What controls the shape of a real Doppler spectrum?

Frederic Tridon University of Leicester, Leicester, United-Kingdom

ERAD Short Course, Sunday 31st August 2014, Garmisch-Partenkirchen, Germany

Research Applications of Radar Doppler Spectra

and contributions from A. Battaglia, P. Kollias, S. Kneifel, E. Luke and M. Maahn

Outline Observed Doppler spectra

How are they produced? Signal processing Estimation - Accuracy

Trade-offs Compromises between different parameters Influence on the Doppler spectra

Simulation of Doppler spectra Microphysical model Air state model Electromagnetic model Instrument model

Observed Doppler spectra

Giangrande et al. (2012)

Traditionally, radar measurements are characterized by the first 3 spectral moments: Reflectivity, which can be related to the weather signal power Mean Doppler velocity, which is essentially the air motion toward or away from the radar Spectrum width, which is a measure of the velocity dispersion, i.e. shear or turbulence within the resolution

volume

Example of measured spectrum (ARM UHF wind profiler)

Zeroth moment: reflectivity factor

First moment: mean Doppler velocity

Second moment: spectrum width

The Doppler spectra contain a wealth of information but how can we get them and why don’t we always get them?

dvvSZ )(

wVdvvS

dvvvSV DSDDop

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2

2

)(

)()(shtDSD

Dop

zdvvS

dvvSVv

Z

VDop

σz2

Pulsed Doppler radars: amplitude and phase sampling

time delay between any transmitted pulse and its echo (range time)sample time

(uniform PRT)

Now, consider the signal V(mTs,s) at a specific s

corresponding to the range r

SS mTttjQtItV ),()()(

1 2, , s s sG

Pulse 1 (m=0)

Pulse 2 (m=1)

Pulse 3 (m=2)

sam

ple-ti

me

range-time

s

1s2s

sT

time series data from the same range (gate)

),( SSmTV Nm ,,2,1

s2-s1: gate spacing

)(tV

Pulse 1 (m=0)

Pulse m where ri = vimTs

with fd=-2vi/

Fluctuations of total signal amplitude and phase

Unambiguous velocity

The Nyquist (or folding) velocity is defined as

All Doppler velocities between ±fN are the principal aliases and the frequencies higher than fN are ambiguous with those between ±fN

So, we cannot relate the phase change to one unique Doppler frequency and the unambiguous limits for radial velocities are

The echo phase e is sampled at intervals Ts and its change e over Ts is a measure of the Doppler frequency. But, it is not possible to determine whether V(t) rotated clockwise or counter clockwise and how many times it circled the origin in the time Ts. a correct Doppler shift could be any of these (where n is a positive or negative integer)

Pulse 1 (m=1)

Pulse m where ri = vimTs

with fd=-2vi/

Fluctuations of total signal amplitude and phase

The different hydrometeors cause individual backscattering (amplitudes) and phase shifts (fall velocity) at the receiver.

Because the position of drops changes with time, radar echoes (coming from the sum of the individual returns of drops) fluctuate V(t,r) is a random variable A single measurement from a given region is not enough and it is necessary to observe a

volume for a certain amount of time (many milliseconds) to obtain a reasonable average reflectivity

In dealing with random signals, the most one can hope to accomplish is to estimate some average statistical parameters. Two useful ones are the mean and autocorrelation

Signal processing Autocorrelation: cross-correlation of a signal

with itself

TIME

SIG

NA

L

Doppler Spectrum

Raw radar data

For a sequence of M+1 uniformly spaced transmitted pulses, the autocorrelation function R ̂(Ts) at lag Ts can be estimated as

Then, the estimate of mean velocity v̂̂ and spectrum width σ̂̂v are

PP processing

the power spectrum decomposes the content of the signal of a stochastic process into the different frequencies present in that process, and helps identify periodicities. In our case, it decomposes the energy scattered by hydrometeors according to their velocity with respect to the radar.

Power spectrum: Fourier Transform of the autocorrelation function frequency distribution of the signal power

similarity between observations as a function of the time lag between them, mathematical tool for finding repeating patterns such as a periodic signal

FFT processingFFT

Precipitation distributed over large regions radar measurements made at a large number of range/time locations

FFT: large computation time and large storage covariance calculations traditionally used

But through the progress of technology, the Fourier transform is more and more used

Uncertainty of Doppler moments (spectral processing)

where vn is the normalized spectrum width:

and Ti is the time to independence:

Only 3 parametersM: number of FFTSNR=S/N: signal to noise ratiovn: normalized spectrum width

At high SNR

(Doviak and Zrnic, 1984)

Uncertainty of Doppler moments (PP vs spectral processing)

The Fourier derived means have variances that increase much more slowly than the autocovariance method

Another advantage of the spectral processing is the ease to identify and eliminate anomalies and system malfunctions

2600m1500m

800m

kazr-kasacr comparisonRange and time kasacr spectrograms

Spurious bulges appear at the sides of kasacr spectra where there should be only

noise (kazr)

15 Sep 2011 19:32 to 19:58

Example of spectra artefact

Trade-offs

8

c

R aa

Time integration – Variability of the scene Accuracy of the 3 moment estimates increases with the number of samples (i.e.

time integration) , it is true also for the spectral density

But, because of the variability of weather signals, the time integration cannot be too long without losing detailed features of the spectra

Example: simulation of time integration over a changing scene Homogeneous DSD (but it can vary as well!) 1 m/s vertical wind gradient within the time integration The W-band spectrum obtained is smoothed (similarly to a broadening effect)

Range ambiguity

For Doppler radars operated with a uniform PRT Ts, the unambiguous range is defined as

If scatterers have range r larger than cTs/2, their echoes for the nth transmitted pulse are received after the (n+1)th pulse is transmitted. Therefore, these echoes are received during the same time interval as echoes returned from scatterers at r < cTs/2 from the (n+1)th pulse

Pow

er

TimeTsTs Ts

The Doppler Dilemma

sa Tv

4

The maximum unambiguous Doppler velocity (Nyquist velocity) is directly proportional to PRF (1/Ts) and λ:

But, the maximum unambiguous range Rmax is inversly proportional

to PRF (proportional to Ts):

2s

a

cTR

Combination of both equations:

Need to find a compromise between maximum range and maximum Doppler velocity.

8

c

R aa

Example: W-band radar

Rmax= 15 km vdop, max = 8 m/s

Spectral resolution The number NFFT of points in a spectrum corresponds to the number of samples

used to perform the FFT. This number is generally kept lower than ~500 for a question of data storage

the spectral resolution v is related to the unambiguous velocity through

In order to keep a sufficient spectral resolution, the unambiguous velocity must not be too large

Example: Simulation of a spectrum with insufficient resolution

16 points FFTs of the same spectrum

Va=6 m/sVa=18 m/s

Liquid vs ice - Aliasing

Typical fall velocity: rain 0-10 m/s, snow-graupel 0-4 m/s

Need better spectral resolution for ice and larger va for rain in order to avoid velocity aliasing

Example on the simulator -> aliasing of a Rain spectrum

40dB 15dB

8 m/s 0.7 m/s

Simulation of Doppler spectra

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Radar Doppler Spectra Modeling

Microphysical ModelParticle Size DistributionPhase, Density Fall Velocity

E/M Radiation ModelScattering/AbsorptionDielectric properties

Instrument ModelTx/Rx ParametersPulse/Antenna PatternSignal Processing

Air state ModelAir motionTurbulence

Microphysics modelling

Shape: sphere, spheroid, other?

Phase: liquid, solid, mixed?

Number concentration: N(D)

Mass: m(D)

Cross section area: A(D)

Density: ρ(D)

Air state modelling Vertical wind w

Air broadening: wind shear, turbulence (eddy dissipation rate)

Air density correction

airhydr swvswvs )()(

Receiver noise: white spectrum (independent of velocity)

Random fluctuation of the signal: the weather signal is exponentially distributed. It can be written as (with x a uniformly distributed random number between 0 and 1):

Signal processing modelling

(Zrnic, 1975), Kollias et al.,

2011)

Example on the simulator -> effect of the number of samples

Break