## 2 Methods
### Projection of Side Points and Elevation Correction  
The elevation of points in the side point cloud Pside is higher than that of the water surface; that is, points in the side point cloud Pside calculated in Section 2.4 are above the water body. Therefore, while maintaining the x and y coordinates of Pside as constant, the elevation coordinates of each point were corrected to ensure that each point is as close to the water surface as possible. Because there are two junction points A and B on the water surface, and the projected point should be close to the elevation of A and B, these points are selected as the start and end points of the elevation difference, respectively, the height difference weighted by distance of each point in Pside is distributed, and points in Pside are sorted according to the x coordinate from small to large, as shown in Figure 5.  
Figure 5: The schematic diagram of optimal estimation of coastline at cliff. Of note, when calculating the distance here, the distance of the first point in \(\text{P}_{\text{side}}\) is the distance between point A and the first point in \(\text{P}_{\text{side}}\), and the distance of the last point is the distance between the last point and B; then, the calculation formula is as follows:  
\[\Delta h_{i}=\frac{d(\text{P}_{\text{side}}(i),\text{P}_{\text{side}}(i+1))}{ \sum\limits_{i=0}^{N}d(\text{P}_{\text{side}}(i),\text{P}_{\text{side}}(i+1))} (H_{A}-H_{B}) \tag{5}\]  
where, \(\Delta h_{i}\) is the elevation difference allocated to each point; \(d(\text{P}_{\text{side}}(i),\text{P}_{\text{side}}(i+1))\) is the distance between two adjacent points; \(N\) is the total number of points in \(\text{P}_{\text{side}}\); and \(H_{A},H_{B}\) are the elevation of points A and B. Assuming that the point cloud after projection and elevation correction is \(\text{P}^{\prime}_{\text{side}}\),  
\[ \begin{array}{l}\text{P}^{\prime}_{\text{side}}(i)\_x=\text{P}_{\text{side }}(i)\_x\\ \text{P}^{\prime}_{\text{side}}(i)\_y=\text{P}_{\text{side}}(i)\_y\\ \text{P}^{\prime}_{\text{side}}(i)\_z=H_{A}-\Delta h_{i}\end{array}  \tag{6}\]  
Calculation of the Points of Optimal Coastline and Integration with the Points of Coastline Outside the Cliff  
The points of the most inwardly inclined coastline were obtained from the projection and elevation correction of side points in Section 2.5; that is, when the specific terrain under the cliff is not clear, the most inwardly inclined coastline is the extreme coastline toward the land side, such that the coastline can no longer deviate toward the land side. Regarding the coastline of the cliff, if the edge of the cliff is almost vertical to the water surface, the collected edge point cloud is the coastline at the cliff, assuming that the coastline in this case is the outermost coastline. The outermost coastline can be obtained by projecting the boundary at the cliff section in Figure 3 onto the water surface, and correcting the elevation according to the elevations of points A and B. The median value between the innermost and outermost coastlines can be obtained by the design method, and this median value can be considered the optimal estimate of the coastline at the cliff. Specifically, the points of the innermost and outermost coastlines are sorted from small to large according to the x coordinate, and the two coastline points are converted into the same elevation surface and then separately connected into lines. Put the point cloud data of the innermost coastline and the outermost coastline into the same plane, and only consider their x and y coordinates.