Investigation of turbulence measurements with a continuous wave, conically scanning LiDAR Abstract 1. Introduction

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1 Investigation of trblence measrements with a continos wave, conically scanning LiDAR Rozenn Wagner 1, Torben Mikkelsen 1, Michael Cortney Risø DTU, PO Box 49,DK4000 Roskilde, Denmark rozn@risoe.dt.dk Abstract LIDAR systems are getting more and more accrate and reliable. It has been shown many times that the mean horizontal wind speed measred by a lidar over flat terrain compares very well with that measred by a cp anemometer. Bt can a lidar measre trblence? Here we investigate the case of a continos wave, conically scanning Zephir lidar. First, the wind speed standard deviation measred by sch a lidar gives on average 80% of the standard deviation measred by a cp anemometer. This difference is de to the spatial averaging inherently made by a cw conically scanning lidar. The spatial averaging is done in two steps: 1) the weighted averaging of the wind speed in the probe volme of the laser beam; ) the averaging of the wind speeds occrring on the circlar path described by the conically scanning lidar. Therefore the standard deviation measred by a lidar resolves only the trblence strctres larger than a length scale depending on the circle diameter and the mean wind speed (range of magnitde: 100m). However, the Zephir lidar gives another trblence qantity, the socalled trblence parameter, which can resolve trblence strctres with a smaller length scale. In this paper, we sggest a volmetric filtering of the trblence to represent the effect of the spatial averaging operated by a lidar when measring the wind speed. We then evalate this model by comparing the theoretical reslts to experimental data obtained with several Zephir systems, for both trblence qantities. 1. Introdction LiDAR systems are more and more attractive for the wind energy indstry, becase of their ability to measre mean wind speed profiles p to abot 00m. Their accracy has increased significantly over the last 5 years. 10 mintes mean wind speed measred by some LiDAR systems compare now very well with cp anemometer measrements [1]. Meanwhile another qantity of great interest for the wind indstry is the trblence. The wind speed standard deviation measred by a remote sensing device is inherently different from the one measred by an anemometer. Indeed an anemometer measrement can be considered as a point measrement, whereas a LiDAR system measres over a volme. In this paper we consider the volme averaging effect obtained with a continos wave (cw) QinetiQ ZephiR LiDAR. Firstly, for sch a system, the measrement at a given height is achieved by focsing the laser beam at the reqired distance. The wind speed component along the laser beam (the radial speed) reslts from a weighted average over the probe volme. Sjöholm et al. [] proposed a volmetric filtering of the trblence to represent the effect of averaging within the probe volme for a ZephIR LiDAR. They have shown a good agreement between this model and experimental data. Secondly, a ZephIR lidar device conically scans the air at a rate of one revoltion per second. The horizontal wind speed reslts from the radial speeds obtained over one or three complete rotations. Therefore there is an averaging of the wind speed over a circlar path centred above the scanning lidar. In this paper we describe the spatial averaging operated by a conically scanning lidar when measring the wind speed and we focs on the effects of the conical scanning. We then refine the model sggested by Mikkelsen & Jørgensen [3] and evalate these improvements by comparing the theoretical reslts to experimental data. To realize sch a comparison, we can look at two qantities given by a ZephIR LiDAR: the wind speed standard deviation and the trblence parameter. These two parameters give information on different scales of trblence. Althogh the first one seems obvios, the standard deviation from a lidar is different from the corresponding standard deviation measred by a cp anemometer becase of the spatial averaging. On the other hand the trblence parameter is not

2 that obvios and we frther explain what this qantity represents in terms of trblence covariances. This investigation therefore has two main objectives: - to define and evalate a simple model for the filtering effect of the conical scanning; - to provide an interpretation of the standard deviation and the trblent parameter measred by a ZephIR lidar.. Volme averaging: theory.1 Staring LiDAR _ Probe length averaging The QinetiQ ZephIR lidar consists of a continos wave (cw) coaxial laser and detector system [4]. In order to explain the way of operation of the system, we first consider a LiDAR staring in one direction. It emits a cw laser radiation, and detects the Doppler shift of the signal backscattered by the particles convected by the air. This Doppler shift is proportional to the radial wind speed, Vr, i.e. the projection of the wind speed vector, = {, v, w }, in the laser beam direction [5]. A cw lidar measres the radial wind speed at a given distance r by focsing the laser beam at that distance. The power of the laser is then distribted along the beam with a distribtion that has its maximm at the focs point. The radial wind speed Vr (r) measred by a cw lidar focsed at a location r can generally be expressed by integration along the laser beam of the wind field projected in the direction of the beam: Vr() r = ϕ().( s n sn + rn) ds (1) ϕ() s where n is a nity vector along the beam and is the spatial volme averaging fnction, which, for a focsed cw coherent Doppler LiDAR, can be approximated by a Lorentzian fnction [6]: 1 zr ϕ() s = π z + ( s r) () R where s is the distance from the focs point along the beam, r the focs distance or range and zr is the so-called Rayleigh length. For the series ZephIR lidar, the Rayleigh length was estimated to be: zr = r. The averaging of the wind speed over the probe length has the effect of a low pass filter on the wind speed power spectrm, which reslts in filtering ot the trblent strctres with a length scale smaller than the probe length (generally defined as z R ). Indeed, assming the beam to be in the direction of the mean wind speed, the conterpart in the wave-nmber domain of φ is: z R k1 LLorentzian = e (3) where k 1 is the wave nmber along the mean wind direction. Sjöholm et al. [] validated the effect of this filter by analysing spectra from wind speed measrements with a staring cw LiDAR and comparing them to sonic measrements. If however the line-of-sight of the lidar does not coincidence with the mean wind direction, the ratio between the volme-averaged lidar-observed wind speed spectrm and the spectrm of the corresponding wind speed component measred by a sonic anemometer, will in general then be dependant on the entire three-dimensional strctre of the trblence []. In this paper we are considering the wind speed standard deviation (see section 3), therefore the integral of the spectrm, and this does not depend on the shape of the spectra. We expect that the spectrm measred with a laser beam not aligned with the wind direction to have a different shape bt the same integral.. Conically scanning lidar_ Spatial averaging over a circle

3 In order to get a 3 dimensional wind vector at a given height, radial wind speeds in different direction are reqired. To do so the QinetiQ ZephIR lidar conically scans the air, with a fixed angle Ф=30 from the vertical (see Figre 1), at the rate of one rotation per second. The cw lidar design allows a Doppler spectra acqisition freqency of 00000Hz. Each grop of consective 4000 Doppler spectra are averaged (corresponding to an average over 0 milliseconds or a conical segment of 7.6m when focsing at 100m) in order to improve the signal to noise ratio, after which the Doppler peak shold stand clearly above a flat shot-noise floor [7]. A radial wind speed is dedced from each averaged spectrm by finding the peak freqency (f peak ) sing the centroid method. The radial speed (v r ) is directly proportional to the peak freqency according to the following relation: v f r peak = where λ is the laser wavelength. Nominally, 50 radial wind speeds are obtained for λ each rotation. Figre 1 Sketch of the conical scanning pattern of the laser beam of a Zephir lidar. The gray scale represents the distribtion (φ) of the power over the laser beam. Here the laser focses at 70m agl. The red dots represent the middle of each 7. angle over which the Doppler spectra are averaged. As the backscattered signal is mixed with the original signal, this lidar system cannot distingish the sign of the Doppler shift (i.e. whether the wind comes towards the lidar or goes away). Therefore the 50 or 150 radial wind speeds as fnction of the azimth angle follow a rectified cosine wave: Vr( θ) = Ai Cos( θ θwind ) + B, where θ is the beam azimth angle (see Figre ). We can then dedce the wind direction: θ wind, the horizontal and vertical components of the wind speed: U = A / Sin( Φ ), w= B/ Cos( Φ) where Φ is the conical angle. Until spring 008, the scanning time was fixed at 3 seconds (3 rotations for one wind speed vector). ZephiR lidars now offer the possibility to choose the scanning time of 1 second or 3 seconds (giving 50 or 150 radial wind speeds respectively). Both configrations are considered in this paper. 6 4 Vr = 7.39 CosH1.86-q wind L Figre Example of reslt from a 3-second scan. The black points are the actal radial wind speed measrements. The red line is the fit fnction (given at the top of the figre).

4 The first step of this investigation is to define a simple filter modeling the effect of the spatial averaging de to the scanning. An effective instantaneos horizontal averaging length scale can be estimated as the combined reslt of time lag and the circlar coverage. For the ZephIR lidar, this means the distance covered by the circle of diameter D convected by the mean wind speed U. Figre 3 displays the path of the laser seen by the wind field, i.e. the path of the laser in the coordinate system attached to the wind field of mean direction the x-axis and mean speed U. According to this figre, a simple effective horizontal length scale, l az, representing an effective filter-averaging length scale from conically scanning for n seconds (n=1 or 3), can to a first approximation be modelled by: 1 laz = D+ U( n 0.5) = z+ ( n 0.5) U = z+ ( n 0.5) U (4) Cos(30) 3 where z is the vertical distance to the grond where the measrement is taken L=0.5U+D L=.5U+D Figre 3 Path of the laser conically scanning at 50m high (then D=55m) and convected by the mean wind speed U=10m/s, for 1 second scanning in green and 3 seconds in ble. From that path we can estimate a representative length scale for the filter de to the combined effect of mean wind advection and conical scanning. The conical scanning has also a low pass filter effect on the wind speed spectrm. Trblence strctres smaller than the length scale l az cannot be effectively measred by the conically scanning lidar. As the magnitde of the radial wind speed varies with the azimth angle following a rectified cosine fnction (see Figre ), the horizontal components of trblence are not filtered homogeneosly over the length scale l az, see Figre 4. Figre 4 Top left and right: Distribtion of the radial wind speed in the wind direction and the cross wind direction respectively (plan view from top of the cone); Bottom left and right: Weighting fnctions for the wind stream component and cross stream component of trblence respectively.

5 In the stream wise direction, the weighting fnction can be modeled by a rectified sine fnction (6) and by a rectified cosine fnction (8) in the cross stream direction. The power spectral transfer fnction corresponding to (6) and (8) are obtained by normalising and sqaring the Forier transform in the wave-nmber domain. π x π x sin if x laz / cos if x laz / f ( x) = laz (5) ; fv ( x) = laz (6) 0 otherwise 0 otherwise In the vertical direction, all variations are taken into accont homogeneosly (see Figre 5), therefore we can model this effect with a simple Box car-like filter fnction (10). The corresponding power spectral transfer fnction in the wave-nmber domain is then a sine cardinal fnction. Figre 5 Left: Distribtion of the radial wind speed in the vertical direction; Right: Weighting fnction for the vertical component of trblence. 1/ laz if x laz / fw( x) = 0 otherwise (7) 3Lidar trblence measrement 3.1 What can a cw conically scanning lidar tell s abot trblence? A ZephIR can measre two different qantities that give information abot trblence. The first one is the wind speed standard deviation over 10 mintes, of the 1 or 3 seconds horizontal wind speeds (obtained from the fitting of the radial speeds). From measrements we know that the standard deviation measred by a LiDAR sally gives only abot 80% of the standard deviation given by a cp anemometer, see Figre 6. This is mainly de to the volme averaging described in section. The standard deviation given by a conically scanning LiDAR does not effectively resolve the part of the trblence having a scale smaller than the length l az defined previosly. The second qantity provided by a ZephIR LiDAR is the so-called trblence parameter (TP), defined as the trblence intensity of the radial wind speed within the scanned circles: where ' v r π TP 1sec ' r π v = (8) 1sec is the variance of the radial wind speed over 1 rotation (π) or for 1 second [8]. We ' v r 6 can define a similar qantity for a 3 second scan, then we average over 3 rotations:. TP π actally qantifies the goodness of fit of the rectified cosine fnction to the radial wind speeds measred over the circlar path. This parameter contains information abot the trblence strctres with a length scale smaller than l az bt larger than the laser beam radial probe length.

6 Figre 6 Example of regression plot of the 10 mintes wind speed standard deviation measred simltaneosly by a Zephir LiDAR (3 second scan) and a cp anemometer at 40 m. 3. Illstration with the Kaimal spectrm To evalate the effect of spatial filtering we se the Kaimal spectrm transformed to wave nmber space. The transformation between the measred freqency and the wave nmber is based on the Taylor s frozen-wave hypothesis. The stream-wise component of the Kaimal spectra in wave nmber space is then [4]: 3 z 10 m F ( k1 ) = * (9) π kz ( ) 5/3 + s π Figre 7 Stream wise component of the Kaimal spectrm (black) shown as a reference. Left: spectrm filtered with the Lorentzian filter (prple), spectrm filtered with the scan filter (green). Right: spectrm of 10 mintes stream-wise trblence obtained from a cw conically scanning LiDAR (red), trblence spectrm obtained with the high-pass filter modeling the trblence nresolved by the conical scanning effect (ble), see eqation (17).The spectra shown correspond to a focs height z=30m (then the circle diameter is 34.6m), for a 3-second scan and z r =0.0017z (z=rcos(30 )). Figre 7 (left) shows the effect of each spatial filter (L Lorentzian and L scan ) on the stream-wise component of the Kaimal spectrm. Both kinds of spatial averaging (over the probe volme and the scanning area) act as low pass filter removing all the small trblence strctres (high wave nmbers). As the effect of the conical scanning is mch stronger than the probe length averaging, the combination of both effects (red spectrm in right-handed plot in Figre 7) reslts in a low pass filter very close to the scanning filter (green spectrm in left-handed plot in Figre 7). Firstly, Figre 7 (right) shows also the first component of the Kaimal spectrm as a reference. Its integral corresponds to the wind speed variance:

7 k1, max σ = F ( k ) dk (10) 0 Secondly, Figre 7 (right) shows the spectrm obtained after applying the combination of both pass filters (red). For an ideal wind speed field experiencing only horizontal flctations and no vertical flctations, then the integration of this spectrm corresponds to the variance measred by a cw conically scanning LiDAR : 1 1 ' ZephIR σ ( 1) 10min ZephIR Scan 1 Lorentzian 1 0 F k L ( k ) L ( k ) dk (11) = = 1 Only large trblence strctres (with a length scale larger than l az ) are left. In reality, both the horizontal and the vertical speeds contribte to the radial wind speed and therefore, the wind speed standard deviation obtained with a LiDAR cannot be derived from the stream-wise trblence spectrm only. However, or objective in this investigation is to make a simple model to comprehend basically the spatial averaging made by a cw conically scanning LiDAR. So we decided to make the rogh assmption that the standard deviation from the LiDAR was coming only from the stream wise trblence, i.e. we assme that eqation (16) is also tre for 3-D trblence. The third spectrm (ble) showed in Figre 7 was obtained by applying the low pass filter corresponding to the Lorentzian fnction and the high pass filter complementary to the scanning filter: ' ZephIR = F ( 1) (1 ( k LScan k1)) LLorentzian ( 1) π 1 10min 0 k dk (1) This integral corresponds to the variance of the radial wind speed (as defined in the appendix) de to the stream-wise component of trblence. As it is an average over 1 (or 3 ) second, it corresponds to the trblence strctres with a length scale smaller than l az bt larger than the length of the probe length (z r ). Since TP is a trblence qantity characteristic of the small eddies, TP is a combination of the high wave nmber part of the spectra in all 3 dimensions: ( ) ' ' ' ' TP = v = + v + w 1sec 1sec r ZephIR ZephIR ZephIR ZephIR π 10min π π π 10min (13) 4 Experiment 4.1 Description of the experiment Combined lidar and mast measrements were made at Risø DTU s Test Station for Large Wind Trbine in Høvsøre, on the west coast of Denmark. The terrain is flat and the wind is mainly blowing from west. The facility comprises an intensively instrmented meteorological mast with measring stations with both cp and 3-D ltrasonic anemometers at 100m, 80m, 60m and 40m. Cp anemometers are monted on the soth and sonic anemometers on the north of the mast. To compare lidar measrements to anemometer measrements we selected two wind sectors where the anemometers are not inflenced by the mast: and The experimental data sed for this investigation were obtained with three ZephiR lidars. The first one was eqipped with the standard software sing three seconds scanning to calclate the wind speed vector. The two others had an pgraded version of the software, giving the possibility of one second scanning. All lidar systems were positioned abot 50m north of the met. Mast. The data were selected for rain free periods. Moreover, a ceilometer placed nearby the mast enables s to measre the height of the clod base. Becase Zephir lidar measrements can be biased by low clods, the periods with clod base below 1600m were removed from the data sets sed for the comparison. The first data set (3 seconds scan) was measred in fall 007. The lidar was measring at 40m, 80m and 100m, so we cold make comparison with the measrements from the sonic

8 anemometers at 40m, 80m, and 100m. The second data set (1 second scan) was obtained in spring 008. The lidar was measring at 60m, 80m and 100m. However a falty sonic anemometer at 80m prevented s from making comparisons at that height. Therefore we have reslts at only heights: 60m and 100m. The third dataset (1 second scan) was obtained in Febrary 009. This lidar was set to measre at only one height: 80m. As trblence measrements can be easily corrpted by noise, we selected the 10 mintes period when the average nmber of points sed to calclate the fit fnction was higher than 65% of the maximm possible nmber of points. 4. Reslts In order to observe the effect of the spatial averaging, we look at the ratio between measrements from the two instrments: lidar and anemometer. For both the wind speed standard deviation and the radial speed variance, the model is compared to experimental data at different heights, ths we can observe the variation of the effect of spatial averaging with height. Moreover, as the length scale l az depends on the nmber of rotations (n) realized dring the scanning, we show the reslts for 1 and 3 revoltions in different plots. 1) Standard deviation: Figre 8 Variation with height of the ratio between wind speed standard deviation measred by the Zephir lidar scanning for 3 seconds (3 rotations for each wind speeds vector) and the wind speed standard deviation measred by the cp anemometer. Comparison of the model presented in part 3 and experimental data. Figre 9 Variation with height of the ratio between wind speed standard deviation measred by the Zephir lidar scanning for 1 second (1 rotation for each wind speeds vector) and the wind speed standard deviation measred by the cp anemometer. Comparison of the model presented in part 3 and experimental data. Figre 10 Comparison of the ratios between wind speed standard deviation measred by a ZephIR LiDAR and a cp anemometer for 8 experiments made in varios types of terrain (flat, flat with bilding, complex). The nmbers displayed in the plot corresponds to the experiment nmber and the color to the type of terrain.

9 Comparison of the wind speed standard deviation measred by a ZephIR LiDAR to cp anemometers measrements were performed in several places, with different ZephIR LiDAR nits in varios types of terrain. Figre 10 sms p the reslts for 8 tests in addition to the measrements we made in Høvsøre. ) Radial speed variance In the follwing plots, we are comparing the qantities: ( ) v = TP ' r ZephIR 1sec 1sec = ' ZephIR vr π 10min and sonic ' ' r = vr sonic 10min 1 sonic sonic sonic 1 ' 1 ' 3 ' = + v + w 4 10min 4 10min 10min v Figre 11 Variation with height of the ratio between radial speed variance measred by the Zephir lidar scanning for 3 seconds (3 rotations for each wind speeds vector) and the radial speed variance measred by the 3- D sonic anemometer. Comparison of the model presented in part 3 and experimental data. Figre 1 Variation with height of the ratio between radial speed variance measred by the ZephIR lidar scanning for 1 second (1 rotation for each wind speeds vector) and the radial speed variance measred by the 3-D sonic anemometer. Comparison of the model presented in part 3 and experimental data. 5 Discssion 5.1 Standard deviation The first observation we can make from Figres 8, 9 and 10 is the wind speed standard deviation measred by a LiDAR gives on average abot 80% of the standard deviation measred by a cp anemometer in flat terrain. The experimental data coincide with the prediction of the model, which favors the proposed model of trblence filtering de to the spatial averaging. Moreover it confirms the fact that the volme averaging is the main reason of the difference of the wind speed standard deviation between a cw conically scanning LiDAR and an anemometer. We can observe that the ratio between the standard deviation measred by the two instrments decreases with height. Indeed the averaging volme increases with height. Finally, the ratio is generally higher for the LiDAR scanning for 1second than for the LiDAR scanning for 3 seconds. Again, this is a direct conseqence of the size of the averaging volme which is smaller for a one second scan than for a three seconds scan. ( TP U) 5. Radial speed variance ( ) The radial speed variance is generally decreasing with height, see Figres 11 and 1. This is de to a combination of different factors inclding the increasing spatial volme of averaging and the decreasing amont of small trblence strctres with height. For both cases (1 sec and 3 sec scanning) we get good agreements between the model and the measrements at the lowest heights (40m and 60m) bt not that good at higher heights. A

10 possible explanation is that TP depends on the goodness of fit of the rectified cosine fnction to the radial speed points which implies that TP also acconts for measrement errors inclding noise, and not only for trblence. As the amont of small trblence strctre decreases with height, v might accont for errors and noise more than trblence at high heights (error that are not r taken into accont in the model). 6 Conclsions We defined a volmetric filtering of the trblence to represent the effect of the spatial averaging operated by a lidar when measring the wind speed. As the objective was to nderstand the principle of the spatial averaging, we decided to keep the model simple, which implied that we have made some rogh approximations. Nevertheless, the averaged experimental data coincide relatively well with the model. This confirms that the difference in wind speed standard deviation measred by a lidar and a cp anemometer is mainly de to the spatial averaging. In average, a cw conically scanning lidar measres 80% of the standard deviation measred by an anemometer, becase sch a lidar cannot resolve the trblence strctres with a length scale smaller than the circle (described by the scanning lidar) diameter approximately. Meanwhile, the trblence parameter given by a ZephIR lidar can give some information abot the smaller eddies. However, as it also qantifies the goodness of fit of the radial wind speed when calclating the wind speed vector, the trblence parameter is easily corrpted by noise and it shold be considered only as an average indicator of the high freqency trblence. References [1] Cortney, Wagner, Lindelöw, Testing and comparison of lidars for profile and trblence measrements in wind energy, ISARS 008 proceedings [] Sjöholm et al.,time series analysis of continos-wave coherent Doppler Lidar wind measrements, IOP Conf. Series: Earth and Environemtnat Science , 008 [3] Mikkelsen, Jørgensen, Trblence measred by the ZephIR TM : The Effects of the Conical scanning and Lorentzian Probe Volme Filtering, IEA Meeting 51, Remote Sensing, Risø, 007 [4] Smith et al., Wind lidar evalation at the Danish wind test site in Høvøsre, Wind Energy , 006 [5] Mikkelsen,Jørgensen: Measring Trblence with the QinetiQ ZephIR Wind Lidar, Upwind progress report, 006 [6] Sonnenschein C M, Horrigan F A. Applied Optics , 1971 [7] M.Harris: Lidar for Trbine control, NREL technical report, 006 [8] Eberhard, Cpp. Doppler Lidar Measrement of Profiles of Trblence and Momentm Flx, Jornal of Atmospheric and Oceanic Technology , 1989 Acknowledgement The athors grateflly acknowledge the interestingdiscssions and sefl data form M Harris and C Abiven from Natral Power. The work was performed as part of the (EU FP6) Upwind project (WP6). R Wagner is spported by the Marie Crie ModObs Network MRTN-CT