Exploration and Modeling of Non-Gaussian Features in The Earthquake Motion Phase

摘要

Many natural phenomena have been elucidated by using the Wiener process and its expanded formula. Especially procedures based on the stochastic differential equation (SDE) and the fractional Brownian motion (FBM) have achieved dazzling results. In SDE and FBM the Weiner process plays a basic role but its probability density function is restricted to the Gaussian distribution, therefore the stochastic processes modeled by SDE or FBM inevitably belong to the Gaussian regime, and those cannot be applied to explain the natural phenomena of which probability density functions at the tail part become very thick and variances do not exist. In this context we introduce a natural phenomenon which cannot be explained by using concepts in the Gaussian regime and develop the new stochastic process of which probability density function does not obey to the Gaussian distribution. In the first part we choose the earthquake motion phase as a target physical phenomenon and show that its randomness cannot be expressed by a stochastic process governed by the Gaussian distribution. For that purpose, we decompose the earthquake motion phase into the linear delay and fluctuation parts, and investigate the stochastic characteristic of phase difference in the fluctuation part. The probability density function of mean phase gradient in the fluctuation part (viz., a quotient of phase difference with, respect to its discrete circular frequency interval) is expressed by a unique stable distribution with fixed parameters, called Levy-flight, for wide range of circular frequency intervals. Because the variance of the Levi-flight distribution cannot be defined it is analytically derived that the earthquake motion phase is a continuous but un-differentiable function with respect to the circular frequency. In the second part we propose the new type of stochastic process which can represent the stochastic characteristic of phase difference in the fluctuation part by the use of Lubesgue-Stieltjes type integral formula. In which the Kernel plays a role to realize the self-affine and auto correlation natures of phase difference and the integration function represents the main stochastic characteristics of phase. We named this stochastic process as the fractional Levy-flight process. Comparison of several numerical simulations with observed earthquake motion phase differences in the fluctuation part results in the efficiency of the newly proposed stochastic process to simulate realistic earthquake motion phases.