**4. Discussion and Results**

A schematic of the numerical model is illustrated in Figure 1. The mesh type of this numerical model is quadrilateral element. A Ricker wavelet with a predominant frequency of 4.0 Hz was employed as the incident wave to investigate the wave patterns, materials and slope geometries based on the seismic responses. The acceleration-time history of the incident wave, which was input from the left bottom of the model, is illustrated in Figure 10b. To analyze the topographic effects, the acceleration amplification ratio was applied, which is the ratio of the peak acceleration of the reflected waves at each point on the surface to that of the incident seismic waves [36,45]. The horizontal and vertical acceleration amplification ratios *r*<sup>h</sup> and *r*<sup>v</sup> are defined in Equations (20) and (21), respectively. For a given wave, *r*<sup>h</sup> and *r*<sup>v</sup> are the values of the acceleration amplification ratio *r* for the horizontal and vertical components, respectively. *r* = *r*<sup>h</sup> <sup>2</sup> + *rv*<sup>2</sup> defines zones of the net amplification of ground acceleration with respect to the energy carried by the incoming waves.

$$r\_{\rm h} = \frac{\max(|a\_{\rm h}|)}{\left| a\_{\rm h,input} \right|} \tag{20}$$

$$r\_{\rm V} = \frac{\max(|a\_{\rm V}|)}{|a\_{\rm v,input}|} \tag{21}$$

where max(|*a*h|) and max(|*a*v|) are the peak horizontal (h) and peak vertical (v) acceleration at the observation points along the slope ridges and slope crests, respectively; and - - *a*h,input - - - and - *a*v,input - are the horizontal (h) and vertical (v) acceleration, respectively.

In this study, the wave patterns were discussed based on the P waves and the SV waves owing to the specific focus on the effects of body waves. In addition, the impact of the materials was investigated in terms of the relative hardness and softness between the slopes and foundations. The shear waves of different materials on the slope topographies are presented in Table 1. Furthermore, the geometries of slopes with varying slope heights, widths and inclinations are discussed. The slope height *H* is a normalized height that is related to the wavelength *λ* [29,36], and it varies among 0.2*λ*, 0.5*λ*, 1.0*λ* and 2.0*λ*. The width of the slope crest *W* varied from 50 to 400 m, and the intermediate values were 100 m and 200 m. The slope inclination *i* varied between 26.6◦, 33.7◦, 45◦, 55◦ and 63.4◦, corresponding to width-depth ratios (defined as cot*i*) of 2.0, 1.5, 1.0, 0.7 and 0.5. All the influencing factors (wave patterns, materials and geometries) were based on the varied directions of incident waves, which were input from the left bottom of the topography with incident angles *θ* in the range of 0◦ to 30◦, sampled at 5◦ intervals.
