Doppler Radar

From Josh Wurman

NCAR S-POL DOPPLER RADAR

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

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

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

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

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

RECALL THE BLOCK DIAGRAM OF A DOPPLER RADAR AND THE “PHASE DETECTOR”

)cos(2

10 tAA

d

)sin(2

10 tAA

d

2210

2QI

AAAmplitude determination:

Phase determination:

I

Qd

1tan

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.

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)

Illustration of the reconstruction of the Doppler frequency from sampled phase values

Dots correspond to the measured samples of phase

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

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

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

Folded velocities

Can you find the folded velocities in this image?

http://apollo.lsc.vsc.edu/classes/remote/graphics/airborne_radar_images/newcastle_folded.gif

Folded velocities in an RHI Velocities after unfolding

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

The Doppler Dilema

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

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

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

NET RESULT: A series of pulses will measure a spectrum of velocities (Doppler frequencies)

Power per unit velocity interval (db)

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

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

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

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

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

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

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.

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

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

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)

Problems complicating process:

1. Noise

2. Folding

3. Clutter

Tends to bias Vr to 0and spectral width to vmax/3

)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

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

210 AA

210 AA

Amplituded

Representation ofI/Q signal on a phaseDiagram in complexspace

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