# Variability of Interpolation Errors and Mutual Enhancement of Different Interpolation Methods
## 2 Materials and Methods
### Research Methodology  
#### 2.2.1 Interpolation Method  
This study employs four interpolation methods, ordinary kriging (OK), universal kriging (UK), inverse distance weighting (IDW), and radial basis functions (RBFs) [39; 40], to spatially interpolate errors in PM\({}_{2.5}\) concentration data for Xinyang City and elevation point data for Yuanmou County, Gaoxiong Yi Autonomous Prefecture, Yunnan Province [41]. The choice of these methods is driven by their individual strengths and [42] suitability for various spatial data conditions. IDW, which assumes that observations closer to the interpolation point have greater influence, is most effective when the data are evenly distributed. It delivers optimal results when data points are dense and exhibit no significant trends [6]. UK, a classical interpolation technique, is appropriate when spatial correlations exist between data points, making it well suited for datasets that display clear spatial trends or variations [43]. Similarly, OK, another spatially correlated method, is highly effective for larger datasets and tends to provide more accurate interpolations. RBF, using radial basis functions, assumes that the value at any point can be interpolated by weighing nearby observations. This method is ideal for datasets with complex or nonlinear distributions, as it captures local variations more effectively [6]. Given these characteristics, the default parameters of these methods were selected to avoid errors that could arise from parameter optimization. The experiment was conducted using ArcMap 10 on a LEGION REN9000K-34 IAZ system equipped with a 12 th Gen Intel(r) Core(tm) i7-12700 KF processor and an NVIDIA GeForce RTX 3070 Ti 8 GB graphics card. PM\({}_{2.5}\) concentration data for Xinyang City were sourced from air quality records published by the Xinyang Municipal People's Government in May 2020. The elevation data for Shadi Village in Yuanmou County were collected in July 2011 using high-precision surveying instruments, including the Leica Total Station TCR802, following standards for large-scale surveying and high-density mixed sampling. To ensure data quality, the PM\({}_{2.5}\) concentration and elevation data were thoroughly examined. Distribution patterns were analyzed using histograms and normal Q-Q plots to determine if transformation was necessary to achieve normality. Trend analysis was conducted to understand the overall data behavior. The spatial autocorrelation and directional influence were assessed using semivariogram and covariance cloud analysis. Points requiring weighting were subsequently denoised and standardized. Finally, optimal weights for interpolation errors were determined using variance and correlation coefficient analysis.  
The point cloud density of PM\({}_{2.5}\) concentration data for Xinyang City is 0.009 points per square kilometer, indicating a uniform distribution that follows a normal distribution. For Yuanmou County's elevation point data, the point cloud density is 0.76 points per square meter, with a dense and normally distributed pattern.