# Dispersion Curve Interpolation Based on Kriging Method
## 5 Discussion  
According to Tables 1 and A1, when ordinary kriging is used in process, \"spherical\" is the best variogram 60% of the period; when universal kriging is used, the ratio is 100%. Thus, the most suitable variogram are determined to be \"spherical\". The results of the first test are shown in Figures A1-A9. The lost point was due to the distance and deviation angle limitations mentioned in Section 3.2. The RMSE was calculated to be 0.06. The max errors were 3% for OK and 4.6% for UK (relative error in Figure 5), which can be considered acceptable [37]. Therefore, we believe that interpolating dispersion curves using kriging is feasible, and this method can be used to obtain an accurate dispersion curve. In addition, combined with the RMSE of the remaining tests mentioned in Section 4.1, the results will be more accurate if we have more input data. It is worth noting that the red lines in Figures A1-A9 have similar shapes. This shows that the method works well for models where the V\({}_{\text{S}}\) structure is not complex. According to Figure 6, the deviation of the dispersion curves interpolated by UK was less than with OK. This is because ordinary kriging requires that the regionalized variable Z(x) be second-order stationary (i.e., E[Z(x)] is a constant); however, the composition of the material in the earth's crust is complex, and some of the material in the volcanic area comes from deeper. Therefore, UK is more suitable for volcanic areas than OK. The results shown in Figures A10-A13 prove that the results were not casual, so we chose UK as the'method' parameter for interpolation. The third test is the sensitivity test about the parameters proposed in Section 3.2. The results demonstrate how the distance (search scope) influences the interpolation accuracy (Figure 7a-d). Figure 7a-d represent the RMSE of the interpolated results with search scopes of 10,000 m, 15,000 m, 20,000 m, and 30,000 m, respectively. Moreover, the mean of each RMSE is shown in the top right-hand corner. The larger the search scope, the higher the time cost; however, it is not necessary to obtain more accurate results. According to Table A2, \"angle\" hardly affects the accuracy of the interpolation, because \"angle\" is proposed to limit the range of interpolation, and \"distance\" mainly determines the amount of data participating in the interpolation. There is no real data to be compared with the predicted outside the array. If not restricted, the predicted results are meaningless. When \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Depth (km)** & **Original** & **Interpolating** & **Expansion Multiple** \\ \hline
1 & 225 & 479 & 2.129 \\
2 & 240 & 479 & 1.996 \\
3 & 239 & 479 & 2.004 \\
4 & 243 & 479 & 1.971 \\
5 & 245 & 479 & 1.971 \\
7 & 234 & 479 & 1.955 \\
10 & 235 & 479 & 2.038 \\
12 & 218 & 479 & 2.197 \\
15 & 198 & 479 & 2.419 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Expansion multiple at each depth. \"angle\" is set larger, more predicted results are obtained, and more results are unacceptable. Therefore, \"distance\" is set to be 15,000 m, and \"angle\" is set to be 265\({}^{\circ}\) in this study. In the application of the interpolation method, different depths have different expansion multiples (Table 3). This is because different original dispersion curves have different period ranges. There are few original data in the long periods (e.g., larger than 10 s). Comparing Figures 4a-c and 4d-f, the interpolation results with grid translation (Figure 4a-c) were very similar to the original results (Figure 4d-f), which provides evidence of the feasibility of dispersion curve interpolation using kriging. For the changes in the depth (e.g., 15 km), on the one hand, direct inversion will obtain more detailed information about areas with high ray density. The results obtained by inverting the predicted dispersion curves will be affected by the forward process and the interpolation process, which makes the results in the depth (e.g., 15 km) smooth.