## 3 Methodology
### Seismic Fluid Mobility Attribute  
A simplified asymptotic representation of low-frequency reflection coefficient \(R\) for the normal-incidence plane P-wave in a fluid-saturated media is expressed as [17; 18]  
\[R(f)=R_{0}+R_{1}\sqrt{m\cdot 2\pi f}, \tag{10}\]  
where \(m\) is the fluid mobility, \(R_{0}\) and \(R_{1}\) are dimensionless coefficients, which depend on neither frequency nor fluid mobility. Taking the derivative of Equation (10) with respect to frequency yields  
\[m=\frac{2}{\pi R_{1}^{2}}\bigg{[}\frac{\partial R(f)}{\partial f}\bigg{]}^{2}f. \tag{11}\]  
Equation (11) implies the fluid mobility is proportional to the product of the square of the first derivative over frequency of the reflection coefficient spectrum and the frequency. According to seismic convolution theory, the seismic amplitude spectrum \(A(f)\) is proportional to the reflection coefficient spectrum \(R(f)\) under the assumption of fixed seismic wavelet. Studies on the seismic amplitude spectrum change in fluid included [10; 29] also show that fluid mobility reflects the first derivative of the seismic amplitude spectrum over frequency. Therefore, we define the seismic fluid mobility \(M\) as  
\[M=\sum_{f\in\Omega}\bigg{[}\frac{A(f+\Delta f)-A(f)}{\Delta f}\bigg{]}^{2}f. \tag{12}\]  
where \(\Delta f\) is the frequency increment, and \(\Omega\) is range of the calculational (low) frequency, which is estimated from the seismic trace data near wells and the logging interpretation data. Choosing a frequency range instead of a single frequency can improve the reliability of fluid mobility.  
Figure 6 displays the seismic amplitude spectra and the corresponding frequency gradients for different lithologies with different gas-bearing conditions near S-2 versus frequency. Noted that the spectrum amplitude of gas shale is higher than that of gas-free shale and siltstone. We also can note that the frequency gradients of the gas zone 13-23 Hz (blue rectangle) are also higher than that of the gas-free zone. This means that the seismic fluid mobility attribute can effectively distinguish the gas zone and the gas-free zone.  
Figure 6: Seismic amplitude spectra (**top**) and the corresponding frequency gradients (**bottom**) for different lithologies with different gas-bearing conditions near well S-2 versus frequency.  
Figures 7-9 display the seismic trace records and the normalized seismic fluid mobility attributes (Equation (12)) near S-2, S-3, and S-4. The seismic fluid mobility attribute extracted by using the MP-EEMD algorithm corresponds to the blue curve and extracted by using the MP algorithm corresponds to the blue dotted curve. From Figure 7, we can see that the high value of seismic fluid mobility attribute indicates the location of gas zone (red rectangle) and the low value corresponds to the gas-free zone. As shown with black arrows in Figures 8 and 9, the seismic fluid mobility based on the MP-EEMD algorithm (blue curves) can exhibit less interference for the gas-free zone at well S-3 and better indicate the gas zone at well S-4 than that based on the MP algorithm (blue dotted curves). These factors not only prove the effectiveness of seismic fluid mobility attribute, but also prove the advanced nature of the MP-EEMD algorithm.