## 2 Planar LiDAR Data  
LiDAR systems are subject to limitations inherent to their principle of operation. Single systems can only measure the so-called line-of-sight (LOS) velocity along the direction of the laser beam, which is calculated from the light backscattered from aerosols [22]. A major source of inaccuracy is the spatial resolution of LiDAR, which depends mainly on the pulse length. When using a pulsed laser in commercial devices, the spatial quantisation of the wind speed in the radial direction is in the range of approximately 30 m [23]. This quantisation limit leads to volume averaging within an assumed cylindrical probe volume. This characteristic of LiDAR measurements has already been studied for its effects in turbulence measurement during the early days of LiDAR wind measurements [23; 24; 25]. Depending on the mode of operation, a second volume-averaging effect can build up in the scanning direction while recording planar measurements. During free flow, the associated error is relatively small [26]. In strongly turbulent flow situations that are very inhomogeneous, such as wind-turbine wakes, stronger deviations arise [26]. To understand the errors in this case, [PERSON] and [PERSON] [27] studied the errors in scanned measurements caused by heterogeneity in temporally averaged wake measurements. A further restriction on LiDAR technology, which also applies to all scanning measurements systems, is that the spatial and temporal resolution of the measurements is in a trade-off relationship. The slower the selected scan speed, the finer the spatial resolution and the longer the scan-repetition time. Within a scan, the velocities measured at each scan angle correspond to different time stamps within a certain interval. Even though a whole scan may be visualised as a single picture or frame, such an illustration is not physically based and may be misleading. To avoid this effect, planar LiDAR data is usually averaged under the assumption that the time shift within a scan is negligible when compared with the averaging time. A serious obstacle to using LiDAR for measurements of wind farms lies in the limited temporal resolution, or rather in the ideal combination of temporal and spatial resolutions that cannot yet be achieved with LiDAR systems. When using a very slow scanning speed, flow measurements are not recorded frequently enough to give a sufficient representation of the fluctuating behaviour, which is why the errors are temporal in nature. When using a very fast scan, the quantisation of the flow is too coarse. In this case, small-scale flow structures cannot be mapped with sufficient accuracy, which leads to errors of a spatial nature. Between these extremes lies an optimal parameter set that can clearly depict spatial structures and temporal fluctuations in the flow. A statistical approach can be used to determine the optimal parameters, and will guarantee that the recorded data is sufficient:  
\[N_{m}\geq\Big{(}\frac{z\,\sigma_{m}}{e}\Big{)}^{2}, \tag{1}\]  
where \(N_{m}\) is the required number of scans, z is the confidence interval, \(e\) is the maximum tolerable error, and \(\sigma_{m}\) is the standard deviation of the measurements [28]. The scan resolution or measurement frequency can be derived from the number of scans required to provide a statistical estimation of the mean value. The basic problem here is that the requisite standard deviation (\(\sigma_{m}\)) is unknown. In practice, this value can be calculated from additional measurements taken by instruments, like sonic or cup anemometers, or it can be estimated from previous LiDAR measurements to select appropriate LiDAR measurement parameters, like the angular-scan resolution. The theoretical consideration of an optimal parameter set clarifies that many planar LiDAR measurements are recorded sub-optimally and may be statistically under-determined. Furthermore, for continuously optimal measurements, the measurement resolution will need to be adapted continuously to the variability in the flow.