Major Research Development of strong motion estimation method for scenario earthquakes beneath metropolitan area

As part of efforts to develop a method for predicting strong ground motions caused by major earthquakes that may occur directly underneath metropolitan areas, we conducted fundamental research on the phase characteristics of seismic motions.
 

The Fourier phase of seismic motions (hereinafter referred to as "seismic motion phase") differentiated by the circular frequency is called group delay time, which is known to be related to the temporal characteristics of seismic motions. In many studies, this relationship is used for waveform synthesis, where waveforms with appropriate temporal characteristics are synthesized by imposing appropriate group delay time. The results of those studies are also used to define design ground motions in a practical manner. However, as the statistical characteristics of seismic motion phase and group delay time are not yet fully understood, studies are still underway for full comprehension. In some of those studies, it is claimed that seismic motion phase cannot be differentiated by the circular frequency anywhere and, therefore, group delay time cannot be defined, as it is a first-order differentiation of seismic motion phase with respect to the circular frequency. If such claim turns out to be true, the very foundation of previous studies that relied on the concept of group delay time as a differentiation of seismic motion phase will come into doubt, threatening any application of the concept for practical use.
 

Therefore, in this study, we revisited the definition of seismic motion phase. We first examined the differentiability of the Fourier transform of seismic motions, and then used the results of the examination to study the differentiability of seismic motion phase. Through the study, we discovered that seismic motion phase cannot be differentiated under limited conditions; not everywhere.
 

The figure below illustrates the conditions in which it is impossible to differentiate seismic motion phase. The arrows indicate the trajectory of the Fourier transform of seismic motions F(ω) on a complex plane. (a) indicates a case where F(ω) moves from the third quadrant to the second quadrant on the complex plane, as ω increases. Here, discontinuity of 2π is seen occurring to the seismic motion phase. Meanwhile, (b) is a case where F(ω) passes through the origin on the complex plane, as ω increases. Here, discontinuity of π is seen occurring to the seismic motion phase. The former issue can be avoided if so-called "unwrapping" is performed properly, while the latter cannot be resolved even with proper unwrapping.
 

From a theoretical standpoint, the only case when phase cannot be differentiated is where F(ω) passes through the origin on a complex plane. However, in numerical computation, we discovered that phase suddenly changes in response to a change in ω when F(ω) passes near the origin, even if it does not pass through the origin, and at such point Δθ/Δω, which is a finite difference approximation of group delay time, becomes numerically unstable. We also identified a method for avoiding such numerical instability.