62 TROPOSPHERIC REFRACTION 



When the effective earth's radius treatment is used, height is 

 calculated as a function of distance, for a ray with 60 = 0, with the 

 equation h = d^/2ka, where d is the distance, k is the effective earth's 

 radius factor, and a is the true radius of the earth ('^6373 km). The 

 errors likely to be incurred when using this equation, assuming as a true 

 atmosphere an exponential N{h) profile as given in the following sub- 

 section will not exceed 5 percent for heights of 1 km or less. 



The preceding background discussion has presented the material neces- 

 sary for the consideration of the suitability of various models of refrac- 

 tivity to describe atmospheric refraction of radio waves. As a guide to 

 what follows, let us ask what a logical sequence of models (or assumptions) 

 would be to describe the effects of atmospheric refraction. 



One such sequence might be: 



(1) Assume an invariant model that is near to the actual average 

 conditions and facilitates the calculation of radio field strengths. This 

 has been done by the 4/3 earth model. 



(2) Assume a variable effective earth's radius factor for the calculation 

 of radio field strengths in various climatic regions. This approach has 

 been followed by Norton, Rice, and Vogler [11]. When it has become 

 apparent that the effective earth's radius approach is inadequate, one 

 might proceed by : 



(3) Correctmg the effective earth's radius model by assuming a more 

 realistic A^ structure in the region where that model is most in error. This 

 "modified effective earth's radius" model would then maintain, for some 

 applications, the advantages of the original model. 



(4) Assume an entirely new model of A'^ structure guided by the 

 average N structure of the atmosphere. 



It is assumed that models (3) and (4) would allow for seasonal and 

 climatic changes of the average refractive index structure of the atmos- 

 phere. 



In the following sections, models (3) and (4) will be developed and 

 tested by their relative agreement with the ray bendings obtained from 

 actual long-term average A^^ profiles. 



The first model of atmospheric refractivity that will be considered is 

 based upon the effective earth's radius concept in the first kilometer. In 

 this atmosphere N is assumed to decay linearly with height from the sur- 

 face hs to 1 km above the surface hs + 1. This linear decay is given by 



N{K) = N, -\- {h - hs) AN,hs < h < hs + I, (3.39) 



where 



-AA^ = 7.32 exp (0.005577 A^.). (3.40) 



