Abstract
Source time functions are essential observable quantities in seismology; they have been investigated via kinematic inversion analyses and compiled into databases. Given the numerous available results, some empirical laws on source time functions have been established, even though they are complicated and fluctuated time series. Theoretically, stochastic differential equations, including a random variable and white noise, are suitable for modeling complicated phenomena. In this study, we model source time functions as the convolution of two stochastic processes (known as Bessel processes). We mathematically and numerically demonstrate that this convolution satisfies some of the empirical laws of source time functions, including non-negativity, finite duration, unimodality, a growth rate proportional to t^3, omega ^{-2}-type spectra, and frequency distribution (i.e., the Gutenberg–Richter law). We interpret this convolution and speculate that the stress drop rate and fault impedance follow the same Bessel process.
Highlights
Uchide and Ide[4] compared the moment functions of Mw 1.7–6.0 events in Parkfield, California, based on multi-scale inversion analyses
The GR law originally means that the probability density function (PDF) of a seismic moment is a power law
We demonstrated that the four empirical laws on STFs, or moment-rate functions, can be reproduced by modeling STFs as the convolution of two Bessel processes with almost the same order of duration
Summary
Uchide and Ide[4] compared the moment functions of Mw 1.7–6.0 events in Parkfield, California, based on multi-scale inversion analyses They pointed out that EL2 holds from the very early to later stages of the source processes. Spatial heterogeneity of fracture energy[18] and temporal fluctuation of dynamic stress transfer[19] introduced in a boundary integral equation play an important role on the rupture complexity While such numerical modelings are developing, mathematical modeling, if available, would contribute to the understanding of complex faulting processes. Wu et al.[22] assumed that the generalized Langevin equation can model the equation of motion for the fault slip rate Their model was based on some physical properties of dynamic friction, their solution was Brownian motion, which cannot satisfy the non-negativeness (EL1) or the ω−2-like spectrum (EL3). A novel approach is needed for SDE-based modeling under EL1–4
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