# 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
#### 3.1.2 Estimated Motion-Induced Variance for 10 min of FLS Measurement  
The laser beam of the WindCube v2\({}^{\infty}\) 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 \((t_{N},t_{E},t_{S},t_{W},t_{Z})\) 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:  
\[\left\{ \begin{array}{l}u(t,h)=\left(\begin{array}{c}u_{10-\min}(h)\\ v_{10-\min}(h)\\ w_{10-\min}(h)\\ \end{array}\right)^{T}\frac{U^{rot}(t)}{2\overset{\rightarrow}{sing}}-\frac{V _{trans}^{pos}(t)}{2\overset{\rightarrow}{sing}\phi}\\ v(t,h)=\left(\begin{array}{c}u_{10-\min}(h)\\ v_{10-\min}(h)\\ w_{10-\min}(h)\\ \end{array}\right)^{T}\frac{V^{rot}(t)}{2\overset{\rightarrow}{sing}}-\frac{V _{trans}^{pos}(t)}{2\overset{\rightarrow}{sing}\phi}\\ w(t,h)=\left(\begin{array}{c}u_{10-\min}(h)\\ v_{10-\min}(h)\\ w_{10-\min}(h)\\ \end{array}\right)^{T}W^{rot}(t)-V_{trans}^{w}(t)\\ \end{array} \right. \tag{20}\]  
with \(\varphi=28^{\circ}\) and :  
\[\left\{ \begin{array}{l}U^{rot}(t)=\overline{Rot}(N,t_{N})-\overline{Rot}(S,t _{S})\text{ and }V_{trans}^{u}(t)=\overline{V_{trans}^{beam}}(N,t_{N})-\overline{V_{trans}^{ beam}}(S,t_{S})\\ V^{rot}(t)=\overline{Rot}(E,t_{E})-\overline{Rot}(W,t_{W})\text{ and }V_{trans}^{v}(t)=\overline{V_{trans}^{beam}}(E,t_{E})-\overline{V_{trans}^{ beam}}(W,t_{W})\\ W^{rot}(t)=\overline{Rot}(Z,t_{Z})\text{ and }V_{trans}^{w}(t)=\overline{V_{trans}^{ beam}}(Z,t_{Z})\\ \end{array} \right. \tag{21}\]  
We note that the effect of translational velocity on the measured LIDAR velocity is subtracted from the wind measurement because, if the two phenomena are oriented in the same direction, the translational velocity tends to absorb a part of the atmospheric wind speed. With the variance estimator defined in Equation (12), we can directly calculate the motion-induced variance using a matrix system. It is first necessary to carry out the development of the equations of \(u(t)^{2}\) and \(v(t)^{2}\) Equations (A1) and (A2) in Appendix A). Then, thanks to the distributive nature of multiplication with respect to addition and to the constant wind speed at altitude \(h\), we identified:\[ \begin{array}{rcl}Var\big{(}V_{hor}^{motion}\big{)}\ \sim&\frac{1}{(2 sin\sigma)^{2}} \Bigg{(}\left(\begin{array}{c}u_{10-\min}(h)\\ v_{10-\min}(h)\\ w_{10-\min}(h)\end{array}\right)^{T^{2}}\frac{1}{{U^{rot}}^{2}+{V^{rot}}^{2}}+2 u_{10-\min}(h)v_{10-\min}(h)\overline{U_{1}^{rot}}\overline{U_{2}^{rot}}+V_{1}^{ rot}V_{2}^{rot}\\ &+2u_{10-\min}(h)w_{10-\min}(h)\overline{U_{1}^{rot}}\overline{U_{3}^{rot}}+V_{ 1}^{rot}V_{3}^{rot}\\ &+2v_{10-\min}(h)w_{10-\min}(h)\overline{U_{2}^{rot}}\overline{U_{3}^{rot}}+V _{2}^{rot}V_{3}^{rot}-2u_{10-\min}(h)\overline{U_{1}^{rot}}\overline{V_{trans} ^{u}}+V_{1}^{rot}V_{trans}^{v}\\ &-2v_{10-\min}(h)\overline{U_{2}^{rot}}\overline{U_{trans}^{u}}+V_{2}^{rot} \overline{V_{trans}^{v}}-2w_{10-\min}(h)\overline{U_{3}^{rot}}\overline{V_{ trans}^{u}}+V_{3}^{rot}V_{trans}^{v}\\ &+\overline{V_{trans}^{u}}^{2}+\overline{V_{trans}^{v}}^{2})-\overline{V_{hor} }(h)^{2}\end{array}  \tag{21}\]  
with \(U_{i}^{rot}\) being the \(i^{th}\) component of the vector \(U^{rot}\).