# Calibration of Mean Wind Profiles Using Wind Lidar Measurements
## 2 Methodology
To explore to what extent Wind Lidars can be used to calibrate mean wind profiles, four scenarios are selected corresponding to different pairs of nominal parameters \(z_{0}\) and \(z_{d}\), representative of different site conditions, as reported in Table 1 and Figure 1. In particular, the four scenarios are defined as such to represent the boundaries of a realistic range of variation of the pair of parameters \(z_{0}\) and \(z_{d}\). Case (a) is representative of an urban area densely covered by low-rise buildings of uniform height. Case (b) is representative of a quite densely built-up area with low-rise buildings having rather heterogeneous height. Case (c) is representative of a densely built-up area with medium-rise buildings of uniform height. Case (d) is representative of a densely built-up area with medium-rise buildings having rather heterogeneous height. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & \(z_{0}\) **[m]** & \(z_{d}\) **[m]** \\ \hline a & 0.05 & 5 \\ b & 1.50 & 5 \\ c & 0.05 & 20 \\ d & 1.50 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Pairs of \(z_{0}\) and \(z_{d}\) parameters used in the simulations. For each sample case, a set of 10,000 pseudo-experimental profiles was generated by adding to the ideal (or \"exact\") logarithmic profile a zero-mean Gaussian noise, characterized at each elevation by a standard deviation equal to 2% of the logarithmic wind speed at that elevation, as representative of realistic measurement errors. The amount of noise added to the profiles was chosen based on the measure error of available Wind Lidars. For each pseudo-experimental profile, the parameters \(z_{0}\) and \(z_{d}\) were identified by means of a least-square error minimization procedure, as follows. Identification of the logarithmic law parameters was first performed by assuming that only Lidar measurements are available, with the lowest measurement point at 30 m above the ground; results are discussed in Section 3. As a first step, identification of the optimum value of the friction velocity \(u^{*}\) can be performed solving a three parameters (\(u^{*},z_{0}\) and \(z_{d}\)) optimization problem applied to the median profile of a set of Wind Lidar profiles measured for a given wind sector. Then, each profile measured in that sector can be scaled to the identified value of \(u^{*}\) and considered as a sample profile. Finally, calibration of the optimum pairs of values \(z_{0}\) and \(z_{d}\) can be performed on the set of scaled measured profiles. In this study, pseudo-experimental profiles were generated by adding a zero-mean Gaussian noise to the same parent logarithmic profile with a given \(u^{*}\), and each of them can be assumed as a single measured profile already scaled to the value \(u^{*}\). For each scaled profile, the parameters \(z_{0}\) and \(z_{d}\) are identified as those minimizing the least-square error between the vector of values \(\overline{u_{i}}\) of the scaled wind speed at the \(n\) available elevations \(z_{i}\) and their logarithmic counterpart \((1/\kappa)ln[(z_{i}-z_{d})/z_{0}]\). Then, the mean values and the coefficients of variation of the pairs of values \(z_{0}\) and \(z_{d}\) identified for different sets of pseudo-experimental profiles are calculated and discussed. Different combinations and elevations of the Lidar measurement points will be analysed and discussed in Section 4.1. Finally, the availability of a further point at a lower elevation, provided by, e.g., a sonic or a cup anemometer, will be considered and discussed in Section 4.2.