# Reducing the Uncertainty of Lidar Measurements in Complex Terrain Using a Linear Model Approach
## 4 Discussion  
The projection of the met mast _USA_ data onto the normal vector of the lidar showed a directional dependence. This is in line with observations that the terrain significantly influences the inflow over the _Albrtauf_(16; 21). CFD analysis (16) does not give a value to quantify the difference and, the CFD analysis assumes neutral stratification. If stratification effects are added as in reality, this could explain the discrepancies between the lidar and the _USA_. The difference in error between the met mast and wind field reconstruction using the _CLS_ and _PRESS_ methods is shown in Figure 11. The results are visualized using the frequency of \(\left|v_{cup}-v_{CLS}\right|-\left|v_{cup}-v_{PRESS}\right|\). Consider a case where the reference or 'true' wind speed is 10 m\(\cdot\)s\({}^{-1}\), and the _CLS_ result is 9.7 m\(\cdot\)s\({}^{-1}\) while the _PRESS_ result is 10.2 m\(\cdot\)s\({}^{-1}\). In this case, the _PRESS_ method has led to a better wind field reconstruction, and the result is 0.3 m\(\cdot\)s\({}^{-1}-\) 0.2 m\(\cdot\)s\({}^{-1}\) = 0.1 m\(\cdot\)s\({}^{-1}\). In this study of 3820 data points, the _CLS_ method resulted in a lower error in 91.1% of the measurement periods. One reason for this is, that the additional measuring point of the _CLS_ method reduces the reconstruction error compared to the _PRESS_ algorithm; the reconstruction error for the wind components is proportional to \(1/\sqrt{N}\) where \(N\) as number of laser beams (30). This is also surprising as it was hoped initially that the _PRESS_ method would be more robust than the _CLS_ method because of the rejection of outliers. In the previous section, the correlation between the presented methods applied to lidar measurements and the horizontal wind speeds of in-situ sensors mounted on a met mast was presented. From those results it can be observed that a suitable measurement strategy can improve the agreement between the lidar and met mast. In this study, using the _linear_ model the agreement with the met mast was considerably improved. The results varies in the magnitude of the CFD simulation of \(\approx\)2% and the offset is nearly zero with 0.03 m\(\cdot\)s\({}^{-1}\). In addition, the width of the 5-95 th quantile is significantly reduced in average of 26.96%. In summary, it is a clear improvement using the _linear_ method compared to the _CLS_ method. Due to the extended _linear_ model, the horizontal wind speed in complex flow (at least at this location) can be determined much more precisely than with the standard VAD method. Whether this model delivers useful results in general for complex locations is still to be verified by further tests. For this location,  
Figure 11: The error in the wind field reconstruction using the _CLS_ method, versus the error due to the _PRESS_ method. Negative values indicate that the _CLS_ method had lower error compared to the met mast than the _PRESS_ for a particular 10-minute period. the correlation is closer to unity and the offset is nearly zero. Furthermore, there are also optimization possibilities for this model in the trajectory configuration as well as in the selection of the angles \(\phi\) and \(\theta\). If this result is related to the distributions of \(R^{2}\) (Figure 10a), then the influence of the used model is clearly visible. For the _PRESS_ method, the \(R^{2}\) distribution shows a very concentrated form with over 75% of the data with over \(R^{2}\geq 0.95\). The results of the correlation (Figure 8b) show, however, the worst matches with the measuring mast data. This has the highest offset, the highest slope and the smallest \(R^{2}\), as well as the widest confidence interval. The results in Figure 10a shows a very good agreement between the data and the used model, however Figure 8b shows that the used model does not match the met mast measurements. These results show that increasing the half opening angle \(\phi\) leads to better correlations between the lidar and mast. The reason for this is that the deterministic error is reduced.