## 2 Materials and Methods
### Methodology
#### 2.3.5 Decomposition of Driving Factors of Carbon Budgets in Territorial Space  
The LMDI method is a widely used analytical tool for evaluating and decomposing the effects of complex factors, particularly in energy consumption studies [30]. The model decomposes total energy consumption into contributions from selected factors, enabling the identification of primary and secondary influences [37]. This study applies the LMDI model from a territorial space perspective, expanding the Kaya identity. The selected factors include carbon emission intensity per unit of territorial space (\(R_{i}\)), land-use structure (\(S_{i}\)), land area per unit GDP (\(\gamma\)), per capita GDP (\(g\)), and population size (\(P\)). The formula is as follows:  
\[C=\sum R_{i}\times S_{i}\times\gamma\times g\times P=\sum\frac{C_{i}}{L_{i}} \times\frac{L_{i}}{L}\times\frac{L}{G}\times\frac{G}{P}\times P \tag{11}\]  
where \(C\) represents the total carbon budget from land use; the formulas for \(R_{i}\), \(S_{i}\), \(\gamma\), \(g\), and \(P\) are shown in Table 4.  
The LMDI additive model is employed to decompose the contributions of various driving factors to changes in territorial space carbon budgets. The formulas are as follows:  
\[\Delta C=C^{T}-C^{0}=\sum_{i=1\cdots 2}R_{i}^{T}\times S_{i}^{T}\times\gamma^{T} \times g^{T}\times P^{T}-\sum_{i=1\cdots 2}R_{i}^{0}\times S_{i}^{0}\times \gamma^{0}\times g^{0}\times P^{0}=\Delta C_{Ri}+\Delta C_{Si}+\Delta C_{\gamma }+\Delta C_{g}+\Delta C_{P} \tag{12}\]  
\[\Delta C_{Ri}=\sum\frac{C_{i}^{T}-C_{i}^{0}}{lnC_{i}^{T}-lnC_{i}^{0}}\times ln \frac{R_{i}^{T}}{R_{i}^{0}} \tag{13}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Driving Factor** & **Formula** & **Unit** \\ \hline Carbon emission intensity per unit of territorial space (\(R_{i}\)) & C\({}_{i}\)/L\({}_{i}\) (carbon emissions per unit of land) & t/ha \\ Land-use structure (\(S_{i}\)) & L\({}_{i}\)/L (proportion of land use) & ha/ha \\ Land area per unit GDP (\(\gamma\)) & L/G (land area per unit GDP) & ha/10,000 yuan \\ Per capita GDP (g) & G/P (per capita GDP) & yuan/person \\ Population size (P) & - & 10,000 people \\ \hline \end{tabular}
\end{table}
Table 4: Formula for the driving factors of carbon emissions in territorial space.  
\[\Delta C_{Si}=\sum\frac{C_{i}^{T}-C_{i}^{0}}{lnC_{i}^{T}-lnC_{i}^{0}}\times ln \frac{S_{i}^{T}}{S_{i}^{0}} \tag{14}\]  
\[\Delta C_{\gamma}=\sum\frac{C_{i}^{T}-C_{i}^{0}}{lnC_{i}^{T}-lnC_{i}^{0}}\times ln \frac{\gamma^{T}}{\gamma^{0}} \tag{15}\]  
\[\Delta C_{g}=\sum\frac{C_{i}^{T}-C_{i}^{0}}{lnC_{i}^{T}-lnC_{i}^{0}}\times ln \frac{g^{T}}{g^{0}} \tag{16}\]  
\[\Delta C_{P}=\sum\frac{C_{i}^{T}-C_{i}^{0}}{lnC_{i}^{T}-lnC_{i}^{0}}\times ln \frac{P^{T}}{P^{0}} \tag{17}\]  
where \(\Delta C_{Ri}\), \(\Delta C_{Si}\), \(\Delta C_{\tau}\), \(\Delta C_{g}\), and \(\Delta C_{P}\) represent the decomposition effects of the driving factors (\(R_{i}\), \(S_{i}\), \(\gamma\), \(g\), \(P\)) from the base period to period t, corresponding to the effects of territorial carbon emission intensity, territorial land-use structure, territorial land-use efficiency, economic development, and population size, respectively.