WEATHER SIGNALSChapter 4

(Focus is on weather signals or echoes from radar resolution volumes filled with

countless discrete scatterers---rain, insects, perturbations in atmospheric

refractive index, etc.)

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Weather Signal Characteristics

• Large dynamic range (100 million!)

• Signals are semi coherent

• Numerous scatterers in the radar resolution volume

The choice instrument for weather surveillance is the pulsed polarimetric Doppler weather radar.

Extracting information (i.e., fields of echo H, V power,Doppler velocity, and correlation of H, V echoes)involves processing of echoes that randomly fluctuate.

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Echoes (I or Q) from Distributed Scatterers (Fig. 4.1)

Weather signals (echoes)

mT s

3

c(s) ≈ t (t = transmitted pulse width)

Resolution Volumes

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Range resolution 150 m

4

Lightning detector

Echo samples from 16 ResolutionVolumes (Fig. 4.3)

Ts = 768 ms

5

Gate 12 Signal Spectrum (Fig. 8.34)

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MTs = 128x0.768 ms = 98.3 ms

12 spectra are averaged

6

Repeat of Fig.

4.3

7

c(mTs) ≈ 2-4 ms

Statistics of Weather Signals

I and Q are uncorrelated zero mean

random variables with

Gaussian probability

density function (pdf)

P has an exponential

pdf

Amplitude|V| has a

Rayleigh pdf

β has a uniform pdf

βThanks to Dr. Sebastian Torres

8

Weather Echo Statistics (Fig. 4.4)

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Weighting Functions for Scatterers and the Weather Radar Equation

22 22 4

3 2 20 0 0

2 3

0

( )( ) | ( ) | ( , )sin

(4 ) ( )

where is the reflectivity (i.e., backscatter cross section

per unit volume-- assumed spatially uniform):

( ) ( ) ( , ) (

or

t or o s

o o

b

Pg rP r W r dr d f d

r r

D N D dD m m

r r

-3 -1

2

s

4

) (4.10)

and ( , ) size distribution (m mm )

( ) the range weighting function;

( , ) theangular weighting function.

N D

W r

f

r

(4.12)

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Drop Size Distributions (Fig. 8.3b)

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(1948)(1943)

(From drop sizes between 1 and 3 mm)

0

3 3 10

( ) exp( ) EXP.DSD

8 10 m mm (MP DSD)

N D N D

N

11

The Angular Weighting Function

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The Measured Range Weighting Function for two Receiver

Bandwidths (Fig. 4.6)

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Range Weighting Function for Echoes Samples at Range Time τs

(Fig.4.7)

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250 m

14

The Resolution Volume V6

Fig. 5.11

Angular weighting function

Range weighting function

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Spectrum of a transmitted rectangular pulse

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If receiver frequency response is matched to the spectrum of the transmitted pulse (an ideal matched filter receiver), some echo power will be lost. This is called the finite bandwidth receiver loss Lr. For and ideal matched filter Lr = 1.8 dB.

16

ft f

1tf

Receiver Loss Factor (Fig. 4.8) for a Gaussian receiver response and rectangular pulse

(in general a matched condition is when B6 =1)

Eq.(4.28)

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B6= 6 dB bandwidth of the receiver’s frequency response. = transmitted pulse width

17

Lr = 1.8 dB for an ideal matched receiver

Reflectivity Factor Z(Spherical scatterers; Rayleigh condition: D ≤ λ/16)

52

m4

6 6

0

52

w e4

2 ow

2i

( ) | | ( ) (4.31)

where

1( ) ( , ) (4.32)

( ) | | ( ) (4.33)

for water drops : | | 0.93 independent of T( C);

for ice particles : | | 0.16 dependent on T and icedensity.

ii

K Z

Z D N D D dDV

K Z

K

K

r r

r r

r r

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Reflectivity Factor of Spheroids(Horizontally Polarized Waves)

p[De,e,ξ] = probability density

σh[De,e,ξ] = backscatter cross section for H pol.De = equivalent volume diametere = eccentricity of the spheroid scattererξ = angle between the symmetry axis and the electric field directionN0 = the number density per unit diameter (m-4)

deddDeDeDpK

NZ eehe

e

oh

],,[],,[0

25

4

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Differential Reflectivity

in dB units:ZDR(dB) = Zh(dBZ) - Zv(dBZ)

in linear units:Zdr = Zh(mm6m-3)/Zv(mm6m-3)

- is independent of drop concentration N0

- depends on the shape of scatterers

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Shapes of raindrops falling in still air and experiencing drag force deformation.

De is the equivalent diameter of a spherical drop. ZDR (dB) is the differential reflectivity in decibels(Rayleigh condition is assumed). Adapted from Pruppacher and Beard (1970)

The Weather Radar Equation

0

w

A form of the weather radar equation for echo power from rain is:

5 17 2 2 2 6 -310 (W) ( s) (deg.) | | (mm m ) t s 1 w

( ) (mW)0 14 2 2 26.75 2 (ln 2) (km) (cm)0 r

( ) Expected peak weather signal power in

(4.35)P g g K Z

r

E P

E P

r

r

s

2

w

t

1 | |

milliwatts;

Peak transmitted pulse power (typically 500 kW)

net power gain of the echo in going from the antenna to the radar output. = pulse width

one-way half-power beamwidth; dielect

P

K

g

6w 0

ric factor of water

reflectivity factor for water spheres; range (in km) to the center of the resolution volume V

one-way loss factor (a number 1) incurred for propagation through a rain fil

Z r

r

led atmosphere.

loss factor due to the finite bandwidth of the receiver; wavelength of the transmitted radiation

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Acquisition, Processing and Display of Weather radar data

1

2

M

Range-time sTime series of M power samples

Base Data: Ph, Pv,v, SW, etc.

Azimuthal Scan(constant elevation)

Volume Coverage PatternProduct displays (e.g., CAPPI, etc.)

Sam

ple-

tim

e

s

Data ra

dial

Radial of reflectivity factor Z

(M power samples are processed to produce one Z estimate at r)

Range r = cs/2

(I2 + Q2)

WSR-88D Thresholding Data Fieldsbased on Signal to Noise Ratios

Locations with non-significant powers are censored:

-Non significant returns have a SNR below a user-defined threshold

-The system allows a different threshold for each spectral moment

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SNR> -3dB

Z (

dBZ

)

SNR> 2dB

Z (

dBZ

)

26

SNR> -0.5 dB

SNR> 3.5 dB

Weather Echo Power vs Range (WSR-88D)

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