# Generic Methodology for Field Calibration of Nacelle-Based Wind Lidars
## 6 Uncertainties of Reconstructed Wind Characteristics
### Example for a Two-Beam Nacelle Lidar
#### 6.2.3 Uncertainty Budget, Results and Scale Analysis  
In this paragraph, LOS velocities \(V_{r1}\) and \(V_{r2}\) are'simulated' according to Equation (22). All results are obtained with a half-opening angle \(\alpha=15^{\circ}\)--typical of commercial systems--and a tilt \(\varphi=-1^{\circ}\). At a coverage factor \(k=1\), the uncertainty budget is:  
1. LOS velocity: taking advantage of the previously observed linearity (see Section 5.6.5), the LOS velocity uncertainty of beam \(i\) is estimated to \(u_{i}=m\cdot V_{ri}+n\) where \(m=0.008\) and \(n=0.0225\). We here obtained \(m\) and \(n\) by approximating the gain and offset value of the expanded uncertainty linear relation (see Table 1 ) and dividing it by the coverage factor;
2. Tilt inclination: \(u_{\varphi}=0.05^{\circ}\) as prescribed by the inclinometers calibration;
3. Opening angle: from the geometry verification, we estimate \(u_{\alpha}=0.1^{\circ}\). In the GUM methodology, the correlation between uncertainty components may have a large impact on the combined uncertainty. The variability of uncertainty results depending on the correlation between \(u_{1}\) and \(u_{2}\) is shown in Figure 12. In the full correlation case, both expanded uncertainties \(U_{V_{x}}\) and \(U_{V_{ri}}\) are \(\sim\)50% higher than in the uncorrelated one. The effect is even more critical for \(U_{V_{y}}\) and \(U_{\theta_{r}}\): the calibration uncertainty is almost zero for \(r_{1,2}=1\). Note that an extra uncertainty quantifying the wind model inadequacy should be added to the calibration uncertainty. In practice, the LOS velocity calibration will probably be performed for both beams at the same site, with the same reference instruments. Moreover, the reference instruments contribute for \(>\)90% to the total \(V_{LOS}\) uncertainty (7.1). \(u_{1}\) and \(u_{2}\) are most likely highly and positively correlated. In the next paragraphs, we arbitrarily choose \(r_{1,2}=0.9\) and \(r_{xy}=0.2\), later shown to be negligible. Figure 13 displays expanded uncertainties as a function of wind speed (with \(\theta_{r}=5^{\circ}\)) and direction (with \(V_{H}=10\) m\(\cdot\)s\({}^{-1}\)). \(U_{V_{H}}\) varies linearly with \(V_{H}\) and is approximately of same magnitude as \(U_{V_{LOS}}\)  
The wind direction uncertainty \(U_{\theta_{r}}\) decreases with the wind speed. The relative wind direction has little impact on \(U_{V_{H}}\) and \(U_{\theta_{r}}\), as long as it stays within a reasonable range. Seeking to identify which terms of Equations (26) and (27) are dominant, a scale analysis is conducted (Table 2) with \(V_{H}\sim 10\) m\(\cdot\)s\({}^{-1}\) and \(\theta_{r}\sim 10^{-1}\)rad \(\approx 5^{\circ}\). Concerning \(u_{V_{x}}\) and \(u_{V_{y}}\), the tilt and opening angle uncertainties are negligible, irrespective of \(r_{1,2}\). Uncertainties \(u_{V_{H}}\) and \(u_{\theta_{r}}\) are governed respectively by the uncertainty on the downstream and transverse component. The correlation term can be neglected irrespective of the value of \(r_{xy}\). Equations (26) and (27) can thus be approximated by:  
\[\left\{ \begin{array}{l}u_{V_{x}}^{2}\approx\frac{u_{1}^{2}+u_{2}^{2}+2r_{1,2 }u_{2}}{(2\cos\varphi\cos a)^{2}}\quad\text{and}\quad\left\{\begin{array}{l}u _{V_{H}}^{2}\approx\frac{V_{x}^{2}u_{V_{x}}^{2}}{V_{x}^{2}}\\ u_{V_{y}}^{2}\approx\frac{V_{x}^{2}u_{V_{y}}^{2}}{V_{x}^{2}}\end{array}\right. \end{array} \right. \tag{28}\]