satisfactory interpolation surface ; this was evident when calculation 

 points for a ray had moved within two grid units of the shore. This is 

 because the negative "land" velocity values, used in derivation of the 

 surface, made the general tilt of the surface steeper than indicated by 

 the velocity values at the central four grid intersections. This is 

 shown in Figure 9B where the rays bend, after moving within two grid 

 units of the shore, more than do the rays produced by the linear-interpola- 

 tion program (Figure 9A) or the rays produced by the graphical-construction 

 method (Figure 9D) . 



Linear-interpolation Surface . The linear-interpolation programs 

 (MAIN 1620, Appendix D; and MAIN 7094, Appendix E) were developed in order 

 to remedy the excessive tilt created by the quadratic-interpolation program. 

 The advantage offered by the linear-interpolation program is that only the 

 velocity values at the central four grid intersections are needed in order 

 to derive the surface. The assumption, presented under the heading "Selec- 

 tion of Input," stating that the grid interval be selected such that bottom 

 contours in any grid cell be represented by straight and parallel lines, is 

 not actually valid. Although bottom topography may be represented by a 

 plane (i.e., it changes linearly) in a given cell, the associated velocity 

 values may be changing exponentially (i.e., velocity is an exponential 

 function of depth) in the same cell. However, for purposes of this program, 

 it is assumed that the velocity values in a given grid cell can be adequately 

 represented by a plane. 



The use of the variable PCTDIF in MAIN 1620 (card number RAYN 19, 

 Appendix D) and MAIN 7094 (card number RAYN 20, Appendix E) serves to give 

 an indication of the relative "fit" of a surface to a given set of velocity 

 values. The output values for PCTDIF (see OUTPUT, Appendix D) give the 

 percent difference between only one of the four velocity values at the 

 nearest four grid intersections and its related value on the plane fit to 

 the same four values. PCTDIF by no means represents the degree of "fit" 

 of the surface to the four values because these percent differences may 

 vary for each of the four values to which a plane is fit. This is especially 

 true in cells where there are great differences between the four velocity 

 values (such as near the shore). PCTDIF does, however, give an estimate of 

 the relative error encountered in the interpolation in a given grid cell. 



Just as the quadratic-interpolation program yielded an excessive tilt 

 when the given grid cell in use fell within two grid units of the shore, 

 the linear-interpolation program yields an excessive tilt when the given 

 cell falls within one grid unit of the shore. Therefore, it is expected 

 that PCTDIF will be large in value when the grid cell being interpolated 

 is located near the shore. In view of this fact, the rays run with the 

 linear surface-fitting program should be considered as rough approximations 

 only, when closer than one grid unit to the shore. 



10 



