## 5 Discussion
However, neglecting soil heat flux (G) in the EF calculation may bring large errors in particular over regions where daily soil heat flux is large. For global application in our study, daily EF was calculated by daily ET and daily values of Rn\(-\)G from the ETMonitor model, in which daily G was estimated by a machine learning algorithm established based on measurements of G and several explanatory variables, and the results of G showed good agreement with the ground measurements at flux tower sites. In addition, ETMonitor could consider the impact of soil water content on ET, which made it able to capture the spatiotemporal variation of EF well [64; 90]. The ETMonitor model was applied to regional scales successfully and proven to have high accuracy as shown in several studies [65; 66; 67]. For example, a study on evaluating satellite-based ET products showed that ETMonitor had the highest R value of 0.83 and the lowest RMSD of 0.92 mm/d, while the other 6 ET products had correlation coefficients of 0.58 to 0.78 and RMSD of 1.13 to 1.41 mm/d [91]. Although the accurate ET from ETMonitor partly ensured the overall good performance of the EF-LUE model for cropland GPP estimation in this study, future work is still needed on systematic analysis of the remaining uncertainties brought by the EF. Carbon and water fluxes are inherently tightly coupled through plant stomatal activity. Carbon gain through photosynthesis and water loss through transpiration are the two dominant processes in the global water and carbon cycles and are simultaneously controlled by plant stomatal behavior in response to environmental conditions [92]. The two processes have similar characteristics to respond to changes in many environmental conditions, such as radiation, temperature, and VPD [93]. Soil moisture affects evapotranspiration and photosynthesis by regulating leave water potential and thereby affecting stomatal conductance that determines carbon and water exchanges between plants and the surrounding atmosphere. Some studies based on the conductance method have used some form of carbon-water coupling to estimate GPP and evapotranspiration [69]. As demonstrated by our study, integrating EF in the LUE type of model for GPP estimation is an effective method to provide satisfactory results in most cases. In future studies, more attentions should be paid to exploring methodologies that tightly couple the carbon and water processes. Moreover, some studies have shown that GPP and LUE increase with increasing diffuse radiation [94]. The diffuse radiation may reach shadowed leaves where the photosynthesis rate is usually limited [95]. A previous study found that the enhancement in LUE by diffuse radiation is significantly different in different ecosystems. However, crops showed less sensitivity to diffuse radiation than natural vegetation, which may be due to cultivation management by farmers [96]. We used GPP derived from flux tower sites to calibrate model parameters and verify the model results. GPP is not directly observed by the eddy covariance system. The partitioning algorithm for calculating GPP from CO\({}_{2}\) flux measurements introduces uncertainty. There are too few cropland flux tower sites globally, and these sites are not evenly distributed in space. Most of the sites are located in temperate and boreal regions (such as Europe and North America), and the semi-arid and tropical regions are underrepresented in the calibration dataset. These also bring large uncertainty to large scale applications. Furthermore, the pixel size (1 km in this study) of remote sensing data is inconsistent with the footprint of ground GPP observations (generally several hundred meters depending on the winds and atmospheric stability), which will inevitably lead to uncertainties in the verification of results. Further research needs to consider the impact of scale mismatch between ground measurements and the pixel size of satellite observations, and combine different spatial resolutions data for verifying model performance.