EXPONENTIAL MODEL 73 



relationship. This agreement is attributed to the fact that the reference 

 atmosphere is a good representation of the A^ distribution below the 

 California elevated inversion and to the fact that a majority of the bend- 

 ing is accomplished below the elevated inversion height of about 500 m. 

 Further, it can be easily shown that the bending integral is increasingly 

 insensitive to strong A'' discontinuities as the height increases. 



Figure 3.11 shows a similar comparison for a high initial elevation angle, 

 00 = 15° and a large height increment, h — hs = 70 km. This compari- 

 son shows that both of the reference atmospheres are in closer agreement 

 with the long-term mean bendings than are the 4/3 earth bendings. Note 

 that, whether r is predicted from A^^ or AN, the 4/3 earth model gives but 

 a single value of bending that is outside the limits of the values of r ob- 

 tained from the long-term mean profiles. 



In considering the comparisons of figures 3.10 and 3.11, one might ask 

 if they reflected the form of the basic equation for bending; namely, at 

 low angles is r determined by the A^ gradient throughout the A'" profile, 

 and at high angles is r essentially a function of the value of A^ at both ends 

 of the A^ profile (i.e., the limits of integration). Thus one might expect 

 the deviations to be smaller if the comparisons were made on the basis of 

 a function of the A^ gradient such as AA^, particularly for small values of 

 do. Such a comparison is given by figures 3.12 and 3.13 for the same 

 initial elevation angles and height increment as before. It is seen that 

 the AA^-specified reference atmospheres improve the agreement for the 

 low-angle case, but decidedly decrease the agreement for the high-angle 

 case. 



A numerical evaluation of the root mean scjuare (rms) deviation of the 

 long-term mean bendings from both the reference atmospheres deter- 

 mined as a function of both AA^ and A''^ was made for a variety of initial 

 elevation angles for the height increments 3 and 70 km. Root mean 

 square deviations were not calculated for the 4/3 earth model since it was 

 felt that this model was obviously in marked disagreement with the long- 

 term mean bendings under these conditions. Figure 3.14 summarizes 

 the rms deviations for the h — hs = 3-km case. It is seen that for 

 do < 10 mrad (about 0.5°), the AA^-specified reference atmospheres have 

 the smaller rms deviations. Also, the exponential reference atmospheres, 

 whether specified by AA'^ or A^^, have smaller rms deviations than the 

 reference atmosphere. 



It is seen for the 70-km case, figure 3.15, that the A^s-specified reference 

 atmospheres have a significantly smaller rms deviation than the AA^- 

 specified atmospheres for do > 5 mrad. Again it is seen that the exponen- 

 tial i-eference atmosphere generally has the smaller rms deviation for 

 values of do less than 10 mrad. Howev(M', the slightly smaller rms devia- 

 tions associated with the reference atmosphere for ^o > 10 mrad reflect 

 that model's closer agreement with the actual A structure of the atmos- 

 phere at high heights. 



