# Study on Microseismic Monitoring of Landslide Induced by Blasting Caving
## 4 Microseismic Monitoring Analysis of Landslide
### Principal Component Analysis of the Orientation of the Slip Surface
Establish the principal component equations to determine the principal component values. Let the unit eigenvectors corresponding to the first two eigenvalues \(\lambda_{i}\) be denoted as follows:  
\[D=(d_{i1},d_{i2},\ldots d_{in}),\ i=1,2 \tag{6}\]  
Figure 11: Principal component directions of microseismic events within the plane. The plane formed by the first and second eigenvectors corresponds to the slip surface defined by the spatial clustering of microseismic events. By determining the plane formed by the first and second eigenvectors in space, the strike and dip angle of the clustered microseismic event surface can be identified. Following the large-scale blasting, the upper slip zone (Zone II) contains a total of 54 clustered microseismic events. These 54 events were selected as the dataset for analyzing the blasting-induced mountain body collapse, as shown in Figures 12 and 13. The 54 clustered microseismic events in Zone II are selected as the study objects for the slip surface analysis. Their spatial coordinate distribution is shown in Table 1.  
The original data are standardized using the Z-score method. The results are presented in Table 2.  
Figure 12: Side view of microseismic event clusters on the slip surface. Figure 13: Plan view of microseismic event clusters on the slip surface. Based on the aforementioned PCA analysis method, a MATLAB 8.0 computational program was developed to calculate the covariance matrix \(R=(r_{ik})_{3\times 3}\) using Equation (4). The covariance matrix is as follows:  
\[R= \begin{bmatrix}1.0000&0.8544&-0.75906\\ 0.8544&1.0000&-0.60758\\ -0.7590&-0.6075&1.0000\\ \end{bmatrix} \]  
The eigenvalues and eigenvectors of the covariance matrix \(R=(r_{ik})_{3\times 3}\) are shown as follows. \[D= \begin{bmatrix}0.1128&0&0\\ 0&0.4018&0\\ 0&0&2.4852\\ \end{bmatrix} \]  
\[V= \begin{bmatrix}0.7806&0.1375&-0.6096\\ -0.5545&0.6023&-0.5742\\ 0.2882&0.7863&0.5464\\ \end{bmatrix} \]  
The non-zero values on the diagonal of matrix \(D\) represent the eigenvalues, arranged in ascending order. Each column in matrix \(V\) corresponds to the eigenvector of the respective eigenvalue in matrix \(D\). In this example, the first two principal components are \(d_{1}=2.4852\), \(d_{2}=0.4018\), with their corresponding eigenvectors as follows: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Event** & & **Coordinate** & & **Event** & & **Coordinate** & \\
**Number** & **North/m** & **East/m** & **Depth/m** & **Number** & **North/m** & **East/m** & **Depth/m** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: Spatial coordinate data of microseismic events. \[\left\{\begin{array}{l}\overrightarrow{v}_{1}=\left[ \begin{array}{cc}-0.6096&-0.5742 &0.5465\\ \overrightarrow{v}_{2}=\left[\begin{array}{cc}0.1375&0.6023&0.7863\\ \end{array}\right]\end{array} \right]\right.\]  
The vectors \(\overrightarrow{v_{1}}\) and \(\overrightarrow{v_{2}}\) are not collinear and span a plane. The normal vector of this plane is:  
\[\overrightarrow{n}=\overrightarrow{v}_{1}\times\overrightarrow{v}_{2}=\left[ \begin{array}{cc}-0.7807&0.5545&-0.2882\end{array} \right]\]  
By setting the \(z\) component of the normal vector \(\overrightarrow{n}=(x,y,z)\) to zero, the projection vector \(\overrightarrow{t}=\left[ \begin{array}{cc}-0.7807&0.5545&0\end{array} \right]\)\(\overrightarrow{n}\) onto the horizontal plane defined by the xy axes can be obtained. Let the direction vector of the \(\overrightarrow{x}=(1,0,0)\), The cosine of the angle between the projection vector \(\overrightarrow{t}\) and the x axis is:  
\[\cos\alpha=\frac{\overrightarrow{t}\cdot\overrightarrow{x}}{\left| \overrightarrow{t}\right|\left|\overrightarrow{x}\right|}=\frac{-0.7807\times 1 +0.5545\times 0+0\times 0}{\sqrt{(-0.7807^{2})+0.5545^{2}+0^{2}}\times\sqrt{1^{2}+0 ^{2}+0^{2}}}= \begin{array}{cc}-0.8153\end{array}