# High-Fidelity OC-Seislet Stacking Method for Low-SNR Seismic Data
## 1 Introduction
Based on the kinematic theory of reflection waves, [PERSON] [3] proposed the Fourier DMO method in the frequency-wavenumber domain from a geometric perspective [16]. This method is theoretically accurate and serves as a benchmark for evaluating the precision of other DMO methods. But it requires numerical integration, which is computationally intensive, and its theory does not effectively illuminate amplitude and phase issues. Some researchers [14; 17], based on [PERSON] and [PERSON]'s theory [15], have analyzed the time-domain impulse response of the DMO operator and developed various Kirchhoff integral DMO algorithms. These algorithms are based on the integral formulation and have been found to have the drawback of causing interface distortions. However, their advantage lies in not requiring regular sampling. For three-dimensional data acquisition with significant variations in source-receiver distance and azimuth, integral methods are often the most practical for DMO calculations. [PERSON] [18] proposed a Radon-domain dip moveout (Radon DMO) processing technique. The Radon DMO operator directly maps NMOed data to the Radon domain DMO wavefield. The forward transform converts a single NMO trace into multiple traces distributed along hyperbolas in the Radon domain. Applying the inverse linear Radon transform to produce DMO images results in a good frequency and amplitude preservation. DMO can also be performed in the shot domain, [PERSON] and [PERSON] [19] designed a shot gather DMO method equivalent to [PERSON]'s common-offset gather DMO, with this operator being equivalent to the one proposed by [PERSON] et al. In more general situations, such as non-linear observation systems [20; 21; 22] and anisotropic media [23; 24; 25], various DMO methods have been developed to meet specific processing requirements. Each of these DMO methods has its own range of applicability and associated advantages and disadvantages. Meanwhile, most DMO stacking methods remain fixed and require the use of L2 norm optimization mathematical theory to scan stacking velocities and implement NMO. The approach of obtaining stacking results based on average amplitude energy has a limited fidelity for diffraction waves. Thus, research into high-fidelity stacking methods for seismic data holds significant industrial value. In terms of algorithm improvements, many scholars [26; 27] have developed efficient DMO algorithms using the logarithmic time-stretching transform method originally proposed by [PERSON] et al. [12]. The computational efficiency of the \"logarithmic stretching\" technique is approximately two orders of magnitude higher than [PERSON]'s DMO. For the accurate calculation of the DMO operator, [PERSON] [28] introduced a new linear OC partial differential equation. This new DMO operator can be obtained by solving a special initial value problem of the OC equation, with known forms of DMO operators [14] appearing as specific cases of the more general OC operators. Compared to earlier equations, the improved OC equation solves the problem of amplitude variation [29], and the new DMO operator also exhibits causality, resolving the issue of the wavelet's impulse response extending into negative time. [PERSON] et al. [30] combined OC theory with the seislet algorithm, proposing an OC-seislet transform theory in the frequency-wavenumber domain after the logarithmic stretching of the time axis. This method shows good performance in reconstructing missing seismic data and denoising in complex structural areas [31]. The seislet transform incorporates a lifting scheme [32] for discrete wavelet transform (DWT), with the zero-scale coefficient analogous to the stacking calculation [33]. The seislet stack leverages the high compression advantage of the discrete wavelet transform [34] and employs prediction and update operators to perform moveout and stacking simultaneously in a single process. The benefits of partial approximation moveout and multiple calculations within the seislet loop enhance the prominence features [35].