# Small Target Radiometric Performance of Drone-Based Hyperspectral Imaging Systems
## Appendix A
The fractional amount of diffuse sky \(f_{sky}\) reflected to an imaging system can be described by taking the ratio of \(\Omega_{i}\) and solid angle of the hemisphere (i.e., \(2\pi\) sr):  
\[f_{sky}=\frac{\Omega_{i}}{2\pi}=1-\cos 2\theta_{m}\] (A7)  
The spectral radiant intensity of a convex mirror deployed outside under nominal field conditions (see Figure A1b) can be described by the following equation:  
\[I_{m}(\lambda) = {}^{1}\!/4\rho_{m}(\lambda)R_{m}^{2}\!\left[E_{solar}(\lambda)+f_ {sky}E_{sky}(\lambda)\right]\] (A8) \[= {}^{1}\!/4\rho_{m}(\lambda)R_{m}^{2}\!\left[1+f_{sky}\frac{E_{sky} (\lambda)}{E_{solar}(\lambda)}\right]\!E_{solar}(\lambda)\]  
where the irradiance is split into two components, the solar and sky contributions. The sky irradiance is modified by the fractional amount of sky irradiance reflected to the imaging system. If \(\theta_{m}=45\) degrees, the convex mirror reflects the entire hemisphere (i.e., \(f_{sky}=1\)). In Section 2.2, the total downwelling irradiance (solar and sky) and the sky irradiance are the only direct measurements using a field spectroradiometer device configured with a cosine corrector. The ratio of the sky and sun's contributions to the total downwelling irradiance, \(G(\lambda)\) and \(1-G(\lambda)\), respectively, can be expressed by the following equations:  
\[G(\lambda)=\frac{E_{sky}(\lambda)}{E_{T}(\lambda)},\qquad 1-G(\lambda)=\frac{E_{solar }(\lambda)}{E_{T}(\lambda)}\] (A9)  
where both equations can be used to further simplify Equation (A8) with two field measurements, the solar irradiance and the Global-to-Diffuse Ratio. The following equation now defines the spectral radiant intensity of a convex mirror under nominal daylight conditions that accounts for both solar and sky irradiances:  
\[\theta_{m}=\sin^{-1}\left(\frac{D_{m}}{2R_{m}}\right) \tag{2}\]0 (A10)  
For imaging systems viewing the point source at altitudes much greater than the focal length of the convex mirror, the spectral radiant intensity can be propagated to distance \(H\) to the imaging system. Assuming isotropic behavior, the entrance aperture-reaching spectral irradiance \(E_{EAR}(\lambda)\) is described by  
\[\theta_{m}=\sin^{-1}\left(\frac{D_{m}}{2R_{m}}\right) \tag{2}\]1  
where the inverse-square law defines the relationship between the spectral radiant intensity and entrance aperture-reaching spectral irradiance. For remote sensing systems recording a pixel-level spectral radiance, it is convenient to describe the mirror's signal in the form of entrance aperture-reaching spectral radiance \(L_{EAR}(\lambda)\) by incorporating the sensor's solid angle:  
\[\theta_{m}=\sin^{-1}\left(\frac{D_{m}}{2R_{m}}\right) \tag{2}\]2  
The factor \(H^{2}\,\Omega_{sensor}\) in the denominator of \(L_{EAR}(\lambda)\) can be rewritten in terms of the directional ground sampling distance (\(GSD\)) of the imaging system. The \(GSD\) is the area of a pixel projected onto the ground and is a standard parameter for Earth observing imaging systems:  
\[\theta_{m}=\sin^{-1}\left(\frac{D_{m}}{2R_{m}}\right) \tag{2}\]3  
where \(IFOV\) and \(p\) define the directional components to the instantaneous field-of-viewand pixel pitch, respectively, and \(f\) is the imaging systems focal length. The entrance aperture-reaching spectral radiance can be generalized for any remote sensing system by incorporating the upwelling atmospheric transmission (from target to aperture) and path radiance, including adjacency effects. The equation assumes that the solar irradiance is measured at the Earth's surface such that downwelling atmospheric transmission is incorporated. Thus, we have,  
\[\theta_{m}=\sin^{-1}\left(\frac{D_{m}}{2R_{m}}\right) \tag{2}\]4  
The equation used in this experiment is a modified form of Equation (14) for drone-based experiments. Due to the altitude (i.e., less than 400 ft or 121.92 m), the upwelling atmospheric transmission \(\tau_{\uparrow}(\lambda)\) is negligible [3].