# An Inter-Comparison Study of Multi- and DBS Lidar Measurements in Complex Terrain
## 3 Results and Discussion
### Radial Velocity Components
#### 3.1.2 Second-Order Statistics
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **MA** & **EE** & **SE** & **SW** \\ \hline No 10 min & & & & \\ periods & 503 & 2592 & 1419 & 1278 \\ \hline  & & **Mean**\(\overline{\upsilon_{\mathit{r}}}\) & & \\ \hline \(m\) & 1.045 & 0.992 & 1.005 & 0.993 \\ \(b\) (m\(\cdot\)s\({}^{-1}\)) & 0.056 & \(-\)0.036 & 0.018 & 0.055 \\ \(R^{2}\) & 0.858 & 0.999 & 0.998 & 1.000 \\ \(RMSD\) (m\(\cdot\)s\({}^{-1}\)) & 0.087 & 0.104 & 0.116 & 0.079 \\ \hline  & & **Variance**\(\overline{\
u^{\prime 2}_{\mathit{r}}}\) & & \\ \hline \(m\) & 0.970 & 0.829 & 0.836 & 0.819 \\ \(b\) (m\(\cdot\)s\({}^{-1}\)) & 0.017 & \(-\)0.025 & \(-\)0.018 & 0.001 \\ \(R^{2}\) & 0.969 & 0.968 & 0.952 & 0.970 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the inter-comparison of the radial velocities of the WindScanners and the sonic anemometer at 188 m; RMSD indicates the root-mean-square deviation; \(R^{2}\) is the goodness of the fit, \(m\) is the slope and \(b\) the intersect of the of the linear regression. (Equation (7)) deviate from the observed shape and the attenuation only starts at wavenumbers higher than theoretically predicted for the SE and SW systems. Figure 4: _Cont._  
Figure 3: Variance of the radial velocity measurements of the WindScanners \(\left(\overline{v^{\prime 2}}_{\tau\;\mathit{lidar}}^{2}\right)\) against the respective variance in the direction of the beam derived from the sonic anemometer \(\left(\overline{v^{\prime 2}}_{\tau\;\mathit{sonic}}\right)\) when the wind direction was close to parallel (\(\pm 15^{\circ}\)) to the azimuth angle of the WindScanner for (**a**) SE; (**b**) SW; (**c**) EE; grey diamonds indicate the unfiltered sonic measurements; blue crosses are the sonic data after the application of (10). The sonic data was filtered with \(\left|\xi\left(k_{1}\right)\right|^{2}\) to separate out the effects of the temporal averaging. Up until now, only a few experimental studies investigating \(\overline{\
u^{2}_{r}}_{\mathit{lidar}}\) and its spectrum against reference measurements have been undertaken. Most interesting in regards to pulsed lidars, [PERSON] and co-authors [14] reported a good agreement between the observed and theoretical forms of \(T_{\
u_{r}}\). However, in contrast to this study, they only used five selected periods and the lidars were set up in the direct vicinity of the reference mast. As this leads to large elevation angles, they had to use a spectral tensor model to derive \(T_{\
u_{r}}\). Also, they used a simpler, triangular-shaped weighting function. The influence of measurement noise at the high-wave-number end of the spectra has also been reported for continuous wave lidars [36; 37] and deserves further investigation for the WindScanner systems. [PERSON] et al. [36] found similar deviations between \(T_{\
u_{r}}\) from theoretical considerations and \(T_{\
u_{r}}\) computed from lidar and sonic spectra for a continuous wave lidar. As in the present investigation the reason for the deviation from theoretical considerations remains unclear. Nevertheless, Figure 4e indicates that the concept of line averaging (9 and 10) can be used to explain some of the differences in the observed second-order statistics between the sonic and the WindScanners.