## 5 Conclusions  
This paper proposes a computational method for modal superposition applicable to two-dimensional soil sites for analyzing the acceleration response of the site under seismic force. The calculation method for modal truncation for acceleration response is investigated. The variation of errors in velocity and acceleration time histories derived from numerical differentiation of displacement time histories is discussed. Based on the theoretical analysis and numerical results, the following conclusions can be drawn:  
1. The contributions of the higher modes to acceleration are larger than to the displacement. Therefore, more modes need to be included in the calculation of acceleration response for the modal superposition method. The cumulative modal contribution factor of acceleration can be used as a modal truncation index for acceleration response. The modal participation mass and displacement-based modal contribution factor can only be used as a modal truncation index for displacement response.
2. The mode number of modal truncations is related to the selected degrees of freedom. For a homogeneous site, the acceleration at the site surface can be regarded as the key response quantity. For scarp topography, the position with a distance of one step height from the toe of the slope at the lower step can be used as the basis for modal truncation.
3. When the cumulative modal contribution factor of acceleration is used as the truncation criterion, the accuracy of velocity and acceleration obtained by the central difference method is essentially consistent with the results from the modal superposition method. Therefore, displacement responses can be computed using the modal superposition method, and then the velocity and acceleration responses at locations of interest can be obtained using the central difference method.
4. The theoretical foundation of this method is the modal superposition approach, which is suitable for linear elastic analysis. However, for nonlinear seismic response analysis of soil layers, the equivalent linearization method is commonly used. Therefore, during each iteration step, the computational method proposed in this paper can be employed to improve the efficiency of equivalent linearization calculations.  
However, in this study, the verification of the calculation error of the modal superposition method is limited to numerical simulations. In the future, further analyses of the nonlinear seismic response of actual complex sites will be conducted by combining equivalent linearization and comparing with measured results to evaluate the influence of soil properties and seismic excitation characteristics on the computational error of seismic response, providing a more comprehensive validation of the effectiveness of the proposed algorithm.  
H.C.: Conceptualization. Y.Y.: Investigation, X.L.: Investigation, Validation, Writing--original draft & editing. D.P.: Investigation, Writing--review. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China grant number 52478121.  
Not applicable.  
The data that support the findings for this study are available from the authors by request.  
Conflicts of Interest: Authors [PERSON] and [PERSON] were employed by the company China First Highway Xiamen Engineering Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.