## 3 Method
### Temporal Correction and Data Synchronisation  
Since planar LiDAR measurements are recorded with temporally distributed scans, each scan maps the evolution of flow over the scanning period. This temporal shift within the scan increases with longer scan duration, making scanned measurements difficult to compare with measurements that are presented in instantaneous form. The temporal correction method discussed in this section is intended to align the temporal shift within a scan so the flow situation at a single moment in time can be expressed. The refinement of the temporal resolution to scales less than that of the measurement, as introduced in Section 3.2, lays the foundation for the temporal correction. The time interval of each scan can be described as \(T_{n}=[t_{n,1},t_{n,2}]\{t_{n,X}\in T_{n}\mid t_{n,1}<t_{n,X}<t_{n,2}\}\), where \(t_{n,1}\) and \(t_{n,2}\) are the beginning and end of the scan. Since the azimuth angle (\(\phi\)) and the elevation angle (\(\theta\)) are functions of time, they can be expressed as \(\phi(t_{n,X})\) and \(\theta(t_{n,X})\). As the time gaps between \(\phi(t_{n,X})\) and \(\phi(t_{n+1,X})\) and between \(\theta(t_{n,X})\) and \(\theta(t_{n+1,X})\) are closed with discrete interpolation steps because of the mixed wind-field propagation, 3D natural-neighbour interpolation can be applied at time \(t\)[40]. The first two dimensions represent the scanned measurement area and the third dimension represents time. Figure 5 illustrates the temporal correction method and the 3D interpolation in 2D form, and the measurement area is represented as the scan angle, \(\phi(t)\), \(\theta(t)\), over time. The propagation steps (light-blue lines) between two consecutive scans (dark-blue lines) are indicated by their slope, such that the measurement result and the refinement of the temporal resolution represent the measurement area over a time interval. By choosing the interpolation time step (\(\Delta t\)) carefully to give an integer number for interpolation (\(\Pi\)), we can exclude the reset time of the scanner (\(t_{r}\)) (green lines), during which no measurements are captured. As shown in Figure 5, 3D interpolation is used to give the values of wind speed over the entire measurement area at the moment (\(t\)) (red line). When a smaller value of \(\Delta t\) is chosen, the temporal correction will have finer spatial resolution, since the spatial coverage is time-dependent. With the application of the temporal correction, measurements taken over a period of time can be interpolated into the same time and can thereby be synchronised to a single moment. Next, the RHI and PPI scans are synchronised to match the temporal resolution of the reference LES wind-field data. Figure 4: Example of the weighting function, \(w_{n}(t)\), for forward propagation (blue) and the function, \(1-w_{n}(t)\), for backwards propagation (red). The possibility of synchronising time-series data is not entirely novel. [PERSON] et al. (2019) shifted a time series of pulsed LiDAR data statically to a certain time using [PERSON]'s approach [PERSON] et al. (2019) to calculate the data's coherence. While this approach retains the time-series dynamics, our method instead gives a new planar time series of data. Figure 6 shows an example scan with \(\omega_{\phi}=1^{\circ}/\mathrm{s}\), from which the effect of the temporal alignment can be seen within the planar scans. The first column (Figure 6a,d,g) shows the original LiDAR data measured in the time interval, \(T_{n}=[160\;\mathrm{s},\;200\;\mathrm{s}]\). In the second column (Figure 6b,e,h), we see the temporal improved and corrected wake velocity data at the start of the scan at \(t=160\;\mathrm{s}\) (Figure 6b), at the middle at \(t=185\;\mathrm{s}\) (Figure 6e) and at the end at \(t=200\;\mathrm{s}\) (Figure 6h). The third column (Figure 6c,f,i) visualises the instantaneous wind-speed deviations of the original LiDAR scan to the corrected data.