## 4 Discussion
### Suitability of the Mathematical Approximations Assumed in the FLEX Ai  
Over the last few years, various SIF retrieval strategies have been developed based on the exploitation of different spectral regions, the solar Fraunhofer lines [2; 4; 5; 6; 7; 8] or the telluric oxygen absorption bands [9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Alternatively, remote sensing SIF retrieval strategies can also be categorized according to those which retrieve SIF directly from TOA level, e.g., [2; 4; 5], or those which retrieve SIF from TOC after the AC of the signal, e.g., [9; 10; 17; 18]. The latter are the so-called two-step approximations and involve: (1) performing the AC, and (2) coupling an SF method to estimate SIF at surface level. Two-step approaches have been used in the past to estimate SIF from airborne data [39; 40], and also a two-step strategy has been proposed in the context of the forthcoming FLEX/Sentinel-3 tandem mission [17; 22]. Although there are many advantages about the use of this approximation,e.g., the derivation of the \(\rho_{app(atm)}\) as an intermediate product to be analysed (discussed in Section 4.2), it requires an accurate AI procedure.  
To start with, when it comes to inverting non-smooth high spectral resolution functions, such as the atmospheric transfer functions with their characteristic absorption regions, an important aspect implies dealing with the convolution to the sensor instrumental spectral response function. On the one hand, all the assumptions considered for the AI procedure of the FLEX mission were evaluated. Although in general terms it follows the classical full physics atmospheric inversion procedures ([PERSON] _et al._, 2010; [PERSON], 2011; [PERSON], 2012), FLEX AI introduces two main approximations: (1) considering \(S\cdot\rho\approx S\cdot\rho_{app}\), and (2) using the second order series expansion. Assuming both, the independent convolution of each of the atmospheric transfer functions is avoided, which would otherwise rapidly derive into errors in the estimated SIF. Results suggest that the impact of these two approximations is at least one order of magnitude lower than the impact on TOA radiance caused by varying 10% of the SIF signal. This implies that FLEX AI formulation becomes suitable for fluorescence retrieval.  
On the other hand, at the surface level, apparent reflectance (\(\rho_{app}\)) must be disentangled from emitted SIF and reflectance. As it was earlier noted by [PERSON] et al. (2010), \(\rho_{app}=\rho+\frac{\pi\cdot F}{E}\) and \(\rho_{app(atm)}\) from the AI are substantially different, particularly in the absorption regions. This bears consequences to the SF method. In essence, SF strategies are based on minimizing the difference between simulated and measured spectra to decouple \(\rho\) and SIF. In the case of spaceborne or even airborne scale, measured data at the surface level is always derived from an AI procedure. Thus, in order to consistently couple the atmospherically inverted and the simulated signal; the formulation used should be consistent in both processes. Conversely, if a different formulation was used in the simulation and in the AI, then the SF method would minimize the differences between two inconsistently defined spectra. As demonstrated in Section 2.2, this would lead to errors depending on the atmospheric transfer functions, i.e., depending on the atmospheric state.  
To close this section, we should remark that reported error values are estimated under the assumption of a Lambertian surface without considering instrument noise. Evaluating the formulation under ideal conditions allows any other source of uncertainty to be discarded. Therefore, the errors identified here must be understood as an error baseline, intrinsic to the AI and to the coupling processes.