## Appendix 1 Deduction of the Mean Footprint Reflectance (HCRF)
### Sensor Response  
A light flux (power) radiated from a small area \(dA\) to another small area \(dA^{\prime}\) is given by [PERSON]'s Cosine Law [55]:  
\[d\,\Phi_{r}=\frac{L_{\theta}\cdot dA\cdot\cos(\theta_{r})\cdot dA^{\prime} \cdot\cos(\theta^{\prime})}{r^{2}}, \tag{1}\]  
where \(L_{\theta}\) [umol\(\cdot\)s\({}^{-1}\cdot\)m\({}^{-2}\cdot\)sr\({}^{-1}\)] is the radiance from \(dA\) with an angle \(\theta_{r}\) to the normal of \(dA\), falling onto the area \(dA^{\prime}\) with an angle \(\theta^{\prime}\) to the normal of \(dA^{\prime}\), and \(r\) is the distance between the two small areas (Figure 1). The projected solid angle subtended by \(dA\) is denoted by \(d\omega^{\prime}\) and the solid angle subtended by \(dA^{\prime}\) is denoted by \(d\omega\), _i.e._, \(d\omega=(dA^{\prime}\cdot\cos\,(\theta^{\prime}\,))/r^{2}\), and \(d\omega^{\prime}=(dA\cdot\cos\,(\theta_{r}))/r^{2}\), then:  
\[d\,\Phi_{r}=L_{\theta}\cdot dA\cdot\cos(\theta_{r})\,d\omega=L_{\theta}\cdot dA ^{\prime}\cdot\cos(\theta^{\prime})\cdot d\omega^{\prime}. \tag{2}\]  
Figure 1: Schematic diagram of a light beam from the ground small area \(dA\) to the sensor aperture area \(dA^{\prime}\).  
The measurement given by a light sensor, _i.e._, the sensor response, is the total integrated flux of \(d\,\varPhi_{r}\) over the footprint area \(A\) and the sensor aperture \(A^{\prime}\)[52]. From the aspect of sensor, the sensor response is:  
\[\varPhi_{r}=\int_{A^{\prime}}\int_{A}\ d\,\varPhi_{r}=\int_{A^{\prime}}\int_{ \Omega_{A}}L_{\theta}\cdot dA^{\prime}\cdot\cos(\theta^{\prime})\,d\omega^{ \prime}. \tag{10}\]  
Note that the integration over the footprint area \(A\) is the same as over the solid angle \(\Omega_{A}\) subtended by the footprint \(A\), or in other words, subtended by the sensor FOV, depending on what variable is used in the integrand. We also assume that the sensor is a simple aperture, with equal response to radiation at any point across its surface and from any direction [53], _i.e._, a cosine-corrected sensor. The mean radiance of the footprint measured by such a sensor is given by the definition of radiance:  
\[\bar{L}_{r}=\frac{\varPhi_{r}}{\int_{A^{\prime}}\int_{\Omega_{A}}d\Lambda^{ \prime}\cdot\cos(\theta^{\prime})\cdot d\omega^{\prime}}=\frac{\int_{A^{\prime }}\int_{\Omega_{A}}L_{\theta}\cdot dA^{\prime}\cdot\cos(\theta^{\prime}) \cdot d\omega^{\prime}}{\int_{A^{\prime}}\int_{\Omega_{A}}d\Lambda^{\prime} \cdot\cos(\theta^{\prime})\cdot d\omega^{\prime}}. \tag{11}\]  
Since \(dA^{\prime}\) and \(d\omega^{\prime}\) are independent of each other, and the projected area \(\cos(\theta^{\prime})\cdot dA^{\prime}\) is independent of the other variables in the equation, we take out the projected area and get:  
\[\bar{L}_{r}=\frac{\int_{\Omega_{A}}L_{\theta}\cdot d\omega^{\prime}}{\int_{ \Omega_{A}}d\omega^{\prime}}. \tag{12}\]  
The projected solid angle is denoted by \(\Omega\), and:  
\[\Omega=\int_{\Omega_{A}}d\omega^{\prime}=\int_{A}\ \ \frac{\cos(\theta_{r})}{r^{2}} \cdot dA\,, \tag{13}\]  
where \(dA\) can be further expressed as \(dA=dxdy\) in a Cartesian coordinate system and \((x,y)\) is a point within the footprint ellipse.