# Calibration of Mean Wind Profiles Using Wind Lidar Measurements
## 3 Problem Statement  
Assuming that only Lidar measurements are available, and according to the technical specifications so far guaranteed by such devices, the lowest measurement point was assumed at 30 m above the ground, and a maximum of 12 measurement points was considered. As a first example, twelve elevations z = 30, 40, 50, 60, 70, 80, 100, 120, 140, 160, 180, 200 m are considered here, denoted as measurement setup #0 in Table 2.  
By means of the least-square error minimization procedure described in Section 2, for the set of pseudo-experimental profiles corresponding to the sample cases of Table 1, the parameters \(z_{0}\) and \(z_{d}\) of the logarithmic law were identified for measurement setup #0. The scatter of the identified parameters with respect to the target pair \(z_{0}\) and \(z_{d}\) used to generate the pseudo-experimental profiles is shown in Figure 2, where the identified parameters of each profile correspond to a green point on the \(z_{0}\)-\(z_{d}\) plane, and a black square indicates their average values, while the red circle indicates their target values. A very large scatter can be observed among the pairs of parameters \(z_{0}\) and \(z_{d}\) identified from the different pseudo-experimental profiles belonging to the same case. However, their average values (black squares) are quite in agreement with the target values of \(z_{0}\) and \(z_{d}\) for Case (d), and with the target value of \(z_{d}\) value for Case (c). These two cases correspond to the larger values of \(z_{d}\). On the other hand, the average values are rather away from the target values for Case (a) and Case (b), corresponding to lower values of \(z_{d}\). According to these results, the quality of the mean identified parameters very much depends on the elevation of the first available measurement point. This explains the poor identification of the logarithmic law parameters (and in particular the zero-plane displacement \(z_{d}\)) when \begin{table}
\begin{tabular}{c c} \hline
**Setup** & **Elevation of Measurement Points [m]** \\ \hline \#0 & z = 30, 40, 50, 60, 70, 80, 100, 120, 140, 160, 180, 200 \\ \#1 & z = 30, 35, 40, 50, 60, 70, 80, 100, 120, 140, 170, 200 \\ \#2 & z = 30, 35, 40, 45, 50, 60, 70, 80, 110, 140, 170, 200 \\ \#3 & z = 30, 35, 40, 45, 50, 55, 60, 70, 90, 120, 160, 200 \\ \#4 & z = 30, 32, 34, 37, 40, 45, 50, 55, 60, 70, 120, 200 \\ \hline \end{tabular}
\end{table} 
Table 2: Elevation of measurement points for different setup. Figure 2: Identified parameters \(z_{0}\) and \(z_{d}\) (green points) and their average values (black squares), and parent values (red circles) used to generate the pseudo-experimental profiles: Case (a) \(z_{0}\) = 0.05 m, \(z_{d}\) = 5 m; Case (b) \(z_{0}\) = 1.5 m, \(z_{d}\) = 5 m); Case (c) \(z_{0}\) = 0.05 m, \(z_{d}\) = 20 m); Case (d) \(z_{0}\) = 1.5 m, \(z_{d}\) = 20 m). Diamonds in diagram (a) indicate the identified pairs \(z_{0}\) and \(z_{d}\) of profiles represented in Figure 3.  
the lowest measurement point (at 30 m) is far from the zero-speed elevation \(z_{0}+z_{d}\) of the wind profile, as for the Case (a) and Case (b) in Figure 2.  
Although the pairs reported in Figure 2 are quite scattered, the corresponding interpolated profiles are very close to each other in the range of measured data, i.e., at elevations above 30 m. On the other hand, the interpolated profiles are very scattered in their extrapolation at lower elevations. Sample of trends are shown in Figure 3 for Case (a) of Table 1; two interpolated profiles derived from two sets of pseudo-experimental points are shown, corresponding to the red and black diamonds in Figure 2a, respectively; they prove to be different from each other and far from the target profile at low elevations. All the profiles are scaled to a velocity value equal to 12 m/s at the elevation of 50 m.  
To assess the statistical significance of the ensemble averages \(\mu_{z0}\) (\(N\)) and \(\mu_{zd}\) (\(N\)) of the identified profile parameters for different values of the sample size \(N\), the following procedure is applied.