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.
Analogy: The Doppler shift for sound waves is the frequency shift that occurs as race cars 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.

3. The electric field of a transmitted wave
The returned electric field at some later time back at the radar
The time it took to travel
Substituting:
The received frequency can be determined by taking the time derivative if the quantity in parentheses and dividing by 2p

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
Red: Receding from radar
Blue: Toward 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 GHz GHz
Radial velocity
62.5 Hz Hz Hz
625 Hz Hz Hz
3125 Hz Hz Hz
1 m/s
10 m/s
50 m/s
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. Amplitude determination:
Phase determination:

9. Why is emphasis placed on phase determination instead of determination of the Doppler frequency?
Typical period of Doppler frequency
= 0.3 to 50 milliseconds
Typical pulse duration
= 1 microsecond
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. 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)
(1)
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

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

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. What is the maximum radial velocity possible before ambiguity in the measurement of velocity occurs?
We need at least two measurements per wavelength to determine a frequency
The phase change between pulses must therefore be less than half a wavelength
From (1)
 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
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. Folded velocities in an RHI
Velocities after unfolding

18. But recall that for a large unambiguous RANGE
Doppler radars must operate at a low PRF
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. 2. Use slightly different PRFs in alternating sequence
For 1st PRF
For 2nd PRF
Solve simultaneously
Example: l = 5.33 cm, F = 900 s-1, F = 1200 s-1
MEASURE fd = -150 hz, fd = 450 hz
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. 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
Melting level
Note ground clutter

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
Where:
Return from 1:
Return from 2:
With a bit of trigonometry….
Where:
Total Echo power proportional to sum of two fields squared
Constant term
Term which depends on particles relative velocities and wavelength

28. For a large ensemble of particles
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. 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:
M = # of samples
f0 = frequency resolution
4. Frequency components (radial velocities) occur at discrete intervals, with M intervals separated by intervals of 1/MTr = fD

30. Signal appear in all M lines
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
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.
Point target, M = 8
fD = 2.5 f0
Signal appear in all M lines
of 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
Signal
Noise

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?
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
Tends to bias Vr to 0
and spectral width to vmax/3
2. Folding
3. Clutter

35. Amplitude determination:
RECALL THE PHASE
DETECTOR IN A
DOPPLER RADAR SYSTEM
Amplitude determination:
Phase determination:

36. Sample of I/Q channel voltage at time 1:
Autocorrelation function:

37. Representation of
I/Q signal on a phase
Diagram in complex
space wd
Amplitude f1

38. Graphical depiction of how average amplitude (returned power)
And phase (radial velocity) are recovered from autocorrelation function
f5-f4
f4-f3
f3-f2
f2-f1
The spectral width can also be recovered from autocorrelation function