## 1 Introduction  
Rock-socketed piles are a prevalent form of pile foundation utilized in underground bedrock to transfer structural loads of large buildings. They exhibit high bearing performance, low settlement, and allow for the full exploitation of the structural strength of the pile body and the surrounding rock mass. The bearing capacity of rock-socketed piles is attributable primarily to two factors: side frictional resistance and pile base resistance. Among these factors, the side resistance shaft resistance is considered the dominant factor under common working loads [1, 2, 3]. The development of side resistance is influenced primarily by the shear stress at the pile-rock contact interface, which is largely contingent on the material strength of the pile and the rock body, as well as the interface morphology [4]. Additionally, extensive laboratory experiments and field observations have demonstrated that the roughness of a hole wall strongly affects the bearing capacity of rock-socketed piles. In a previous study, [PERSON] et al. [5] examined the roughness of the sleeves formed via various drilling techniques in Sydney sandstone. They classified the hole wall roughness into four grades on the basis of the size of the concavity and convexity of the inner wall of the borehole. Subsequently, [PERSON] et al. [6], [PERSON] et al. [4], [PERSON] et al. [7] investigated the correlation and influencing factors between the side resistance and roughness from different perspectives. [PERSON] et al. proposed a relationship between the ultimate side friction resistance and hole wall roughness, and [PERSON] et al. noted that the degree of roughness was influenced by the rock type, the drilling method employed, and the drilling speed. Furthermore, [PERSON] et al. proposed a factor, \(RF\), for the quantitative characterization of roughness, which accounts for the socket length, hole radius, and undulation of the sidewall. [PERSON] et al. [8] proposed a shaft resistance coefficient SRC that can describe the side resistance-settlement relationship by idealizing the undulating rock surface as multiple short lines of constant length, and assuming that the bump heights are approximated as Gaussian distributions, as shown in Figure 1.  
The aforementioned studies were typically founded upon experimental and idealized analyses. With the swift advancements in computer technology and numerical modeling, the ability to construct intricate pile-rock models, and rapidly solve and compute them has become possible. Numerical simulations, including the finite element method (FEM) [9; 10; 11; 12], discrete element method (FDM) [13], and boundary element method (BEM), have been employed by numerous researchers to examine the side resistance [14; 15; 16] and shear response of rock-socketed piles [17; 18; 19; 20]. For example, [PERSON] et al. [21] utilized a finite element approach to investigate the influence of pile surface roughness on the pile load-carrying capacity and elucidate the associated damage mechanism. Similarly, [PERSON] et al. [15] conducted load tests using a numerical discrete element model of a rock-socketed pile to analyze the effects of different roughnesses on the settlement response and lateral shear capacity. [PERSON] et al. [22] solved the load-displacement curves using the finite difference method and assumed that the contact interface was circular and wavy, with the goal of considering the effect of roughness on the bearing capacity. The roughness data employed in the aforementioned methods is typically measured by a mechanical profilometer [23]. The amount of data measured by profilometers is relatively small, and the accuracy is relatively low. [PERSON] et al. [24] utilized an umbrella-shaped aperture gauge to measure the aperture of bored piles and designed the roughness of the pile bodies based on the aperture. Nevertheless, in reality, the aperture cannot completely reflect the roughness of the hole wall.