A rig for calibrating a continuous-wave coherent Doppler wind lidar has been constructed. The rig consists of a rotating flywheel on a frame together with an adjustable lidar telescope. The laser beam points toward the rim of the wheel in a plane perpendicular to the wheel's rotation axis, and it can be tilted up and down along the wheel's periphery and thereby measure different projections of the tangential speed. The angular speed of the wheel is measured using a high-precision measuring ring fitted to the periphery of the wheel and synchronously logged together with the lidar speed. A simple geometrical model shows that there is a linear relationship between the measured line-of-sight speed and the beam tilt angle, and this is utilised to extrapolate to the tangential speed as measured by the lidar. An analysis of the uncertainties based on the model shows that a standard uncertainty on the measurement of about 0.1 % can be achieved, but also that the main source of uncertainty is the width of the laser beam and its associated uncertainty. Measurements performed with different beam widths confirm this. Other measurements with a minimised beam radius show that the method in this case performs about equally well for all the tested reference speeds ranging from about 3 to 18 m s

Wind lidars are often referred to as being “absolute” instruments which means that, given only the parameters of the laser wavelength and the frequency at which we sample the backscattered light, we are able to calculate the measured line-of-sight (LOS) speed through the well-known equation

The current practice for calibrating wind lidars is to use cup or sonic anemometers

Inspired by a similar concept commonly used for calibrating laser Doppler anemometers (LDAs)

It might seem strange to use a rotating steel wheel as the measurement target; after all, the lidar is intended for measuring small aerosols carried by the wind and not a solid metal target. On the other hand, the lidar fundamentally measures a frequency shift in the backscattered light due to a relative motion, and it is this frequency shift measurement and the subsequent conversion to a speed we wish to calibrate. The origin of the backscatter is, in this connection, of less importance. One could, however, envision a scenario where a lidar calibrated in this fashion is used to calibrate another lidar, e.g. in a wind tunnel or the free atmosphere. This would lead to a calibration procedure resembling that of the current practice but where the limiting accuracy of a cup anemometer is alleviated.

The paper is organised in the following way. First, the calibration rig and lidar are described. Then the model describing the relation between the measured line-of-sight speed and tilt angle is gradually developed, beginning from a simple 1-D model to a more realistic 2-D model and finally a 3-D model. This model forms the basis of the following suggestion for a calibration procedure and analysis of the various uncertainty contributions. Finally, calibrations performed with different laser beam widths and at different reference speeds are presented.

The calibration rig consists of an aluminium frame on which a stainless-steel wheel is mounted together with the transceiver, or telescope, of the lidar. The wheel has a radius of 286.76 mm with a measured eccentricity of about 0.01 mm, and it is coupled directly to a servo motor to control its rotational speed. The telescope is mounted at approximately the same height as the top of the wheel and in such a way that it can be tilted around a horizontal axis parallel to the wheel's rotational axis using a fine-threaded adjustment screw; see Fig.

Photo of the calibration rig. To the left is the flywheel with cables through which the motor is controlled and to the right the lidar telescope with optical cables connecting it with the laser and detectors. The inclinometer is not visible in the photo.

In order to measure the rotational speed, a high-precision measuring ring is fitted to the periphery of the wheel together with a corresponding measurement head sitting near the bottom of the wheel

Physical properties of the calibration rig.

The lidar is a direction sensitive continuous-wave coherent Doppler lidar operated with a 1565 nm fibre laser

Physical properties of the lidar.

Real-time signals from lidar, inclinometer and rotation encoder are streamed to a measurement computer which synchronises at 100 Hz and stores the data for post-processing.

A rotating wheel for calibration has been used with LDAs for many years

In the following subsections, the model is derived: first under the approximation that the laser beam has no transverse component (i.e. it is infinitely narrow), during which we establish the relationship between the beam tilt angle,

Figure

Schematic drawing of the calibration rig illustrating the basic geometry of the rig. The laser beam, illustrated in red, can be tilted using an adjustment screw on the telescope mount.

Now, to find the relation between

Instead of tilting the beam, the centre of the wheel is rotated around the lens centre.

It is thus seen that for small tilt angles there is a linear relationship between the speed ratio

Now, a real laser beam is of course not infinitely narrow but has a transverse profile of finite width; e.g. the laser used in this study has a Gaussian profile. We therefore expand the model to include the beam width radius,

For a beam of finite width, a finite part of the wheel perimeter will be illuminated by the laser and thus a range of line-of-sight speeds be measured; see Fig.

Schematic drawing used to derive the 2-D thick beam model. Notice that

Even for a beam of finite width, Eq. (

As mentioned above, Eq. (

In the previous section, we modelled the laser beam intensity profile as a 2-D top-hat shape. This is in conflict with the physical reality in two ways; firstly, confining the model to two dimensions effectively means that we are assuming the beam cross section to be square and not round, and secondly the real laser beam has a Gaussian intensity profile and not a top-hat shape. To take these facts into account, we must therefore expand the model to three dimensions.

Still assuming that the beam is collimated, we can model the beam as a cylinder of radius

In order to find the ratio

In order to compare the different models from the simple narrow beam approximation and 2-D top-hat beam to the full 3-D Gaussian beam, a numerical evaluation of each has been performed for a beam radius of 2.5 mm (

Comparison of the different models evaluated for tilt angles from 0 to 1

Starting from angles larger than

For angles smaller than

As written in Sect.

Figure

Schematic drawing of the influence of the wheel not rotating around its centre point. The wheel rotates around the point (

As we have seen above, our model predicts that there is a linear relationship between the ratio

The difficulty with the method lies in establishing the angles

The second angle,

Another complication to the calibration arises due to the offset introduced by the finite beam width, as explained in Sect.

In this section, we will give an estimate of the uncertainties associated with the various parameters going into the calibration and of course the overall calibration uncertainty. First, the uncertainty on the reference speed is estimated, then the uncertainty on the tangential speed measured by the lidar, and finally we combine it into a total calibration uncertainty.

From Eq. (

The calibration flywheel is made of stainless steel which has a thermal expansion of the order of

With the calibration procedure suggested here, the laser beam is slowly tilted more and more, covering a wide range of projected speeds. Since the response in

Let us begin by looking at the uncertainty of the non-compensated intercept

We know that

We will assume that the measurement of

Now,

We can finally find the overall measurement uncertainty. The lidar estimate of the wheel speed is the compensated calibration constant multiplied by the reference wheel speed:

In this section, calibration measurements made with different beam widths and for different reference speeds will be presented.

It is clear from the uncertainty analysis in Sect.

Following the procedure outlined in Sect.

Comparison between the measured beam widths and the theoretically calculated

Theoretical and measured beam widths together with estimated uncertainties for the different focus distances used.

Figures

Example of calibration measurement made with a focus setting of 1.53 m, meaning that the waist of the beam is placed very close to the top of the wheel. The black curve indicates the measurement data and the red curve indicates a least-squares fit of a straight line to the data. Panel

Example of calibration measurement made with a focus setting of 2.53 m. The black curve indicates the measurement data and the red curve indicates a least-squares fit of a straight line to the data. Panel

Figure

Fit slope and intercept as a function of focus distance. The intercept values show a clear minimum at 1.53 m, where the beam width at the wheel is smallest, clearly illustrating the need for compensating these results.

More interesting for the calibration purpose is of course the fit intercept, which is shown in red in Fig.

Compensated regression intercepts together with the estimated standard uncertainties.

In this section, we present the results of calibrations made with different reference speeds but a fixed focus distance of 1.53 m, i.e. with the smallest possible beam width at the wheel.

The tested reference speeds range from about 3.3 to 17.3 m s

Example of calibration measurement made with a reference speed of 5.44 m s

Example of calibration measurement made with a reference speed of 13.89 m s

Figure

Results of fits for different reference speeds. Fit intercept is not compensated for

Again, the regression intercepts have been compensated and the result is shown in Fig.

Compensated fit intercepts together with the estimated standard uncertainties.

It can seem paradoxical to use a flywheel to calibrate a Doppler wind lidar when the parameter we want to measure, the peripheral speed, is the one thing the lidar cannot measure. Nevertheless, the presented measurements and analysis show that the proposed calibration method is not only practically feasible but could actually lead to a significant reduction in calibration uncertainty compared to the current practice. However, there could also be other methods for achieving a similar calibration result. For instance, it seems more straightforward to measure a linear motion along the direction of the beam. This might very well be the case because besides directly measuring the desired parameter, the uncertainties introduced by the angle measurement and assessing the zero point of the angle scale together with the beam width can be alleviated. On the other hand, there are also arguments for using the flywheel, as discussed earlier, by scanning a range of speeds and fitting, the inherent uncertainty due to discretisation is reduced. The symmetrical nature of the wheel makes it easy to obtain a very stable reference speed, whereas with a linear motion the target would probably have to be moved back and forth and thus accelerated up to a known speed repeatedly. This would then require the position of the reference target to be logged together with its speed, which again demands a more complicated geometrical model. Another idea could be to measure in a range of angles covering the direction toward the centre of the wheel. In this way, a zero point defined as the angle where the beam is perpendicular to the wheel surface could be established as a place where no speed is measured, thus alleviating the problems seen above with finding

The uncertainty analysis shows that the main uncertainty contributor is

Inspired by a similar concept commonly used for calibrating LDAs, we have constructed a setup for calibrating coherent Doppler wind lidars based on a spinning flywheel with the lidar beam skimming the wheel's periphery. The setup is made in such a way that the laser beam can be tilted and thus probe different projections of the wheel's tangential speed. A simple model shows that there is a linear relation between the beam tilt angle and the measured LOS speed. This can be utilised to extrapolate back to the true tangential speed at zero tilt, which is the one angle otherwise impossible to measure at because the physical overlap between wheel surface and laser beam disappears. The model takes into account the finite width of the laser beam but only under the assumption that the beam is collimated, while in reality the beam used in the tests is actually focused in order to control the beam radius. The model also forms the basis of the uncertainty analysis which concludes that a total calibration standard uncertainty of about 0.1 % can be achieved with this setup, which is approximately an order of magnitude better than current practice. The uncertainty analysis reveals that the main contributor to the total uncertainty is the finite radius of the laser beam, and in order to reduce the uncertainty it is essential to determine this better than we have been able to achieve so far. Calibration measurements performed at different reference speeds and with different beam widths all show good agreement with the model and confirm that the lowest calibration uncertainty is achieved when the beam width is minimised.

In this section, we discuss how the ratio

If we apply Taylor's expansion to the third order to Eq. (

The result of Eq. (

The theoretical beam width at the top of the wheel can calculated by an appropriate combination of the following two equations:
the width of an untruncated Gaussian beam at a distance

The measurements and scripts for data analysis can be found at

ATP was responsible for measurements, data processing, data analysis, model development and manuscript writing. MC was responsible for the conceptual idea, model development and manuscript writing.

The authors declare that they have no conflict of interest.

Torben Mikkelsen and Steen Andreasen are gratefully acknowledged for initiating, designing and constructing the calibration rig. All members of the DTU Wind Energy, TEM, measurement systems engineering team are thanked for their great help and support. The anonymous reviewers are gratefully thanked for thorough and constructive criticism.

This research has been supported by the Danish EUDP (grant no. project TrueWind (64015-0635)).

This paper was edited by Gerd Baumgarten and reviewed by three anonymous referees.