## 2 Materials and Methods
### Methods  
#### 2.2.1 Simulating FPAR Based on the PROSAIL Model  
The PROSAIL model, which consists of the SAIL canopy bidirectional reflection model and the PROSPECT leaf optical property model, is a widely used model in vegetation parameter inversion [26]. The PROSPECT model is a radiation-transfer model based on a plate model modified by [PERSON] (1969) for simulating the directional hemispherical reflectance and transmissivity of various green monocots and dicots and senescent leaves, expressing leaf optical properties of 400-2500 nm. On the other hand, the SAIL model applies to the SUIT model to simulate the reflectivity of the leaf canopy's direction under certain observational conditions by inputting the LAI, the leaf angle distribution, and other parameters [27; 28]. As a secondary parameter of the PROSAIL model, FPAR needs to be converted into direct input parameters (such as LAI) or calculated between radiation transmissions. From the perspective of energy balance, the expression of FPAR is  
\[FPAR\ =\ \Big{(}PAR_{\mathcal{IAC}}-PAR_{\mathcal{IAC}}+PAR_{\mathcal{IBC}}- PAR_{\mathcal{IBC}}\Big{)}/PAR_{\mathcal{IAC}}. \tag{1}\]  
Equation (1) can also be converted to:  
\[FPAR\ =\ 1-a-t_{PAR}+t_{PAR}a_{s}, \tag{2}\] \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Product** & **Sensor** & **Time Range** & **Spatial Resolution** & **Time Resolution** & **Main Algorithm** & **Valid Range** & **Scale Factor** \\ \hline MOD15A2H & Terra & 2002.7–Present & 500 m & 8 d & 3D-RTmodel\(\backslash\)LUT & 0–100 & 0.01 \\ GEOV1-FPAR & PROBA-V & 2014.1–Present & 300 m & 10 d & Neural Networks & 0–235 & 1/250 \\ GLASS09A01 & \(\backslash\) & 2000–2014 & 1 km & 8 d & Neural Networks & 0–250 & 0.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The mean characters of fraction of absorbed photosynthetically active radiation (FPAR) products. where \(a\) is the visible albedo at the top of the canopy and \(a_{S}\) is the soil albedo, \(t_{PAR}~{}=~{}PAR_{\downarrow BC}/PAR_{\downarrow AC}\), which is the transmittance of PAR emitted from the bottom of the canopy. This algorithm ignores \(t_{PAR}a_{S}-a\) and results in the following FPAR expression (Equation (3)) [28]:  
\[FPAR~{}=~{}1-t_{PAR}. \tag{3}\]  
The PAR incident on the surface canopy of the blade includes direct radiation and scattering, so the transmittance corresponding to the PAR incident on the soil portion is further expressed as  
\[t_{PAR}~{}=~{}t_{PAR}^{dir}-\Big{(}t_{PAR}^{dir}-t_{PAR}^{dif}\Big{)}/f_{skyl}, \tag{4}\]  
where \(t_{PAR}^{dir}\) and \(t_{PAR}^{dif}\), respectively, refer to the transmittance of direct incident PAR and scattered incident PAR and \(f_{skyl}\) refers to the sky light scale factor. The transmittance of the canopy is related to various parameters, including sensor parameters and plant parameters. If the absorptivity of the leaves is \(a\), the transmittance of the direct incident PAR can be approximated by the exponential model (Equation (5)) from [PERSON] (1998). If the scattered radiation comes from all directions, the transmittance of the scattered incident PAR can be obtained by integrating the transmittance of the direct incident PAR in each incident direction (Equation (6)) [29]. \[t_{PAR}^{dir}~{}=~{}e^{-\sqrt{\alpha}\alpha K(\theta)\cross\Omega\times LAl} \tag{5}\]  
\[t_{PAR}^{dif}~{}=~{}2\int_{0}^{\Omega I/2}t_{PAR}^{dir}\sin\theta\cos\theta d\theta, \tag{6}\]  
where \(\Omega\) is the clumping index (CI), angle \(\theta\) is the solar zenith angle, and \(K(\theta)\) is the extinction coefficient of the canopy for PAR. The clumping index is determined by the spatial distribution pattern of the leaves, and the global clumping index of different vegetation types is shown in Table 3 (for shrubs, broadleaf forests, and needleleaf forests). Another algorithm for simulating FPAR is based on LAI (alternate algorithm). LAI is a direct input parameter in the PROSAIL model, which can be directly obtained from the model inversion.