## 2 Methods
### Point Cloud Extraction at the Cliff  
The point cloud extraction at the cliff is now defined. Here, P represents a point cloud dataset, P\({}_{\text{text}}\) represents a point cloud dataset named text, and P\({}_{\text{text}}(i)\) represents the point with number of \(i\) in the dataset, P\({}_{\text{text}}(i)\_x,P_{\text{text}}(i)\_y,P_{\text{text}}(i)\_z\) represents the coordinates of the point, P\({}_{\text{text}}(i)\_I\) represents the echo intensity of the point, and P\({}_{\text{text}}(i)\_N\) is the number of echoes of the point. For the coastal zone data form, as shown in Figure 1b, the boundary point cloud dataset was obtained using the extraction method, provided in the literature [5], in which edge points were extracted based on the distribution rule of acquired data during airborne lidar scanning. The distribution rule is roughly as follows: the x or y coordinates of the points will change from small to large, then from large to small, and then from small to large with the scanning line. The point cloud dataset of the boundary was set as P\({}_{\text{border}}\), which contains both the boundary points of the normal coastal zone and those near the sea at the cliff. There were two special transition points at the intersection of these two types of boundary point clouds, points A and B (Figure 3). These two points connect the boundary point set of the normal coastal zone to the edge point set of the cliff, and then again to the boundary point set of the normal coastal zone. For the boundary point cloud of the normal coastal zone, the elevation difference between two adjacent points was relatively small, and the maximum elevation difference was approximately 10 cm. As such, when point A was reached, the elevation of the point next to A was much higher than that of point A, and then the elevation of the point increased. When the point reached the top, the elevation decreased. Once it was reduced to junction point B, the elevation change of the point was small, and it transited to the boundary point cloud of the normal shoreline. Because the magnitude of the elevation changes of points was not easy to determine, the change ratio of the difference in elevation between the rear and current points, and that between the  
Figure 2: The methodological flow chart. front and current points, was evident. Therefore, the corresponding judgment rules can be set according to this feature to find points A and B. First, the original coastal zone data is imported into existing point cloud processing software to determine the primary information of the approximate trend of the coastline. If the coastline is mainly east-west or west-east, the points in \(\mathrm{P_{border}}\) are sorted from small to large by the x coordinate; if the x coordinate is the same, the points are sorted from small to large by the y coordinate; if x and y are the same, the points are sorted from small to large by the z coordinate. If the coastline is mainly north-south or south-north, the points in the \(\mathrm{P_{border}}\) are sorted from small to large by y-coordinate. If the y-coordinate is the same, the points are sorted from small to large using the x-coordinate. If x and y are the same, the points are sorted from small to large according to the z-coordinate, and the first and last points are removed. The following relevant processing methods take the coastline which is roughly east-west or west-east, as a model. If the coastline is roughly north-south or north-south, the x- and y-coordinates are accordingly adjusted. The following equation was applied and calculation was performed:  
\[H_{\mathrm{ratio}}=\left|\frac{\mathrm{P_{border}}(i+1)\_z-\mathrm{P_{border}}(i)\_z}{ \mathrm{P_{border}}(i)\_z-\mathrm{P_{border}}(i-1)\_z}\right|\ i=2\cdots,N-1 \tag{1}\]  
where, \(N\) is the total number of points in \(\mathrm{P_{border}}\). When the value of \(H_{\mathrm{ratio}}\) is close to one, it can be regarded as the boundary point of the normal coastline.