4 6 8 10 12 14 16 



Grazing Angle (degl 



Figure 1 1 Estimated specular reflection coefficients are 

 compared with those determined by accumulating rays The 

 upper estimate is based on evenly spaced, similar keels. The 

 bottom estimate is based on randomly spaced, similar keels. 



SCATTERING FUNCTIONS 



The scattering functions are a sum of the reflected and refracted fields. A full analytical 

 description of neither of these fields is attempted in the following discussion. However, some 

 equations and plots are given to clarify some aspect of the fields. 



Returning to Fig. 10. the slope of the parabola *t the point a ray stnkes can be 

 determined. The ray is defined by the value of s or distance from the center of the wavefront 

 that intersects the keel. The angle of the wave front. 0, is defined as the angle the rays make to 

 the horizontal, where the angles of rays arriving from below the horizontal are positive. The 

 slope of the parabola 4> is gjven by 



r> = Un-'[-2ir (b 2 )] (5) 



where the intercept range, r, is given by 



r = { (tan 6)!(d/lr) - [ (tan 2 9), {(Pitt) * Md - j/cos 6)t(dib l )^}l2 



Figure 1 2 summarizes the use of these last tw> equations for the keel of Fig. 10. It is a 

 plot of certain slope values at points of intersection between a ray and the keel. The rays are 

 determined by ray angle 8 and s, the posnon on the wave front as defined in Fig. 10. The two 

 lines labeled "limiting ray" are the tangent rays, and they intersect the parabola where its slope 

 is -B. The angle between ray and parabola is 6 * <t> or zero at the point of tangency. The width 



12 



