## 1 Introduction  
As a natural material, rocks exhibit random distributions of various internal defects. The deterioration of a rock's macroscopic mechanical properties due to changes in these internal defects under external loads and environmental conditions is termed damage [1, 2]. The propagation of microcracks is often seen as a process of damage accumulation in rocks, ultimately leading to their macroscopic destruction [3, 4, 5, 6, 7]. The deformation and damage mechanisms of laminated rock bodies are influenced by the rock's fine structure, the orientation of laminations, and ambient pressure, resulting in significant anisotropy [8, 9]. Changes in local stress and storage environment further complicate the damage characteristics and mechanisms of layered rock [10, 11, 12, 13]. Therefore, understanding the damage evolution process of layered rock masses is crucial for engineering stability control and safety prediction. In recent years, numerous scholars have conducted theoretical and experimental studies on the damage evolution process of layered rock bodies. [PERSON] et al. [6] and [PERSON] et al. [14] performed impact tests on shale, using the HJC model to define damage variables and establishing a finite element model that considers confining pressure and inclination angle. [PERSON] et al. [7] used digital imaging techniques to investigate the time damage evolution of laminated hard sandstone. [PERSON] et al. [15] defined an intrinsic model of layered rock damage characterized by a second-order tensor, verified through numerical simulations. [PERSON] et al. [1] conducted biaxial loading tests on layered sandstones with a 15\({}^{\circ}\) dip angle, analyzing the statistical damage evolution considering lateral pressure. [PERSON] et al. [16] examined the relationship between the critical damage and laminar inclination angle of laminated sandstone under cyclic loading, analyzing the change in fatigue sensitivity with laminar inclination angle. [PERSON] et al. [17] explored the effect of laminar inclination on the damage evolution of yellow sandstone based on energy dissipation, establishing a uniaxial damage principal model with logistic function to account for initial damage. Currently, research on rock damage has advanced significantly, and these studies have laid a solid foundation for analyzing the damage-deformation-destruction process of layered rocks, employing a diverse array of tools and methods. At the macroscopic scale, numerous damage variables have been established to characterize the evolution of rock damage. These variables can be derived from experimental measurements, statistical analyses, or geometric damage theories. However, despite the progress, existing damage variables for rocks exhibit certain limitations. The primary issue lies in the fact that the definition and description of initial damage remain unclear. To avoid negative damage, it is essential to define a suitable initial damage that reflects pre-existing defects or those present at the onset of external force application [18; 19; 20]. [PERSON] et al. [21; 22] modeled the acoustic properties and damage evolution of rocks under loaded conditions, defining the longitudinal wave velocity of rocks under unloaded conditions as the initial damage. [PERSON] et al. [23] used elastic longitudinal wave velocity to define initial damage and analyze the damage evolution process of surrounding rock during excavation. [PERSON] et al. [24] defined initial damage using acoustic emission ringing counts as a covariate to study the damage process of layered composite rocks under uniaxial compression. In summary, defining the initial damage of rock based on ultrasonic wave velocity effectively reflects its damage evolution characteristics [25]. However, this approach primarily focuses on longitudinal wave velocity at specific moments during loading or stressing, without examining real-time wave velocity changes.