# Small Target Radiometric Performance of Drone-Based Hyperspectral Imaging Systems
## 3 Methodology
### Data Processing and Analysis  
For each mirror, we predict the entrance aperture-reaching spectral radiance, using Equation (3), which is then compared to the collected HS, radiance calibrated, imagery. The uncertainty of Equation (3) can be derived and evaluated by following the Guide to Uncertainty Measurement framework [25]. This requires taking the partial derivative of Equation (3) with respect to all variables assuming small variations about the mean and negligible non-linear behavior. When all the variables have an estimated absolute uncertainty, the combined uncertainty for the entrance aperture-reaching spectral radiance from a point target can be estimated. The derivation of uncertainties, related to Equation (3), can be found in Appendix B. Table 1 provides an example of the maximum relative uncertainty of all input variables as well as the predicted entrance aperture-reaching spectral radiance. Figure 4: The total averaged downwelling irradiance (**a**) and averaged sky irradiance (**b**) were measured during the experiment at 1 s intervals. The black curves are variation estimates of both signals over the 8 min experiment. The predicted entrance aperture-reaching spectral radiance (Figure 6a) and its respective uncertainty (Figure 6b) are plotted for each mirror. Mirror-to-mirror variations are less than 4% across the VNIR spectrum. It was determined that the mirror reflectance and radius of curvature were the main contributors to this difference. In this investigation, when comparing two different mirrors, we should not observe differences in the radiometric signal greater than the mirror-to-mirror variation. The assumed 2% uncertainty in the radiometric calibration of the ASD field spectroradiometer results in the uncertainty resembling the point target signal. The relative combined uncertainty of the spectral radiance predictions is around 8% as seen in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{u(\rho_{m})}\) & \(\mathbf{u(R_{m})}\) & \(\mathbf{u(D_{m})}\) & \(\mathbf{u(G)}\) & \(\mathbf{u(E_{T})}\) & \(\mathbf{u(GSD)}\) & \(\mathbf{u_{c}(L_{EAR})}\) \\ \hline \(\mathbf{\rho_{m}}\) & \(\mathbf{R_{m}}\) & \(\mathbf{D_{m}}\) & \(\mathbf{G}\) & \(\mathbf{E_{T}}\) & \(\mathbf{GSD}\) & \(\mathbf{L_{EAR}}\) \\ \hline
3\% & 2\% & 2\% & 2.06\% & 2.05\% & 3\% & 8.01\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of relative uncertainties for all input variables, left of vertical bar, along with the estimated relative combined uncertainty (i.e., predicted entrance aperture-reaching spectral radiance), right of vertical bar. Figure 5: The total downwelling irradiance and the Global-to-Diffuse Ratio are used to make an estimate of the point target spectral radiance. The corresponding uncertainties are shown as vertical error bars for each plot. Figure 6: The predicted entrance aperture-reaching spectral radiance (**a**) and the combined uncertainty (**b**) for all mirrors examined in this experiment. The point target entrance aperture-reaching spectral radiance represents the energy reflected to the sensor. The Permanflect panels shown in Figure 7 demonstrate two key concepts about the experimental conditions, consistency and repeatability. The consistency and repeatability of the spectral radiance reflecting off the panels from the 18 overpasses demonstrate the stability of the atmosphere and the HS instrument. It is important that the stability of the atmosphere is well understood with measurements from the ASD field spectroradiometer and reflected signals off the Lambertian targets. The measurement variation observed in Figure 4 (i.e., black curve) and the tight signal variations reflected off the Lambertian targets over the 18 overpasses (Figure 7) demonstrate this stability. It is critical that the HS instrument and atmosphere are stable during this investigation so as to eliminate these factors that could potentially contribute to additional variation in the predicted entrance aperture-reaching spectral radiance of the point targets.