THEORY AND OBSERVATION OF RADAR STORM DETECTION 
By RAYMOND WEXLER 
Imperial College of Science and Technology 
Theory of Storm Detection 
The basic problem in radar storm detection is the 
absorption and scattering of a plane electromagnetic 
wave by a sphere. This problem was first studied by 
Mie [8] who analyzed the absorption and scattering of 
light from gold particles suspended in a liquid. Applica- 
tion to the theory of the detection of precipitation was 
first made by Ryde [11]. The Canadian Army Opera- 
tional Research Group showed that the theory was 
valid, at least in respect to order of magnitude, by 
measurement at the ground of the drop size im rain- 
storms observed by radar. More exact laboratory veri- 
fication is presently being undertaken at Harvard Uni- 
versity. 
The equation for detection of a particle of scattering 
cross section o at distance R by electromagnetic energy 
propagated isotropically from its source is given by 
z ° ee =) (1) 
where P, is the power received by the radar and P is 
the peak power emitted by the radar antenna of effect- 
ive area A. If the radiation is directional, a gain factor 
must be included in the numerator of the right-hand 
side of the equation. This gain is derived from the fact 
that the radiation is propagated as a beam from the 
antenna and is G times the radiation from an isotropic 
source. Because of the fact that the beam is normally 
completely intercepted by the cloud at ranges under 75 
miles, the directional aspect need not be considered. 
The term P/4rR? will be recognized as the power 
density of the propagated radiation, and P,/A the 
power density of radiation received at the antenna. The 
radar cross section o is defined by the equation. It is 
the area normal to the radiation such that if all the 
power incident on it were scattered isotropically it 
would give, at the receiver, the same power as the ob- 
served echo. 
For complete interception of the beam by the cloud, 
the equation must summate the total number of scat- 
terers in the volume of air illuminated by the pulse. 
This volume V may be considered to be a spherical 
shell at radius R of half a pulse-length thickness: 
V = 4cR'h/2, (2) 
where h is the pulse length of the radar. The reason for 
the factor 1/2 is that energy is emitted from a time 
t = 0 to t = h/c, the pulse duration, where c is the 
velocity of light. In that time the energy must make a 
1. For a derivation using the directional aspect see, for 
example, Wexler and Swingle [16]. 
round-trip distance 2R. The maximum minus the mini- 
mum distance illuminated by one pulse is then 
2(R + AR) — 2R = (+ h/c)e — te, 
whence the distance illuminated by the pulse is AR 
= h/2. 
If within a unit volume there are n; scatterers of cross 
section o; all located at random within the beam, then 
the equation becomes, by combining (1) and (2), 
Pi ee aia:, (3) 
TL 4 
If the beam is not completely intercepted by the cloud, 
the percentage of the beam so intercepted must be in- 
serted on the right-hand side of equation (8). 
The equation holds for scatterers randomly located 
within the beam on the assumption that the distances 
between drops are large compared to the diameter of 
the drops and the wave length of radiation. If there were 
forces causing the drops to fall coherently in pairs, the 
power received would be twice that indicated by equa- 
tion (3). However, the assumption of random positions 
of the drops in a rain cloud appears to be justified. 
The equation for storm detection including the effect 
of attenuation is 
IP, = fee 3 NO, lO paaes (4) 
8rh? F 
where K is the attenuation in decibels per unit distance. 
If P, is the minimum detectable signal of the radar, 
a figure known for most sets, the distance R in (4) re- 
presents the maximum range of detection for the tar- 
get Unio; . 
From Mie’s study of the diffraction of electromag- 
netic waves by a sphere, it is possible to determine the 
radar cross section in the form of an infinite series of 
spherical Bessel functions of parameter mmD/), where 
m is the complex index of refraction for the precipita- 
tion at a given wave length d, and D is the diameter of 
the drop. For values of D/X < 0.05, the Rayleigh law 
of scattering is approximately valid: 
en: 
os = (<3) ye? (5) 
where e, the dielectric constant, is equal to the square 
of the index of refraction. At D/X = 0.08, the value of 
c is about 0.8 that of Rayleigh; it then rises to a value 
twice that of Rayleigh at D/X = 0.2, beyond which 
it undergoes rapid and large fluctuations about a con- 
stant value near +D2/4, the radar cross section of a 
large conducting sphere. This behavior varies somewhat 
according to the wave length, but qualitatively it char- 
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