## 1 Introduction
### Purpose of the Study  
In this paper, we discuss how to define the phase spectrum and appropriate initial Fourier amplitude for simulating DRSCEMs that reflect site conditions and source mechanisms. Because the earthquake motion phase has a significant effect on the nonlinear response of structures (as described in the above section), much effort has been made to clarify the stochastic characteristics of the phases, but the uncertainty of the earthquake motion phase still cannot be evaluated properly. Recently, [PERSON] [33] studied the fractal characteristic of the earthquake motion phase and showed that it is possible to simulate a phase with this characteristic by fBm. The fBm is a continuous stochastic process that has a long memory with respect to the circular frequency. Therefore, it is essential to detect the Hurst index and a parameter, which controls the strength of the power law defined by using an observed earthquake motion phase. One of our main research purposes is to provide regression equations for these two parameters as functions of the earthquake magnitude, epicenter distance, and bias index, which can be treated as a type of site-specific effect. After the occurrence of severe damage to infrastructure systems caused by recent strong earthquakes in several countries, a movement to revise seismic design codes from the classical strength demand concept to a new displacement capacity demand concept has been initiated. To this end, it is essential to conduct nonlinear dynamic analyses of structural systems. The simulation of DRSCEMs then become a common topic for earthquake engineers to maintain continuity between classical and new design codes because the seismic forces used in the classical codes were usually defined based on response spectra. To simulate DRSCEMs, we must assign a proper phase spectrum. This requirement is one of the main topics of this paper. Using many simulated DRSCEMs, we conducted a comprehensive statistical study of nonlinear response spectra, termed \"YSC demand spectra\". Based on our statistical analysis, we propose a method to define YSC demand spectra for seismic design purposes. To simulate DRSCEMs, the target response spectrum used here is defined in the Chinese seismic design code (GB50011-2010) as: earthquake intensity 8 (peak ground acceleration of 0.2 or 0.3 g), a rare earthquake event (the probability of exceedance during 50 years is 2-3% with a return period of 1640-2475 years), site class II (medium firm), design earthquake group 2 and a damping ratio of 0.05 [34]. \begin{table}
\begin{tabular}{c c l l c} \hline
**No** & **NGA** & **Event, Year** & **Station** & **Magnitude** & **Epic. Distance** \\  & **Number** & & & & **(km)** \\ \hline
1 & 1752 & Northwest China-03, 1997 & Jiashi & 6.10 & 19.11 \\
2 & 6 & Imperial Valley-02, 1940 & El Centro Array \#9 & 6.95 & 12.99 \\
3 & 942 & Northridge-01, 1994 & Alhambra—Fremont School & 6.69 & 40.15 \\
4 & 42 & Lytle Creek, 1970 & Cedar Springs Pumphouse & 5.33 & 22.51 \\
5 & 101 & Northern Calif-07, 1975 & Cape Mendocino & 5.2 & 30.54 \\
6 & 1151 & Kocaeli, Turkey, 1999 & Balikesir & 7.51 & 218.64 \\
7 & 143 & Tabas, Iran, 1978 & Tabas & 7.35 & 55.24 \\
8 & 1108 & Kobe, Japan, 1995 & Kobe University & 6.9 & 25.4 \\
9 & 864 & Landers, 1992 & Joshua Tree & 7.48 & 13.67 \\
10 & 855 & Landers, 1992 & Fort Irvin & 7.48 & 120.99 \\
11 & 586 & New Zealand-02, 1987 & Maraenui Primary School & 6.60 & 72.62 \\
12 & 825 & Cape Mendocino, 1992 & Cape Mendocino & 7.01 & 10.36 \\
13 & 1749 & Northwest China-01, 1997 & Xiker & 5.90 & 51.75 \\
14 & 688 & Whittier Narrows-01, 1987 & Riverside Airport & 5.99 & 59.59 \\
15 & 142 & Tabas, Iran, 1978 & Sedeh & 7.35 & 177.9 \\
16 & \(\sim\) & Off-Kushiro, Japan, 1993 & Ishino & 7.60 & \(\sim\) \\
17 & \(\sim\) & Hyogoken-Nanbu, 1995 & Amagasaki & 7.20 & \(\sim\) \\
18 & \(\sim\) & Tangshan, China, 1976 & Beijing Hotel Department & 7.60 & \(\sim\) \\ \hline \end{tabular}
\end{table} 
Table 1: Main information of recorded earthquake ground motions used in the study.