A Unified Formulation for the Computation of the Six-Degrees-of-Freedom-Motion-Induced Errors in Floating Doppler Wind LiDARs

Abstract

This work presents an analytical formulation to assess the six-degrees-of-freedom-motion-induced error in floating Doppler wind LiDARs (FDWLs). The error products derive from the horizontal wind speed bias and apparent turbulence intensity. Departing from a geometrical formulation of the FDWL attitude and of the LiDAR retrieval algorithm, the contributions of the rotational and translational motion to the FDWL-measured total error are computed. Central to this process is the interpretation of the velocity–azimuth display retrieval algorithm in terms of a first-order Fourier series. The obtained 6 DoF formulation is validated numerically by means of a floating LiDAR motion simulator and experimentally in nearshore and open-sea scenarios in the framework of the Pont del Petroli and IJmuiden campaigns, respectively. Both measurement campaigns involved a fixed and a floating $ZephI{R}^{\mathrm{TM}}$ 300 LiDAR. The proposed formulation proved capable of estimating the motion-induced FDWL horizontal wind speed bias and returned similar percentiles when comparing the FDWL with the fixed LiDAR. The estimations of the turbulence intensity increment statistically matched the FDWL measurements under all motional and wind scenarios when clustering the data as a function of the buoy’s mean tilt amplitude, mean translational-velocity amplitude, and mean horizontal wind speed.

1. Introduction

The more homogeneous and stronger wind fields found on open-sea environments make offshore wind energy a dependable source of electricity generation [1,2]. In the last few decades, the wind energy industry has made important investments in offshore wind energy [3]. Specifically, a cumulative offshore wind capacity of 35 GW was deployed around the world in 2020, with Europe and China standing out in the market [4]. However, it is still one of the most expensive sources of renewable energy [5], and cost reduction is the main goal of the industry [6,7]. One of the main considerations is to obtain reliable data to assess the viability of a future wind farm based on the location. Usually, offshore wind farms are deployed in shallow waters, but there is a tendency to go further offshore and to deeper waters [8], where a higher wind capacity can be obtained. At those locations, the traditional technology used for atmospheric assessment, namely meteorological masts planted on the seabed, is no longer an option. This motivates the use of alternative remote sensing sensors such as satellites [9,10], radars [11], sodars [12,13], and combined methods [14,15]. Nevertheless, due to the high requirements of the industry in terms of the measurement accuracy, temporal and spatial resolution, and data availability, Doppler Wind LiDARs (DWLs) have become the most trusted devices for wind remote sensing [7].

In general, two types of DWLs are used for wind remote sensing: pulsed and focusable continuous-wave (CW) DWLs [16]. Pulsed DWLs use the time of flight to build up a vertically resolved wind profile, whereas focusable CW DWLs achieve the same by focusing the emitted beam on a particular height by changing its optics. Among pulsed DWLs, two main detection techniques are used: direct detection and heterodyne detection. Direct detection utilizes an optical frequency analyzer, typically an interferometer, and the receiver directly measures the signal photoelectrons [17], whereas in heterodyne detection, the backscattered signal is mixed with the frequency-offset version of the beam of a local oscillator laser, and the receiver measures the resultant beating frequency [18].

When placed over floating platforms or buoys, DWLs are able to measure the wind profile in a cost-effective way [7,19,20]. Moreover, in contrast to metmasts, they can easily be redeployed and thus cover large areas [21]. On the other hand, FDWLs suffer 6 degrees of freedom (DoF) motion, induced by the waves [22,23,24], which increases the variance on the reconstructed wind vector by the LiDAR [25,26]. However, in wind energy standard averaging periods, typically 10 or 30 min, the motion-induced error on the retrieved mean wind vector can be neglected, as it is averaged out [25,27,28,29].

The motion-induced variance in FDWL measurements is encountered at higher wind statistical moments, such as the turbulence intensity ($TI$) [28]. The $TI$ is defined as the ratio between the standard deviation of the horizontal wind speed ($HWS$) and the mean $HWS$. In the presence of the wave motion, FDWLs measure an apparent added turbulence due to the motion-induced variance, which corrupts the $TI$ measurements [25,26]. $TI$ is one of the parameters of main importance for wind farm design and operation, as wrong estimates of $TI$ could lead to turbine overdesign or malfunctioning [30] and thus higher costs.

In the state of the art, the study of the motion-induced error on FDWLs is twofold: error estimation and error compensation. This paper addresses the former topic, specifically, error estimation in focusable CW FDWLs, which is carried out using a thorough analytical formulation of the system mechanics and the wind vector retrieval algorithm. The error estimation in pulsed FDWLs has been addressed elsewhere [31,32]. The latter topic, i.e., error compensation, using either mechanical [27] or signal processing techniques [25,28,33,34] is outside the scope of the present work.

So far, multiple studies have addressed the study of the motion-induced error on focusable CW FDWLs on both the scan-time and 10 min levels:

On the scan-time level, Tiana-Alsina et al. [35] proposed a basic CW FDWL motion simulator, in which the roll and pitch motions were simulated by means of Euler’s rotation matrices. This simulator provided insights on the $HWS$ error in relation to the wind direction ($WD$) and the buoy tilt but was unable to simulate the effects of the translational motion on the $HWS$ error. Bischoff et al. [36] presented a simulation environment to estimate the uncertainties in the measurements by CW FDWLs using a simplified buoy model. However, the error estimates could not match the measurements for the high-resolution (1 Hz) data. Kelberlau et al. [28] provided a thorough geometrical description of the FDWL dynamics taking into account the 6 DoF motion of the FDWL buoy and provided a method for compensating the apparent $TI$ measured by the SEAWATCH CW FDWL buoy by Fugro${}^{\mathrm{TM}}$. This geometrical description permitted a preliminary study of the 6-DoF-motion-induced error in a LiDAR scan but was limited by the oversimplification of assuming a constant value for the initial scan phase. Departing from this geometrical description, Salcedo et al. numerically simulated the 6 DoF CW DWL measurement error [37] and provided a first 6 DoF motion correction algorithm using an unscented Kalman filter [33]. The simulator [37] modeled each of the six DoF as a sinusoidal signal with a given amplitude, frequency, and phase and enabled full understanding of the motion-induced error in a LiDAR scan through the principle of error superposition. Most of the CW FDWL motion-related error studies in the literature resorted to numerical simulations due to the inherent complexity of the velocity–azimuth display (VAD) algorithm, because it involves a least squares fit as a nonlinear operation. These simulations are computationally expensive and difficult to implement [35,36,37,38].

On the 10 min level, Kelberlau et al. [38] recently presented a study on the motion-induced error in the measurement of the mean $HWS$ using a motion simulator. The study theoretically proved the reduced sensitivity of the mean $HWS$ error to the LiDAR motion and its relation to the buoy motion frequency. However, the study was limited to the mean $HWS$ and provided error values up to one order of magnitude smaller than the typically measured values. The study by Bischoff et al. [36] was also limited to studying the bias in the $HWS$ mean, and difficulties in validating the estimated values with the experimental data were encountered. Gutiérrez-Antuñano et al. [25] presented a method to estimate the $TI$ measurement error based on a 2 DoF (roll and pitch) motion simulator as a way to compensate for this error. Overall, it showed good estimation of the motion-induced error, but the performance of the method largely depended on the motion scenario because only two DoF were taken into account. In addition, the simulation was computationally demanding.

The above considerations motivate the study of a unified 6 DoF (rotational and translational) method to quantify the motion-induced error in focusable CW FDWLs: In this work, we provide, for the first time, a complete analytical formulation and calculus of the 6-DoF-motion-induced error on the $HWS$ retrieved by the focusable CW FDWL in a scan. The formulation relies on the VAD LiDAR wind retrieval algorithm and motion parameterization using the characteristic amplitude, frequency, and phase of each of the six DoF (i.e., three rotational and three translational components). Methods to statistically quantify the 10 min $HWS$ bias and the $TI$ increment in response to the floating LiDAR motion are also derived. The framework is the focusable continuous-wave $ZephI{R}^{\mathrm{TM}}$ 300 LiDAR, using experimental data gathered during the “Pont del Petroli” and IJmuiden measurement campaigns.

This paper is structured as follows: Section 2 presents the “Pont del Petroli” and IJmuiden measurement campaigns and describes the instrumentation used; Section 3 describes the analytical motion model of the FDWL and presents the motion estimation method; Section 4 validates the analytical model by means of the simulation [37] and the experimental measurements from the “Pont del Petroli” and IJmuiden campaigns; Section 5 gives some concluding remarks.

2. Materials

2.1. Pont del Petroli Campaign

The Pont-del-Petroli (PdP) campaign took place in June 2013 at the “Pont del Petroli” pier (Badalona, Barcelona, Spain) [39]. It aimed to assess the wind measurement capabilities of a proof-of-concept FDWL buoy in the context of the NEPTUNE project by comparing its performance against a fixed LiDAR [27]. Both the fixed and the FDWL were ZephIR 300 focusable continuous-wave DWLs. The fixed LiDAR was situated at the PdP pier in a standstill configuration. The FDWL was deployed offshore 50 m away from the fixed LiDAR. Both ZephIR 300s were identical instruments and were calibrated onshore one beside the other for a period of 3 h. They were configured to measure the wind at a height of 100 m above sea level (a.s.l.), and the calibration ensured identical measurements at 1 s and 10 min intervals. The experiment surroundings were dominated by an open-sea topology in the south and east directions and an urban topology in the west and north directions. The coastline followed the southwestern to northeastern direction.

ZephIR 300 DWL. The ZephIR 300 is a continuous-wave focusable Doppler LiDAR specially conceived for offshore operation [40]. It is able to measure the wind at user-defined heights from 10 m up to 200 m in steps of 1 m [41]. In order to retrieve the wind vector for a particular measurement height, the ZephIR 300 measures the Doppler radial speed along 50 lines of sight (LoSs) every second over a scanning cone of a 30 deg aperture from the zenith. A wedge prism rotating with a uniform circular motion of one revolution per second is used to steer the laser beam in the directions of the 50 LoSs. In uniform wind, the projection of the wind vector along the 50 LoSs, as a function of the azimuth scanning angle (i.e., the so called VAD function), takes the shape of a cosine. The LiDAR uses the VAD algorithm to retrieve the wind vector from the LoSs by fitting a cosine function to the LoS measurements [16]. The ZephIR 300 is a homodyne detection LiDAR, which means that it cannot discern the sign of the Doppler shift. As a result, the VAD function takes the form of a rectified cosine wave, creating an ambiguity of ±180 deg for the $WD$ measurement [42]. The ZephIR 300 is equipped with a wind vane to disambiguate the $WD$.

FDWL buoy. The FDWL buoy is a proof-of-concept experimental buoy specially conceived to host a ZephIR 300 LiDAR in an offshore operation [43,44]. It weighs 3 tons and has a width of 3.77 m. It consists of a four-floater structure designed to conform with the wind energy measurement requirements as well as to monitor multiple sea-related parameters. Apart from the ZephIR 300, it hosts a MicroStrain 3DM-GX3-45 inertial measurement unit (IMU) that was used to monitor the 6 DoF motion of the lidar.

3DM-GX3-45 IMU. The IMU consists of a GPS unit, an accelerometer, a magnetometer, and a gyro [45], measuring the LiDAR’s tilt and velocity at a 10 Hz sampling frequency. The IMU was located on the bottom of the FDWL buoy structure. We defined, at the location of this FDWL buoy bottom, an inertial right-handed clockwise north–east–down (NED) frame of reference. The gyro separately measured the angular accelerations around the north, east, and down axes, and the accelerometer measured the translational accelerations along these axes. In order to avoid incurring an accumulated error when integrating the accelerations, the 3DM-GX3-45 uses an extended Kalman filter, which permits the monitoring of the buoy’s attitude by considering the GPS and magnetometer measurements. The 3DM-GX3-45 is able to measure the buoy’s velocity and tilt with an accuracy of ±0.1 m/s and ±0.35 deg, respectively.

2.2. IJmuiden Campaign

The IJmuiden campaign took place between 1 April and 1 June 2015 at the IJmuiden metmast test facilities (North Sea, 85 km offshore of the Netherlands coast). The goal of the campaign was to validate the precommercial FDWL buoy EOLOS${}^{\mathrm{TM}}$ against the reference IJmuiden metmast, located 300 m apart. The EOLOS${}^{\mathrm{TM}}$ FDWL was similar to the FDWL used in the PdP campaign but prepared for harsher scenarios and higher endurance. The FDWL buoy hosted a ZephIR${}^{\mathrm{TM}}$ 300 DWL and a 3DM-GX3-45 IMU as well. The FDWL was configured to sense the wind profile at four different heights (25, 38, 56, and 83 m) coincidental with the measurement heights of the metmast anemometers. An identical ZephIR${}^{\mathrm{TM}}$ 300 DWL was installed on the metmast platform (21 m) sensing the wind above the metmast top from 90 m up to 315 m every 25 m.

3. Methods

3.1. Basic Definitions

The instantaneous wind vector $\overrightarrow{u}$ ($\overrightarrow{u}=\overrightarrow{U}+\overrightarrow{u^{\prime }}$, where $\overrightarrow{U}$ denotes the mean wind and $\overrightarrow{u^{\prime }}$ the turbulent component) is defined as the three-dimensional vector $(u_1,u_2,u_3)$ computed here from the $HWS$, $WD$, and vertical wind speed ($VWS$) as

$$\overrightarrow{u}=[HWS·cos\left(WD\right),HWS·sin\left(WD\right),VWS].$$

(1)

In wind energy, the 10 min mean wind vector is the usual standard, and hereafter, $HWS$, $WD$, and $VWS$ refer to the 10 min mean values unless otherwise indicated.

The $TI$ is indicative of the $HWS$ variations with respect to the mean $HWS$, and it is computed as

$$TI=\frac{{\sigma }_{HWS}}{HWS},$$

(2)

where ${\sigma }_{HWS}$ is the 10 min $HWS$ standard deviation.

Instantaneous error. To study the motion-induced FDWL measurement error in the retrieved $HWS$ at the i-th lidar scan, we defined the “instantaneous” error (in practice, the 1 s error considering a scan rate of 1 scan/s) as

$${ϵ}_{HWS,i}=\widehat{HW{S}_i}-HW{S}_i,$$

(3)

where $\widehat{HWS}$ denotes the $HWS$ “estimated” by the VAD algorithm, and $HW{S}_i$ denotes the “true” $HWS$ at the i-th scan.

10 min error. The wind energy industry is interested in quantifying both the $HWS$ measurement bias and the $TI$ increment due to the wave-induced motion on FDWLs at a 10 min level.

The bias during a sequence of N measurement scans (i.e., the sample statistical interval; usually $N=600$ for 10 min measurements when measuring at a single height) is defined as the difference between the mean $HWS$ measured by the FDWL and the mean $HWS$ measured by the reference fixed LiDAR. Formally,

$$bia{s}_{meas}=\frac{1}{N}\sum _{i=1}^N{\widehat{HWS}}_i-\frac{1}{N}\sum _{i=1}^{N}HW{S}_i=\frac{1}{N}\sum _{i=1}^N{ϵ}_{HWS,i}.$$

(4)

Statistically, it can be expressed as

$$bia{s}_{est}=E\left[\widehat{HWS}\right]-HWS=E\left[{ϵ}_{HWS,i}\right],$$

(5)

where $E[·]$ denotes the expectation operator, $\widehat{HWS}$ is the 10 min horizontal wind speed estimated by the FDWL, and $HWS$ is the “true” 10 min horizontal wind speed.

The $TI$ increment, $\Delta T{I}_{meas}$, is defined as the $TI$ measured by the FDWL minus the $TI$ measured by the reference fixed LiDAR. Formally,

$$\Delta T{I}_{meas}=\frac{{\sigma }_{HW{S}_{FDWL}}}{{\overline{HWS}}_{FDWL}}{|}_{bin}-\frac{{\sigma }_{HW{S}_{fixed}}}{{\overline{HWS}}_{fixed}}{|}_{bin},$$

(6)

where the overbar denotes the “mean” over the 10 min $HWS$ samples in the calculus bin.

On the other hand, the $TI$ increment can be estimated as

$$\Delta T{I}_{est}=\frac{{\sigma }_{{ϵ}_{HWS,i}}}{HWS},$$

(7)

where ${\sigma }_{{ϵ}_{HWS,i}}$ is the standard deviation of the instantaneous error (Equation (3)).

3.2. Reconciling the Estimated and the Measured TI

Next, we related Equation (7) to Equation (6) from a statistical and probabilistic theoretical perspective. Toward this goal and in order to aid the mathematical notation, we introduce the short-hand notation F-R-W as follows: we changed the mnemonics “FDWL” into “F” and “fixed” into “R” (representing “reference” fixed LiDAR). We also introduce the subscript “W” to denote the wind that a nonmoving DWL would measure at the same place as the moving FDWL. In other words, subscript “W” denotes the “true” wind measurement but includes the 1 s average and spatial smoothing inherent to the DWL scanning procedure. To simplify the notation, the “true” wind HWS, $HW{S}_W$, is simply denoted as $HWS$.

3.2.1. On the Estimated TI

According to probability theory, the variance in the difference of two random variables (i.e., ${ϵ}_{F,W}=HW{S}_F-HW{S}_W$, which reproduces Equation (3)) is equal to the sum of each of their variances minus twice their covariance [46],

$${\sigma }_{{ϵ}_{F,W}}^2={\sigma }_F^2+{\sigma }_W^2-2\rho {\sigma }_F{\sigma }_W,$$

(8)

where $\rho$ is the correlation coefficient between the “F” and “W” variables, and $cov(F,W)={\rho }_{F,W}{\sigma }_F{\sigma }_W$.

At this point, we introduce the assumption that the FDWL-measured and the “true” $HWS$ (subscripts “F” and “W”, respectively) are linearly correlated so that ${\rho }_{F,W}\approx 1$. By experiment, Tiana-Alsina et al. [35] reported correlation coefficients of ${\rho }_{F,R}=0.9$, ${\rho }_{F,R}=0.86$, and ${\rho }_{F,R}=0.66$ between the co-located moving and reference DWLs in a controlled-motion experiment for tilt motion amplitudes of 10, 16, and 25 deg, respectively. In past experimental campaigns involving different FDWL buoy topologies [23,25,28,38], the most frequent buoy tilts were reported to range from 0 up to 10 deg. These low tilts are also supportive of high F-to-W correlation coefficients, ${\rho }_{F,W}\approx 1$, at the FDWL location. Inserting ${\rho }_{F,W}=1$ in Equation (8) above and rearranging the terms, we obtain

$${\sigma }_{{ϵ}_{F,W}}^2\simeq {({\sigma }_F-{\sigma }_W)}^2.$$

(9)

By taking the square root of Equation (9) above and substituting it into Equation (7), the estimated turbulence intensity can be written as

$$\Delta T{I}_{est}\simeq \frac{{\sigma }_F-{\sigma }_W}{HWS}.$$

(10)

3.2.2. On the Measured TI

As in the preceding subsection, one can assume that the 10 min FDWL- and the reference-LiDAR-measured $HWS$ are virtually identical so that $HW{S}_{FDWL}\approx HW{S}_{fixed}\approx HWS$ [20,29]. Introducing this approximation into Equation (6) and using the shorthand notation F-R-W introduced above, it can be rewritten as

$$\Delta T{I}_{meas}=\frac{{\sigma }_F-{\sigma }_R}{HWS}.$$

(11)

If the FDWL and the fixed LiDAR carried out exactly co-located measurements, the estimated $TI$ (Equation (10)) and the measured $TI$ (Equation (11)) would be identical (i.e., ${\sigma }_R={\sigma }_W$). In practice, the fixed and the FDWL were located 50 m apart in the PdP campaign and 300 m apart in the IJmuiden campaign (see Section 2); therefore, small differences up to 1% [33] were expected to arise between the estimated and the measured $TI$ increment when different wind flows were measured by the reference and floating LiDARs.

3.3. FDWL Geometrical Model

We defined the right-handed Cartesian XYZ “moving body” coordinate system of the LiDAR buoy (hereafter, the “moving” coordinate system) and the right-handed Cartesian north–east–down (NED) “fixed” inertial frame of reference (hereafter, the “fixed” system, Figure 1). The IMU (see Section 2) measured the rotation and translation of the “moving” coordinate system with respect to the “fixed” system. The vectors $\widehat{n}$, $\widehat{e}$, and $\widehat{d}$ were the unitary vectors aligned with the N, E, and D axes of the fixed coordinate system, respectively. $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$ were the unitary vectors aligned with the X, Y, and Z axes, respectively, of the moving coordinate system. $\widehat{h}$ was a unit vector in the outbound direction of the LiDAR scanning cone axis and is defined as the opposite of $\widehat{z}$. The half-angle cone aperture was ${\theta }_0$ = 30 deg.

The unitary vector $\widehat{r}$ defined the LiDAR beam pointing direction in each LoS measurement during the LiDAR scan. In what follows, $\widehat{r}$ is given in spherical coordinates relative to the moving coordinate system. Specifically, $\widehat{r}$ is given by the zenith angle relative to $\widehat{h}$, which is denoted ${\theta }_0$ (i.e., the cone aperture), and by the azimuth angle relative to $\widehat{x}$, which is denoted $\varphi$ (Figure 1). As mentioned, ${\theta }_0$ is the constant value of 30 deg corresponding to the scanning-cone aperture. $\varphi$ is a time-variable angle representing the LiDAR scanning phase (positive towards $\widehat{y}$).

During the LiDAR scan, the vector $\widehat{r}$ rotated around $\widehat{h}$ with a uniform circular motion at a rate of 1 revolution per second, i.e., $\varphi \left(t\right)=2\pi \times t$ [rad]. In what follows, for simplicity, we use the variable $\varphi$ instead of the usual time variable t. The variable $\varphi$ ranged from 0 to $2\pi$ [rad] during a LiDAR scan. Therefore, $\widehat{r}$ can be formulated in the moving coordinate system as follows:

$$\widehat{r}\left(\varphi \right)=sin\left({\theta }_0\right)·cos(\varphi -{\varphi }_0)\widehat{x}+sin\left({\theta }_0\right)·sin(\varphi -{\varphi }_0)\widehat{y}-cos\left({\theta }_0\right)\widehat{z},$$

(12)

where ${\varphi }_0$ is the LiDAR initial scan phase or, hereafter, the “initial phase”.

3.4. The VAD Algorithm as a First-Order Fourier Series

Focusable CW DWLs use the VAD algorithm to retrieve the wind vector from the measured along-LoS-projected wind velocity during the LiDAR scan, which is computed as the so-called VAD function,

$$f\left(\varphi \right)=\overrightarrow{u}·\widehat{r}\left(\varphi \right).$$

(13)

In a motionless uniform nonturbulent wind scenario, the wind vector can be decomposed as $\overrightarrow{u}=HWS\widehat{\rho }+w\widehat{h}$, where $\widehat{\rho }=cos\left(WD\right)\widehat{x}+sin\left(WD\right)\widehat{y}$, with $WD$ as positive counterclockwise. Then, the ideal VAD function $f\left(\varphi \right)$ takes the shape of the model cosine function [16,47],

$$f_{model}\left(\varphi \right)=A·cos(\varphi -B)+C,$$

(14)

where A is the amplitude, B is the phase, and C is the offset term. From Equation (14) above, the $HWS$, $WD$, and $VWS$ components of the wind vector are obtained as [28]

$$\begin{array}{cc}\hfill HWS& =\frac{A}{sin\left({\theta }_0\right)},\hfill \\ \hfill WD& =B,\hfill \\ \hfill VWS& =\frac{C}{cos\left({\theta }_0\right)}.\hfill \end{array}$$

(15)

In Equation (15) above, the $HWS$, $WD$, and $VWS$ refer to the values retrieved at the i-th scan.

The VAD algorithm uses the least squares (LSQ) algorithm to fit the model cosine function $f_{model}$ (Equation (14)) to the measured function $f\left(\varphi \right)$ in order to obtain the sought-after parameters A, B, and C:

$$min{|}_{A,B,C}\left\{\left|\right|f\left(\varphi \right)-[A·cos(\varphi -B)+C]|{|}^2\right\}.$$

(16)

The LSQ fit is usually carried out by means of numerical methods. Yet, they do not allow the explicit analytical formulation of solving coefficients A, B, and C with a view to understanding the propagation of the motion-induced error. A way out of this problem is to treat the error-norm minimization of Equation (16) in the Hilbert space of the square integrable functions $L^2[0,2\pi ]$, in which $\left\{1,cos\left(m\varphi \right),sin\left(n\varphi \right)\right\}$, $m,n=1\dots \infty$ form an orthogonal basis. Equivalently, the Fourier series is the LSQ projection of a target function to an orthogonal space defined by the sum of the sine and cosine functions [48]. Therefore, the solution of Equation (16) is the first-order Fourier series of $f\left(\varphi \right)$ in the basis function $\left\{1,cos\left(\varphi \right),sin\left(\varphi \right)\right\}$. The first-order Fourier series can be rewritten in the form of Equation (14), in which the A, B, and C parameters can be expressed as a function of the Fourier coefficients as [49]

$$\begin{array}{cc}\hfill A& =\sqrt{a_1^2+b_1^2},\hfill \\ \hfill B& =arctan\left(-\frac{b_1}{a_1}\right),\hfill \\ \hfill C& =\frac{a_0}{2},\hfill \end{array}$$

(17)

where $a_0$, $a_1$, and $b_1$ are the first-order Fourier coefficients of $f\left(\varphi \right)$, obtained as [50]

$$\begin{array}{cc}\hfill a_0& =\frac{2}{P}{\int }_0^{P}f\left(\varphi \right)d\varphi ,\hfill \\ \hfill a_1& =\frac{2}{P}{\int }_0^{P}f\left(\varphi \right)cos\left(\varphi \right)d\varphi ,\hfill \\ \hfill b_1& =\frac{2}{P}{\int }_0^{P}f\left(\varphi \right)sin\left(\varphi \right)d\varphi ,\hfill \end{array}$$

(18)

where $P=2\pi$.

From Equations (15) and (17) above, it emerges that the Fourier coefficients $a_1$ and $b_1$ propagate errors to the $HWS$ and $WD$ (i.e., to the wind horizontal components), while the coefficient $a_0$ propagates errors only to the retrieved $VWS$.

As mentioned, homodyne-detection LiDARs, as in the case of the ZephIR 300, are only able to measure the magnitude of the along-LoS radial velocities but not the sign. This is equivalent to saying that $f\left(\varphi \right)$ takes the shape of a rectified cosine, i.e., the absolute value of $f\left(\varphi \right)$[28]. When applying the VAD algorithm to $\left|f\right(\varphi \left)\right|$, parameters A and C retain the same value as for the heterodyne case (VAD applied to $f\left(\varphi \right)$), but B has a $\pm \pi$ rad ambiguity. The latter can be disambiguated by the wind vane co-located with the LiDAR buoy.

3.5. Estimation Error Methodology

In the absence of buoy motion, the X, Y, and Z axes of the moving coordinate system would coincide with the N, E, and D axes of the fixed system [23]. In the presence of waves, the FDWL experiences angular and translational motion around the three NED axes. In this section, we formulate analytical expressions for the motion influence on the VAD-retrieved wind vector.

In order to simplify the obtained analytical expressions, the following assumptions are considered: (i) the motion in each of the six DoF is considered to be a zero-mean simple harmonic motion; (ii) the rotational and translational motions are considered to be independent error sources; and (iii) under motion, the $VWS$ contribution to the $HWS$ error is considered to be null as compared to the $HWS$ contribution [51].

3.5.1. Rotational Motion Model

The FDWL buoy undergoes roll, pitch, and yaw motions around the three fixed coordinate axes, i.e., N, E, and D, respectively. As observed experimentally [23], roll and pitch motions can be modeled using a first-order small-angle approximation as simple harmonic motions. The yaw motion, which shows larger and more constant values with time, can be modeled by the constant tilt.

We define roll and pitch tilts (in units of radians) as sinusoidal signals of the form

$${\mathsf{\Omega }}_x\left(\varphi \right)=A_x·sin(f_x\varphi -{\alpha }_x),$$

(19)

where $A_x$, $f_x$, and ${\alpha }_x$ denote the amplitude, frequency, and motional phase, respectively. Subindex $x=roll,pitch$ denotes the $roll$ and $pitch$ motion, respectively. Note that, for convenience, Equation (19) is expressed as a function of the LiDAR scan-phase variable $\varphi$ ($\varphi =2\pi t$) instead of time variable t.

As mentioned, the yaw tilt is considered a constant value ${\mathsf{\Omega }}_{yaw}\left(\varphi \right)=A_{yaw}$. This is conducted without limiting the general formulation of the problem. Thus, the reader could extend the formulation of the yaw angle to the harmonic case by defining the yaw motion as a sinusoidal signal (Equation (19)) plus an offset term equal to its mean value during the interval under study, and by following the formulas in Section 3.5.1 and Appendix A.1.

First, in order to compute the rotated LiDAR pointing direction, ${\widehat{r}}_{rot}$, in the fixed NED coordinate system given the roll, pitch, and yaw rotations, we used the Euler rotation matrix. Euler’s rotation theorem states that any rigid-body rotation can be defined by three rotation angles [52]. There are many different conventions regarding the rotation angles and their order of application. Here, we considered the NED convention, i.e., three elemental rotations were carried out sequentially: first, a rotation around the N axis (roll motion); second, a rotation around the E axis (pitch motion); and finally, a rotation around the D axis (yaw motion). The rotations were positive counterclockwise. Therefore, the Euler rotation matrix can be written as [53]

$$\mathit{R}={\mathit{R}}_{\mathit{D}}·{\mathit{R}}_{\mathit{E}}·{\mathit{R}}_{\mathit{N}},$$

(20)

where ${\mathit{R}}_{\mathit{N}}$, ${\mathit{R}}_{\mathit{E}}$, and ${\mathit{R}}_{\mathit{D}}$ are the elemental rotation matrices around the N, E, and D axes, respectively. They can be formulated as follows:

$$\begin{array}{cc}\hfill {\mathit{R}}_{\mathit{D}}& =\left[\begin{array}{ccc}cos\left({\mathsf{\Omega }}_y\right)& -sin\left({\mathsf{\Omega }}_y\right)& 0\\ sin\left({\mathsf{\Omega }}_y\right)& cos\left({\mathsf{\Omega }}_y\right)& 0\\ 0& 0& 1\end{array}\right],\hfill \\ \hfill {\mathit{R}}_{\mathit{E}}& =\left[\begin{array}{ccc}cos\left({\mathsf{\Omega }}_p\left(\varphi \right)\right)& 0& sin\left({\mathsf{\Omega }}_p\left(\varphi \right)\right)\\ 0& 1& 0\\ -sin\left({\mathsf{\Omega }}_p\left(\varphi \right)\right)& 0& cos\left({\mathsf{\Omega }}_p\left(\varphi \right)\right)\end{array}\right],\hfill \\ \hfill {\mathit{R}}_{\mathit{N}}& =\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\left({\mathsf{\Omega }}_r\left(\varphi \right)\right)& -sin\left({\mathsf{\Omega }}_r\left(\varphi \right)\right)\\ 0& sin\left({\mathsf{\Omega }}_r\left(\varphi \right)\right)& cos\left({\mathsf{\Omega }}_r\left(\varphi \right)\right)\end{array}\right],\hfill \end{array}$$

(21)

where subindexes y, p, and r stand for the “yaw”, “pitch”, and “roll”, respectively.

The small-angle approximation for roll and pitch angles [23,28] translates into a first-order Taylor approximation for the sine and cosine functions (i.e., $sin\left(x\right)\simeq x$ and $cos\left(x\right)\simeq 1$). These first-order approximations were incorporated into Equation (A1), Appendix A.1, in order to simplify the rotation matrix $\mathit{R}$.

The second step was to compute the rotated version of the LiDAR pointing vector, ${\widehat{r}}_{rot}\left(\varphi \right)$, in the NED fixed coordinate system:

$${\widehat{r}}_{rot}\left(\varphi \right)=\mathit{R}·\widehat{r}\left(\varphi \right).$$

(22)

Third, the FDWL-measured wind under rotational motion was computed by projecting the wind vector over the rotated LiDAR pointing vector for each LoS. This was expressed by means of the dot product as follows:

$$f_{rot}\left(\varphi \right)=\overrightarrow{u}·{\widehat{r}}_{rot}\left(\varphi \right).$$

(23)

Eventually, the motion-corrupted FDWL wind vector was derived by applying the VAD algorithm to the function $f_{rot}\left(\varphi \right)$ above. With this aim, the first-order Fourier coefficients of $f_{rot}\left(\varphi \right)$ (i.e., $a_0^{rot}$, $a_1^{rot}$, and $b_1^{rot}$) were computed through Equation (18) by substituting $f_{rot}\left(\varphi \right)$ in place of $f\left(\varphi \right)$. The $HWS$ retrieval error was computed as the difference between the “true” $HWS$ (i.e., without motion influence) and the FDWL-retrieved $HWS$ under rotational motion:

$${ϵ}_{HWS,i}^{rot}=HWS,i-\frac{1}{sin\left({\theta }_0\right)}\sqrt{{\left(a_1^{rot}\right)}^2+{\left(b_1^{rot}\right)}^2},$$

(24)

where $a_1^{rot}$ and $b_1^{rot}$ are the Fourier coefficients of $f_{rot}\left(\varphi \right)$. See Appendix A.3 for the mathematical results in expanded form.

3.5.2. Translational-Motion Model

Waves also induce translational motion to the FDWL in the N, E, and D directions, i.e., surge, sway, and heave motions, respectively. Similar to the derivation of the rotational motion components in Equation (19), sinusoidal variation was assumed for each translational-motion component. They were formulated as follows:

$$t_x\left(\varphi \right)=A_x·sin(f_x\varphi -{\alpha }_x),$$

(25)

where $A_x$, $f_x$, and ${\alpha }_x$, are the amplitude, frequency, and phase of the translational motion x, where x denotes the surge, sway, or heave components. We also defined the translational-velocity vector $\overrightarrow{t}\left(\varphi \right)$ in the NED fixed coordinate system as the three-component vector,

$$\overrightarrow{t}\left(\varphi \right)=[t_{su}\left(\varphi \right),t_{sw}\left(\varphi \right),t_{he}\left(\varphi \right)],$$

(26)

where the subscripts $su$, $sw$, and $he$ referred to the surge, sway, and heave, respectively.

First, to compute the FDWL-measured wind under translational motion, we computed the apparent wind vector. The apparent wind vector measured by the FDWL was the difference between the wind vector and the translational-velocity vector:

$${\overrightarrow{u}}^{trans}\left(\varphi \right)=\overrightarrow{u}-\overrightarrow{t}\left(\varphi \right).$$

(27)

Second, the translational-velocity contribution to the LiDAR-measured LoS, $f_{trans}\left(\varphi \right)$, was obtained by projecting the apparent wind vector, ${\overrightarrow{u}}^{trans}$, on the LiDAR pointing direction $\widehat{r}\left(\varphi \right)$ as follows:

$$f_{trans}\left(\varphi \right)={\overrightarrow{u}}^{trans}\left(\varphi \right)·\widehat{r}\left(\varphi \right).$$

(28)

The LiDAR-retrieved wind vector was the result of applying the VAD algorithm over the motion-corrupted LoS, $f_{trans}\left(\varphi \right)$. The Fourier coefficients $a_0^{trans}$, $a_1^{trans}$, and $b_1^{trans}$ were obtained through Equation (18) by substituting $f\left(\varphi \right)$ by $f_{trans}\left(\varphi \right)$. The Fourier coefficients are given in expanded form in Appendix A.4 of Appendix A.

Finally, the $HWS$ measurement error due to the FDWL’s translational motion ${ϵ}_{HWS,i}^{trans}$ became

$${ϵ}_{HWS,i}^{trans}=HW{S}_i-\frac{1}{sin\left({\theta }_0\right)}\sqrt{{\left(a_1^{trans}\right)}^2+{\left(b_1^{trans}\right)}^2}.$$

(29)

3.5.3. Total Error Model

In order to estimate the total error, we departed from the assumption that the rotational and translational motion were independent error sources (see Section 3.5). Therefore, the total error in the retrieved $HWS$ in a LiDAR scan is the superposition of errors:

$${ϵ}_{HWS,i}(\overrightarrow{u},{\varphi }_0)={ϵ}_{HWS,i}^{rot}(\overrightarrow{u},{\varphi }_0)+{ϵ}_{HWS,i}^{trans}(\overrightarrow{u},{\varphi }_0).$$

(30)

In Equation (30), the total error, ${ϵ}_{HWS,i}(\overrightarrow{u},{\varphi }_0)$, is parameterized by the instantaneous wind vector $\overrightarrow{u}$ and initial phase ${\varphi }_0$.

Figure 2a illustrates the estimated $HWS$ total error in a LiDAR scan parameterized as a function of the wind direction ($WD$ ranging from 0 to 360 deg) and the initial phase ${\varphi }_0$ in a roll-only motion scenario ($HWS=10$ m/s, $VWS=0$ m/s). As can be observed, the $HWS$ error was highly dependent on both input parameters (WD and initial phase), ranging from −1.5 up to 1.5 m/s (the $HWS$ errors for other input $HWS$s are scalable through simple direct proportionality). If the $WD$ is known, the $HWS$ error in a LiDAR scan is dependent only on the initial phase. This is represented in Figure 2b, where the $HWS$ error in the LiDAR scan in Figure 2a is represented as a function of the initial phase ${\varphi }_0$ for the constraint settings $HWS=10$ m/s, $WD=275$ deg, and $VWS=0$ m/s.

However, the initial phase is the manufacturer’s undisclosed parameter for the ZephIR 300 LiDAR. As the instantaneous value of the initial phase at each LiDAR scan cannot be known nor, consequently, the total error, we assumed the initial phase was a random variable with a uniform distribution between 0 and 2$\pi$ rad [25]. As a result, $HWS$ error moments rather than instantaneous values were derived.

The 10 min $HWS$ bias (Equation (5)) was computed as the expectation (first raw moment) of the $HWS$ total error function (Equation (30)), ${ϵ}_{HWS,i}(\overrightarrow{u},{\varphi }_0)$, with respect to the initial phase, ${\varphi }_0$, constrained to the mean wind vector $\overrightarrow{U}$ as follows:

$$bia{s}_{est}=E_{{\varphi }_0}\left[{ϵ}_{HWS,i}(\overrightarrow{U},{\varphi }_0)\right].$$

(31)

In practice, the $bia{s}_{est}$ is simply the mean value of ${ϵ}_{HWS,i}(\overrightarrow{U},{\varphi }_0)$ computed as $bia{s}_{est}=\frac{1}{N}{\sum }_{i=1}^N{ϵ}_{HWS,i}(\overrightarrow{U},{\varphi }_{0,i})$, where the discrete random variable ${\varphi }_0$ is defined by a finite list $\{{\varphi }_{0,1},\dots ,{\varphi }_{0,N}\}$ of equally likely outcomes (vertical line in Figure 2a).

Analogously, the 10 min TI increment of Equation (7) is estimated as follows:

$$\Delta T{I}_{est}=\frac{ST{D}_{{\varphi }_0}\left[{ϵ}_{HWS,i}(\overrightarrow{U},{\varphi }_0)\right]}{\widehat{HWS}},$$

(32)

where $STD$ stands for the standard deviation, which is the square root of the variance of the total error (${\sigma }_{{ϵ}_{HWS,i}}$), and $\widehat{HWS}$ is the 10 min mean $HWS$ measured by the FDWL.

3.6. Bias and TI-Increment Estimation Procedure

Figure 3 summarizes the procedure described in Section 3.5 to compute the estimated 10 min bias and $TI$ increment. The 10 min mean wind vector ($\overrightarrow{U}$), as well as the characteristic amplitude, frequency, and phase of each of the 6 DoF, were the inputs. In practice, the yaw frequency and phase were set to zero. The Fourier coefficients $a_1^{rot}$ and $b_1^{rot}$ were obtained through Equations (A3) and (A4), respectively, which estimated the $HWS$ measurement error due to the rotational motion (${ϵ}_{HWS,i}^{rot}$, Equation (24)). Similarly, the Fourier coefficients $a_1^{trans}$ and $b_1^{trans}$ were obtained through Equations (A5) and (A6), respectively, which estimated the $HWS$ measurement error due to the translational motion (${ϵ}_{HWS,i}^{trans}$, Equation (29)). The following step was to sum the rotational- and translational-error expressions to yield the total error as a function of the wind vector and initial scan phase (Equation (30)). Then, it was constrained to the 10 min mean wind vector, $\overrightarrow{U}$, by substituting the $\overrightarrow{u}$ in Equation (30) with $\overrightarrow{U}$. Finally, the bias and $TI$ increment were estimated by means of Equations (31) and (32), respectively.

Alternatively, for a specific 1 s error model validation (refer to Section 4.1), the total error (Equation (30)) was obtained as a function of the instantaneous wind vector and the LiDAR initial scan phase.

3.7. Sinusoidal Characterization of the Measured Motion Time Series

In order to estimate the FDWL motion-induced error with the presented methodology based on Equations (19) and (25), there is a requirement that the motion time series in each of the DoF be modeled as a sinusoidal signal with a characteristic amplitude, frequency, and phase (A, f, and $\alpha$, respectively). The estimation procedure for these parameters is described next through a case example.

Figure 4a shows a record of the IMU-measured roll time series during the PdP campaign. As can be noticed, the signal was composed of multiple frequency components. To analyze the signal spectrum, the power spectral density (PSD) was computed in batches of 10 min segments using the Blackman–Tukey method [54]. In Figure 4b, it emerges that the signal had a broad spectrum with relevant frequency components from approximately 0.25 to 0.5 Hz. The characteristic frequency f was computed as the frequency corresponding to the peak PSD maximum (≃0.3 Hz).

In order to estimate the characteristic amplitude A, we made use of the fact that the amplitude of a sinusoidal signal is related to its power, P, through the relationship $P=A^2/2$ [54]. Moreover, the mean power $\overline{P}$ of a stochastic signal $s\left(t\right)$ (i.e., the IMU-measured motion) with duration T is computed as [54]

$$\overline{P}={\int }_0^T{\left|s\left(t\right)\right|}^{2}dt.$$

(33)

The mean power, $\overline{P}$, takes into account all the frequency components of the signal spectrum [55]. Therefore, the characteristic amplitude was estimated from the mean signal power as $A=\sqrt{2\overline{P}}$. Finally, the characteristic phase $\alpha$ was obtained from the first-order Fourier decomposition of $s\left(t\right)$ in a similar fashion to the calculation of phase term B in Equation (17), given the first-order coefficients $a_1$ and $b_1$ in Equation (18) (see Section 3.4).

Figure 4a shows the IMU-measured roll time series against the “fitted” sinusoidal signal with the characteristic amplitude, frequency, and phase of $A=1.3$ deg, $f=0.3$ Hz, and $\alpha =1.1$ deg, respectively. It can be observed that the fitted sinusoidal signal reproduced the measured time series under a first-harmonic approximation with reasonable accuracy.

3.8. A Note on Appendix A and the Supplementary Materials Math Formulations

Provided in Appendix A is a formulation compendium to compute the 6-DoF-motion-induced error (Section 3.5.3). A MATLAB${}^{\mathrm{TM}}$ R2020a code to compute the total error (Section 3.5.3) is also included as part of the Supplementary Materials of this manuscript.

The mathematical derivations of this Appendix A were checked both algebraically and numerically. Algebraically, the manually derived math expressions were validated using the MATLAB${}^{\mathrm{TM}}$ R2020a symbolic toolbox. Numerically, these expressions were coded in MATLAB and compared with the outputs of the FDWL simulator [35] (Figure 5, Figure 6 and Figure 7) for quality assurance.

4. Results and Discussion

4.1. Error Model Validation

In order to validate the presented analytical-error formulation (Section 3.5 and Appendix A; in what follows, “the unified-error formulation”), the estimated total error in the FDWL-retrieved $HWS$ (Equation (30)) was compared with the LiDAR motion simulator as a reference [37]. The study was carried out by inputting the same reference wind vector ($HWS=10$ m/s and $VWS=0$ m/s) into both the unified formulation and the simulator under three different motion scenarios: (i) rotational motion only, (ii) translational motion only, and (iii) 6 DoF motion. In all cases, $f_x=0.3$ Hz and ${\alpha }_x=0$ deg, $x=roll,pitch$ were used. In each scenario, the performance of the analytically estimated error was evaluated as a function of the $WD$ and initial scan phase ${\varphi }_0$.

In roll-only scenario #1 (Figure 5), the analytically estimated error (Figure 5a, Equation (24)) was virtually identical to the simulator-estimated error (Figure 5b) over the whole span of the $WD$s and the initial-phase values. From Figure 5c, all the error differences between panels (a) and (b) were lower than 0.3 m/s, which is evidence of the satisfactory accuracy attained using the first-order approximation in the formulation. The $RMSE$ between the error estimates of Figure 5a,b (equivalently, the square root of the mean of the squares of the error values shown in Figure 5c) was 0.04 m/s.

When addressing the translational-only scenario #2 (Figure 6), null differences (0 m/s) were found in Figure 6c between the analytically estimated and the simulator-estimated error. The exact 0 m/s error difference was explained by the fact that no approximations were used when deriving the analytical expressions for the translational-motion component.

Opposite to other floating remote-sensing devices such as high-frequency surface-wave radars, which measure the horizontal propagation of waves and thus are not sensitive to heave-induced Doppler effect [56,57], the ZephIR${}^{\mathrm{TM}}$ 300 FDWL is sensitive to all three translational motion components due to its inherent conical scanning pattern [16].

Finally, the successful performance of the unified formulation presented for the analytically estimated error was re-encountered in Figure 7, where both the translational and rotational motion were combined into the 6 DoF motion scenario #3. Despite the greater complexity of this scenario, the error differences (Figure 7c) remained below $0.7$ m/s. The $RMSE$ was 0.22 m/s.

4.2. Experimental Results

The unified-error formulation (Section 3.5 and Appendix A) was also validated by comparing the analytically estimated $HWS$ mean bias and $TI$ increment (Equations (31) and (32), respectively) against their experimental values (Equations (4) and (6), respectively). The experimental values were computed from the 10 min errors between the FDWL and the reference fixed LiDAR used in both the nearshore (“Pont del Petroli”) and open-sea (IJmuiden) scenarios. For the IJmuiden campaign, the measurements of the FDWL at 83 m and the measurements of the metmast DWL at 92 m were considered. Towards this purpose, 848 data records (6 to 30 June 2013) from PdP campaign and 3893 data records (1 April to 7 May 2015) from the IJmuiden campaign were used (see Section 2). The data were filtered for quality assurance following the outlier-rejection criteria [33]: the measured $HWS$ values outside the 1–80 m/s range, the rain-flagged data, the spatial variation (SV) values higher than 0.2, and the backscatter coefficients lower than 0.02 were removed. The SV is an indicator of the goodness of the VAD fitting of the measured LoS in a LiDAR scan [58]. The backscatter coefficient is indicative of the power of the received LiDAR echo.

4.2.1. The Performance according to the Estimation of the $HWS$ Bias

In the PdP campaign, using statistical analysis, the measured 10 min $HWS$ bias (Equation (4)) showed virtually nil values in accordance with the previous results in nearshore locations in the state of the art [27,28]. The median of the measured 10 min $HWS$ bias was −0.02%, with 25th and 75th percentiles of −0.89% and 0.76%, respectively. On the other hand, the analytically estimated 10 min $HWS$ bias (Equation (5)) showed values of the same order of magnitude as the measured ones, thus validating the method. Thus, the median of the estimated bias was 0.06%, and the 25th and 75th percentiles were −0.66%, and 0.64%, respectively. The median bias was in accordance with the results obtained in other state-of-the-art studies [38].

In the Ijmuiden campaign, the measured 10 min $HWS$ showed higher values than the observed ones in the “Pont del Petroli” campaign: the median of the measured 10 min $HWS$ bias was −0.53%, with 25th and 75th percentiles of −2.30% and 1.08%, respectively (and notice the asymmetrical distribution of these percentiles around 0%). These values were in accordance with the previous open-sea measurement campaigns [32,59]. This is due to the fact that in an open-sea scenario there is stronger wave motion, which induces a higher error in the FDWL measurements. The abovementioned asymmetry on the measured biases accounts for the wind shear between the fixed DWL and FDWL measurement heights (92 and 83 m, respectively). The analytical model estimated biases of the same order of magnitude as the measured ones with a median of 0.05%, and 25th and 75th percentiles of −1.48% and 1.47%, respectively, which further validated the analytical model.

4.2.2. Performance according to the Estimation of the $TI$ Increment (I): Case Example

As part of the validation study, in the next case example, the analytically estimated 10 min $TI$ increment, $\Delta T{I}_{est}$ (Equation (32)), was compared with the measured one, $\Delta T{I}_{meas}$ (Equation (7)). The different motion conditions studied are described as a function of the mean tilt amplitude and mean translational-velocity amplitude of the LiDAR buoy.

The mean tilt amplitude was computed as the 10 min mean of the quadratic sum of roll and pitch tilts (simple harmonic motion, Equation (19)) [28]. It can be formulated as follows:

$$\overline{A_{tilt}}=\frac{{\sum }_{k=1}^N\sqrt{{\mathsf{\Omega }}_{roll}{\left(k\right)}^2+{\mathsf{\Omega }}_{pitch}{\left(k\right)}^2}}{N},$$

(34)

where ${\mathsf{\Omega }}_x$, $x=rollpitch$ is the IMU-measured roll/pitch tilt, $N=6000$ is the number of samples in a 10 min interval at a 10 Hz IMU-sampling frequency, and k is a reminder of the discrete time $t_k$.

The mean translational-velocity amplitude was computed as the 10 min mean norm of the FDWL translational-velocity vector,

$$\overline{A_{v_{trans}}}=\frac{{\sum }_{k=1}^N\left|\right|\overrightarrow{t}\left(k\right)\left|\right|}{N},$$

(35)

where $\overrightarrow{t}\left(k\right)$ is the translational-velocity vector defined by Equation (26).

Figure 8a compares the estimated $TI$-increment time series ($\Delta T{I}_{est}$) with the measured one ($\Delta T{I}_{meas}$) in the context of the PdP campaign. The motion-induced error on the FDWL-measured $TI$ manifested as $TI$ increments between 0.5 and 5%. The highest $TI$ increments occurred during the daytime (from 10 a.m. to 4 p.m., approximately), corresponding to time periods with high HWSs. These periods were also related to the high wave motion [27,60]. This was corroborated by Figure 8b, where the mean tilt amplitude and mean translational-velocity amplitude of the floating LiDAR are depicted as time series. It can be observed that the high TI-increment values in Figure 8a were linked to high-motion scenarios, i.e., to the time periods with high tilt and velocity amplitudes in Figure 8b.

As shown in Figure 8a, the estimated $TI$ increment exhibited similar values to the measured ones over the period under study. The 1 h averaged time series further showed the goodness of the TI-increment estimates. However, a small underestimation of the peak values occurred. Thus, while the measured $TI$ increment peaked at 5% close to noon, the estimated one only reached 3.5%. Larger differences were observed in one-to-one comparisons of the 10 min estimates with the 10 min measurements. A suitable explanation for this is that in the unified formulation presented in Section 3.5.3, the wind flow during 10 min periods was assumed to be uniform, and only the motion influence was considered. As the LiDARs were located 50 m apart, wind scenarios with high spatial variability (e.g., turbulent winds coming from the urban area, Section 2) usually led to different instantaneous measurements between the floating and the reference LiDAR. Therefore, the correlation coefficients ${\rho }_{F,R}<1$ (Section 3.2.1) may limit the validity of the estimates, $\Delta T{I}_{est}$. This effect may also occur in low-motion scenarios [33].

4.3. The Performance of the Estimation of the $TI$ Increment (II): Statistical Analysis

The performance of the analytical-error formulation when estimating the 10 min $TI$ increment was also assessed with reference to the FDWL-measured $TI$ increment under nearshore and open-sea motional scenarios (the PdP and IJmuiden campaigns, respectively). The statistical sample consisted of 848 data records (from 6 to 30 June 2013) for the PdP campaign and 3893 data records (from 1 April to 7 May of 2015) for the IJmuiden campaign. The estimated and the measured $TI$ increments ($\Delta T{I}_{est}$ and $\Delta T{I}_{meas}$, respectively) for the whole campaign were clustered as a function of the wind intensity, using the HWS as the binning variable, and as a function of the type of motion (i.e., rotational or translational), using the mean tilt amplitude and the mean translational-velocity amplitude as the respective binning variables.

The estimation accuracy was quantitatively assessed according to the root mean square error ($RMSE$), which is formulated as follows:

$$RMSE=\sqrt{\frac{{\sum }_{p=1}^N{[\Delta T{I}_{est}\left(p\right)-\Delta T{I}_{meas}\left(p\right)]}^2}{N}},$$

(36)

where N is the number of samples in the bin under study, $\Delta T{I}_{est}\left(p\right)$ is the p-th $TI$-increment estimate in the bin, and $\Delta T{I}_{meas}\left(p\right)$ is the corresponding p-th measurement.

Figure 9 depicts the statistical results for the PdP (left panels) and IJmuiden (right panels) campaigns. In all panels, it can be observed that the $TI$ increment, $\Delta TI$, was always positive, evidencing the motion-induced additive turbulence [26,28]. Moreover, it increased with increasing wave motion, i.e., with increasing mean tilt amplitude (panels (a), (b)) and mean translational-velocity amplitude (panels (c), (d)). $\Delta TI$ ranged from median values of $\Delta TI\simeq 1\%$ in low-motion scenarios to $\Delta TI\simeq 2\%$ in high-motion ones. The green bars illustrating the statistically significant bins (i.e., bins containing greater than or equal to 5% of the population of the PdP campaign, i.e., ≃50 samples) showed mean tilt amplitudes between 1 and 3.5 deg for the PdP campaign and between 1 and 5.5 deg for the IJmuiden campaign (notice the different X-axis ranges used). The observed mean translational-velocity amplitudes ranged between 0.1 and 0.2 m/s for the PdP campaign and between 0.1 and 0.65 m/s for the IJmuiden campaign (notice the different X-axis ranges used). The $HWS$ ranged between 2 and 8 m/s in the PdP campaign and between 2 and 22 m/s in the IJmuiden campaign (notice the different X-axis ranges used). The higher motion and wind magnitudes evidenced the harsher scenario experienced by the FDWL during the IJmuiden campaign.

On one hand, Figure 9a,b plot the $TI$ increment as a function of the mean tilt amplitude in 0.5 deg bins, which was representative of the rotational-motion component. The median of the estimated $\Delta TI$ values virtually matched that of the measurements, thus validating the overall analytical-error formulation. In the PdP campaign (Figure 9a), the error bars showed a larger dispersion for the measured values ($\Delta T{I}_{meas}$) than for the estimated ones ($\Delta T{I}_{est}$). This is attributable to the urban topology surrounding the experimental area, which creates a high spatial variation in the wind field. In contrast, this was not found in IJmuiden. Thus, Figure 9b,d,f shows estimated error bars virtually coincidental with the measured ones due to the more homogeneous wind fields found over the ocean.

On the other hand, Figure 9c,d represent the $TI$ increment as a function of the FDWL mean translational velocity in 0.025 m/s bins for the PdP campaign and 0.05 m/s bins for the IJmuiden campaign, respectively. Similar to Figure 9a,b, the median of the estimated $\Delta TI$ values almost ideally matched the measurements in all the significant bins (dark green), Figure 9c,d. The largest difference was $\Delta T{I}_{meas}-\Delta T{I}_{est}=0.26\%$ for the 0.175 m/s bin. Similar comments apply to those for Figure 9a,b regarding the measurement error bars.

Finally, Figure 9e,f depict the $TI$ increment as a function of the mean $HWS$ in 1 m/s bins for the PdP campaign and 2 m/s for the IJmuiden campaign, respectively. Similar concurrent results were observed, with the estimated $\Delta TI$ median values matching the measurements for all $HWS$ bins except for the 2 m/s one. This was due to the fact that the ZephIR${}^{\mathrm{TM}}$ 300 $HWS$ measurements below 3 m/s were tagged as “unreliable”, according to the manufacturer’s specs for the LiDAR [40]. Similar discussion comments apply for the error bars shown in Figure 9e,f.

The $RMSE$ between the estimated and the measured $\Delta TI$ (Equation (36)) is depicted in red for each of the tilt, velocity, and $HWS$ bins. Similar values were obtained in all the statistically significant bins, ranging from $RMSE=0.004$ (0.4%) up to $RMSE=0.013$ (1.3%), which was comparable to (therefore, consistent with) the size of the measurement error bars obtained for $\Delta T{I}_{meas}$. As can be observed, the $RMSE$ was slightly correlated with the tilt, so that the larger the buoy tilt, the larger the $RMSE$ in rough approximation, which in turn accounted for the first-order approximation used in Equation (A1) (refer to Appendix A.1).

5. Conclusions

A unified analytical formulation for the computation of the 6-DoF-motion-induced error in focusable CW FDWLs was presented (Figure 3). The total error in the retrieved $HWS$ was computed as the superposition of the rotational- and translational-motion errors. The formulation proved to be capable of estimating the $HWS$ bias (Section 4.2.1) and $TI$ increment (Figure 8 and Figure 9).

The analytical model departed from a thorough formulation of the FDWL buoy geometry and dynamics in order to derive the rotational- and translational-motion influence on the LiDAR-measured LoS measurements in a CW DWL scan. The well-known VAD algorithm was computed as a first-order Fourier series, which allowed derivation of the sought-after analytical expressions describing the FDWL-measured wind-vector error as a function of the buoy attitude and the “true” wind vector. The first-order approximation was retained when computing the buoy rotational matrix. The assumption of a uniformly distributed random initial scan phase permitted the estimation of the $HWS$ measurement bias and $RMSE$ over 10 min time intervals.

The wave-induced motion in each of the six DoF was modeled as a simple harmonic motion with a characteristic amplitude, frequency, and phase recomputed every 10 min for each DoF. A method to estimate these variables based on spectral analysis of the LiDAR-buoy motional time series was presented in Section 3.7.

The proposed formulation was numerically validated by comparing the $HWS$ error figures obtained as a function of the $WD$ and the initial scan phase with those output by the FDWL motion simulator [37]. A virtually perfect match between both methods was found except for the second-order approximation errors.

The methodology was further validated using experimental data from the “Pont del Petroli” (Badalona, Barcelona) and IJmuiden (North sea, 85 km offshore the Netherlands) campaigns, in which an FDWL and a reference fixed LiDAR were used. The estimated 10 min bias using the analytical formulation presented here yielded similar results to the measured one. Thus, for the PdP campaign, the median values obtained for the estimated and the experimental $HWS$ bias were −0.02% and 0.06%, respectively, and the 25th and 75th percentiles were −0.66% (estimated) and −0.89% (measured) and 0.76% (measured) and 0.64% (estimated), respectively. For the IJmuiden campaign, median values of −0.53% (measured) and 0.05% (estimated) were found, and the 25th and 75th percentiles were −2.30% (measured) and −1.48% (estimated) and 1.08% (measured) and 1.47% (estimated), respectively. The inherent asymmetry between the positive and negative biases was due to the wind shear between the fixed DWL and FDWL measurement heights. The slight differences between the measured and estimated values were inconsequential and were due to the harsher sea conditions at the IJmuiden site. With regard to the estimation of the $TI$ increment, the analytical estimates matched the measured values in statistical terms under all motion and wind scenarios for both the PdP and IJmuiden campaigns (Figure 9). Thus, the root mean square differences between the medians of $\Delta T{I}_{est}$ and $\Delta T{I}_{meas}$ were 0.16% (PdP) and 0.11% (IJmuiden) for the rotational motion (Figure 9a,b), 0.13% and 0.16% for the translational motion (Figure 9c,d), and 0.14% and 0.21% for the $HWS$ (Figure 9e,f), giving mean values of about 0.015 (1.5%). The experimental validation was limited by the fact that the compared LiDARs were a minimum of 50 m apart and subject to wind flows that were not always uniform. This caused moderate dispersion in the turbulence values measured by the two LiDARs, which deteriorated in one-to-one comparison of the 10 min $TI$.

All in all, the unified formulation presented here proved to be a straightforward and accurate tool for quantitatively assessing the 6-DoF-motion-induced error in focusable CW FDWLs in terms of both the $HWS$-bias and $TI$-increment estimation, in addition to the real one. This information is increasingly being recognized as critical to the acceptance of FDWLs and ship-borne LiDARs in the Wind Energy industry as compared to fixed LiDARs and reference metmasts [61,62]. To the best of our knowledge, the proposed unified analytical formulation is the first in the state of the art able to estimate the $TI$ increment at a 10-min basis from solely FDWL measurements under virtually all motional and wind scenarios. In addition, the novel analytical expressions obtained are less computationally demanding than numerical simulators and unveil a key understanding of the involved FDWL attitude and LiDAR-retrieval-algorithm error sources. These findings have important implications for the accurate measurement of wind speed and turbulence in FDWLs, particularly when they are being deployed as standalone measurement instruments. $Matla{b}^{\mathrm{TM}}$ R2020a codes are also provided to the reader as part of the Supplementary Materials of this manuscript. Further steps could include the study of multimodal motion as well as the wind flow variability in a LiDAR scan. Finally, experimental validation would benefit from tests at different LiDAR sounding heights.