## 2 Phase Spectrum Simulation by Modified Fractional Brownian Motion
### The Validation of the Developed Method to Generate Earthquake Motion Phase  
To demonstrate the efficiency of the algorithm to simulate earthquake motion phase, we used the Hurst index and the variance, which were obtained in Section 2.2 for Landers ground motion. Then we simulated an ensemble of Fourier phases according to the modified fBm process. Based on Fourier inverse analysis we can simulate an ensemble of time histories by using the simulated phases and the Fourier amplitude of Joshua Tree time history. After we have conducted studies of the elastic and inelastic response using original and simulated time histories, we find that the usage of the modified fBm gives almost similar elastic response over the wide range of natural periods, but in inelastic response, we can recognize many fluctuations. In Figure 4a, the full blue line shows the elastic response of the original Joshua Tree time history and the red dots lines show elastic response of 15 simulated time histories using phases of modified fBm process. Figure 4b shows the inelastic response spectra of the original and simulated time histories. Those fluctuations in the inelastic response are especially important to evaluate the nonlinear dynamic performance of a structure. Furthermore, to show the efficiency of the proposed method, we simulated another an ensemble of time histories again by using recorded ground motions phases and the Fourier amplitude of Joshua Tree time history following to the previous studies [6; 25; 26]. Here we use ten different ground motions phases (Northwest China-03, Imperial Valley-02, Northridge-01, Tabas, Sedeh, Kobe, Kushiro-oki, and Whetiter) detail information about these records is given in Table 1. We then calculate the elastic and inelastic responses with which we compare the results obtained by using phases simulated based on the modified fBm. Figure 5a shows the elastic response of the Joshua Tree and the ten simulated time histories, and the inelastic response in Figure 5b. Comparing Figures 4b and 5b, we can see that the fluctuation of nonlinear response calculated by using the phase from observed ground motions is higher than those of obtained by using phase simulated based on the modified fBm. There is no doubt that reducing the fluctuations and uncertainty is a popular demand in the earthquake design field. Because the fluctuation in Figure 4b is lesser than that of Figure 5b, it seems better to use the results shown in Figure 4b for discussing the effect of earthquake motion phase uncertainty on the inelastic response spectrum. To deduce Figure 5b we assume that ten sample phases are chosen from the mother set of phase composed of whole time histories of observed earthquake motions with wide range of earthquake magnitudes and extensive epicenter distance ranges as well as the variety of local soil conditions. That means, we consider that each earthquake motion phase  
Figure 3: Flowchart of the phase difference simulation. is extracted from a sample time history of the mother set in which as much as possible observed earthquake motion time histories are collected. This is the reason the fluctuation in Figure 5b is large. Moreover, we developed a method to extract the variance feature of phase with the power law from one sample time history and proposed an algorithm to simulate earthquake motion phases according to probability density function defined by Equation (9) with a power law. This is the reason that the fluctuation in Figure 4b becomes smaller than that in Figure 5b. Therefore, we can say that the usage of modified fBm is better than the classic methods to evaluate the uncertainty of Fourier phase, in which the uncertainty of phase is extracted by direct use of many observed earthquake motions. In our method we assume that each observed earthquake motion phase itself has an uncertainly and developed a method to evaluate this uncertainly in the Sections 2.1-2.3 and in Section 3.