## 2 The AI Process: From TOA Radiance to SIF through the Apparent Reflectance Inversion
### Assessment of Mathematical Approximations
Figure 2: Reference reflectance (\(\rho\)), fluorescence (\(F\)), and apparent reflectance (\(\rho_{app}\)) spectra marked as black dashed line, red solid line and black solid line, respectively. High resolution (HR) \(\rho_{app}\), at 0.1 cm\({}^{-1}\), is also shown as a thin grey solid line. The MODerate resolution atmospheric TRANSmission (MODTRAN) inputs parameters used in the simulation and values used for the spectral convolution are detailed in Appendix B. Reflectance and Solar-Induced Chlorophyll FLuorescence (SIF) spectra are mean reference values defined in an internal FLEX technical note [28]. An additional factor to consider is that the radiance acquired by an instrument, \(L_{sen}\), is spatially and spectrally convolved by its instrumental response function. Consequently, the isolation of \(\rho_{app}\) term from Equation (6) would imply the convolution of each atmospheric transfer function independently, which would derive a wrong apparent reflectance estimation:  
\[\rho_{app}\
eq\frac{L_{sen}-\langle L_{0}\rangle}{\langle\frac{E.T^{\dagger}}{ \pi}\rangle+(L_{sen}-\langle L_{0}\rangle)\langle S\rangle}. \tag{7}\]  
This inequality exists because the convolution of the product of two given functions is not mathematically equivalent to the product of these functions convolved, i.e., \(\langle a\cdot b\rangle\
eq\langle a\rangle\cdot\langle b\rangle\). This fact becomes especially relevant when working with non-smooth functions, such as the atmospheric transfer functions within the absorption regions and at high spectral resolution. In order to derive \(\rho_{app}\) and thereby avoid errors associated with the convolution of each individual atmospheric transfer function, we can take the original series expansion in Equation (8). As such, the high spectral resolution atmospheric transfer functions provided by a radiative transfer model (e.g., MODTRAN [26], 6S [29]) can now be multiplied and then convolved with the corresponding Instrument Spectral Response Function (ISRF):  
\[L_{sen}=\langle L_{0}\rangle+\langle\frac{E}{\pi}\cdot T^{\dagger}\rangle\cdot \rho_{app}+\langle\frac{E}{\pi}\cdot S\cdot T^{\dagger}\rangle\cdot\rho_{app} ^{2}+\langle\frac{E}{\pi}\cdot S^{2}\cdot T^{\dagger}\rangle\cdot\rho_{app}^{ 3}+\ldots \tag{8}\]  
Hence, it is necessary to evaluate until which order the series expansion in Equation (8) must be taken so that the errors in the inverted apparent reflectance are below a required threshold. In this case, the threshold has been defined as the relative difference in \(\rho_{app}\) caused by increasing the SIF emission 10% of its value. Figure 3c,d evaluates the relative difference (%) on TOA radiance between taking the series expansion expression (Equation (8)) until the 2 nd and 3 rd order and its convergence (Equation (6)) in the O\({}_{2}\)-B and the O\({}_{2}\)-A regions, respectively. As can be observed, the error derived from the 2 nd order approximation is one order of magnitude lower than the impact caused by the defined threshold. Consequently, FLEX AI inverts \(\rho_{app}\) by resolving a second order equation:  
\[E=E_{dir}\cdot\cos(\theta)+E_{dif}. \tag{2}\]0 Relative difference on Top-Of-Atmosphere (TOA) radiance (%) between Equations (4) and (6) for a range of aerosol loads in the O\({}_{2}\)–B (**a**) and the O\({}_{2}\)–A (**b**) spectral regions, i.e. \(100\cdot(L_{sen(E_{d})}-L_{sen(E_{d},6)})/L_{sen(E_{d},6)}\). Relative difference on TOA radiance (%) between using the series expansion (2 nd or 3 rd order) and the TOA radiance expression (Equation (6)) in the O\({}_{2}\)–B (**c**) and the O\({}_{2}\)–A (**d**) spectral regions, i.e. \(100\cdot(L_{sen(2 nd/3 rd)}-L_{sen(E_{d},6)})/L_{sen(E_{d},6)}\).