1. What controls the shape of a real Doppler spectrum?
ERAD Short Course, Sunday 31st August 2014, Garmisch-Partenkirchen, Germany
Research Applications of Radar Doppler Spectra
What controls the shape of a real Doppler spectrum?
Frederic Tridon
University of Leicester, Leicester, United-Kingdom
and contributions from A. Battaglia, P. Kollias, S. Kneifel, E. Luke and M. Maahn

2. Outline Observed Doppler spectra Trade-offs
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

3. Observed Doppler spectra
Giangrande et al. (2012)

4. Example of measured spectrum (ARM UHF wind profiler)
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
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? Z
σz2
VDop

5. Pulsed Doppler radars: amplitude and phase sampling
time delay between any transmitted pulse and its echo (range time)
sample time
(uniform PRT)
Pulse 1 (m=0)
Pulse 2 (m=1)
Pulse 3 (m=2)
sample-time
range-time
time series data from the same range (gate)
ts2-ts1: gate spacing
Now, consider the signal V(mTs,ts) at a specific ts corresponding to the range r

6. Fluctuations of total signal amplitude and phase
Pulse 1 (m=0)
where Dri = vimTs
Pulse m
with fd=-2vi/l

7. Unambiguous velocity
The echo phase ye is sampled at intervals Ts and its change Dye 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)
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

8. Fluctuations of total signal amplitude and phase
Pulse 1 (m=1)
where Dri = vimTs
Pulse m
with fd=-2vi/l
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

9. Signal processing PP processing FFT processing SIGNAL
Autocorrelation: cross-correlation of a signal with itself
For a sequence of M+1 uniformly spaced transmitted pulses, the autocorrelation function R̂(Ts) at lag Ts can be estimated as
 similarity between observations as a function of the time lag between them, mathematical tool for finding repeating patterns such as a periodic signal
Power spectrum: Fourier Transform of the autocorrelation function  frequency distribution of the signal power
Then, the estimate of mean velocity v̂ and spectrum width σ̂v are
 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.
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
FFT processing
FFT
Raw radar data
Doppler Spectrum
SIGNAL
TIME

10. Uncertainty of Doppler moments (spectral processing)
Only 3 parameters
M: number of FFT
SNR=S/N: signal to noise ratio
svn: normalized spectrum width
At high SNR
where svn is the normalized spectrum width:
and Ti is the time to independence:
(Doviak and Zrnic, 1984)

11. 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

12. Example of spectra artefact
kazr-kasacr comparison
Range and time kasacr spectrograms
2600m
1500m
Spurious bulges appear at the sides of kasacr spectra where there should be only noise (kazr)
800m
15 Sep :32 to 19:58

13. Trade-offs

14. 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)

15. Range ambiguity Power TimeTs Ts Ts
Time
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

16. The Doppler Dilemma
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):
Combination of both equations:
 Need to find a compromise between maximum range and maximum Doppler velocity.
Example: W-band radar
Rmax= 15 km vdop, max = 8 m/s

17. Spectral resolution Va=18 m/s Va=6 m/s
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 Dv is related to the unambiguous velocity through
16 points FFTs of the same spectrum
Va=18 m/s
Va=6 m/s
In order to keep a sufficient spectral resolution, the unambiguous velocity must not be too large
Example: Simulation of a spectrum with insufficient resolution

18. Liquid vs ice - Aliasing
40dB
15dB
0.7 m/s
8 m/s
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

19. Simulation of Doppler spectra?

20. Radar Doppler Spectra Modeling
Microphysical Model
Particle Size Distribution
Phase, Density
Fall Velocity
Air state Model
Air motion
Turbulence
E/M Radiation Model
Scattering/Absorption
Dielectric properties
Instrument Model
Tx/Rx Parameters
Pulse/Antenna Pattern
Signal Processing

21. 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)

22. Air state modelling Vertical wind w
Air broadening: wind shear, turbulence (eddy dissipation rate)
Air density correction

23. Signal processing modelling
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):
(Zrnic, 1975), Kollias et al., 2011)
Example on the simulator -> effect of the number of samples

24. Break