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ReviewDoppler Radar (Fig. 3.1)

A simplified block diagram

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Complex plane(Phasor diagram)

A

jQ(t)

I(t)ψe

ii

( , )exp 2 t

rE j f t j

r c

A

o o exp 4 / tV I jQ A j πr λ jψ

If the range r of the scatterer is fixed, phasor A is fixed.But if scatterer has a radial velocity, Phasor A rotatesabout the origin at the Doppler frequency fd.

rr 2

( , ) 2exp 2 t

rj f t j

r c

AE

Electric field incident on scatterer

Reflected electric field incident on antenna

i i

4exp 2 t

rV A j ft j j

Voltage input to the synchronous detectors;This pair of detectors shifts the frequency f to 0

Echo phase at the output of the detectors and filters

e ( 4 / ) tψ πr λ ψ

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Range-Time

Stationaryscatterers

Moving scatterer

1 μs

0

0

(A)

(B)

1

1

2

2

3

34

45

5

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Pulsed Radar Principle

cτ

λ

c = speed of microwaves = ch for H and = cv for V wavesτ = pulse lengthλ = wavelength = λh for H and λv for V wavesτs = time delay between transmission of a pulse and reception of an echo.

r=cτs/2

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For typical atmospheric conditions (i.e., normal) the propagation pathis a straight line if the earth has a radius 4/3rds times its true radius.

Normal and Anomalous Propagation

Super refraction (trapping)

(anomalous propagation:

Unusually cold moist air near the ground)

Sub refraction

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Free space

Normal atmosphere: Theray’s radius of curvature Rc ≈ constant)

2 2d d w1 w w2 w1n C P / T C P / T C P / T

refractive index n c / v

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Angular Beam Formation(the transition from a circular beam of constant diameter to

an angular beam of constant angular width)

22 / 1.5 km;

WSR-88D: 8.53 m; =10 cm

D

D

22 /D θE

φEFresnel zone

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Far field region

θ1= 1.27 λ/D (radians)

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Eq. (3.4)

Antenna (directive) Gain gt

The defining equation:

(W m-2) = power density incident on scatterer

r = range to measurement

),(2 f = radiation pattern (= 1 on beam axis)

tP = transmitted power (W)

),(4

22

fgr

PS t

ti

iS

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Wavenumbers for H, V WavesHorizontal polarization:

Vertical polarization: where k = free space wavenumber = 3.6x106 (deg./km)

for λ = 10 cm (e.g., for R= 100 mm h-1, = 24.4okm-1, = 20.7o km-1) Therefore: ch < cv ; λh < λv ; kh > kv

Specific differential phase: (for R = 100 mm h-1)

(KDP: an important polarimetric variable related to rainrate)

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h h h( ) = 2 /k k k π λ

v v v( ) = 2 /k k k π λ

hk vk

1DP h v 3 7(deg. km )K k k .

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Specific Differential Phase

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Echo phase of H = h = 2khr

(Fig.6.17)

Eq. (6.60)

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Backscattering Cross Section, σb

for a Spherical Particle

2

2

1

2

Rayleigh condition on a spherical particle of diameter

16 wavelength

;

; Dielectric Factor of the medium filling the sphere; Eq.3.6

the complex index

m

mK

m

D :

D λ / ; λ

m n jnκ

52 6

b m4

πσ = |K | D

λ

2 2w

2 2i

of refraction

0 90 0 93 for water, and

-30 18 for ice (density = 0.917 g m )

m

m

| K | | K | . .

| K | | K | .

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Backscattered Power Density Incident on Receiving Antenna

2

2 2

0

( , ) 1( , , ) (

4 4

( ) ( )

3.13a)

where is the loss factor (due to attenuation)

exp 3.13b

t tr b

r

g

P g fS r

r r

k k dr

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iS

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4/),( 22rre fgA

Echo Power Pr Received

Ae is the effective area of the receiving antenna for radiation from the θ,φ direction. It is shown that:

(3.20)

(3.21)

If the transmitting antenna is the same as the

receiving antenna then:

),(),,( err ArSP

),(),(),( 222 gffgfg ttrr 10/29-11/11/2013

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The Radar Equation(point scatterer/discrete object)

2 2 2

2 2

4 -14

6 4

b

3 244 4 4

Example:

0 1m 20km 2x10 m) min) 10 W);

10 W; peak); 3 10 1 no path loss)

Calculating the minimum detectable backscattering

(min) 2 10

t br

r

t

b

Pgf ( θ,φ ) σ gλ f ( θ ,φ )P ( . )

πr πr π

λ . ; r ( ; P ( (

P ( g x ; (

σ :

σ x

7 2m for a 6.3 mm drop!bσ

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Unambiguous Range ra• If targets are located beyond ra = cTs/2, their echoes from the nth transmitted pulse are

received after the (n+1)th pulse is transmitted. Thus, they appear to be closer to the radar than they really are!– This is known as range folding

• Ts = PRT

• Unambiguous range: ra = cTs/2

– Echoes from scatterers between 0 and ra are called 1st trip echoes,

– Echoes from scatterers between ra and 2ra are called 2nd trip echoes, Echoes from scatterers between 2ra and 3ra are called 3rd trip echoes, etc

time

True delay > Ts

(n+1)th pulse

nth pulse

TsApparent delay < Ts

ra

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Ambiguous Doppler Shifted Echoes(Fig. 3.14)

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Unambiguous Velocity

A pulsed Doppler radar measures radial Doppler velocity by keeping track of phase changes between samples that are Ts

(pulse repetition time) apart

Recall that echo phase shift is e = 4r/ . Then, the phase change from pulse to pulse is De = 4Dr/ = 4vrTs/ Note that only phase changes between – and can be unambiguously

resolved

Therefore, the unambiguous velocity is: 4vaTs/ = va = /4Ts

This is related to the Nyquist sampling theorem: Doppler velocities outside the ±va interval will be aliased!

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Δr = vrTs is the change in range of the scatterer between successive transmitted pulses

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Another PRT Trade-Off

• Correlation of pairs:– This is a measure of signal coherency

• Accurate measurement of power requires long PRTs–

– More independent samples (low coherency)

• But accurate measurement of velocity requires short PRTs–

– High correlation between sample pairs (high coherency)– Yet a large number of independent sample pairs are required

2( ) exp 8 /s v sT T

lim ( ) 0s

sTT

0

lim ( ) 1s

sTT

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Signal Coherency• How large a Ts can we pick?

– Correlation between m = 1 pairs of echo samples is:

– Correlated pairs:

(i.e., Spectrum width must be much smaller than unambiguous velocity va = λ/4Ts)

• Increasing Ts decreases correlation exponentially– also increases exponentially!

• Pick a threshold:– – Violation of this condition results in very large errors of estimates!

) = ex pT T 2

s v s( 8 ( / ) v s

s vs

( ) 1 1T

TT

20.5( ) 8 / 0.5 /s v s v aT e T v

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v vVar[ ˆ] and Var[ ˆ ]

v

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Spec

trum

w

idth

σv

Signal Coherency and Ambiguities• Range and velocity dilemma: rava=cl/8

• Signal coherency: sv < va /p

• ra constraint: Eq. (7.2c)

– This is a more basic constraint on radar parameters than the first equation above

• Then, sv and not va imposes a basiclimitation on Doppler weather radars

– Example: Severe storms have a median sv ~ 4 m/s and 10% of the time sv > 8 m/s. If we want accurate Dopplerestimates 90% of the time with a 10-cmradar (l = 10 cm); then, ra ≤ 150 km. This will often result in range ambiguities

8a

v

cr

10/29-11/11/2013Unambiguous range ra

150 km

8 m s-1

Fig. 7.5

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

Weather signals (echoes)

mT s

20

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

t

Weather Echo Statistics (Fig. 4.4)

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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 (ℓr= 1.5).

22

ft f

1tf

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

D

r r

r r

r r

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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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