## 1 Introduction  
In recent years, there has been significant research focused on renewable and clean wind energy as a more effective solution to addressing resource limitations and mitigating climate change. To harness more stable and potent wind resources, there has been a continuous expansion in the performance of offshore structures [1]. In terms of design, manufacturing, and installation, single-pile foundations are the most prevalent and cost-effective choice. In fact, approximately 80% of offshore wind farms in Europe opt for this basic form of foundation [2, 3]. Under prolonged exposure to water flow, the presence of a single pile can alter the initial flow field and the dynamics of water flow, leading to local scour. This phenomenon causes the erosion and degradation of the surrounding soil near the pile foundation in offshore wind turbines and other engineering projects. As a result, the lateral bearing performance and stiffness of the pile foundation are significantly affected, posing a serious risk to the stability and safety of the pile foundation and its supporting structure [4]. Therefore, it is necessary to study the influence of scouring on a single pile. Scholars both domestically and internationally have made noteworthy advancements in investigating erosion-related concerns pertaining to pile foundations, aiming to unveil its impact on the lateral bearing performance of a single pile. [PERSON] et al. [5] conducted tests on model piles, both with and without scour, to unveil the impact of scouring on the design of laterally loaded piles. [PERSON] et al. [6] calibrated the numerical model for a single pile in marine soft soil using field test data without scour and established the correlation between scour dimensions and the ultimate lateral bearing performance of a single pile. [PERSON] et al. [7] revealed the impact of scour dimensions on the bearing performance and displacement behavior of a single pile using the p-y curve method. Through the improved p-y curve method based on [PERSON]'s solution, [PERSON] et al. [8] revealed the influence of the soil stress history and three-dimensional scour geometry on the lateral mechanical properties of a single pile in soft clay. Utilizing a tailored strain wedge method for scour holes, [PERSON] et al. [9] disclosed how scour hole sizes affect the lateral load response of a single pile in sand. Based on the established earthquake-wind-wave-vehicle-bridge (EWWVB) dynamic coupling system, [PERSON] et al. [10] computed the load-displacement relationship of a pile foundation under different scour degrees using the p-y curve method. In current pile design practices, the local scour effect is typically addressed by removing the sediment layer within the local scour depth range when analyzing the lateral response of the pile. However, the contribution of the sediments surrounding the pile is often disregarded in this process [11; 12]. If the soil layer is completely scoured to the scour depth, the pile's side resistance will be significantly reduced compared to considering the geometry of the scour hole. As a result, there will be a substantial increase in deflection and the maximum bending moment at the top of the pile, which can make the design overly conservative. To facilitate numerical analysis, crater geometry is usually idealized as an inverted truncated cone [7; 13]. However, the physical model test clearly shows that the scour hole is asymmetric in reality [14]. For the pile foundation in the asymmetric scour hole, the soil around the pile is eroded to varying degrees, resulting in unequal lateral forces generated via the self-weight of the overlying soil on both sides of the pile foundation along the flow direction, resulting in the pile foundation in a dangerous state. Therefore, it is necessary to study the influence of asymmetric scour holes on the bearing characteristics of horizontally loaded piles. Through an array of centrifugal model tests on single model piles in saturated sand, [PERSON] et al.