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\begin{document}

\title{Source time functions of earthquakes based on a stochastic differential
equation}



\author[1,1,1]{Shiro Hirano}%
\affil[1]{Ritsumeikan University}%


\vspace{-1em}



  
  \date{June 14, 2024}


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\selectlanguage{english}
\begin{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 fluctuate along time series.
Theoretically, stochastic differential equations, which include a random
variable and white noise, are suitable for modeling such 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\selectlanguage{ngerman}³, \selectlanguage{greek}ω\selectlanguage{english}-\selectlanguage{ngerman}²-type spectra, and frequency distribution. We interpret this
convolution and speculate that the stress drop rate and fault impedance
follow the same Bessel process.%
\end{abstract}\selectlanguage{ngerman}%



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