## 3 Spatial Characterization Methods
### Theoretical Distribution Pattern Space  
With the rock specimen in mechanical equilibrium, we assume that a virtual unit crack space exists in the \(n\) dimension in the form of a sphere space, which also remains in equilibrium in all directions. Therefore, it can be represented by Equation (7) as  
\[\left\|\mathbf{r}\right\|^{2}={r_{1}}^{2}+{r_{2}}^{2}+\cdots+{r_{n}}^{2}=1 \tag{7}\]  
where the vector \(\mathbf{r}\) has a length of 1 in all directions. Deformation occurs when the unit crack space is exposed to an applied stress \(\sigma\). In general, the stress \(\sigma\) can be decomposed into \(n\) components in different directions, such as \(\sigma_{1},\ \sigma_{2},\ \cdots,\ \sigma_{n}\), and the orientations of the components are \(a_{1},a_{2},\cdots,a_{n}\), respectively. When the crack space is balanced, the stress conditions in the deformed crack space can be written as follows:  
\[\frac{{r_{1}}^{2}}{\left|\sigma_{1}\right|^{2}}+\frac{{r_{2}}^{2}}{\left| \sigma_{2}\right|^{2}}+\cdots+\frac{{r_{n}}^{2}}{\left|\sigma_{n}\right|^{2}}=1 \tag{8}\]  
When the stress values in all orientations are equal, the crack space still maintains a sphere. In 3D space, the crack space is thereby assumed to have three orientations perpendicular to each other. As we know, the internal defects are misaligned when the materials are subjected to stress, causing microcracks, and they are detected by AE monitoring. Moreover, it was found that the crack propagation surface produced is perpendicular to the minor principal stress direction [28,32]. Since the three decomposed stress orientations are perpendicular to each other, the microcracks should propagate along the other two stress orientations. Therefore, microcracks are assumed to be distributed along the stress orientations. It can be inferred that the orientations of the microcrack distribution (theoretically, crack space), the stress, and the real crack should be consistent. Therefore, the distribution characteristics of the microcracks can theoretically be used to characterize either the stress or the real crack. At the same time, the scale of microcrack propagation varies in all directions due to the different values of stress components. Assuming a linear relationship between the stress value and the length of microcrack propagation, the theoretical distribution pattern space can be established as  
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \tag{9}\]  
where \(a\), \(b\), and \(c\) are the scales of microcrack propagation of the three decomposed stresses \(\sigma_{a}\), \(\sigma_{b}\), and \(\sigma_{c}\), respectively. On the basis of the above theoretical analysis, the optimal three directions for characterizing the spatial distribution of microcracks \(\mathcal{S}(\mathbf{x},\mathbf{y},z)\) can be found. Mathematically, the variance \(\text{Var}(\mathcal{S}^{*})\) of the projection of microcracks in the target direction should be the largest. Second, every two directions need to be perpendicular to each other to avoid correlation. The covariance \(\text{Cov}\big{(}\mathbf{s}_{i}^{*},\mathbf{s}_{j}^{*}\big{)}\) between the projected data in each of the two directions should be zero. The variance and the covariance are in the form:  
\[\text{Var}(\mathbf{s}^{*})=\frac{1}{m}\sum_{i=1}^{m}\big{(}\mathbf{s}_{i}^{*}-\overline {\mathbf{s}^{*}}\big{)}^{2}=\frac{1}{m}\sum_{i=1}^{m}{\mathbf{s}_{i}}^{*2} \tag{10}\]\[\text{Cov}(\textbf{s}_{i}^{*},\textbf{s}_{j}^{*})=\frac{1}{m}\sum\limits_{i=1}^{m} \textbf{s}_{i}^{*}\textbf{s}_{j}^{*} \tag{11}\]  
It is found that two conditions can be satisfied by diagonalizing the covariance matrix \(\textbf{C}_{S^{*}}\):  
\[\textbf{C}_{S^{*}}=\frac{1}{m}\textbf{S}^{*}\textbf{S}^{*T}