# Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach
## 4 Discussion
This aspect has been shown by the determination of the second statistical moment by lidar data using an analytical model [(23)]. The optimum angle of a four point VAD method was estimated with an analytical solution to \(\phi=54.7356^{\circ}\)[(30)]. This correlates very well with the result in this study. The condition number \(\kappa\) of the \(\mathbf{A}\) solution matrix illustrates this. The \(\kappa\) describes the sensitivity of the solution to changes in the data. This means that in worst case, the input error is amplified by this \(\kappa\). The condition number \(\kappa\) of the different solutions tested here is shown in Table 3. The largest \(\kappa\) occurs in the _CLS_ procedure for the SWE Scanner with a \(\phi=15^{\circ}\). The smallest \(\kappa\) is found using the _CLS_ method with a \(\phi=55^{\circ}\). The ratio between largest and smallest \(\kappa\) (4.6) is not directly reflected in difference between the results, but the ability of \(\kappa\) to identify improved scan geometry for the same analysis methods is confirmed by the results. The question that naturally arises is, how representative are measurements with a \(\phi=55^{\circ}\) at a measuring height of 100 m? After all, the radius of the circle for this configuration is 76.5 m. This question cannot be answered without further investigation. For load simulations of wind turbines according to IEC 61400-01 [(31)], coherent wind fields are created using the exponential coherence model. This model depends on the average wind speed, distance of the observed points and the considered frequency. The validity of the coherence model is related to structures in the wind fields and the wind evolution over time and space, and should be investigated in future studies. Another aspect that needs to be addressed here is the influence of the different scan durations on the measurement results. An overview of the detailed scan duration for the different trajectories is listed in Table 3. The SWE Scanner measures 5.57 times more data in 10 min than the Galion. This should lead to a reduction of the random error. This can be seen to a lesser extent in the comparison of Figures 8a and 9b which measured with a very similar \(\phi\). The characteristic of the different trajectories are listed in Table 3. The differences between these two half opening angles are marginal (the offset differs by 0.074 m\(\cdot\)s\({}^{-1}\), the slope by 0.074 and \(R^{2}\) by 0.019). At the second position the scan duration was reduced by a factor of 1.84, but there are no noticeable improvements. The averaged error over the wind speed range between 0 m\(\cdot\)s\({}^{-1}\) and 20 m\(\cdot\)s\({}^{-1}\) is for _CLS_ at the first position 1.034 m\(\cdot\)s\({}^{-1}\) and at the second position is a bit less with 1.00 m\(\cdot\)s\({}^{-1}\). Unfortunately, the question of the influence of the scan duration cannot be answered on the basis of the results. It is quite clear that the random error is reduced by a higher data availability, but other effects (e.g., opening angle) have a greater influence. Conducting \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**SWE Scanner**} & \multicolumn{6}{c}{**Galion**} \\ \cline{2-7}  & **VAD 5+1** & **VAD 18** & \(\boldsymbol{\phi_{20}}\) & \(\boldsymbol{\phi_{39.2}}\) & \(\boldsymbol{\phi_{55}}\) & _linear_ \\ \hline \(\phi\) [\({}^{\circ}\)] & 15 & 15 & 20, 39.2, 55 & 20 & 39.2 & 55 & 20, 39.2, 55 \\ scan duration [m\(\cdot\)s\({}^{-1}\)] & 8.8 & 4.78 & 49 & 16.3 & 16.3 & 16.3 & 49 \\ \(\kappa\) [-] & 5.78 & 5.78 & 2.02 & 4.30 & 2.00 & 1.26 & 120.66 \\ Nb. of Points [-] & 6 & 6 & 18 & 6 & 6 & 6 & 18 \\ Location [-] & 1 & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics for the different configurations. additional experiments in flat terrain with homogeneous flow conditions would be a possible solution there to answer the question on the scan duration. Further investigations are also necessary into the suitability of the _linear_ model.