1. Atmospheric InstrumentationM. D. Eastin Fundamentals of Doppler Radar Mesocyclone WER Hook Echo Radar ReflectivityRadar Doppler Velocities

2. Atmospheric InstrumentationM. D. Eastin Outline Fundamentals of Doppler Radar Basic Concept Single Radar Pulse Multiple Radar Pulses Maximum Doppler Velocity Range Doppler Dilemma Doppler Spectra of Weather Targets

3. Doppler Effect: Generic Definition A frequency shift (cycles per second → Hertz) of any electromagnetic wave pulse due to the “target” moving toward or away from the observer Atmospheric InstrumentationM. D. Eastin Basic Concept

4. Doppler Effect: Sound Waves The Doppler shift for sound waves is the change in sound that one hears as race cars or airplanes approach and then recede from a stationary observer Atmospheric InstrumentationM. D. Eastin Basic Concept

5. Doppler Effect: Meteorology A frequency shift between the transmitted radar pulse and the return echo pulse due to hydrometeors moving either toward or away from the radar antenna Atmospheric InstrumentationM. D. Eastin Basic Concept

6. Doppler Effect: Meteorology The sign [ + / – ] of the frequency shift is used to determine the color of the radial velocity data plotted on Doppler velocity displays A negative shift (called a “red shift” in optics) occurs as targets move away from the radar Lower frequency= Positive radial velocity =Outbound flow A positive shift (called a “green shift” in optics) occurs as targets move toward the radar Higher frequency= Negative radial velocity =Inbound flow These “color” shift conventions are then translated to radar displays: Atmospheric InstrumentationM. D. Eastin Red: Moving away from radar Green: Moving toward radar Basic Concept

7. Doppler Effect: Meteorology The magnitude of the frequency shift is used to determine the brightness of the radial velocity data plotted on Doppler velocity displays Much lower frequency = Strong outbound flow Slightly lower frequency = Weak outbound flow Much higher frequency = Strong inbound flow Slightly higher frequency = Weak inbound flow Atmospheric InstrumentationM. D. Eastin Basic Concept

8. Radial Velocity: Along-Beam Motion The measured radial velocity is that portion of the actual 3-D wind vector oriented along the radar beam [ most radial velocities contain horizontal and vertical motions ] Any single radar only measures one component of the three dimensional wind vector, so to obtain a more practical estimate of two or three-dimensional flow: 1. Users must learn how to interpret single radar Doppler imagery (next lecture…) 2. Multiple nearby Doppler radars can be used to “re-construct” the 3-D flow (later…) Atmospheric InstrumentationM. D. Eastin Basic Concept Actual Wind Radial component seen by radar RADAR

9. Doppler Effect: Frequency Shift The relationship between the return echo frequency to the transmitted pulse frequency is: (1) where: f R = return echo frequency (s -1 ) f T = transmitted frequency (s -1 ) v R = “along-beam” radial velocity (m s -1 ) c = speed of light (m s -1 ) The frequency difference between the return echo and transmitted pulse (after a little algebra): (2) where: f DOP = Doppler frequency (s -1 ) Atmospheric InstrumentationM. D. Eastin Single Radar Pulse Stationary Target Moving Away Doppler Frequency Shift Moving Toward

10. Doppler Effect: Phase Shift The frequency difference can also be expressed as a phase shift (0 → 2 π ) between the return echo and the transmitted pulse (3) where: f DOP =Doppler frequency (s -1 ) f T =transmitted frequency (s -1 ) Δφ =phase shift (radians) Δt =elapsed time (s) c =speed of light (m s -1 ) If we use the relationship between transmitted frequency and wavelength, we can define the fractional phase shift for a single return echo: (4) Atmospheric InstrumentationM. D. Eastin 0 0 π2π2ππ2π2π π/3 Time Single Radar Pulse Amplitude

11. Doppler Effect: Phase Shift Magnitude Since we know the transmitted wavelength ( λ ), we can estimate the maximum fractional phase shift ( Δφ/2π ) of a single return echo using the pulse period ( Δt = T R ) for a typical range of Doppler radial velocities ( v R ) PROBLEM:The maximum fractional phase shift ( Δφ/2π ) returned by a single pulse is much smaller than the full phase shift cycle required to reconstruct the return echo “wave form” and determine the Doppler frequency Atmospheric InstrumentationM. D. Eastin Phase Shift ( Δφ/2π )Transmitted Pulse Wavelength ( λ ) [ for T R = 1×10 -3 s ]X-bandC-bandS-band Radial Velocity ( v R )( 3-cm )( 5-cm )( 10-cm ) 1 m/s0.0670.0400.020 5 m/s0.3330.2000.100 10 m/s0.6670.4000.200 Single Radar Pulse

12. Method to Overcome: Transmit a rapid-fire “train” of multiple pulses → increase the pulse repetition frequency Each pulse in the “train” will return a slightly different phase ( φ 1, φ 2, φ 3, φ 4, … φ N ) The multiple phase shifts are used to reconstruct (estimate) the full Doppler shift cycle Doppler radial velocity ( v R ) is then computed from the mean phase shift along the train Atmospheric InstrumentationM. D. Eastin Multiple Radar Pulses Time Phase shift from a single pulse in a pulse train Full Doppler Frequency Cycle Amplitude

13. ANOTHER PROBLEM No unique solution More than one Doppler frequency (or waveform) will “fit” a finite sample of phase shifts Minimum Phase Criteria: A minimum of two phase observations are required to determine a waveform of a Doppler frequency ( N ≥ 2 ) OR The phase change between any two successive pulses must be less than half a wavelength( Δφ ≤ π ) Other Criteria: Since the maximum number of phase observations is set by the pulse repetition frequency and the minimum number is set by the wavelength, there is a range of possible radial velocities than can be unambiguously determined (next few slides…) Atmospheric InstrumentationM. D. Eastin Amplitude Time Multiple Radar Pulses

14. Nyquist Velocity: Starting with (4) and using the pulse period ( T R ) – or time between sequential pulses – for the maximum elapsed time ( Δt ): (5) We next re-arrange and apply the “half wavelength criteria” (0 → π ) OR(6) Solving (6) for radial velocity ( v R ) and using the relationship between pulse period ( T R ) and pulse repetition frequency (F) (7) The Nyquist velocity represents the maximum (or minimum) radial velocity a Doppler radar can measure unambiguously [ function of wavelength and pulse repetition frequency ] Atmospheric InstrumentationM. D. Eastin Maximum Radial Velocity Range

15. Nyquist Velocity: The maximum (or minimum) radial velocity a radar can measure unambiguously Any actual radial velocities larger (or smaller) than this value will be “aliased” back into another unambiguous range → multiple aliases can occur Example:Assume a radar with a Nyquist velocity of ±10 m/s observes an area of rainfall moving away from the radar at 15 m/s [ ReportedActual ] M. D. Eastin Maximum Radial Velocity Range 0-1010-550-1010-550-1010-55 Unambiguous Velocity Range 0-10-20-30102030 Actual Radial Velocity Atmospheric Instrumentation Aliased Velocities

16. M. D. Eastin Can you find the aliased velocities in this image? Atmospheric Instrumentation Maximum Radial Velocity Range Radar Reflectivity (DBZ)Radial Velocity (VR)

17. Atmospheric InstrumentationM. D. Eastin Doppler Dilemma Maximizing the Nyquist Velocity: This table shows that Doppler radars capable of measuring a large range of radial velocities unambiguously have a longer wavelength (λ) and a large pulse repetition frequency (F) Problem: Recall that in order for radars to maximize their range, a small pulse repetition frequency is required Nyquist VelocityPulse Repetition Frequency (F) Wavelength (λ)200 s -1 500 s -1 1000 s -1 2000 s -1 3 cm1.53.757.515.0 5 cm2.56.2512.525.0 10 cm5.012.525.050.0 Which do we choose? They are inversely related

18. M. D. Eastin Doppler Dilemma Maximizing the Nyquist Velocity: Atmospheric Instrumentation

19. M. D. Eastin How to Circumvent the Dilemma: Radar transmits pulses at alternating low and high pulse repetition frequencies Lower frequencies are used for surveillance (reflectivity) Higher frequencies are used for velocities (radial winds) A version of this technique has been used regularly by the WSR-88D radars 1992–2008→ Alternating pulse repetition frequencies (lower two elevations scans) → Doppler winds determined out to 120 km range → Reflectivity determined out to 240 km range 2008–now→ Separate lower elevation scans (different pulse repetition frequencies) → Doppler winds determined out to 300 km range → Reflectivity determined out to 360 km range Doppler Dilemma Measure reflectivityMeasure radial velocity Pulse 1 1 2 3 4 5 6 7 8 9 10 Time Atmospheric Instrumentation

20. M. D. Eastin Impact of the Dilemma: The impact of determining radial velocities at a closer range than the radar reflectivity is the “purple haze” (or range folding) often seen at far ranges on Doppler radar imagery This results from echoes returning after the next pulse is transmitted (i.e., r > r MAX ) Doppler Dilemma Atmospheric Instrumentation

21. M. D. Eastin Doppler Spectra of Weather Targets Variations in Radial Velocity: A series of rapid-fire pulses in a pulse train will measure a “spectrum” of Doppler frequencies (or radial velocities) from a which a “mean” and “standard deviation” can be computed Radial velocity (v R )=mean value of the spectrum Spectral width (σ)=standard deviation of the spectrum =measure of the “spread” in velocities observed within the sampling volume –V MAX +V MAX 0VRVR σ spectrum

22. Atmospheric InstrumentationM. D. Eastin Doppler Spectra of Weather Targets Variations in Radial Velocity: Despite short time periods between each pulse of a rapid-fire pulse train, variations in the computed mean radial velocities exist due to (1) changes in air motions, and (2) variability in the drop size distribution within the contributing volume Reasons for Variability: 1. Wind shear 2. Turbulence 3. Differential fall velocity 4. Antenna rotation 5. Curvature in the main lobe

23. Atmospheric InstrumentationM. D. Eastin Doppler Spectra of Weather Targets Variations in Radial Velocity: Doppler spectra observed by a vertically pointing radar during passage of a winter storm with mixed-phase precipitation Notice how the spectra at individual heights vary by 1-2 m/s as a result of variable drop diameters and their associated fall speeds Archived Spectral Widths: Spectral widths observed by the WSR-88D radars are not archived due to the large amount of storage space required Forecasters can observe it in real-time to help identify: 1. Small tornadoes at far ranges 2. Intense turbulence that may impact aircraft operations

24. Atmospheric InstrumentationM. D. Eastin Summary Fundamentals of Doppler Radar Basic Concept Single Radar Pulse Multiple Radar Pulses Maximum Doppler Velocity Range Doppler Dilemma Doppler Spectra of Weather Targets

25. Atmospheric InstrumentationM. D. Eastin References Atlas, D., 1990: Radar in Meteorology, American Meteorological Society, 806 pp. Crum, T. D., R. L. Alberty, and D. W. Burgess, 1993: Recording, archiving, and using WSR-88D data. Bulletin of the American Meteorological Society, 74, 645-653. Doviak, R. J., and D. S. Zrnic, 1993: Doppler Radar and Weather Observations, Academic Press, 320 pp. Fabry, F., 2015: Radar Meteorology Principles and Practice, Cambridge University Press, 256 pp. Reinhart, R. E., 2004: Radar for Meteorologists, Wiley- Blackwell Publishing, 250 pp.