## 3 Method
### LAI Inversion by the PROSAIL Radiative Transfer Model  
Numerous models have been developed to invert the LAI, but all models are associated with a rather limited number of input variables [26]. The canopy radiative transfer simulation model, PROSAIL (a combination of the PROSPECT leaf optical properties model and SAIL (scattering by arbitrarily inclined leaves) canopy reflectance model), is one of the most commonly-used models, and it has been widely validated in the scientific community in terms of its stability and robustness in the study of plant canopy spectral reflectance and vegetation bio-physical variables [70,71].Therefore, PROSAIL was selected to retrieve LAI in this study.  
The main input variables of the PROSAIL model are summarized in Table 4, and the estimation of LAI is based on a look-up table (LUT) approach in this study [72]. To build the LUT, input parameter combinations were randomly generated and used in the forward calculation of the PROSAIL model to simulate the broadband reflectance in the visible and near-infrared wavelengths associated with the multispectral sensors. To achieve an appropriate representation, a total number of 200,000 reflectance data for LUT has been generated in this study. To overcome an ill-posed inversion problem resulting from measurement errors and a similar response of radiometric signals to different variable combinations in the LUT, the solution should be the average of the parameter combinations that yield the smallest values of the cost function between the remotely-sensed reflectance and the simulated reflectance [73,74]. In this study, the 10% threshold of the sorted smallest value in the LUT size of 200,000 was used to select the entries from the LUT. The 10% threshold is feasible for balancing computer resources and the accuracy of the variable estimates and is consistent with the optimal number adopted by previous studies [74, 75, 76, 77].  
To eliminate inversion distortions from clouds, moisture, aerosols, sensor errors and mixed pixels, a Savitzky-Golay (S-G) filter [78,79] was used to smooth the noise of LAI time series imagery, and the adequate LAI trajectory from the phenological stage of green-up to maturity was received.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Variables** & **Abbr.** & **Unit** & **Min** & **Mean** & **Max** & **Std.Dev** & **Distribution** \\ \hline Leaf structural parameter & \(N_{i}\) & - & 1.2 & 1.5 & 1.8 & 0.3 & Gauss \\ Chlorophyll1 a + b content & C\({}_{ab}\) & \(\mu\)g\(\cdot\)cm\({}^{-2}\) & 30 & 50 & 70 & 7.5 & Gauss \\ Equivalent water thickness & C\({}_{w}\) & cm & 0.60 & 0.75 & 0.85 & - & Uniform \\ Dry matter content & C\({}_{w}\) & g\(\cdot\)cm\({}^{-2}\) & 0.003 & 0.007 & 0.011 & 0.002 & Gauss \\ Brown pigments content & C\({}_{bp}\) & \(\mu\)g\(\cdot\)cm\({}^{-2}\) & 0 & 0 & 0.2 & 0.3 & Gauss \\ Leaf area index & _LAI_ & - & 0 & 5 & 8 & - & Uniform \\ Average leaf inclination angle & _ALIA_ & \(\circ\) & 30 & 60 & 80 & 5 & Gauss \\ Hot spot parameter & _lspot_ & - & 0.1 & 0.3 & 0.5 & 0.2 & Gauss \\ Soil brightness parameter & _psoil_ & - & 0.5 & 1.2 & 3.5 & 2.0 & Gauss \\ Fraction of diffuse incoming solar radiation & _skyl_ & \% & - & 10 & - & - & Fixed \\ Solar zenith angle & _tts_ & \(\circ\) & 25 & 46 & 70 & - & Fixed \\ View zenith angle & _tto_ & \(\circ\) & 0 & 32 & 80 & - & Fixed \\ Relative azimuth angle & _psi_ & \(\circ\) & \(-120\) & 90 & 120 & - & Fixed \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ranges and distributions of the PROSAIL input parameters used to generate the LUT.