## 2 T* Revise Attenuated Travel Time Inversion  
Q value can be calculated by using the inverted attenuated travel time [45]. The misfit of attenuated travel time between the travel time from the recorded data and synthetic data can be used for Q value calculation; Q value is updated by minimizing the misfit:  
\[\delta{t_{t}}^{*}=t_{r}^{*}-t_{m}^{*}=\sum{t_{ij}\delta{Q_{j}}^{-1}}, \tag{1}\]  
where \({t_{r}}^{*}\) and \({t_{m}}^{*}\) are the attenuated travel times calculated from real data and synthetic data, respectively. \(\delta{t_{i}}^{*}\) is the difference between the attenuated travel time from real data and \(\delta{Q_{j}}^{-1}\) the attenuated travel time from synthetic data, and is the Q value needed to be updated towards the true value. The attenuated travel time calculated from real data \({t_{r}}^{*}\) can be calculated by the logarithmic spectral ratio (LSR) inversion method:  
\[b=ln(\frac{A_{(t_{2},f)}}{A_{(t_{1},f)}})=ln(G)-\frac{\pi f(t_{2}-t_{1})}{Q}=B -\pi ft_{r}^{*}, \tag{2}\]  
where A is amplitude spectra of seismic wave; at travel times t1 and t2, the amplitude spectra of seismic waves are \(A_{(t_{1},f)}\) and \(A_{(t_{2},f)}\), respectively. The frequency independent attenuation is G. B is the intercept term and equals to \(ln(G)\). The proposed method can inverse both \(ln(G)\) and \({t_{r}}^{*}\) using multi traces to avoid the effect of frequency range selection in the conventional LSR method. When the seismic events are close to each other in the vertical direction, the time-frequency spectrum can be obtained by time-frequency transformation of the seismic signal. The common time-frequency analysis methods include short-time Fourier transform, continuous wavelet transform, Wigner-Ville distribution, S-transform, generalized S-transform, etc. Short-time Fourier transform is limited by the Heisenberg inaccuracy and the window function, and the time-frequency resolution is fixed and cannot be improved simultaneously. Continuous wavelet transform needs to transform time-scale distribution into time-frequency distribution, which can adjust the window function adaptively with frequency and solve the shortcoming of fixed short-time Fourier time-frequency resolution. The Wigner-Ville distribution portrays the time-frequency distribution of the signal from the perspective of the energy spectrum or power spectrum and does not use the window function, thus avoiding the interaction between the time and frequency domains and has a high time-frequency resolution, but when dealing with multi-component signals, the interference of cross terms is generated. The S-transform uses a Gaussian window function to directly convert the signal from the time domain to the time-frequency domain, with variable time-frequency resolution. The window function of S-transform varies with frequency in a fixed trend, which lacks flexibility. The generalized S-transform is able to adjust the window function by introducing the adjustment factor, which can adjust the time-frequency resolution according to the signal characteristics and has good practicality, as follows:  
\[GST(\tau,f)=\int_{-\infty}^{+\infty}f(t)\frac{\lambda|f|^{p}}{\sqrt{2\pi}}e^{ (-\frac{\lambda^{2}p(t_{r}-\tau)^{2}}{2})}e^{(-i2\pi f)t}dt \tag{3}\]  
where \(\lambda\) and \(p\) are the scale adjustment factors of the Gaussian window function. The different ray travel paths from the selected reflections will cause error in \({t_{r}}^{*}\) results. In each layer, the different travel times of the selected reflections is taken into account. Thus, the overburden effect can be effectively avoided by the proposed method. As shown in Figure 1, a two-layer model is used here to test the performance of the proposed method. By using the reference reflections \(A_{1}\), \(A_{3}\),..., \(A_{2n-1}\) from the first layer and target reflections \(A_{2}\), \(A_{4}\),... \(A_{2n}\) from the second layer, the proposed method calculates attenuated travel time simultaneously.