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**Research article**
27 Nov 2018

**Research article** | 27 Nov 2018

Comparison of methods to derive radial wind speed from a continuous-wave coherent lidar Doppler spectrum

^{1}DTU Wind Energy, Roskilde 4000, Denmark^{2}Windar Photonics A/S, Taastrup 2630, Denmark

^{1}DTU Wind Energy, Roskilde 4000, Denmark^{2}Windar Photonics A/S, Taastrup 2630, Denmark

**Correspondence**: Dominique P. Held (domhel@dtu.dk)

**Correspondence**: Dominique P. Held (domhel@dtu.dk)

Abstract

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Continuous-wave (cw) lidar systems offer the possibility to remotely sense wind speed but are also affected by differences in their measurement process compared to more traditional anemometry like cup or sonic anemometers. Their large measurement volume leads to an attenuation of turbulence. In this paper we study how different methods to derive the radial wind speed from a lidar Doppler spectrum can mitigate turbulence attenuation. The centroid, median and maximum methods are compared by estimating transfer functions and calculating root mean squared errors (RMSEs) between a lidar and a sonic anemometer. Numerical simulations and experimental results both indicate that the median method performed best in terms of RMSE and also had slight improvements over the centroid method in terms of volume averaging reduction. The maximum, even though it uses the least amount of information from the Doppler spectrum, performs best at mitigating the volume averaging effect. However, this benefit comes at the cost of increased signal noise due to discretisation of the maximum method. Thus, when the aim is to mitigate the effect of turbulence attenuation and obtain wind speed time series with low noise, from the results of this study we recommend using the median method. If the goal is to measure average wind speeds, all three methods perform equally well.

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Held, D. P. and Mann, J.: Comparison of methods to derive radial wind speed from a continuous-wave coherent lidar Doppler spectrum, Atmos. Meas. Tech., 11, 6339–6350, https://doi.org/10.5194/amt-11-6339-2018, 2018.

1 Introduction

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Remote sensing is an attractive alternative to traditional in situ measurements of wind speed. For wind turbines, light detection and ranging (lidar) devices can replace the installation of large meteorological masts hosting cup or sonic anemometers in order to meet the constantly increasing measurement height requirements. This flexibility led to a large variety of applications of lidars spanning from lidar-assisted yaw and pitch control (Schlipf, 2015) to site assessment (Sanz Rodrigo et al., 2013) and power curve validation (Borraccino et al., 2017). However, one has to keep in mind that there is one important principal difference between measurements from a lidar device and sonic or cup anemometers, namely the averaging over a rather large measurement volume.

Lidars can be operated in two different modes: laser light can either be emitted in continuous-wave (cw) or pulsed form. For a cw lidar, a laser beam is focused on the desired point in space and measures the backscattered light. The radial speed of the aerosols can be estimated from the induced Doppler shift of the backscattered light. However, there is an ambiguity in the definition of the dominant frequency of the Doppler spectrum. In early systems, simply the maximum value of the power spectral density (PSD) was used. But since this gives integer multiples of the frequency step (which depends on the fast Fourier transform set-up) it has the disadvantage of returning a noisy signal. Thus, nowadays most commercial cw lidar systems use the centroid of the PSD above a certain noise level (Harris et al., 2006), while in research instruments (e.g. short-range WindScanner) the median method is implemented (Angelou et al., 2012).

On the other hand, pulsed lidars emit a light pulse of finite length. This allows the atmosphere to be probed at several positions along the laser beam based on the current location of the light pulse as it propagates. For pulsed lidars significantly more investigations have been done on how to determine the radial speed from a Doppler spectrum. For example, Frehlich (2013) presents an investigation on the maximum likelihood (ML) algorithm and minimum mean squared error method. Both estimators had similar performance and the latter was chosen because of computational efficiency. In addition, Dolfi-Bouteyre et al. (2016) examined two simple methods (maximum and centroid), the previously mentioned ML approach, polynomial fitting and an adaptive filter method. The polynomial fitting method was found to perform best in laminar flow but it is suggested that in more complex flow, advanced estimators are needed. However, for a cw lidar these more complex methods are unavailable because they rely on an underlying model for the shape of the Doppler spectrum. According to the current knowledge of the authors no such model exists for cw lidars.

One of the measurement differences of a lidar compared to cup or sonic
anemometers is its large probe volume, which leads to turbulent fluctuation
attenuation. While the effect of the probe volume on turbulence attenuation
can be modelled by theory for the centroid method, no theories exist for the
median or maximum methods. Several studies aimed at validating the theory for
the centroid method by comparing the spectral transfer function between a
lidar and sonic anemometer, i.e. the ratio between the power density spectra
of lidar and sonic measurements. An early study by Lawrence et al. (1972)
compared a cw carbon dioxide (CO_{2}) lidar to a cup anemometer
around 10 m above the ground. Due to the short focus distance (approximately
30 m), the Rayleigh length was less than 0.5 m and analysis of the PSD
showed no effect of the lidar's volume averaging. Another cw
CO_{2} lidar at focus distances up to 200 m close to the ground
was used in Banakh and Smalikho (1999). High-frequency fluctuation attenuation due to
volume averaging was observed. A power spectral analysis showed significant
deviations between the lidar and sonic power spectrum, starting at 0.4 Hz and
becoming more severe at increasing frequencies. The experimental results
support the theory developed in the paper.

A cw lidar focused close to a sonic anemometer mounted 78 m above ground was used in Sjöholm et al. (2009). Data gathered at 20 Hz were used to investigate the spatial averaging, and good agreement between theory and experiment was found. However, during periods with low-level clouds the measurements were affected negatively. The same objective was followed in Angelou et al. (2012), in which the transfer function of a tower-mounted horizontally staring lidar was determined against a mast-mounted sonic anemometer allowing for horizontal measurements. The study was limited to when the lidar beam was aligned with the wind direction, and during these periods the spatial averaging effect from measurements agreed perfectly with the theory for the centroid method attenuation. Further, this study presented a method to reduce the influence of noise at high frequencies by estimating the spectral transfer function using the cross-spectrum between lidar and sonic measurements and the auto-spectrum of the sonic anemometer.

Two studies investigated the spatial averaging of a long-range pulsed lidar compared to mast-mounted sonic anemometers (Fuertes et al., 2014; Mann et al., 2009). Both studies showed the feasibility of measuring 3-D wind vectors by synchronised lidars focused at one point in space. It was also shown that the attenuation from spatial averaging can be predicted by the theory for pulsed lidar systems, which is presented in the two studies.

A slightly different approach was followed in Peña et al. (2017). In this study a cw nacelle lidar and a pulsed nacelle lidar were compared against both a cup and sonic anemometer. Turbulence statistics was calculated by fitting a spectral tensor model including a lidar volume averaging model to the sonic measurements and an average Doppler spectrum method, which was also used in Branlard et al. (2013). The first method enabled the retrieval of filtered turbulence statistics, while the second yielded measures not affected by spatial averaging. These results reiterated that predictions of the spatial averaging effect are consistent with theory.

A machine-learning approach to produce unfiltered wind speed variances from pulsed lidar signals was used in Newman and Clifton (2017). Besides a model for spatial averaging, the algorithm also includes automatic noise removal. Comparisons to sonic anemometers showed improvements when using the algorithm under all stability classes, but the results are highly dependent on the input variables and the training sets.

From the studies mentioned above it can be seen that the effect of the lidar's spatial averaging can be predicted theoretically, which has also been confirmed experimentally. In contrast to pulsed lidars, little work has been done on the effect of how the radial wind speed is calculated from a Doppler spectrum for cw lidars. Thus, the objective of this study is to investigate the influence of using different methods of determining the dominant frequency in a lidar Doppler spectrum (maximum, median, centroid) and its influence on the volume-averaging effect of lidar measurements. This is important because the lidar's probe volume has an attenuating effect on the measurement of turbulent fluctuations. As a consequence, estimates of wind speed variances will be biased if the lidar is used for site characterisation. Also for lidar-assisted control integration an accurate measurement of the turbulent fluctuations is important. Since no theory has been formulated for the median and maximum method yet, the study was motivated by initial numerical simulations that showed improved performance for these two methods compared to the centroid method. In this study the numerical simulations are extended and compared to data gathered during a field experiment.

2 Materials and methods

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Statistically, the fluctuating part of an incompressible,
homogeneous wind field ** u**(

A cw lidar measurement can be modelled by the convolution of the projected
radial component ** n**⋅

$$\begin{array}{}\text{(1)}& {v}_{r}\left(\mathit{r}\right)=\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}\mathit{\phi}\left(s\right)\mathit{n}\cdot \mathit{u}(s\mathit{n}+\mathit{r})\mathrm{d}s,\end{array}$$

where *z*_{R} is the so-called Rayleigh length that characterises the probe
volume, *s* is the distance from the focus point along the beam,
** n** is the beam unit vector and

Equation (1) assumes the definition of the centroid of the Doppler
spectrum; in the following we will refer to it as *cen*. Another
method of determining the dominant frequency in a Doppler spectrum is by
simply taking the frequency bin where the peak occurs (*max*) or by
treating it as a probability density function (PDF) and taking the median
value (*med*). The estimated radial wind speed using these three
methods will be compared to the laser-line-projected sonic wind velocity
${v}_{\mathrm{s}}=\mathit{n}\cdot \mathit{u}\left(\mathit{x}\right)$. This will be done
for the numerical simulations and the experiment.

To evaluate how well lidar and sonic measurements correlate in the wave number domain, an estimation of the transfer function between the two signals is used:

$$\begin{array}{}\text{(2)}& G\left({k}_{\mathrm{1}}\right)={\left|{\displaystyle \frac{{\mathit{\chi}}_{r,\phantom{\rule{0.125em}{0ex}}\mathrm{s}}\left({k}_{\mathrm{1}}\right)}{{F}_{\mathrm{s}}\left({k}_{\mathrm{1}}\right)}}\right|}^{\mathrm{2}},\end{array}$$

where *χ*_{r, s}(*k*_{1}) refers to the cross-spectrum between the lidar and
sonic signal and *F*_{s}(*k*_{1}) refers to the auto-spectrum of the sonic
signal. The closer *G*(*k*_{1}) is to unity, the smaller the effect of volume
averaging of the lidar is. We prefer to use the transfer function defined in
Eq. (2) to the more traditional $G\left({k}_{\mathrm{1}}\right)={F}_{r}\left({k}_{\mathrm{1}}\right)/{F}_{\mathrm{s}}\left({k}_{\mathrm{1}}\right)$ because
the auto-spectrum *F*_{r}(*k*_{1}) may be affected by noise whereas the
cross-spectrum *χ*_{r, s}(*k*_{1}) is not (Angelou et al., 2012), assuming the
sonic measurements contain considerably less noise than the lidar
measurements.

When using $\mathcal{F}\left[\mathit{\phi}\right(s\left)\right]\left(k\right)=\mathrm{exp}(-{z}_{\mathrm{R}}|k\left|\right)$, Eq. (1) can be expressed in wave number space by

$$\begin{array}{}\text{(3)}& {F}_{r}\left({k}_{\mathrm{1}}\right)=\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}{\mathbf{\Phi}}_{ij}\left(\mathit{k}\right)\mathrm{exp}(-\mathrm{2}{z}_{\mathrm{R}}|\mathit{n}\cdot \mathit{k}\left|\right)\mathrm{d}{k}_{\mathrm{2}}\mathrm{d}{k}_{\mathrm{3}},\end{array}$$

where ℱ[.] refers to the Fourier transformation. The integration
in Eq. (3) can only be solved analytically for simple forms of
**Φ**_{ij}(** k**). For the line-of-sight projected
sonic measurement

$$\begin{array}{}\text{(4)}& {F}_{\mathrm{s}}\left({k}_{\mathrm{1}}\right)=\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}{\mathbf{\Phi}}_{\mathrm{ij}}\left(\mathit{k}\right)\mathrm{d}{k}_{\mathrm{2}}\mathrm{d}{k}_{\mathrm{3}}.\end{array}$$

The cross-spectrum between the lidar and sonic measurements can then be written as

$$\begin{array}{}\text{(5)}& {\mathit{\chi}}_{r,\phantom{\rule{0.125em}{0ex}}\mathrm{s}}\left({k}_{\mathrm{1}}\right)=\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}{\mathbf{\Phi}}_{ij}\left(\mathit{k}\right)\mathrm{exp}(-{z}_{\mathrm{R}}|\mathit{n}\cdot \mathit{k}\left|\right)\mathrm{d}{k}_{\mathrm{2}}\mathrm{d}{k}_{\mathrm{3}}.\end{array}$$

It should be noted that ${lim}_{{k}_{\mathrm{1}}\to \mathrm{0}}G\left({k}_{\mathrm{1}}\right)=\mathrm{1}$ is only true when the lidar beam is aligned with the mean wind direction. In misaligned cases ${lim}_{{k}_{\mathrm{1}}\to \mathrm{0}}G\left({k}_{\mathrm{1}}\right)<\mathrm{1}$ holds (Kristensen and Jensen, 1979). Thus, for misaligned cases we do not expect the transfer function to tend towards unity for small wave numbers.

Another measure used to evaluate the performance of the different methods is the root mean squared error (RMSE):

$$\begin{array}{}\text{(6)}& \text{RMSE}\left({v}_{r,\phantom{\rule{0.125em}{0ex}}\text{method}}\right)=\sqrt{\stackrel{\mathrm{\u203e}}{({v}_{r,\phantom{\rule{0.125em}{0ex}}\text{method}}-{v}_{\mathrm{s}}{)}^{\mathrm{2}}}},\end{array}$$

where method refers to either centroid, median or maximum and the overbar indicates averaging in time. In contrast to the transfer function estimate mentioned previously, for this measure, noise in lidar measurements will affect the performance and gives an indication of the difference between lidar and sonic measurements in the time domain.

Numerical simulations illustrate results in an environment where no noise is present. The methodology used to perform these simulations was developed in Mann (1998). First it is described how a Doppler spectrum is obtained from a simulated wind time series. We narrowed our investigation to the (horizontal) 2-D case in which the cw lidar measures horizontally only. We furthermore assumed Taylor's frozen turbulence hypothesis:

$$\begin{array}{}\text{(7)}& \mathit{u}(x,\phantom{\rule{0.125em}{0ex}}y,\phantom{\rule{0.125em}{0ex}}t=\mathrm{0})=\mathit{u}(x+\mathrm{Ut},\phantom{\rule{0.125em}{0ex}}y,\phantom{\rule{0.125em}{0ex}}t),\end{array}$$

where *U* is the mean wind speed, so the wind field at any given time can be
obtained by translating the wind field at *t*=0. We did not consider any
sources of noise. In this case the Doppler spectrum *S*(*v*, *t*) can be written
as

$$\begin{array}{}\text{(8)}& S(v,\phantom{\rule{0.125em}{0ex}}t)=\underset{-\mathrm{\infty}}{\overset{\mathrm{\infty}}{\int}}\mathit{\phi}\left(s\right)\mathit{\delta}(v-\mathit{u}(s)\cdot \mathit{n})\mathrm{d}s,\end{array}$$

where *δ* is the Dirac delta function. Notice that Eq. (8) is a
convolution of the weighting function *φ* and the delta function
$\mathit{\delta}(v-\mathit{u}\cdot \mathit{n})$. If *φ* was
disregarded, Eq. (8) could be viewed as a histogram of wind
velocities. The discretisation of the histogram is chosen to match the
typical velocity bin resolution of a lidar system, which in this simulation
case was 0.1 ms^{−1} bin^{−1}.
When *φ* is included, the
wind velocities are weighted, such that the velocities around the focus point
count most. Due to the finite length of the simulated turbulent boxes, the
integration in Eq. (8) needs to be truncated. Here we chose a
distance of *M*=12*z*_{R} along the beam after which the truncation is applied,
where the Lorentzian weighting function has a value of
$\approx \mathrm{1.5}\cdot {\mathrm{10}}^{-\mathrm{4}}$ (or 0.69 % relative to the maximum value at the
focus point):

$$\begin{array}{}\text{(9)}& S(v,\phantom{\rule{0.125em}{0ex}}t)=\underset{-M}{\overset{M}{\int}}\mathit{\phi}\left(s\right)\mathit{\delta}(v-\mathit{u}(s)\cdot \mathit{n})\mathrm{d}s.\end{array}$$

To generate the wind time series we assumed for simplicity that the turbulent
fluctuations in the direction of the mean wind can be described by the model
by Mann (1994)^{1}. A
turbulent wind box was created with a horizontal 2-D wind vector at each grid
point. The dimensions of the box are 4096×4096 grid points with a
separation of 0.732 m. A total of 20 turbulent boxes with different
initial turbulence seeds have been simulated, resulting in 20 different
realisations of the turbulent field. Note that here we only simulate the
fluctuating part, and the mean wind speed is zero. An illustration of the
simulation set-up and an example of a simulated Doppler spectrum including the
radial wind speed estimates using the three different methods is shown in
Fig. 1.

In this section the experiment conducted at Risø campus with a lidar
system by Windar Photonics A/S will be presented. The WindEYE is a
commercial Doppler wind lidar that uses an all-semiconductor laser source
with a wavelength of 1553 nm; see Hu (2016). Because the purpose of the
product is wind direction measurement, it can focus in two positions by
deflecting the beam through two different lenses. The switching occurs every
half second, which means that the lidar focuses on one position for 0.5 s
and then a liquid crystal will bend the beam towards the second focus point
for another 0.5 s. Usually the device is mounted on the nacelle of wind
turbines, but for this experiment it was installed on a tower to be able to
focus on the location of two sonic anemometers around 10 m above the
ground; see Fig. 2. The lidar beams are aligned
horizontally to measure horizontal components only. The focus distance is
90 m and the Rayleigh length *z*_{R} is 14.5 m. The angle between the two beams is
60^{∘}, but since the beams are compared individually it is of no
importance here.

The sonic anemometers are two USA-1 anemometers by Metek GmbH, which were mounted on a tower at the exact position of the focus points. The focus distances have been verified experimentally in an optical laboratory, and the alignment of the lidar to the sonic anemometers was checked using an infrared sensor card; see Dellwik et al. (2015). The laser beams pass approximately 1 m above the sonic devices.

The sonic anemometers were sampled at 35 Hz and have a transducer distance of 0.175 m implying that the device retrievals approximate a point measurement compared to the averaging volume of the lidar. For all measurements the standard 2-D flow correction has been removed and instead a 3-D correction was used (Bechmann et al., 2009). Due to the inherent switch mechanism of the WindEYE lidar the data had to be combined to a rather low sampling frequency of 1 Hz. The experiment extended from 9 January to 23 March 2014 but due to synchronisation problems only the periods in February and March could be used.

3 Results and discussion

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In this section we first present an example of the numerical simulation and experimental results and then we will compare both to the analytical results. An example of the numerical simulation and the experiments can be found in Fig. 3.

At first the 10 min averages of the lidar measured wind speed component
*v*_{r} and the 3-D sonic wind vector projected on the line of sight,
** n**⋅

The wind rose derived from the two sonic anemometers is shown in Fig. 5.
Mean wind speed and direction were calculated over 10 min periods and then the average of both anemometers was taken. Two main
wind directions can be identified, of which one is aligned with the northern
beam direction. Thus, more data were gathered for a misalignment of 0^{∘}
for the northern beam compared to the southern beam.

For wind lidar systems using a homodyne detection method, there is an
ambiguity in the wind direction (whether the wind blows towards or away from the
lidar). Further the limitation to the line-of-sight component of the wind vector leads
to an ambiguity in the misalignment (whether the wind direction is misaligned
towards the left or right side of the beam). In all these cases the radial
wind speed measurements will be the same. For example, a case of a wind
direction misaligned by 10^{∘} is equivalent to a misalignment of
170, 190 and 350^{∘}. Thus, it is possible to reduce the
full 360^{∘} to one quadrant ranging from 0 to 90^{∘}. In the
following analysis the data have been binned into 10^{∘} sectors ranging
from 0 to 90^{∘}.

To create numerical simulations as close as possible to the experimental
conditions, the Mann spectral tensor has been fitted, following the procedure
in Mann (1994), to the *u* and *v* spectra obtained from sonic
measurements, where the mean wind speed was above 6 ms^{−1}.
The tensor model has three parameters: *α**ϵ*^{2∕3}, where
*ϵ* is the rate of viscous dissipation of turbulent kinetic energy and
*α* is the spectral Kolmogorov constant; *L*, which is a length scale; and
Γ, which is an anisotropy parameter. For details, see Mann (1994). This has been
done sector-wise for sectors of 10^{∘} for each beam, and the results can
be seen in Table 2.

In order to reduce the computational effort, we have taken the average value
of each parameter over all sectors and both beams and found the following
parameter: $\mathit{\alpha}{\mathit{\u03f5}}^{\frac{\mathrm{2}}{\mathrm{3}}}=\mathrm{0.58}\cdot {\mathrm{10}}^{-\mathrm{2}}$m^{4∕3}s^{−2}, *L*=22.3 m,
Γ=2.26. These parameters have been used to perform the numerical
simulations mentioned in Sect. 2.2.

In this section we will present the combination of experimental and
simulation results together with the numerical integration of
Eq. (2) (using Eqs. 3–5). In the following plots
theoretical results are represented by a solid black line, simulation results
(*Simu*) by coloured solid lines and experimental
results (*Exp*) by coloured dashed lines. The different colours stand
for the three methods to derive the radial speed from the Doppler spectrum:
centroid (cen), median (med) and maximum (max).

First, we present the simplest case when the lidar beam is aligned with the
wind direction. In this case Eq. (2) can be solved analytically
because the exponential in Eqs. (3) and (5) does not
depend on either *k*_{2} or *k*_{3} and can be moved outside the integral. This
results in

$$\begin{array}{ll}{\displaystyle}G\left({k}_{\mathrm{1}}\right)& {\displaystyle}={\left|{\displaystyle \frac{{\mathit{\chi}}_{r,\phantom{\rule{0.125em}{0ex}}\mathrm{s}}\left({k}_{\mathrm{1}}\right)}{{F}_{\mathrm{s}}\left({k}_{\mathrm{1}}\right)}}\right|}^{\mathrm{2}}\\ {\displaystyle}& {\displaystyle}={\left|{\displaystyle \frac{\mathrm{exp}(-{z}_{\mathrm{R}}|\mathit{n}\cdot \mathit{k}\left|\right){\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{\mathbf{\Phi}}_{\mathrm{ij}}\left(\mathit{k}\right)\mathrm{d}{k}_{\mathrm{2}}\mathrm{d}{k}_{\mathrm{3}}}{{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{\mathbf{\Phi}}_{ij}\left(\mathit{k}\right)\mathrm{d}{k}_{\mathrm{2}}\mathrm{d}{k}_{\mathrm{3}}}}\right|}^{\mathrm{2}}\\ \text{(10)}& {\displaystyle}& {\displaystyle}=\mathrm{exp}(-\mathrm{2}{z}_{\mathrm{R}}{k}_{\mathrm{1}}).\end{array}$$

Note that in Eq. (10) the radial speed is defined by the
centroid method. For the aligned case the transfer function only depends on
*z*_{R} and not on the turbulence model parameters.

The results for the aligned case are shown in Fig. 6, where Eq. (10) is shown as the black solid line. It can be seen that both red lines (solid and dashed) agree well with the theoretical results. Some significant deviations can be observed for the simulation results at small wave numbers, which can be explained by the truncation of the Doppler spectrum in Eq. (9).

It can also be seen that the transfer functions when using the median or maximum method lie above the results for the centroid method. This indicates that the turbulence attenuation is less severe for these two methods compared to the centroid method. Thus, fluctuations which have been measured by the sonic and are attenuated when using the centroid method due to volume averaging can indeed be sensed when using the median and maximum method. The improved performance is stronger for the numerical simulation due to the absence of noise. The median method seems to perform slightly better than the centroid method, and the maximum method has an even bigger improvement.

As an example for a misaligned case, we will now focus on a misalignment of
40^{∘}; see Fig. 7. The transfer functions for the
remaining misalignments can be found in the Appendix. For a misalignment of
40^{∘}, Eq. (10) is not valid anymore. However, Eq. (2) can be integrated numerically. It is shown again as the solid
black line. Again we can see that the simulation results using the centroid
method matches the theoretical results well for large wave numbers, but there
are deviations at low wave numbers. Similar to the aligned case, it can be
observed that the transfer functions for the median and maximum method lie
above that of the centroid method. Again, this implies that turbulent
fluctuations that were not measured using the centroid method can be sensed
using the median and maximum method. The maximum method again shows the best
performance.

Examples of these improvements can also be identified in the time domain when
looking at Fig. 3. For the numerical simulations (panel a) the improved fluctuation measurements using the maximum method are
very clear, while the median method is also able to slightly enhance the
measurements compared to the centroid method, which performs worst. A similar
tendency is also observed from the experiment (panel b). Just before
120 s we can note a very good agreement between sonic and maximum-method
*v*_{r}, whereas the other two methods are not able to detect this fluctuation.
In other periods the observed improvement is very small.

Next we consider the RMSE results. Since it was seen previously that both the median and maximum method outperformed the centroid method, the RMSE of the two methods normalised by the RMSE of the centroid is compared now:

$$\begin{array}{}\text{(11)}& \mathrm{1}-\text{RMSE}\left({v}_{r,\phantom{\rule{0.125em}{0ex}}\text{method}}\right)/\text{RMSE}\left({v}_{r,\phantom{\rule{0.125em}{0ex}}\text{cen}}\right),\end{array}$$

where method is either the median or the maximum method. Thus positive numbers indicate better performance compared to the centroid method and vice versa.

The results can be seen in Fig. 8 and show that the median method consistently outperforms the centroid method. Improvements of up to 4 % from the experimental data was observed, while the simulations showed performance increases between 3 % and 5 %. In contrast, the maximum method has persistent disadvantages compared to the centroid method. The shortcoming increases with increasing misalignment and reaches values of approximately −9 % for the experiments and close to −10 % for the numerical simulations. This indicates that the improved performance of the maximum method in reducing the effect of spatial averaging has the consequence that more signal noise is introduced. This is the result of selecting the maximum of the Doppler spectrum since the maximum position can vary depending on the frequency bin width and Doppler spectrum noise. The other two methods are more robust against noise contamination.

4 Conclusions

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In this study we compared a cw wind lidar to sonic measurements, where the sonic anemometers are mounted exactly at the focus positions of the lidar system. The lidar measurements are affected by their large probe volume, which leads to an attenuation of turbulence. The objective of the paper was to study how different methods of determining the dominant frequency in a Doppler spectrum affect wind speed measurements by a cw lidar. We used an estimation of the transfer function to evaluate the lidar's attenuation of turbulent fluctuations and the RMSE to give a metric to the general performance of the methods. Theoretical analysis, numerical simulation and data from a 2-month-long experiment have been used, and three different methods for deriving the radial speed were applied: the centroid, median and maximum method.

The analysis was able to show that the simulations, as well as the experiments, agree well with the theoretical results for the centroid method. Further, the median and maximum methods performed better both in simulations and experiments compared to the centroid method in reducing the effect of spatial averaging. Interestingly the maximum method had the highest reduction of the effect of spatial averaging. However, it also showed the highest RMSE values out of all methods due to the discretisation of picking the maximum value of the Doppler spectrum. Thus, from this study we recommend, if one's aim is to mitigate the effect of turbulence attenuation by the lidar and retrieve time series with low noise levels, using the median method as it shows slight improvements of reducing the volume effect compared to the centroid method and has the best RMSE performance. When comparing 10 min averages all methods performed equally well.

The method of using average Doppler spectra (typically 10 or 30 min averages) has also been studied to derive turbulence statistics (Branlard et al., 2013). However, using this approach, only statistics can be derived, namely the wind speed PDF and its statistical moments. What is presented here shows how carefully choosing the method of radial speed retrieval from a Doppler spectrum can partly alleviate the inherent volume averaging effect of lidar systems to provide time series information.

It should be noted that these conclusions only apply to cw lidars and not to pulsed systems as the method of deriving radial velocities is different for the latter.

Code and data availability

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Code and data availability.

The computer code to generate synthetic turbulence fields can be found at http://www.wasp.dk/weng#details__iec-turbulence-simulator (WAsP Engineering, 2018).

Appendix A

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Author contributions

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Author contributions.

DPH performed the research work and prepared the manuscript. JM conceived the research plan and supervised the research work and the manuscript preparation.

Competing interests

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Competing interests.

The work of Dominique P. Held was partly funded by Windar Photonics A/S through an industrial PhD stipend (project number: 5016-00182).

Acknowledgements

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Acknowledgements.

This study was supported by Innovationsfonden Danmark in the form of an
industrial PhD stipend (project number: 5016-00182). The authors want to
thank Ebba Dellwik and Antoine Larvol for their support with the sonic
anemometer and lidar data.

Edited by: Ad
Stoffelen

Reviewed by: two anonymous referees

References

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Short summary

In this paper we study the effect of different methods to derive the radial wind speed from a lidar Doppler spectrum. Numerical simulations and experimental results both indicate that the median method has slight improvements over the centroid method in terms of turbulent attenuation and also showed the lowest root mean squared error. Thus, when the aim is to reduce the volume averaging effect and obtain time series with a high temporal resolution, we recommend using the median method.

In this paper we study the effect of different methods to derive the radial wind speed from a...

Atmospheric Measurement Techniques

An interactive open-access journal of the European Geosciences Union