# Quantification and Correction of Wave-Induced Turbulence Intensity Bias for a Floating LIDAR System

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}mounted on a 12-ton anchored buoy as a function of met-ocean conditions, and we construct a subsequently applied correction method suitable for 10-min wind LIDAR data storage. To this end, we build a model to simulate the effect of buoyancy movements on the LIDAR’s wind measurements. We first apply the model to understand the mechanisms responsible for the wind LIDAR measurement error. The effect of the buoy’s rotational and translational motions on the radial wind speed measurements of the individual beams is first studied. Second, the temporality induced by the LIDAR operation is taken into account; the effect of motion subsampling and the interaction between the different measurement beam positions. From this model, a correction method is developed and successfully applied to a 13-week experimental campaign conducted off the shores of Fécamp (Normandie, France) involving the buoy-mounted WindCube v2

^{®}compared with cup anemometers from a met mast and a fixed WindCube v2

^{®}on a platform. The correction improves the linear regression against the fixed LIDAR turbulence intensity measurements, shifting the offset from ~0.03 to ~0.005 without post-processing the remaining peaks.

## 1. Introduction

^{®}LIDAR from Leosphere

^{®}. The contribution of this paper consists in a less computationally intensive model-based correction estimating the atmospheric turbulence from the time series of an IMU and the 10-min statistics of a pulsed floating LIDAR. By grouping the corrections of each radial measurement of the LIDAR in global matrices, this method reduces the need for time synchronization between the two devices and reduces the amount of data to be stored. The use of a simple model also allows to study and understand the influence of the buoy’s movements on the floating wind LIDAR measurement. In Section 2, we explain the theory from the WindCube v2

^{®}LIDAR as well as the contribution of an IMU for buoy motion measurement. In Section 3, we present the model and its implementation for estimating the atmospheric turbulence intensity from 10-min floating LIDAR statistics. Then, we present the experimental conditions from the campaign conducted off the shores of Fécamp (Normandie, France), and its measurements were used to evaluate the correction method. In Section 4, we apply the model to understand buoyancy motions of increasing complexity; from the effect of a sinusoidal motion with one degree of freedom on a single measurement radius to the motions of the experimental validation campaign.

## 2. LIDAR Theory

^{®}: WindCube v2

^{®}.

#### 2.1. WindCube Coordinate System and Calculated Quantities

^{®}emits laser pulses into five fixed directions, four inclined at ${28}^{\xb0}$ and one vertical. The directions are defined by the angles $\theta ,$ and $\phi $, respectively the azimuth angle measured clockwise from the north of the LIDAR, and the elevation angle measured from the zenith position (Figure 1) In the LIDAR reference frame, the angles defining the five line-of-sight (LOS) directions ${D}_{LOS}=\left(N,E,S,W,Z\right)$ are (in degrees): $\theta \left({D}_{LOS}\right)=\left(0,{90}^{\xb0},{180}^{\xb0},{270}^{\xb0},{0}^{\xb0}\right)$ and $\phi \left({D}_{LOS}\right)=\left({28}^{\xb0},{28}^{\xb0},{28}^{\xb0},{28}^{\xb0},{0}^{\xb0}\right)$.

#### 2.2. WindCube Principle

#### 2.2.1. Line-of-Sight Radial Wind Speed Measurement

^{®}are maintained for a duration of 0.8 s for the non-vertical beams and 1 s for the vertical position. During a LOS measurement, repeated laser pulses are sent from the LIDAR to the particles in the atmospheric boundary layer. As the time of light propagation is the unit used to evaluate the distance from the LIDAR to the particle backscattering the laser, the duration of the laser pulse induces an uncertainty in the position of the particle within the probe volume [19]. In the case of the WindCube, the pulses have a duration of 175 ns. Therefore, for each pulse, the backscattered signal is temporally sliced into range gates and a weighting function is applied to each of the sliced signals to take into account the filtering effect exerted by the probe volume on the measured radial wind speed [14]. At the end of a LOS measurement, a mean radial wind speed and a carrier to noise ratio (CNR) are determined at several altitudes using the maximum likelihood estimator (MLE) method to the averaged spectrum of all the pulses [20].

#### 2.2.2. Doppler Beam Swinging (DBS) Principle

#### 2.3. Floating LIDAR Systems (FLS)

## 3. Method

#### 3.1. A Model-Based Method for Correcting the 10-min TI Measurements from the Wind LIDAR

^{®}in a turbulence-free (or constant) atmospheric wind for a measurement period of 10 min. The model requires the temporal data from an IMU and the 10-min statistics of the floating LIDAR measurements (mean horizontal wind speed, mean wind direction, mean vertical wind speed). The considered velocity vector is the 10-min average wind measurement from the floating LIDAR, it is defined for each altitude $h$ by:

^{®}LOS directions as ${t}_{LOS}=\left(0.8s,0.8s,0.8s,0.8s,1s\right)$, as estimated from the experimental measurement files. By default, the LOS direction at t=0 is the north axis of the LIDAR reference frame. After its LOS measurement time, the orientation of the beam moves according to the vector ${D}_{LOS}=\left(N,E,S,W,Z\right)$. The choice of the first beam orientation has a strong impact on the 10-min wind statistics, its influence is studied in detail in Section 4.2.3. The time step of the model needs to be small enough to consider all the effects induced by the motion on the LIDAR measurements, but not too large to keep a low computational cost. We define the temporal discretization at 10 Hz.

#### 3.1.1. Measurement from One Line-of-Sight Subjected to Wave-Induced Motion

#### 3.1.2. Estimated Motion-Induced Variance for 10 min of FLS Measurement

^{®}measures only one position at a time. It uses the average radial velocities previously measured at the other positions to calculate the velocity vector at the end of a LOS measurement (Equation (7)). Therefore, at a time $t$ subsequent to the times $\left({t}_{N},{t}_{E},{t}_{S},{t}_{W},{t}_{Z}\right)$ corresponding to the end of the line-of-sight measurements in the directions ${D}_{LOS}$, the three components of the wind speed vector are calculated at altitude $h$ by:

#### 3.2. Experimental Campaign

^{®}. The FLS measurements are compared with data measured by an anemometer and an additional WindCube v2

^{®}located on a fixed platform. The experimental campaign took place off the shores of Fécamp (Normandie, France) from 20 April 2018 to 31 October 2018 (6-month trial period). The test site is located at an approximate distance of 14 km from the coast. The distance between the measuring buoy and the platform is 250 m (Figure 2).

^{®}) is equipped with two anemometers (Risoe P2546 A horizontal boom-mounted) at altitudes 33.8 m and 54.8 m to MSL. Only the measurements from the cup anemometer located at altitude 54.8 m are used in this study. For all the results presented in Section 4.3, the data measured by a WindCube at the desired height for comparison with the anemometer are linearly interpolated from the twelve evaluated altitudes. In the comparisons of the floating LIDAR to the fixed LIDAR, since the fixed LIDAR is taken as the reference, we interpolate the floating LIDAR measurements to the altitudes evaluated by the fixed LIDAR.

^{®}(France). The 12-ton buoy (Figure 2a) is powered by solar panels, a turbine in the wave tank, methanol cartridges and batteries. Through the wave tank, the system uses passive mechanical stabilization to limit the motion of the buoy. The buoy is anchored to the ground and the mooring line is connected to an intermediate buoy to limit the effect of line tension on the buoy movement. A wave buoy is associated with the WINDSEA_02, providing 30-min statistics of waves behavior (height and period). The WindCube v2

^{®}installed on the buoy is associated with an IMU measuring the six degrees of freedom ($\Phi ,\beta ,\Psi ,{V}_{u},{V}_{v},{V}_{w}$) at a frequency rate of 10 Hz. We note that the dynamic behavior of the buoy in response to wave motion depends on its geometry, weight and anchorage. The IMU measurements are directly related to a particular buoy, and are not representative of all experimental setups. The use of inertial measurements to correct the wind LIDAR measurements captures all the physics of the buoy motion perceived by the wind LIDAR. We also note that the yaw measurements from an IMU tend to drift over time. In our study, the model used for the turbulence intensity correction relies on the relative temporal dynamics of the yaw motion in a 10-min interval. Since the drift occurs on time scales longer than 10 min, we keep the yaw measurement from the IMU unchanged.

## 4. Results and Discussion

^{®}measurements.

#### 4.1. IMU Measurements as a Function of Sea Conditions

#### 4.2. Theoretical Impact of Buoy Movements on the FLS Horizontal Velocity Estimates

^{®}measurement beams are symmetrical in the horizontal plane, roll and pitch have a similar effect on the beams having the same relative position to their rotation axis. Therefore, for the sake of simplicity, we combine these two motions and decompose the rotational motions into two effects: Tilt and yaw. The motions are represented by zero-centred sinusoids, defined by an amplitude $a$ and period $P$, of the form:

#### 4.2.1. Horizontal Velocity Bias from an Isolated LIDAR Beam Subjected to a Period of Rotation

#### 4.2.2. Tilt Motion

^{®}is studied in Figure 4. The horizontal velocity is decomposed into two components: ${V}_{1}$ component in the A–C beams plane and the ${V}_{2}$ component in the B–D beams plane.

^{®}. However, in a real case, the measurements are not simultaneous and this effect has a strong influence on the LIDAR measurement.

- The effect of a variation of $\Delta {V}_{1}$ in the measurement of the component measured by A-C rays:$$\begin{array}{c}\Delta {V}_{ho{r}_{A\u2013C}}=\sqrt{{\left({V}_{1}\pm \Delta {V}_{1}\right)}^{2}+0}-{V}_{hor}=\sqrt{{\left(1\pm {\Delta V}_{1}\right)}^{2}{V}_{hor}^{2}}-{V}_{hor}=\pm \Delta {V}_{1}\end{array}$$
- The effect of a variation of $\Delta {V}_{1}$ in the measurement of the component measured by B-D rays:$$\begin{array}{c}\Delta {V}_{ho{r}_{B\u2013D}}=\sqrt{{V}_{1}^{2}+{\left(\pm \Delta {V}_{1}\right)}^{2}}-{V}_{hor}=\sqrt{\left(1+\frac{\Delta {{V}_{1}}^{2}}{{V}_{hor}^{2}}\right){V}_{hor}^{2}}-{V}_{hor}=\sqrt{1+\frac{\Delta {{V}_{1}}^{2}}{{V}_{hor}^{2}}}-1\end{array}$$

#### 4.2.3. Yaw Movement

^{®}reference frame (Figure 1).

#### 4.2.4. Effect of Motion Subsampling for an Isolated Beam

#### 4.2.5. Influence of a 10-min Sinusoidal Motion on the Measurement of the Entire LIDAR

^{®}operation, we compute the lidar measurement statistics from the recorded data from each LOS measurement. We normalize the velocities and standard deviations evaluated throughout this section by the incident horizontal wind speed ${V}_{ho{r}_{0}}$. All the simulations in this section are carried out under similar wind conditions (${V}_{ho{r}_{0}}=8,{W}_{dir}={90}^{\xb0}$).

#### 4.3. Influence of Wave-Induced Motion on the Experimental TI Measurement of the WindCube v2^{®} in Fécamp

#### 4.3.1. Study of Wave-Induced TI Experimental Error

#### 4.3.2. Model-Based Correction of Turbulence Intensity Measurements

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

^{®}and AKROCEAN

^{®}for providing the experimental data from the campaign of Fécamp. We would like to thank all the people involved in the MATILDA project: Phillipe Baclet (WEAMEC

^{®}), Samy Kraiem (CSTB), Romain Barbot and Lise Mourre (VALOREM

^{®}), Benoit Clauzet (EDF Renouvelables

^{®}), Maxime Bellorge (AKROCEAN

^{®}) We also like to thank Paul Mazoyer and Andrew Black from Leosphere

^{®}, that took the time to answer our questions on the WindCube v2

^{®}operation.

## Conflicts of Interest

## Appendix A

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**Figure 1.**WindCube v2

^{®}pulsed LIDAR principle and the six degrees of freedom of buoy-induced motion.

**Figure 2.**Presentation of the experimental means deployed on the site 14 km off the coasts of Fécamp. (

**a**) Photograph of the WINDSEA_02 buoy (in the foreground) accompanied by the wave buoy (in red). The fixed platform is visible in the background. (

**b**) Diagram of the experimental means, the dimensions are not respected and the LIDAR is not to scale.

**Figure 3.**Measurements of the six degrees of freedom from the IMU on board of the WINDSEA_02 buoy. Rotational angles are presented for calm; and (

**a**) rough; (

**c**) sea conditions. Translational velocities are presented for calm; and (

**b**) rough; (

**d**) sea conditions.

**Figure 4.**Diagram representation showing the influence of a tilt change $T$ (roll or pitch) on the angles $\phi $ and $\theta $ of the five measurement directions of the WindCube v2

^{®}denoted A, B, C, D, Z (side view). The suggested times are added to recall the temporality of the WindCube measurement. The horizontal velocity is decomposed into two components: ${V}_{1}$ and ${V}_{2}$.

**Figure 5.**Response of the beam A to a sinusoidal motion of tilt $T$ as a function of the incident wind component ${V}_{1}$. (

**a**) radial wind speed as a function of time relative to the period of the sinusoid. (

**b**) Time-averaged value of the radial wind speed as a function of the sinusoid amplitude.

**Figure 6.**Response of the beam B to a sinusoidal motion of tilt $T$ as a function of the incident wind components ${V}_{1}$ and ${V}_{2}$. The results are plotted as a function of the time relative to the period of the sinusoid. (

**a**) Motion-induced variation of the angle $\phi $. (

**b**) Motion-induced variation of the angle $\theta $. (

**c**) Variation of the component according to ${V}_{1}$ of the radial wind speed. (

**d**) Variation of the ${V}_{2}$ component of the radial wind speed.

**Figure 7.**Diagram representation showing the influence of a yaw variation on the angles $\theta $ of the five measurement directions of the WindCube v2

^{®}(top view). The suggested times are added to recall the temporality of the WindCube measurement. The horizontal velocity is decomposed into two components: ${V}_{1}$ and ${V}_{2}$.

**Figure 8.**Influence of a sinusoidal yaw motion on the radial wind speed measurement of the beam N as a function of the incident wind components ${V}_{1}$ and ${V}_{2}$. All the results are plotted as a function of the amplitude of the motion and the time relative to the period of the sinusoid. (

**a**) Variation of the ${V}_{{r}_{N;{V}_{1}}}$ component. (

**b**) Variation of the ${V}_{{r}_{N;{V}_{2}}}$ component.

**Figure 9.**Example of the subsampling and averaging effect on the radial wind speed measurement of beam A described in Section 4.2.2 subjected to a sinusoidal tilt $T$ of amplitude ${a}_{tilt}={10}^{\xb0}$. The radial wind speed measured by the ray (in blue) is normalized by the value measured in a motionless case ${V}_{{r}_{0}}$. Different periods of tilt are evaluated: P = 1 s (

**a**), 2 s (

**b**), 4 s (

**c**), 8 s (

**d**), 10 s (

**e**), 40 s (

**f**). The tilt motion is displayed (in grey), the scale of the motion is displayed on the right hand side of the figures.

**Figure 10.**Influence of the period of a sinusoidal motion on the mean horizontal velocity and its standard deviation estimated by the model and normalized by the incident horizontal velocity ${V}_{ho{r}_{0}}$. The periods having the greatest impact on the LIDAR measurement, called resonance period, are specified. (

**a**,

**b**) LIDAR subjected to a sinusoidal tilt motion for an incident wind along the component perpendicular to the axis of rotation. (

**c**,

**d**) LIDAR subjected to a sinusoidal yaw motion. (

**e**,

**f**) LIDAR subjected to a sinusoidal heave motion.

**Figure 11.**Influence of direction of the initial beam within the model on the mean horizontal wind speed and its standard deviation estimated by the model and normalized by the incident horizontal speed ${V}_{ho{r}_{0}}$. (

**a**,

**b**) LIDAR subjected to a sinusoidal tilting motion of period ${P}_{tilt}$ = 4 s for an incident wind along the component perpendicular to the rotation axis. (

**c**,

**d**) LIDAR subjected to a sinusoidal yaw motion of period ${P}_{yaw}$ = 3 s. (

**e**,

**f**) LIDAR subjected to a sinusoidal heave motion of period ${P}_{heave}$ = 4 s.

**Figure 12.**Linear regression of turbulence intensity measurements by the fixed (

**a**) and floating (

**b**) LIDARs in comparison with the cup anemometer measurements, at altitude 54.8 m. The ticks are normalized by the anemometer mean TI from all the 10-min samples.

**Figure 13.**Distribution of the floating LIDAR TI absolute measurement error as a function of the 30-min wave statistics measured by the wave buoy. The turbulence intensity measurement error (absolute discrepancy) of the FLS is evaluated on the 10-min data in comparison to the fixed LIDAR at the altitude of 94.4 m. The 10-min data is averaged over 30 min for correlation with the 30-min wave statistics. Quartiles are displayed as dash lines for visualizing the statistical distribution of the data. (

**a**) 30-min absolute FLS TI error as a function of the mean period of all waves. (

**b**) Colormap of the 30-min absolute FLS TI measurement error with respect to the wave mean period and height.

**Figure 14.**Study of the dominant phenomenon in the turbulence intensity measurement error of the floating LIDAR on the Fécamp site. (

**a**) Origin of the winds in the floating LIDAR reference frame, the color scale designates the normalized horizontal speed. (

**b**) Colormap of the absolute TI measurement error of the floating LIDAR with respect to the fixed LIDAR as a function of roll standard deviation (in degrees) and normalized mean wind speed.

**Figure 15.**Linear regression of the 10-min turbulence intensity measurement from the uncorrected and corrected floating LIDAR with respect to the fixed LIDAR measurement at two altitudes: 94.4 m and 134.4 m. All ticks are normalized by the fixed LIDAR’s mean TI from all the 10-min samples. (

**a**,

**c**) Raw floating LIDAR measurement at the two evaluated altitudes. (

**b**,

**d**) Floating LIDAR measurement corrected by the model at the two evaluated altitudes.

**Figure 16.**Effect of the model-based correction of the turbulence intensity measurement for the harshest week of the overall 13 (from 06/24/18 to 07/01/18). FLS measurements are compared with the fixed LIDAR measurements. All samples are normalized by the mean value from the fixed LIDAR. (

**a**) Uncorrected floating LIDAR measurement. (

**b**) Corrected measurement using the 10-min model-based correction presented in Section 3.1.

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**MDPI and ACS Style**

Désert, T.; Knapp, G.; Aubrun, S.
Quantification and Correction of Wave-Induced Turbulence Intensity Bias for a Floating LIDAR System. *Remote Sens.* **2021**, *13*, 2973.
https://doi.org/10.3390/rs13152973

**AMA Style**

Désert T, Knapp G, Aubrun S.
Quantification and Correction of Wave-Induced Turbulence Intensity Bias for a Floating LIDAR System. *Remote Sensing*. 2021; 13(15):2973.
https://doi.org/10.3390/rs13152973

**Chicago/Turabian Style**

Désert, Thibault, Graham Knapp, and Sandrine Aubrun.
2021. "Quantification and Correction of Wave-Induced Turbulence Intensity Bias for a Floating LIDAR System" *Remote Sensing* 13, no. 15: 2973.
https://doi.org/10.3390/rs13152973