## 3 Results
### Gray Correlation Analysis Results  
#### 3.3.1 Correlation between Carbon Emissions and Urban Land Area  
The GM (1,1) model was run in MATLAB software, and correlation indices were calculated between the carbon-emission-increase rate and the urban-land expansion intensity on \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Content** & **Units** & **1949** & **1979** & **1996** & **2003** & **2008** & **2013** & **2016** \\ \hline Total carbon emissions (TCE) & t-CO\({}_{2}\)/10\({}^{4}\) & 1.73 & 5.88 & 5.21 & 6.96 & 12.32 & 17.88 & 18.13 \\ Per capita emissions (PCE) & 10\({}^{4}\) t-CO\({}_{2}\)/per. & 66.24 & 601.73 & 851.07 & 1392.75 & 2963.87 & 5404.60 & 6095.21 \\ Emission growth rate (EG) & \% & / & 26.95 & 2.44 & 9.09 & 22.56 & 16.47 & 4.26 \\ Residential sub-model (\(F_{\rm{ES}}\)) & \% & 6.42 & 25.32 & 50.95 & 132.37 & 208.77 & 347.48 & 410.2 \\ Commercial-service sub-model (\(F_{\rm{com}}\)) & & 5.02 & 52.24 & 65.89 & 240.39 & 471.61 & 784.93 & 1125.6 \\ Industrial sub-model (\(F_{\rm{ind}}\)) & 10\({}^{4}\) t-CO\({}_{2}\) & 42.35 & 472.27 & 587.47 & 641.26 & 1813.72 & 2147.59 & 2333.6 \\ Transportation sub-model (\(F_{\rm{TIn}}\)) & & 12.52 & 51.96 & 146.81 & 378.79 & 469.81 & 2124.64 & 2225.8 \\ Sequestation sub-model (\(F_{\rm{Gas}}\)) & & \(-695.95\) & \(-690.3\) & \(-592.44\) & \(-552.02\) & \(-427.69\) & \(-343.41\) & \(-231.54\) \\ Carbon-emission intensity (EI) & t-CO\({}_{2}\)/CNY10\({}^{4}\) & 23.08 & 28.14 & 2.05 & 1.29 & 0.9 & 0.76 & 0.72 \\ Emission per unit land (EPL) & 10\({}^{6}\) t-CO\({}_{2}\)/Km\({}^{2}\) & 9.89 & 11.62 & 8.11 & 6.72 & 14.14 & 13.72 & 12.80 \\ \hline \end{tabular}
\end{table} 
Table 4: Total carbon emissions, carbon-emission intensity, per capita emissions and emissions per unit of land. Figure 5: Model checking. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Year**} & \multicolumn{2}{c}{**Carbon Emissions**} \\ \cline{2-3}  & **Base Scenario** & **COVID-19 Scenario** \\ \hline
2017 & 6431.31 & 6824.57 \\
2018 & 6903.27 & 7773.32 \\
2019 & 7398.59 & 8393.766 \\
2020 & 7917.81 & 9084.231 \\
2021 & 8461.51 & 9571.036 \\
2022 & 9030.24 & 10,070.21 \\
2023 & 9624.56 & 10,581.52 \\
2024 & 10,245.02 & 11,104.73 \\
2025 & 10,892.20 & 11,550.77 \\
2026 & 11,566.66 & 12,000.61 \\
2027 & 12,268.95 & 12,453.83 \\
2028 & 12,999.66 & 12,910.04 \\
2029 & 13,759.34 & 13,375.11 \\
2030 & 14,548.55 & 13,836.29 \\ \hline \end{tabular}
\end{table} 
Table 5: Carbon emission prediction of Changsha. the one hand and between the carbon-emission intensity and the land-expansion intensity per unit of urban land area on the other hand, as shown in Table 6. The correlations between both the total carbon emissions and the per capita carbon emissions on the one hand and the area of each type of land on the other hand are very high, all above 0.9. The correlation between industrial-land expansion, which is the largest source of carbon emissions, and the carbon emission growth rate is the lowest, only 0.168, reflecting the decoupling trend between industrial lands and urban carbon emissions in Changsha. The correlation between the increase rate of carbon emissions and the expansion intensity of road lands is weak (\(-0.148\)). Road lands themselves do not produce significant carbon emissions, but the increase in residential lands and commercial lands is caused by the increase in road lands. Indeed, the correlation of residential and commercial lands with the increase rate of carbon emissions is considerably strong (0.788 and 0.549, respectively). There was a negative correlation between carbon-emission intensity and all types of land area. It is strongly correlated with the amount of commercial service land, followed by roads and public-square lands. The GDP output from the expansion of commercial-service land, as well as transportation and logistics lands, accounts for a larger proportion of the urban GDP growth. The correlation of carbon-emission intensity is low for residential lands and industrial lands (0.621 and 0.634, respectively).