## 3 Methodology  
### Spectral Decomposition Technology without Frequency Interference  
The key of using low-frequency seismic attribute to characterize hydrocarbon reservoir is the TF resolution of spectral decomposition algorithm. MP integrated with WVD is a high TF resolution spectral decomposition technique without the limit of [PERSON]'s uncertainty principle, where MP is the crucial signal decomposition algorithm and WVD is the spectral calculation algorithm. MP decomposes a seismic signal into a linear superposition of a series of pre-defined wavelet atoms. \[\mathbf{s}=\text{Re}\Bigg{\{}\sum_{n=1}^{N}c_{n}\mathbf{m}_{n}+\mathbf{r}_{N} \Bigg{\}}, \tag{1}\]  
where \(\mathbf{s}\) is the (real-valued) seismic signal, \(n\) is the iteration number, \(c_{n}\) is the complex coefficient at the \(n\)th iteration, \(\mathbf{m}_{n}\) is the optimal wavelet atom at the \(n\)th iteration, \(\mathbf{r}_{N}\) is the residual error at the \(N\)th iteration, which is only noise and all significant features of seismic signal are recorded in the first term on the right-hand side of Equation (1), and \(\text{Re}[\bullet]\) is the real part operator. The wavelet atom \(\mathbf{m}_{n}\), containing the time and frequency features of seismic signal, is extracted from the atomic dictionary. Here, we choose the Morlet wavelet as the atom because it is suitable for frequency quantification and attenuation studies of seismic signal [21]. \[\mathbf{m}_{n}=\exp\Big{[}-4\ln 2f_{n}^{2}k_{n}^{-2}(t-u_{n})^{2}+i2\pi f_{n}(t -u_{n})\Big{]}, \tag{2}\]  
where \(t\) is the time (s), \(u_{n}\) is the center time (s), \(k_{n}\) is the (dimensionless) scale, \(f_{n}\) is the center frequency (Hz), and \(i\) is the imaginary unit. The key parameters (\(u_{n}\), \(k_{n}\),\(f_{n}\)) of \(\mathbf{m}_{n}\) are estimated by the following equations [10]  
\[u_{n}=\arg\max_{t}[\mathbf{r}_{n-1}(t)]^{\alpha\
u}, \tag{3}\]  
\[(k_{n},f_{n})=\arg\max_{k\in D_{k},f\in D_{f}}\frac{\left|\mathbf{r}_{n-1}(t) \cdot\mathbf{m}_{n}(t,u_{n},k,f)\right|}{\left\|\mathbf{m}_{n}(t,u_{n},k,f) \right\|_{2}}, \tag{4}\]  
where \(\mathbf{r}_{n-1}\) is the residual error at the (\(n\)-1)th iteration, \(\mathbf{r}_{0}=\mathbf{s}+i\mathbf{s}^{H}\), superscript \(H\) is the Hilbert transform operator, superscript \(env\) is the envelope operator, \(\bullet\) is the dot product operator, \(D_{k}\) and \(D_{f}\) is the searching interval of \(k_{n}\) and \(f_{n}\), respectively. In general, \(D_{k}\) and \(D_{f}\) are given empirically, such as \(D_{k}=\left[0.3,6\right]\) and \(D_{f}=\left[f_{u}-R_{f},f_{u}+R_{f}\right]\), where \(f_{u}\) is instantaneous frequency of \(\mathbf{r}_{n-1}\) at the time \(u_{n}\), and \(R_{f}\) is the search radius of \(f_{n}\), which is always set to 20 Hz [10; 21]. The complex coefficient \(c_{n}\), containing the amplitude and phase features of seismic signal, is expressed as follows  
\[c_{n}=\Big{(}\mathbf{m}_{n}^{T}\mathbf{m}_{n}+\varepsilon\Big{)}^{-1}\mathbf{ m}_{n}^{T}\mathbf{r}_{n-1}. \tag{5}\]  
where superscript \(T\) is the transpose operator and \(\varepsilon\) is the damping constant. According to the WVD, the TF spectrum of seismic signal is expressed as [10]  
\[\mathbf{A}(t,f)=\sum_{n=1}^{N}\frac{4\big{[}(\text{Re}\{c_{n}\})^{2}+(\text{ Im}\{c_{n}\})^{2}\big{]}\exp\Bigl{[}-\frac{\pi^{2}}{2\ln 2}\frac{k_{n}^{2}}{f_{n}}(f-f_{n})^{2}-8\ln 2 \frac{f_{n}^{2}}{k_{n}^{2}}(t-u_{n})\Bigr{]}}{1+\exp\Bigl{(}-\frac{\pi^{2}k_{n }^{2}}{2\ln 2}\Bigr{)}\cos\bigl{(}\arctan\frac{\text{Im}\{c_{n}\}}{\text{Re}\{c_{n} \}}\Bigr{)}}, \tag{6}\]  
where \(\text{Im}\{\bullet\}\) is the imaginary part operator. As follows from Equation (4), the accuracy of \(f_{n}\) affects the frequency resolution of MP algorithm. However, the interference of multi-frequency components in nonstationary seismic signals causes inaccuracy in searching \(f_{n}\), and then introduce error in the TF spectrum. To reduce the effect of frequency interference, a modified MP scheme based onEEMD (MP-EEMD) is proposed in this work. MP-EEMD mainly includes four stages.