## 3 Results and Discussion
### The Influence of Meteorological Conditions on GPP
However, their distribution of population density is statistically less evident than that in the 0.05\({}^{\circ}\) data due to the sparse number of the samples. The GPP-TCI and GPP-PCI relationships in the 0.05\({}^{\circ}\) data and the county-level data spread out widely, indicating a prominent regional dependency (Figure 4a,b). The state-level data reveal that GPP in which state is the most sensitive to the meteorological condition  
Figure 4: Joint histogram of TCI-GPP relationship (x-axis) and PCI-GPP relationship (y-axis) at (**a**) satellite-, (**b**) county-, and (**c**) state-level data for the average of July and August. In (**a**,**b**), the \(x\) and \(y\) axes are separated by a 0.02 interval for each bin. Figure 3: Detailed correlation analysis of annual variation between TCI and PCI, TCI and GPP, and PCI and GPP for pixel-level, county-level, and state-level 0.05\({}^{\circ}\) grids for the average of July and August over North Dakota (ND), South Dakota (SD), Nebraska (NE), and Kansas (KS). The dotted circle line represents the region of Black Hills National Forest (BHNF). (Figure 4c). For example, GPP values in the Great Plains (KS, NE, and SD) exhibit a stronger relationship with meteorological conditions than that in the Midwest. This regional dependency, consistent with the results shown in Figure 2, can be associated with the background summer climate in each region. The background climate in the Great Plains is arid, while the Midwest has a relatively cool and wet climate (Figure 1a,b). To investigate the influence of background climate on regional dependency of the relationship between GPP and meteorological variable, the correlation values at the 0.05\({}^{\circ}\) data in the 12 states are sorted by long-term mean temperature and precipitation during summer (Figure 5). Figure 5 shows the relationship between GPP and the meteorological variables as a function of the regional background climate. The long-term mean climate determines GPP sensitivity to TCI and PCI each year. The relationship between GPP and the meteorological condition tends to be strong in a higher (lower) long-term mean temperature (precipitation). GPP in a region is more sensitive to changes in meteorological conditions if a regional background climate is warm and dry in summer. The more sensitive GPP response to TCI and PCI in these regions is likely due to the non-linear relationship between GPP and temperature, particularly evident in the semi-arid area [34]. Since GPP tends to decrease exponentially with increasing temperature above a certain threshold, a larger decrease in GPP with an anomalous warming is expected in the regions of higher background temperature [34]. This prominent role of background temperature might be one of the reasons for the aspect that GPP is more sensitive to TCI than PCI. Note that this non-linear relationship with increasing temperature also appears in comparison with the crop yield [16]. The results in this section strongly suggest that the temperature and precipitation sensitivity of GPP is much more prominent in the Great Plains due to the warm and dry background conditions. If the variation of GPP is consistent with crop yield, the regionally dependent sensitivity to meteorological conditions would be valid in the annual crop yield. Figure 5: Correlations of GPP-TCI (red) and GPP-PCI binned by long-term mean summer (**left**) temperature and (**right**) precipitation using the 0.05\({}^{\circ}\) data in the 12 states of major maize and soybean production. Top rows indicate the average of correlation between July and August, and bottom rows indicate a ratio of the number of grid points in each bin. In the top rows, standard deviation of correlations in each bin is indicated as vertical ranges.