## 3 Methodology
### Spectral Decomposition Technology without Frequency Interference
In stage 1, performing the EEMD algorithm over seismic signal yields  
\[\mathbf{s}=\sum_{j=1}^{J}\boldsymbol{\varphi}_{j}+\boldsymbol{\eta}, \tag{7}\]  
where \(j\) is the serial number of intrinsic mode function (IMF), \(\boldsymbol{\varphi}_{j}\) is the \(j\)th IMF, and \(\boldsymbol{\eta}\) is the residual. In stage 2  
To further verify the feasibility of MP-EEMD, the seismic trace data near well S-2 (Figure 4a) is selected for real data test. Figure 4b,c displays the components decomposed by EEMD and their corresponding frequency spectra obtained by FFT. It can be seen that the five components have different dominant frequency and frequency band. By comparing with the frequency spectra of the real seismic signal \(\mathbf{s}\) and that of the five components, we regard the component \(\boldsymbol{\varphi}_{1}\) as a high-frequency (>200 Hz) low-amplitude noise and remove it in the subsequent steps. In practice, the frequency band of seismic signal is limited and narrow (about 10-60 Hz), so we set the threshold of dominant frequency as 80 Hz in this study. The components with dominant frequency greater than this threshold are considered to be noises, and the others are the valid signals. Therefore, MP-EEMD can eliminate certain noise from the real seismic signal by analysis the spectral features of the components decomposed by EEMD. Then, after adaptive stacking (Equation (8)), the four valid components are stacked to be two sub-signals (Figure 4d), which represent the high-frequency and low-frequency information of the real seismic signal, respectively. Figure 5 displays the TF spectra of the real seismic signal using MP and MP-EEMD. As indicated in the white rectangles, MP-EEMD can avoid the obvious average effect, resulting from frequency interference of \(\psi_{1}\) and \(\psi_{2}\), and obtain a more focused TF spectrum than MP. In addition, the dominant frequency of MP-EEMD-based TF spectrum decreases gradually from shallow to deep, which is conforms to the propagation theory of seismic waves. Therefore, the TF spectrum obtained by MP-EEMD performs better. Figure 3: Synthetic signal (**a**), TF spectrum obtained by MP algorithm (**b**), and TF spectrum obtained by MP-EEMD algorithm (**c**). In panel (**a**), the synthetic signal (blue) is obtained by superimposing four wavelets W1–W4 (black), and the specific parameters of wavelets W1–W4 are shown in Table 1. In panels (**b**,**c**), the white dotted line is the true TF location of W1, the three white points are the true TF centers of W2–W4, the three black points are the TF centers of W2–W4 estimated by spectral decomposition algorithm, the cool colors represent small spectral amplitudes, and the warm colors represent large spectral amplitudes. Figure 4: Real seismic signal (**a**), five components decomposed by EEMD (**b**), frequency spectra of the real seismic signal **s** and the five components (**c**), and the sub-signals obtained by adaptive stacking (**d**). \[\mathbf{m}_{n}=\exp\Big{[}-4\ln 2f_{n}^{2}k_{n}^{-2}(t-u_{n})^{2}+i2\pi f_{n}(t -u_{n})\Big{]}, \tag{2}\]0 Real seismic signal (**left**), TF spectrum obtained by MP algorithm (**middle**), and TF spectrum obtained by MP-EEMD algorithm (**right**). The cool colors represent small spectral amplitudes, and the warm colors represent large spectral amplitudes.