## 3 Results
### Single Band Regression
#### 3.2.1 Analysis of Errors  
Validation metrics are computed across all experiments and reported in Tables 3 and 4 for L1C and L2A data, respectively. The tables present the root mean square error (RMSE), the mean absolute error (MAE), the relative error (RE) and the coefficient of determination (\(R^{2}\)). The rows of the tables are sorted by increasing values of RE for L1C and RMSE for L2A. In Section 1.2, we concluded that 3% error for L1C and 0.01 in surface reflectance values for L2A were good targets for band reconstruction. Of course, we are measuring reconstruction errors using data which itself may have errors, even if they are below the radiometric specifications. Therefore, the error bounds need not to be taken very strictly. Finally, Sentinel-2 can be considered to be over-specified in terms of radiometric quality for most applications, which makes using these error bands rather conservative from our point of view. We see that for L1C only the reconstruction of B7 has an RE lower than 3%, although the other red edge and NIR bands are below 3.8%. For L2A, B5, B6 and B7 have an RMSE lower than 0.01, and B8A is only slightly above this level. Estimating the noise in surface reflectances using the RMSE can suffer from strong outliers. The MAE gives a measure that is robust to these cases and shows that even B12 could be considered for reconstruction. The error values presented in Tables 3 and 4 are averages over the validation samples and do not show the proportion of pixels that do not fulfill the radiometric requirements. For this purpose, Tables 5 and 6 show the percentage of pixels whose error is lower than a given threshold. Table 5 presents, for each L2A band, the percentage of pixels whose error is larger than a given threshold (from 0.01, which is the accuracy of the L2A processor, up to 0.025). We see that even for the best-predicted bands (in the red edge), less than 90% of the pixels fulfill the requirements. However, lowering the requirement accuracy to 0.015, 95% compliance is achieved for these three bands. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Band** & **RMSE** & **MAE** & **RE** & \(\mathbf{R^{2}}\) \\ \hline B5 & \(7.33\times 10^{-3}\) & \(4.96\times 10^{-3}\) & \(2.07\times 10^{-1}\) & \(9.95\times 10^{-1}\) \\ B6 & \(8.26\times 10^{-3}\) & \(5.04\times 10^{-3}\) & \(1.28\times 10^{-1}\) & \(9.96\times 10^{-1}\) \\ B7 & \(8.42\times 10^{-3}\) & \(5.02\times 10^{-3}\) & \(1.18\times 10^{-1}\) & \(9.97\times 10^{-1}\) \\ B8A & \(1.11\times 10^{-2}\) & \(6.14\times 10^{-3}\) & \(2.23\times 10^{-1}\) & \(9.95\times 10^{-1}\) \\ B12 & \(1.49\times 10^{-2}\) & \(9.49\times 10^{-3}\) & \(2.41\times 10^{-1}\) & \(9.75\times 10^{-1}\) \\ B11 & \(2.06\times 10^{-2}\) & \(1.36\times 10^{-2}\) & \(4.05\times 10^{-1}\) & \(9.81\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Single-band regression results for L2A. The colors in the RMSE (relative error) column indicate whether the specification is fulfilled (light gray), nearly fulfilled (middle gray) or unfulfilled (dark gray). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Band** & **RMSE** & **MAE** & **RE** & \(\mathbf{R^{2}}\) \\ \hline B07 & \(7.17\times 10^{-3}\) & \(3.90\times 10^{-3}\) & \(2.96\times 10^{-2}\) & \(9.96\times 10^{-1}\) \\ B06 & \(1.82\times 10^{-2}\) & \(4.77\times 10^{-3}\) & \(\mathbf{3.61\times 10^{-2}}\) & \(9.88\times 10^{-1}\) \\ B8A  
Table 6 shows the same results for L1C data. The performance seem to be much better than for L2A, but we must remember that the requirements for L1C are given as relative errors (the error must not exceed 3%). Table 7 shows the percentage of validation pixels compliant with different error thresholds. We see that the requirement has to be lowered from 3% to 10% in order to get 95% compliance for the red edge and NIR bands. This poor performance is mainly due to high relative errors in the low reflectances. Tables 8-12 show the compliance with relative error thresholds for different intervals of reflectances.