# Variability of Interpolation Errors and Mutual Enhancement of Different Interpolation Methods
## 3 Results
### Spatial Pattern of Error Obtained Using Different Interpolation Methods  
Figure 6 illustrates significant variations in the hotspot distribution of PM\({}_{2.5}\) concentration (0.009 particles/km\({}^{2}\)) across the four interpolation methods, highlighting spatial variability in interpolation errors. Specifically, the interpolation errors in hotspot areas for IDW are concentrated in Huangchuan, Xixian, Luoshan, and Guangshan counties, while the radial basis function (RBF) method is primarily concentrated in Huangchuan, Xixian, and Luoshan counties. OK and UK errors are mainly concentrated in Huangchuan and Xixian counties. Although all four methods have shown errors in Huangchuan and Xixian counties, the greatest differences in error distribution are observed between IDW and UK, as well as between IDW and OK, within these hotspot areas. Regarding coldspot areas, IDW errors are primarily located in Huangchuan County, while RBF, OK, and UK errors are more concentrated in Luoshan County. This suggests that the distribution of interpolation errors for RBF, UK, and OK in coldspot areas is relatively similar, while IDW shows a distinct error pattern. The distribution patterns of interpolation errors between IDW and RBF, UK, and OK are notably different in both hotspot and coldspot areas. These observations suggest that the error distribution characteristics of IDW and the OK and UK methods are not closely aligned.  
Figure 6: PM\({}_{2.5}\) error hotspot distribution by the four methods.  
As shown in Figure 7, the spatial distribution of the four interpolation hotspots for the elevation data (756,409.05 points/km\({}^{2}\)) varies. In hotspot areas (confidence level \(\geq\) 90%), errors from the inverse distance weighting (IDW) method are mainly found on the slope surfaces at the source and near slope surfaces at channel bifurcations; errors from the radial basis function (RBF) method primarily occur on the slope surfaces at the bottoms of gullies; errors from the ordinary kriging (OK) method are concentrated on the slope surfaces at the bottoms of gullies; and errors from the universal kriging (UK) method are predominantly located on the slope surfaces at the source, near channels, and around bifurcations. For coldspot areas (confidence level \(\geq\) 90%), IDW interpolation errors are mainly located on the slope surfaces at river bifurcations and bends; RBF interpolation errors are primarily found on the slope surfaces at the heads of gullies and turns in channels; OK interpolation errors are mainly located on the slope surfaces at the heads of gullies and turns in channels; and UK interpolation errors are primarily found on the slope surfaces at river bends. Based on the distribution of hot- and coldspot areas for the four interpolation methods, the distributions of hot- and coldspot areas for RBF and UK are the most dissimilar, indicating that the error distribution characteristics of these two interpolation methods are not similar. Therefore, different interpolation methods not only exhibit certain differences in data but also show significant variations in spatial distribution.  
Figure 7: Hotspot distribution of elevation errors of the four methods.  
Comparing Figures 6 and 7 reveals that while the errors of different interpolation methods share some spatial distribution similarities, substantial differences also exist. Additionally, the spatial distribution of interpolation errors differs significantly between the two datasets with varying densities (PM\({}_{2.5}\) concentration and elevation data).