## 3 Apparent Reflectance Error Analysis and Its Predictive Power  
### Spectral Error Analysis on Apparent Reflectance  
This section analyses the spectral distortions in \(\rho_{app(atm)}\) caused by an inaccurate AC, fundamentally due to errors in the estimation of aerosol optical properties. We emphasized the aerosols' optical properties since aerosol characterization can be considered as one of the main sources of uncertainty when retrieving SIF from space [13]. However, it must be noted that distortions in \(\rho_{app(atm)}\) can also be caused by other factors such as: an inaccurate instrumental spectral characterization [31], instrumental radiometric calibration errors or an inaccurate surface pressure estimation. While the characterization of other key atmospheric parameters such as the Water Vapour (WV) or the ozone (O\({}_{3}\)) are also crucial in any AC algorithm [32; 33; 34], their radiometric effect can be neglected in the O\({}_{2}\) absorption regions, whereas the aerosol radiometric effect cannot. Thus, we studied the spectral distortions derived from an inaccurate estimation of:  
* the AOT, which is directly related to the total aerosol load in a vertical atmospheric column, becoming a greater aerosol load the higher the AOT value. * the Angstrom exponent (\(\alpha\)) [35], which accounts for the AOT spectral variation and is associated with the aerosol size, becoming a higher (\(\alpha\)) exponent with a smaller aerosol size. * the asymmetry parameter (\(g\)) of the Henyey-Greenstein (HG) scattering phase function [36], which indicates the anisotropy of the scattering pattern, this parameter being limited to \([-1,1]\)  
Figure 6: Relative error (\(\epsilon_{r}\)) between retrieved and reference SIF values at the bottom of the O\({}_{2}\)–A absorption band, 760.7 nm, covering different values of AOT at 550 nm and wavelength intervals (R1–R4) for an inconsistent coupling process **(a)** and a consistent coupling process **(b)**. The following error statistics are provided: median (horizontal red line), 25 th and 75 th percentiles (blue boxes) and extreme min/max values (black dashed lines). The \(g\) parameter takes the value \(-1\) and \(1\) for a full backward and forward scattering respectively, while taking the value \(0\) for an isotropic scattering pattern. These three possible causes of distortions are evaluated over three different surfaces types (see Figure 7): (a) a bare soil, (b) a mixture of vegetation and bare soil and (c) a dense vegetation. We selected this variety of spectra to ascertain if errors derived from the AC produced a systematic error in the surface apparent reflectance, \(\rho_{app(atm)}\), regardless of the nature and shape of the underlying surface. This section thus performs an assessment of how an erroneous estimation of aerosol optical properties leads to spectral distortion on the \(\rho_{app(atm)}\). In particular, we want to quantify (a) how the spectral distortion pattern depends on the evaluated aerosol optical variable, and more specifically, (b) if these distortions can be totally disentangled from the underlying surface reflectance and SIF emission. To account for that, TOA radiance spectra were simulated following Equation (4) using the three surface reflectance and SIF emission spectra from Figure 7, and assuming a reference atmospheric state defined by AOT = 0.16, \(\alpha\) = 1.39 and \(g\) = 0.75. Afterwards, by using Equation (9), the AI was performed by overestimating and underestimating the AOT, the \(\alpha\) and the \(g\) values used as a reference in the simulation. In Figures 8 and 9, the over-/under-estimation is represented as a percentage of the initial values on the \(y\)-axis, while the differences in \(\rho_{app(atm)}\) are shown in a color scale.