Modeling of earthquake phase spectrum using a stochastic deferential equation and development of a simulation method of earthquake motion form phase spectrum

地震动的幅值和相位特性为评估设计结构的抗震能力提供了关键信息。设计地震反应谱的模拟方法是设计规范中具有挑战性的课题之一，用于生成与设计地震反应谱相一致的设计地震动。提出了一种基于群延迟时间和小波分析的地震运动相位特性的简单建模方法。首先对近、远源区的相位谱进行建模。利用已有的地震动观测数据和小波分析方法计算了各次地震动的群延迟时间。得到了平均群延迟时间和群延迟时间标准差随震级、震中距和几个震源参数变化的回归方程。模拟样本相谱假设高斯性质的群延迟时间，地震运动兼容的设计反应谱，然后模拟和他们的特点进行了研究，使用延性需求谱。其次，我们发展了一种从相谱模拟真实地震运动的方法。非平稳时间函数（如地震动）的本质要求是因果关系。由于地震运动的这种性质，它的振幅和相位不能单独分配，以模拟现实的地震运动。然而，当其傅立叶振幅被先验地给定时，对于模拟因果地震运动，对相位谱的限制还没有明确的定义。我们发展了一种方法来模拟地震运动从一个给定的相位谱考虑到因果关系的限制。第三，讨论了利用随机微分方程模拟群时延的新方法。

Both the amplitude and phase characteristics of earthquake motion provide key information needed to evaluate the aseismic capacity of designed structures. A method to simulate phase spectra is one of challenging topic in the design standard which is used to generate design earthquake motions compatible with the design spectra. We present a simple method for modeling the phase characteristic of earthquake motion based on group delay time and wavelet analysis. First the phase spectra are modeled for near and far source regions. Existing data sets of observed earthquake motions and wavelet analysis are used to calculate the group delay time of each earthquake motion. Regression equations for the mean group delay time and standard deviation of group delay time are obtained as functions of the earthquake's magnitude, epicenter distance and several source parameters. Simulating sample phase spectra assuming Gaussian nature of group delay time, earthquake motions compatible with the design response spectra then are simulated, and their characteristics investigated using ductility demand spectra. Second we develop a method to simulate realistic earthquake motions from phase spectra. The essential requirement of non-stationary time function such as earthquake motion is the causality. Because of this nature of earthquake motion its amplitude and phase cannot be assigned independently to simulate realistic earthquake motions. The restriction to be place on its phase spectra, however, has not been clearly defined to simulate a causal earthquake motion when its Fourier amplitude is given apriori. We develop a method to simulate earthquake motions from a given phase spectrum taking into account the causality restriction. Third we discuss a new method to simulate group delay time using the stochastic differential equation.