doppler effect

1. Doppler Radar
From Josh Wurman
NCAR S-POL DOPPLER RADAR

2. Doppler Shift: A frequency shift that occurs in electromagnetic
waves due to the motion of scatterers toward or away from the
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.
Analogy: The Doppler shift for sound waves is the frequency shift that occurs as race
cars approach and then recede from a stationary observer

3. ( ) ( )00 2cos φπ += tfEtE ttThe electric field of a transmitted wave
The returned electric field at some later
time back at the radar ( ) ( )( )11 2cos φπ +∆+= ttfEtE tt
The time it took to travel
c
r
t
2
=∆
Substituting: ( ) 





+





+= 11
2
2cos φπ
c
r
tfEtE tt
The received frequency can be determined by taking the time derivative if the
quantity in parentheses and dividing by 2π
dt
rt
t
t
ttr ff
c
vf
f
dt
dr
c
f
f
c
r
tf
dt
d
f +=+=+=





+





+=
222
2
2
1
1φπ
π

4. Sign conventions
The Doppler frequency is negative (lower frequency,
red shift) for objects receding from the radar
The Doppler frequency is positive (higher frequency,
blue shift) for objects approaching the radar
These “color” shift conventions are typically also used on radar
displays of Doppler velocity
Blue: Toward radar
Red: Receding from radar

5. Note that Doppler radars are only sensitive to the radial motion of objects
Air motion is a three dimensional vector: A Doppler radar can only measure one
of these three components – the motion along the beam toward or away from the radar

6. Magnitude of the Doppler Shift
Transmitted Frequency
X band C band S band
9.37 GHz 5.62 GHz 3.0 GHz
Radial velocity
1 m/s
10 m/s
50 m/s
62.5 Hz 37.5 Hz 20.0 Hz
625 Hz 375 Hz 200 Hz
3125 Hz 1876 Hz 1000 Hz
These frequency shifts are very small: for this reason, Doppler
radars must employ very stable transmitters and receivers

7. RECALL THE BLOCK DIAGRAM OF A DOPPLER RADAR AND THE
“PHASE DETECTOR”

8. )cos(
2
10
φω += t
AA
d
)sin(
2
10
φω += t
AA
d
2210
2
QI
AA
+=Amplitude determination:
Phase determination: 





=+ −
I
Q
d
1
tanφω

9. Typical period of Doppler frequency
df
1
= = 0.3 to 50 milliseconds
Typical pulse duration = 1 microsecond
Why is emphasis placed on phase determination instead of
determination of the Doppler frequency?
Only a very small fraction of a complete Doppler frequency cycle is contained within a pulse
Alternate approach: one samples the Doppler-shifted echo
with a train of pulses and tries to reconstruct, or estimate,
the Doppler frequency from the phase change that occurs
between pulses.

10. rrvTd =
λπ
φφ rrvT2
2
12
=




 −





 −
=
π
φφλ
22
12
r
r
T
v
We can understand how the phase shift can be related to the radial
velocity by considering a single target moving radially.
Distance target moves radially in one pulse period Tr
The corresponding phase shift of a wave between two
Consecutive pulses (twice (out and back) the fraction
of a wavelength traversed between two consecutive
pulses)
Solving for the radial velocity
In practice, the pulse volume contains billions of targets moving at
different radial speeds and an average phase shift must be
determined from a train of pulses
(1)

11. Illustration of the reconstruction of the Doppler
frequency from sampled phase values
Dots correspond to the measured samples of phase φ

12. PROBLEM
More than one Doppler frequency (radial velocity) will always
exist that can fit a finite sample of phase values.
The radial velocity determined from the sampled
phase values is not unique

13. 




 ∆
=
π
φλ
22 r
r
T
v
π
λ
π
φ <=∆ rrTv4
max
42
v
F
T
v
r
r ===
λλ
What is the maximum radial velocity possible before
ambiguity in the measurement of velocity occurs?
From (1)
The phase change between pulses must therefore be less than half a wavelength
We need at least two measurements per wavelength to determine a frequency
 vmax is called the Nyquist velocity and represents the maximum (or
minimum) radial velocity a Doppler radar can measure unambiguously – true
velocities larger or smaller than this value will be “folded” back into the
unambiguous range

14. EXAMPLE VALUES OF THE MAXIMUM
UNAMBIGUOUS DOPPLER VELOCITY
Wavelength Radar PRF (s-1
)
cm 200 500 1000 2000
3 1.5 3.75 7.5 15
5 2.5 6.25 12.5 25
10 5.0 12.5 25.0 50
Table shows that Doppler radars capable of measuring a
large range of velocities unambiguously have long
wavelength and operate at high PRF

15. Folded velocities

16. Can you find the folded velocities in this image?

17. http://apollo.lsc.vsc.edu/classes/remote/graphics/airborne_radar_images/newcastle_folded.gif
Folded velocities in an RHI Velocities after unfolding

18. F
c
r
2
max =
8
maxmax
λc
vr =
But recall that for a large unambiguous RANGE
Doppler radars must operate at a low PRF
4
max
F
v
λ
=
THE DOPPLER DILEMA: A GOOD CHOICE OF PRF TO ACHIEVE A
LARGE UNAMBIGUOUS RANGE WILL BE A POOR CHOICE TO
ACHIEVE A LARGE UNAMBIGUOUS VELOCITY

19. The Doppler Dilema

20. Ways to circumvent the ambiguity dilema
1. “Bursts” of pulses at alternating low and high pulse repetition frequencies
Measure reflectivity Measure velocity
Low PRF used to measure to long range, high PRF to measure velocity

21. max2
2
nv
f
V d
r ±=
λ
max2
2
vn
f
V d
r
′′±
′
=
λ
( )maxmax
4
nvvnff dd −′′±=′−
λ
( )nFFnff dd −′′±=′−
2. Use slightly different PRFs in alternating sequence
For 1st
PRF
For 2nd
PRF
Solve simultaneously
Example: λ = 5.33 cm, F = 900 s-1
, F′ = 1200 s-1 1
max 12
4
−
== ms
F
v
λ 1
max 16
4
−
=
′
=′ ms
F
v
λ
MEASURE fd = -150 hz, f′d = 450 hz
( )nn 9001200300 −′±=
1==′ nn Data is folded once

22. Real characteristics of a returned signal from a distributed target
Velocity of individual targets in contributing volume vary due to:
1) Wind shear (particularly in the vertical)
2) Turbulence
3) Differential fall velocity (particularly at high elevation angles)
4) Antenna rotation
5) Variation in refraction of microwave wavefronts

23. NET RESULT: A series of pulses will measure a
spectrum of velocities (Doppler frequencies)
Power per unit velocity interval (db)

24. ( ) ( )dvvSdffSP
v
v
rdr ∫∫
+
−
∞
∞−
==
max
max
( )
( )
( )
r
v
v
r
v
v
r
v
v
r
r
P
dvvvS
dvvS
dvvvS
v
∫
∫
∫
+
−
+
−
+
−
==
max
max
max
max
max
max
( ) ( )
( )
( ) ( )
r
v
v
rr
v
v
r
v
v
rr
v
P
dvvSvv
dvvS
dvvSvv ∫
∫
∫
+
−
+
−
+
−
−
=
−
=
max
max
max
max
max
max
22
2
σ
The moments, or integral properties, of the Doppler Spectrum
Average returned power
Mean radial velocity
Spectral width

25. Example of Doppler spectra
As a function of altitude
measured in a winter
snowband. These spectra
were measured with a
vertically pointing Doppler
profiler with a rather wide (9
degree) beamwidth
Note ground clutter
Melting level

26. The Doppler spectrum represents the echo from a single contributing region
Mean Doppler frequency (or velocity) Related to the reflectivity weighted mean
radial motion of the particles
Spectral width Related to the relative particle motions
RECALL: Fluctuations in mean power from pulse to pulse occur due
to interference effects as the returned EM waves superimpose upon
one another.
Fluctuations are due to the relative motion of the particles
between pulses and therefore to the spectral width

27. Effects of relative particle motion:
Consider two particles in a pulse volume
Return from 1: ( ) ( )[ ]1111 cos φωω ++= tEtE D
Return from 2: ( ) ( )[ ]2222 cos φωω ++= tEtE D
tfπω 2=
λ
π
ω 2,1
2,1
4 r
D
v
−=
Where:
Total Echo power proportional to sum of two fields squared
With a bit of trigonometry….
( ) ( ) ( ) ( ) ( ) ( )[ ]ttEtEtE DD 11121 sinsincoscos ωαωα −=+
( ) ( ) ( ) ( )[ ]ttE DD 112 sinsincoscos ωβωβ −+
Where: 1φωα += t 2φωβ += t
( )[ ]2121
2
2
2
1
cos
22
DDr EE
EE
P ωω −++∝
Constant term Term which depends on particles relative velocities and wavelength

28. For a large ensemble of particles
( )[ ]∑∑∑ −+∝
j
DjDi
ii
i
r EE
E
P ωωcos
2
21
2
To determine the echo power, one must average over a
large enough independent samples that the second term
averages to zero
HOWEVER!!
To determine the Doppler frequency (and velocity) from
consecutive measurements of echo phase, the samples must
be DEPENDENT (more frequent) than those required to
obtain the desired resolution in reflectivity

29. Determining the Doppler Spectrum
1. Doppler spectrum is measured at a particular range gate (e.g. at )
2
tc
r
∆
=
2. Must process a time series of discrete samples of echo Er(t) at intervals
of the pulse period Tr
3. Analyze the sampled signal using (fast) Fourier Transform methods:
( ) [ ]r
M
m
r mTkfkfF
M
mTE 0
1
0
0 2cos
1
)( π∑
−
=
=
( ) [ ]r
M
m
rr mTkfmTEkfF 0
1
0
0 2cos)( π∑
−
=
=
4. Frequency components (radial velocities) occur at discrete intervals, with M
intervals separated by intervals of 1/MTr = fD
M = # of samples
f0 = frequency resolution

30. Discrete Doppler spectra
computed for a point
target, with M = 8. Dots
represent the discrete
frequency components of
the spectra.
Point target, M = 8
fD = 2 f0
Point target, M = 8
fD = 2.5 f0
Signal appear in all M lines
of the spectrum
If Doppler frequency is not an
integral multiple of the
frequency resolution (normally
not the case), the discrete
Fourier transform will “smear”
power into all of the frequencies
across the spectrum.

31. With a distributed target, which has a spectrum of Doppler frequencies, the
discrete Fourier transform will always produce power in all frequencies.
The power will be relatively uniform at frequencies not associated with the true
Doppler frequencies, and peak across the range of true Doppler frequencies.
Noise NoiseSignal

32. In most applications (such as the operational NEXRADs), the Doppler
spectra are not needed.
Recording the entire Doppler spectra at each range gate takes an
enormous amount of data storage capability, quickly exceeding the
capacity of current electronic storage devices.
What are needed are the moments of the spectra – the average
returned power, the mean Doppler velocity, and the spectral width

33. How can the moments be obtained from the series of discrete samples?
1. Record time series at each range gate and Fourier analyze Doppler
Spectra. Calculate the moments. Discard Spectral data.
(Computationally inefficient, given that these calculations must be
done for every range gate on every beam!
or…
2. Calculate moments as the time series is recorded using the
Autocorrelation function (see below), and discard data continuously
following the calculation (little data storage required and
computationally efficient)

34. Problems complicating process:
1. Noise
2. Folding
3. Clutter
Tends to bias Vr to 0
and spectral width to vmax/3

35. )cos(
2
10
φω += t
AA
d
)sin(
2
10
φω += t
AA
d
2210
2
QI
AA
+=Amplitude determination:
Phase determination: 





=+ −
I
Q
d
1
tanφω
RECALL THE PHASE
DETECTOR IN A
DOPPLER RADAR
SYSTEM

36. Sample of I/Q channel voltage at time 1:
( )[ ]1
10
1 exp
2
φω += ti
AA
R D
Sample of I/Q channel voltage at time 2:
( )[ ]2
20
2 exp
2
φω += ti
AA
R D
Autocorrelation function:
( )[ ]12
21
2
0*
21 exp
4
φφ −= i
AAA
RR
*
1
1
1
+
=
∑= n
M
n
n RR
M
C

37. 2
10 AA
2
10 AA
Amplitude
ωd
φ1
Representation of
I/Q signal on a phase
Diagram in complex
space

38. φ2−φ1
φ3−φ2
φ4−φ3
φ5−φ4
4
21
2
0 AAA
4
32
2
0 AAA
4
43
2
0 AAA
4
54
2
0 AAA
Graphical depiction of how average amplitude (returned power)
And phase (radial velocity) are recovered from autocorrelation function
The spectral width can also be recovered from autocorrelation function