# Rapid Scale Wind Profiling with Autoregressive Modeling and L-Band Doppler
Radar

Baptiste Domps, Julien Marmain, and Charles-Antoine Guérin B. Domps and J.
Marmain are with the Radar and Scientific Applications Department, Degreane
Horizon, 83390 Cuers, France (e-mail<EMAIL_ADDRESS>julien.marmain@degreane-horizon.fr). C.-A. Guérin is with the Mediterranean
Institute of Oceanography (MIO), Université de Toulon, Aix-Marseille
Université, CNRS, IRD, Toulon, France (e-mail: guerin@univ-tln.fr).

###### Abstract

Radar Wind Profilers (RWP) are well-established instruments for the probing of
the atmospheric boundary layer, with the immense advantage of long-range and
all-weather operation capability. One of their main limitations, however, is a
relatively long integration time compared to other instruments such as lidars.
In the context of L-band RWP we show that the use of Autoregressive (AR)
modeling for the antenna signals combined with the Maximum Entropy Method
(MEM) allows for a correct estimation of radial wind velocity profiles even
with very short time samples. A systematical analysis of performance is made
with the help of synthetic data. These numerical results are further confirmed
by an experimental dataset acquired near the landing runways of Paris Charles
de Gaulle (CDG) Airport, France, and validated using a colocated optical lidar
at the Aerological station of Payerne, Switzerland. It is found that the AR-
MEM approach can successfully derive reliable wind estimates using integration
times as short as $\displaystyle 2.5\text{\,}\mathrm{s}$ where the classical
spectral approach can barely provide any measurement. In addition, the
technique helps mitigating the impact of uncooperative flyers in the
processing of the atmospheric echo, leading to a more successful application
of the wind estimation algorithm.

##  1 Introduction

Radar wind profilers (RWP) measure vertical profiles of wind speed in the
lower atmosphere under all kinds of meteorological conditions and are widely
used in the context of atmospheric research (e.g. [14, 26]), operational
meteorology (e.g. [31, 4, 13]), data assimilation [39] or instruments
validation ([19, 8]). They generally operate in the VHF or UHF frequency bands
in pulse-Doppler mode [1], even though there have been isolated attempts to
develop frequency modulated continuous wave (FMCW) systems (e.g. [38, 24]).
They work in a coherent integration mode in order to record not only the
strength of the echo but also its Doppler shift. Contrarily to optical
measurement systems (lasers and lidars) they are not hindered or attenuated by
clouds or fog nor necessitate the presence of aerosols as backscattering
elements; contrarily to microwave weather radars, they are not bound to the
sole backscattering of hydrometeors but also have the unique capability of
capturing a “clear-air” return even in the far range, up to a few dozen of
kilometers in L-band [43]. In the classical electromagnetic theory, this is
explained by a Bragg scattering mechanism from the small turbulent
fluctuations of the refractive index in the air volume which follow the wind
motion (e.g. [36, 25]). The radial speed of atmospheric echoes is
characterized by a Doppler frequency peak which can be measured by a coherent
integration of the backscattered signal [43]. It is in general assumed that
the mean velocity of the scattering structures in the atmosphere (obtained
with the first moment of the Doppler spectrum) coincide with wind speed while
its dispersion (quantified with the second moment) quantifies the turbulence.
The wind vector can be recombined from the simultaneous measurement of radial
speeds obtained with several oblique antennas, a technique often called
Doppler Beam Swinging (DBS, [3]). Hence RWP can provide all-weather, long-
range reliable estimates of the wind vector with a typical range resolution of
$\displaystyle 150\text{\,}\mathrm{m}300\text{\,}\mathrm{m}$ corresponding to
the transmitted pulse duration.

Now, detecting the very weak clear-air echoes and accurately evaluating their
speed dictates the use of “long” coherent integration time, that is a large
number of emitted pulses. This is necessary for both increasing the Signal-to-
Noise Ratio (SNR) and improving the estimation of the Doppler spectrum.
Doppler spectra are then incoherently averaged over integration times of the
order of $\displaystyle 20\text{\,}\mathrm{s}$ per beam which are far beyond
the temporal scale of many rapidly fluctuating phenomena such as planes and
birds echoes, or vortex and wakes caused by flying objects. In part because of
this limitation but also for practical reasons, RWP has been competed since
the early 2000s with Doppler lidars which are compact and versatile
instruments. These active laser-based instruments measure wind speed and
direction at smaller spatio-temporal scales. Wind lidars indeed use much
shorter integration times, of the order of $\displaystyle
1\text{\,}\mathrm{s}$ per beam (see, e.g., [18]), together with smaller range
cell resolution (typically $\displaystyle 50\text{\,}\mathrm{m}$). This has
allowed for a number of studies of the turbulent atmosphere in the context of
emerging industrial applications, mostly linked to the monitoring of turbulent
wakes from planes or wind turbines (e.g. [2, 32, 20]). However, this is at the
expense of a much shorter range, typically a few kilometers.

Nevertheless, the unique ability of RWP to study atmospheric turbulence in
all-weather and the much longer range they achieve as compared to optical
sensors make them irreplaceable if the limitation of long integration time can
be overcome. The aim of this paper is to show that the reduced number of
available samples for short integration time can be mitigated by the use of
non-spectral signal-processing techniques, which bypass the time-frequency
trade-off of the Fourier analysis. One such technique, based on an
autoregressive (AR) representation of the backscattered signal associated with
a maximum entropy method (MEM), has been successfully applied in the context
of oceanographic radars [15] to obtain Doppler spectra at rapid scale and
capture some transient phenomena such as tsunamis-induced currents [17]. In
the atmospheric context, the AR-MEM approach was successfully used for the
time-frequency analysis of various transient phenomena at rapid time scale
(see Figure 3 of [16]). Here, we apply the same method to RWP for wind
measurement and demonstrate its potential on an experimental set of L-band
data.

This article is organized as follows. Section 2 describes the mathematical
framework used to process the atmospheric backscattered signal with the AR
representation and the MEM for evaluating its coefficients. The performances
of the method are assessed with synthetic RWP data and compared to a classical
Fourier analysis (Section 3). Section 4 presents a first application of this
methodology to an actual RWP dataset. Finally, the validity of the short-time
estimations is evaluated against lidar measurements (Section 5).

##  2 AR Modeling of the Atmospheric Doppler Spectrum

We briefly recall the backscattering mechanisms at play in the turbulent
atmosphere and the few signal processing steps which are conventionally
employed to extract the spectral moments from the antenna voltage time-series.
The reader is referred to [41, 5, 7] for a complete review of the processing
scheme.

###  2.1 Atmospheric Doppler Spectrum

The theoretical Doppler spectrum $\displaystyle\sigma(\omega)$ is the limiting
form of the power spectral density (PSD) of the backscattered range-resolved
complex time-series $\displaystyle s(t,r)$ for large integration time and a
given detection volume (the range index $\displaystyle r$ is implicit and will
be omitted in the following). To an amplitude factor, given by the radar
equation, the Doppler spectrum writes

$\sigma(\omega)\sim\Bigg{<}\bigg{|}\int_{0}^{T}e^{-i\omega
t}s(t)W(t)\,dt\bigg{|}^{2}\Bigg{>}$ (1)

where $\displaystyle W(t)$ is a windowing function over the integration time
$\displaystyle T$ and $\displaystyle\omega$ is the circular frequency. As it
is well known, the Doppler frequency shift is proportional to the radial
velocity $\displaystyle V_{r}$ of scatterers, $\displaystyle\omega=2\pi f=4\pi
V_{r}/\lambda$, where $\displaystyle\lambda$ is the radar wavelength. Hence
the Doppler spectrum can be interpreted as a power-weighted distribution of
the radial velocities within the resolution volume [43], where the weighting
function depends on the scatterers reflectivity and velocity. In the following
we will therefore express the Doppler spectra as a function of the radial
speed $\displaystyle v$ rather than Doppler frequency Due to the random motion
of scatterers during the observation time, the shape of the weighting function
is generally assumed to be Gaussian and centered about the average radial wind
speed. Its width is related to the distribution of the individual scatterer
velocities which is spread by shear or turbulence. It can happen that the
assumption of Gaussian distribution of velocities is violated when some strong
intermittent clutter echoes (in particular from flyers) is superimposed to the
atmospheric return. Such parasitic echoes should be filtered or at least
ignored when inferring radial wind speed from Doppler spectrum. Therefore in
the absence of noise and clutter, a Gaussian-shaped atmospheric Doppler
spectrum has the generic form [41]:

$S(v)=\frac{S_{0}}{\sigma_{v}\sqrt{2\pi}}\exp\left(-\frac{(v-V_{r})^{2}}{2\sigma_{v}^{2}}\right)$
(2)

Under this form, the Doppler spectrum is fully characterized by its first
three moments:

${\cal M\mit}_{n}=\int_{-\infty}^{+\infty}v^{n}S(v)\,dv,\ n=0,1,2$ (3)

The zeroth-moment $\displaystyle S_{0}={\cal M\mit}_{0}$ is the weather signal
power, the normalized first-moment $\displaystyle V_{r}={\cal M\mit}_{1}/{\cal
M\mit}_{0}$ is the mean radial velocity and the normalized second-moment
$\displaystyle\sigma_{v}^{2}={\cal M\mit}_{2}/{\cal M\mit}_{0}$ is the radial
velocity variance.

Complex voltage time-series are coherently recorded at the output of a
quadrature demodulator and sorted into range cells. For each gate, around 50
to 100 samples (that is 50 to 100 emitted pulses) are coherently summed which
is equivalent to a time-domain averaging. After digital decimation the
sampling frequency is of the order of 150 to $\displaystyle
200\text{\,}\mathrm{Hz}$ giving a typical Nyquist interval of the order of
$\displaystyle\pm$\displaystyle
10\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$. Note that the sampling
strategy can be adapted by the radar operators depending on the atmospheric
conditions. The calculation of the Doppler spectrum (1) is usually performed
with a Fast Fourier Transform (FFT) over short subseries (typically,
$\displaystyle N=128$ time steps):

$P(\omega_{k})=(\Delta t)^{2}\left|\sum_{n=0}^{N-1}e^{-i\omega_{k}n}s(n\Delta
t)W(n\Delta t)\right|^{2}$ (4)

where $\displaystyle\omega_{k}=k/T$ for $\displaystyle k=0,\dotsc,N-1$ is the
discrete set of dual frequencies and $\displaystyle\Delta t$ is the sampling
period after coherent integration and digital decimation.

The resulting sample spectra are further incoherently summed over a certain
number $\displaystyle\operatorname*{NSPEC}$ of half-overlapping sequences to
obtain the ensemble average (1). Incoherent averaging is performed using the
Statistical Averaging Method (SAM) of Merritt [29]. In the SAM approach, it is
assumed that the atmospheric signal has Gaussian statistics whereas
intermittent echoes (birds, planes, uncooperative flyers, etc.) are larger,
non-Gaussian signals. Temporal variations of each range-Doppler bin (or,
equivalently, range-radial velocity bin) should therefore comply to an
exponential distribution test as proposed by Hildebrand and Sekhon (HS) [22].
Hence the SAM makes only use of the frequency bin passing the exponential
energy distribution test of HS, every other being considered as non-
atmospheric echoes and discarded from the average. The main strength of the
SAM is that it needs no assumption on the nature of the intermittent echoes.
However, as it is based on a statistical criterion, the SAM assumes that the
atmosphere can be observed free of parasitic echoes at each range gate and
frequency bin for at least one sample Doppler spectrum.

Once the “clean” Doppler spectra are obtained, a Multipeaks Picking Procedure
(MPP) is used (see e.g. [11, 30, 40]) to identify at most three peaks for each
range gate as the averaged spectra may still contain some ground-clutter and
flyers echoes in addition to the atmospheric signal. Selection of one single
peak would therefore lead to wrong estimates as non-atmospheric echoes may
remain after the SAM processing. The first three spectral moments of the
identified peaks are computed and used to remove those having specific non-
atmospheric signatures. Finally, some empirical criteria on the spectral
moments are used along with the constraint of velocity continuity with respect
to range to select for the most likely atmospheric peak. In the following, we
will refer to this standard processing scheme as the “FFT method”.

A typical RWP Doppler spectrum for a resolution volume in fair weather is
shown in Figure 1. It is obtained from an incoherent summation of
$\displaystyle\operatorname*{NSPEC}=61$ instantaneous spectra of
$\displaystyle 128$ samples each. The sampling frequency is $\displaystyle
204\text{\,}\mathrm{Hz}$, leading to an unambiguous (or Nyquist) velocity of
$\displaystyle N_{v}=$\displaystyle
12\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$. Two echoes, labeled (a) and
(b), can be observed in the spectrum. The zero-velocity peak corresponds to
reflections from the ground or “fixed-echoes”, while the broader peak is a
Gaussian-shaped atmospheric echo. Here, fixed-echoes are of greater energy
than the atmospheric return, a situation commonly encountered in the lowest
range gates. The mean radial velocity is estimated to
$\displaystyle\hat{V}_{r}=$\displaystyle
4.50\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$ and the spectrum width to
$\displaystyle\hat{\sigma}_{v}=$\displaystyle
1.36\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$. Note that the spectrum
width is often normalized by twice the Nyquist velocity, so that
$\displaystyle\hat{\sigma}_{vn}=\hat{\sigma}_{v}/2N_{v}=0.057$.

Figure 1: Example of typical Doppler spectrum ($\displaystyle\mathrm{dB}$)
obtained using an L-band RWP and standard processing on range gate 14 of “low
mode” ($\displaystyle 1050\text{\,}\mathrm{m}$ range). Blue crosses are
centered on two peaks identified by the multipeaks picking procedure (MPP):
(a) Ground clutter; (b) Atmospheric echo. For each peak, length of the
horizontal bars correspond to the estimated spectral width. Dashed line marks
the noise level obtained by the Hildebrand and Sekhon (HS) method.

The temporal resolution of the inverted wind velocity is constrained by the
minimal integration time $\displaystyle T=N\Delta t$ of each individual
Doppler spectra necessary to ensure a satisfactory frequency (hence radial
velocity) resolution and SNR and a sufficient number
$\displaystyle\operatorname*{NSPEC}$ of incoherent spectra to obtain a
meaningful ensemble average. With the typical experimental values
$\displaystyle T=128\times$\displaystyle 5\text{\,}\mathrm{ms}$=$\displaystyle
640\text{\,}\mathrm{ms}$$ for the coherent integration time and $\displaystyle
50$ for the number of incoherent spectra, this leads to a temporal resolution
of about 16 seconds for the wind velocity, an acquisition rate that might be
too coarse for the observations of some transitory phenomena. Due to the time-
frequency dilemma, the integration time cannot be further reduced in the frame
of the FFT method. This limits the accuracy of the velocity estimation and
calls for an alternative, non-spectral method.

###  2.2 AR-MEM Representation of Time Series and PSD

In the AR approach, the instantaneous received complex signal at the sample
rate $\displaystyle\Delta t$ is modeled as a linear combination of its past
values and a white noise $\displaystyle\varepsilon$

$s(n\Delta t)=-\sum_{k=1}^{p}a[k]s\left(s(n-k)\Delta t\right)+\varepsilon[n]$
(5)

The number $\displaystyle p$ of values in the past which are involved in the
summation is termed the AR order and the $\displaystyle\\{a[k]\\}_{k=1}^{p}$
are the AR coefficients. After determination of the latter, the signal PSD is
given by [35]:

$P(\omega)=P_{\varepsilon}\left|1+\sum_{k=1}^{p}a[k]e^{-i\omega k\Delta
t}\right|^{-2}$ (6)

where $\displaystyle P_{\epsilon}$ is the white noise PSD. It must be
underlined that this derivation of the PSD presents important differences with
respect to the expression (4) obtained with a classical FFT. In the AR
approach, the signal is not assumed to be periodic and the spectral estimation
does not suffer from windowing nor truncation. Furthermore, the PSD (6) can be
evaluated at arbitrary frequency $\displaystyle\omega$ and is not bound to a
discrete set of dual frequencies as imposed by the FFT. As pointed out by
[23], this is particularly suitable for short time series, when the FFT would
neither ensure sufficient frequency resolution nor statistical convergence.

A key point in estimating the AR spectra is the choice of the optimal order
$\displaystyle p$, a problem that has been extensively discussed in the
literature (e.g. [33, 12]). Small values of $\displaystyle p$ with respect to
the number of samples $\displaystyle N$ generally leads to a poor frequency
resolution while large values can induce spurious spectral peaks. It has
therefore been suggested by [37] to select $\displaystyle N/3-1\leq p\leq
N/2-1$. In the specific context of radar applications, the authors proposed to
retain $\displaystyle p=N/2$ [15] on the basis of empirical tests.

According to (6), estimating the AR PSD is equivalent to finding the AR
coefficients. Multiple methods have been proposed, namely the Yule-Walker
equations, the Levinson-Durbin algorithm, or the Burg method (we refer to [27]
for a review). This last method is known to give PSD estimates with high
resolution, especially for short records [34, 9], a property which motivated
our choice in the present context. In the Burg method [6] the AR coefficients
are obtained by minimizing the square sum of the forward and backward
prediction errors, considering the AR model (5) as a linear prediction filter.
Multiple implementations of the Burg’s method are nowadays available in most
digital signal processing programming languages. The Burg’s method can further
be interpreted in terms of signal entropy as it maximizes the randomness of
the unknown time series. In agreement with the radar literature [21], we will
hereinafter refer to the Burg’s method as the Maximum Entropy Method (MEM) and
its combination with an AR process as the AR-MEM. The standard processing
scheme described in Section 2.1 with the replacement of the FFT with the AR-
MEM for the computation of the Doppler spectrum will be referred to in the
following as the “AR-MEM method”.

##  3 Evaluation With Synthetic Data

To assess the performances of the AR-MEM in the context of rapid scale wind
profiling, we went through a series of tests using synthetic RWP-like time-
series with a known radial wind velocity. We used the method described in [42]
for constructing synthetic data with realistic statistical properties. We
could therefore generate “weatherlike” voltage complex time series with
prescribed radial wind speed $\displaystyle V_{r}$, spectral width
$\displaystyle\sigma_{vn}$ and SNR. To do this we first construct the signal
PSD as:

$P(\omega)=-\big{(}\sigma(\omega)+C(\omega)+\mathcal{N}\big{)}X(\omega)$ (7)

where $\displaystyle\sigma$ is a centered Gaussian-shaped atmospheric PSD (2),
$\displaystyle C$ is a narrow Gaussian-shaped PSD accounting for ground-
clutter echo, $\displaystyle\mathcal{N}$ is the uniform normalized PSD of
white noise and $\displaystyle X$ is an exponentially distributed random
variable. The complex spectral components lead to the complex voltage time
series through an inverse Fourier transform:

$s(t)=\sum_{j}\sqrt{P(\omega_{j})}e^{i(\omega_{j}t+\varphi_{j})}e^{i\Phi_{D}(t)}$
(8)

where $\displaystyle\omega_{j}=2\pi j/T$ and $\displaystyle\varphi_{j}$ are
uniform independent random phases. By construction, the amplitudes
$\displaystyle\sqrt{P}$ are Rayleigh distributed and the individual frequency
components are complex Gaussian variables. The deterministic varying phase
$\displaystyle\Phi_{D}(t)$ accounts for the phase shift induced by the
velocity of the inhomogeneities carried by the wind
$\displaystyle\Phi_{D}(t)=4\pi/\lambda V_{r}t=\omega_{D}t$.

Figure 2: Influence of the SNR ($\displaystyle\mathrm{dB}$) on the RMSE
($\displaystyle\mathrm{m}\text{.}{\mathrm{s}}^{-1}$) in estimating the first-
order moment (radial wind speed $\displaystyle U_{r}$) from synthetic RWP
time-series with $\displaystyle U_{r}=$\displaystyle
4\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$ and using: (a) FFT; (b) AR-
MEM. The radar signal is modeled by a random stationary process with
prescribed Doppler spectrum (8) and a representative value of spectral width
($\displaystyle\sigma_{vn}=0.313$). For each SNR value, 100 time series are
generated and split into $\displaystyle 60$ overlapping sub-series of
$\displaystyle N$ time steps.

Figure 3: Same as Figure 3 but for the RMSE
($\displaystyle\mathrm{m}\text{.}{\mathrm{s}}^{-1}$) in estimating the second-
order moment (spectral width $\displaystyle\sigma_{v}$).

We performed a series of numerical tests according to the model (7), (8) and
(2) for some typical values of the radial wind speed $\displaystyle V_{r}$ and
spectral width $\displaystyle\sigma_{v}$ and for a wide range of samples sizes
$\displaystyle N$ and SNR. The instantaneous PSD was recalculated using either
the AR-MEM or the classical FFT approach by processing half overlapping series
of variable length. The sample size was increased from $\displaystyle N=16$ to
$\displaystyle N=128$ corresponding to an integration time from $\displaystyle
T=80$ ms to $\displaystyle T=640$ ms. While the number of available frequency
bins with the FFT is equal to the sample length, the AR-MEM makes it possible
to evaluate the PSD using (6) over an arbitrary large number of frequency bins
(here 128). For each test case, a total of 100 random time series were
generated and the root-mean-square error (RMSE) was calculated between
estimated and simulated radial wind speed. Figure 3 shows the influence of the
SNR on the accuracy of the first moment for a radial wind speed of
$\displaystyle 4\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$ and a typical
spectral width ($\displaystyle\sigma_{vn}=0.0313$ [28]) when using FFT (Figure
3(a)) and AR-MEM (Figure 3(b)). As expected, the RMSE is smaller for “long”
than “short” integration times in both FFT and AR-MEM. Furthermore, in the
case of “short” integration times ($\displaystyle N=16$), the RMSE in AR-MEM
is twice lower than in FFT ($\displaystyle
0.56\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$ versus $\displaystyle
1.01\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$, respectively). Note that
the quality of the estimation is robust to noise, as seen from the quasi-
constant value of the RMSE as the SNR is increased. This is due to the
barycentric method used by the MPP to identify the precise peak location
within the Doppler spectrum. In the same manner we also assessed the
robustness of the second-order moment (Figure 3). For “long” time series
($\displaystyle N=128$) FFT and AR-MEM provide accurate estimates, with an
RMSE close to $\displaystyle 25\text{\,}\mathrm{cm}\text{.}{\mathrm{s}}^{-1}$.
However, the RMSE of the estimation obtained with FFT increases significantly
when the sample size is decreased (up to $\displaystyle
1.5\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$ for $\displaystyle N=16$). As
previously stated the frequency resolution of the FFT is insufficient for such
short samples, causing the atmospheric echo to spread over multiple frequency
bins and mixing with the ground-clutter echo. In AR-MEM, the frequency
resolution remains constant regardless of the integration time and the
atmospheric echo is adequately described in the Doppler spectrum.

Finally we assessed the computational time needed to produce AR-MEM Doppler
spectra with respect to the conventional FFT method. The latter is known to
require $\displaystyle O(N\log N)$ operations for a single sample of
$\displaystyle N$ time steps. In AR-MEM, the authors estimated that the
computational time with an optimal order $\displaystyle p=N/2$ is of the order
of $\displaystyle O(N^{2})$ [15]. This should not prevent from using the AR-
MEM as the average computational time per beam for 60 range gates (that is,
the average computational time requested for computing Doppler spectra as
Figure 6(c)) was found to be $\displaystyle 21.40\text{\,}\mathrm{s}$ on a
standard desktop computer, while being only of $\displaystyle
2.18\text{\,}\mathrm{s}$ in FFT. In practice, this computational time can be
further decreased by diminishing the order ($\displaystyle N/3$ instead of
$\displaystyle N/2$) without significantly deteriorating the spectral
estimation.

##  4 Experimental assessment with a L-band RWP

The potential of the AR-MEM to estimate wind velocities at a rapid time scale
in experimental conditions was evaluated on the basis of the measurements
provided by the PCL1300 UHF RWP manufactured by company Degreane Horizon,
operating at $\displaystyle 1.274\text{\,}\mathrm{GHz}$. The data set under
consideration was acquired on September 24, 2012, near the landing runways of
Paris Charles de Gaulle Airport, Roissy-en-France, France ($\displaystyle
49\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{N}$, $\displaystyle
2.75\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{E}$, $\displaystyle
107\text{\,}\mathrm{m}$ above sea level). The weather was mostly cloudy with
an average relative humidity of $\displaystyle 76\text{\,}\%$. Four radar
beams were pointed $\displaystyle\theta=$\displaystyle
17\text{\,}\mathrm{\SIUnitSymbolDegree}$$ from the zenith to the north, east,
west and south and a fifth beam pointed to the zenith, using the classical DBS
technique. Uninterrupted time series of 3264 samples ($\displaystyle
17.5\text{\,}\mathrm{s}$) were collected for each beam and range gate. In
order to achieve a trade-off between the range resolution and the available
range, which are impacted in contradictory ways by the pulse duration, two
pulsing modes were used alternatively. The first sweep cycle was realized on
the 5 antennas in “high mode”, that is with a pulse duration of
$\displaystyle\tau=$\displaystyle 2.5\text{\,}\mathrm{\SIUnitSymbolMicro s}$$;
a second sweep cycle was run in “low mode” corresponding to a shorter pulse
duration of $\displaystyle\tau=$\displaystyle
1\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. The high mode provides measurements
to a maximal theoretical range of $\displaystyle 8.5\text{\,}\mathrm{km}$ with
a range resolution of $\displaystyle 375\text{\,}\mathrm{m}$ while the low
mode give access to a finer range resolution ($\displaystyle
150\text{\,}\mathrm{m}$) over a shorter range interval (up to $\displaystyle
4.5\text{\,}\mathrm{km}$). Hereinafter, the “reference” processing scenario
consists in using the FFT method over “long” half overlapping samples
($\displaystyle N=128$) and applying the SAM over the resulting
$\displaystyle\operatorname*{NSPEC}=61$ individual PSD. This corresponds to a
total integration time of $\displaystyle 17.5\text{\,}\mathrm{s}$ and provides
a frequency resolution $\displaystyle\Delta f=$\displaystyle
1.61\text{\,}\mathrm{Hz}$$ allowing for a velocity resolution
$\displaystyle\Delta V_{r}=$\displaystyle
0.19\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$, assuming stationary wind
over this duration.

Figure 4: Data recorded by the Paris CDG RWP on September 24, 2012, on the
range gate 3 of “low mode ” ($\displaystyle 225\text{\,}\mathrm{m}$ range) and
processed using the FFT method from “short” time-series ($\displaystyle N=16$
time steps). (a) Time-frequency spectrogram ($\displaystyle\mathrm{dB}$,
colorscale); (b) SAM-averaged Doppler spectra ($\displaystyle\mathrm{dB}$)
obtained using the Merritt algorithm applied over
$\displaystyle\operatorname*{NSPEC}=61$ individual spectra. Figure 5: Same as
Figure 4, but using the AR-MEM. IQ-imbalance artifacts can be seen in the
negative radial velocities of the panels three to five of the time-frequency
spectrogram.

We first assessed the performances of the AR-MEM in computing the Doppler
spectrum from “short” time-series. In this fast-time scenario the voltage
time-series were split into 487 overlapping sub-series of $\displaystyle N=16$
time steps from which the individual PSD were computed using the FFT and the
AR-MEM. As an example the temporal variations of the instantaneous PSD are
shown in the time-Doppler plane in Figs. 4(a) and 5(a) using $\displaystyle
17.5\text{\,}\mathrm{s}$ of data. The FFT spectrogram yields a coarse
frequency resolution $\displaystyle\Delta f=$\displaystyle
13.6\text{\,}\mathrm{Hz}$$ (or velocity resolution $\displaystyle\Delta
V_{r}=$\displaystyle 1.6\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$), while
the short-time AR-MEM computed with optimal order ($\displaystyle p=8$) on a
finer frequency grid (128 bins) exhibits a frequency resolution
$\displaystyle\Delta f=$\displaystyle 1.76\text{\,}\mathrm{Hz}$$ (that is
$\displaystyle\Delta V_{r}=$\displaystyle
0.21\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$$). In the former case
(Figure 4(a)), the wind echo in the positive radial velocities is mixed with
the ground-clutter. A non-cooperative flyer, likely a bird, produces a strong
echo between the 5th and 9th second on the time axis and cannot be
distinguished from the atmospheric return. However, the AR-MEM time-Doppler
spectrogram (Figure 5(a)) preserves an accurate description of the atmospheric
echo, even in presence of the flyer.

Figure 6: Normalized Range-Doppler power spectra ($\displaystyle\mathrm{dB}$;
colorscale), inferred wind profiles (—-,
$\displaystyle\mathrm{m}\text{.}{\mathrm{s}}^{-1}$) and first-order spectral
width (—-, $\displaystyle\mathrm{m}\text{.}{\mathrm{s}}^{-1}$). The
acquisition was performed on September 24, 2012, at 19:03:27 UTC along the
southern beam of the Paris CDG RWP operating in low mode, and processed using:
(a) FFT and long sub-series ($\displaystyle N=128$); (b) FFT and short sub-
series ($\displaystyle N=16$); and (c) AR-MEM and short subseries
($\displaystyle N=16$). Incoherent averaging is set to a maximum of
$\displaystyle 60$ subseries. Solid lines represent the first-order spectral
moments obtained using MPP and horizontal bars mark the second-order spectral
moment. Figure 7: Scatter plots (colorscale, counts) of estimated first-order
moments (or radial velocity $\displaystyle U_{r}$,
$\displaystyle\mathrm{m}\text{.}{\mathrm{s}}^{-1}$) along every range gate of
the southern beam of the Paris CDG RWP on September 24, 2012, operating in low
mode and during a 24-h interval. (a) FFT and long subseries ($\displaystyle
N=128$) versus AR-MEM and long subseries ($\displaystyle N=128$). (b) FFT and
long subseries ($\displaystyle N=128$) versus FFT and short subseries
($\displaystyle N=16$). (c) AR-MEM and long subseries ($\displaystyle N=128$)
versus AR-MEM and short subseries ($\displaystyle N=16$). The RMSE (with
respect to the reference long-time estimates) using short-time FFT was found
to be $\displaystyle 4.83\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$, while
being only of $\displaystyle 2.10\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$
in short-time AR-MEM.

To confirm these observations we further used the SAM from 8 groups of
$\displaystyle\operatorname*{NSPEC}=61$ individual PSD (delimited by dashed
lines on Figs. 4(a) and 5(a)). The resulting Doppler spectra are shown in
Figs. 5(a) and 5(b) for the FFT and AR-MEM methods, respectively. In the fast-
time scenario using FFT (Fig. 5(b)), the atmospheric echo is mixed with the
ground-clutter for the near-zero velocities. The MPP therefore fails to
retrieve the wind peak as the resolution of the Doppler spectrum is clearly
insufficient. However the AR-MEM Doppler spectra (Fig. 4(b)) is able to
identify unambiguously two distinct peaks corresponding to the ground-clutter
and the atmospheric return.

We now assess the performances of AR-MEM with respect to the FFT method by
comparing the corresponding range-Doppler profiles. Fig. 6 depicts such
profiles along the southern beam for the high mode data using either method.
Again, the PSD was computed using $\displaystyle\operatorname*{NSPEC}=61$ half
overlapping time-series followed by the SAM as previously described. The MPP
is applied on each range gate to identify 3 peaks and a consensus algorithm is
used to retain the most probable wind peak. Both the aforementioned reference
and fast-time scenarios are tested with the FFT and AR-MEM methods. In the
former scenario (Fig. 6(a)), a typical wind echo is clearly identifiable up to
$\displaystyle 3\text{\,}\mathrm{km}$ with a radial velocity close to
$\displaystyle 5\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$ and mean
spectral width of $\displaystyle
2.5\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$. In the fast-time scenario
using FFT (Fig. 6(b)), the atmospheric echo is mixed with the ground-clutter
in the lowest range gates. The spectral moments could not be properly
retrieved as the resolution of the Doppler spectrum is very insufficient,
leading to obviously erroneous wind estimates. However, the short-time AR-MEM
(Fig. 6(c)) computed with optimal order ($\displaystyle p=8$) on a finer
frequency grid (128 bins) leads to a wind profile which is fully consistent
with the reference scenario.

To confirm this observation, a systematic comparison of the two processing
methods using long and short integration time was run using $\displaystyle
24\text{\,}\mathrm{h}$ of the high-mode data acquired along the 5 beams. We
performed consistency tests between the velocity estimates obtained with long
and short integration times and obtained using the different methods. The
scatterplots in Fig. 7 show that the FFT and AR-MEM method are perfectly
consistent for long integration times but quite different for short
integration time, where only the AR-MEM is capable of reproducing the
performances of the long time counterpart.

In a similar way we investigated the accuracy of the second-order moment
$\displaystyle\sigma_{v}$. As shown in Fig. 8(a), the spectral width parameter
is significantly overestimated using the short-time FFT when compared to the
reference long-time FFT estimates, with a RMSE of $\displaystyle
6\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$. Again, the AR-MEM estimates
benefit from a finer frequency resolution of the PSD and have a much smaller
RMSE ($\displaystyle 0.88\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$).

Figure 8: Scatter plots (same colorscale as Fig. 7) of estimated second-order
moments (in terms of velocity spread $\displaystyle\sigma_{v}$,
$\displaystyle\mathrm{cm}\text{.}{\mathrm{s}}^{-1}$) from the same dataset as
Figure 7 and processed using: (a) FFT and long subseries ($\displaystyle
N=128$) versus AR-MEM and long subseries ($\displaystyle N=128$). (b) FFT and
long subseries versus FFT and short subseries ($\displaystyle N=16$). (c) AR-
MEM and long subseries versus AR and short subseries. In any case, incoherent
averaging is set to a maximum of $\displaystyle 60$ subseries.

##  5 Validation with colocated Lidar measurements

The validity of the short-time AR-MEM to infer radial wind velocities from RWP
experimental measurements has first been assessed by self-consistency tests
with long-time estimates (Figures 7 and 8). We now provide a further
validation of the method by comparison with a colocated wind Doppler lidar.
The comparison of radar and lidar profiler is always challenging due to the
very different spatio-temporal scales entering in play,. However, a good
agreement between the two types of instruments has already been reported in
the literature. For example using the NOAA high-resolution Doppler lidar
(HRDL) and a $\displaystyle 915\text{\,}\mathrm{MHz}$ boundary layer RWP,
[10], an excellent correlation between lidar and RWP observations
($\displaystyle R^{2}=0.99$) and a small mean absolute error ($\displaystyle
0.15\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$) were reported.

The integration times required with lidars are by at least one order of
magnitude smaller than those typically employed with RWP; this is therefore
adapted to evaluate the accuracy of short-time measurements with RWP. Both
instruments were operated at the Aerological station of Payerne
($\displaystyle 7\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{E}$,
$\displaystyle 47\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{N}$,
$\displaystyle 491\text{\,}\mathrm{m}$ above sea level), $\displaystyle
45\text{\,}\mathrm{km}$ North-East of Lausanne, Switzerland. This station is
part of a network owned by the Federal Office of Meteorology and Climatology
(MeteoSwiss) intended to monitor the atmosphere above the Swiss Plateau. The
RWP of Payerne has the same characteristics as the aforementioned Paris CDG
RWP (see Section 4), except that its oblique beams are rotated by
$\displaystyle 45\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to North,
hence pointing to North-East, South-East, South-West and North-West,
respectively, with an angle $\displaystyle\theta=$\displaystyle
17\text{\,}\mathrm{\SIUnitSymbolDegree}$$ to the zenith. The lidar is of type
WindCube WLS200S manufactured by company Vaisala. An overview of the operating
characteristics of this lidar, compared to the RWP, is shown in Table 1. The
measurements were performed under clear weather from 12:00 to 23:59 UTC on
March 2, 2021.

Table 1: Characteristics for the Degreane Horizon PCL1300 RWP and the Vaisala WindCube WLS200S Doppler lidar Feature | Wind profiler | Doppler lidar
---|---|---
Wavelength | $\displaystyle 23\text{\,}\mathrm{cm}$ | $\displaystyle 1.54\text{\,}\mathrm{\SIUnitSymbolMicro m}$
Vertical resolution | $\displaystyle 150\text{\,}\mathrm{m}$ | $\displaystyle 30\text{\,}\mathrm{m}$
Pulse duration | $\displaystyle 1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ | $\displaystyle 200\text{\,}\mathrm{ns}$
Typical integration time | $\displaystyle 15\text{\,}\mathrm{s}$ | $\displaystyle 0.5\text{\,}\mathrm{s}$
Figure 9: Scatter plots of estimated radial wind velocity during
$\displaystyle 12\text{\,}\mathrm{h}$ on March 2, 2021, using the RWP of
Payerne and a colocated lidar as reference. (a) FFT, long sample
($\displaystyle N=128$ time steps). (b) AR-MEM, long sample ($\displaystyle
N=128$ time steps). (c) FFT, short sample ($\displaystyle N=16$ time steps).
(d) AR-MEM, short sample ($\displaystyle N=16$ time steps). In any case,
incoherent averaging is set to a total of 60 subseries.

For the purpose of the comparison with the RWP, the lidar data under
consideration were processed as follows. First, the zonal, meridional and
vertical wind components $\displaystyle(u,v,w)$ were projected along the four
oblique radar beams as

$U_{r,\varphi}=u\sin\theta\sin\varphi+v\sin\theta\sin\varphi+w\cos\theta$ (9)

where $\displaystyle\varphi$ is the beam azimuth with respect to North. Then
the projected radial velocities were interpolated on the radar range gates and
the lidar measurements contained within each radar integration time were
averaged. The comparisons were made using FFT and AR-MEM methods, using long
($\displaystyle N=128$ time steps) and short ($\displaystyle N=16$ time steps)
time series. We chose to present only the measurements performed on two
opposite radar beams, here the North-East and the South-West, but similar
results were obtained using the two remaining beams.

The results of the comparisons for each different processing are shown in
Figure 9 using the lidar measurements as reference. Additionally, the
performances of the different methods when compared with the reference lidar
measurements are recap in Table 2. For long time-series, the FFT and AR-MEM
estimates (Figs. 9(a) and (b), resp.) show similar performances (although a
slightly improvement is obtained with AR-MEM) with a global RMSE of about
$\displaystyle 0.45\text{\,}\mathrm{m}\text{.}{\mathrm{s}}^{-1}$ and a high
correlation coefficient. At short integration time (Figs. 9(c) and (d),
resp.), however, the use of the AR-MEM allows for an important increase in the
number of available measurements with respect to the conventional FFT method
(from 167 to 522) as well as more accurate estimates, with a RMSE reduced by
about $\displaystyle 20\text{\,}\%$ and a higher correlation coefficient.

Table 2: Comparison with Doppler lidar

Method | RMSE ($\displaystyle\mathrm{m}\text{.}{\mathrm{s}}^{-1}$) | Number of estimates
---|---|---
FFT long | 0.45 | 64
AR-MEM long | 0.43 | 71
FFT short | 1.07 | 167
AR-MEM short | 0.81 | 522

Note: RMSE (in $\displaystyle\mathrm{m}\text{.}{\mathrm{s}}^{-1}$) between the
colocated RWP radial velocity and the Doppler lidar measurements (see Figure
9), together with the number of compared estimates for each method.

##  6 Discussion and Conclusion

We have presented an application of the AR modeling with MEM to the wind
profiling of the lower atmosphere using L-band Doppler radars. While the
common operational approach relies on the computation of the Doppler spectrum
using the FFT algorithm, we made use of an AR representation of the complex
antenna voltage time-series. This allowed to bypass the time-frequency dilemma
encountered when using the FFT. The MEM was further found to be efficient to
estimate the AR coefficients in the context of short radar samples. We have
first tested the proposed AR-MEM approach on the basis of noisy synthetic RWP
time-series and quantified its augmented performances with respect to the
classical spectral estimation in terms of robustness to noise when inverting
the first- and second-order spectral moments from short samples.

The AR-MEM approach was applied to a $\displaystyle 24\text{\,}\mathrm{h}$-set
of data recorded with a RWP during an experiment in the vicinity of the
landing runways of Paris CDG Airport. It was shown that the AR-MEM can be used
to visualize the temporal variations of the Doppler spectrum at high temporal
rate, whereas the FFT fails to accurately represents the atmospheric echo in
such conditions. A statistical analysis of the performances of wind profiling
using either method, AR-MEM and FFT, has been made for short-time samples
using long-time samples as reference. The main outcome is that the use of AR-
MEM allows for a significant decrease in the RMSE of both first- and second-
order spectral moments estimates. To further validate the AR-MEM wind
profiling we analyzed a $\displaystyle 12\text{\,}\mathrm{h}$-dataset of
colocated RWP and lidar measurements in Payerne. Again, the AR-MEM velocity
estimates were found in closer agreement to the lidar for short-time samples.

RWP measurements are often contaminated by the spurious and frequent returns
of uncooperative flyers such as birds or planes. The robustness of the AR-MEM
method to these parasitic echoes even with short time series has been
assessed. This paves the way to new strategies to mitigate the unwanted
presence of uncooperative flyers in wind profiling. Another useful application
of the rapid wind profiling is the spectral analysis of clear-air turbulence
echoes which requires a large variety of temporal scales. For both
applications, dedicated measurement campaigns will be carried out in the
future.

## Acknowledgment

The authors would like to thank Dr Maxime Hervo and the people from the
Aerological station of Payerne (Federal Office of Meteorology and
Climatology), Switzerland, for performing and providing RWP and lidar
measurements. The authors are also indebted to Dr Philipp Currier, retired
from Degreane Horizon, for his thorough advices on radar signal processing.

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