## 3 Basic Theory of Slope Characterization Models
### Basic Theory of Micrometric Weibull Distribution Modeling of Nonhomogeneous Slopes
#### 3.2.2 Parametric Randomized Distribution Methods  
The Weibull distribution parameter assignment method is a widely used method in numerical simulation to describe the inhomogeneous properties of rocks. In numerical simulations, the parameters of the fine structural units of the rock (elastic modulus, friction angle, cohesion, etc.) are assumed to obey the Weibull distribution according to the different properties of the material, and the assignment process is realized by the Monte Carlo method. The probability distribution Equation (11) and the probability density Equation (12) of the Weibull distribution have the following expressions:  
\[F(x)= \begin{cases}1-\text{e}^{-(\frac{x}{\varepsilon_{0}})^{\beta}}&x>0\\ 0&x\leq 0\end{cases}  \tag{11}\]  
\[f(x)= \begin{cases}\frac{\beta}{\varepsilon_{0}}\left(\frac{x}{\varepsilon_{0 }}\right)^{\beta-1}\text{e}^{-(\frac{x}{\varepsilon_{0}})^{\beta}}&x>0\\ 0&x\leq 0\end{cases}  \tag{12}\]where \(F(x)\) is the probability distribution function of random variable \(x\), \(f(x)\) is the probability density function of random variable \(x\), \(\beta\) is the shape parameter, \(\varepsilon_{0}\) is the scale parameter. Figure 5 is a graph of the probability distribution function of the Weibull distribution, according to the Monte Carlo method, first produce a set of random number sequences \(\xi_{i}\) obeying a uniform distribution in the interval [0, 1], so that the probability distribution function \(F(x_{i})=\xi_{i}\), the use of its inverse function of the corresponding \(x_{i}\). For \(x_{i}\), can be calculated according to the probability density function \(f(x_{i})\). In this way, by the uniform random number sequence \(\xi\) mapped to a special sequence \(x_{i}\), by the large number theorem, when the number of random numbers to obtain a large enough, \(x_{i}\) obey the probability density of \(f(x)\) distribution, that is, by the [0, 1] interval of uniform random number mapping to obtain a set of random numbers obeying a given distribution. When using the Weibull distributed parameter assignment method to assign the unit parameters, the mapping from uniform random numbers to random numbers obeying the Weibull distribution is realized sequentially for all the fine-grained units, and the Weibull-distributed random number is defined as the parameter of the unit, thus realizing the non-uniform parameter assignment of the whole numerical computation model, and the parameters of all the units in general obey the given Weibull distribution. Different \(\beta\) values of Weibull distribution parameters represent non-uniform materials with different degrees of homogeneity. For the same Weibull distribution parameters \(\beta\) and \(\varepsilon_{0}\), the spatial distribution of the material parameters is also different for each randomly generated sample. The value of \(\beta\) reflects the degree of non-uniformity of the material structure in the statistical model, and when it varies from small to large, the distribution function of the strength of the material's fine-scale units varies from flat and wide to high and narrow, the strength of the fine-scale units tends to be homogeneous, and most of the fine-scale units contained within the material are close to the given parameter \(\varepsilon_{0}\). In general, when using the Weibull distribution to assign non-homogeneous parameters, under the same scale parameter \(\varepsilon_{0}\), the degree of homogeneity of the overall parameters of the rock can be described by the shape parameter \(\beta\). The shape parameter \(\beta\) can be used to describe the overall homogeneity of the rock. As shown in Figure 5b, the smaller \(\beta\) is, the greater the dispersion of the physical and mechanical parameters of the fine-scale units of the rock, and the worse the overall homogeneity of the rock, and conversely, the larger \(\beta\) is, the better the overall homogeneity of the rock. Figure 5: The probability distribution function and probability density function curve of Weibull distribution.