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 avg

First Assumption: Particles are all spheres

Small raindrops and cloud droplets: SphericalLarge raindrops: EllipsoidsIce crystals Varied shapesGraupel and rimed particles Can be sphericalHail 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

2a (~ 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

incidentplanewave

DielectricSphere

(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.

p

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).

2

30 incEKD

p

mFarads /1085.8 120

(1)

And the intensity of the scattered electric field at the location of the particle is:

r

pEr

02

(2)

Combining (1) and (2) we get:

r

EKDE incr 2

32

2

To determine the radar cross section we inc

r

S

Sr 24

(a) divide (3) by Einc

(b) Square both sides of the resulting equation(c) Multiply by 4r2

(3)

inc

r

inc

r

S

S

E

E2

4

625

DK

(4)

What is K?

K is a complex number representing the scattering (real part) and absorption characteristics of the medium

2

1

r

rK

where

0

1

r Permittivity of medium

Permittivity of vacuum

Values of 2

K

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

Water

Ice

0.176 for ice particles (0.208 is used when snowflake sizesare expressed as the diameters of water drops obtained by melting the ice).

4

625

DK

(4)The radar cross section

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

j

jj

j DK 64

25

(5)

Now we determine the radar reflectivity:

c

jj

c

jj

V

DK

V

6

4

25

(6)

The quantity is of utmost importance in radar meteorology

c

jj

V

D 6

It is designated with the symbol Z, and is called the

radar reflectivity factor

In logarithmic units:

)log(10 ZdBZ

It is the quantity that is displayed on a radar screen.

Recall the radar equation for a distributed target:

2

2

2

23

)2ln(1024 r

ZKGPcP tr

4

25

ZK

Vc

jj

Relationship between the radar reflectivity and the radar reflectivity factor:

(7)

Combining:

222

2 )2ln(1024 rGP

cP tr

2

2

2

2

3

)2ln(1024

K

rP

GPcZ r

t

THE RADAR EQUATION FOR WEATHER TARGETS

constants Radarcharacteristics

Targetcharacteristics

where Z in normally expressed in logarithmic units

36 /1log10

mmm

ZdBZ

2

2

2

2

3

)2ln(1024

K

Pr

GPcZ r

t

The Weather radar equation: review of the assumptions

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.

2

2

2

2

3

)2ln(1024

K

Pr

GPcZ r

t

The Weather radar equation: review of the assumptions

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

= 5 cm

= 3 cm

= 0.8 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)

Raindrops: 0.01 – 0.5 cm (all rain)Snowflakes: 0.01– 1 cm (small snowflakes)Hailstones: 0.5 – 0.75 cm (small hail)

Raindrops: 0.01 – 0.5 cm (all rain)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)Ice crystals: 0.01– 0.15 cm (single crystals)

Validity of the Rayleigh Approximation for weather targets

Invalid

= 10 cm

= 5 cm

= 3 cm

= 0.8 cm

Hailstones: > 2.0 cm (large hail)

Snowflakes > 1 cm (large snowflakes)Hailstones: > 0.75 cm (moderate to large hail)

Raindrops: 0.01 – 0.5 cm (all rain)Snowflakes > 0.5 cm Hail and large graupel

Drops > 100 micronsAll 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

2

2

2

2

3

)2ln(1024

K

Pr

GPcZ r

te

Units of Z

c

jj

V

D

Z

6

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-88DPrecipitation

Mode

WSR-88DClear Air

Mode

36 /1log10

mmm

ZdBZ

75 dbZ = giant hail 5.7log Z777,622,3110 5.7 Z

-28 dbZ = haze droplets 8.2log Z

001585.010 8.2 Z

45-50 dbZ = heavy rain

5log Z000,100105 Z

25 dbZ = snow

5.2log Z31610 5.2 Z

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