Abstract
Decomposing the earthquake motion phase into the linear delay part and the fluctuation part from it, we investigate the stochastic characteristics of the phase difference in the fluctuation part. The probability density function of the Newton's difference quotient of phase (approximation of the group delay time), which is defined as the quotient of phase difference with respect to the discrete circular frequency interval, is expressed by a unique stable distribution for any arbitrary circular frequency intervals. Because the variance of the stable distribution cannot be defined it is analytically derived that the group delay time, as well as the phase difference, are the discontinuous function with respect to the circular frequency. The earthquake motion phase, therefore, is the continuous but undifferentiable function with respect to the circular frequency. We propose a new type of stochastic process being able to represent these stochastic characteristics of phase difference by the use of Lebesgue-Stieltjes type integral formula. In which the Kernel plays a role to realize the self-affine and auto correlation nature of phase difference and the integration function represents the main stochastic characteristics of earthquake motion phase. Comparison of several numerical simulations with observed earthquake motion phase differences results in the efficiency of the newly proposed stochastic process to simulate a realistic earthquake motion phase.
