# The Fractal Characteristics of Ground Subsidence Caused by Subway Excavation
## 4 Fractal Interpolation Simulation
### The Effect  
Figure 2 illustrates the surface deformation to have undergone two distinct stages, namely the \"obvious deformation period\" and the \"unclear deformation period\". Over the course of 100 days, the surface deformed significantly, with a maximum deformation of 6.44 mm. Over the subsequent 100-290 days, the surface deformation tended to stabilize.  
Following the principle of random midpoint [57; 58], we separately selected nine interpolation points in the \"obvious deformation period\" and \"unclear deformation period\", and used the Matlab program to calculate the fractal interpolation of the curves in the above two periods. the results are shown in Figures 7 and 8.  
Figure 6: Comparison between the original monitoring data and de-noised data.  
As illustrated in Table 2, 291 data points were derived from 18 interpolation points through fractal interpolation in this study, representing a 2.94-fold increase in the original monitoring data information, with approximately 18% of the original monitoring data utilized. This equates to the daily monitoring of surface deformation. Furthermore, as illustrated in Figure 9, the mean relative error between the fractal interpolation results and the monitoring data is approximately 13%, thereby demonstrating that the fractal interpolation methodology employed in this study is highly accurate. It is important to note that the relative errors between the fractal interpolation results and the monitoring data are greater when the monitoring periods are 13-26 days, 194-222 days, and 244-264 days, with values of 58.31%, 33.87%, and 29.99%, respectively. This is due to the fact that the raw monitoring curves in these three time periods exhibit pronounced spikes and significant fluctuations in the data (as illustrated in Figure 6), which increases the error.  
Figure 7: The fractal interpolation effect in the “obvious deformation period”. (**a**) Interpolation effect of the first iteration; (**b**) interpolation effect of the second iteration; (**c**) interpolation effect of the third iteration; (**d**) interpolation effect of the fourth iteration.  
Figure 8: The fractal interpolation effect in the “unclear deformation period”. (**a**) Interpolation effect of the first iteration; (**b**) interpolation effect of the second iteration; (**c**) interpolation effect of the third iteration; (**d**) interpolation effect of the fourth iteration.  
\begin{table}
\begin{tabular}{c c c} \hline
**Time Period/d** & **Monitoring Data (Quantity)** & **Interpolation Results (Quantity)** \\ \hline
1–13 & 0.0324 (2) & 0.0384 (13) \\
13–26 & 0.0818 (5) & 0.1295 (13) \\
26–40 & 1.8905 (6) & 2.0088 (14) \\
40–50 & 3.8863 (6) & 4.3266 (10) \\
50–63 & 4.6987 (6) & 4.8323 (13) \\
63–75 & 5.2813 (4) & 5.6643 (12) \\
75–89 & 2.5576 (8) & 2.5132 (14) \\
89–100 & \(-\)0.4147 (9) & \(-\)0.3984 (11) \\
100–124 & 0.2445 (16) & 0.2314 (25) \\
124–148 & 0.8864 (9) & 0.9232 (24) \\
148–170 & 0.8730 (6) & 0.9480 (22) \\
170–194 & 0.3345 (6) & 0.3196 (24) \\
194–222 & 0.4895 (3) & 0.6553 (28) \\
222–244 & 0.3798 (3) & 0.3528 (22) \\
244–264 & 0.3385 (4) & 0.2370 (20) \\
264–290 & 0.1087 (7) & 0.1184 (26) \\ \hline \end{tabular}
\end{table}
Table 2: Comparison between the fractal interpolation results and the monitoring data.  
It can be found that the slope of In(r) and In(N(1/r)) are 1.3406. That is, the fractal dimension of the Von Koch curve is 1.3406, which differs from the theoretical value of 1.26 [59] by 6.4%. It is considered that the algorithm to calculate fractal dimension is accurate.