## 1 Introduction  
The rapid development of urban rail transit brings the environmental vibration problem [1; 2; 3]. To predict and evaluate the train-induced ground-borne vibrations, mathematical models are useful. The vibrations can be calculated using empirical or theoretical approaches, or a combination of both. For analyzing the ground vibrations induced by underground train loads, the tunnel-soil dynamic interaction is a significant part of the prediction model. Different analytical, semi-analytical, and numerical models have been developed to consider this interaction. [PERSON] and [PERSON] [4] proposed the \"beam in the ground\" model, which simulated the tunnel as an Euler beam in the elastic stratum (fixed lower surface). This model was used to study the surface vibration response under moving constant force, moving harmonic load, and fixed random load. [PERSON] et al. [5] improved the model proposed by [PERSON] and [PERSON] [4]. The elastic formation was transformed into an elastic half-space, and the wavelet technology was employed to perform the inverse Fourier transform to obtain the vibration response in the space domain. In order to consider the influence of tunnel lining structure and boundary shape and establish a more refined tunnel model, [PERSON] and [PERSON] [6] established the pipe-in-pipe (PIP) model. In this model, a thin-walled cylindrical shell of infinite length was used to simulate the tunnel lining structure, and a thick-walled cylinder with an infinite outer diameter was used to simulate the soil outside the lining. [PERSON] et al. [7] extended the PIP model to parallel double-hole tunnels to solve the fundamental solution of surface displacement under point load in tunnel. Based on Flugge thin-shell theory and infinite period theory, [PERSON] et al. [8] established a semi-analytical tube-in-tube model to predict the three-dimensional dynamic response of periodic tunnels in soil. [PERSON] and [PERSON] [9] investigated the tunnel-soil-pile dynamic interaction based on the tube-in-tube model. [PERSON] and [PERSON] [10] used the direct stiffness method to solve the dynamic response of a multi-layer half-space under a spatial periodic moving load, which can be used as the fundamental solution of the periodic boundary element formula. [PERSON] et al. [11] proposed a novel periodic tunnel-soil model with a track slab to simulate the propagation of train-induced vibrations, and its validity was verified through the literature and in situ measurement. In order to consider the reflection effect of surface boundaries on waves, [PERSON] et al. [12] obtained the fundamental displacement solution of a circular tunnel in a three-dimensional layered half-space by using the transformation characteristics of planar cylindrical waves and the transfer matrix method. [PERSON] et al. [13] also proposed an analytical solution for the dynamic response of multiple scattering effects between tunnels, taking into account the conversion characteristics of plane cylindrical waves and the case of two adjacent tunnels. Based on the concept of analytic layer elements with circular holes, [PERSON] et al. [14] adopted the wave function method to solve dynamic responses under moving loads on the ground or within tunnels. In practical engineering, soil layer parameters tend to change with depth. Treating foundation soil as a uniform elastic body cannot fully reflect the actual situation. Therefore, some scholars simplified the foundation soil model into a layered model. The thin-layer method (TLM), a semi-analytical and semi-numerical method, is suitable for the solution of layered foundations. The TLM was first proposed by [PERSON] and [PERSON] [15] in 1972 as a transfer boundary to represent the infinite extension of the horizontal direction of the foundation. [PERSON] [16] proposed the paraxial boundary to solve the propagation problem of TLM-simulated waves in the infinite domain. [PERSON] [17] discussed the accuracy and convergence of the TLM in computational dynamic problems.