## 1 Introduction  
The earthquake-triggered landslide represents a dangerous natural hazard causing significant damage to property and infrastructure and personal casualties [1, 2]. There have been many reports on the earthquake-triggered landslides in the past few years [3, 4, 5]. For example, on 8 October 2005, the Kashmir earthquake triggered a large number of landslides in the northern Pakistan. The most disastrous landslide destroyed three villages and caused nearly 1000 deaths [6]. A total of 60,104 landslides induced by the 2008 Wenchuan earthquake (Ms = 8.0) directly killed over 20,000 people [7]. Less than five years later, the 2017 Jiuzhaigou earthquake shocked the Sichuan province, China, and induced as many as 179 landslides [8, 9]. In spite of their significant damage to property and human casualties, the cause and effect of earthquake-triggered landslides is not well understood [10, 11]. The process of earthquake-triggered landslides is complex, which is also called the dynamic response of the landslide [12]. When the seismic wave reaches the ground surface, the dynamic behavior of the seismic wave will be affected and modified by the soil. Thus, it appears possible to reduce the seismic response by intervening and modifying the characteristics of the soil [13, 14, 15]. The failure behavior of the earthquake-triggered landslide is also closely related to its dynamic response under the seismic waves. The earliest study on the effect of the seismic waves can be traced back to 1965, when [PERSON] proposed a method permitting a rapid estimate of the dams and embankments displacement in an earthquake [16]. After that, many studies focusing on the effects of seismic waves on the earthquake-triggered landslide dynamic response and failure behavior have been published [17, 18, 19, 20, 21]. For example, [PERSON] et al. developed a nowcasting model for estimating the probability of landslides induced by earthquakes [22]. [PERSON] et al. analyzedthe controlling factors of the distribution of the coseismic landslides [23]. [PERSON] et al. investigated that larger landslides were associated with higher ground motion and explained why stronger shaking could induce larger landslides [24]. [PERSON] et al. modeled a symmetrical slope by using the DDA (discontinuous deformation analysis) method, and the results showed that pulse-like ground motion (PLGM) might be the main factor for the earthquake-triggered landslides [25]. In spite of the extensive studies on the earthquake-triggered landslides, how to consider the effect of the seismic wave on the landslide dynamic process is not well understood. 3 DEC (the three-dimensional discrete element code) is a powerful tool in numerically modeling the dynamic response and failure behavior of the earthquake-triggered landslides [26]. The input of the seismic wave in 3 DEC mainly depends on the seismic wave data measured in the field. However, in an earthquake, the ground moves randomly in all directions, both horizontally and vertically. The earthquake accelerations are available for motion in two horizontal directions (East-West (EW) direction and North-South (NS) direction) as well as in the vertical direction (Up-Down (UD) direction). Therefore, researchers have not reached the same conclusion on which one of the seismic waves should be selected as the input of the seismic wave [27; 28]. For example, [PERSON] et al. assumed that the vertical seismic wave has no apparent effect on landslides [29]. However, [PERSON] et al. analyzed the effect of the vertical seismic waves on the failure mechanism of a practical earthquake-triggered rock avalanche and concluded that the vertical seismic force played an important role. [PERSON] et al. obtained that the horizontal acceleration from P-waves played a dominant role in the failure of the slope with the discrete element method [30]. [PERSON] et al. used the DDA method to simulate the failure process of the Daguangbao landslide under the action of multidirectional seismic waves.