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Weather radar equations
To convert equations for distributed targets into weather radar
equations, we must determine the radar reflectivity of arrays of
precipitation particles.
This problem can be divided into three parts:
(a) Finding the radar cross of a single particle;
(b) Finding the total radar cross section for the entire contributing
region
(c) Dividing the total cross section by the effective volume of the
contributing region to obtain the average radar reflectivity havg
First Assumption: Particles are all spheres
Small raindrops and cloud droplets:
Large raindrops:
Ice crystals
Graupel and rimed particles
Hail
Spherical
Ellipsoids
Varied shapes
Can be spherical
May or may not be spheres
The scattering properties and radar cross
sections of spherical particles have been
calculated and are well understood.
Second assumption: The particles are sufficiently small compared to
the wavelength of the impinging microwaves that the scattering can
be described by Raleigh Scattering Theory
How small is small? From the figure above, the radius of the
particle, a, must be

a
2
(~ 1/6 of the wavelength)
What is the fundamental difference between the Rayleigh, Mie,
and Optical regimes?
With Rayleigh scattering,
the electric field is
assumed to be invariant
in the vicinity of the
particle
Einc
incident
plane
wave
p
Dielectric
Sphere
(water drop)
A plane wave with electric field Einc induces an electric
dipole p in a small sphere. The induced dipole is
parallel to the direction of Einc which is also the
direction of polarization of the incident wave.
The angular patterns of the scattered intensity from particles of
three sizes: (a) small particles, (b) large particles, and (c) larger
particles
Rayleigh scattering pattern
From Rayleigh scattering theory, the dipole moment p induced in
a spherical particle is proportional to the particle’s volume (D3),
the material the particle is made of (K: ice or water) and the
magnitude of the incident electric field (Einc).
p
0 KD Einc
3
2
(1)
 0  8.85 1012 Farads / m
And the intensity of the scattered electric field at the location of
the particle is:
p
Er  2
  0r
(2)
Combining (1) and (2) we get:
 2 KD3 Einc
Er 
2
2 r
(3)
Sr
To determine the radar cross section   4r
we
Sinc
2
(a) divide (3) by Einc
(b) Square both sides of the resulting equation
(c) Multiply by 4r2
 K D6

4
5
2
(4)
2
 Er   S r 

  

 Einc   Sinc 
What is K?
K is a complex number representing the scattering (real part) and
absorption characteristics of the medium
 1 Permittivity of medium
 r  1 where
 
K
r  2
Values of
K
r
0
Permittivity of vacuum
2
Water
 = 10 cm
 = 3.21 cm
 = 1.24 cm
 = 0.62 cm
20C
0.9280
0.9275
0.9193
0.8926
10C
0.9340
0.9282
0.9152
0.8726
0C
0.9340
0.9300
0.9055
0.8312
Temperature
Ice
0.176 for ice particles (0.208 is used when snowflake sizes
are expressed as the diameters of water drops obtained by melting the ice).
 K D6

4
2
5
The radar cross section
(4)
For an array of particles, we determine the average radar cross section
 K
   j 
4

j
5
2
6
D
 j
(5)
j
Now we determine the radar reflectivity:
h 
 j
j
Vc
5 K

4
2
6
D
 j
j
(6)
Vc
The quantity
6
D
 j
j
is of utmost importance in radar meteorology
Vc
It is designated with the symbol Z, and is called the
radar reflectivity factor
In logarithmic units:
dBZ  10 log( Z )
It is the quantity that is displayed on a radar screen.
Relationship between the radar reflectivity and the radar reflectivity factor:
h 

j
Vc
j
5 K Z

4
2
(7)
Recall the radar equation for a distributed target:
c
Pr  2
 1024 ln( 2)
PG  
2 2
t
Combining:
2
2



PtG   K Z 
c


Pr 
2
 r 2 
1024 ln( 2) 



3
h 
 r 2 
THE RADAR EQUATION FOR WEATHER TARGETS
2
2 



1024 ln( 2)

Pr r 



Z
2
3
2



c
 PtG   K 
constants
Radar
characteristics
Target
characteristics
where Z in normally expressed in logarithmic units


Z

dBZ  10 log 
6
3 
 1 mm / m 
The Weather radar equation: review of the assumptions
 r 2 Pr 
1024 ln( 2) 
2


Z
2
3
2


c
P

G

K
 t


1. The precipitation particles are homogeneous dielectric spheres
with diameters small compared to the radar wavelength
2. The particles are spread throughout the contributing region. If
not then the equation gives an average reflectivity factor for the
contributing region.
3. The reflectivity factor Z is uniform throughout the contributing
region and constant over the period of time needed to obtain the
average value of the received power.
The Weather radar equation: review of the assumptions
 r 2 Pr 
1024 ln( 2) 
2


Z
2
3
2


c
P

G

K
 t


4. All of the particles have the same dielectric factor; that is, they
are all either water droplets or ice particles.
5. The main lobe of the antenna is adequately described by a
Gaussian function.
6. Microwave attenuation over the distance between the radar
and the target is negligible.
7. Multiple scattering is negligible. Multiple scattering and
attenuation are related so if one is true the other is too.
8. The incident and back-scattered waves are linearly polarized.
Validity of the Rayleigh Approximation for weather targets
Valid
 = 10 cm
Raindrops: 0.01 – 0.5 cm (all rain)
Snowflakes: 0.01– 3 cm (most snowflakes)
Hailstones: 0.5 – 2.0 cm (small to moderate hail)
 = 5 cm
Raindrops: 0.01 – 0.5 cm (all rain)
Snowflakes: 0.01– 1 cm (small snowflakes)
Hailstones: 0.5 – 0.75 cm (small hail)
 = 3 cm
Raindrops: 0.01 – 0.5 cm (all rain)
Ice crystals: 0.01– 0.5 cm (single crystals)
Graupel: 0.1 -- 0.5 cm (graupel)
 = 0.8 cm
Raindrops: 0.01 – 0.15 cm (cloud and drizzle drops)
Ice crystals: 0.01– 0.15 cm (single crystals)
Validity of the Rayleigh Approximation for weather targets
Invalid
 = 10 cm
Hailstones: > 2.0 cm (large hail)
 = 5 cm
Snowflakes > 1 cm (large snowflakes)
Hailstones: > 0.75 cm (moderate to large hail)
 = 3 cm
Raindrops: 0.01 – 0.5 cm (all rain)
Snowflakes > 0.5 cm
Hail and large graupel
 = 0.8 cm
Drops > 100 microns
All ice particles except small crystals
When the assumptions built into the radar equation are not
satisfied, the reflectivity factor is referred to as:
The Equivalent Radar Reflectivity Factor, Ze
 r 2 Pr 
1024 ln( 2) 
2


Ze 
2
3
2

c
P

G

 t
 K 
Units of Z
Z
6
D
 j
j
Vc
One would think the standard units of Z
would be m6/m3 = m3
But no…
The standard units for Z are mm6/m3
If these units are not used, you will be off by 10-18
Range of radar reflectivity factor in weather echoes
WSR-88D
Precipitation
Mode
WSR-88D
Clear Air
Mode


Z

dBZ  10 log 
6
3 
 1 mm / m 
75 dbZ = giant hail
log Z   7.5
Z  107.5  31,622,777
45-50 dbZ = heavy rain
log Z   5
Z  105  100,000
25 dbZ = snow
log Z   2.5
Z  10 2.5  316
-28 dbZ = haze droplets
log Z   2.8
Z  10 2.8  0.001585
Nebraska record hailstorm 2003 75 dBZ
Heavy rain in Hurricane Andrew’s Eyewall = 45 dBZ
Snowstorm over Great Lakes: ~ 25-30 dBZ
Clear air echoes (few small insects) -12 dBZ