## 3 Method
### Wind-Field Propagation  
For most wind speed analyses that aim to give wind speed averages, the recorded scans can be assumed to be quasi-instantaneous. The time shift within a scan is relatively small when compared to the total averaging time. In contrast, if certain dynamics will be calculated or derived from the  
Figure 3: Illustration of the measurement geometry, including the wind vector, \(\overset{\rightarrow}{u}\), the azimuth angle, (\(\phi\)), and the elevation angle, (\(\theta\)), of (**a**) a horizontal PPI scan and (**b**) a vertical RHI scan. The red line indicates the LiDAR laser beam. wind-speed measurements, a high-resolution time series will be needed. If the measurement frequency of points within the measurement region is higher, the conclusions about the dynamics will be more accurate. When analysing scanned wake measurements, special attention must be paid to the temporal shift within a scan, as this temporal shift means that the scan does not show the flow situation at one moment, but instead covers a time interval. When scanning cross-wise in the flow direction, characteristic flow structures are distorted depending on their expansion and flow velocity. If the scan direction is parallel to the flow direction, flow structures may be distorted or may be imaged several times in a single scan, depending on the flow dynamics and scan velocity. This effect is relevant when determining the wake deficit of a wind turbine or when determining the dynamics at the centre of the wake in the downstream direction. Two aspirations are joined with this effect: (1) Our aim is to improve on the temporal resolution and (2) to give a temporally realistic representation of the scanned data. As Section 3.3 will show, the improvement of the temporal resolution improves the physical realism of the data. Table 1 shows that planar scans are typically taken over intervals from 1.2 s to 40 s. The scan-repetition time limits the temporal scale of the data that can be analysed and synchronised with simultaneously measured data. As mentioned in Section 1, the correlation of flow structures with atmospheric or turbine-measured data is essential for identifying specific synergies in the LiDAR measurements. The approach used in the following space-time conversion is chosen to meet the requirement that a characteristic flow situation be measured several times in sequence. Depending on the flow velocity and the measurement range, this limits the region in which wind-field propagation can be applied. According to the requirement that a flow situation be measured in two or more consecutive scans, we can use the laws of fluid dynamics to convert the spatial wind velocity information collected in a scan into temporally expressed information. This general idea has been used in other studies to interpolate or extrapolate flow situations from measurements or defined states. [PERSON] and [PERSON] (2010) reconstructed instantaneous flow fields from time-resolved volumetric particle tracking velocimetry (PTV) measurements using multidimensional velocity measurements and their derivatives. Their vortex-in-cell plus (VIC+) method relies on the calculation of vorticity, which in turn requires at least 2D velocity information, which cannot be collected with a single LiDAR system. Like the approach we describe below, [PERSON] et al. (2014) used a semi-Lagrangian advection scheme and a stepwise flow solver to estimate the available power during the curtailment of a wind farm by forecasting the flow dynamics and aerodynamic interactions between the turbines. [PERSON] et al. (2015) applied the hypothesis of frozen turbulence [PERSON] et al. (2015) with local topographic corrections to forecast wind speeds at specific downstream positions from LiDAR measurements taken over the very-short term. In this section, we propose a model that describes how the flow evolves between subsequent scans, which we then apply to achieve continuous closure of planar LiDAR data over time at a specific downstream position.