## 2 Materials and Methods
### Methodology
#### 2.3.1 Data Preprocessing
Surface radiation and energy is used to heat the air, evaporate water, and heat the soil. The radiative energy balance can be expressed as follows:  
\[Rn=H+LE+G. \tag{4}\]  
\(Rn\) is the net radiation with the unit of W/m\({}^{2}\), \(H\) is the sensible heat flux (which can change the temperature of the air) with the unit of W/m\({}^{2}\), \(LE\) is the latent heat flux (which can evaporate water) with the unit of W/m\({}^{2}\), and \(G\) is the ground heat flux (which can heat the soil) with the unit of W/m\({}^{2}\). In our study, we focus on the heat flux to the atmosphere, which can be calculated as follows:  
\[Q=Rn-LE=H+G. \tag{5}\]  
We can see from this formula that the heat flux to the atmosphere is the sum of the sensible heat flux and ground heat flux. To obtain the net radiation forcing and heat flux to the atmosphere from 2000 to 2015 in northern China, we calculated the monthly albedo, emissivity, land surface temperature, and latent heat flux data from the 8 d origin data at first. To obtain accurate estimates of the surface energy radiation balance due to cropland expansion, we must ensure that the value of the surface parameter data for each pixel represents the real value of one land use class. Therefore, we extracted the unchanged land use class pixels from 2000 to 2015, where the land use property in the pixel was more than 80% [38]. Those pixels represented the pure pixels of some land use class, and the stable pixels were an unchanged land use class. We compared the albedo, emissivity, and other biogeophysical parameters of the typical land-use types found nearby the cropland expansion. The forest, grassland and cropland adjacent to each other were chosen for pairwise comparison analyses. Then, we used the stable pixels to extract the monthly albedo, emissivity, latent heat flux, and land surface temperature. We statistically analyzed those parameters for different climate zones, and formed the best-parameters datasets. The best-parameter datasets included the monthly parameters of each climate zone for each land use class, such as \(Alb_{i,z,m}\), \(Emis_{i,z,m}\), \(LST_{i,z,m}\), \(LE_{i,z,m}\), where the abbreviations referred to the albedo, emissivity, land surface temperature, and latent heat flux in the zone \(z\) of \(i\) land use class in the \(m\) month. Next, we calculated the net shortwave radiation, net longwave radiation, net radiation, and heat flux to the atmosphere on a monthly basis. Net shortwave radiation (\(Rns\)) was calculated based on the radiation data and the albedo data; the formula [39] is:  
\[Rns_{i,z,m}=(1-Alb_{i,z,m})*Ssrd_{z,m} \tag{6}\]  
Here, \(Rns_{i,z,m}\) is the monthly net shortwave radiation in the zone \(z\) of \(i\) land use class in the \(m\) month with the unit of W/m\({}^{2}\); \(Alb_{i,z,m}\) is the monthly albedo in the zone \(z\) of \(i\) land use class in the \(m\) month; and \(Ssrd_{z,m}\) is the monthly downward shortwave radiation data in the zone \(z\) in the \(m\) month. Net longwave radiation (Rnl) was calculated based on the emissivity data, LST data, and radiation data; the formula is:  
\[Rnl_{i,z,m}=Emis_{i,z,m}*Strd_{z,m}-*\left(5.67*10^{-8}\right)*Emis_{i,z,m}*LST _{i,z,m}\lx@note{footnote}{Here, $Rnl_{i,z,m}$ is the monthly net longwave radiation in the zone $z$ of $i$ land use class in the $m$ month, with the unit of W/m\({}^{2}\); \(Emis_{i,z,m}\) and \(LST_{i,z,m}\) are the monthly emissivity and land surface temperature in the zone \(z\) of \(i\) land use class in the \(m\) month, respectively; \(Strd_{z,m}\) is the monthly downward longwave radiation in the zone \(z\) in the \(m\) month; \(\sigma\) is the Boltzmann Constant (\(5.67\times 10^{-8}\)W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-4}\cdot\)).  
Net radiation (\(Rn\)) was calculated by \(Rns\) and \(Rnl\); the formula is:  
\[Rni_{i,z,m}=Rnl_{i,z,m}+Rns_{i,z,m} \tag{8}\]  
Here, \(Rn_{i,z,m}\) is the monthly net radiation in the zone \(z\) of \(i\) land use class in the \(m\) month with the unit of W/m\({}^{2}\).