334 REFRACTION AND REFRACTIVE INDEX MODELS 



as elevation angle errors. Many methods have been proposed to take 

 into account these refraction effects for the purpose of improving measure- 

 ments by removing systematic bias. One of these involves the use of 

 the surface value of the radio refractivity, A''^, a quantity which can be 

 measured directly with a microwave refractometer, or calculated from 

 the ordinary meteorological variables of temperature, pressure, and 

 humidity, to predict values of either range error or elevation angle error; 

 this method has been shown theoretically to be useful, with the accuracy 

 increasing with increasing initial elevation angle [17, 18, 19]. It is the 

 purpose of the present note to compare recent experimental determina- 

 tions of atmospheric refractive effects with values estimated theoretically 

 from surface meteorological conditions. 



8.3.2. Theory 



The operation of a radio tracking system depends on the measurement, 

 in some manner, of radio signals received from the target. The radio 

 signals are transmitted in the form of radio waves which travel from the 

 target to the tracking system. The form of these radio waves is distorted 

 by the presence of the earth's atmosphere. Since solutions of the wave 

 equation are extremely difficult to obtain for the case of general atmos- 

 pheric propagation over a spherical earth, it is common practice to 

 evaluate refraction effects by means of ray tracing, a process which is 

 based on the use of Snell's law. 



One of the two types of refraction errors considered in this appendix 

 is the elevation angle error, c, which is the difference between the apparent 

 direction to a target, as indicated by the angle of arrival of a normal to 

 the radio wave front, and the true direction. This error is primarily a 

 function of the refraction, or bending, of the radio ray. For targets 

 beyond the atmosphere, the two quantities are asymptotically equal (with 

 increasing range). . The values of e and r at any point on the ray path 

 obey the following inequality: 



r/2 ^ e ^ T. 



Recalling that (chapter 3) the bending of a radio ray may be expressed 

 by an equation of the form 



T = a -\-hNs (8.18) 



where a and b would be functions of the initial elevation angle of the ray, 

 do, and the height (or range) along the ray path at which the bending is to 

 be calculated. Such an assumption can be checked by examining the 

 behavior of values of r, ray traced for a number of observed height profiles 

 of radio refractive index, plotted against the corresponding values of N s- 



