## 1 Introduction  
The immersed tunnel has the characteristics of good adaptability to stratum conditions and high water tightness of prefabricated construction. At present, it is more and more used in cross-river and cross-sea tunnels. With the continuous construction of infrastructure, the underwater immersed tunnel is likely to be built near the earthquake zone, which may lead to disastrous consequences in the event of earthquake damage. Therefore, it is of great significance for the safety evaluation and reasonable design to study the seismic response of immersed tunnels. Immersed tunnels are usually built in weak strata, with long lengths and shallow burial depths. Under the action of an earthquake, there are dynamic interactions between soil and structure, the coupling of water and structure, etc. These make the seismic analysis of immersed tunnels a complex problem. Numerical methods (NM) such as the finite element method (FEM), finite difference method, and boundary element method are widely used in the dynamic analysis of immersed tunnels due to their versatility and reliability. NM can effectively solve the seismic response problem under actual working conditions [1]. Researchers often use a simplified calculation method [2, 3, 4] and two-dimensional (2D) cross-section model [5, 6] to analyze the dynamic performance of immersed tunnels. The simplified calculation method is to introduce the mass-spring model [7] into the finite element (FE) model, and cut the soil around the tunnel into several slices. The mass of each slice is concentrated in a lumped mass point, and the lumped points are connected with each other and connected to the tunnel beam. Meanwhile, the bedrock is connected to the lumped points of soil. These connections are composed of springs and viscous dampers in parallel, and the parameters of each slice are calculated by the basic shear mode of the soil layer. However, vertical shear cannot be transmitted between adjacent slices, and the model cannot properly simulate the dual role of soil as both the propagation medium of the seismic wave and the supporting medium of the structure. Hence, considering the soil-structure interaction (SSI) is essential to check the seismic performance of the underground structure by using a global model. During an earthquake, the dynamic response of underground structures is controlled by the deformation of surrounding soil [8; 9], and the type and strength of soil [10; 11; 12] and the selection of the soil-structure interface [12; 13] have significant effects. Furthermore, the SSI factor can even be used to evaluate the parameter values in the design [14]. In addition, models considering SSI can reduce the peak acceleration of the soil surface [15], allowing more accurate results to be obtained for seismic response studies of underground structures [16; 17; 18; 19]. The 2D global model simplifies the tunnel into a 2D plane and simply decomposes the dynamic response of the tunnel into in-plane and out-of-plane for solution. This model cannot effectively simulate the cooperative work among various components of the tunnel, and it is difficult to fully reflect the structural performance, which subjects the tunnel structure to potential safety hazards in the service process. Thus, it is appropriate and desirable to utilize a three-dimensional (3D) FE numerical model of SSI that consider the most comprehensive factors. The traveling wave effect is one of the characteristics of the spatial variation of ground motion. For buried structures with large dimensions such as long tunnels, the impact of the traveling wave effect should be properly considered, and the oblique incidence of seismic waves is the main physical mechanism of the traveling wave effect. Most of the existing studies on seismic action assume that earthquakes are vertically incident.