## 2 Methodology
### Time-Series Analysis
#### 2.4.1 Calculation of the Energy under Gravity  
The first step in the time-series analysis is to calculate the energy of the rock slope under gravity. More specifically, the energy of all rock blocks under gravity are calculated according to the data achieved in Section 2.3. For a three-dimensional rock slope, when the slope is under gravity, some rock blocks on the slope will slide. Figure 3 illustrates the change in displacement of a rock block from \(T_{0}\) to \(T_{n}\). As illustrated in Figure 3, the total displacement of the rock block from \(T_{0}\) to \(T_{n}\) is \(\Delta d\). The total displacement \(\Delta d\) could be decomposed into \(\Delta x\) in the X direction, \(\Delta y\) in the Y direction, and \(\Delta z\) in the Z direction, thus \(\Delta d\) could be expressed as Equation (1)  
\[\Delta d=\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}} \tag{1}\]  
Energy could be expressed as the force times the displacement in the direction of the force. Thus, the energy under gravity is equal to gravity times \(\Delta z\). \(\Delta z\) is equal to the Z coordinate of the block center at \(T_{n}\) minus the Z coordinate of the block center at \(T_{0}\), and the Z coordinate is obtained in Section 2.3. In addition, density is a mechanical parameter determined before DEM modeling, and the energy of all rock blocks under gravity could be calculated according to Equation (2)  
\[Energy=\rho_{n}v_{n}g\Delta z_{n} \tag{2}\]  
Energy--Energy under the gravity (N-m)  
\(\rho_{n}\)--Density (kg/m\({}^{3}\))  
\(v_{n}\)--Volume (m\({}^{3}\))  
\(g\)--Acceleration of gravity (N/kg)  
\(\Delta z_{n}\)--Displacement in Z direction (m)  
The energy-based method proposed in this paper is a generalized form of the displacement-based method [22]; this is because the proposed method considers not only the displacement of each block but also the weight of each block. The proposed energy-based method is applicable to slopes without an obvious potential sliding surface. The displacement-based method considers that the weight of each block is one; however, the weight of each block obtained by the proposed energy-based method is the actual value that is calculated by volume and density. Thus, the proposed energy-based method is more applicable. Figure 2: A simple illustration of a rock slope changing from stability to failure. 2.4.2. Calculation of the Coefficient of Variations Based on Energy  
The second step in the time-series analysis is to calculate the coefficient of variation (CV) of the energy for all rock blocks of a jointed rock slope. The CV, which is also known as the dispersion coefficient, is a normalized measure of the degree of dispersion of the probability distribution. CV has no dimension. Its size is affected not only by the degree of dispersion of variable values but also by the average level of variable values. It is defined as the ratio of the SD to the mean. In this paper, the CV based on energy is used as an important index to determine when the iterative process would be terminated  
\[Mean=\frac{\sum_{i=1}^{n}\rho_{i}v_{i}\delta\Delta z_{i}}{n} \tag{3}\]  
\[SD=\sqrt{\frac{\sum_{i=1}^{n}(\rho_{i}v_{i}\delta\Delta z_{i}-Mean)^{2}}{n}} \tag{4}\]  
\[CV=\frac{SD}{Mean}=\frac{\sqrt{\frac{\sum_{i=1}^{n}(\rho_{i}v_{i}\delta\Delta z _{i}-Mean)^{2}}{n}}}{\frac{\sum_{i=1}^{n}\rho_{i}v_{i}\delta\Delta z_{i}}{n}} \tag{5}\]  
In this section, the mean of energy and the SD of energy for all rock blocks are achieved by conducting statistical analysis; and the CV of energy corresponding to each iteration step is obtained. Then, a curve of the relationship between the CV of energy and the number of iterations is plotted. The mean of energy is calculated by Equation (3), the SD of energy is calculated by Equation (4) and the CV of energy is calculated by Equation (5). Every CV of energy and the corresponding number of iterations constitute a time series.