## 2 Planar LiDAR Data
### Synthetic LiDAR Data
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(\omega_{\phi}\), & \(\Delta\phi\), & N\({}_{\phi}\), & \(n_{pnt}\) & \(n_{\theta}\), & \(\mathcal{R}_{\phi\prime}\) & T\({}_{\phi}\) & \(f_{\mathrm{s}}\) & \(\eta_{m}\) \\ \(\omega_{\phi}\) & \(\Delta\theta\) & N\({}_{\theta}\) & \(n_{\theta}\) & \(n_{\theta}\) & \(\mathcal{R}_{\theta}\) & T\({}_{\theta}\) & & \\ \hline \(1^{\circ}\)/s & 40\({}^{\circ}\) & 15 & 36000 & 200 & 0.2\({}^{\circ}\) & 40.0 s & 0.024 Hz & 97.2 \% \\ \(2^{\circ}\)/s & 40\({}^{\circ}\) & 29 & 18000 & 100 & 0.4\({}^{\circ}\) & 20.0 s & 0.047 Hz & 94.2 \% \\ \(3^{\circ}\)/s & 40\({}^{\circ}\) & 42 & 11880 & 66 & 0.6\({}^{\circ}\) & 13.3 s & 0.068 Hz & 91.0 \% \\ \(4^{\circ}\)/s & 40\({}^{\circ}\) & 54 & 9000 & 50 & 0.8\({}^{\circ}\) & 10.0 s & 0.089 Hz & 89.2 \% \\ \(5^{\circ}\)/s & 40\({}^{\circ}\) & 66 & 7920 & 40 & 1.0\({}^{\circ}\) & 8.0 s & 0.011 Hz & 86.8 \% \\ \(6^{\circ}\)/s & 40\({}^{\circ}\) & 77 & 6840 & 33 & 1.2\({}^{\circ}\) & 6.7 s & 0.127 Hz & 84.6 \% \\ \(8^{\circ}\)/s & 40\({}^{\circ}\) & 97 & 4500 & 25 & 1.6\({}^{\circ}\) & 5.0 s & 0.161 Hz & 80.6 \% \\ \(12^{\circ}\)/s & 40\({}^{\circ}\) & 133 & 2880 & 16 & 2.5\({}^{\circ}\) & 3.3 s & 0.221 Hz & 73.4 \% \\ \(19.11^{\circ}\)/s & 40\({}^{\circ}\) & 183 & 1800 & 10 & 4.0\({}^{\circ}\) & 2.1 s & 0.303 Hz & 63.4 \% \\ \(26.22^{\circ}\)/s & 40\({}^{\circ}\) & 221 & 1260 & 7 & 5.7\({}^{\circ}\) & 1.5 s & 0.370 Hz & 55.8 \% \\ \(33.33^{\circ}\)/s & 40\({}^{\circ}\) & 250 & 1080 & 6 & 6.7\({}^{\circ}\) & 1.2 s & 0.417 Hz & 50.0 \% \\ \hline \end{tabular}
\end{table} 
Table 1: Simulated LiDAR trajectories of cross-measurements for \(n_{r}=180\) and T \(=600\) s.  
Figure 2: Illustration of the polar measurement grid of a PPI scan in the Cartesian \(\chi\)-\(\gamma\) coordinate system. Each sub-figure plots the grid density for eight out of a total of eleven different angular velocities (\(\omega_{\phi}\)). the LiDAR at \(\theta_{acc}=0.2\) s compared to the temporal resolution of the LES at 1 Hz. This difference is critical, so we cannot ensure the absolute transferability of the results presented in Section 4 to real LiDAR measurements taken from a free field. Nevertheless, our analysis of the standard deviations gives some indication of trends in the wake behaviour. However, to give simulated behaviour that is most similar to the full-scale measurements, we considered the reset time (t\({}_{r}\)), which is needed by the LiDAR device to restart the trajectory. During this time, the LiDAR returns to the initial scanner position without recording any measurements. The repetition time of a scan is, therefore, the sum of the scan time (T\({}_{\phi}\)) and the reset time (t\({}_{r}\)). The reset time (t\({}_{r}\)) was derived from real LiDAR measurements and was set to t\({}_{r}=1.2\) s for an opening angle of 40\({}^{\circ}\). The following equations explain the formal relationship of the PPI and RHI trajectories. For the sake of brevity, only the PPI case is written out below:  
\[\text{T}_{\phi}=\frac{\Delta\phi}{\omega_{\phi}}, \tag{4}\]  
\[n_{\phi}=\frac{\text{T}_{\phi}}{\theta_{acc}}, \tag{5}\]  
\[\mathcal{R}_{\phi}=\frac{\Delta\phi}{n_{\phi}}, \tag{6}\]  
\[\text{N}_{\phi}=\bigg{\lceil}\frac{\text{T}}{\text{T}_{\phi}}\bigg{\rceil}, \tag{7}\]  
\(\text{T}=600\)\(s\) in the present study. \[n_{pnt}=n_{\phi}\cdot n_{r}, \tag{8}\]  
\[\eta_{m}=1-\frac{\text{t}_{r}}{\text{T}_{\phi}}, \tag{9}\]  
\[f_{s}=\frac{1}{\text{T}_{\phi}+\text{t}_{r}}, \tag{10}\]  
For the calculation of N\({}_{\phi}\) and N\({}_{\theta}\), we rounded up the scans that began in the 10-min interval, but did not finish within the interval. This rounding was not applied when determining \(\eta_{m}\) to visualise how the measurement time efficiency is changed by the different angular velocities. We also normalised the coordinate system to the wind turbine rotor diameter (\(D\)). We used the defined measurement trajectories listed in Table 1 when running the LIXIM LiDAR simulator developed at ForWind by [PERSON] (2017), which was used by [PERSON] et al.