## 5 Results
### Preliminary Evaluation Results
#### 5.4.2 Per-Band Absolute Error  
The Mean Absolute Percentage Error (MAPE), the Root Mean Square Error (RMSE) of absolute percentage errors and other statistics of Pre-F and I0 calibration parameter sets are presented in Table 7. The RMSE and MAPE, with respect to RapidEye, were obtained using the following expressions (Equations (10) and (11)):  
\[\mathrm{RMSE} = \sqrt{\frac{1}{n}\ \sum_{i=1}^{n}\left(R_{i}-F_{i}\right)^{2}} \tag{10}\] \[\mathrm{MAPE} = \frac{1}{n}\sum_{i=1}^{n}\left|\frac{R_{i}-F_{i}}{R_{i}}\right| \times 100 \tag{11}\]  
where \(n\) is the number of evaluation samples; \(R\) and \(F\) are the magnitudes derived from the common samples collected by RapidEye and FASat-C, respectively (radiance or reflectance). Considering the skewness of the distribution of the APE of the I0 samples (Figure 9), the first and third quartile values are also provided. As seen, after cross-calibration, the shape of the distribution of all bands has changed, with the samples being concentrated at the lower absolute errors, especially for B1 and B2. For B3, the trend is similar to B1 and B2, in the sense that the distribution of the absolute errors is left-skewed; however, absolute errors are concentrated below 3%, and not below 2%, as in the case of B1 and B2. In the particular case of B4, the distribution turns into a more uniform one. Overall, once the updated coefficients are applied, the level of dispersion in the sample was reduced by 32%, 21%, 6% and 12% for B1, B2, B3 and B4, respectively. For B1, the MAPE decreased from 6.01% down to 1.51% and the RMSE from 6.27% down to 1.94%; besides the median value of 1.2% must be observed in the case of I0, since it would be more representative. In addition, the maximum error diminished from 13.4% down to 8.1%. According to B1 histograms (Figure 9a), using Pre-F gains, just 32% of the sample presented an absolute error of up to 5%, while 66% of the sample had absolute errors in the range 5%-10%. After the cross-calibration, 71% of the samples have errors of up to 2%; 27% are in the range 2%-5%; and just 2% of the sample present errors higher than 5%. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Band** & **Calibration** & **RMSE** & **MAPE** & **St Dv** & **Min.** & **1 st Quartile** & **Median** & **3 rd Quartile** & **Max.** \\ \hline \multirow{2}{*}{B1} & Pre-F & 6.27\% & 6.01\% & 1.81\% & 3.03\% & 4.80\% & 5.53\% & 7.35\% & 13.36\% \\  & I0 & 1.94\% & 1.51\% & 1.23\% & 0.00\% & 0.56\% & 1.20\% & 2.15\% & 8.12\% \\ \hline \multirow{2}{*}{B2} & Pre-F & 5.00\% & 4.61\% & 1.93\% & 0.79\% & 3.29\% & 4.26\% & 5.88\% & 13.10\% \\  & I0 & 2.28\% & 1.71\% & 1.52\% & 0.01\% & 0.53\% & 1.23\% & 2.53\% & 9.90\% \\ \hline \multirow{2}{*}{B3} & Pre-F & 5.18\% & 4.93\% & 1.59\% & 1.40\% & 3.90\% & 4.86\% & 5.79\% & 9.98\% \\  & I0 & 2.75\% & 2.31\% & 1.50\% & 0.00\% & 1.20\% & 2.14\% & 3.09\% & 7.40\% \\ \hline \multirow{2}{*}{B4} & Pre-F & 7.80\% & 7.52\% & 2.08\% & 0.24\% & 5.97\% & 7.50\% & 9.25\% & 12.66\% \\  & I0 & 3.56\% & 3.05\% & 1.84\% & 0.00\% & 1.53\% & 3.04\% & 4.43\% & 9.60\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Absolute errors for FASat-C at-sensor radiances and TOA reflectances with respect to RapidEye. Results for B2 indicate that RMSE decreased from 5% down to 2.28% after cross-calibration, and MAPE dropped from 4.61% down to 1.71%. Besides, the maximum error diminished from 13.1% down to 9.9%, and the median value of the errors moved from 4.26% down to 1.23%. Prior to the calibration (Figure 9b), 62.5% of the sample had an error of up to 5% and 36.9% an error of up to 10%. Just 0.7% of the sample presented absolute errors higher than 10%. After cross-calibration, 66.9% of the sample presents errors of up to 2%; 29.7% of the sample has errors in the range 2%-5%; and just 2.1% has absolute errors larger than 5%. For B3, the RMSE decreased from 5.18% down to 2.75% and MAPE from 4.93% down to 2.31%, while the median value of errors is 2.14%. The maximum error lessened from 9.98% down to 7.4% after cross-calibration.