# Study on Microseismic Monitoring of Landslide Induced by Blasting Caving
## 4 Microseismic Monitoring Analysis of Landslide
### Principal Component Plane Ellipticity Analysis  
Theoretically, the principal component method can identify the corresponding clustering plane for any aggregated event, whether the events are randomly dispersed or exhibit a distinct clustering tendency. Consequently, the principal component plane derived from microseismic event analysis does not entirely represent the true slip plane of the microseismic event cluster. When microseismic events are in a relatively random and dispersed state, the principal component plane merely reflects the mathematical clustering tendency of the event distribution. In contrast, when microseismic events form a clear clustering pattern, the principal component plane corresponds to the slip plane. The clustering degree of microseismic event clusters is generally quantified using the ellipticity ratio of the principal component ellipsoid. Figure 14 is relationship between the clustering degree of microseismic events (red dots) and the ellipticity ratio (blue dotted circle). The more dispersed the distribution of microseismic events is, the smaller the ellipsoid rate will be, conversely, the more concentrated the distribution of microseismic events is, the larger the ellipsoid rate will be. 1 Moment of Inertia Method [27]  
By applying principal component analysis (PCA), the principal plane is defined by the first and second principal components. All microseismic events are projected onto this principal plane. Consequently, the key step in the calculation involves determining the size and shape of the ellipse that encloses the spatial distribution of all microseismic events. This requires simultaneous consideration of the spatial distribution distances from the center of mass and the weighted contributions of each event. To address this, the concept of moment of inertia from physics is introduced to compute the ellipticity of dispersed microseismic events. Figure 14: Relationship between the clustering degree of microseismic events and the ellipticity ratio: (**a**) low ellipticity: random distribution of microseismic events, (**b**) high ellipticity: clustered activity along the fault plane. The moment of inertia quantifies the resistance of a rigid body to rotational motion about an axis, representing the tendency of an object to maintain uniform circular motion or remain stationary. It is denoted by the symbol II and is determined solely by the shape, mass distribution, and the position of the axis of rotation. A schematic illustration is provided in Figure 15.  
The expression for the moment of inertia is:  
\[I=\sum_{i}^{n}m_{i}r_{i}^{2}\]  
2 Determination of Ellipse Major and Minor Axes  
Taking 54 microseismic events clustered on the slip plane after a large-scale blasting as in the case study, the ellipticity of this cluster is analyzed. In Figure 16, the plane is defined by the first and second principal component eigenvectors obtained through principal component analysis (PCA). By applying the coordinate transformation formula (Equation (2)), spatial points are dimensionally reduced and projected onto the two-dimensional plane formed by the first and third principal component eigenvectors. This results in a new set of spatial coordinates along the directions of the first and third principal components, as illustrated in Figure 17. By solving for the spatial orientation of the principal component plane, all events are projected onto this plane, transforming the problem into calculation of the ellipticity ratio within a single planar domain. Figure 16: Schematic illustration of microseismic event clusters and fault plane relationship using MATLAB software. Figure 15: Schematic diagram of moment of inertia calculation. To quantitatively evaluate the clustering degree of these events, a conventional method involves constructing an ellipse based on the spatial distribution of points, then calculating the ellipticity \(e\) using the major and minor axes of the ellipse. The expression is defined as:  
\[e=