## 6 Conclusions  
In this paper, an analytical model based on the Hilbert's best approximation problem (BAP) has been proposed to analyze the dynamic response of large-scale slopes under non-uniform seismic excitation. The analysis of the model of slopes has extended our knowledge of the non-uniform spatio-temporal dynamic response and amplification effect. Particularly, the seismic waves arriving at the large-scale slope base from the hypocenter at the same moment are non-uniform. This is very different from the uniform incidence based on the plane-wave hypothesis. The ratio of the slopes length and the focal depth (\(L/h\)) is the foundation for determining whether the slope model is consistent with BAP or the plane-wave hypothesis, as well as the parameter that affects the degree of non-uniformity. A non-uniform P-SV type of surface wave, designated the Stoneley wave, is designed to illustrate the model of non-uniform wave incidence. The value of the displacement on the interface at the base of the slope varies between different locations at the same moment, and the variation patterns are subjected to \(L/h\). Such a non-uniform spatio-temporal dynamic response is caused by non-uniform seismic wave incidence. Meanwhile, the scattering model of elastic waves at rock interfaces is proposed to study the dynamic amplification effect of a large-scale slope subjected to non-uniform seismic waves. Considering the non-uniform seismic wave incidence on the interface at the base of the slope, the Knot equation is obtained. Eventually, the acceleration amplification factor is proposed. Numerical models of the Finite Difference Method (FDM) and the artificial seismic wave have been applied to simulate the non-uniform incidence of seismic waves. The results of the numerical simulations provide an intuitive representation of the dynamic response of the slope under seismic incidence, as proposed by the analytical model. In the case of non-uniform incidence, the displacement curves of each observation point exhibit a range of frequencies and phases. The degree of dispersion is observed to increase with \(L/h\)indicating that the non-uniformity of the dynamic response maintains a consistent tendency with \(L/h\). The amplification effect at the surface of the slope exhibits a concentration of distribution in the dynamic response subjected to non-uniform seismic wave incidence. It is imperative to consider the non-uniform dynamic response when \(L/h\) is greater than 4:20. Slopes of this scale, when subjected to seismic wave incidence, satisfy the BAP principle. Conversely, the dynamic response is observed to be uniform when \(L/h\) is smaller than 1:20, which is in accordance with the plane wave assumption of Huygens' principle. In this paper, only one geological model is studied, which has some limitations. In future studies, attention should be paid to large slopes with different geological conditions, such as open-pit mine slopes, highway trench slopes and reservoir bank slopes, to improve the results of the study. Conceptualization and methodology, C.S. ; writing--original draft preparation and software, S.W. ; writing--review and editing, G.C. ; supervision, Y.F. ; project administration, Q.H. All authors have read and agreed to the published version of the manuscript. This study was supported by the National Natural Science Foundation of China (Grant No. 52379100), and China Huaneng Group Co., Ltd. (Grant No. HNKJ22-H107). Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author(s). We thank PowerChina, Kunming Engineering Corporation Limited for their valuable assistance in data collection and technical support. Additionally, we appreciate the constructive comments from the reviewers and editors, which significantly improved the quality of this paper.