# 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  
Principal Component Analysis (PCA) is a statistical method based on dimensionality reduction. It transforms multiple variables into a smaller set of synthetic variables (principal components), where each principal component is a linear combination of the original variables. These principal components are mutually uncorrelated, allowing them to capture most of the information contained in the original variables without redundancy [24; 25]. The spatial distribution of clustered microseismic events forms a three-dimensional variable sample, with each event characterized by \(x\), \(y\), and \(z\) coordinates. By applying PCA to the 3D coordinate data of these clustered microseismic events, the first and second principal component directions can be extracted. The plane formed by the first and second principal components can determine the orientation of the plane defined by the clustered microseismic events. PCA effectively reduces the dimensionality of the spatial coordinates (from 3D to 2D), enabling precise characterization of the strike and dip angles of the slip surface formed by the spatial clustering of microseismic events. Figure 11 illustrates an example of randomly distributed microseismic events in a plane, where \(Xp\) represents the first principal component direction and \(Yp\) represents the second principal component direction. Let \(x_{1},x_{2},\ldots,x_{m}\) denote m microseismic location events, and let \(I_{1},I_{2},\ldots,I_{n}\) represent \(n\) features used to evaluate the events. Each object \(x_{i}\) has n feature observations, so the microseismic event \(x_{i}\) can be expressed as an n-dimensional vector:\(x_{i}=(x_{1},x_{2},\ldots,x_{jn})\), since microseismic events are inherently three-dimensional, n = 3, the variables of microseismic events can be represented by the following feature matrix:  
\[X= \begin{pmatrix}x_{11}&x_{12}&\ldots&x_{1n}\\ x_{21}&x_{22}&\ldots&x_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ x_{m1}&x_{m2}&\ldots&x_{mn}\end{pmatrix} =\left(x_{ij}\right)_{m\times n} \tag{1}\]  
Figure 10: Photographs of the upper mountain body collapse process: (**a**) before landslide (time: 07:21:00), (**b**) onset of landslide (time: 07:23:13), (**c**) during landslide (time: 07:23:20), (**d**) after landslide (time: 07:24:23). Apply the Z-score method [26] to standardize the original data, let  
\[Z_{ij}=(x_{ij}-\overline{x}_{j})/s_{j} \tag{2}\]  
In the formula  
\[\overline{x}_{j}=\frac{1}{m}\sum_{i=1}^{m}x_{ij_{i}}\ s_{j}^{2}=\frac{1}{m} \sum_{i=1}^{m}\ (x_{ij}-\overline{x}_{j})^{2},\ j=1,2,\ldots,n\]  
After the above transformation, the new data vector \(z_{ij}=(z_{1j},z_{2j},\ldots,z_{mj})^{T}\), \((j=1,2,\ldots,n)\) has a mean of 0 and a variance of 1. The transformed feature matrix is as follows:  
\[Z= \begin{pmatrix}z_{11}&z_{12}&\ldots&z_{1n}\\ z_{21}&z_{22}&\ldots&z_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ z_{m1}&z_{m2}&\ldots&z_{mn}\end{pmatrix} =\left(z_{ij}\right)_{m\times n} \tag{3}\]  
Construct the covariance matrix of the standardized data vectors \(R\):  
\[R=(r_{ik})_{m\times n}r_{ik}=\frac{1}{m-1}\sum_{j=1}^{m}\ (z_{ji}-z_{jk}),(i,j=1,2, \ldots,n) \tag{4}\]  
Here, \(r_{ik}\) is the correlation coefficient between the vector of m transformed values for the \(i\)-\(th\) feature and the vector of m transformed values for the \(k\)-\(th\) feature. That is, \(r_{ik}\) represents the correlation coefficient of the random variables formed by the \(i\)-\(th\) and \(k\)-\(th\) columns of matrix \(Z\). Solve for the eigenvalues \(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\) of matrix \(R\). Since \(R\) is an \(n\times n\) real symmetric matrix, all its eigenvalues are real numbers. Arrange them in descending order as follows:  
\[\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n} \tag{5}\]  
Since microseismic events are three-dimensional variables, to reduce their dimensionality to two dimensions, the first two principal components are primarily computed.