## 2 Planar LiDAR Data
### Synthetic LiDAR Data
(2017) to calculate velocity data from an LES wind field. In total, 2334 synthetic scans representing 11 different angular velocities were simulated within the same 10-min interval of the LES. As with the physical LiDAR measurements, the resulting velocity data are given in radial coordinates. The limitations and peculiarities of numerically simulated LiDAR measurements have been investigated before. [PERSON] et al. (2017) studied the errors that affect a simulated dual-Doppler LiDAR system. They found that the error in the determination of the radial velocity consists of a random error due to measurement inaccuracies caused by the speckle effect and detector noise, a systematic error due to the frequency shift of the laser, non-linear amplifiers, digitising errors and non-ideal noise statistics, and direction errors due to the imperfect adjustment of the LiDAR system and/or scanner movement. Together, these errors cause a projection error, like that described in Equation (14). [PERSON] et al. (2017) used simulated data to investigate the ability of dual-Doppler LiDAR systems to estimate turbulence length scales. The use of a LiDAR simulator in the present study is meant to represent an ideal LiDAR which, apart from volume averaging, does not consider any other interference, as described by [PERSON] et al. (2017). This ideal simulation emphasises the peculiarities of the space-time conversion method discussed below. Within the LIXIM software, we used the weighted average of the LOS velocity over the sample volume we considered. To this end, we defined a linear coordinate (\(s\)) in the beam direction and varied its orientation as shown in Figure 2 with different azimuth/elevation angles. The linear coordinate, \(s\), represents the radial distance from the LiDAR to the measurement point. The range gate length corresponding to the spatial extension (\(\Delta p=36\) m) and the Gaussian-formed laser pulse with intensity, \(I_{p}\), characterised by a full-width at half-maximum (\(\Delta r\)) of 30 m [36] were used to calculate the LOS velocity:  
\[I_{p}(s)=\frac{1}{\sqrt{\pi}\Delta r}e^{(-\frac{s^{2}}{\Delta r^{2}})}. \tag{11}\]  
The estimated LOS velocity is therefore:  
Figure 2:0  
where:  
Figure 2:1  
with \(I=\frac{I_{p}(s)}{\int_{-\infty}^{\infty}I_{p}(s)ds}\) and \(k_{p}=2.56\). We are aware that the pulse intensity of a real LiDAR differs from the Gaussian representation in Equation (11), but [PERSON] used a similar Gaussian distribution [24]. [PERSON] et al. [34] and [PERSON] et al. [35] used a similar weighting function. In other studies, the pulse shape was modelled with other approaches. [PERSON] et al. [37] used an axisymmetric function that, according to [PERSON] [38], reasonably approximates the distribution of the pulses emitted by the WindCube LiDAR system. [PERSON] et al. [39] built on [PERSON] et al. [37] for the pulse-weighting of a single LiDAR, but they used a 3D Gaussian function to evaluate three synchronised LiDAR measurements. The choice of the laser pulse function directly affects the calculation of the LOS velocity. As can be deduced from Equation (11), the radial velocity component is a weighted average of the pulse geometry. This averaging in the beam direction affects the resulting velocity, as it is affected by the velocity shear within the pulse geometry. In a wind field with constant laminar wind speed, volume averaging would have no effect. The stronger and faster the velocity shear within the probe volume, however, the greater the error in the accumulated speed. This influences the representation of the mean value and the standard deviation, which will thereby be under-estimated in relation to the reference. Depending on the pulse shape that is assumed, volume averaging will have different effects. As we used it, LIXIM was only able to calculate the spatial average in the radial direction, but not in the scanning direction. This provided deeper insight into step-and-stare measurements, in which the scanner stops and accumulates data for each measurement.