## 2 Theory
### The Adaptive VMD-AIC Method  
The result of VMD is a collection of oscillation components from high to low frequency. For obtaining a good noise reduction effect, it is necessary to choose the appropriate sub-signals,which are the basis of reconstruction. The Pearson correlation coefficient (CC) is introduced to determine the boundary position of the noise sub-signal and the valuable sub-signal, which is defined as follows [38; 39]:  
\[CC(j)=\frac{E(x_{t}*IMF_{t}(j))-E(x_{t})*E(IMF_{t}(j))}{\sqrt{E(x_{t} ^{2})-E(x_{t})^{2}}*\sqrt{E(IMF_{t}(j)^{2})-E(IMF_{t}(j))^{2}}} \tag{12}\]  
where \(E\) is the mathematical expectation, \(IMF_{t}(j)\) is the \(j\)-th sub-signal, and \(CC(j)\) is the \(j\)-th correlation coefficient. It is an indicator to measure the relevance of different data sets. The decomposed IMF components have different correlations with the original signal. The dominant MS component usually has a large \(CC\) value, while the disordered noise component corresponds to a small \(CC\) value. In the transition component from noise to signal, there will be a sudden increase in the value of \(CC\), which is used to distinguish the above two types of components.  
A typical noisy MS signal can be roughly divided into three parts: the front noise segment, the middle MS signal segment, and the rear noise segment, just as shown in Figure 1. Only the middle MS part, reflecting the rock fracture and stress level, is what the researchers need. If the noise in the front and rear sections are removed, the MS characteristics can be further highlighted. After removing the high-frequency noise component through the correlation coefficient, we use the AIC method to determine the distribution interval of the MS waveform. The first arrival position of the MS waveform is relatively accurately determined because of its obvious fluctuation. We only need to make adjustments in a small range on the basis of initial positioning to get a more exact starting position. The end position of MS is not so prominent due to its low amplitude and its mixing with noise. Therefore, it is necessary to reuse AIC on the rear section separately to obtain a better positioning result. After locating, those data points that are considered to be noise are thresholded to zero while leaving the rest unchanged. The multiple window denoising method combining VMD and AIC is processed by scheme in Figure 2, and the detailed implementation steps are as follows:  
Figure 1: Three parts of a typical microseismic (MS) signal.  
Figure 2: The VMD-AIC algorithm flow chart.  
Step 2: Calculate \(CC\) of each component and its difference between its neighbors, \(\triangle CC\).  
Step 3: Determine the boundary component with the largest \(\triangle CC\) and select the components after the boundary component to reconstruct the signal.  
Step 4: Calculate the AIC value of the entire time window of the reconstructed signal and determine the first start position S1 and the first end position E1 of the AIC minimum values before and after the maximum value of the waveform, respectively.  
Step 5: In the interval section \([\text{S1}-(\text{R}-\text{S1})/4,\text{S1}+(\text{R}-\text{S1})/4]\), the AIC method is applied again to obtain a more accurate starting position S, where R is the position where the maximum value is located.  
Step 6: In the rear noise section \([\text{E1}-(\text{R}-\text{S1})/4,\text{L}]\), apply the AIC method to determine the corrected minimum position E2, where L is the time window length.  
Step 7: Threshold the waveform in window \([1,\text{S}]\) and \([\text{E2},\text{L}]\) to zero, and obtain the processed signal.