## 2 The AI Process: From TOA Radiance to SIF through the Apparent Reflectance Inversion  
### Assessment of Mathematical Approximations  
The radiance reaching the sensor at satellite level can be simplified as the contribution of three different fluxes (Figure 1) [27]:  
1. The scattered light from the atmosphere or path radiance (thick dashed-dotted line). 2. The reflected light from the observed target which is transmitted to the sensor (solid line). 3. The light coming from multiple reflections between the surface and the atmosphere, not necessarily produced at the observed target, but finally reaching the sensor (thin dashed line). Assuming a Lambertian surface reflectance \(\rho\), the radiance reaching the sensor (\(L_{sen}\)) can consequently be formulated as the following (Equation (1)):  
\[L_{sen}=L_{0}+\frac{E}{\pi}\cdot T^{\dagger}\cdot\rho+\frac{E}{\pi}\cdot S \cdot T^{\dagger}\cdot\rho^{2}+\frac{E}{\pi}\cdot S^{2}\cdot T^{\dagger}\cdot \rho^{3}+..., \tag{1}\]  
where \(L_{0}\) is the path radiance (scattered light from the atmosphere), \(E\) is the total solar irradiance reaching the surface, \(T^{\dagger}\) is the total upward transmittance and \(S\) is the atmospheric spherical albedo, which accounts for the reflections occurred from the atmosphere to the surface. Likewise, \(E\) can be understood as the contribution of the solar irradiance diffusely transmitted to the surface (\(E_{dif}\)) and the solar irradiance directly transmitted to the surface (\(E_{dir}\)), corrected by the solar zenith angle (\(\theta\)):  
\[E=E_{dir}\cdot\cos(\theta)+E_{dif}. \tag{2}\]  
Equation (1) is mathematically known as a geometrical series expression, which converges into Equation (3):  
\[L_{sen}=L_{0}+\frac{E\cdot\rho\cdot T^{\dagger}}{\pi(1-S\cdot\rho)}. \tag{3}\]  
Furthermore, in the particular case of vegetation, the SIF emission (\(F\) in the formulation for brevity) can be included in Equation (3) as an additional source of energy. Since SIF is orders of magnitude lower than the TOA radiance reaching the sensor, it can be physically considered as a small perturbation of Equation (3):  
\[L_{sen}=L_{0}+\frac{(\frac{E}{\pi}\cdot\rho+F\cdot)T^{\dagger}}{(1-S\cdot\rho )}=L_{0}+\frac{E\cdot\rho_{app}T^{\dagger}}{\pi(1-S\cdot\rho)}, \tag{4}\]  
Figure 1: Contribution of distinct fluxes acquired at satellite level assuming a Lambertian surface reflectance behaviour. The thick black area at the Top-of-Canopy (TOC) level represents the observed target. where \(\rho_{app}\) is the so-called apparent reflectance defined as Equation (5), which accounts for the reflectance and the SIF emission normalized by the solar irradiance at surface level (Figure 2):  
\[\rho_{app}=\rho+\frac{\pi\cdot F}{E}. \tag{5}\]  
However, the impossibility of retrieving the reflectance (\(\rho\)) and the apparent reflectance (\(\rho_{app}\)) terms simultaneously from Equation (4) should be noted. To overcome this limitation, FLEX AI assumes that \(S\cdot\rho\approx S\cdot\rho_{app}\), which leads to the following Equation (6):  
\[L_{sen}=L_{0}+\frac{E\cdot\rho_{app}T^{\uparrow}}{\pi(1-S\cdot\rho_{app})}. \tag{6}\]  
Thus, assuming the atmospheric transfer functions (\(L_{0}\), \(S\), \(E\) and \(T^{\uparrow}\)) as known, it becomes now possible to invert the apparent reflectance from the radiance acquired by a sensor, \(L_{sen}\). This approximation, however, comes with an associated error, which is illustrated in Figure 3a,b. Here, the relative difference on TOA radiance (%) between using Equations (4) and (6) is evaluated for the reference fluorescence and reflectance spectra used in Figure 2 and various atmospheric conditions (low to high aerosol load). As it is demonstrated, the impact of this approximation on TOA radiance is one order of magnitude lower than the impact caused by an increase of 10% in the SIF emission. The selection of such a threshold for comparison, 10% of the emitted SIF, follows the criteria defined by the FLEX mission [22].