## 3 Soil Erosion Area Prediction Model
### Development of Prediction Model  
In order to predict the S-shaped growth curve framework based on different soil materials under different rainfall intensities, we used the previous observation values to train the parameters of a model to predict the object in the next time domain and then repeat the process in the following time domain, which is similar to the dynamically updated method. The modified S-shaped growth curve framework is used to predict the eroded area \(s_{t+1}\) at time \(t\)+1, as in Equation (2):  
\[s_{t+1}=\frac{k^{\prime}}{1+b_{t}e^{-\alpha_{t}(x_{t}-x_{t-1})}},\alpha_{t}>0, k^{\prime}>0,b_{t}>0 \tag{2}\]  
where \(\alpha_{t}\), related to material, is the parameter affecting the erosion rate at time \(t\). It is considered a variable as time elapses because the particles of the soil change. \(b_{t}\) is considered as the variable which depends on the cumulative eroded area and rainfall intensity at time \(t\). \(x_{t}\) and \(x_{t-1}\) are the cumulative rainfall at time \(t\) and \(t\)-1, respectively. In Equation (1), the initial value of \(b_{0}\) can be calculated when \(t\) is zero. Therefore,  
\[b_{0}=\big{(}\frac{k^{\prime}}{s_{0}}-1\big{)} \tag{3}\]  
where \(k^{\prime}\) is the \"saturation value\" or \"carrying capacity\" of the eroded area. From the results in Table 1, it is found that \(k\) is mainly affected by rainfall intensity \(m\) linearly, rather than the material parameter \(a\), which is read as \begin{table}
\begin{tabular}{c c c c} \hline Name & \(k\) & \(a\) & \(b\) \\ \hline Soil Sample 1-25 mm/h & 3000 & 1.088 \(\times\) e\({}^{4}\) & 10.4 \\ Soil Sample 1-45 mm/h & 6350 & 4.75 \(\times\) e\({}^{5}\) & 15.2 \\ Soil Sample 1-65 mm/h & 9250 & 4.215 \(\times\) e\({}^{5}\) & 1.176 \\ Soil Sample 2-45 mm/h & 6350 & 4.4 \(\times\) e\({}^{5}\) & 20.6 \\ Soil Sample 3-45 mm/h & 6350 & 4.267 \(\times\) e\({}^{5}\) & 16.9 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of five S-shaped growth curves. Figure 12: S-growth curve fitting: (**a**) soil Sample 1-25 mm/h; (**b**) soil Sample 1-45 mm/h; (**c**) soil Sample 1-65 mm/h; (**d**) soil Sample 2-45 mm/h; and (**e**) soil Sample 3-45 mm/h. \[k^{\prime}=\lambda m-\mu,m>3.44 \tag{4}\]  
where \(m\) is the rainfall intensity and the coefficients \(\lambda\) (147.5) and \(\mu\) (507.5) of a linear relationship can be obtained by experimental data fitting. The minimum value of \(m\) is 3.44, meaning the threshold of erosion beginning (in order to make \(k^{\prime}\) positive). If the detachment capacity is significantly lower than the transport capacity, the process is referred to as \"detachment-limited erosion\" [21]. Materials are transported only after being detached. Thus, it should have a threshold of the rainfall amount, which is the start point of soil erosion. This agrees with the fact that the shear force induced by runoff over the resistance force is a precondition of soil loss, as reported by The Water Erosion Prediction Project [22] and SHETRAN [23]. The final model is as follows after putting Equations (3) and (4) into Equation (5):  
\[s_{t+1}=\frac{\lambda m-\mu}{1+(\frac{\lambda m-\mu}{s_{t}}-1)e^{-\alpha_{t}(x _{t}-x_{t+1})}},\alpha_{t}>0,k^{\prime}>0,s_{t}>0 \tag{5}\]  
For the proposed model framework, we just need to input the rainfall intensity and the calculated cumulative rainfall at time \(t\) without other parameters in order to obtain the eroded soil area at time \(t\)+1. When new data at time \(t\)+1 is inputted, this model can predict the eroded soil area at time \(t\)+2. Figure 13 indicates the five predictions of the S-growth curve model based on the physical experiments. The prediction results match the observation values well and the average errors are between 3% and 9.7%.