# Scaling of FAPAR from the Field to the Satellite
## 3 Methodology
### Temporal Normalization
#### 3.2.3 FAPAR Model Integrating FAPAR-P and 6S Models  
Using Equations (2)-(4), the skylight proportions estimated from SZA and VIS can be substituted into the FAPAR-P model. Thus the effects of skylight proportions on instantaneous FAPAR were integrated into the effects of SZA. If only the skylight proportion determined by SZA and VIS were considered, the diurnal variations of FAPAR for a given LAI were induced by changing SZAs. Figure 3: Modeling skylight proportions from solar zenith angle (SZA) and visibility (VIS) modeled from the second simulation of a satellite signal in the solar spectrum (6S) (blue signals represent 6S model outputs, whereas red lines represent the fitted equations). Using the integrated FAPAR model, the effects of SZA and the associated skylight proportions on instantaneous FAPAR were firstly investigated (Figure 4). As shown in Figure 4a, larger SZA increases FAPAR values, especially at low LAI values. At LAI = 1, the increase in FAPAR from 0\({}^{\circ}\) to 70\({}^{\circ}\) can reach as high as 50%. This effect could be explained by the lengthening of the photons' paths going through the canopy which increase their probabilities of being absorbed or scattered. The higher the value of LAI, the less is the effect of SZA on FAPAR. At extremely high LAI, _i.e._, LAI = 8, FAPAR is very close to 1 and the effect of SZA is minimal. In Figure 4b, the FAPAR increases with LAI increasing, exhibiting an exponential shape. The increase in FAPAR with large SZA could also be observed at the same LAI. Further, the combined effects of LAI, SZA and associated skylight proportions on instantaneous FAPAR were investigated, as shown in Figure 5. It can be seen that the VIS does affect the derived FAPAR values, but in a subtle way at low SZA angles or in dense vegetation canopies with high LAI values (e.g., LAI = 8). At large SZA angles (>50\({}^{\circ}\)), the effects of VIS are amplified in sparse vegetation. Figure 4: Effects of SZA and leaf area index (LAI) on FAPAR simulated from FAPAR-P model under VIS = 30 km: (**a**) effects of SZA on FAPAR with LAI = 1, 2, 4 and 8; and (**b**) effects of LAI on FAPAR with given SZA. Figure 5: Relationship of FAPAR and LAI/cos(\(\theta\)) derived from the FAPAR model integrating FAPAR-P and 6S model. To correct the effects of SZA, the key is the slope of FAPAR varying with independent variable LAI/cos(\(\theta\)). The steeper the slope is, the more the correction is needed. It could be inferred that such slope is much steeper at lower LAI values; given the same LAI, the slope decreases with the increasing SZAs. Thus the effects of SZA are intertwined with both LAI and VIS. However, for each case with given LAI and VIS, the FAPAR can be fitted by an exponential formula, as:  
\[\mathrm{FAPAR}=\mathrm{k_{1}}-\mathrm{k_{2}}\times\exp\left(-G\times LAI/ \mathrm{cos}\left(\theta\right)\times\mathrm{k_{3}}\right) \tag{5}\]  
where \(k_{1}\), \(k_{2}\) and \(k_{3}\) are fitting parameters. The parameters of \(k_{1}\) and \(k_{2}\) are associated with the composition of direct solar radiation and diffuse skylight, as well as the multi-scattering within the canopy layer. The parameter \(k_{3}\) is related to the vegetation structure. It is worth noting that when \(k_{1}\), \(k_{2}\) and \(k_{3}\) all equal to 1, Equation (5) becomes \(\mathrm{FAPAR}=1-\mathrm{e}^{-G\cdot LAI/\mathrm{cos}\left(\theta\right)}\), as shown in the solid purple line in Figure 5. This is in accordance with the form of the FAPAR model based on gap probability [45]. The equation only stands under the assumption of \"black leaves\" (\(\mathrm{r_{1}}\left(\lambda\right)=\mathrm{\tau_{1}}\left(\lambda\right)=0\)) that makes \(p\) equal to zero and only the direct solar radiation is considered. At SZA lower than 40\({}^{\circ}\), simple SZA correction could be done based on this simpler form of FAPAR model.