Doppler Radar From Josh Wurman NCAR S-POL DOPPLER RADAR.
Transcript of Doppler Radar From Josh Wurman NCAR S-POL DOPPLER RADAR.
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Doppler Radar
From Josh Wurman
NCAR S-POL DOPPLER RADAR
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Doppler Shift: A frequency shift that occurs in electromagneticwaves 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
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00 2cos tfEtE ttThe electric field of a transmitted wave
The returned electric field at some later time back at the radar 11 2cos ttfEtE tt
The time it took to travel
c
rt
2
Substituting:
11
22cos
c
rtfEtE tt
The received frequency can be determined by taking the time derivative if the quantity in parentheses and dividing by 2
dtrt
tt
ttr ffc
vff
dt
dr
c
ff
c
rtf
dt
df
2222
2
11
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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
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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 oneof these three components – the motion along the beam toward or away from the radar
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Magnitude of the Doppler Shift
Transmitted Frequency
X band C band S band
9.37 GHz 5.62 GHz 3.0 GHzRadial 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
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RECALL THE BLOCK DIAGRAM OF A DOPPLER RADAR AND THE “PHASE DETECTOR”
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)cos(2
10 tAA
d
)sin(2
10 tAA
d
2210
2QI
AAAmplitude determination:
Phase determination:
I
Qd
1tan
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Typical period of Doppler frequencydf
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.
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rrvTd
rrvT2
212
2212
rr 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 twoConsecutive 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)
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Illustration of the reconstruction of the Doppler frequency from sampled phase values
Dots correspond to the measured samples of phase
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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
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22 rr T
v
rrTv4
max42v
F
Tv
rr
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
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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
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Folded velocities
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Can you find the folded velocities in this image?
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http://apollo.lsc.vsc.edu/classes/remote/graphics/airborne_radar_images/newcastle_folded.gif
Folded velocities in an RHI Velocities after unfolding
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F
cr
2max
8maxmax
cvr
But recall that for a large unambiguous RANGEDoppler radars must operate at a low PRF
4max
Fv
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
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The Doppler Dilema
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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
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max22
nvf
V dr
max22
vnf
V dr
maxmax
4nvvnff dd
nFFnff dd
2. Use slightly different PRFs in alternating sequence
For 1st PRF
For 2nd PRF
Solve simultaneously
Example: = 5.33 cm, F = 900 s-1, F = 1200 s-1 1max 12
4 ms
Fv
1max 16
4
ms
Fv
MEASURE fd = -150 hz, fd = 450 hz
nn 9001200300
1 nn Data is folded once
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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
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NET RESULT: A series of pulses will measure a spectrum of velocities (Doppler frequencies)
Power per unit velocity interval (db)
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dvvSdffSPv
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
The moments, or integral properties, of the Doppler Spectrum
Average returned power
Mean radial velocity
Spectral width
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Example of Doppler spectraAs 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
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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
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Effects of relative particle motion:
Consider two particles in a pulse volume
Return from 1: 1111 cos tEtE D
Return from 2: 2222 cos tEtE D
tf 2
2,12,1
4 rD
v
Where:
Total Echo power proportional to sum of two fields squared
With a bit of trigonometry….
ttEtEtE DD 11121 sinsincoscos
ttE DD 112 sinsincoscos
Where: 1 t 2 t
2121
22
21 cos
22 DDr EEEE
P
Constant term Term which depends on particles relative velocities and wavelength
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For a large ensemble of particles
j
DjDiii
ir EE
EP cos
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
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Determining the Doppler Spectrum
1. Doppler spectrum is measured at a particular range gate (e.g. at )2
tcr
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
mr mTkfkfF
MmTE 0
1
00 2cos
1)(
r
M
mrr mTkfmTEkfF 0
1
00 2cos)(
4. Frequency components (radial velocities) occur at discrete intervals, with M intervals separated by intervals of 1/MTr = fD
M = # of samplesf0 = frequency resolution
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Discrete Doppler spectra computed for a point target, with M = 8. Dots represent the discrete frequency components of the spectra.
Point target, M = 8fD = 2 f0
Point target, M = 8fD = 2.5 f0
Signal appear in all M linesof 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.
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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
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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
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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)
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Problems complicating process:
1. Noise
2. Folding
3. Clutter
Tends to bias Vr to 0and spectral width to vmax/3
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)cos(2
10 tAA
d
)sin(2
10 tAA
d
2210
2QI
AAAmplitude determination:
Phase determination:
I
Qd
1tan
RECALL THE PHASEDETECTOR IN A
DOPPLER RADAR SYSTEM
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Sample of I/Q channel voltage at time 1:
110
1 exp2
tiAA
R D
Sample of I/Q channel voltage at time 2:
220
2 exp2
tiAA
R D
Autocorrelation function:
1221
20*
21 exp4
iAAA
RR
*1
1
1
n
M
nnRR
MC
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210 AA
210 AA
Amplituded
Representation ofI/Q signal on a phaseDiagram in complexspace
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421
20 AAA
432
20 AAA
443
20 AAA
454
20 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