## 5 Discussion  
The LUE-based models are widely used in remote sensing-based methods for GPP estimation. Actual LUE highly varies and is determined by complex processes of crop physiology and environmental conditions, thus leading to difficulty in GPP estimation with high accuracy at the global scale [83]. Actual LUE is determined by potential LUE (\(\varepsilon_{max}\)) and environmental conditions. Previous studies suggest that the potential LUE (\(\varepsilon_{max}\)) is related to different vegetation types and is relatively more stable than actual LUE. However, a study based on flux tower measurements showed the large spatial variability in \(\varepsilon_{max}\) within individual biome types [38]. Unlike the MOD17 GPP method which assumed a constant \(\varepsilon_{max}\) for all crops, in the EF-LUE model proposed in this paper, the model parameters were optimized using available cropland flux tower data and took into account the parameter differences in different climate zones when applied at global scale. This overcomes the serious underestimation in MOD17 GPP. Based on a literature survey, the range of \(\varepsilon_{max}\) was between 0-4 g C MJ\({}^{-1}\)[83, 84]. Our optimized value clearly showed the large range of \(\varepsilon_{max}\) values for different crops in different climate zones, with the largest \(\varepsilon_{max}\) value (3.9 g C MJ\({}^{-1}\)) found at maize (C4 crop) site. Maize is a C4 crop with a greater LUE than C3 crops, while the value of \(\varepsilon_{max}\) for maize derived from ground observations in previous studies was 3.84 \(\pm\) 0.08 g C MJ\({}^{-1}\)[85], close to our estimation. The great variation of cropland LUE is associated with crop types and environmental and water supply conditions, and therefore is dependent on the actual crop cultivation. Compared with natural ecosystems, crop types may change rapidly due to management practices by farmers (such as crop rotation plans), which means the values of \(\varepsilon_{max}\) will change with the different crop rotation cycles. There are some regions adopting rotation  
Figure 11: The percentage of anomalies of monthly mean GPP relative to monthly GPP of multi-year mean values for four major heat-wave and drought events: (**a**) 2010 (June–August) Russia heat-wave; (**b**) 2009 (April–June) India heat-wave; (**c**) 2013 (January–June) China drought; (**d**) 2012 (June–December) US Corn Belt drought. or intercropping system, e.g., planting maize and soybeans in different years or in the same year by intercropping. Due to different photosynthesis pathways, the \(\varepsilon_{max}\) of maize is much higher than that of soybean. Optimizing the model parameters using the flux tower data without distinguishing the two crops may lead to underestimation of maize GPP and overestimation of soybean GPP. For the cropland GPP estimation based on the LUE type models using remote sensing data at regional and global scales, further attention should be paid to the impact of different crop types (at least distinguishing C3 and C4 crops) and crops phenological periods on the optimization of model parameters. The model parameters can be estimated more precisely if detailed crop types are known. In the EF-LUE model, the water availability constraint factor, represented by EF, modulates the influence of soil water supply on crop productivity and can avoid overestimation of GPP when crop experiencing water stress. As in previous studies carried out at site-level, EF can be used to quantify soil water availability in LUE models and can improve the accuracy of GPP estimation [23; 86; 87], and many studies have tried to revise the GPP model based on remote sensing EF as an indicator of water stress at regional scales [88; 89]. Soil heat flux was neglected in the calculation of available energy in the study by [88], instead Rn was used in the denominator of the equation for EF calculation in their LUE model for GPP estimation in regional scale application.