## 2 Analysis of Water Accumulation in Rutted Roads
### Road Surface Water Morphology Under the Influence of Ruts  
Within half of the rut, the accumulated water from the grids will be integrated and filled from the bottom of the rut to the top until it is flat with the road surface. Therefore, based on the 3D contour of the rut and the principle of volume equality, we let the accumulated water volume of the rut (\(V_{w}\)) under current rainfall conditions be equal to the volume of the rut (\(V_{R}^{\prime}\)) used to store the accumulated water. As shown in Figure 5, according to the volume calculation principle, it is possible to find the area of accumulated water (\(S_{w}\)) of the vehicle rut under current working conditions and the distance (\(\Delta Z_{w}\)) between the plane and the next nearest calculated surface.  
\[V_{R}^{\prime}=\frac{1}{3\cdot\Delta Z}\sum_{j=1}^{n}\left(S_{j}+S_{j-1}+ \sqrt{S_{j}S_{j-1}}\right)+\frac{1}{3\cdot\Delta Z_{w}}\sum_{j=1}^{n}\left(S_ {w}+S_{n}+\sqrt{S_{w}S_{n}}\right)=V_{w} \tag{8}\]  
According to \(\Delta Z_{w}\) and the superposition of the various contoured line intervals, the maximum depth of water accumulation (\(H_{w}\)) within half of the rut can be determined under current rainfall conditions.  
\[H_{w}=n\Delta z+\Delta Z_{w} \tag{9}\]  
where \(H_{w}\) is the maximum depth of water accumulation within the rut, \(\Delta z\) is the calculated spacing of rut volume, and \(\Delta Z_{w}\) is the height difference between the surface of the rut and the nearest rut surface.  
After identifying the elevation of the lowest point of the rut, we can find the elevation of water surface (\(Z_{w}\)) based on the maximum depth of water accumulation.  
\[Z_{w}=\min\left\{Z_{i,j}\right\}+H_{w} \tag{10}\]  
where \(Z_{w}\) is the elevation of the water surface, \(Z_{i,j}\) is the elevation of any point within the rut, and \(H_{w}\) is the maximum depth of water accumulation.  
Figure 5: Schematic diagram of finding depth distribution of half rut in accumulated water.  
Then, based on the difference between the elevation of the plane and any point within the outline surface inside the rut, the depth of water (\(h_{w,i,j}\)) at any point in the accumulated water area can be calculated.  
\[h_{w,i,j}=Z_{w}-Z_{i,j} \tag{11}\]  
where \(h_{w,i,j}\) is the depth of water accumulation at any point in the rut, \(Z_{w}\) is the elevation of accumulated water surface, and \(Z_{i,j}\) is the elevation of any point within the rut.  
By projecting the identified \(Z_{w}\) contour lines onto the \(XOY\) plane, it is possible to calculate the width (\(W_{w}\)) and length (\(L_{w}\)) of the accumulated water in half of the rut based on the maximum horizontal and vertical coordinates. The specific method is the same as that of the rut width and length.  
If the 3D morphology of the left and right half of the rut is different, the state of water accumulation is different as well. On the basis of 3D road surface reconstruction and rut water accumulation recognition, the road surface is divided into a water surface and a non-water surface by identifying the elevation and location of the left and right half of the rut. For the non-water surface, a prediction model is used to calculate the average water film thickness of the road surface. For the water surface, the water accumulation identification method proposed in this section is used to identify the average water film thickness within each grid (Equation (12)). When calculating the water film thickness, if the grid has both a water surface and non-water surface at the same time, the water film thickness is calculated using the method for a larger area within the grid.  
\[h_{k}=\sum_{i=1}^{m}\sum_{j=1}^{n}h_{w,i,j} \tag{12}\]  
where \(h_{k}\) is the average thickness of water film within the \(k\)-th grid of the road surface, \(h_{w,i,j}\) is the thickness of water film at any point within the grid, and \(m\) and \(n\) are the number of point clouds in the driving direction and the horizontal direction of the road, respectively.