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.

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

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

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,

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

A cw lidar focused close to a sonic anemometer mounted 78 m above ground was
used in

Two studies investigated the spatial averaging of a long-range pulsed lidar
compared to mast-mounted sonic anemometers

A slightly different approach was followed in

A machine-learning approach to produce unfiltered wind speed variances from
pulsed lidar signals was used in

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.

Statistically, the fluctuating part of an incompressible,
homogeneous wind field

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

Equation (

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:

When using

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

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

Numerical simulations illustrate results in an environment
where no noise is present. The methodology used to perform these
simulations was developed in

Example data for the numerical simulation (

10 min average sonic radial component speed versus lidar wind speed. The lidar wind speed has been calculated using the centroid method.

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

The software can be downloaded free of charge at

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

Parameters of the line fit to the 10 min correlation between the sonic radial component and lidar.

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

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

Wind rose derived from 10 min averages of sonic anemometer wind speed and direction of the 2-month-long experiment. The beam directions are indicated as dashed lines.

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.

At first the 10 min averages of the lidar measured wind speed component

The wind rose derived from the two sonic anemometers is shown in Fig.

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

To create numerical simulations as close as possible to the experimental
conditions, the Mann spectral tensor has been fitted, following the procedure
in

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:

Sector-wise fitted parameters of the Mann model for each beam to data obtained from sonic measurements at 10 m over 2 months.

Transfer function

Transfer function

In this section we will present the combination of experimental and
simulation results together with the numerical integration of
Eq. (

First, we present the simplest case when the lidar beam is aligned with the
wind direction. In this case Eq. (

The results for the aligned case are shown in Fig.

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

RMSE value for the median and maximum method normalised by

Examples of these improvements can also be identified in the time domain when
looking at Fig.

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:

The results can be seen in Fig.

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

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.

The computer code to generate synthetic turbulence
fields can be found at

Transfer function

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

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

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