# Quantification and Correction of Wave-Induced Turbulence Intensity Bias for a Floating LIDAR System
## 4 Results and Discussion
### Theoretical Impact of Buoy Movements on the FLS Horizontal Velocity Estimates
#### 4.2.2 Tilt Motion
Since the angles \(\varphi\) and \(\theta\) are modified by the tilt, the measurements of the radius B are impacted by both wind components \(V_{1}\) and \(V_{2}\):  
\[V_{r_{B}}=V_{r_{B}V_{1}}+V_{r_{B}V_{2}} \tag{25}\]  
The measurement of the \(V_{r_{B}V_{1}}\) component of the radial wind speed is impacted by the tilt motion (Figure 6c). Since beam B is originally perpendicular to the wind component \(V_{1}\), it is not supposed to measure it. However, the tilt modifies \(\theta\) and the measurement of beam B is impacted by the \(V_{1}\) component. The measurement of the \(V_{r_{B}V_{2}}\) component is not affected by the tilt movement (Figure 6d). The effects induced by the simultaneous changes in the angles \(\varphi\) and \(\theta\) compensate each other to remove this error. We note that the same tilting movement has an equivalent effect on the radial wind speed measurement of beam D. If the measurements were made simultaneously at both  
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. directions B and D, the effects of the tilting movements on the two opposite beams would be additive. We can now determine the dominant effect induced by tilting on the horizontal wind measurement. For a tilt along a given axis, only the \(V_{1}\) component of the incident wind, perpendicular to the axis of rotation, has an influence on the measurement of the four non-vertical rays. We consider here the case of a whole period of sinusoidal motion of amplitude \(a_{tilt}=10^{\lx@math@degree}\) and we evaluate the order of magnitude of the influence of this motion on the measurement of the different rays. While the average measurement is barely affected by the tilting motion, a large amount of variability is introduced into the time series of the measurement. The effects on the rays parallel (beams B-D) and perpendicular (beams A-C) to the axis of rotation introduce an equivalent standard deviation on the radial wind speed measurement, of the order of \(\sigma_{V_{r}}=0.108V_{1}\). However, the influence on the horizontal velocity variation is not equivalent. In order to estimate this influence, let us assume an incident wind following the \(V_{1}\) component (\(V_{2}=0\)) and a variation \(\Delta V_{1}\) introduced by a tilt on the measurement of the wind component in the plane of each pair of rays. Considering \(V_{hor}=V_{1}\), using a Reynolds decomposition and the Equation (2), we compare:  
* The effect of a variation of \(\Delta V_{1}\) in the measurement of the component measured by A-C rays: \[\Delta V_{hor_{A-C}}=\sqrt{\left(V_{1}\pm\Delta V_{1}\right)^{2}+0}-V_{hor}= \sqrt{\left(1\pm\Delta V_{1}\right)^{2}V_{hor}^{2}}-V_{hor}=\pm\Delta V_{1}\] (26)  
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 \(\varphi\). (**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. * The effect of a variation of \(\Delta V_{1}\) in the measurement of the component measured by B-D rays: \[\Delta V_{hor_{B-D}}=\sqrt{V_{1}^{2}+\left(\pm\Delta V_{1}\right)^{2}}-V_{hor}= \sqrt{\left(1+\frac{\Delta V_{1}^{2}}{V_{hor}^{2}}\right)V_{hor}^{2}}-V_{hor}= \sqrt{1+\frac{\Delta V_{1}^{2}}{V_{hor}^{2}}}-1\] (27)  
There is an order of magnitude of discrepancy between the two estimates: \(\Delta V_{hor_{A-C}}\gg\Delta V_{hor_{B-D}}\).