## 3 Examples  
Before testing the proposed wavelet extraction method, we synthesize the depth-domain seismogram as input. For a depth-domain heterogeneous viscoacoustic model with a sampling number, \(N\), and depth sampling interval, \(\Delta t\), the wavenumber spectrum, \(F_{i}(k)\), of the attenuated wavelet at point \(i\) can be written as follows:  
\[F_{i}(k)=F_{0}(k)\cdot\underset{j=1}{\overset{i}{\Pi}}g_{0,3D_{j}}(x|x_{0},k_{ j}), \tag{27}\]  
Figure 6: Flowchart of depth-domain generalized seismic wavelet extraction method. where \(F_{0}(k)\) is the wavenumber spectrum of the source wavelet, \(\Pi\) means accumulative multiplication, \(g_{0,3D_{j}}(x|x_{0},k_{j})\) represents the wavenumber-domain 3D Green function at point \(j\), and \(k_{j}\) denotes the wavenumber at point \(j\), which is related to the velocity at point \(j\). Additionally, the distance, \(|x-x_{0}|\), equals the depth sampling interval, \(\Delta{h}\). Therefore, the depth-domain attenuated wavelet at point \(i\) can be obtained by the inverse Fourier transform of \(F_{i}(k)\). Substituting the depth-domain attenuated wavelets and the reflectivity into Equation (20) yields the depth-domain synthetic seismogram. A depth-domain weak attenuation model with sampling number 885 and a depth sampling interval of 0.0025 km is presented in Figure 7, which is used to elaborate the workflow of our depth-domain generalized seismic wavelet extraction method. Figure 7a,b depict the velocity and \(Q\), respectively. When a 20 Hz Ricker wavelet propagates through the weak attenuation model, the depth-domain attenuated wavelets computed by Equation (27) are displayed in Figure 7c. It can be seen that with increasing depth, the waveform of the attenuated wavelet is no longer symmetrical. The energy of the right sidelobe decreases, and the energy of the left sidelobe increases. Figure 7e is the AI calculated through the density shown in Figure 7d and the velocity shown in Figure 7a. Figure 7f is the reflection coefficient. Substituting the depth-domain attenuated wavelets and the reflectivity into Equation (20) yields the depth-domain synthetic seismogram, as demonstrated in Figure 7g. Next, the proposed method extracts the depth-domain generalized seismic wavelets from the depth-domain synthetic seismogram. Figure 8a shows the overcomplete dictionary established by the depth-domain generalized seismic wavelets during the first OMP-Gd. The search range of the fractional derivative is 1.5-2.1 with an interval of 0.05, the search range of the reference wavenumber is 6-26 km\({}^{-1}\) with an interval of 1 km\({}^{-1}\), and the depth delay ranges from 0 to 2.21 km with an interval of 0.0025 km. Figure 8b shows the first 2000 atoms enlarged in Figure 8a. The depth-domain generalized seismic wavelet extraction test of Figure 7g is demonstrated in Figure 9. Figure 9a shows the first 30 optimal depth-domain generalized seismic  
Figure 8: (**a**) The overcomplete dictionary during the first OMP-Gd, and (**b**) an enlarged view of the first 2000 atoms of (**a**). Figure 7: A depth-domain weak attenuation model, including (**a**) velocity; (**b**) \(Q\); (**c**) AI; (**d**) density; (**e**) acoustic impedance; (**f**) reflectivity; (**g**) synthetic seismogram. wavelet atoms obtained from the first OMP-G\({}_{\text{d}}\). The red dotted line in Figure 9b represents the reconstructed seismogram of the first OMP-G\({}_{\text{d}}\) obtained by adding these optimal atoms, and the black solid line is the actual seismogram. It can be seen that the primary waveforms of the reconstructed seismogram match those of the real seismogram well. Figure 9c,d show the extracted fractional derivative and reference wavenumber, respectively, where the solid black line represents the reference value calculated according to the analytical wavenumber spectrum of Equation (26).