## 1 Introduction  
### Background  
Due to the severe damage to infrastructure that has occurred during recent earthquakes, engineers have recognized that the use of only classical seismic design methods based on the strength capacity criterion are not appropriate, and that the dynamic behavior of a structural system, especially the nonlinear response characteristics, should be precisely evaluated. Therefore, evaluation of the seismic performance of new or existing structures is an important issue in earthquake engineering [1; 2; 3]. In the current design practice, seismic design forces are represented by smooth design-response spectra. These design-response spectra are convenient for calculating the maximum response of structures if the building is assumed to behave linearly during earthquakes. For severe ground motions, a building will exceed its elastic limit, in which case it is desirable to directly evaluate its displacement capacity through dynamic analysis of the structural system, and therefore, an input regarding the time history of ground acceleration is required [4; 5]. Additionally, an observed accelerogram cannot provide the necessary pieces of information that are specific to the site where a structure is located, such as the surficial topography, underground layer conditions, irregular profiles of primary layers and crustal conditions and existence of active faults and lineaments. Furthermore, the use of earthquake motion as an input for dynamic analyses may be affected by uncertainty in the location of the seismological sources and be controlled by fault rupture mechanisms and propagation path effects. Because of these reasons DRSCEMs are commonly used to analyze structural responses, not only to keep a kind of continuity between the codes based on elastic and inelastic design criteria but also to consider the effect of a high uncertainty characteristic of earthquake ground motion used for seismic design purpose [6; 7; 8]. International seismic design codes do not provide any details about the procedure to scale or simulate earthquake ground motion for design purposes, only impose the requirement that the recalculated response spectra obtained using the modeled earthquake motion should match the response spectrum of the target [9; 10]. Consequently, simulations of DRSCEMs have been conducted by numerous researchers. However, a proper method to simulate DRSCEMs should be developed that can incorporate newly determined characteristics of earthquake ground motion [11]. Recently to simulate realistic earthquake motion time histories, many methods such as, Empirical Green's Function [12], Stochastic Simulation [13] and numerical methods [14] are often used in engineering and research practices. It is a new trend in the field of seismic design of structures to take into account uncertainty of fault mechanisms, such as the uncertainty of the rupture starting point, uncertainty of asperity distribution and rupture propagation direction [15] to simulate earthquake motions to be used for dynamic analysis of the structural systems. However, both the Fourier amplitude and phase spectra of earthquake motion are basic pieces of information that are needed to simulate a DRSCEM model. Following studies conducted by [PERSON] [16] to investigate phase characteristics show that the modeling of the phase spectrum plays an essential role in simulating earthquake ground motion. Much research effort has therefore been performed to identify the phase spectrum characteristics of earthquake motion. [PERSON] and [PERSON] [17] used the concept of group delay time (GDT) and showed that the average arrival time of the earthquake energy and duration of the earthquake motion could be evaluated by the mean and standard deviation of the GDT. [PERSON] et al. [18] used phase difference distributions as functions of the magnitude, distance to the fault, and local soil conditions. [PERSON] et al.