# Quantification and Correction of Wave-Induced Turbulence Intensity Bias for a Floating LIDAR System
## 3 Method  
### A Model-Based Method for Correcting the 10-min TI Measurements from the Wind LIDAR  
The variance of a floating LIDAR horizontal wind speed measurement has two main sources. The first is the natural variance induced by the turbulent structures encountered in the atmospheric boundary layer (\(Var\big{(}V_{hor}^{atm}\big{)}\)), the one we seek to assess. The second is the variance induced by the motion of the LIDAR in its six degrees of freedom (\(Var\big{(}V_{hor}^{motion}\big{)}\)), the one we estimate using the model. The theory of the method is based on the hypothesis that, at a given average wind speed, wave-induced motion and atmospheric turbulence are independent events. Therefore, according to the properties of independent variables, we can decompose the variance of the FLS uncorrected measurements (\(Var\big{(}V_{hor}^{mes}\big{)}\)) into:  
\[Var\big{(}V_{hor}^{mes}\big{)}=Var\big{(}V_{hor}^{atm}\big{)}+Var \big{(}V_{hor}^{motion}\big{)} \tag{9}\] \[and\ :\ \sigma_{V_{hor}^{atm}}=\sqrt{Var\big{(}V_{hor}^{mes}\big{)}- Var\big{(}V_{hor}^{motion}\big{)}}\]  
The atmospheric turbulence intensity can be deduced from the standard deviation calculated by the model:  
\[TI=\frac{\sigma_{V_{therm}}}{V_{hor}^{atm}} \tag{10}\]  
For the purpose of estimating \(Var\big{(}V_{hor}^{motion}\big{)}\), the model simulates the operation of a WindCube v2\({}^{\otimes}\) 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 winddirection, 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:  
\[V_{10-\min}(h)=\left( \begin{array}{c}u_{10-\min}(h)\\ v_{10-\min}(h)\\ w_{10-\min}(h)\end{array} \right)^{T} \tag{11}\]  
with \(u_{10-\min}(h)\), \(v_{10-\min}(h)\) and \(w_{10-\min}(h)\) the 10-min means of the three wind components. The model is based on an incremental temporal discretization. At each time step, the IMU measurements are interpolated and the theoretical measurement from the evaluated beam subjected to the motion is calculated. At the end of a line-of-sight measurement, the evaluated measurements are averaged to deduce the theoretical _LOS_ wind speed measurement. At the end of the evaluated 10-min period, the 10-min statistics from the model provide the variance induced by the motion of the LIDAR. The model principle for calculating the theoretical radial wind speed measurement in a given direction is explained in details in Section 3.1.1 and the method for estimating the 10-min motion-induced variance is detailed in Section 3.1.2. We define the horizontal velocity variance estimator for \(n\)_LOS_ measurements during the 10 min as follows:  
\[ \begin{split} Var(V_{hor}(h))~{}\sim~{}\frac{1}{n}\sum_{i=1}^{n}V _{hor}(t_{i},h)^{2}-\overline{V_{hor}(h)^{2}}\\ =\frac{1}{n}\sum_{i=1}^{n}(u(t_{i},h)^{2}+v(t_{i},h)^{2})- \overline{V_{hor}(h)^{2}}\end{split}  \tag{12}\]  
with \(V_{hor}\) the horizontal velocity (Equation (2)) and \(t_{i}\) the time corresponding to the end of the \(i^{th}\)_LOS_ measurements. We note that the variance formulation is referred to as an estimator as the samples are not uniformly distributed over time, as the _LOS_ measurement time is not constant. In this study, we define the measurement time for each of the WindCube v2\({}^{\text{\textregistered}}\)_LOS_ directions as \(t_{LOS}=(0.8\,s,0.8\,s,0.8\,s,1\,s)\), 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 \(D_{LOS}=(N,E,S,W,Z)\). 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.