## 1 Introduction  
Remote sensing measurement of Solar-Induced chlorophyll Fluorescence (SIF) provides a new optical mean to track plant photosynthesis and gross primary productivity (GPP) of terrestrial ecosystems [1]. SIF consists of photons of red and near infra-red light (650-850 nm) that are emitted by chlorophyll foliar pigments in response to absorption of photosynthetically active radiation. At a satellite level, the SIF signal is around two orders of magnitude lower than the reflected radiance, which makes its detection by remote sensing instruments challenging. However, SIF emission can still be detected by exploiting the fact that SIF is a proportionally larger fraction of the total radiance within dark lines and bands of the atmospheric spectrum [2]. These dark features include both solar Fraunhofer and telluric absorption regions such as the O\({}_{2}\)-B and the O\({}_{2}\)-A bands. Taking advantage of this fact, multiple SIF retrieval strategies have been developed in the last decade [3]. For instance, some SIF retrieval strategies exploit the Fraunhofer solar lines, which are generally weak and narrow but not as influenced by atmospheric effects as the oxygen bands [2; 4; 5; 6; 7; 8]. Other SIF retrieval strategies use the O\({}_{2}\)-A absorption band, which is deeper and wider but is more affected by the atmospheric effects [9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Some other strategies exploit both, the O\({}_{2}\)-A and some Fraunhofer lines in the retrieval scheme [19; 20], and recently, the O\({}_{2}\)-\(\gamma\) band has also been used in combination with some solar Fraunhofer lines to anchor the O\({}_{2}\)-B band and to provide additional information on red SIF [21]. At global scale, first SIF maps were retrieved by the exploitation of the Fraunhofer lines from atmospheric chemistry satellite missions (e.g., [2; 4; 5; 6; 7; 8]). However, since these missions were not designed for vegetation monitoring, SIF was detected with a coarse spatial resolution e.g., \(\sim\)10 km for GOSAT-2 and 2.25 km for OCO-2. Conversely, the future ESA's Fluorescence Explorer (FLEX) mission [22], specifically designed to monitor the health status of terrestrial vegetation, is dedicated to retrieve the full SIF signal from 650-800 nm at a spatial resolution of 300 m. FLEX will particularly exploit the information at the oxygen absorption bands by measuring these regions at a very high spectral resolution, i.e., 0.1 nm of Spectral Sampling Interval (SSI) and 0.3 nm of Spectral Resolution (SR). See Appendix A for more details about the FLuORescence Imaging Spectrometer (FLORIS) on board FLEX. Although measurements in the O\({}_{2}\)-B and the deep O\({}_{2}\)-A absorption bands possess the maximum sensitivity to detect SIF [22], these bands are also highly affected by atmospheric effects, especially by multiple scattering of aerosols and molecules [23]. SIF at satellite level is about 1-2 orders of magnitude (depending on the spectral region) lower than the reflected radiance, which implies that any inaccuracy in the Atmospheric Correction (AC) can rapidly make SIF estimation prone to errors [3]. Thus, given the importance of the AC as a prior step to the application of any SIF retrieval method, e.g., Spectral Fitting (SF) [24], it is important to analyse how errors and assumptions on AC modify the inverted apparent reflectance. Consequently, this paper begins with evaluating the mathematical formulation of the FLEX data processing scheme [22], from the Atmospheric Inversion (AI) and to its coupling with the at-surface level SF method. In the following, the spectral distortions on the apparent reflectance caused by an inaccurate estimation of aerosol optical properties are also evaluated and analysed using simulated data from the Soil Canopy Observation Photochemistry and Energy (SCOPE) [25] and the MODerate resolution atmospheric TRANSmission (MODTRAN) [26] radiative transfer models. As a result, this work proposes the use of the apparent reflectance as quality indicator to refine the AC.