## 1 Introduction  
Seismic wave propagation follows [PERSON]'s law [1]; if the velocities of adjacent layers are equal, transmitted waves pass through without refraction or reflection. Assuming that the medium is uniform or horizontally layered, the vertex of the diffraction time-distance curve (imaging point) and the diffraction point (subsurface reflection point) have the same position in the horizontal direction, which is a prerequisite for the application of time misalignment. However, when the velocity changes laterally, the transmitted wave produces refraction time difference, and there is a deviation in the real imaging position of the underground reflection point. Pre-stack depth misalignment imaging can ensure the amplitude fidelity of seismic waves while realizing misalignment imaging and no longer needs to consider the influence of velocity on the time misalignment profile, so it is a research hotspot [2, 3]. The development of seismic migration dates back to the 1920s, when foundational concepts such as [PERSON]'s law and wavefront geometry were applied to manual reflection mapping [4]. A pivotal shift occurred in the 1970s with [PERSON]'s wave equation-based migration methods, which replaced ray-tracing approaches and enabled efficient wavefield reconstruction via finite difference solutions [5]. [PERSON] further advanced wave equation migration by incorporating amplitude preservation and boundary conditions [6], while [PERSON] and [PERSON] introduced frequency-wavenumber domain solutions [7; 8]. Despite theoretical progress in full-wave and reverse-time migration during the 1980s, computational limitations hindered practical adoption. The 1990s marked a turning point with the application of Kirchhoff pre-stack depth migration in complex exploration settings (e.g., Gulf of Mexico subsalt imaging) [9], alongside advancements in amplitude fidelity and velocity model optimization [10; 11]. Recent innovations, such as frequency division migration and Q-compensation techniques, have enhanced resolution in challenging reservoirs [12; 13], solidifying migration as a cornerstone of modern seismic imaging. At present, there are two kinds of pre-stack depth migration methods commonly used in China: Kirchhoff integral pre-stack depth migration based on a wave equation integral solution and pre-stack depth migration based on wave equation micro-decomposition [14; 15]. [PERSON] used a space-varying frequency-selective integral operator to suppress the spurious frequency of the migration operator [16]. [PERSON] et al. discussed the theoretical expression of the amplitude weighting of the integral method, but it is not applicable to variable-speed media [17]. [PERSON] et al. applied the fast-marching method (FMM) to the travel time calculation of Kirchhoff integral pre-stack depth migration and found that this method improves the migration speed, and the migration result is good [18]. [PERSON] et al. proposed a new seismic travel time algorithm based on quasi-differential operator theory and the Lie algebra integral method, which is suitable for Kirchhoff pre-stack depth migration in laterally varying media. The effectiveness and accuracy of the algorithm were verified by numerical experiments [19]. [PERSON] et al. studied the Kirchhoff integral vector migration formula based on the homogeneous isotropic elastic wave equation, realized wavefield separation, improved the migration imaging accuracy of multi-component seismic data, and effectively eliminated the wave mode leakage phenomenon [20]. [PERSON] et al. studied the Kirchhoff pre-stack depth migration method based on model constraints and compared it with pre-stack time migration. The results show that pre-stack depth migration imaging has a higher signal-to-noise ratio and resolution [21]. [PERSON] et al.