## 3 Model and Methods
### Description of the TL-CLUE Model  
The purpose of the study is to improve the accuracy of GPP estimation by considering seasonal differences in the CI (Figure 2), and includes four parts: (1) analysis of seasonal variation in the CI and the total sensitivity of GPP to CI changes based on the TL-LUE model; (2) division of the OYC into different seasons and estimation of the CI in each season; (3) description of seasonal canopy changes depending on the different \(\Omega\)s of different seasons (TL-CLUE); and (4) evaluation of the performance of the TL-CLUE model at site and spatial scales and exploration of the spatial and temporal pattern of GPP simulation by the TL-CLUE model across North America during 2002-2020. There is potential for the seasonal differences in the CI to reflect the changes in the vegetation status [32; 43; 52]. When ignoring the seasonal variability and interannual variation in the CI, the use of only a constant CI over time will lead to larger uncertainties in modeling studies [43]. In this study, it was expected that the uncertainties of GPP estimation would be reduced by replacing the constant \(\Omega\) currently used in the TL-LUE model with separate \(\Omega\)s for different seasons (TL-CLUE). The key to the TL-CLUE model is to acquire an appropriate CI value (i.e., \(\Omega_{\text{i}}\)) that not only represents the basic trend of the foliage clumping state within a canopy in different seasons but also reduces the potential uncertainty that is probably caused by the upper-level data products (i.e., MODIS/BRDF) and model inversion. In theory, the constant \(\Omega\) currently adopted in the TL-LUE model is not consistent with the fact that the CI varies in different seasons. Therefore, it is appropriate to improve the estimation of GPP in the TL-LUE model by updating Equation (8) with different \(\Omega\)s from different seasons instead of a constant \(\Omega\) as follows:  
\[LAI_{\text{su}}=2\times\cos(\theta)\times\left(1-e^{\frac{-0.5\times\Omega_{ \text{i}}\times L\Delta t}{\cos(\theta)}}\right) \tag{13}\]  
where \(\Omega_{i}\) is the \(\Omega\) in the ith season (Table 2), and i represents different seasons, including the leaf-on season, leaf-off season, leaf-scattering season, and leaf-gathering season. The CI of remote sensing data with a high temporal resolution, such as the daily and eight-day data, currently remains a large uncertainty in the intra-annual changes caused by surrounding and background reflectance changes resulting from various  
Figure 2: Flow chart of the improvement of GPP estimation by considering seasonal differences in the CI (TL-CLUE). The total sensitivity of GPP to the CI is explored by the EFAST method. The OYC for leaf growth is divided into LFS and LSS, consisting of LOS and LGS, labeled according to MODIS LC (land cover) data (MCD12Q1.061) and MODIS LSP (land surface phenology) data (MCD12Q2 V061). Then, the corresponding average CI (\(\Omega\)) of each season for each vegetation type is estimated based on MODIS CI data and published literature. environmental factors, such as brief rainstorms and windstorms. Additionally, although the use of monthly CI data reduces some effects originating from low quality and ephemeral background reflectance, it will make the model complicated to run. Relatively, there are apparent seasonal differences in the CI, with higher values during the dormancy period and lower values during the growing season [42], and the lowest values of the CI occur at the peak of the growing season because the mature dense canopy likely leads to a strong aggregation effect [43]. Therefore, there is a great advantage to using the seasonal \(\Omega\) values with obvious seasonal differences to improve GPP estimation in the TL-LUE model. Global land surface phenology (LSP) with a spatial resolution of 500 m from 2001 to 2021 is provided annually by the Terra and Aqua combined Moderate Resolution Imaging Spectroradiometer (MODIS) Land Cover Dynamics (MCD12Q2 V061) Version 6.1 data product.