DIFFERENTIABILITY OF PHASE SPECTRUM OF EARTHQUAKE GROUND MOTION AND IMPROVEMENT OF NUMERICAL CALCULATION OF GROUP DELAY TIME: A SUPPLEMENTARY WORK

The differentiability of the phase spectrum of earthquake ground motion is discussed and a new numerical scheme is proposed for the calculation of the group delay time of earthquake ground motion. In addition to the discontinuity due to the “unwrapping” procedure, the phase spectrum can theoretical-ly be non-differentiable when the Fourier transform of the earthquake ground motion is located at the origin of the complex plane. This implies that the group delay time cannot be defined for frequencies for which the Fourier amplitude spectrum is zero. From such a point of view, the conventional theoretical link between the group delay time and the temporal characteristics of earthquake ground motion is re-viewed. It is confirmed that the non-differentiability does not affect the link. Even when the Fourier transform is not located precisely at the origin of the complex plane, the numerical calculation of the phase increment and the group delay time can be unstable when the Fourier transform is located close enough to the origin. To mitigate the difficulty, a new numerical scheme is proposed for the calculation of the group delay time of earthquake ground motion. The new method uses the ‘modified’ group delay time, which is defined as the multiplication of the conventional group delay time with the squared Fourier amplitude spectrum. The ‘modified’ group delay time can be defined even for frequencies for which the Fourier amplitude spectrum is zero. In addition, the ‘modified’ group delay time is numerically stable.