## 1 Introduction  
With the accelerating process of urbanization, the construction of underground transportation facilities such as subways is becoming increasingly popular, and shield tunneling technology has become one of the important means of underground tunnel construction. However, with the continuous increase in tunnel diameter and the diversification of usage scenarios, the problem of surface subsidence caused by shield tunneling construction is becoming more and more prominent. In practical construction, it is necessary to predict and control the surface subsidence caused by tunnel excavation reasonably and strictly. The research methods for studying surface settlement caused by shield tunneling mainly include empirical formula methods [1; 2; 3], numerical simulation methods [4; 5; 6; 7; 8], model test methods [9; 10; 11], among which the empirical formula methods have advantages such as simple and fast algorithms, selection of different parameters according to geological conditions and different tunnel sections, suitability for preliminary research and quantitative evaluation of surface settlement deformation, and reductions in engineering costs and time. Therefore, they have been widely applied in practical engineering. Based on a large amount of measured data, Professor [PERSON] proposed the cross-sectional Peck formula for surface settlement troughs [12]:  
\[S_{x}=S_{\text{max}}\text{exp}\left[-\frac{x^{2}}{2i^{2}}\right] \tag{1}\]Here, \(S_{\rm max}\) is the maximum settlement value of the ground surface above the tunnel after excavation; \(i\) is the width of the settlement trough in meters; and \(x\) is the horizontal distance between the settlement point of the ground surface and the tunnel axis in meters. Different scholars have made targeted modifications to the Peck formula through analysis and experiment, in order to make it more applicable to different geological regions and engineering conditions. These modifications include adjusting formula parameters and introducing new correction coefficients. These modifications enable more accurate predictions of surface subsidence caused by tunnel construction, providing important theoretical references for the design and construction of tunnels. Experts and scholars have analyzed the laws of surface subsidence caused by shield construction in different regions of China, and have modified the Peck formula based on the results of their analysis to make it more applicable to regional conditions [13; 14; 15; 16]. [PERSON] et al. [17] modified the traditional Peck formula through a regression analysis of measurement data, making it suitable for predicting surface subsidence in soil-rock composite strata with hard layers that have a significant impact on surface subsidence compared with soil-rock composite strata. [PERSON] et al. [18] studied the laws of ground subsidence caused by different construction sequences in a tunnel, and modified the Peck formula to more accurately predict ground subsidence. [PERSON] et al. [19] studied the deformation characteristics of soil caused by the construction of a double-decker shield tunnel and modified the Peck formula to investigate the effect of changes in tunnel depth and ground loss rate on surface subsidence. [PERSON] et al. [20] modified the width coefficient of the settling trough in the Peck formula based on a large amount of measurement data. [PERSON] et al. [21] studied the laws of surface subsidence in the case of asymmetric geological conditions and proposed a modified Peck formula for asymmetric river terrain conditions. [PERSON] et al. [22] analyzed the surface deformation caused by the construction of a large-diameter slurry shield and obtained the range of key parameters in the Peck formula. Modifying the parameters of the Peck formula for special conditions and geological conditions often cannot meet actual prediction needs and introducing new correction coefficients can often achieve better results. [PERSON] et al.