## 3 Estimating the Hurst Index and Variance of the Phase Differences from Observed Ground Motions  
### Regression Analysis of the Observed Ground Motions  
As mentioned in Section 1.1, we only consider the effect of the Fourier phase uncertainty to evaluate the nonlinear seismic performance of a structural system. Therefore, we require numerous sample DRSCEMs to conduct a series of stochastic analyses that take into account the effect of earthquake motion phase uncertainty on the nonlinear response of a structural system. To do this, we need to assign the correct combination of parameters that control the modified fBm. For this purpose, we developed nonlinear regression equations to define the Hurst index and variance in the modified fBm process as functions of the earthquake magnitude \(M\), epicenter distance \(\Delta\) (km) and bias index. The regression analysis was conducted using 250 free-field acceleration time histories recorded at observation sites distributed throughout the world (the Peer NGA Strong Motion Database and a database offered by the Japanese Meteorological Agency) with different magnitudes \(M\) and epicenter distances \(\Delta\). The data were chosen from 17 different events for the regression analysis. Figure 6 shows the relationship between \(M\) and \(\Delta\) of the recorded accelerograms used in this analysis. Except of 6 data points (shown by red circles), all data are located within a distance of 7 to 800 km from the epicenter, which means the nonlinear regression equations are valid for use in this range because all of the data used in the present study are within this range. Additionally, most of the records selected are for earthquakes with magnitudes between 6 and 8.1. Earthquakes with a magnitude larger than 6 were chosen because our motivation is to use phase spectra to simulate DRSCEMs using the response spectra defined in the Chinese Seismic Design Code GB50011-2010 (targeting earthquakes of intensity 8, rare earthquake events, site class 2 and earthquake group 2) for engineering purposes. The Hurst index and variances for each observed earthquake motion were computed. Equation (16) is the form of the basic nonlinear regression equation used to develop the final form of Hurst index. Equation (17) is also used to determine the variance. We evaluate the final form of the regression equation based on the R-squared value, which is a measure of expressing the proximity of data to a regression line, and the Akaike Information Criterion (AIC), which is a method for selecting the best model. Table 2 shows some of the nonlinear regression results.  
\[\widehat{H}=\beta_{1}M+\beta_{2}\Delta+\beta_{3}\text{M}\Delta+\beta_{4}M^{2} +\beta_{5}\Delta^{2}+\beta_{6} \tag{16}\]  
\[\widehat{\sigma_{0}^{2}}=10^{\beta_{1}M}\times\Delta^{\beta_{2}}\times 10^{ \beta_{3}} \tag{17}\]  
where \(\beta_{1}\), \(\beta_{2}\), \(\beta_{3}\), \(\beta_{4}\), \(\beta_{5}\) and \(\beta_{6}\) are regression coefficients.  
Figure 6: Distribution of magnitudes and epicenter distances of observed ground motions used in this study.  
According to the AIC, the best model has a minimum AIC value. Thus, the expected values of the Hurst index \(\widehat{H}\) and variance \(\widehat{\sigma_{0}^{2}}\) can be estimated using the following regression equation:  
\[\widehat{H}\ =0.89M+1.616\times 10^{-3}\Delta-2.353\times 10^{-4}\text{M}\Delta\ -5.749\times 10^{-2}M^{2}+1.827\times 10^{- 7}\Delta^{2}-2.547 \tag{18}\]  
\[\widehat{\sigma_{0}^{2}}=10^{(0.3809M-1.8964)}\times\Delta^{0.5515} \tag{19}\]  
The association between original Hurst index and expected Hurst index are significantly correlated with each other (\(R=0.603\)), and the common logarithm of original variance and expected variance were significantly correlated with each other (\(R=0.563\)), respectively as presented in Figure 7a,b.