# Scaling of FAPAR from the Field to the Satellite
## 4 Results and Analysis
### Temporal Corrections
#### 4.1.2 Validating Temporal Normalization Method  
To validate the proposed temporal normalization method, a theoretical analysis of the accuracy of the model and a validation using real _in-situ_ FAPAR were conducted. From a theoretical perspective, the errors of the temporal normalization were identified to be sourced from: (i) the modeling errors from the FAPAR-P and the 6S models; (ii) the residual errors of fitting the modeled FAPAR in exponential forms and of fitting the correction coefficients to LAI; and (iii) the measurement errors of input data (e.g., LAI and FAPAR). Firstly, the modeling errors from the FAPAR-P and the 6S model were found to be both within 1% in [6,56], respectively. As a 1% change in extra skylight proportions has shown negligible effects on FAPAR (Figure 6), the overall modeling errors of integrating 6S and FAPAR-P model was estimated as within 1%. Secondly, the derivation of the temporal normalization method used two-round fittings: the fitting of the FAPAR to \(k_{1}\), \(k_{2}\) and \(k_{3}\) using Equation (5) and the fitting of \(k_{1}\) and \(k_{2}\) to LAI under varying VIS cases. The RMSE values for the two-round fittings ranged 0.0002-0.0017 and 0.0002-0.0100, respectively, which were testified as very small. To further investigate the effects of the residual errors from the fitting of the integrated FAPAR model, the FAPAR values derived from \(k_{1}\) and \(k_{2}\) estimated by LAI (from the two-round fittings) were compared to those fitted FAPAR values (from the first-round fitting) and those directly derived from the integrated model. The comparisons were made for varying LAI values at varying SZAs given VIS = 30 km. RMSEs were calculated for varying SZAs from 0\({}^{\circ}\) to 60\({}^{\circ}\) for each LAI value, as shown in Table 3. It was found that the errors from the second-round fitting of \(k_{1}\) and \(k_{2}\) were less than 0.008. The combined errors of the two-round fittings were also less than 0.008. This validated that the propagation of errors from the fitting of \(k_{1}\) and \(k_{2}\) in estimating FAPAR was negligible. The overall errors from the two-round fittings were within 0.01. Thirdly, the effects of measurement errors on temporal normalization were investigated by a sensitivity analysis. The measurement errors were mainly sourced from the measured LAI that determines the correction coefficients \(k_{1}\) and \(k_{2}\) and the FAPAR at specific time that is to be corrected. The sensitivity analysis was performed by adding random errors to both LAI and FAPAR measurements and then correcting FAPAR values at varying SZAs to 0\({}^{\circ}\). In implementation, the input LAI and FAPAR values were generated randomly with normal distribution, defined in the form of mean \(\pm\) standard deviation (STD) (blue error bars in Figure 7). The STD is an indicator for uncertainty [57]. For both LAI and FAPAR, the STD was set to about 20% of their mean values. The mean LAI values were set to 1, 2, 3 and 4, while the mean FAPAR values were set to the true FAPAR values at varying SZAs derived from the integrated model given VIS = 30 km. As shown in Figure 7, the mean values of the corrected FAPAR values (red error bars) were very close to the true FAPAR values at SZA = 0\({}^{\circ}\) (grey dashed line), with small values of STD. The correction errors were found to be related to and were less than the errors in the input FAPAR. Significant reduction of uncertainties has been observed in each step.