## 2 Materials and Methods
### Data-Processing Method
#### 2.4.2 Wavelet Analysis  
Wavelet analysis is a time-frequency analysis method developed from the Fourier transform in the 1980s, with the significant features of self-adaptivity and mathematical microscopy. It essentially splits signals into multiple scales via shifting and scaling of the wavelet basis function [51; 52]. A given wavelet function \(h(x)\) is transformed to the following after expansion and shifting:  
\[h_{j,k}\left(x\right)=2^{j/2}h(2^{j}x-k) \tag{1}\]  
where \(j\) is the scale coefficient and \(k\) is the shift coefficient. The smaller scale corresponds to the higher frequency and vice versa. At an arbitrary scale, the wavelet transformation can be expressed as follows:  
Figure 1: Experimental scene: (**a**) schematic of the instrument arrangement, (**b**) photo of the experimental system under outdoor conditions at night.  
\[C(j,k)=\left\langle h_{j,k}\ (x),f\right\rangle=2^{j/2}\int\overline{h_{j,k}(x)}f(x)dx \tag{2}\]  
where \(C(j,k)\) is the wavelet coefficient and \(\overline{h_{j,k}(x)}\) is the complex conjugate of \(h_{j,k}(x)\).  
For a square-integrable function \(L^{2}(R)\), any given signal \(f(x)\in L^{2}(R)\) can be decomposed into different frequency bands by scale as follows:  
\[f(x)=\sum_{j,k=-\infty}^{\infty}\ C(j,k)h_{j,k}(x)=\ldots+g_{-1}(x)+g_{0}(x)+g _{1}(x)+\ldots+g_{j}(x)+\ldots \tag{3}\]  
where \(g_{j}(x)\) is the component of \(f(x)\) at scale \(j\). If \(h\) is an orthogonal wavelet, the components of Equation (3) are orthogonal, i.e.,  
\[\left\langle g_{i}(x),g_{j}(x)\ \right\rangle=0(i\
eq j) \tag{4}\]  
where \(<\!\!g_{i}(x),g_{j}(x)\!\!>\) denotes the inner product of the two components \(g_{i}(x)\) and \(g_{j}(x)\). If it is 0, then the components are not related and can be arbitrarily combined without being affected by other components. The wavelet analysis method has been a common signal-processing method to extract weak signals from strong noisy backgrounds, and it is applied to many practical applications [53].  
During the experiment, the TIR from the rock surface is affected by the atmosphere, the sun, and stress, etc. In terms of frequency, there are three main, significant components [17; 18; 19]: (1) high frequency: the frequency of TIR changes in this component is larger than that of the stress, which is mainly related to the fluctuant change of the environment and the instrumental noise; (2) middle frequency: the frequency of TIR changes is consistent with that of the stress, which indicates that the TIR changes are induced by stress; (3) low frequency: the frequency of TIR changes is smaller than that of the stress, and the TIR changes are mainly in response to the continuous and slow changes in the environment in a relatively long period [53; 54]. Considering the above frequency characteristics, the \(\Delta\)AIRT signals during rock loading can be divided into three components, as shown in Equation (5):  
\[\Delta\text{AIRT}=\Delta\text{AIRT}_{\text{high}}+\Delta\text{AIRT}_{\text {mid}}+\Delta\text{AIRT}_{\text{low}} \tag{5}\]  
where \(\Delta\)AIRT is the original AIRT change and \(\Delta\text{AIRT}_{\text{high}}\), \(\Delta\text{AIRT}_{\text{mid}}\), and \(\Delta\text{AIRT}_{\text{low}}\) denote the components in high, middle, and low frequency, respectively. The orthogonal Coiflet 5 wavelet basis was selected to decompose \(\Delta\)AIRT. According to the loading period and the sampling frequency of AIRT, the suitable decomposition level for the wavelets is 11. \(\Delta\text{AIRT}_{\text{mid}}\) belonged to the detailed component at level 11, and the \(\Delta\text{AIRT}_{\text{low}}\) belonged to the approximation component at level 11. The remaining detailed components from level 1 to 10 belonged to \(\Delta\text{AIRT}_{\text{high}}\). The absolute error of the postdecomposition synthesis is on the order of \(10^{-9}\).