## 2 The AI Process: From TOA Radiance to SIF through the Apparent Reflectance Inversion
### Coupling Spectral Fitting and Apparent Reflectance  
In order to couple consistently the AI with the SF, it is necessary to understand the difference between \(\rho_{app}\) (introduced in Equation (4)) and \(\rho_{app(atm)}\) (after the AI from Equation (8)) [30]. Even in the ideal case of a perfectly characterized atmosphere, the mathematical approximations presented in Section 2.1 introduce discrepancies between these two apparent reflectance expressions (see Figure 4). SF methods such as [17; 24] are generally based on the minimization between: (1) the sensor acquired radiance, measured at surface level; and (2) the simulated radiance, built from \(L_{loc}=\frac{E\rho}{\pi}+F\) with parametric expressions for \(\rho\) and \(F\). However, in the case of airborne or satellite scale, radiance at surface level comes from an AI procedure. Thus, in order to couple consistently AI with SF, two alternatives can be considered (Figure 5):  
* _Option_ 1: To perform the SF method at TOA level, thereby minimizing the difference between the radiance acquired by the spaceborne instrument and the simulated radiance using Equation (4). * _Option_ 2: To perform the SF method at TOC level, thereby minimizing the difference between the atmospherically corrected radiance (or apparent reflectance) and the simulated radiance (or apparent reflectance) accounting for the AI procedure. Note that in Figure 5 the inconsistent coupling process is also described and labelled as Wrong Option 2. In this case, the discrepancy between the simulated, \(\rho_{app}\), and the inverted, \(\rho_{app(atm)}\), shown in Figure 4 would therefore lead to an inconsistent minimization process, and hence to an erroneous SIF estimation. Figure 4: Apparent reflectance \(\rho_{app}\) (black) computed as \(\rho+\pi F/E\), and apparent reflectance \(\rho_{app(atm)}\) (red) from an atmospheric inversion process where all the atmospheric parameters are perfectly known for the O\({}_{2}\)–B (**a**) and O\({}_{2}\)–A (**b**) spectral regions. HR \(\rho_{app}\), at 0.1 cm\({}^{-1}\), is also shown as a thin grey solid line. MODTRAN inputs parameters used in the simulation and the values used for the spectral convolution are detailed in Appendix B. Here, we focussed on analysing and assessing the coupling process performed at TOC. While both options are valid (_Option_ 1 and _Option_ 2), coupling at TOC implies the retrieval of the \(\rho_{app(atm)}\) as an intermediate product, which has the additional advantage that it can be used as a quality indicator of the atmospheric correction process (see Section 3.2). Therefore, we evaluated the impact of coupling consistently (_Option_ 2) and inconsistently (_Wrong Option_ 2, from Figure 5) the AI process and the SF method into the retrieved SIF in the O\({}_{2}\)-A absorption region. This evaluation was conducted in different wavelength intervals labelled as R1, R2, R3 and R4 corresponding to (759.3-765) nm, (759.3-768) nm, (759.3-770) nm and (759.3-772) nm, according to [24]. In addition, a range of atmospheric aerosol load was simulated to test its influence under both coupling processes. Surface reflectance and SIF spectra, \(\rho\) and \(F\), were modelled as quadratic functions. The first guess of the \(\rho\) polynomial coefficients was estimated by fitting the \(\rho_{app}\) with a quadratic function in the spectral region around the O\({}_{2}\)-A but avoiding the absorption band. The first guess of the \(F\) polynomial coefficients was estimated by fitting the reference \(F\) spectrum to a quadratic function. Figure 6 shows the relative error between the retrieved and the reference SIF spectrum evaluated at the bottom of the O\({}_{2}\)-A absorption band for:  
* An inconsistent coupling process, i.e., modelling \(\rho_{app}\) at TOC as \(\rho_{app}=\rho+\frac{\pi\cdot F}{E}\). The inconsistent coupling causes a large relative error between the retrieved and the reference SIF values for all of the minimization wavelength intervals considered (R1-R4).