## 4 Results
### Biophysical Variables Retrieval  
GPR models were developed for the variables Cab, Cw and LAI using the _retrieval_ training data with noise added for nine bands at 20 m (without B8). These models were then validated against simulated _reference_ data for: (1) the TOC scale, (2) TOA reference data, and (3) TOC dataset with noises according to Sen2 Cor atmospheric correction errors. At the TOC scale, it was no surprise that retrievals against the TOC reference dataset led to excellent validation results with for Cab and Cw an R\({}^{2}\) of 0.94-0.97 (Table 5). LAI was poorer validated, with a R\({}^{2}\) of 0.68, due to saturation for higher LAI values, i.e., above 3. This suggests that the LAI model is suboptimally trained. What is more of importance in the context of this study is that results only degrade slightly when upscaling to TOA data, thus with the inclusion of atmospheric variables in the LUT. Excellent results are again obtained for Cab and Cw (R\({}^{2}\) of 0.91-0.95), while poorer yet consistent results are achieved for LAI (R\({}^{2}\) of 0.62). When moving back to the TOC scale, but now with adding noise levels according to Sen2 Cor atmospheric correction errors, the results tend to degrade further. Given that this exercise is conducted with simulated data, the latter scenario is considered closer to reality. Comparison of these results may suggest that retrieving biophysical variables directly from TOA radiance data can be more beneficial, however, that is yet to be evaluated when applying to real data. Figure 5: Total sensitivity (\(S_{Tl}\)) results of TOA radiance using a GPR emulator of a 12 variables PROSAIL-MODTRAN model. See Tables 2 and 3 for the full names of the variables. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Retrieval:** & **TOC** & **TOA** & **TOC-ATM** \\ \hline \multicolumn{4}{c}{**R\({}^{2}\):**} \\ - Cab & 0.972 & 0.948 & 0.907 \\ - Cw & 0.942 & 0.908 & 0.813 \\ - LAI & 0.684 & 0.623 & 0.520 \\ \multicolumn{4}{c}{**RMSE:**} \\ - Cab & 3.312 & 4.586 & 6.077 \\ - Cw & 0.003 & 0.004 & 0.006 \\ - LAI & 1.120 & 1.223 & 1.381 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Retrieval performance results against 5’000 LUT _reference_ datasets for biophysical variables retrieval from surface reflectance (TOC), TOA radiance (TOA) and surface reflectance with noise levels after atmospheric correction (ATM). For the TOC and TOA _retrieval_ datasets 2% Gaussian noise was added, while for the TOC-ATM _retrieval_ datasets noises are added according to [50]. An easy way to gain insight into the functioning of the GPR models at TOC and TOA scale is by means of inspecting the sigmas (\(\sigma_{b}\)), i.e., the band relevance, of the trained GPR models. They have been plotted in Figure 6 for the three biophysical variables. The lower the \(\sigma_{b}\) is, the more important the band. Accordingly, the \(\sigma_{b}\) reveal the most important wavelengths with spectral information used for the development of the models. The following observations can be made:  
* Overall, no systematic differences between TOC and TOA \(\sigma_{b}\) can be observed. About the same patterns appeared with low \(\sigma_{b}\) for the majority of bands. This suggests that models can be developed from both TOC and TOA data sources with about the same degree of retrieval success. * A closer inspection towards Cab and LAI reveals that TOA data led to considerably higher \(\sigma_{b}\) for some bands (i.e., 490, 783 and 865 nm for Cab; 490 and 740 nm for Cw). This suggests that for these variables the TOA data has more difficulty to develop the retrieval algorithms. Conversely, the \(\sigma_{b}\) similarity between the TOC and TOA bands for the LAI models suggests that the role of atmosphere is of marginal importance for LAI retrieval. * For all variables, the band in the blue is evaluated as poorly contributing, both for TOC and TOA. For TOA this can be explained by the influence of aerosols, while at TOC scale this may be rather due to the remaining impact of the aerosols in the atmospheric correction.