## 3 Methodology
### Verification and Evaluation of Elevation Change Accuracy  
According to the principle of statistics, the error of elevation difference (\(\sigma_{c}\)) is generally expressed by:  
\[\sigma_{c}=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}, \tag{1}\]  
where \(\sigma_{1}\) and \(\sigma_{2}\) are the errors of the first DSM dataset and the second DSM dataset, respectively. According to the previous study [52; 58; 53], for most of the two datasets, the probability density function is well characterized by a Gaussian distribution. It is known that the relative height errors of SRTM DSM and AW3D30 DSM in the study area are 4 m and 5 m, respectively. Therefore, based on the principle of Statistics (Formula (1)), the joint error of the height difference between the two DSMs is 6.4 m.  
Figure 3: 100 landmark invariable buildings were selected as control points. In order to verify the consistency between the elevation change of the two DSMs and the increase of the actual vertical height of the building, 200 randomly distributed checkpoints were selected manually, as shown in Figure 5. The observation items included (1) verifying whether there was some historical information about building demolition or construction during that period, and (2) whether the calculated elevation change was consistent with the height change of building demolition or construction. Generally, if the difference between the two was less than 6 m, was is considered that the two were consistent, and the conclusion of judgment was set as T (true); otherwise, was is F (false). Through building history retrieval and Google Map time-series images analysis, the authenticity of the corresponding relationship between the elevation changes and the  
Figure 4: (**a**) Elevation change (m) obtained from the DSMs in the study area; (**b**) Enlarged view of elevation change in central urban area of Beijing. Figure 5: Distribution of verification points for the authenticity of elevation change. building expansion is verified accurately. Google Map time-series images analysis consists of several steps (Table 3). First, the point to be verified on Google map was located, then the changes of scenery and the height of building facilities on the historical images from 2000 to 2009 were compared, and combined with relevant construction information. Then, it was determined whether the difference between the actual height of building facilities and the elevation difference derived from DSMs was greater than 7 m, as well as whether it belonged to the true height change of buildings. The verification result is shown in Figure 6. The point above 0 indicates where the height increased, and the size of its value indicates the amount of growth. The point below 0 indicates where the height decreased, and the size of its value indicates the amount of reduction. The verification result shows that the calculated elevation difference is inconsistent with the actual building height change in 7 random checkpoints, and the other 193 elevation changes are consistent with the building growth height. Although these tests are not enough to draw a conclusion on this consistency probability, some trends can be basically found, that is, among them, the probability of inconsistency in negative value \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Text Information** & **Height Change** & **Google Image before Change** & **Google Image after Change** \\ \hline Beijing Yintai Center & & & \\ Built in 2008 & & & \\ Increased by 141 m & & & \\ True & & & \\ \hline Chengji building & & & \\ Build in 2007 & & & \\ Increased by 60 m & & & \\ True & & & \\ \hline Tian an men & & & \\ No record of & & & \\ expansioin & & & \\ But increased by 17 m & & & \\ False & & & \\ \hline Old building & & & \\ demolished & & & \\ Reduced 30 m in 2005 & & & \\ True & & & \\ \hline Old building & & & \\ demolished & & & \\ Reduced 53 m in 2005 & & & \\ True & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Verification and evaluation of elevation change accuracy.