# Small Target Radiometric Performance of Drone-Based Hyperspectral Imaging Systems
## Appendix B  
The following appendix outlines the methodology used to derive the uncertainty for Equation (15). The uncertainty derivation involves taking first-order partial derivativeswith respect to all the variables. Equation (A15) requires the substitution of the mirror's geometric angle \(\theta_{m}\). Equation (A16) defines the main function for uncertainty propagation:  
\[L_{EAR}(\lambda)=\
icefrac{{1}}{{4}}\varphi_{m}(\lambda)R_{m}^{2}\bigg{[}1-G( \lambda)\cos\bigg{(}2\sin^{-1}\bigg{(}\frac{D_{m}}{2R_{m}}\bigg{)}\bigg{)} \bigg{]}\frac{E_{T}(\lambda)}{GSD^{2}}\] (A16)  
The uncertainty propagation follows the Guide to Uncertainty Measurement framework [25], where the combined uncertainty for a generalized function is \(y=f(x_{1},x_{2},...,x_{N})\). Each variable is a mean estimate \(\overline{x_{i}}\) with an associated uncertainty \(u(x_{i})\). The generalized form of the combined uncertainty \(u_{c}^{2}(y)\) with a covariance \(u(x_{i},x_{j})\) is associated to each pair of correlated variables:  
\[u_{c}^{2}(y)=\sum_{i=1}^{N}\bigg{(}\frac{\partial f}{\partial x_{i}}\bigg{)}^{ 2}u^{2}(x_{i})+2\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\frac{\partial f}{\partial x_ {i}}\frac{\partial f}{\partial x_{j}}u(x_{i},x_{j})\] (A17)  
For the uncertainty analysis performed for this investigation, the generalized combined uncertainty equation is reduced by assuming all variables are uncorrelated. Equation (A18) defines the uncertainty propagation for uncorrelated variables. The partial derivatives \(\partial f/\partial x_{i}\) define the sensitivity of \(y\) to small changes in each variable:  
\[u_{c}^{2}(y)=\sum_{i=1}^{N}\bigg{(}\frac{\partial f}{\partial x_{i}}\bigg{)}^{ 2}u^{2}(x_{i})\] (A18)  
It should be understood that Equation (A18) has limitations and approximation to the uncertainty. In the assessment of the combined uncertainty, \(u_{c}(y)\), the first-order partial derivative assumes linear behavior over the mean estimate of each variable. Any significant deviation from linearity requires either the addition of high-order terms or a Monte Carlo approach to estimate the combined uncertainty. The following equations outline the partial derivatives of each variable with Equation (A16). The equations were derived by using the Python symbolic mathematics library SymPy [27]:  
\[\frac{\partial L_{EAR}}{\partial\rho_{m}}=\
icefrac{{1}}{{4}}R_{m}^{2}\bigg{[} 1-G(\lambda)\cos\bigg{(}2\sin^{-1}\bigg{(}\frac{D_{m}}{2R_{m}}\bigg{)}\bigg{)} \bigg{]}\frac{E_{T}(\lambda)}{GSD^{2}}\] (A19)  
\[\frac{\partial L_{EAR}}{\partial R_{m}}=\
icefrac{{1}}{{4}} \varphi_{m}(\lambda)\Bigg{[}2R_{m}\bigg{(}1-G(\lambda)\cos\bigg{(}2\sin^{-1} \bigg{(}\frac{D_{m}}{2R_{m}}\bigg{)}\bigg{)}\bigg{)}\] (A20) \[-\frac{D_{m}G(\lambda)\sin\bigg{(}2\sin^{-1}\bigg{(}\frac{D_{m}}{ 2R_{m}}\bigg{)}\bigg{)}}{\sqrt{1-\bigg{(}\frac{D_{m}}{2R_{m}}\bigg{)}^{2}}} \Bigg{]}\frac{E_{T}(\lambda)}{GSD^{2}}\] (A21)  
\[\frac{\partial L_{EAR}}{\partial D_{m}}=\
icefrac{{1}}{{4}}\varphi_{m}(\lambda )R_{m}\Bigg{[}\frac{G(\lambda)\sin\bigg{(}2\sin^{-1}\bigg{(}\frac{D_{m}}{2R_{ m}}\bigg{)}\bigg{)}}{\sqrt{1-\bigg{(}\frac{D_{m}}{2R_{m}}\bigg{)}^{2}}} \Bigg{]}\frac{E_{T}(\lambda)}{GSD^{2}}\] (A22)  
\[\frac{\partial L_{EAR}}{\partial G}=\
icefrac{{-1}}{{4}}\varphi_{m}(\lambda )R_{m}^{2}\cos\bigg{(}2\sin^{-1}\bigg{(}\frac{D_{m}}{2R_{m}}\bigg{)}\bigg{)} \frac{E_{T}(\lambda)}{GSD^{2}}\] (A23)  
\[\frac{\partial L_{EAR}}{\partial E_{T}}=\
icefrac{{1}}{{4}}\varphi_{m}(\lambda )R_{m}^{2}\bigg{[}1-G(\lambda)\cos\bigg{(}2\sin^{-1}\bigg{(}\frac{D_{m}}{2R_{ m}}\bigg{)}\bigg{)}\bigg{]}\frac{1}{GSD^{2}}\] (A24)\[\frac{\partial L_{EAR}}{\partial GSD}=-\
icefrac{{1}}{{2}}\rho_{m}(\lambda)R_{m}^{2} \bigg{[}1-G(\lambda)\cos\bigg{(}2\sin^{-1}\bigg{(}\frac{D_{m}}{2R_{m}}\bigg{)} \bigg{)}\bigg{]}\frac{E_{T}(\lambda)}{GSD^{3}} \tag{100}\]  
The combined uncertainty is expressed by evaluating the above sensitivity coefficients at the mean estimates for all variables.