# High-Fidelity OC-Seislet Stacking Method for Low-SNR Seismic Data
## 3 Results  
### Synthetic Model Data Tests  
Firstly, the OC-seislet stacking method is tested on the benchmark French model, and Figure 2 illustrates the 2-D slice of the French model (Figure 2a) and the pre-stack CMP gathers (Figure 2b). To evaluate the effectiveness of DMO under inaccurate velocity conditions, NMO is performed using an inaccurate velocity. The NMO results are shown in Figure 3. Figure 3a displays a data slice of the reflection events at the cross point of the CMP gather (CMP location 1.04 km), while Figure 3b presents a data slice of the reflection events with a dipping layer (CMP location 1.32 km). From Figure 3, it is evident that the reflection events have not been properly leveled, indicating that the initial NMO is inaccurate. Additionally, due to the presence of diffraction points in the model, distinct diffraction waves are evident in Figure 3. The results of the time-space domain DMO processing are shown in Figure 4. The DMO for the large offset traces is quite noticeable; the events become more horizontal compared to the NMO results. However, due to inaccuracies in velocity analysis, the events at the large offset position cannot be fully aligned to horizontal, which affects the final stacking accuracy. We use the low-order OC-seislet transform based on Haar wavelets to approximate the coefficient compression of conventional frequency-wavenumber-scale domain DMO stacking; Figure 5 shows a comparison between high-order OC-seislet coefficients and low-order OC-seislet coefficients. The results indicate that the high-order OC-seislet transform provides a higher compression ratio, where the coefficients of the high-order seislet transform are compressed in a small dynamic range along the scale axis. A compressibility comparison is shown in Figure 6, in which we can see that the coefficient magnitude of the high-order OC-seislet transform decreases more rapidly, resulting in less leakage of the frequency energy of the signal information. After the zero scale of the OC-seislet is transformed back to the time-space domain, Figure 7 shows a comparison between NMO stacking (Figure 7a), conventional DMO stacking (Figure 7b), and the high-order OC-seislet stacking (Figure 7c). It can be observed that the proposed method performs better in handling dipping layers and intersecting events, effectively  
Next, strong random noise is added to the model data, as shown in Figure 9, to test the denoising effectiveness of the OC-seislet stacking. The noise has a significant impact on low-amplitude signals, such as the diffraction waves and reflection waves from dipping layers in this model. To quantitatively analyze the noise suppression performance of different stacking methods, we select the global _SNR_ of the stacked results to evaluate the denoising ability. \[SNR=10 log_{10}\frac{\|x\|_{2}^{2}}{\|\overline{x}-x\|_{2}^{2}} \tag{18}\]  
where \(x\) represents the noise-free stacked signal, while \(\overline{x}\) is the noisy stacked signal or the noisy stacked signal after noise suppression. The OC-seislet transform with zero-scale soft-thresholding denoising (a percentile threshold of 70%) is compared with the results of conventional stacking for analysis. The zero-offset trace and traces from different stacking methods at the dipping layer position are shown in Figure 10a-d. Due to the strong denoising ability of the stacking, Figure 10b-d exhibit lower noise amplitudes, with effective  
Figure 4: DMO results after NMO with inaccurate velocity, the events still not horizontal. (**a**) Position of the intersected seismic events (CMP location 1.04 km); (**b**) position of the dipping layer (CMP location 1.32 km). Figure 5: OC-seislet coefficients of the input data from Figure 3 along the offset axis using different types of transform basis. (**a**) Coefficients based on Haar basis, to approximate the coefficient compression of conventional DMO stack; (**b**) coefficients based on high-order CDF 9/7 basis. wave frequencies and amplitudes being greatly enhanced.