# Small Target Radiometric Performance of Drone-Based Hyperspectral Imaging Systems
## Appendix A  
The following appendix derives the entrance aperture-reaching spectral radiance originating from a point source and defines the radiometric signature of a point target used in field calibration efforts [15]. A point source is defined by light originating from an unresolved object. The natural radiometric quantity that defines a point source is radiant intensity \(I\) and has units of power per solid angle (\(W/sr\)). Figure 14 provides two visualizations of convex mirrors used to create point source targets for remote imaging systems. Under the assumption of plane waves (i.e., originating from an irradiance source) fully illuminating a convex mirror, a virtual image is formed at half the radius of the curvature or at the focal length. The mirror's spectral radiant intensity \(I_{m}(\lambda)\) leaving the surface can be described as  
\[I_{m}(\lambda)=\frac{\rho_{m}(\lambda)E(\lambda)A_{m}}{\Omega_{i}} \tag{1}\]  
where \(\rho_{m}\) is the mirror's specular reflectance, \(E(\lambda)\) is the irradiance striking the mirror's surface, \(A_{m}\) is the mirror's projected area, and \(\Omega_{i}\) is the solid angle of the virtual image form by the mirror. The mirror's geometric angle \(\theta_{m}\)b is defined as the angle from the optical axis to the mirror's edge when viewed from the center of curvature:  
\[\theta_{m}=\sin^{-1}\left(\frac{D_{m}}{2R_{m}}\right) \tag{2}\]  
The mirror's geometric angle is a physical quantity that can be easily measured. The mirror's projected area \(A_{m}\) is a circle defined by the viewable diameter \(D_{m}\):  
\[A_{m}=\pi D_{m}^{2}=\pi(R_{m}\sin\theta_{m})^{2}=\frac{\pi R_{m}^{2}}{2}(1-\cos 2 \theta_{m}) \tag{3}\]  
The solid angle \(\Omega_{i}\) is defined by the virtual image since this is viewed by an imaging system. The focal point of a convex mirror creates an angle with respect to the optical axis that is twice as large as the geometric angle \(\theta_{m}\). The solid angle for a unit spherical cone can be evaluated over the \(2\pi\) azimuthal angle and \(2\theta_{m}\) polar angle:  
\[\Omega_{i}=\int_{0}^{2\pi}d\phi\int_{0}^{2\theta_{m}}\sin\theta\,d\theta=2\pi (1-\cos 2\theta_{m}) \tag{4}\]  
For simplification purposes, it is convenient to rewrite \(A_{m}\) in terms of \(2\theta_{m}\) using the double angle trigonometric identity. This will be apparent when formulating the radiant intensity equation. Before continuing with the derivation of the mirror's spectral radiant intensity, the mirror's field-of-regard \(FOR\) should be discussed. The \(FOR\) defines the total angle from which the virtual image can be viewed. This quantity is used in pre-planning to ensure that the point source can be viewed during an overpass. That is,  
\[FOR=4\theta_{m}=4\sin^{-1}\left(\frac{D_{m}}{2R_{m}}\right)\] (A5)  
The area (\(A_{m}\)) and solid angle (\(\Omega_{i}\)) terms within \(I_{m}\) can be simplified as  
\[I_{m}(\lambda)=\frac{\rho_{m}(\lambda)E(\lambda)A_{m}}{\Omega_{i}}=\rho_{m}( \lambda)E(\lambda)\frac{R_{m}^{2}}{4}\] (A6)  
where the spectral radiant intensity of a convex mirror is dependent on the surface reflectance, radius of curvature, and the irradiance striking the mirror's surface. When mirrors are deployed outside, there are multiple sources of energy striking the mirror's surface, being a combination of both direct solar and diffuse sky irradiance. The solar component is the primary irradiance and has a small angular extent such that all of that energy is reflected. The diffuse sky contribution has a dependence on the solid angle of the virtual image formed at the mirror's focal point.