## 1 Introduction  
The demand for underground structures is increasing with the rapid growth of the economy and population, especially for projects such as subways, underpasses, underground shopping streets, etc. However, due to limited space within cities, many underground projects are adjacent to the operating metro structures. Therefore, soil deformation induced by the braced excavation should be carefully controlled so as to minimize the impact on the existing metro system. Numerical simulation analysis is an important method for assessing and predicting the soil deformation in foundation pit engineering. Finite element modeling (FEM) is widely used to assess foundation pits in the design stage [1; 2; 3; 4; 5; 6; 7]. The selection of the constitutive model and its parameters play a key role in the accuracy of numerical simulation. Currently, the commonly used constitutive models for braced excavation include the Mohr-Coulomb (MC) model, the hardening soil (HS) model, the hardening soil-small (HSS) model, etc. Among them, the HSS model contains all the model parameters of the HS model, as well as two parameters to control the stiffness of the soil in a small-strain state, i.e., the initial shear modulus \(G_{0}^{ref}\) and shear strain \(\gamma_{0.7}\). The HS and HSS models have been widely adopted due to their ability to accurately simulate the stress-strain relationship of soil [8; 9; 10; 11; 12; 13]. However, to accurately predict the soil deformation caused by braced excavation during the design stage of pit engineering, so as to carry out a safe and economical design that retains the structure of the pit, the selection of constitutive model parameters appears to be crucial. The values of the constitutive parameters of a soil are usually determined by laboratory tests. A large number of scholars have carried out experimental studies to determine the parameters of constitutive models and the proportional relationships among the parameters. For example, [PERSON] et al. [14] and [PERSON] et al. [15; 16] obtained the parameters and the proportional relationship of the HSS model and HS model for typical soft soil layers in Shanghai City through indoor experiments. However, the significant number of experiments required to determine the constitutive parameters of the HS and HSS models, coupled with the lengthy test period and substantial uncertainties involved, poses a challenge for achieving an effective and accurate deformation assessment of foundation pits. In contrast, the inverse analysis method can accurately invert soil parameters by using only the field-measured data without the need to conduct complicated indoor tests. This approach is widely recognized as an effective technique for determining the parameters of soil constitutive models [17; 18]. Many scholars have used inverse analysis to determine soil parameters, such as [PERSON] et al. [19], who proposed an inverse analysis method to quantify the three-dimensional effect of excavation, and corrected the prediction based on plane strain. [PERSON] et al. [20] proposed an inverse analysis method that could predict the three-dimensional displacement of soil caused by braced excavation. [PERSON] et al. [21; 22] used the HSS and HC models to invert the constitutive parameters for the sidewall deformation of a deep foundation pit in Chicago, and analyzed the stress-strain characteristics of the soil layer during the excavation of the foundation pit. [PERSON] et al. [23] proposed a three-dimensional inverse analysis method for foundation excavation and successfully predicted the deformation of the sidewalls. [PERSON] et al. [24] used a general selection method for the parameters of the HSS model using the inverse analysis method, and compared the applicability of the MC and the HSS models under different excavation depths. In addition, it has been shown that the compressive modulus of the soil has a good linear relationship with the standard penetration test (SPT) blow counts [25; 26; 27].