# Pseudo-3D Receiver Deghosting of Seismic Streamer Data Based on \(l_{1}\) Norm Sparse Inversion
## 3 Results
### Synthetic Example  
To ensure the completeness of the synthetic data, we choose the wave equation-based simulation method, i.e., the fine difference algorithm, to conduct forward modeling. The synthetic multi-shot synthetic records include 256 CSGs, 256 COGs, 512 samplings, and a sampling interval of 0.004 s. Then, we apply the conventional 2D method and the sparse inversion method to obtain the deghosting results, and we compare the results of the two methods from the CSGs, COGs, \(f\)-\(k\) spectrum, etc. Figure 6 shows the original data before deghosting. Figure 6a-c shows the data cube, the zoom section of the COG, and the corresponding \(f\)-\(k\) spectrum of the CSG. From Figure 6a, we can observe that when the sinking depth is large, ghost is easy to recognize. The separation of the primaries and the ghosts can be clearly observed in the near-offset section. Figure 6b shows that the frequency bandwidth of the deghost-interfered datasets is narrowed, with a lack of low frequency (5-10 Hz) and a main frequency notch (25-50 Hz). Therefore, in order to obtain a frequency-rich seismic record, it is necessary to deghost. Figures 7 and 8 respectively demonstrate the deghosting results of the conventional 2D algorithm and the proposed method. Comparing Figures 7a and 8a, it can be seen that both methods have suppressing effects on deghosting. However, the conventional algorithm has some artifacts in the deep position (blue arrow). The reason is that the complexity of the underground structure increases and the inversion process is unstable. These artifacts in the sparse inversion result are effectively suppressed, which proves that the introduction of the proposed method improves the deghosting effect. From Figures 7b and 8b, we see that the notch frequency is mainly 25-50 Hz. Both results compensate for the notch effect, but some energy leakage can still be observed in the conventional results. The sparse inversion results have the complete frequency band and more balanced high-frequency energy, which proves the effectiveness of the proposed method. Figures 7c and 8c show the zoomed  
Figure 5: The demonstration of the saltdome model. section of the COGs. The difference in the deep events (1.5 s, 150-250 th shots) is obvious. The conventional method has artifacts but the sparse inversion results do not. Figure 6: The synthetic data before deghosting. (**a**) 3D data cube including 256 CSGs and 256 COGs. (**b**) _f-k_ spectrum in which the black oval marks the effect of ghost. (**c**) COG wherein the events are similar to the underground interface. Figure 7: The conventional deghosting results. (**a**) 3D data cube including 256 CSGs and 256 COGs. (**b**) _f-k_ spectrum in which the black oval marks the effect of ghost. (**c**) COG wherein the events are similar to the underground interface. To further demonstrate the effectiveness of the proposed method, we calculate the frequency spectrum in the time window (1.25-1.5 s) to quantitatively compare the frequency compensation ability of the conventional deghosting method and the pseudo-3D deghosting method (Figure 9). Figure 9b clearly shows the energy comparison of the three curves. Compared with the _f-k_ spectrum shown above, we can observe a good correspondence between them. It is shown in Figure 6b that the frequency bandwidth of deghost-interfered datasets is narrowed at the main frequency notch (25-50 Hz), and in Figure 9b, the original data has the lowest energy (near 0.25) at approximately 40 Hz. Both the red curve and the yellow curve in Figure 9b have higher energy than the original data in the main frequency notch (25-50 Hz), which is consistent with the phenomenon seen in Figures 7b and 8b, where both methods compensate for the notch effect. Similarly, the energy leakage caused by traditional methods also leads to a large difference between the 2D method and the pseudo-3D method in a quantitative comparison of the main frequency energy. The results of the quantitative comparison also prove the effectiveness of the proposed method.