## 2 Optimization Theory of the Microseismic Network  
The spatial array of sensors is one of the factors that affect the reliability of microseismic data. Considering the reasonable arrangement density, installation horizon, and other conditions, a small system positioning error can be guaranteed. Because of the above problems, the method of analyzing the positioning accuracy error is usually adopted to optimize the layout scheme of the combined system. The \(D\)-value theory holds that the size of the determinant of the covariance matrix of source parameters is proportional to the volume of the error ellipsoid. The method has been successfully applied to various fields [14; 24].  
The microseismic source is H (\(t_{0}\), \(x_{0}\), \(y_{0}\), \(z_{0}\)), and the \(i^{\text{th}}\) station is S\({}_{i}\) (\(t_{i}\), \(x_{i}\), \(y_{i}\), \(z_{i}\)). For the uniform and isotropic velocity models, the shortest time \(T_{i}\) from the source H to the \(i^{\text{th}}\) station S\({}_{i}\) can be described by Equation (1):  
\[T_{i}=\frac{\sqrt{\left(x_{0}-x_{i}\right)^{2}+\left(x_{0}-x_{i}\right)^{2}+ \left(x_{0}-x_{i}\right)^{2}}}{V_{p}} \tag{1}\]  
In the equation, H (\(t_{0}\), \(x_{0}\), \(y_{0}\), \(z_{0}\)) are the time and three-dimensional coordinates of the microseismic source, respectively. S\({}_{i}\) (\(t_{i}\), \(x_{i}\), \(y_{i}\), \(z_{i}\)) are the time and three-dimensional coordinates of the \(i^{\text{th}}\) sensor, respectively. \(V_{p}\) is the uniform microseismic propagation velocity, \(i=1\), \(2\), \(\ldots\), \(n\), where \(n\) is the number of stations installed in the mine. [PERSON] [9; 10] considered that the optimization of the sensor station location depends on the covariance matrix \(C_{x}\) of \(x\), as shown in Equation (2):  
\[\begin{array}{c}C_{x}=k(A^{T}A)^{-1}\\ A=\begin{bmatrix}1&\frac{\partial T_{1}}{\partial x_{0}}&\frac{\partial T_{1}}{ \partial y_{0}}&\frac{\partial T_{1}}{\partial z_{0}}\\ \vdots&\vdots&\vdots&\vdots\\ 1&\frac{\partial T_{1}}{\partial x_{0}}&\frac{\partial T_{1}}{\partial y_{0}}& \frac{\partial T_{1}}{\partial z_{0}}\\ \end{bmatrix}\end{array} \tag{2}\]  
In Equation (2), \(A\) is the calculated partial differential matrix with the corresponding earthquake arrival time, and \(k\) is a constant. This covariance can be graphically explained with the confidence ellipsoid, i.e., the eigenvalues of the covariance matrix constitute the length of the principal axis of the confidence ellipsoid. Finding the station arrangement with the smallest volume of the ellipsoid is called the optimal design of the \(D\) value. The volume of the ellipsoid is proportional to the product of the covariance eigenvalues, that is, to the determinant of \(C_{x}\). As shown in Equation (3), det [\(C_{x}\)] is minimized to satisfy the \(D\)-value optimization criterion:  
\[obj=\min\left(\sum_{i=1}^{ne}M_{e}(h_{i})\lambda x_{0}(h_{i})\lambda y_{0}(h_{ i})\lambda z_{0}(h_{i})\lambda t_{0}(h_{i})\right) \tag{3}\]  
In the equation, \(ne\) is the number of hypocenter points calculated in the monitoring area. \(M_{e}(h_{i})\) is the microseismic event impact factor index. \(\lambda_{x0}(h_{i})\), \(\lambda_{y0}(h_{i})\), \(\lambda_{z0}(h_{i})\), and \(\lambda_{t0}(h_{i})\) are the eigenvalues of \(C_{x}\).