## 2 Phase Spectrum Simulation by Modified Fractional Brownian Motion  
### Method for Calculating the Continuous Phase with Respect to the Circular Frequency  
If the time history \(f\left(t\right)\) is an Riemann integrable function with respect to the time we can obtain the real and imaginary parts of Fourier transform of \(f\left(t\right)\) as \(R\left(\omega\right)\) and \(I\left(\omega\right)\) that are continuous functions with respect to the circular frequency \(\left(\omega\right)\). We usually calculate the Fourier phase by \(\tan\widetilde{\phi}\left(\omega\right)=I\left(\omega\right)/R\left(\omega\right)\) but we could not distinguish the value of \(\widetilde{\phi}\left(\omega\right)\) at where \(F\left(\omega\right)\) is located in the first and third quadrants as well as the second and the fourth quadrants. Therefore, the ordinary Fourier phase \(\widetilde{\phi}\left(\omega\right)\) is given by  
\[\widetilde{\phi}\left(\omega\right)=Arg\left(F\left(\omega\right)\right) \tag{2}\]  
where \(Arg\) is the function taking argument of the complex value \(F\left(\omega\right)\). \(\widetilde{\phi}\left(\omega\right)\) is a partially continuous function with respect to the circular frequency within a principal value of \(\left[-\pi,\pi\right]\). To obtain the continuous phase \(\phi\left(\omega\right)\) from \(\widetilde{\phi}\left(\omega\right)\) we need the operation to unwrap a phase value near \(\pi\) to a value near \(-\pi\). Because \(F\left(\omega\right)\) is a multi-variable complex function for a fixed circular frequency and we must carefully identify the complex sheet number on which \(F\left(\omega\right)\) is located. This requests to define the value of \(n\) that is equal to the number of complex sheets being passed by to reach the current complex value of \(F\left(\omega\right).\) Due to this complexity, there is no rigorous algorithms for this purpose, so we used Equation (3) which was developed by [PERSON] (1986) to calculate the phase increment \(d\phi\left(\omega\right)\) of the continuous phase \(\phi\left(\omega\right)\) with respect to the circular frequency.  
\[d\phi\left(\omega\right)=\frac{RdI-IdR}{R^{2}+I^{2}} \tag{3}\]  
The continuous phase \(\phi\left(\omega\right)\) can be obtained by integrating \(d\phi\left(\omega\right)\) with respect to the circular frequency. The Fourier phase of earthquake motion can usually be de-convoluted into linear delay and fluctuation parts as follows:  
\[\phi\left(\omega\right)\ =\ -\omega\,t_{0}+\psi\left(\omega\right) \tag{4}\]  
where \(-\omega t_{0}\) corresponds to the linear delay part and \(\psi\left(\omega\right)\) corresponds to fluctuation part. The linear delay expression only shifts the position of the time history and does not change its configuration, which is only influenced by the phase corresponding to the fluctuation part. Therefore, throughout this paper, we refer to this fluctuation part as the phase. Importantly, attention must be paid is an asymmetric (odd) characteristic of the phase with respect to the circular frequency, it is expressed as \(\phi\left(\omega\right)=-\phi\left(-\omega\right)\). It follows that the fluctuation part of phase \(\psi\left(\omega\right)\) and its increment \(d\psi\left(\omega\right)\) are of the same nature:  
\[\psi\left(\omega\right)=-\psi\left(-\omega\right),\ d\psi\left(\omega\right) =-d\psi\left(-\omega\right) \tag{5}\]  
After [PERSON] (1986) found the power law hidden in the earthquake motion phase, he recognized that the phase spectrum can also be expressed by the fBm. However, it was recognized that direct application could not generate a sample phase process because the phase process is an asymmetric function with respect to the circular frequency, as defined by Equation (5).