## 4 Results
### Calculation of Synthetic LiDAR Data
#### 4.1.1 Error Case Discrimination  
We determine the deviation from the LES by defining an average velocity error:  
\[\varepsilon_{\overline{u}}=\frac{\overline{u_{lidar}}-\overline{u_{LES}}}{ \overline{u_{LES}}}, \tag{22}\]  
and the standard-deviation error:  
\[\varepsilon_{\sigma_{u}}=\frac{\sigma_{lidar}-\sigma_{LES}}{\sigma_{LES}}, \tag{23}\]  
where \(N_{lidar}\) is the number of velocity points included in the original LiDAR data, while \(N_{LES}\) is the number of corresponding velocity points in the LES data. Note that \(u\) is normalised before this step, so that \(\varepsilon_{\overline{u}}\) and \(\varepsilon_{\sigma_{u}}\) are dimensionless and are given in percentages. For a more detailed analysis, we distinguished three regions of errors:  
* Error at the centreline in the wake: \(\gamma D^{-1}=0\) and \(\zeta D^{-1}=0\). * Planar wake error: \(-1\leq\gamma D^{-1}\leq 1\) and \(-1\leq\zeta D^{-1}\leq 1\). * Error in the free stream outside of the wake (opposite the planar wake case). We introduce these three groups of errors to clarify the added volume-averaging error in the scanning direction. Since this effect is not reproduced in the LiDAR simulator, we attempted to account for it by posteriorly averaging the planar error. The aim of this process was not exactly to recreate the scan-wise volume averaging in the measured data, but instead to reproduce a similar trend. As a result, the first case, considering data on the centreline, reproduces the step-and-stare measurement behaviour. The second case, planar wake error, resembles the behaviour of on-the-fly measurements. The third case considers free-stream error in the measurements of flow outside of the wake. The plots in Figure 9 show the behaviour of \(\varepsilon_{\overline{u}}\) and \(\varepsilon_{\sigma_{u}}\) for each of the three types of error, depending on the scan speeds, \(\omega_{\phi}\) and \(\omega_{\theta}\). As mentioned above in Section 1, the largest errors occur at the extremes of very slow and very fast measurements. In general, similar behaviour is seen in all the errors in the PPI and RHI measurements. The errors are high at the scanning speed of \(1^{\circ}/\mathrm{s}\), decrease as the scan speed increases until the minima are reached, and increase again up to the speed of \(33.33^{\circ}/\mathrm{s}\). Mean-value error (\(\varepsilon_{\overline{u}}\)) and the standard-deviation error (\(\varepsilon_{\sigma_{u}}\)) emerge along the centreline in the RHI scans, \(\zeta D^{-1}=0\). At the scanning speed of \(\omega_{\theta}=8^{\circ}/\mathrm{s}\), these errors are more scattered than the other errors. In addition, the centreline errors in the PPI scans show an unexpectedly low value at the scanning speed of \(\omega_{\phi}=33.33^{\circ}/\mathrm{s}\), which we are unable to explain. The wind-speed data along the centreline is clearly affected the least by the change between the polar and Cartes  
Figure 9: Velocity error (\(\varepsilon_{\overline{u}}\), blue) and standard-deviation error (\(\varepsilon_{\sigma_{u}}\), red) of synthetic LiDAR data as compared against the LES for the planar section (crosses), centreline (circles), and free-flow region (squares) of (**a**) PPI scans and (**b**) RHI scans. since the scan geometry and the resulting distribution of angular measurements cover this line more consistently than other positions with measurement points. While all errors in the RHI and PPI scans decrease along with increasing angular speeds, the ensemble minimum occurs at approximately 5\({}^{\circ}\)/s. The centreline error increases dramatically after 5\({}^{\circ}\)/s. The logarithmic representation in these plots makes it difficult to recognise the point symmetry of the error behaviour. The PPI error is symmetric around the minimum at \(\omega_{\phi}=4^{\circ}\)/s and \(\varepsilon_{\overline{\mu}}=0.46\) %. The minimum of the standard deviation is found at \(\omega_{\phi}=5^{\circ}\)/s and \(\varepsilon_{\sigma_{u}}=2.29\) %.