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2. 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. 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. 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. 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. 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. 7. RECALL THE BLOCK DIAGRAM OF A DOPPLER RADAR AND THE “PHASE DETECTOR”
8. 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. 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. 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. 11. Illustration of the reconstruction of the Doppler frequency from sampled phase values Dots correspond to the measured samples of phase φ
12. 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. 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. 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. 15. Folded velocities
16. 16. Can you find the folded velocities in this image?
17. 17. http://apollo.lsc.vsc.edu/classes/remote/graphics/airborne_radar_images/newcastle_folded.gif Folded velocities in an RHI Velocities after unfolding
18. 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. 19. The Doppler Dilema
20. 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. 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. 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. 23. NET RESULT: A series of pulses will measure a spectrum of velocities (Doppler frequencies) Power per unit velocity interval (db)
24. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 34. Problems complicating process: 1. Noise 2. Folding 3. Clutter Tends to bias Vr to 0 and spectral width to vmax/3
35. 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. 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. 37. 2 10 AA 2 10 AA Amplitude ωd φ1 Representation of I/Q signal on a phase Diagram in complex space
38. 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
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