# Variability of Interpolation Errors and Mutual Enhancement of Different Interpolation Methods
## 3 Results  
### Similarity of Interpolation Results  
The parameters for the four interpolation methods were set to default, and the resulting RMSE values are presented in Table 1. For both elevation and PM\({}_{2.5}\) concentration data, the OK and UK methods exhibited similar accuracy. In the case of elevation data, all four methods showed comparable performance, with only minimal differences in the RMSE. However, the interpolation accuracy for elevation data was consistently higher than that for PM\({}_{2.5}\) data. Four interpolation methods were used to generate PM\({}_{2.5}\) concentration and gully elevation maps, and RDC was employed to assess the similarity between different interpolation results, as shown in Table 2. For PM\({}_{2.5}\) concentration, the highest similarity was observed between OK and UK, with an RDC value of 0.9938. The least similar results were from IDW and UK, with an RDC of 0.9433. For gully elevation, the highest similarity was again between OK and UK (RDC = 0.7524), while the lowest similarity was observed between RBF and UK (RDC = 0.4398). \begin{table}
\begin{tabular}{c c c} \hline \hline
**RMSE** & **PM\({}_{2.5}\)** & **Elevation Data** \\ \hline IDW & 0.366 & 0.006 \\ RBF & 0.368 & 0.005 \\ UK & 0.374 & 0.006 \\ OK & 0.372 & 0.006 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSE of four kinds of interpolation of PM\({}_{2.5}\) and elevation data (unit: \(\mu\)g\(\cdot\)m\({}^{-3}\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **IDW** & **OK** & **RBF** & **UK** \\ \hline IDW & & (0.9467; 0.4649) & (0.9897; 0.7167) & (0.9460; 0.4952) \\ OK & (0.9447; 0.4541) & (0.9606; 0.4737) & (0.9938; 0.7672) \\ RBF & (0.9896; 0.7362) & (0.9614; 0.5197) & (0.9614; 0.5194) \\ UK & (0.9433; 0.4522) & (0.9937; 0.7524) & (0.9600; 0.4398) 1 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relative difference coefficient from different interpolation methods for PM\({}_{2.5}\) and elevation point data. The similarity between the results from different interpolation methods varies, but the trend remains consistent. For example, if Dataset A (PM\({}_{2.5}\) concentration interpolation errors in Xinyang City, Henan Province) shows similar results between IDW and RBF, then Dataset B (interpolation errors of gully data in Shadi Village, Yuanmou County, Yunnan Province) is likely to show similar results between these methods, although the degree of similarity may vary, as illustrated in Figure 5.  
An interesting phenomenon emerges from the differing point densities between the PM\({}_{2.5}\) concentration and gully elevation data. When point density is higher, interpolation errors tend to be smaller, but the similarity between the results from different methods is lower. Conversely, when point density is lower, the overall interpolation results become more similar, even though the error across methods may be larger. The functional relationships between PM\({}_{2.5}\) concentration and gully elevation, as modeled by scatter plots using different interpolation methods, are presented in Table 3. The coefficient of determination (R\({}^{2}\)) from linear regression serves as the evaluation criterion. A higher R\({}^{2}\) indicates a better model fit and a stronger correlation between the two interpolation methods. The relationships between the different interpolation methods for the PM\({}_{2.5}\) concentration and gully elevation data are summarized in Table 3.  
As shown in Table 3, the scatter plots of interpolation errors for the different methods are relatively clustered. In the PM\({}_{2.5}\) concentration sample area, the R\({}^{2}\) values for the scatter functions vary. Specifically, the R\({}^{2}\) values for the universal kriging (UK) and inverse distance weighing (IDW) methods are 0.9472, indicating the largest error difference between UK and IDW. Meanwhile, the R\({}^{2}\) value between OK and IDW is 0.9473, suggesting a similar level  
Figure 5: Similarity in interpolation error results: (**a**) is the PM\({}_{2.5}\) concentration and (**b**) is the elevation data.