REFRACTION IN BI-EXPONENTIAL MODEL 321 



8.1.4. Refraction in the Bi-Exponential Model 



The test of a model of atmospheric refractive index is the degree to 

 which it represents the average A^ structure of the atmosphere. A 

 further critical test is the degree to which the refraction, or bending, of a 

 radio ray is represented by this atmosphere. The bending is given as 

 the angular change of a radio ray as it passes from rii to n-i in a spherically 

 stratified atmosphere and is adequately represented by 



Ti,2 = - / cot ddn (8.6) 



where 6 is the local elevation angle and is determined at any point from 

 Snell's law, 



nr cos 6 = rioro cos ^o, (8.7) 



where r is the radial distance from the center of the earth and zero sub- 

 script denotes the initial conditions. It is customary to evaluate (8.6) 

 numerically [11] since the integral is intractable for all but the most 

 simple models of n versus r. The bending was obtained for the mean A^ 

 profiles for one-half of the 22 U.S. weather stations mentioned earlier. 

 The values of r predicted by the bi-exponential model for these same 

 stations were obtained by preparing U.S. maps of Hd and H,j, from the 

 other half of our data, selecting values oi Hd and Hu, for the test stations, 

 calculating the bending and obtaining rms differences between these 

 values and those obtained from the mean A'' profiles. These rms differ- 

 ences are shown on figure 8.9. Also, for comparison, the rms errors 

 obtained from the CRPL exponential atmosphere are shown. This 

 atmosphere, based upon a single exponential curve passing from the 

 surface value, N s, to the value at 1 km, A^i, is founded upon the expression 



A^i = A^, - 7.32 exp {0.005577 A^^} (8.8) 



which has been found to be applicable in the United States [1]. 



An obvious advantage of the bi-exponential model is that the scale 

 heights do reflect the physical properties of the atmosphere in a much 

 clearer way than does the single exponential model. 



8.1.5. Extension to Other Regions 



The present study is based upon data from the continental United 

 States and one wonders if the same approach might be of utility in other 

 regions. As a brief check, the D and W term scale heights were deter- 

 mined for conditions typical of the long arctic night (Isachsen, Northwest 



