## 2 Physically Based Dynamic Models for Modeling Complex Landslides  
### A Multi-Block Model for Slowly Sliding Landslide  
#### 2.1.1 Basic Assumptions  
The displacement of sliding masses with high integrity develops slowly in the early stages, and the deterioration of their integrity can be ignored. Thus, at this stage, the landslide can be approximated as a series of rigid blocks that are close to each other (Figure 1a). Each block is subjected to a combined action of sliding and anti-sliding forces. A block (\(C_{i,j}\)) in physical space (Figure 1a) corresponds to a grid cell in the Digital Elevation Model (DEM) (Figure 1b). The entire landslide can be discretized into a numerical model with fixed-size cells, each of which contains physical information, such as the mass (\(m\)), fluid height (\(h\)), elevation of basal terrain (\(z_{b}\)), velocity, and stress about the landslide (Figure 1c,d).  
In Figure 2, the red arrow marked by \(\mathbf{n}\) is the normal direction of the tangent plane, and its intersection angle with the positive direction of the \(z\)-axis is denoted by \(\theta\). The intersection angle \(\theta\) between the normal vector \(\mathbf{n}\) and the global \(Z\)-axis is less than \(90^{\circ}\) for most terrain, and the corresponding value of \(n_{z}\) will be positive in these cases. The angles between the tangent plane and the \(x\)-axis and \(y\)-axis are denoted by \(\alpha_{x}\) and \(\alpha_{y}\) in the vertical plane. The elevation function of the basal terrain can be defined as \(f\left(x,y,z\right)=z-z_{b}\left(x,y\right)=0\), which can obtain the elevation values of the cells from the digital elevation model (DEM) in applications. The elevation, as a scalar, is defined at the center of the cells. The accuracy of terrain function data generally depends on the resolution of the DEM.  
Figure 1: Schematic diagram of landslide. (**a**) Landslide in physical space; (**b**) a top view of a group of blocks; (**c**) a side view of a group of blocks; (**d**) stresses acting on a typical block.  
The normal vector \(\mathbf{n}\) of the tangent plane passing through any point on the terrain (Figure 2) can be expressed as [19]  
\[\mathbf{n}=\left(n_{x},n_{y},n_{z}\right)=\frac{\left(-\partial z_{b}/\partial x,-\partial z_{b}/\partial y,1\right)}{\left[\left(\partial z_{b}/\partial x \right)^{2}+\left(\partial z_{b}/\partial y\right)^{2}+1\right]^{1/2}}, \tag{1}\]  
From Equation (1), we can obtain  
\[n_{z}=\cos\theta=\frac{1}{\left[\left(\partial z_{b}/\partial x\right)^{2}+ \left(\partial z_{b}/\partial y\right)^{2}+1\right]^{1/2}}, \tag{2}\]  
After substituting the geometric relationship, namely \(\tan\alpha_{x}=\partial z_{b}/\partial x\) and \(\tan\alpha_{y}=\partial z_{b}/\partial y\), into Equation (2), we obtain  
\[\tan^{2}\theta=\tan^{2}\alpha_{x}+\tan^{2}\alpha_{y}, \tag{3}\]  
Therefore, intersection angle \(\theta\) can be determined via the tangent values of \(\alpha_{x}\) and \(\alpha_{y}\), which can be obtained directly using the digital elevation model (DEM). The central difference method can be used to calculate these tangent values in the \(x\) and \(y\) directions:  
\[\tan\alpha_{x}^{i,j}=\frac{\partial z_{b}}{\partial x}=\frac{z_{b}^{i+1,j}-z_ {b}^{j-1,j}}{2\Delta x}, \tag{4}\]  
\[\tan\alpha_{y}^{i,j}=\frac{\partial z_{b}}{\partial y}=\frac{z_{b}^{i,j+1}-z_ {b}^{i,j-1}}{2\Delta y}, \tag{5}\]  
where \(\Delta x\) and \(\Delta y\) are the cell sizes of DEM in the \(x\) and \(y\) directions, respectively. The superscripts \(i\) and \(j\) denote the cell indexes in the \(x\) and \(y\) directions (Figure 1b).