## 4 Results
### Influence of the Interpolation Time Step, \(\Delta\)t, on the Statistical Error  
As we discussed in Section 3.3, the number of propagation steps (II) between two measured scans influences the temporal accuracy of the mapping of a scan interpolated at a certain time (\(t\)). If \(\Pi\) is too low, the temporal changes in the flow structures are not resolved sufficiently, which increases the temporal error. Since our interpolation method employs a simplistic approach to represent fluid dynamics, the propagation will make the representation diverge somewhat from a realistic physical picture of the flow field. As more steps are interpolated between two scans, the temporal aspect of the reproduction of the flow between each scan will improve. At the same time, the reconstruction error will increase. Since some inaccuracy will arise in the speed of advection between pairs of scans, the position of the flow structures will be shifted somewhat. To investigate the effects of different interpolation steps (\(\Delta t\)) and to determine the parameter configuration that has the lowest statistical errors, we have evaluated different values of \(\Pi\) for each scanning speed, \(\omega_{\phi,\theta}\). The corresponding results for RHI scans are presented in Appendix D.  
Applying the definition of the averaged error (\(\varepsilon_{\overline{n}}\)) in Equation (22) and the standard-deviation error, \(\varepsilon_{\omega_{\phi}}\), in Equation (23) to the temporally improved wind fields with different combinations of \(\omega_{\phi}\) and \(\Pi_{\phi}\) (respectively, \(\omega_{\theta}\) and \(\Pi_{\theta}\)), Figure 12 shows the resulting three error cases that were reduced above:  
* Inside the wake for \(-1\leq\gamma D^{-1}\leq 1\) and \(-1\leq\zeta D^{-1}\leq 1\). * Along the centreline for \(\gamma D^{-1}=0\) and \(\zeta D^{-1}=0\). * In the free flow. The errors are shown in colour, and the iso-lines represent the power of the ratio of the error of the propagated data to the error of the original LiDAR data, \(\eta_{\overline{n}}=\frac{\varepsilon_{propated}}{\varepsilon_{\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n },\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{ n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n}, \overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n}, \overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n},\overline{n}, \overline{n},\overline{n},\overline{n},\overline{n},\overline{n}  
The results in Figure 12 show that an optimal interpolation time step exists for each angular velocity. If we compare these results with those in Figure 9, we see that a single optimal angular velocity yields the smallest mapping errors for a set of wake speed and atmospheric conditions. The resulting \(\eta_{\overline{u}}\) and \(\eta_{e_{u}}\) are plotted as iso-lines in Figure 12 to show which value of \(\Pi_{\phi}\) most improves the temporal resolution and minimises the absolute error. These values show the exponent of the decimal expression of \(\eta_{\overline{u}}\) and \(\eta_{e_{u}}\). The zero line is a significant feature as it indicates the boundary at which the combination of \(\omega_{\phi}\) and \(\Pi_{\phi}\) shifts from reducing to amplifying the mapping error. In all three regions, both the average wind-speed error and standard-deviation error are reduced over the entire range of angular velocities. The most significant improvements are achieved at scanning speeds slower than the optimum, but they do not reflect the absolute smallest error. The results in Figure 12 show that for the test case we considered, the error is reduced to the minimum of \(\eta_{\overline{u}}\) = 30.5% of the original \(\epsilon_{\overline{u}}\) and to \(\eta_{e_{u}}\) = 42% of the original \(\epsilon_{\overline{u}}\).