# 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
#### 3.1.1 Measurement from One Line-of-Sight Subjected to Wave-Induced Motion  
The radial wind speed measured by a line-of-sight from a moving LIDAR is impacted by two major phenomena: The alteration of the measurement cone geometry induced by the rotational angles, and the influence of the translational velocity. At time \(t\), the local roll \(\Phi(t)\), pitch \(\beta(t)\) and yaw \(\Psi(t)\) angles as well as the surge \(V_{u}(t)\), sway \(V_{v}(t)\) and heave \(V_{w}(t)\) translational velocities are estimated by linear interpolation of the IMU measurements. The effect of beam angular velocity is not considered here as it is applied in the beam direction's normal plane, so it has little or no influence on the measurement. We also note that the tilt of the LIDAR implies a change in the measurement altitude in addition to the change in the geometry of the measurement cone. The two phenomena have opposite effects: Increasing the tilt angle \(\varphi\) tends to decrease the measurement altitude and thus to encounter lower wind speeds because of wind shear, but the decrease in the angle relative to the horizontal wind tends to increase the radial wind speed measurement. In this study, we did not include this effect in the model. However, its influence is expected to be quite small. [PERSON] et al. (2016) estimated from their offshore dataset the factor \(\mathrm{k}=5.8\%\), by which the effect of changed scanning geometry is reduced by the effect of wind shear. We first study the effect of the alteration of the measurement cone geometry. At time \(t\), the LIDAR measurement cone is defined by the angles \(\varphi\) and \(\theta\) for each beam direction. In order to estimate the effect of the geometric transformation, we need to calculate the actual angles (in the non-rotated reference frame) of the laser beam at each discretized time in the model. We apply the rotation \((\Phi(t),\beta(t),\Psi(t))\) to the non-rotated normalized vector \(\overrightarrow{n}(D_{LOS})\) defined for each direction (Equation (5)), through the rotation matrix \(T_{rot}\) (Equation (8)):  
\[\overrightarrow{n^{rot}}(D_{LOS},t)=T_{rot}(\Phi(t),\beta(t),\Psi(t))\cdot \overrightarrow{n}(D_{LOS}) \tag{13}\]  
Then, we calculate the angles \(\theta(D_{LOS},t)\) and \(\varphi(D_{LOS},t)\) from the rotated vector \(\overrightarrow{n^{rot}}(D_{LOS},t)\):  
\[\theta(D_{LOS},t)=mod\big{(}\pi+atan2\big{(}n_{2}^{rot}(D_{LOS},t),n_{1}^{rot }(D_{LOS},t)\big{)},2\pi\big{)} \tag{14}\]  
\[\varphi(D_{LOS},t)=\mathrm{acos}\left(\overrightarrow{n^{rot}}(D_{LOS},t)^{ T}\cdot\left( \begin{array}{c}0\\ 0\\ -1\end{array} \right)\right) \tag{15}\]  
with \(n_{1}^{rot}(P_{LOS},t)\) and \(n_{2}^{rot}(P_{LOS},t)\) being respectively the first and second component of the rotated normalized beam vector. Finally, according to Equation (6), the radial wind speed theoretically measured by the beam at position \(D_{LOS}\) at time \(t\) is calculated by:  
\[ \begin{array}{l}V_{r}(D_{LOS},\,t)=\left(\begin{array}{c}u_{10-\min}\\ v_{10-\min}\\ w_{10-\min}\end{array}\right)^{T}Rot(D_{LOS},t)\\ Rot(D_{LOS},t)=\left(\begin{array}{c}sing\varphi(D_{LOS},t)cos\theta(D_{ LOS},t)\\ sing\varphi(D_{LOS},t)sin\theta(D_{LOS},t)\\ cos\varphi(D_{LOS},t)\end{array}\right)\end{array}  \tag{16}\]  
During a line-of-sight measurement in a given direction (approximate time \(1\) s), there is sufficient time for the roll, pitch and yaw angles to vary significantly (Figure 3). The set of discrete points contained in the time of a \(LOS\) measurement are combined and averaged to estimate the overall effect of the geometry change. Therefore, for a \(LOS\) measurement at position \(D_{LOS}\) of duration \(t_{LOS}\) performed between times \(t_{1}\) and \(t_{2}\), the average radial wind speed including the effect of the geometry modification of the measurement cone \(\overrightarrow{V_{r}^{beam}}\) is calculated by:  
\[