# Scaling of FAPAR from the Field to the Satellite
## 3 Methodology
### Temporal Normalization
#### 3.2.1 FAPAR-P Model  
Physical models characterize the effects of SZA and skylight proportions on the diurnal variations of instantaneous FAPAR and are thus the basis for normalizing _in-situ_ FAPAR measured at varying time. Previous studies [6; 7; 8] clearly show the effects of skylight proportion and SZA on the instantaneous FAPAR. However, to correct the effects, a simple and easy-to-retrieve physical model is preferred.  
The FAPAR-P model developed by [PERSON] _et al._[6] was used in the study as the theoretical basis for temporal normalization. It considers the multiple scattering of incident photons in the canopy by introducing the concept of recollision probability, a parameter only dependent on canopy structure as defined in [44]. Based on energy conservation law, the fraction of absorption is computed as the probability of intercepting photons minus the total probability of scattering. By using a complete consideration of canopy structure, incident solar radiation and spectral characteristics, the model is capable of capturing the temporal variations at different time. By using effective LAI instead of LAI, the FAPAR-P model can be extended to row crops (e.g., maize) [6]. The relevant input parameters and variables of the FAPAR-P model were summarized in Table 2, as well as the data sources to run the model. The leaf reflectance and transmittance were acquired from LOPEX database corresponding to maize, while the soil reflectance were measured with an ASD spectrometer in the field. The basic assumption of G = 0.5 was inherited from the development of the FAPAR-P model and has been verified by Monte Carlo simulation. More details can be found in [6].  
According to the study of [PERSON] _et al._[6], the diurnal variations of FAPAR were mainly attributed to LAI, SZA and skylight proportions based on the sensitivity analysis of the FAPAR. The other parameters in Table 2, including the spectra of soil, leaf reflectance and transmittance, as well as p and G can be given as constants under the normal condition of the given scene, which can simplify the problem of temporal normalization of FAPAR. Thus, the use of the FAPAR-P model is to investigate the effects of LAI, \(\theta_{i}\) and \(\beta\) on FAPAR, while \(\theta_{i}\) and \(\beta\) mainly account for the temporal discrepancies among the _in-situ_ measurement at varying time.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Values or Data Sources** \\ \hline Leaf reflectance & \(r_{1}\left(\lambda\right)\) & LOPEX93 corresponding to maize \\ Leaf transmittance & \(\tau_{1}\left(\lambda\right)\) & \\ Soil reflectance & \(r_{8}\left(\lambda\right)\) & Field measurement \\ Recollision probability & P & \(P_{LC}=P_{LC}\left(1-\exp(-\mathrm{kLA}^{\mathrm{b}})\right)\), where \\  & & \(P_{LC_{max}}=0.88\), k = 0.7 and b = 0.75 [44] \\ Projection of leaves per unit ground area on the & \(G\) & 0.5 \\ plane perpendicular to the incident solar beam & \(\theta_{i}\) & 0–70 at 10\({}^{\circ}\) interval \\ Solar zenith angle (SZA) & \(\theta_{i}\) & 0–70 at 10\({}^{\circ}\) interval \\ Effective leaf area index & LAI & 0–8 at interval of 1 \\ Portion of diffused skylight in total & \(\beta\) & 0–1 \\ incident radiation & \(\beta\) & 0–1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input parameters and variables of FAPAR-P model.