## 4 Results and Analysis
### Quantification of Uncertainty in the DoD  
The uncertainty of the data must be considered to distinguish between real changes in the terrain and the noise generated by the various sources of error. The interpretation of the uncertainty in the DoD typically requires two steps:  
1. Propagating the uncertainty in an individual DEM to DoD  
The uncertainty sources in the DEM generated by interpolation of the data for the ground points include the system error of the UAV, the accuracy and density of the point cloud, the filtering error, the surface composition, the sampling interval, and the interpolation method. Uncertainty is expressed as \(\delta z\), and ignoring the horizontal component, the relationship between \(\delta z\) and the actual elevation \(Z_{Actual}\) is as follows:  
\[Z_{Actual}=Z_{DEM}\pm\delta_{Z} \tag{4}\]  
where \(Z_{Actual}\) is the true elevation value, and \(Z_{DEM}\) is the actual elevation value. Use of the RMSE value based on the checkpoint data is one of the most common methods to estimate the DEM uncertainty \(\delta z\). Two methods can be used to evaluate the vertical accuracy of data: comparison with field observation data and comparison with two point clouds [23]. For rigorous estimation of the error, we need to collect the complete checkpoint data for the ground, which clearly is difficult to achieve in mountainous terrain. Figure 8: Time series subsidence curve for the main section: (a) strike subsidence curve fitting; and (b) dip subsidence curve fitting. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data Set** & **Measured Value (m)** & **Fitting Value (m)** & **Relative Error (\%)** \\ \hline Line A 07.20-06.14 & \(-\)1.43 & \(-\)1.44 & 0.70 \\ Line A 09.07-06.14 & \(-\)2.17 & \(-\)2.43 & 11.98 \\ Line A 11.15-06.14 & \(-\)3.30 & \(-\)3.50 & 6.06 \\ Line A 07.31-06.14 & \(-\)3.60 & \(-\)3.66 & 1.67 \\ Line B 07.20-06.14 & \(-\)1.53 & \(-\)1.46 & 4.58 \\ Line B 09.07-06.14 & \(-\)1.86 & \(-\)2.19 & 17.74 \\ Line B 11.15-06.14 & \(-\)2.33 & \(-\)2.78 & 19.31 \\ Line B 07.31-06.14 & \(-\)2.86 & \(-\)2.90 & 1.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of maximum subsidence values between measured values and curve fitting. Therefore, the standard deviation of the M3C2 distance for evaluation in multi-temporal point cloud co-registration is used as an index of the DEM uncertainty \(\delta z\). When calculating the value of the differential DEM, the error will propagate to DoD, and the propagation error (\(\delta\mu\)) is determined as follows [24]:  
\[\delta uDoD=t\sqrt{\left(\delta z_{new}\right)^{2}+\left(\delta z_{old}\right)^ {2}} \tag{5}\]  
where \(\delta uDoD\) is the propagation error, and \(\delta z_{new}\) and \(\delta z_{old}\) are the individual errors in the two DEMs which have been subtracted, respectively, assuming that the error in each grid is random and independent, t is the critical Student's \(t\)-value at a user-chosen confidence interval (i.e., \(t=1\) for 68% confidence interval) [25,26].If the spatially explicit estimation of \(\delta z_{new}\) and \(\delta z_{old}\) does not exist, the combined error can be calculated as a single value of the entire DoD [27]. 2. Assessment of the significance of the propagated uncertainty  
An assessment of the significance of the change in the uncertainty of the elevation in the DoD depends on the threshold selected and whether we discard or apply lower weights for changes in height below the minimum level of detection (minLoD). The propagation uncertainty (i.e., \(\delta uDoD\)) is used to define the threshold for the change in elevation or the minLoD. In the DoD data of 06.14-07.20, the standard deviation values for the M3C2 distance calculated by the airborne LiDAR data of 06.14 and 07.20 were 0.13 m and 0.15 m, respectively, such that a minLoD of 0.20 m was calculated. The change between \(-0.2\) m and 0.2 m in the DoD was considered insignificant and discarded from the results.