Chapter 9 
THEORETICAL ANALYSIS OF ERRORS IN RADAR 
DUE TO ATMOSPHERIC REFRACTION 
PURPOSE 
HIS REPORT is a theoretical evaluation of errors 
in altitude, azimuth, and range caused by atmos- 
pheric refraction. These errors are compared with 
the error tolerance specified in military characteris- 
ties for fire control radar equipment. Regional 
climatological data are utilized to determine probable 
refractive index gradients used in the determination 
of the error. Errors in heightfinding resulting from 
ducts are also treated. An Evans Signal Laboratory 
[ESL] report now under preparation discusses errors 
which may occur during specific meteorological 
situations and which may exceed the errors indicated 
in this report. 
PROCEDURE 
The variation of the index of refraction perpen- 
dicular to the path of a radio wave results in a 
curvature of the ray toward the higher index. The 
curvature of the ray is approximately equal to the 
rate of decrease of the index of refraction with 
altitude. Errors due to atmospheric refraction will 
therefore depend on the rate of decrease of the 
index of refraction perpendicular to the ray path 
and to the range. A simplified equation for the error 
in azimuth and altitude is derived below and is 
utilized in this report. This method has been found 
to check to within a thousandth of a degree with 
more accurate methods” of ray tracing.° 
The rate of decrease of the index of refraction in 
a standard atmasphere is 12 X 10% unit per 1,000 
ft up to 4,000 ft above mean sea level. Thig corres- 
ponds to a curvature of the path of the ray approxi- 
mately one fourth the curvature of the earth.’ The 
standard atmosphere represents average conditions 
in temperate zones. In tropical air such as exists in 
equatorial regions and southeast Asia and southeast — 
United States in summer, the average rate of decrease 
of the index of refraction is approximately 18 x 10° 
unit per 1,000 ft up to 6,000 ft corresponding to a 
curvature of the ray % that of the earth. Over trade 
wind regions of the ocean (latitude 10° to 30°) dry 
*By Raymond Wexler, Signal Corps Ground Signal Agency. 
>Errors in angle of altitude due to a duct with a standard 
atmosphere above the duct have been computed by members 
of Group 42 of the Radiation Laboratory. Values computed by 
the method outlined below under Derivation of Formulas 
have been found to agree with their results. 
°For a more detailed analysis of ray tracing methods, see 
reference 75. 
subsiding air exists over a moist tropical layer. The 
rate of decrease of the index of refraction in these 
regions is approximately 24 X 107% unit per 1,000 ft 
corresponding to a curvature of the ray one-half that 
of the earth. Within layers of atmosphere designated 
as “ducts” the curvature of the ray may exceed the 
earth’s curvature and may result in a trapping of the 
ray within the duct. Errors due to atmospheric 
conditions in each of the above atmospheres are 
analyzed. In Table 1 are tabulated values of the 
index of refraction at selected levels for the standard 
atmosphere, tropical atmosphere, and tropical dry 
atmosphere as utilized in this report. 
TaBLeE 1. Values of the index of refraction for selected 
levels in different air masses.* 
(n — 1) 108; n = index of refraction. 
Elevation above 
mean sea level Standard Tropical Tropical air 
(feet) atmosphere atmosphere dry air above 
0 324 394 348 
2,000 300 358 300 
4,000 276 322 260 
6,000 255 286 242 
8,000 236 255 227 
10,000 219 234 216 
15,000 191 195 179 
20,000 151 159 146 
30,000 105 107 105 
*Aerological data for Miami and San Diego for July 1943 were utilized to 
compute the indices of refraction for the tropical atmosphere and the trop- 
ical atmosphere with dry air above, respectively. 
APPLICATION TO GROUND 
RADAR EQUIPMENTS 
GuNLAYING (ANTIAIRCRAFT) RADAR 
Military characteristics for gunlaying radar call 
for a tolerance of 50-yd error in a range of 29,000 
yd and an angle of 1.5 mils in azimuth and elevation. 
Initial angles of sight are between 10° and 90°. 
RESULTS 
In a standard atmosphere, errors in angle of eleva- 
tion for a range of 29,000 yd and an initial angle of 
sight 10° are 0.5 mil. A maximum error is obtained 
at 0.9 mil. For an initial angle of sight of 20° the 
maximum error is about 0.6 mil as compared to an 
error in a standard atmosphere of 0.4 mil. Errors in 
azimuth and range are negligible.. 
EarLy WARNING HEIGHTFINDING RADAR 
Military characteristics call for the following 
tolerances in heightfinding radars. 
