## 4 Multi-Source Monitoring Function Based on DIC-AE Fusion
### Establishment of Multi-Source Monitoring Function  
The failure process of red sandstone specimens is accompanied by the synergistic response of multiple sources, such as surface strain fields and AE signals. Considering different dimensions of these information parameters, the multi-source information is processed by the normalization method to eliminate the dimensional differences. After being normalized, the multi-source data are converted into dimensionless scalars, which makes it easier to comparatively analyze different units and indexes of different orders of magnitude.  
In this study, max-min normalization is adopted to normalize multi-source data, as expressed by the following:  
\[x_{k}^{*}=\frac{x-\min}{\max-\min}, \tag{1}\]  
where \(x_{k}^{*}\) is the normalized function value of the monitoring data, k is the ordinal number of parameters of different multi-source information, and [min, max] is the value range of the monitoring data.  
For the purpose of characterizing the dynamic evolution of the multi-source information of red sandstone over time, the multi-source monitoring data in the failure process are fitted and analyzed, and a polynomial function is constructed. The fitting function is:  
\[f_{i}(x)=\sum_{j=1}^{5}a_{ij}x^{j}+a_{0}, \tag{2}\]  
where \(i\) in \(f_{i}(x)\) can be 1, 2...7, respectively;\(f_{1}\)(x) is the standard deviation function of the horizontal strain field;\(f_{2}\)(x) is the standard deviation function of the vertical strain field; \(f_{3}\)(x) is the standard deviation function of the shear strain field; \(f_{4}\)(x) is the cumulative AE count;\(f_{5}\)(x) is the cumulative AE RA/AF value;\(f_{6}\)(x) is the cumulative AE dominant frequency;\(f_{7}\)(x) is the amplitude corresponding to the cumulative AE dominant frequency; and \(a_{ij}\) is the coefficient of the fitting function.  
The fitted multi-source information curves are illustrated in Figure 15.  
The value \(a_{ij}\) of the fitting function is:  
Figure 15: Function curves of the multi-source information.  
\[a_{ij}^{T}=\begin{bmatrix}-9.71\times 10^{-2}&-1.06\times 10^{-1}&1.56\times 10^{-2}& -1.31\times 10^{-1}&-1.57\times 10^{-1}&-1.54\times 10^{-1}&-1.59\times 10^{-1}\\ 6.74\times 10^{-3}&6.12\times 10^{-3}&2.82\times 10^{-4}&6.82\times 10^{-3}&8.42\times 10^{ -3}&8.10\times 10^{-3}&8.38\times 10^{-3}\\ -8.94\times 10^{-5}&-6.12\times 10^{-5}&2.87\times 10^{-5}&-8.76\times 10^{-5}&-1.14 \times 10^{-4}&-1.07\times 10^{-4}&-1.10\times 10^{-4}\\ 4.57\times 10^{-7}&2.82\times 10^{-7}&-2.14\times 10^{-7}&4.51\times 10^{-7}&6.05 \times 10^{-7}&5.56\times 10^{-7}&5.73\times 10^{-7}\\ -1.01\times 10^{-9}&-5.40\times 10^{-10}&5.63\times 10^{-10}&-9.93\times 10^{-10}&-1.3 7\times 10^{-9}&-1.23\times 10^{-9}&-1.27\times 10^{-9}\\ 8.03\times 10^{-13}&3.74\times 10^{-13}&-4.79\times 10^{-13}&7.89\times 10^{-13}&1. 11\times 10^{-12}&9.83\times 10^{-13}&1.02\times 10^{-12}\end{bmatrix} \tag{3}\]  
The parameters of the fitting function for the seven multi-source information indexes are listed in Table 2, where \(R^{2}\) characterizes the fitting degree of the fitting function. The closer the value of \(R^{2}\) is to 1, the higher the fitting degree is. In Table 2, the \(R^{2}\) values of \(f_{1}(x)-f_{7}(x)\) are 0.9869, 0.9954, 0.9415, 0.9749, 0.9108, 0.9358, and 0.9693, respectively, indicating that the polynomial function has an ideal fitting degree.