# 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  
In this section, care has been taken to make all the results independent of the nature of the tilt (pitch or roll), so the directions of the rays are defined by A, B, C, D and Z for zenith. The effect of tilt on the angles \(\varphi\) and \(\theta\) of the beams in the different directions of the WindCube v2(r) is studied in Figure 4. The horizontal velocity is decomposed into two components: \(V_{1}\) component in the A-C beams plane and the \(V_{2}\) component in the B-D beams plane. For beams A and C, the tilt only affects the angle \(\varphi\). And for beams B and D, the tilt modifies \(\theta\) and \(\varphi\). The effect on the measurements is a composition of two angular effects. In the Z case, the ray is at the zenith and the tilt has an effect only on \(\varphi\). As the horizontal velocity measurement is only affected by the measurements of the beams A, B, C and D, we do not study here in detail the zenith beam. We first focus on the rays in the normal plane to the tilt axis (beams A-C). In this plane, only the angle \(\varphi\) is affected by the rotational movement, it evolves in a linear way with the tilt: \(\Delta\varphi=T\). Therefore, only the \(V_{1}\) component of the incident horizontal wind affects the radial wind speed measurement. We present here the effect of motion on the measurement of beam A. The radial wind speed is calculated according to the Equation (6):  
\[V_{r_{A}}=V_{r_{A};V_{1}}=\sin(\varphi)V_{1} \tag{24}\]  
The tilt of the LIDAR has the effect of increasing or decreasing the value of \(\varphi\) which directly affects the value of the measured radial velocity (Figure 5a). The larger the value  
Figure 4: Diagram representation showing the influence of a tilt change \(T\) (roll or pitch) on the angles \(\varphi\) and \(\theta\) of the five measurement directions of the WindCube v2(r) denoted A, B, C, D, Z (side view). The suggested times are added to recall the temporality of the WindCube measurement. The horizontal velocity is decomposed into two components: \(V_{1}\) and \(V_{2}\). of \(\varphi\), the more aligned the beam is with the horizontal velocity and the greater the radial wind speed measurement. In Figure 5b, it is observed that this effect is not uniformly distributed around the value \(\varphi=28^{\lx@math@degree}\). The properties of the sine function around the angle \(\varphi=28^{\lx@math@degree}\) bring more influence to the decrease of the angle than to its increase. Thus, the average measurement of the beam A is affected by the tilting motion. For an entire period of tilt of amplitude \(a_{tilt}=10^{\lx@math@degree}\), the average measurement of the beam is decreased by less than 1%. We note that the same tilting movement has an opposite effect on the radial wind speed measured by beam C. Thus, if the measurements were carried out simultaneously at both directions A and C, the tilting movement would have very little influence on the measurement of the velocity component \(V_{1}\) by the WindCube v2(r). However, in a real case, the measurements are not simultaneous and this effect has a strong influence on the LIDAR measurement. Now, we focus on the beams in the plane parallel to the tilt axis (beams B-D). In this plane, the motion simultaneously affects the angles \(\theta\) and \(\varphi\) of the beams. Here, we present the effect of the tilt on the radial wind speed measurement from beam B. Figure 6a,b show the influence of a period of sinusoidal motion on the variation of angles \(\varphi\) and \(\theta\) of beam B: \(\Delta\varphi_{B}\) and \(\Delta\theta_{B}\). The variation is defined as the difference of the angle subjected to the motion with respect to its original position. The angle \(\theta\) evolves counterclockwise for a positive inclination. While the angle \(\varphi\) is accentuated whatever the sign of the tilt \(T\).