in the 

 future program to transfer a ray to a larger scale grid when approaching 

 the shore in order to allow better grid control (i.e., better representation 

 of depth values) in an area where the velocity values are rapidly changing. 

 There is, however, a limit to the maximal scale of a grid as beach and 

 nearshore topography are constantly changing, especially during each storm. 

 Thus the selection of grid interval is a question to be pursued in additional 

 studies. 



12 



