## 4 River Blockage Criteria  
### An Improved Probability Model  
Landslides may induce two types of river blockage: complete blockage and partial blockage. The secondary disasters induced by these blockage types have completely different developing processes and impact ranges and offer different degrees of hazard. Compared with partial blockage, complete blockage is induced by a larger landslide dam and could form a more dangerous dammed lake. If the massive dammed lake abruptly ruptures, it may produce severe mudslides or flood disasters downstream. Because the dangers of complete blockages are much greater than those of partial blockages, it is critical to evaluate the degree of river blockage caused by landslides. All experimental results were used to explore an assessment model of river blockage under different landslide conditions. Because many factors could affect the degree of river blockage, it is difficult to obtain a quantitative assessment that approximates reality due to the limited experimental factors. Thus, a logic evaluation was carried out by simply assuming complete blockage and partial blockage as 1 and 0 to enhance the applicability of the following method. Here, logistic regression analysis was used to establish the strongest relationships between the river blockage state and the investigated experimental factors. This could also be applied to more complicated cases by taking more factors into consideration. According to the linear probability model, the probability function of complete river blockage can be expressed as:  
\[\hat{y}=P(y=1|\mathbf{x})=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots+\beta_{n} x_{n}, \tag{1}\]  
where \(y\) indicates the river blockage state, \(\hat{y}\) indicates the probability of complete river blockage, and \(x_{i}\) and \(\beta_{n}\) indicate factors that could affect the river blockage state and the corresponding fitting coefficients. The linear probability model has two drawbacks. First, it violates the assumption of the Gauss-Markov theory that makes the covariance nonzero, thus leading to poor regression. Second, \(\hat{y}\) may indicate two meaningless results where the probability of complete river blockage is negative or greater than 1. Thus, the sigmoid function was selected as a connection function to overcome these drawbacks, and the improved probability model is expressed as:  
\[S(\mathbf{x},\mathbf{\beta})=P(y=1|\mathbf{x})=\frac{1}{1+e^{-\hat{y}}}, \tag{2}\]  
The result of \(S(\mathbf{x},\mathbf{\beta})\) is a number within the range [0, 1]. If a result is less than 0.5, it means a partial blockage; otherwise, it means a complete blockage. The coefficients \(\mathbf{\beta}\) could be estimated by the least squares method. Three factors were considered in this experiment, i.e., the slide angle \(\alpha\), the landslide volume \(V\), and the particle size \(D\). Because the particle size of each group was randomly distributed in a range, the average particle size (\(D_{50}\)) was used to determine the fitting coefficients. It can be calculated as the average value of the maximum and minimum particle sizes, so the particle size ranges of 0.1-0.2 cm, 0.2-0.5 cm, and 0.5-1.0 cm are represented by 0.15 cm, 0.35 cm, and 0.75 cm, respectively. Finally, the improved probability model for the experimental results was fitted as:  
\[S(\alpha,V,D_{50})=\frac{1}{1+e^{37.2-0.6\alpha-9.1D_{50}-156.6V}},\] (Figure 18 presents the discriminant results of the improved probability model as a scatter plot. These points are divided into two categories by the line \(S=0.5\). If a point is located below this line, the discriminant result is partial blockage; otherwise, the discriminant result is complete blockage. The experimental results of blockage states are also presented in Figure 18 in different colors. The red color indicates complete blockage in the experiment, and the blue color indicates partial blockage in the experiment. Compared with the experimental results, only four cases are not correctly classified by the improved probability model.