## 2 Materials and Methods
### Methodology
#### 2.3.4 Exploring Spatiotemporal Variation Patterns of Carbon Budgets  
(1) Grid sampling method  
The grid sampling method partitions the study area into equal-sized grids, selecting one sample point per grid. This method effectively covers the study area while ensuring a uniform distribution of sample points [33]. To visualize spatiotemporal carbon budget patterns, this study applies a grid-based approach to the Dongting Lake region. The carbon emission coefficient for construction land is 8.3102 kg(C)\(\cdot\)m\({}^{-2}\cdot\)a\({}^{-1}\). Using the grid sampling method from landscape ecology to capture spatial patterns and considering the Dongting Lake region's size, this study employs 5 km \(\times\) 5 km grids, generating 3087 sample units. Carbon sequestration, emissions, and budgets for each grid are assigned to its central point, and Kriging interpolation is used to visualize the spatial evolution of the Dongting Lake region. The ordinary Kriging interpolation formula is as follows:  
\[\hat{Z}(S_{0})=\sum_{i=1}^{n}\lambda_{i}Z(S_{i}) \tag{7}\]  
where \(Z(S_{i})\) is the measured value at location \(i\); \(\lambda_{i}\) is the unknown weight assigned to the measured value at location \(i\); and \(n\) is the number of measured values. (2) SDE analysis method  
The SDE analysis method, initially proposed by [PERSON] [34], has been refined over time and is now widely used to study the spatial distribution of geographic phenomena. It accurately depicts the primary directional distribution of carbon budgets and quantitatively describes spatiotemporal evolution trends. In the SDE analysis, the ellipse's long axis represents the dominant directional distribution of carbon budgets, while the short axis reflects the degree of dispersion or clustering. The centroid coordinates of the ellipse indicate the trajectory of the carbon budget's spatial center over time. The formula for calculating the centroid coordinates is as follows:  
\[\overline{X}=\frac{1}{n}\sum_{i=1}^{n}x_{i},\ \overline{Y}=\frac{1}{n}\sum_{i=1}^{n}y_ {i} \tag{8}\]  
where \(x_{i}\) and \(y_{i}\) are the center coordinates of each administrative unit in the study area; \(\overline{X}\) and \(\overline{Y}\) are the centroid coordinates; and \(n\) is the total number of grids. \begin{table}
\begin{tabular}{c c} \hline \hline
**Territorial Space Carbon Sink Type** & **Carbon Sequestration Coefficient (tC/hm\({}^{2}\cdot\)a)** \\ \hline Forest ecological land & \(-\)0.0586 \\ Grassland ecological land & \(-\)0.0210 \\ Water ecological land & \(-\)0.0459 \\ Potential ecological land & \(-\)0.0005 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Carbon sequestration coefficients of territorial space types in the Dongting Lake region. (3) Spatial autocorrelation analysis  
Spatial autocorrelation is widely used to examine geographic phenomena' spatial distribution patterns and relationships [35]. This study uses the grid sampling method to generate data on carbon sinks, carbon sources, and net carbon emissions. Global (Moran's I) and local (Local Moran's I) spatial autocorrelation are applied to evaluate the spatial correlation of carbon budgets within the territorial space [36]. The formulas for global and local Moran's I indices are as follows:  
\[I=\frac{n\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}(x_{i}-\overline{x})\big{(}x_{j}- \overline{x}\big{)}}{\left(\sum_{i=1}^{n}\sum_{j=1}^{n}w_{ij}\right)\sum_{i=1} ^{n}\left(x_{i}-\overline{x}\right)^{2}} \tag{9}\]  
\[I_{L}=\frac{n(x_{i}-\overline{x})\sum_{j=1}^{n}w_{ij}\big{(}x_{j}-\overline{x }\big{)}}{\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}} \tag{10}\]  
where \(I\) is the [PERSON]'s I index; \(n\) is the total number of grid cells; \(w_{ij}\) is the spatial weight matrix; \(x_{i}\) and \(x_{j}\) are the carbon emissions of the ith and jth grid cells, respectively; \(\overline{x}\) is the mean carbon emissions. The I value ranges from \(-1\) to \(1\). An III value less than \(0\) indicates a negative correlation, while an \(I\) value greater than \(0\) indicates a positive correlation.