eo 
ANALYSIS OF ERRORS IN RADAR 67 
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 %o, and let the Y axis 
coincide with this line. Since the ray curves toward 
higher index of refraction m, according to Snell’s law: 
ncosB = n , (1) 
where 8 is the angle the ray makes with the line 7. 
Then from trigonometric relations: 
dX _ Vn? — 1? 
= mn 
tan B = 
Q 
nt 
_ Vn= mH Vn Fr 
No 
(2) 
Since 7 and 7» are extremely close to unity no appre- 
ciable error will result if we.assume that m + n) = 2,. 
hence 
n— MN. 
dX v2 
a RAN 
Assuming a linear variation of the index of refraction 
in the X direction, m = 7) + wX, and 
if =+|,4 aX ny W2X 
Vu eV 
y2 = 2 ni X 
(3) 
2) 
Equation (3) indicates that the ray follows a para- 
bolic path. Let us convert into polar coordinates by 
the transformation X =r sin @ and Y =r cos ¢, 
where r is the actual range. Then the equation of 
the path becomes 
= 2h 
r 
tan ¢ sec ¢. (4) 
Since in actual practice, ¢ is extremely small and np 
is extremely close t6 unity, equation (4) can be 
written as 
tan ¢ = me (5) 
Here w represents the rate of change of the index of 
refraction perpendicular to the ray, and @ 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 w 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 ¢ = = cosa. (6) 
Equation (6) has been utilized to compute errors in 
azimuth and angle of elevation. 
