## 4 Landslide Volume Prediction Model
### Predictive Model Development  
To predict (S-growth curve model) the landslide volume in the deformation zone, we used two consecutive observations (Vt, Vt+1) to train the parameters of the model so as to predict the landslide volume Vt+2 at moment t + 2, and then repeated this process in the following time domain, which is similar to the dynamic update method. The landslide volume Vt+2 at moment t + 2 was predicted using a modified S-shaped growth curve model as shown in Equation (4). \[\mathrm{V_{t+2}}=\frac{\mathrm{K^{\prime}}}{1+\mathrm{b_{t}e^{-a_{t}(x_{t+2}- x_{t})}}},\ a_{t}>0,\ b_{t}>0,\mathrm{K^{\prime}}>0 \tag{4}\]  
where at and bt are the parameters that affect the erosion rate at the time of t. xt+2 and xt are the cumulative rainfall at time t + 2 and t.  
Assuming that the cumulative rainfall xt corresponding to the first observation selected is zero, according to the observations Vt and Vt+1, the value of bt can be calculated by Equation (1) and then at can be calculated as follows:  
\[\mathrm{b_{t}}=\frac{\mathrm{K^{\prime}}}{\mathrm{V_{t}}}-1 \tag{5}\]  
\[\mathrm{a_{t}}=-\frac{\ln\frac{\frac{\mathrm{K^{\prime}}}{\mathrm{b_{t+1}}}-1} {\mathrm{b_{t}}}}{\mathrm{x_{t+1}}-\mathrm{x_{t}}} \tag{6}\]  
where K^\(\prime\) is the environmental capacity, which here refers to the percentage of landslide volume after stabilization. As can be seen from the results in Table 2, K^\(\prime\) has a linear relationship with the seven factors we examined, namely:  
\[\mathrm{K^{\prime}}=\lambda\mathrm{i}+\gamma \tag{7}\]  
where i is the influence factor of the final damage volume of the landslide, and for experiments S11-S16, here, i refers to the vegetation cover density and typhoon wind speed, respectively. The coefficients \(\lambda\) and \(\gamma\) can be found by linear regression of the experimental data. Since the rainfall intensity was constant during the experiment according to the  
Figure 5: S-growth curve fitting (S11–S16). begin to appear. When the internal fracture of the slope is fully penetrated and a complete sliding surface is formed, the first sliding of the middle and low part of the slope provides space for larger sliding to occur in the upper part, which will subsequently produce larger sliding damage, which also corresponds to the peak of the area change rate and volume change rate. The development of the deformation zone in the process of slope instability failure is mainly divided into three stages which can be called the preslip stage, the sliding failure stage, and the stabilization stage. The preslip stage refers to the interval from stability to near the appearance of damage on the slope; the performance of this stage is that the slope, under the action of rainfall, gradually increases in moisture content, increasing the slope's weight. The sliding force gradually increases close to the antislip force, and the landslide area and volume in this process have almost no value. The sliding failure phase refers to the interval between the beginning of the slope and the end of the failure. As shown in Figure 6, the slide failure stage divides the landslide into two types: progressive type and progressive backward type according to the type of spatial failure [34]. The spatial damage types of all experiments almost always fit these two patterns (Figure 7a). The progressive failure is manifested as follows: Under the action of a large sliding force, the trailing edge of the landslide first produces tensile cracks, and then the soil in the back section continues to develop forward and inside the slope, and the soil in the antislip section in front of the push is deformed, forming a rising crack on the leading edge; this type of landslide failure is faster, the landslide area and volume increase exponentially and then tend to be stable, and the landslide area and volume change rate curve have a large peak (Figure 7b).