Doppler Radar
From Josh Wurman
Radar Meteorology
M. D. Eastin
Doppler Radar
Outline
• Basic Concepts
• Doppler Radar Components
• Phase Shifts and Pulse Trains
• Maximum Range of Radial Velocity
• Doppler Dilemma
• Doppler Spectra of Weather Targets
Radar Meteorology
M. D. Eastin
Basic Concepts
Doppler Shift:
• A frequency shift in electromagnetic waves due to the motion of scatters
toward or away from the observer
Analogy: The Doppler shift for sound waves is the change in frequency one
detects as race cars or airplanes approach and then recede from
a stationary observer
Doppler Radar:
• A radar that can determine the frequency shift through measurement of the phase change
that occurs in electromagnetic waves during a series of pulses
Radar Meteorology
M. D. Eastin
Basic Concepts
Doppler Shift from a Single Radar Pulse:
• Recall the electric field of a transmitted wave:
Et t   E0 cos2ft t  0 
(1)
• The returned electric field at some later time:
Et t   E1 cos2ft t  t   1 
(2)
• Time it took to travel to and from the object(s):
2r
t 
c
(3)
• Substituting:


 2r 
Et t   E1 cos 2f t  t    1 
c 



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(4)
M. D. Eastin
Basic Concepts
Doppler Shift from a Single Radar Pulse:


 2r 
Et t   E1 cos 2f t  t    1 
c 



(4)
• The received frequency can be determined by taking the time derivative of the quantity
in parentheses and dividing by 2π:
2 f t dr
f r  ft 
c dt
f r  ft 
2 f t vr
c
f r  ft  f d
Radar Meteorology
where:
(5)
vr = Radial velocity of target
fd = Doppler shift
 2 f t vr  2vr
fd 

c

(6)
M. D. Eastin
Basic Concepts
Sign Conventions:
 2 f t vr  2vr
fd 

c

• Doppler shift is negative (lower frequency, red shift) for objects
moving away from the radar (positive vr)
• Doppler shift is positive (higher frequency, blue shift) for objects
moving toward the radar (negative vr)
• These “color” shift conventions are often translated to radar displays:
Red: Moving away from radar
Blue/Green: Moving toward radar
Radar Meteorology
M. D. Eastin
Basic Concepts
Component of Motion:
• The observed radial velocity is the component of three-dimensional air motion that
is along the radar beam
• In essence, the Doppler radar only measures one component of the full wind field
Radar Meteorology
M. D. Eastin
Basic Concepts
Magnitude of a Doppler Shift:
Transmitted Frequency
X-band
C-band
S-band
Radial Velocity
9.37 GHz
5.62 GHz
3.0 GHz
1 m/s
62.5 Hz
37.5 Hz
20.0 Hz
10 m/s
625 Hz
375 Hz
200 Hz
50 m/s
3125 Hz
1875 Hz
1000 Hz
• These frequency shifts are very small: Thus Doppler radars must employ very stable
transmitters and receivers in order to detect Doppler shifts with high accuracy
(i.e. resolve vr to within 1 m/s or less)
Radar Meteorology
M. D. Eastin
Doppler Radar Components
Block Diagram:
• STALO generates local frequency (fL)
• COHO generates a known phase (fC)
• Mixer combines fC with fL to get
transmitted frequency (fT)
• Klystron amplifies
• Antenna transmits
• Frequency of received echo is the
transmitted (fT) plus Doppler shift (fD)
• Receiver uses STALO signal to
remove local frequency
• Signal amplified
• Phase detector use COHO signal to
estimate the Doppler shift from the
original phase
Radar Meteorology
M. D. Eastin
Doppler Radar Components
Block Diagram:

A0 A1
cos( d t   )
2

A0 A1
sin(d t   )
2
Amplitude of Doppler signal:
A0 A1
 I 2  Q2
2
Phase of the Doppler signal:
d    tan1 
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Q

I
 
M. D. Eastin
Pulse Shifts and Pulse Trains
Why Emphasis is on Phase and not Frequency?
• Typical period of a Doppler shift cycle → 1/fD → 1 millisecond
• Typical pulse duration → τ → 1 microsecond
Problem:
• Only a very small fraction of an entire Doppler shift cycle is contained in a single return
Method to Overcome:
• Transmit a “rapid-fire” train of pulses
• Each pulse will return a slightly different phase (φ1, φ2, φ3, φ4, …)
• The multiple phase shifts are then used to reconstruct, or estimate, the Doppler shift cycle
(see next slide)
• The Doppler frequency (i.e. radial velocity, vr) can then be estimated from the mean
difference between successive phases returned by the train of pulses
(see the slide after next)
Radar Meteorology
M. D. Eastin
Pulse Shifts and Pulse Trains
Reconstructing the Doppler shift cycle from multiple phase shifts:
Dots correspond to the measured samples of phase φ
from a “train” composed of 16 pulse returns
Radar Meteorology
M. D. Eastin
Pulse Shifts and Pulse Trains
Relating Phase Shifts to Radial Velocity:
• Consider a single target moving radially along the radar beam
• Distance target moves in one pulse period (Tr):
d  Tr vr
(7)
• Corresponding phase shift between two successive pulses is equal to the
the fraction of a wavelength traversed between two consecutive pulses:
 2  1  2Tr vr



 2 
(8)
• Solving for radial velocity:
vr 
  2  1 


2Tr  2 
(9)
• In practice, the radial velocity must be determined from the mean phase shift
from all successive pulses in the train
Radar Meteorology
M. D. Eastin
Pulse Shifts and Pulse Trains
Problem: No Unique Solution
• More than one Doppler frequency (i.e. radial velocity) will fit a finite sample of phase values
• In essence a determined radial velocity is not unique
• However, the possible radial velocities are multiples of a common value determined
by the radar transmission characterisiics (see next slide…)
Radar Meteorology
M. D. Eastin
Maximum Range of Radial Velocity
What is the maximum possible radial velocity before ambiguity occurs?
• We need at least two measurements per wavelength to determine phase
• Thus, the phase change between successive pulses must be less than half a wavelength:
• Starting with (9):
vr 
  2  1 


2Tr  2 
• Re-arranging and applying the criteria above:
 
4vrTr


(10)
• Solving for radial velocity in the extreme case [right side of (10)]:
vr  max 
where:
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
4Tr

F
4
(11)
F = sampling rate (or the PRF for the pulse period)
M. D. Eastin
Maximum Range of Radial Velocity
Nyquist velocity (vr-max):
vr  max 

4Tr

F
4
• Represents the maximum (or minimum) radial velocity a Doppler radar can measure
unambiguously
• True radial velocities larger (or smaller) than this value will be “folded” back into the
unambiguous range → multiple folds can occur
Unambiguous Velocity Range
-10
-5
0
5
10
-10
-5
0
5
10
-10
-5
0
5
10
Actual Radial Velocity
-30
-20
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-10
0
10
20
30
M. D. Eastin
Maximum Range of Radial Velocity
Folded Radial Velocities:
Folded
Velocities
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M. D. Eastin
Maximum Range of Radial Velocity
Can you find the folded velocities in this image?
Radar Meteorology
M. D. Eastin
Doppler Dilemma
Maximizing your Nyquist Velocity :
Radar PRF (s-1)
Wavelength
(cm)
200
500
1000
2000
3
1.5
3.75
7.5
15.0
5
2.5
6.25
12.5
25.0
10
5.0
12.5
25.0
50.0
• Table shows that Doppler radars capable of measuring a large range of radial velocities
unambiguously have long wavelengths and large PRFs
Problem:
• Recall that in order for radars to maximize their range, a small PRF is required
rmax 
c
2F
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vmax 
F
4
vmaxrmax 
c
8
Which do we choose?
They are inversely related
M. D. Eastin
Doppler Dilemma
Maximizing your Nyquist Velocity :
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M. D. Eastin
Doppler Dilemma
How to Circumvent the Dilemma: Alternating PRFs
• Radar transmits burst of pulses at alternating low and high frequencies
• Lower PRF for reflectivity with higher PRF for radial velocities
Measure reflectivity
Measure velocity
• This technique is regularly used by the NEXRAD radars
• The result → Doppler winds are determined out to 120 km range
→ Reflectivity determined out to 240 km range
Radar Meteorology
M. D. Eastin
Doppler Spectra of Weather Targets
Variability in Vr:
• Despite small time periods between each pulse in a train, changes in air motions and
the drop size distribution within the contributing volume will occur
• As before, we need to account for this variability
Reasons for Variability:
1. Wind shear (especially in the vertical)
2. Turbulence
3. Differential fall velocity (more relevant at large elevation angles)
4. Antenna rotation
5. Curvature of microwave wave fronts (e.g. Gaussian main lobe)
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M. D. Eastin
Doppler Spectra of Weather Targets
Variability in Vr: Result
• A series of pulses will measure a spectrum of velocities (or Doppler frequencies)
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M. D. Eastin
Doppler Spectra of Weather Targets
Variability in Vr: First Three Moments
Zero Order

Pr 
 S  f  df
d


 vmax
 S v  dv
r
 vmax
• Average returned power from pulse train
• Area under the curve (see previous slide)
• Related to equivalent radar reflectivity factor Ze
Radar Meteorology
M. D. Eastin
Doppler Spectra of Weather Targets
Variability in Vr: First Three Moments
First Order
 vmax
 vS v  dv
 vmax
r
vr 
 vmax
 vmax
 S v  dv
 vS v  dv
r

 vmax
Pr
r
 vmax
• Mean radial velocity
• Associated with peak in the power spectrum (see previous slide)
• Reflectivity weighted (i.e. large drops have greater influence on mean radial velocity)
Radar Meteorology
M. D. Eastin
Doppler Spectra of Weather Targets
Variability in Vr: First Three Moments
Second Order
 
2
v
 vmax
 vmax
 vmax
 vmax
2


v

v
 r S vr  dv
 vmax
 S v  dv

2


v

v
 r S vr  dv
Pr
r
 vmax
• Spectral width
• Associated with the variation in observed radial velocities (see previous slide)
• Influenced by turbulence and wind shear
Radar Meteorology
M. D. Eastin
Doppler Spectra of Weather Targets
Example:
• Vertically pointing Doppler radar
with a large beam width (8 degs)
during a spring storm
Snowflakes
Freezing Level
Ground Clutter
Small Raindrops
Radar Meteorology
M. D. Eastin