54 TROPOSPHERIC REFRACTION 



For initial elevation angles less than about 5°, the errors inherent in this 

 method exceed 10 percent (except near the surface) and rise quite rapidly 

 with decreasing ^o- 



3.4. The Statistical Method 



Another method for determining high-angle bending is the statistical 

 linear regression technique developed by Bean, Gaboon, and Thayer [5]. 

 It has been found that for normal conditions and all heights the righthand 

 integral of (3.6) is approximately a linear function of A''^ (^o, r constant) 

 for 00 < 17 mrad ('~1°) and that the second term of (3.8) tends to be 

 constant. Thus (3.6) reduces to a linear equation, 



ri,2 = hNs + a, (3.10) 



where h and a are constants (as in tables 9.1 to 9.9) and N s is the surface 

 refractivity. 



The form of (3.10) is very attractive, since it implies two things: 



(1) The value of n ,2 may be predicted with some accuracy as a function 

 only of N s (surface height and ^o constant), a parameter which may be 

 observed from simple surface measurements of the common meteorological 

 elements of temperature, pressure, and humidity. 



(2) The simple linear form of the equation indicates that, given a 

 large number of observed ri,2 versus N s values for many values of h and 

 da, the expected (or best estimate) values of h and a can be obtained by 

 the standard method of statistical linear regression. 



This is what was done to obtain values listed in tables 9.1 to 9.9. 



Tables 9.1 to 9.9 also show the values of the standard error of estimate, 

 SE, to be expected in predicting the bending, and the correlation coeffi- 

 cients, r, for the data used in predicting the lines. Linear interpolation 

 can be used between the heights given to obtain a particular height that 

 is not listed in the tables. For more accurate results, plot the values of 

 r from the tables (for desired N s) against height, and then plot the values 

 of the standard error of estimate on the same graph. Then connect these 

 points with a smooth curve. This will permit one to read the r value and 

 the SE value directly for a given height. 



3.5. Schulkin's Method 



Schulkin has presented a relatively simple, numerical integration 

 method of calculating bending for A-pro files obtained from ordinary 

 significant-level radiosonde (or "RAOB") data [6]. The A profile ob- 

 tained from the RAOB data consists of a series of values of A for different 



