## 3 Examples
The magenta, blue, and green solid lines represent the extraction results of the first, second, and third iterations of OMP-G\({}_{\text{d}}\), while the red dotted line is the result of the fourth OMP-G\({}_{\text{d}}\) (the final output). It can be seen that with the increase in the number of iterations, the extracted parameters of depth-domain generalized seismic wavelets move closer to the reference values. Figure 9e shows the corresponding depth-domain generalized seismic wavelets, which are very similar to the depth-domain attenuated wavelets in Figure 7c. We compare our method with the pseudo-depth-domain SPE method. Figure 10 shows the extraction results of the pseudo-depth-domain SPE method, where the vertical ordinate is the pseudo-depth. Figure 10a shows the pseudo-depth-domain seismogram obtained by converting the depth-domain seismogram with the depth-domain velocity model and the constant velocity value of 3 km/s into the pseudo-depth domain. Similarly, Figure 10b shows the pseudo-depth-domain reflection coefficient. Figure 10c,d show the extracted constant wavelet and the synthetic record in the pseudo-depth domain, respectively. Figure 11 displays the depth-domain synthetic records obtained by the pseudo-depth-domain SPE method and the depth-domain generalized seismic wavelet extraction method, where the former is returned to the depth domain from the result in Figure 10d. Due to the resampling in the pseudo-depth-domain conversion, the reflection coefficients are stretched or compressed. Therefore, the synthetic record of the pseudo-depth-domain SPE method poorly matches the real seismogram. As the green arrows show, it exhibits distortion and dislocation of the waveform and changes the relative energy relationship between waveforms. In contrast, the synthesized record obtained by the proposed method agrees with the actual seismogram in terms of the waveform and amplitude. The PCCs between the synthetic records and the real seismogram are 85.7% and 98.52%, respectively, showing that our method can accurately extract the depth-domain attenuated wavelets in the case of weak attenuation. Figure 9: An example of depth-domain generalized seismic wavelet extraction from Figure 7g. (**a**) The decomposed optimal atoms by OMP-Gd; (**b**) a comparison of the original (black solid line) and reconstructed (red dashed line) seismic trace, where the latter is the superposition of the optimal atoms; (**c,d**) are extracted fractional derivatives and the reference wavenumber, respectively; (**e**) extracted depth-domain generalized seismic wavelets. In practice, some media exhibit strong attenuation, namely small \(Q\), such as weathering zones and deserts, etc. Therefore, based on the weak attenuation model, the value of the quality factor is reduced to build a strong attenuation model in the depth domain, as shown in Figure 12a. Also, let a 20 Hz Ricker wavelet propagate through the depth-domain strong attenuation model. The depth-domain attenuated wavelets computed through Equation (26) are displayed in Figure 12b. It can be found that with increasing depth, the energy of the right sidelobe decreases more than that of the weak attenuation model. Figure 12c demonstrates the depth-domain synthetic seismogram by convoluting the depth-domain attenuated wavelets with the reflectivity shown in Figure 7f. By comparing the seismograms in Figures 7g and 12c, it is evident that the deep waveform becomes significantly wider in the case of strong attenuation, marked by the cyan dashed ellipse. Next, the proposed method extracts the depth-domain generalized seismic wavelets, as seen in Figure 12c. Figure 12d shows the estimated depth-domain generalized seismic wavelets, which are highly similar to the depth-domain attenuated wavelets in Figure 12b. Figure 12f is the estimated pseudo-depth-domain constant wavelet by the pseudo-depth SPE method. Figure 12e compares depth-domain synthetic records obtained by the two methods.