THEORY 337 



From table 8.5 it can be seen that the geometric range error, ARg, does 

 not represent a significant portion of the total range error except at very- 

 small initial elevation angles, between zero and about 3°. This being so, 

 the behavior of the total range error will be primarily a function of the 

 first integral in (8.19) for elevation angles greater than about 3°. The 

 integral expression 



ARe ^ 10~' / ' N csed dh, 



may be rewritten as 



ARe ^ CSC ^0 X 10"' 



N dh 



1 - 2 sin' ( — T-^ ) + cot 00 sin {d - do) 



or 



ARe ^ CSC do N dh 



+ Z (-l)''-"^ N 



cot 00 sin (0 - do) -2 sin' 



^)] 



dh, (8.20) 



for sin d < 2 sin do, < d < 7r/2. This expression is analogous to that 

 derived for the case of ray bending, (3.6) of chapter 3, and, similarly, the 

 integral series on the right-hand side of (8.20) contributed only 3 percent 

 or less to the value of ARe for do larger than about 10°. From (8.20) one 

 would thus suspect that the radio range error might be well e.stimated as 

 a linear function of the integral of N with respect to height. In treating 

 this integral, it is informative to note that any given N(h) profile may be 

 "broken up" into three primary components: 



N(h) = N'{Ns, h) + N"ih + hs) + 8N(h) 



where A'^' is that part of the profile which can best be expressed as a 

 function of A^^ and height, N" is a standard distribution of refractivity 

 with resjiect to altitude above mean sea level (h -\- hs) which is independ- 

 ent of N s, especially above the tropopause, and 5N represents a random 

 component of the profile which cannot in general be accounted for 

 a priori. 



