# Generic Methodology for Field Calibration of Nacelle-Based Wind Lidars
## 6 Uncertainties of Reconstructed Wind Characteristics
### Example for a Two-Beam Nacelle Lidar  
#### 6.2.1 Reconstruction Algorithm  
We here use a simple reconstruction algorithm applied to a two-beam nacelle lidar, see Figure 11. Although the flow model described hereafter is expected to be strongly inadequate for multiple-beam profiling nacelle lidars, the example corresponds to the technique employed by the first generation of \begin{table}
\begin{tabular}{l l c c c c c c c} \hline \hline \multirow{2}{*}{**Lidar**} & \multirow{2}{*}{**LOS**} & \multicolumn{4}{c}{**Calibration Relation**} & \multicolumn{4}{c}{**Expanded Uncertainties (\(k=2\))**} \\ \cline{3-10}  & & \(\theta_{los}\) & \(a\) & \(R^{2}\) & \(Npts\) & \(U\) (m\(\cdot\)s\({}^{-1}\)) & \(U\) at 4 m\(\cdot\)s\({}^{-1}\) & \(U\) at 16 m\(\cdot\)s\({}^{-1}\) \\ \hline \multirow{4}{*}{5B} & LOS 0 & 286.03\({}^{\circ}\) & 1.0058 & 0.9999 & 742 & 0.0171 \(\cdot\)\(V_{LOS}+0.0367\) & 2.72\% & 1.94\% \\  & LOS 1 & 285.99\({}^{\circ}\) & 1.0072 & 0.9999 & 502 & 0.0156 \(\cdot\)\(V_{LOS}+0.0460\) & 2.73\% & 1.84\% \\  & LOS 2 & 285.99\({}^{\circ}\) & 1.0084 & 1.0000 & 1087 & 0.0157 \(\cdot\)\(V_{LOS}+0.0449\) & 2.73\% & 1.85\% \\  & LOS 3 & 286.06\({}^{\circ}\) & 1.0090 & 0.9999 & 446 & 0.0152 \(\cdot\)\(V_{LOS}+0.0476\) & 2.73\% & 1.82\% \\  & LOS 4 & 285.99\({}^{\circ}\) & 1.0059 & 1.0000 & 1508 & 0.0155 \(\cdot\)\(V_{LOS}+0.0457\) & 2.68\% & 1.84\% \\ \hline \multirow{2}{*}{ZDM} & 179\({}^{\circ}\)–181\({}^{\circ}\) & 287.44\({}^{\circ}\) & 1.0050 & 0.9998 & 2140 & 0.0162 \(\cdot\)\(V_{LOS}+0.0437\) & 2.75\% & 1.89\% \\  & azimuth & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of LOS velocity calibration results. commercial systems--such as the two-beam _Avent Wind Iris_ and pair-derived wind characteristics that can be extracted from the _ZephIR Dual Mode_ lidar--hence its relevance. At one specific range, the hypotheses of the flow model are:  
1. two-dimensional wind vector: the vertical component \(V_{z}\) is 0, downstream and transverse components are denoted \(V_{x}\) and \(V_{y}\) respectively;
2. horizontal flow homogeneity: \(V_{x}\) and \(V_{y}\) are independent of the \(y_{h}\) coordinates (see Figure 11) of the beams position;
3. the probe volume averaging is neglected: time-averaged lidar \(V_{LOS}\) are considered as point-like quantities;
4. the lidar roll inclination is 0\({}^{\circ}\): both beams sense winds at the same height. The LOS velocities are given by:  
\[\left\{ \begin{array}{l}V_{r1}=V_{H}\cos\varphi\cos\left(\alpha+\theta_{r} \right)\\ V_{r2}=V_{H}\cos\varphi\cos\left(\alpha-\theta_{r}\right)\end{array} \right. \tag{22}\]  
where \(V_{H}\) is the horizontal wind speed, \(\theta_{r}\) the wind direction relative to the turbine's yaw position, \(\alpha\) the lidar half-opening angle, \(\varphi\) the lidar tilt inclination. The wind vector components are then:  
\[\left\{ \begin{array}{l}V_{x}=V_{H}\cos\left(\theta_{r}\right)\\ V_{y}=V_{H}\sin\left(\theta_{r}\right)\end{array} \right.\equiv\left\{ \begin{array} []{l}V_{x}=\frac{V_{r1}+V_{r2}}{2\cos\varphi\cos\alpha}\\ V_{y}=\frac{-V_{r1}+V_{r2}}{2\cos\varphi\sin\alpha}\end{array} \right. \tag{23}\]  
Finally, the wind speed and relative direction are derived as follows:  
\[\left\{ \begin{array}{l}V_{H}=\sqrt{V_{x}^{2}+V_{y}^{2}}\\ \theta_{r}=\text{atan}\left(\frac{V_{y}}{V_{z}}\right)\end{array} \right. \tag{24}\]  
In normal operations, the turbine yaws according to the wind direction, \(\theta_{r}\) is expected to be close to 0. Hence, for simplicity, we used the arctangent function (atan) instead of its four-quadrant extension (atan2). \(u_{\theta_{r}}\) (Equation (27)) is only the calibration uncertainty of the relative direction. Practically, the alignment of the lidar to the nacelle centreline is an additional source of uncertainty--its magnitude will depend on the mounting procedure - that must be accounted for by quadratically adding it to \(u_{\theta_{r}}\)  
Figure 11: Two-beam nacelle lidar schematics for wind field reconstruction.