## 5 Results and Discussion
### Uncertainty Analysis  
Although the distinction between the aleatory and the epistemic uncertainties has a large impact on the seismic hazard assessment and how users interpret and apply the obtained ground-motion intensities, in practice, the distinction can be fuzzy. In this study, the uncertainties were induced by the simulation method and the selection of dip angle, SP, \(\Delta\sigma\), and \(\kappa_{0}\). We assumed that the only aleatory uncertainty was induced by the simulation method. Figure 13 illustrates the aleatory uncertainty related to the iteration times. The residual is defined as the logarithmic difference between each simulation and the average response spectrum. In general, the residuals increased with increasing frequency. Moreover, it should be noted that the SD of 10 iteration times was the smallest because the aleatory uncertainty was not fully captured. The results show that for iteration times larger than 40, the SD tends to be stable. Thus, instead of 30 iteration times suggested in the group standard for performing multi-SFFSM in practice, the results indicate that at least 40 iteration times should be performed to reach a stable aleatory uncertainty [60]. The epistemic uncertainties of \(\Delta\sigma\) and \(\kappa_{0}\) were embedded in the data distribution and the assigned weight coefficient. In general, the adopted \(\Delta\sigma\) values and its weight control the systemic amplitude level and those of \(\kappa_{0}\) control the amplitude level at frequencies above 1 Hz in multi-SFFSM. Herein, we analyzed the influence of the variation in dip angle and SP. Figure 14 shows the residuals between dip angles of \(70^{\circ}\), \(75^{\circ}\), and \(80^{\circ}\), respectively. The residuals varied in a small range of \(-0.00716\)-\(0.00726\). This result indicates that the influence of a dip angle variance of 10 degrees is ignorable and should be attributed to the well-documented geological survey data. For cases lacking geological survey data or needing to consider complex fault geometry, larger between-dip uncertainty should be expected. Figure 13: Illustration of the aleatory uncertainties induced by the simulation’s iteration times: (a) 10 times, (b) 30 times, (c) 40 times, (d) 50 times, and (e) 80 times. (f) The average residual standard deviation of different iteration times. Figure 15a shows the residuals of the four SPs, in which the rupture directivity of SP1 and SP2 point toward the target site and that of SP3 and SP4 point toward the negative direction. Although the residuals vary in a narrow range of \(-\)0.023-0.016, in the studied case, the influence of SP is larger than that of the dip angle. This result indicates that the residuals are reversed at a crossover period of 0.35 s. [PERSON] et al. [61] analyzed the rupture directivity effects in the 2022 Lushan \(M_{\rm S}6.1\) earthquake. The strong-motion stations 51 QLY and 51 TQL were in the forward direction and backward direction of the ruptured fault. The \(V_{\rm S30}\) and rupture distance of 51 QLY and 51 TQL were 505 and 498 m/s and 32 and 30 km, respectively. Thus, the difference in the observed PSAs at the two stations can be mainly attributed to the rupture directivity effects. The salient feature of rupture directivity effect during the Lushan earthquake is illustrated in Figure 15b. Comparing the residuals of forward and backward SPs with the observed rupture directivity effect, it can be concluded that the residual variation in the four SPs captured the salient feature of the rupture directivity effect. Moreover, the residuals of SP2 were larger than those of SP3 for the whole period range, although their location was the same. [PERSON] and [PERSON] [62] investigated the rupture directivity effects observed in the 2008 Wenchuan \(M_{\rm W}7.9\) earthquake, and they found that the PSA of stations in the forward direction were systemically larger than those in the backward direction. Figure 14: Variation in frequency-dependent residual for different dip angles.