CONCH SIONS 



I Oil 



Azimuth 



Elrrors in azimuth arise from horizontal variations 

 in the index of refraction in the atmosphere, (ienerally 

 these variations are of insufficient magnitude to cause 

 such errors to be appreciable. In order to obtain an 

 error of 0.05° in 50,000 yd it can be shown that a 

 change in the index of refraction of 1.5 X 10~ 6 unit 

 in 44 yd perpendicular to the path of propagation 

 is required. This corresponds to an increase of 1C 

 temperature and a decrease of 0.1 mb in vapor 

 pressure. Such changes within 44 yd may occur in 

 propagation parallel to a sea coast or to a sharp 

 cold front, or in isolated regions such as between 

 forest and meadow, valley and plain, or land and 

 water surfaces. Except in the vicinity of a cold front 

 or sea coast it is unlikely that such horizontal gradi- 

 ents of the index of refraction exist along the entire 

 path of the ray. 



Range 



Errors in range due to vertical refraction within 

 a duct are approximately of the order of 1 yd in 

 50,000 yd. The error in range corresponding to an 

 azimuth error of 0.05° is estimated at less than 0.2 

 yd in 50,000 yd. 



13.4 



CONCLUSIONS 



1. For gunlaying (antiaircraft) radar, the maxi- 

 mum error in angle of elevation at 29,000-yd range 

 is 0.9 mil, as compared to a military tolerance of 

 1.5 mil. 



2. In early warning heightfinding radar, errors of 

 1,000 ft absolute altitude at 75 miles range may be 

 exceeded even with the application of a standard 

 atmosphere (% earth radius) correction. Because of 

 ducts, errors may be as much as 2,000 ft at 50 miles. 

 Errors in relative altitude may likewise exceed 500 

 ft in 75 miles. 



3. Errors in azimuth may exceed 0.05° in 50,000 

 yd in propagation parallel to a sea coast or a cold 

 front. Errors of this magnitude will, however, be rare. 



4. Errors in range are negligible for all possible 

 meteorological situations. 



Derivaton of Formulas 



Let the origin of the coordinate system be the 

 point where a ray is initially tangent to a line of 

 constant index of refraction n , and let the Y axis 



coincide with this line. Since the ray curves toward 

 higher index of refraction n, according to Snell's law: 



ii cos (8 = n B , (1) 



where /3 is the angle the ray makes with the line n. 

 Then from trigonometric relations: 



rlX y/n- — n% 



tan j3 = 



d\ 



■\A 



n 



Wo y/n + n 



n 



(2) 



Since n and n are extremely close to unity no appre- 

 ciable error will result if we assume that n + n B = 2, 

 hence 



clX 

 dY 



V2 



y/r, 



n 



Assuming a linear variation of the index of refraction 

 in the X direction, n = n -(- uX, and 



Y = 



Y 2 = 



n» 



dX 

 y/2u) a y/X 



2 n% X 



= n 



y/2X 

 y/o> 



(3) 



Equation (3) indicates that the ray follows a para- 

 bolic path. Let us convert into polar coordinates by 

 the transformation X = r sin <j> and Y = r cos <j>, 

 where r is the actual range. Then the equation of 

 the path becomes 



2», ! 





tan 4> sec 4> 



(4) 



Since in actual practice, <j> is extremely small and n 

 is extremely close to unity, equation (4) can be 

 written as 



tan <f> = 



(5) 



Here u represents the rate of change of the index of 

 refraction perpendicular to the ray, and <j> is the 

 error. If the ray were initially at an angle a to the 

 line of equal index of refraction, then the rate of 

 change of the index of refraction perpendicular to 

 the ray would be u cos a. Hence, more generally, the 

 equation for the path of a ray at a mean angle a to 

 the lines of index of refraction can be written as 



tan (j> 



cos a 



(6) 



Equation (6) has been utilized to compute errors in 

 azimuth and angle of elevation. 



