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Review Doppler Radar (Fig. 3.1) A simplified block diagram

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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

Echo phase at the output of the detectors and filters

ψe

I(t)

If the range r of the scatterer is fixed, phasor A is fixed.

But if scatterer has a radial velocity, Phasor A rotates

about the origin at the Doppler frequency fd.

Complex plane

(Phasor diagram)

1

μ

s

Range-Time

2

3

1

0

4

5

4

3

5

0

2

1

Stationary

Moving

(A)

scatterers

scatterer

(B)

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r=cτs/2

cτ

λ

c = speed of microwaves

= ch for H and = cv for V waves

τ = pulse length

λ = wavelength

= λh for H and λv for V waves

τs = time delay between

transmission of a pulse and

reception of an echo.

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For typical atmospheric conditions (i.e., normal) the propagation path

is a straight line if the earth has a radius 4/3rds times its true radius.

Sub refraction

Free space

Normal atmosphere: The

ray’s radius of curvature

Rc≈ constant)

Super refraction (trapping)

(anomalous propagation:

Unusually cold moist air near the ground)

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Fresnel zone

Far field region

θ1= 1.27 λ/D (radians)

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Antenna (directive) Gaingt

The defining equation:

Eq. (3.4)

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

r = range to measurement

= radiation pattern (= 1 on beam axis)

= transmitted power (W)

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Wavenumbers for H, V Waves

Horizontal 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, = 20.7okm-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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Specific Differential Phase

(Fig.6.17)

Echo phase of H = h = 2khr

Eq. (6.60)

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Echo Power PrReceived

(3.20)

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

(3.21)

If the transmitting antenna is the same as the

receiving antenna then:

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nth pulse

(n+1)th pulse

Ts

time

Apparent delay < Ts

True delay > Ts

- 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 raare 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

ra

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

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- 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

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- 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!

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8 m s-1

- Range and velocity dilemma: rava=cl/8
- Signal coherency: sv<va /p
- raconstraint: 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

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

Fig. 7.5

150 km

Spectrum width σv

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Unambiguous range ra

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

t

c(s)≈ t

(t = transmitted

pulse width)

mTs

Weather signals (echoes)

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ftf

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).

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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)

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