# Variability of Interpolation Errors and Mutual Enhancement of Different Interpolation Methods
## 3 Results
### Accuracy Evaluation After Fusion of Interpolation Results  
The analysis further highlights that IDW and UK interpolations exhibit the largest differences in PM\({}_{2.5}\) data. Spatial analysis shows that IDW differs most from UK and OK, which are more similar interpolation methods, resulting in minimal differences between the latter two. In fusion accuracy analysis, the fusion results for OK and UK interpolations are consistent in PM\({}_{2.5}\) data. To improve fusion accuracy, IDW and UK interpolations, which exhibit significant differences, were weighted and fused. The global weighting fusion weight for IDW is 0.705, as shown in Table 4.  
The global weighted fusion results indicate that the RMSE of the fused data is approximately 0.004 \(\upmu\)g/m\({}^{3}\), which is lower than the RMSE of IDW and UK before fusion, showing clear improvement. The fused data's minimum value (MIN) is about \(-0.173\)\(\upmu\)g/m\({}^{3}\) and its maximum value (MAX) is approximately 0.248 \(\upmu\)g/m\({}^{3}\), both of which represent significant improvements. The MIN for IDW after fusion improves by approximately 15.564 \(\upmu\)g/m\({}^{3}\) compared to before fusion, while the accuracy of UK improves by around 15.899 \(\upmu\)g/m\({}^{3}\). The average value (MEAN) and standard deviation (STD) have also improved significantly. The MEAN after fusion is about \(-0.001\)\(\upmu\)g/m\({}^{3}\), a reduction of approximately 0.189 \(\upmu\)g/m\({}^{3}\) for IDW and 0.078 \(\upmu\)g/m\({}^{3}\) for UK. The STD after fusion is about 0.063 \(\upmu\)g/m\({}^{3}\), a decrease of approximately 4.726 \(\upmu\)g/m\({}^{3}\) for IDW and 4.813 \(\upmu\)g/m\({}^{3}\) for UK. Overall, all key indicators show substantial improvement, confirming the efficacy of fusion for PM\({}_{2.5}\) concentration data with an interpolation density of 0.009 points/km\({}^{2}\). The difference analysis revealed that the elevation data discrepancies between RBF and UK interpolation errors were the largest. To further enhance fusion accuracy, a weighted fusion of RBF and UK interpolations was performed on the elevation data. The global fusion weight for RBF interpolation was set at 0.603, while the weight for UK interpolation was calculated as the complement (\(1-\) the RBF weight). The dataset contained 8055 sample points, with an interpolation point density of 756,409.05 points/km\({}^{2}\). The results are presented in Table 5. \begin{table}
\begin{tabular}{c c c c} \hline  & **IDW** & **UK** & **Weighted Fusion** \\ \hline RMSE & 0.366 & 0.374 & 0.004 \\ Min. & \(-15.737\) & \(-16.072\) & \(-0.173\) \\ Max. & 16.487 & 15.448 & 0.248 \\ Mean & \(-0.190\) & \(-0.079\) & \(-0.001\) \\ STD & 4.789 & 4.876 & 0.063 \\ \hline \end{tabular}
\end{table} 
Table 4: PM\({}_{2.5}\) concentration feature values based on IDW interpolation and globally weighted fusion using the UK method (unit: \(\upmu\)g/m\({}^{3}\)). \begin{table}
\begin{tabular}{c c c c} \hline  & **RBF** & **UK** & **Weighted Fusion** \\ \hline RMSE & 0.005 & 0.006 & 0.004 \\ Min. & \(-8.417\) & \(-7.987\) & \(-3.380\) \\ Max. & 4.440 & 4.409 & 6.376 \\ Mean & 0.007 & 0.006 & \(-0.001\) \\ STD & 0.511 & 0.527 & 0.376 \\ \hline \end{tabular}
\end{table} 
Table 5: Elevation data feature values based on RBF interpolation and globally weighted fusion using the UK method (unit: km\({}^{3}\)). The global weighted fusion improved certain performance metrics. The RMSE after fusion was approximately 0.004 m\({}^{3}\), lower than the RMSE for UK and RBF before fusion, although the improvement was modest. For UK, the RMSE after fusion was 4.607 m\({}^{3}\) lower than before fusion. The MAX value post-fusion increased to 6.376 m\({}^{3}\), showing no improvement but a slight rise in value. For RBF, the MAX value after fusion was 1.936 m\({}^{3}\) higher than before, and for UK, it increased by 1.967 m\({}^{3}\). The average value (MEAN) and standard deviation (STD) also showed slight improvements.