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

Atmospheric Instrumentation M. D. Eastin

Fundamentals of Doppler Radar

Mesocyclone

WER

HookEcho

Radar Reflectivity Radar Doppler Velocities


Outline


• Basic Concept• Single Radar Pulse• Multiple Radar Pulses• Maximum Doppler Velocity Range• Doppler Dilemma• Doppler Spectra of Weather Targets

Doppler Effect: Generic Definition

• A frequency shift (cycles per second → Hertz) of any electromagnetic wave pulsedue to the “target” moving toward or away from the observer


Basic Concept

Doppler Effect: Sound Waves

•The Doppler shift for sound waves is the change in sound that one hears as race carsor airplanes approach and then recede from a stationary observer


Basic Concept

Doppler Effect: Meteorology

• A frequency shift between the transmitted radar pulse and the return echo pulse dueto hydrometeors moving either toward or away from the radar antenna


Basic Concept


• The sign [ + / – ] of the frequency shift is used todetermine the color of the radial velocity dataplotted 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 translatedto radar displays:


Red: Moving away from radar

Green: Moving toward radar

Basic Concept


• The magnitude of the frequency shift is used todetermine the brightness of the radial velocity data plotted on Doppler velocity displays

Much lower frequency = Strong outbound flowSlightly lower frequency = Weak outbound flow

Much higher frequency = Strong inbound flowSlightly higher frequency = Weak inbound flow


Basic Concept

Radial Velocity: Along-Beam Motion

• The measured radial velocity is that portion of the actual 3-D wind vector oriented alongthe 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…)


Basic Concept

Actual Wind

Radial component

seen by radar

RADAR

Doppler Effect: Frequency Shift

• The relationship between the return echo frequencyto the transmitted pulse frequency is:

(1)

where: fR = return echo frequency (s-1) fT = transmitted frequency (s-1)vR = “along-beam” radial velocity (m s-1) c = speed of light (m s-1)

• The frequency difference between the return echoand transmitted pulse (after a little algebra):

(2)

where: fDOP = Doppler frequency (s-1)


Single Radar Pulse

c

vffff RT

DOPTR

cv

cvff

R

RTR 1

1

Stationary Target

Moving Away

Doppler Frequency Shift

Moving Toward

Doppler Effect: Phase Shift

• The frequency difference can also be expressedas a phase shift (0 → 2π) between the returnecho and the transmitted pulse

(3)

where: fDOP = Doppler frequency (s-1) fT = 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 definethe fractional phase shift for a single return echo:

(4)


c

vf

tf RT

DOP

4

c

fT

Rvt

2

2

0 0 π 2ππ 2π

π/3

Time

Single Radar Pulse

Am

pli

tud

e

Doppler Effect: Phase Shift Magnitude

• Since we know the transmitted wavelength (λ), we canestimate the maximum fractional phase shift (Δφ/2π)of a single return echo using the pulse period (Δt = TR) for a typical range of Doppler radial velocities (vR)

PROBLEM: The maximum fractional phase shift (Δφ/2π) returned by a single pulse ismuch smaller than the full phase shift cycle required to reconstruct thereturn echo “wave form” and determine the Doppler frequency


Phase Shift (Δφ/2π) Transmitted Pulse Wavelength (λ)

[ for TR = 1×10-3 s ] X-band C-band S-band

Radial Velocity (vR) ( 3-cm ) ( 5-cm ) ( 10-cm )

1 m/s 0.067 0.040 0.020

5 m/s 0.333 0.200 0.100

10 m/s 0.667 0.400 0.200

Rvt

2

2

Single Radar Pulse

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 (φ1, φ2, φ3, φ4, … φN)• The multiple phase shifts are used to reconstruct (estimate) the full Doppler shift cycle• Doppler radial velocity (vR) is then computed from the mean phase shift along the train


Multiple Radar Pulses

AVGR t

v

4

Time

Phase shift froma single pulsein a pulse train

Full Doppler Frequency Cycle

Am

plit

ud

e

ANOTHER PROBLEM

• No unique solution• More than one Doppler frequency

(or waveform) will “fit” a finitesample of phase shifts

Minimum Phase Criteria:

A minimum of two phase observationsare required to determine a waveformof a Doppler frequency (N ≥ 2)

OR

The phase change between any twosuccessive pulses must be less thanhalf 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…)


Am

plit

ud

e Time

Multiple Radar Pulses

Nyquist Velocity:

• Starting with (4) and using the pulse period (TR) – 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 (vR) and using the relationship between pulse period (TR)and pulse repetition frequency (F)

(7)

•The Nyquist velocity represents the maximum (or minimum) radial velocity a Doppler radarcan measure unambiguously [ function of wavelength and pulse repetition frequency ]


Maximum Radial Velocity Range

RR T

v

4

RRTv4

RRTv4

FTR

1

RR T

v4

4

FvR

Nyquist Velocity:

• The maximum (or minimum) radial velocitya 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 velocityof ±10 m/s observes an area of rainfallmoving away from the radar at 15 m/s

[ Reported Actual ]

M. D. Eastin


4

Fvv RMAX

0-10 10-5 5 0-10 10-5 50-10 10-5 5

Unambiguous Velocity Range

0-10-20-30 10 20 30

Actual Radial Velocity

Atmospheric Instrumentation

Aliased Velocities

M. D. Eastin

Can you find the aliased velocities in this image?



Radar Reflectivity (DBZ) Radial Velocity (VR)


Doppler DilemmaMaximizing 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 frequencyis required

Nyquist Velocity Pulse Repetition Frequency (F)

Wavelength (λ) 200 s-1 500 s-1 1000 s-1 2000 s-1

3 cm 1.5 3.75 7.5 15.0

5 cm 2.5 6.25 12.5 25.0

10 cm 5.0 12.5 25.0 50.0

F

crMAX 2

4

FvMAX

8

crv MAXMAX Which do we choose?

They are inversely related

M. D. Eastin

Doppler DilemmaMaximizing the Nyquist Velocity:


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 reflectivity Measure radial velocity

Pulse 1 1 2 3 4 5 6 7 8 9 10

Time


M. D. Eastin

Impact of the Dilemma:

• The impact of determining radial velocities at a closer range than the radar reflectivity isthe “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 > rMAX)

Doppler Dilemma



Doppler Spectra of Weather TargetsVariations 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 (vR)= mean value of the spectrum

Spectral width (σ) = standard deviation of the spectrum= measure of the “spread” in velocities observed

within the sampling volume

–VMAX +VMAX0 VR

σspectrum



• Despite short time periods between each pulse of a rapid-fire pulse train, variations in thecomputed mean radial velocities exist due to (1) changes in air motions, and (2) variabilityin the drop size distribution within the contributing volume

Reasons for Variability:

1. Wind shear 2. Turbulence3. Differential fall velocity4. Antenna rotation5. Curvature in the main lobe



• Doppler spectra observed by a verticallypointing radar during passage of a winterstorm 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 theirassociated fall speeds

Archived Spectral Widths:

•Spectral widths observed by the WSR-88Dradars are not archived due to the largeamount of storage space required

•Forecasters can observe it in real-time tohelp identify:

1. Small tornadoes at far ranges2. Intense turbulence that may

impact aircraft operations


Summary


• Basic Concept• Single Radar Pulse• Multiple Radar Pulses• Maximum Doppler Velocity Range• Doppler Dilemma• Doppler Spectra of Weather Targets


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.

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

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