## 2 Methodology
### Concept of Depth-Domain Generalized Seismic Wavelet  
A generalized seismic wavelet is a compact and flexible wavelet model that describes the actual seismic wavelet well. The generalized seismic wavelet is defined by the following [36]:  
\[\Phi(\omega|u,\omega_{0})=\Big{(}\frac{u}{2}\Big{)}^{-u/2}\frac{\omega^{u}}{ \omega_{0}^{u}}\exp\left(-\frac{\omega^{2}}{\omega_{0}^{2}}+\frac{u}{2}\right) \exp\left(-i\omega\tau_{0}+i\pi\Big{(}1+\frac{u}{2}\Big{)}\right), \tag{17}\]  
where \(u\) represents the fractional derivative, which controls the sidelobe and phase, \(\omega_{0}\) denotes the reference frequency, controlling the width, and \(\tau_{0}\) denotes the time delay, which controls the central position of the generalized seismic wavelet. Figure 3 shows the generalized seismic wavelet with different fractional derivatives. The reference frequency of these generalized seismic wavelets is 20 Hz, and the time delay is 0 s. It can be seen that the larger the fractional derivative, the greater the sidelobe, and the larger the phase rotation. When the fractional derivative equals 2, the generalized seismic wavelet is equivalent to the Ricker wavelet. Substituting Equation (14) into Equation (17) yields the following:  
\[\Phi(k|u,k_{0})=\Big{(}\frac{u}{2}\Big{)}^{-u/2}\frac{k^{u}}{k_{0}^{u}}\exp \left(-\frac{k^{2}}{k_{0}^{2}}+\frac{u}{2}\right)\exp\left(-ikl_{0}+i\pi\Big{(} 1+\frac{u}{2}\Big{)}\right), \tag{18}\]  
Figure 3: Comparison of generalized seismic wavelets with varied fractional derivatives. These wavelets have a common reference frequency of 20 Hz and a time shift of 0 s.  
Figure 2: Phase velocity of a homogeneous viscoacoustic medium with different \(Q\) values, the velocity equals 2 km/s. where \(u\) represents the fractional derivative, \(k_{0}\) denotes the reference wavenumber, controlling the width, and \(l_{0}\) denotes the depth delay, which controls the central position of the depth-domain generalized seismic wavelet. It is necessary to investigate the similarity between the depth-domain generalized seismic wavelet and the depth-domain attenuated wavelet before using the former to characterize the latter. Hereinafter, the term \"attenuated wavelet\" will refer to the wavelet of attenuation media. Figure 4 compares the depth-domain generalized seismic wavelets and attenuated wavelets shown in Figure 1, where the \(Q\) value differs between rows, and the distance varies between columns. The parameters of the depth-domain generalized seismic wavelet used in Figure 4 are shown in Table 1, where the reference wavenumbers correspond to the peak wavenumbers of the wavenumber spectrum of the depth-domain attenuated wavelets shown in Figure 1, and the fractional derivatives are obtained according to experience. Additionally, the depth delay equals the distance. As can be seen from Figure 4, the depth-domain generalized seismic wavelet and the depth-domain attenuated wavelet always have high similarity. Although the error of the two gradually increases with the decrease in the \(Q\) value or the increase in the propagation distance, the error is still tiny, even in the case of a 1.0 km propagation in the strong attenuation media with \(Q=20\). Figure 5 demonstrates the Pearson correlation coefficients (PCCs) between the depth-domain generalized seismic wavelets and the depth-domain attenuated wavelets, from which, it can be seen that the similarity between the two remains high. In particular, when propagating 1.0 km in a strongly attenuated medium with \(Q=20\), the PCC between the depth-domain generalized seismic wavelet and the depth-domain attenuated wavelet is still greater than 98.5%. It has been proved that the depth-domain generalized seismic wavelet can be used to characterize the depth-domain wavelet of attenuation media.