## 3 Apparent Reflectance Error Analysis and Its Predictive Power
### Spectral Distortions in the Apparent Reflectance as Quality Indicator of the Atmospheric Correction  
In Section 3.1, we demonstrated that errors in the retrieved aerosol optical properties lead to specific spectral distortions in the \(\rho_{app(atm)}\). In this section, we explore the idea of using the retrieved \(\rho_{app(atm)}\) spectra to infer errors in the aerosol optical properties, i.e., \(\Delta\)AOT, \(\Delta\alpha\) and \(\Delta g\). In this way, for a given known instrumental response, \(\rho_{app(atm)}\) could be used to refine the characterization of atmospheric parameters. As a proof of concept, a database (DB) containing a set of simulated TOA radiance spectra was created using MODTRAN and SCOPE. At the surface level, five reflectance and SIF spectra were used (three of them shown in Figure 7 and two additional spectra between the mixed vegetation and bare soil cases (b) and (c)). The corresponding LAI and \(Chl\) of the five spectra selected are detailed in Appendix C. For the atmospheric transfer functions simulations, we covered all possible combinations of reference values of the atmospheric parameters AOT = 0.16, \(\alpha\) = [1.39, 1.54, 1.74], \(g\) = [0.75, 0.8], WV = [1.2, 2.4] g/cm\({}^{2}\) and surface elevation \(h\) = [100, 1500] m, assuming the viewing and illumination geometric conditions detailed in Appendix B. The motivation of these reference values is as follows:  
* to select a medium AOT value to evaluate the aerosols effect but avoiding extreme cases [37];
* to range variations around the \(\alpha\) and \(g\) values of 1.54 and 0.8, respectively, which corresponds to typical values found for continental aerosol types [38];
* to simulate the effect of the surface pressure by including two different altitudes;
* to determine if the different WV content impacts the predictive power of the \(\rho_{app(atm)}\) on the O\({}_{2}\)-B absorption region. Figure 9: Apparent reflectance spectral distortion caused by an over-/under-estimation of the Aerosol Optical Thickness (AOT), \(\alpha\) and \(g\) aerosol optical properties for three different surfaces at the O\({}_{2}\)–A absorption band for the FLORIS spectral resolution. The dashed horizontal black line over the \(g\) parameter figures delimits the area, where the \(\rho_{app}\) spectral distortions have been computed by extrapolating \(g\) values higher than 1. For each TOA radiance spectra, the surface apparent reflectance was atmospherically inverted by over-/under-estimating the corresponding reference values of AOT, \(\alpha\) and \(g\) by a factor of 0%, \(\pm\)5%, \(\pm\)15%, \(\pm\)25% and \(\pm\)50% (all possible combinations). Accordingly, the DB comprised a collection of more than \(\sim\)8\(\cdot\)10\({}^{4}\) wrongly corrected \(\rho^{\prime}_{app}(atm)\) spectra and its corresponding wrongly estimated values of AOT\({}^{\prime}\), \(\alpha^{\prime}\) and \(g^{\prime}\). As an extra parameter, the Normalized Difference Vegetation Index (NDVI') was also computed from \(\rho^{\prime}_{app}\) and stored in the DB. The NDVI' values were included to add an extra source of information related to the surface reflectance spectral shape out of the oxygen absorption regions. Therefore, following the multivariate linear regression formulation, \(Y=MX\), the used input values (\(X\)) were: \(\rho^{\prime}_{app}\), AOT\({}^{\prime}\), \(\alpha^{\prime}\), \(g^{\prime}\), and the NDVI' index. The output values (\(Y\)) were the \(\Delta\)AOT, \(\Delta\alpha\) and \(\Delta g\), i.e., the over-/under-estimation of each aerosol parameter in percentage [%]. In order to accelerate the regression process, the spectral DB dimensionality was reduced by means of applying a Principal Component Analysis (PCA) technique, taking the first 50 components. Then, 30% of the DB was randomly selected to train, i.e., to obtain the multivariate linear regression coefficient matrix \(M=(X^{\top}\cdot X)^{-1}\cdot X^{\top}\cdot Y\).