Weather radar equation

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: Spherical

Large raindrops: Ellipsoids

Ice crystals Varied shapes

Graupel and rimed particles Can be spherical

Hail 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 (~ 1/6 of the wavelength)

2

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

p

Einc





Dielectric

incident Sphere

plane (water drop)

wave









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 (D 3),

the material the particle is made of (K: ice or water) and the

magnitude of the incident electric field (Einc).



 0 KD Einc3

p (1)  0  8.85  10 12 Farads/ m

2



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 KD 3 Einc

Er  (3)

22 r

Sr

To determine the radar cross section   4r

2

we

S inc



(a) divide (3) by Einc 2

 Er   S r 

E  S 

(b) Square both sides of the resulting equation    

(c) Multiply by 4r2  inc   inc 





 K D6

5 2



 (4)

4

What is K?



K is a complex number representing the scattering (real part) and

absorption characteristics of the medium

 r 1  1 Permittivity of medium

K where  

r  2 0

r

Permittivity of vacuum

2

Values of K

Water

Temperature  = 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



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

5 2



The radar cross section  (4)

4



For an array of particles, we determine the average radar cross section



 K 5 2



   j   j

D6 (5)

j 4 j







Now we determine the radar reflectivity:



 j 5 K

2  j

D6

h  j



j (6)

Vc 4 Vc

The quantity  j

j

D6

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:



 j

5 K Z

2



h  j

 (7)

Vc 4



Recall the radar equation for a distributed target:



h 

Pr  2

c

 1024 ln( 2)

PG   

t

2 2

r2 

 



Combining:



 PtG 2  K Z 

2

c 3

  

Pr 

1024 ln( 2) 

 2  r 2 

 

THE RADAR EQUATION FOR WEATHER TARGETS





1024 ln( 2)  2

 Pr r 2 

Z   2 

c 3  PtG 2  K 

  

constants Radar Target

characteristics 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



1024 ln( 2)  2  r 2 Pr 

Z   2 

c 3  PtG 2  K 

  





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



1024 ln( 2)  2  r 2 Pr 

Z   2 

c 3  PtG 2  K 

  



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

Raindrops: 0.01 – 0.5 cm (all rain)

 = 10 cm Snowflakes: 0.01– 3 cm (most snowflakes)

Hailstones: 0.5 – 2.0 cm (small to moderate hail)



Raindrops: 0.01 – 0.5 cm (all rain)

 = 5 cm Snowflakes: 0.01– 1 cm (small snowflakes)

Hailstones: 0.5 – 0.75 cm (small hail)



Raindrops: 0.01 – 0.5 cm (all rain)

 = 3 cm Ice crystals: 0.01– 0.5 cm (single crystals)

Graupel: 0.1 -- 0.5 cm (graupel)



Raindrops: 0.01 – 0.15 cm (cloud and drizzle drops)

 = 0.8 cm

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)





Snowflakes > 1 cm (large snowflakes)

 = 5 cm

Hailstones: > 0.75 cm (moderate to large hail)



Raindrops: 0.01 – 0.5 cm (all rain)

 = 3 cm Snowflakes > 0.5 cm

Hail and large graupel



Drops > 100 microns

 = 0.8 cm

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





1024 ln( 2)  2  r 2 Pr 

Ze    2 

c 3  PtG 2  K 

  

Units of Z

 j

D6 One would think the standard units of Z

Z

j

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 WSR-88D

 Z 

Precipitation Clear Air

 1 mm6 / m3 

dBZ  10 log 

Mode Mode

 

75 dbZ = giant hail log Z   7.5

Z  10 7.5  31,622 ,777

45-50 dbZ = heavy rain

log Z   5

Z  10 5  100 ,000



25 dbZ = snow

log Z   2.5

Z  102.5  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