360 REFRACTION AND REFRACTIVE INDEX MODELS 



Because the difference between Re and the true slant range, Rq, is small 

 compared to the height error, the slant range and radio range are assumed 

 to be identical to the geometric range, R. 



The apparent height of the target, in figure 8.28, is obtained by solving 



(ro + haY = ro^ + R"" + 2r, R sin 6^ (8.32) 



for ha. The following form is useful for numerical calculations: 



K = ^(^ + 2rosin^o) _ ^g 33^ 



ro + Vro^ -\- R{R -\- ro sm do) 



The height error for a target at height h is found by 



€, = ha - h (8.34) 



which will always be positive if n decreases with height. 



If the refractive index is known as a function of height the foregoing 

 procedure is useful for determining the height error when the true height 

 and the arrival angle of the ray are hypothesized. Unfortunately, it is 

 not applicable for obtaining the height error from the apparent position 

 of the target. 



8.4.4. Use of the Effective Earth's Radius 



The inaccuracy of the "four-thirds earth" correction stems mainly 

 from the assumption that all radio rays have the same constant curvature. 

 The accuracy would be greatly enhanced if an "average" effective radius 

 could be determined for each ray path. 



The following expression, with the effective earth's radius denoted by 

 Te in figure 8.28, 



(r, + hy = r,2 -\- R^ -\- 2re R sin ^o, (8.35) 



can be combined with (8.32) and (8.34) to obtain 



« + ^ - f = (~^)(- - -). (8.36) 



Ve 2re \ 2 /Vo Ve/ 



The difference between the curvature of the actual earth and the "average" 

 effective earth for the ray path represents the "average" curvature of the 

 ray. Thus, if the ray curvature can be determined as a function of the 



