Accurate estimation of input bedrock motion is essential for the seismic design of underground structures. However, most seismic observations are recorded at the top surface, making direct measurement of bedrock motion exceedingly challenging. Consequently, deriving actual bedrock seismic motion holds substantial engineering significance. In recent years, the usage of inversion methods to calculate bedrock input motion has garnered significant attention. Inversion involves evaluating bedrock input motion based on surface observations, primarily to address input excitation for site seismic response or soil-structure interaction analyses. However, the inversion analysis of bedrock horizontal seismic motion primarily relies on surface horizontal seismic records in engineering. It is usually assumed that seismic waves are vertically incident, disregarding the incident angle of bedrock motion and treating it as a one-dimensional （1D） inversion problem. In reality, seismic waves at bedrock often have a certain angle of incidence, thus studying the inversion of bedrock motion under oblique incidence has important significance for engineering applications.

It is worth noting that current inversion research is limited to the assumption of isotropic media. Nevertheless, natural soils generally exhibit obvious anisotropy due to processes such as weathering and sedimentation, and the horizontal modulus of natural soil is generally different from its vertical modulus. Transversely isotropic （hereinafter referred to as TI） media are commonly employed to describe these anisotropic geomaterials. These models primarily account for differences in mechanical properties between vertical and horizontal directions. Therefore, it is more in line with the actual conditions to describe soil anisotropy by using TI medium mechanical model. Moreover, the existing studies on the layered TI half-space primarily focus on forward modeling of site surface responses, with limited attention to the inversion analysis of bedrock seismic motion in the layered TI sites.

Building on the authors’ previous researches on bedrock motion inversion in the layered isotropic half-space, this study extends the approach to more realistic transversely isotropic （TI） site conditions, proposing a two-dimensional （2D） inversion method for bedrock motion in the layered TI half-space. The proposed method is based on the exact dynamic stiffness matrix of TI soil layers and the bedrock half-space. Utilizing seismic horizontal and vertical acceleration records of a certain point on the surface, an inversion objective function is formulated, an inversion objective function is established, and the control parameters in the inversion process are updated by employing optimization methods to achieve the inversion of bedrock motion and incident angle. Furthermore the method was validated using three different TI parameter site models as case studies.

The research results demonstrated that bedrock motion and incident angle of the TI half-space can be accurately inverted based on the surface acceleration records and TI site characteristics. The Pearson correlation coefficient was selected to evaluate the overall accuracy of the two-dimensional inversion method for the layered TI half-space. An average Pearson correlation coefficient of 0.999 was obtained between the inverted bedrock motion and the actual values for qP and qSV wave incidence. The relative error of peak acceleration was used to evaluate the local accuracy of the inversion results, yielding a total average relative error of 0.998% for peak ground acceleration between the inverted bedrock motions and the actual input waves under qP and qSV wave incidence. These findings quantitatively confirm the accuracy and precision of the proposed inversion method for calculating bedrock motion of the layered TI half-space.

Moreover, the traditional isotropic （ISO） bedrock motion inversion model assumes the vertical material parameters of the site equivalent to the horizontal ones. In this study, the ISO model is applied to evaluate the error of the two-dimensional inversion method for the TI half-space. When seismic waves are incident at a small angle, the inverted acceleration time history and response spectrum of the bedrock motion for the layered TI half-space, used in traditional ISO inversion model, closely approximate the actual input waves. However, as the incident angle increases, errors in inverted bedrock motion for the layered TI site using the ISO model become increasingly pronounced. The inverted response spectrum of bedrock motion exhibits significant differences from the actual values in terms of curve shape, peak amplitude, and peak period, with the inversion accuracy decreasing significantly. This is because the traditional ISO model ignores the influence of the differences in the horizontal and vertical properties of the TI medium on the wave velocity, resulting in certain errors in the model and significant errors in the inversion calculation results. This indicates the necessity of developing a two-dimensional inversion method for the layered TI half-space.