# High-Fidelity OC-Seislet Stacking Method for Low-SNR Seismic Data
## 1 Introduction  
Random noise is one of the most common types of interference in seismic data; in complex media areas such as the piedmont zone, high-steep structures, and volcanic rock cap layers, there is often a significant loss of energy in the effective reflection information [1]. The task of enhancing weak seismic signals becomes increasingly important, because random noise interference often hinders the accurate delineation of deep-target oil and gas zones during onshore and offshore exploration. The classic horizontal stacking technique leverages the advantages of multiple coverage geometry in seismic exploration [2], playing a crucial role in enhancing the SNR of seismic data. Although numerous methods for suppressing random noise in pre-stack data have been developed by many researchers, seismic data stacking remains the most stable and effective high-SNR processing technique in the industry. Since the horizontal stacking method is based on a horizontal layering model, it is essential to perform normal moveout (NMO) before stacking to eliminate the normal time differences at various offset positions of the traces [2]. However, relying solely on time difference moveout is insufficient, as stacking results can be inaccurate in the presence of dipping layers [3]: On the one hand, the CMP gathers after NMO donot correspond to the data recorded from the same reflection point, so the stack cannot reflect the true zero-offset profile. On the other hand, the stacking process based on NMO enhances reflections with specific moveout velocities, while attenuating wave propagation events with different moveout velocities (such as multiple reflections), so this leads to the enhancement of reflection layers with specific slopes during the NMO stacking process, while reflections and diffraction with different slopes, such as those at layer breaks, are attenuated [3]. This phenomenon is particularly pronounced in low-SNR data. Since 1978, DMO has been employed in seismic data processing and has significantly improved the imaging accuracy of seismic data [4]. DMO technology can shift the NMOed data to the zero-offset position, thereby reducing the mutual attenuation effects of wavefields with different reflection slopes. DMO, also known as partial pre-stack migration, is often one of the most overlooked steps in time-domain imaging, yet it provides more accurate stacking velocities [5]. DMO processing is essential before post-stack migration imaging, as it significantly improves the imaging quality of stacked profiles in complex geological structures. Compared to pre-stack migration, which demands a high accuracy in velocity modeling and considerable computational resources, DMO offers a practical compromise between cost and precision and has achieved widespread application in the industry. Many researchers have proposed different DMO processing methods to achieve accurate imaging. For instance, [PERSON] and [PERSON] [6] introduced a finite-difference DMO method [7; 8; 9; 10; 11] in common-offset seismic gathers. This method uses a high-order approximations of the wave equation, offering mature theoretical and practical applications, along with advantages in denoising. However, finite-difference methods require calculations at equally spaced intervals and often struggle to accurately represent large offsets and steep reflector dips [7; 9; 10], even in constant velocity scenarios. [PERSON] et al. [12] were the first to suggest extending the concept of offset continuation (OC) and DMO using partial differential equations, describing them as continuous wavefield propagation processes. [PERSON] and [PERSON] [13] proposed representing migration continuation with a complete set of pre-stack continuation operators. [PERSON] [14] later defined DMO as an continuation to zero offset, but the limitation of the continuation theory lies in its use of many approximations, which leads to distortions in the impulse response compared to the true elliptical DMO impulse response [15].