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Transcript
What does radar measure?
Hydrometeors: rain drops, ice particles
Other objects: e.g. birds, insects.
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.
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
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
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)
p
Einc
incident
plane
wave
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.
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
(1)
2
 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)
[The electric dipole moment for
a pair of opposite charges of
magnitude q is defined as the
magnitude of the charge times
the distance between them and
the defined direction is toward
the positive charge.]
Combining (1) and (2) we get:
 2 KD3 Einc
Er 
2
2 r
(3)
Sr
  4r
Sinc
2
To determine the radar cross section
(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
 r 1
K
r  2
Values of
K
where
1
r 
0
Permittivity of medium
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
Vc
(6)
The quantity
6
D
 j
is of utmost importance in radar meteorology
j
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.
Review of the impact of the 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)
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
Reflectivity calculation for hail storms from
different radar frequencies
Reflectivity calculation for hail storms from
different radar frequencies
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
One would think the standard units of Z would be
m6/m3 = m3
Vc
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