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Review Doppler Radar (Fig. 3.1) A simplified block diagram. Electric field incident on scatterer. Reflected electric field incident on antenna. Voltage input to the synchronous detectors; This pair of detectors shifts the frequency f to 0. jQ (t). A o. - PowerPoint PPT Presentation
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METR 5004 1
ReviewDoppler Radar (Fig. 3.1)
A simplified block diagram
10/29-11/11/2013
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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