What does radar measure?

Hydrometeors: rain drops, ice particlesOther 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 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.

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

2a (~ 1/6 of the wavelength)

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

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)

[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:

r

EKDE incr 2

32

2

To determine the radar cross section

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.

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

2a (~ 1/6 of the wavelength)

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

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

2

2

2

2

3

)2ln(1024

K

Pr

GPcZ r

te

Units of Z

c

jj

V

D

Z

6One 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

Echoes in clear air from insects

Common is summer. Watch for echoes to expand area as sun sets and insects take off for their nocturnal travels

Echoes from birds and bats

Birds take off radially from roosts in morning producing concentric rings of echo. Bats do the same at night. Migrating birds will produce small echo clusters that move across screen.

Birds departingAt 1114 UTC

Blocked beams (topography or buildings)

Blockedbeam

D istan c e (km )

R ef le ct iv ity fa cto r (d B Z )

S tr a t i fo rm a rea C on v ec t io n

B BAltitude (km)

The bright band an enhancement of radar reflectivity at the melting level when snowflakes fallingFrom above aggregate and develop wet surfaces.

Note convectiveand stratiform regionsof squall line. Precipitationestimates in stratiformregion must be carefullyexamined because of bright band effects

Why there is a bright band?

Recall |K2| for water is about 0.93 and for ice is about 0.176

When ice particles falling below 0oC, they start to melt, beginning from the surface. Therefore, they keep the size of the ice particle but coated with water. So the radar received power difference would be:

P ice coated with water / P ice = 0.93/0.176 = 5.284 dBZ Zice coated with water = 10log Zice *5.284 = 10logZice + 10log(5.284) = dBZ Zice + 7.2 dBZ

After totally melted, the ice particles becoming rain drops. The density increased from ~0.4 g/cm3 to 1 g/cm3 and the particle size decreased, so the reflectivity decreased.

The bright band appears as a ring on PPI displays where the radar beam crosses the melting level

An extreme example of bright band contamination of precipitation estimation – radar estimates 6 inches of precipitation in a winter storm on January 31, 2002!

Different radar displays

NEXRAD (WSR-88D)Operations and Scanning

Procedures

Plan-Position Indicator (PPI) Scanning procedure

Echoes close to the radar are at a low elevation

Echoes far from the radar are at a high elevation

Data collected on a cone are projected onto a plane

Precipitation mode scan geometry Severe weather scan geometry

Saves time…fewer elevations

WSR-88D Volume Scanning Procedures

Clear air mode: Fewer elevations, slower antenna rotation toachieve greater sensitivity for sensing clear air turbulence, insectsand clouds, light drizzle or light snowfall.

65 55 45 35 25 15 5

Reflectivity Factor (dBZ)

Radar reflectivity: A measure of the power scattered back to the radar from objects in the path of a radar beam. Proportional to the sum of the sixth power of the diameter of all the particles illuminated by a pulse provided the particles are much smaller than the radar wavelength.

Base reflectivity: Echo intensity at the lowest scan level (0.5o) measured in dBZ

Composite reflectivity: Maximum echo intensity at any scan level measured in dBZ

“Precipitation” mode

“Clear air” mode(often used for snow)

Radar parameters set to increase sensitivity

Storm Total Precipitation: Measured in inches fallen after an NWS selected start time.

Storm total precipitation is the primary tool for flash flood forecasting

Vertically integrated liquid (VIL): VIL is the integration of reflectivity within a column of air. Once thought to be related to potential hail size, but not really a good predictor.

Radial velocity: Velocity component along the radar beam direction (knots)

-51 -37 -24 -10 3 17 30 44

Wind Speed (knots)

12

24

0180 225 270 315 360

Hei

ght(

kft)

Hei

ght(

kft)

12

24

0

Wind speed (kt)0 20 40 60 80

12

24

0180 225 270 315 360

Hei

ght(

kft)

Hei

ght(

kft)

12

24

0

Wind speed (kt)0 20 40 60 80

-51 -37 -24 -10 3 17 30 44

Wind Speed (knots)

Radial Velocity

A measure of thecomponent of thewind along thedirection of theradar beam

Radial Velocity is important for detection of rotation in storms

Storm relative radial velocity: Radial velocity with the component of storm motion along the beam subtracted. Best display to see mesoscyclones, tornado vortex signatures and microburst signatures.

Radar reflectivity

Radial VelocityRadar reflectivity together withRadial velocity provide meansto identify potential tornadoes

NEXRAD PRODUCTS:

LEVEL 2 DATA: Radar reflectivity and radial velocity at original sampling resolution.

LEVEL 3 DATA: NEXRAD Information and Dissemination

System (NIDS) Products

Individual radarsBase reflectivity, Composite reflectivityStorm total precipitation, Vertically integrated liquidRadial velocity, Storm relative radial velocity

National or regional reflectivity composites

Other radar displays

Range-Height Indicator (RHI) Scanning procedure

Radar is scanned in elevation at a fixed azimuth

Volume scans are accomplished by rotating slowly in azimuthwhile scanning rapidly in elevation

A sequence of RHI scans showing the

development of precipitation in a small

cumulus cloud over Florida

Note ground clutter and echo from tall buildings (echo from radar side lobe)

Time-Height Display (Radar beam pointed vertically as storms move over the radar)

Winter snowbands move across vertically pointing radar

“Reconstructed” RHI displays

Vertical cross section through a mesoscale convective system reconstructed from scans from a PPI volume from a NEXRAD

reflectivity

Radial velocity

Horizontal cross sections

Radar data are interpolated from cylindrical coordinates to Cartesian coordinates and displayed in Cartesian space

If more than one Doppler radar scans the storm, winds can be derived using multiple-Doppler analysis techniques that we will discuss later in the course.

Note location of cross section

Vertical cross sections

Cartesian data can be used to construct vertical cross sections similar to the RHI

If more than one Doppler radar scans the storm, vertical winds can also be derived using multiple-Doppler analysis techniques.

Radar composites: Data from PPI scans from many radars are projected onto a single image to give a map of precipitation distribution.

On the display below, surface data are used to recolor the echo to show where snow and mixed precipitation are occurring. Radar data are not used to make this distinction.

National Weather Service composite image

Assignment #2 If air with precipitation water content of 1 g/m3, assuming mono size distribution (all hydrometeors have same size) How many raindrops would be in 1 m3 (total number density N) if all raindrops have diameter (D) of 1 mm? Calculate 10log(N*D6), units in mm6/m3. How many snowflakes would be in 1 m3 if all snowflakes have diameter of 5 mm and density in 0.3 g/cm3? Calculate 10log(N*D6), units in mm6/m3. How many hails would be in 1 m3 if all hails have diameter of 10 mm and density in 0.9g/cm3? Calculate 10log(N*D6), units in mm6/m3.