## 5 Conclusions  
In this paper, we propose a DSM generation method for satellite images based on a double-penalty bundle adjustment optimization algorithm. The proposed method can automatically generate 3D point clouds and DSM without ground control points. The DSM generated by the proposed method has a good visual quality in the building area, when the acquisition interval is less than 15 days, the sun angle difference is less than 20\({}^{\circ}\), and the intersection angle is from 3\({}^{\circ}\) to 15\({}^{\circ}\). The experimental results show that the completeness and the accuracy of the DSMs generated by the proposed method are better than those of the DSMs generated by the traditional BA method and the S2P method. The visual quality experiment and the quality evaluation experiment demonstrate the effectiveness and the robustness of the proposed method. However, for satellite image pairs with the sun angle difference larger than 20\({}^{\circ}\), the intersection angle larger than 15\({}^{\circ}\), or an area with more vegetation, the DSMs generated by the proposed method need to be improved. This is what we're going to research in the future work.  
Conceptualization, H.L. and J.Y.; methodology, H.L. and L.J.; software, H.L.; validation, H.L. and J.Y.; formal analysis, H.L., L.J. and J.Y.; investigation, H.L. and L.J.; resources, H.L.; data curation, H.L.; writing--original draft preparation, H.L.; writing--review and editing, H.L. and L.J.; visualization, H.L. and L.J.; supervision, J.Y.; project administration, J.Y.; funding acquisition, H.L. and J.Y. All authors have read and agreed to the published version of the manuscript.  
The work has been supported by Major Program of National Natural Science Foundation of China NSFC (nos.12292980, 12292984); by National Key R\D Program of China (nos. 2023 YFA100900, 2023 YFA1009004, 2020 YFA0712203, 2020 YFA0712201); by Key Projects of National  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**AOI** & **Method** & **CP (\%)** & **ME (m)** \\ \hline \multirow{3}{*}{Site 1} & Multi-date S2P [20] & 71.87 & **0.60** \\  & Traditional BA [32] & 71.99 & 0.68 \\  & Proposed method & **72.60** & 0.65 \\ \hline \multirow{3}{*}{Site 2} & Multi-date S2P [20] & 65.67 & **0.42** \\  & Traditional BA [32] & **66.98** & 0.55 \\  & Proposed method & 66.91 & 0.47 \\ \hline \multirow{3}{*}{Site 3} & Multi-date S2P [20] & **71.47** & 0.52 \\  & Traditional BA [32] & 61.58 & 0.78 \\  & Proposed method & 68.61 & **0.46** \\ \hline \multirow{3}{*}{Site 4} & Multi-date S2P [20] & 50.49 & 0.99 \\  & Traditional BA [32] & 52.71 & 0.94 \\ \cline{1-1}  & Proposed method & **62.66** & **0.66** \\ \hline \hline \end{tabular}
\end{table}
Table 4: The quality evaluation results of DSMs generated from multi-date satellite images.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**AOI** & **Method** & **Number of Iterations** & **Optimize Time (s)** \\ \hline \multirow{3}{*}{Site 1} & Traditional BA [32] & 59 & 19.73 \\  & Proposed method & 56 & **18.21** \\ \hline \multirow{3}{*}{Site 2} & Traditional BA [32] & 61 & 16.21 \\  & Proposed method & 63 & **15.91** \\ \hline \multirow{3}{*}{Site 3} & Traditional BA [32] & 58 & 19.94 \\  & Proposed method & 65 & **19.72** \\ \hline \multirow{3}{*}{Site 4} & Traditional BA [32] & 61 & 15.22 \\  & Proposed method & 62 & **14.68** \\ \hline \hline \end{tabular}
\end{table}
Table 5: The optimization performance of the traditional BA method and the proposed method.  
Natural Science Foundation of China NSFC (no. 12031016); by Beijing Natural Science Foundation (no. BNSF-Z210003); and by the Department of Science, Technology, and Information of the Ministry of Education (no. 8091B042240); by the Fundamental Research Funds for the Central Universities (no. 241202 QD024).  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors declare no conflict of interest.