# Numerical Modeling of 3D Slopes with Weak Zones by Random Field and Finite Elements
## 2 Methodology  
### Method for Generating 3D Random Fields  
The 3D random field is generated using the modified linear estimation method. The basic principle of this method is to generate a stationary Gaussian random field, where the mean is zero, and the unit variance and the spatial correlation length are \(\sqrt{n}\)[12]. Spatially correlated variables can be expressed by a linear equation combining fixed effects and random effects as follows:  
\[t=\mu+e \tag{1}\]  
where \(t\) represents the spatial variation of rock material properties at different locations, and fixed effects and random effects are represented by the mean value \(\mu\) and the residual \(e\), respectively. \(e\) is often assumed to be a constant, with deviations from this having a mean value \(u\). The covariance matrix \(V\) of the residual \(e\) can be described as follows:  
\[V=\sigma^{2}C \tag{2}\]  
where \(\sigma\) is the standard deviation of the random variable at each node, and \(C\) is the spatial autocorrelation matrix. To avoid generating negative values for the strength parameters of rock and weak zones, the strength parameters of some rock and weak zones usually adopt log-normal distribution. In this case, the random field can be modeled by the following method conversion:  
\[t=\exp(\mu_{\ln\ z}+\sigma_{\ln\ z}\varepsilon) \tag{3}\]  
where \(\varepsilon\) is the vector of correlated random variables; \(\mu_{\ln z}\) is the mean of the logarithm of the material parameters; and \(\sigma_{\ln z}\) is the standard deviation of the logarithm of the rock property.  
The modified linear estimation method needs to generate an n-dimensional m-vector attribute field with spatial cross-correlation matrix \(C\). Then, vector attribute field stretch modeling is used to generate a random field with any specified spatial correlation length. The specific steps are as follows:  
First, the n-dimensional space containing position vector \(s\) is discretized into n-dimensional grids with unit grid spacing. In addition, the grid nodes are filled using a random vector \(r\) with \(m\) random numbers following the standard Gaussian distribution.  
Second, Cholesky decomposition is performed on the spatial cross-correlation matrix \(C\) according to Equation (4).  
\[C=L\times L^{T} \tag{4}\]  
where \(L\) is the lower triangular matrix.  
Third, calculate the position of \(y\) in s according to Equation (5).  
\[s=J\times y+\varepsilon \tag{5}\]  
where \(y\) is the attribute field containing the position vector; \(s\) is the random field containing the position vector; \(J\) is the direction vector and \(\varepsilon\) is a translation vector with n components. \(J\) and s are independent random variables, and they have specific values. In this study, the components of \(J\) and \(\varepsilon\) are uniformly distributed in the range of [0, \(\pi/2\)] and [0,1],respectively. According to Equation (2), the attribute field \(y\) will be rotated and translated to obtain the random field \(s\).  
Finally, the continuous attribute field of any coordinates can be calculated by Equation (6), as shown below.  
\[f^{j}(y)=\sum_{i=1}^{2^{n}}\sqrt{N_{i}(y)}\cdot f^{j}_{i,k^{\prime}},\ j=1,2 \ldots m. \tag{6}\]  
where \(f^{j}(y)\) and \(f^{j}_{i,k}\) are the \(j\)th component of the attribute field containing the position vector \(y\) and the \(i\)th node of element \(k\), respectively; \(N_{i}(y)\) is a form function with \(2^{n}\) nodal elements, which is taken from the library of form functions established by the finite element method.