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ReviewDoppler Radar (Fig. 3.1)

A simplified block diagram

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Complex plane(Phasor diagram)

A o

jQ(t)

I(t)ψe

ii

( , ) exp 2 trE 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, the phasor (Ao, ψe) is fixed (i.e., no change in Ao nor ψe. But if scatterer has a radial velocity, phasor (Ao, ψe) rotates about the origin at the Doppler frequency fd.

rr 2

( , ) 2exp 2 trj f t j

r c

AE

Electric field incident on scatterer

Reflected electric field incident on antenna

i i4exp 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 voltage Vo at the output of the detectors and filters .The echo amplitude is Ao and phase is

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

3445

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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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

DD

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 gtThe defining equation:

(W m-2) = Power density incident on a scatterer

r = range to measurement (m)

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

tP = transmitted power (W)

),(4

22

fg

rPS t

ti

iS

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Backscattering Cross Section, σb

for a Spherical Particle

2

2

12

Rayleigh condition on a spherical particle of diameter16 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 | Dλ

2 2w

2 2i

of refraction

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 rr 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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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 1TT

T

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=c/8• Signal coherency: v < va /

• ra constraint: Eq. (7.2c)– This is a more basic constraint on radar parameters than the first equation above

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

– Example: Severe storms have a median v ~ 4 m/s and 10% of the time v > 8 m/s. If we want accurate Dopplerestimates 90% of the time with a 10-cmradar ( = 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

16

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

t

Weather Echo Statistics (Fig. 4.4)

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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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