## 3 Results
### Spatiotemporal Variation of Carbon Budgets in Territorial Space
#### 3.2.2 Spatiotemporal Differentiation and Clustering Effects of Carbon Budgets in Territorial Space  
To visually explore the spatiotemporal correlation effects of the carbon budget pattern in the Dongting Lake region, this section presents SDEs and centroid migration trajectories of carbon budgets, revealing differentiation patterns for the region as a whole and for individual cities (Figure 7). The left side of Figure 7 illustrates the overall spatiotemporal variation in carbon budgets: (1) For centroid migration, the carbon budget centroid is situated near the Yueyang-Yiyang boundary and follows a clockwise trajectory from west to east. From 2005 to 2010, the centroid shifted towards the southeast (11.32 km), and since 2010, it has continued to migrate south (4.69 km). (2) For ellipse shapes, the SDE of carbon budgets extends along the north-south axis. The increasing ratio of the ellipse's major to minor axes each year further reflects the flattening of carbon budgets along the north-south direction. (3) For directional changes, the azimuth angle shows a progression of \"initial contraction followed by expansion\". These results underscore the strong correlation between the north-south spatial arrangement of the Dongting Lake basin and the region's carbon budget distribution. Moreover, the carbon budget differentiation pattern's dependence on the water system increases annually. Compared to the overall differentiation characteristics, the spatiotemporal differentiation in individual cities is more pronounced, as illustrated on the right side of Figure 7: (1) For centroid migration, the carbon budget centroids of the five cities predominantly shift toward internal river systems, underscoring the significant influence of the Dongting Lake region's water ecological resources on these trajectories. (2) For ellipse shapes, the SDE flattening ratios for Yueyang and Yiyang increased by 0.035 and 0.093, respectively, reflecting a specific spatial spillover effect in carbon budgets. In contrast, the flattening ratios for Jingzhou, Yiyang, and Changsha decreased by 0.023, 0.109, and 0.241, respectively, indicating increasing concentration of carbon budgets along their primary directions. Changshashows the highest degree of concentration, with a particularly pronounced polarization of the carbon budget pattern toward central construction land. (3) For directional changes, Yueyang exhibits the most excellent azimuth rotation (2\({}^{\circ}\)46\({}^{\prime}\)34\({}^{\prime\prime}\)), driven by fluctuations in Dongting Lake's water body area. This further highlights the instability of Yueyang's carbon budget dynamics and its pivotal role in shaping the region's overall carbon budget pattern. In conclusion, river system orientation and construction land distribution strongly shape the carbon budget patterns of individual cities. This section utilizes Moran's I index to examine the spatiotemporal clustering characteristics of territorial space carbon budgets. Table 6 displays the spatial correlation of carbon budgets in the Dongting Lake region throughout the study period. Overall, the Moran's I indices for the four periods are 0.595, 0.645, 0.655, and 0.661. The Z-values increase annually, reaching 45.20, 49.00, 49.69, and 50.08--well above the significance threshold of 2.58. Furthermore, the p-values consistently remain below 0.01. These findings confirm a strengthening positive spatial correlation in territorial space carbon budgets in the Dongting Lake region. Figure 8 illustrates the visualized clustering effects. (1) H-H clusters declined from 7.5% to 5.5% between 2005 and 2020, with Jingzhou consistently holding the highest proportion (11.68%). These clusters are primarily located in economically active central urban areas. Carbon budgets remain high overall due to construction land's high carbon emissions and low sequestration capacity. From 2005 to 2020, H-H cluster areas in Jingzhou and Changde exhibited a shrinking trend.