StochasticSTF

Shiro Hirano

Summary

In fortran90+, julia, and matlab code,

x = StochasticSTF(n, 1.0)

returns a stochastic Source Time Function (STF) of length n that satisfies the following properties:

• The STF starts from and terminates at zero (i.e., x(1) = x(n) = 0).
• The STF amplitude is non-negative.
• The Fourier amplitude spectrum follows $\omega^{-2}$-model.
• The moment function $M_0(t) = \displaystyle\int_0^t \textrm{STF}(s) \ ds$ is propotional to $t^3$.

Running

x = StochasticSTF(n, r)

with a floating-point number r $(> 1.0)$ results in more complicated STFs as in the above figure.

Arguments and behavior

The function StochasticSTF(n,r) or StochasticSTF(n,r,d) returns a stochastic Source Time Function (STF) of length n. The floating-point number r $(\ge 1.0)$ is the ratio of two corner frequencies and determines roughness of the STF, where r=1.0 results in STF with the $\omega^{-2}$-type spectrum, and STFs get rougher as r increases. Integer d $(\ge 2)$, dimension of the Bessel bridge, is optional; the default value is d=2, and STFs get smoother as d increases. The result is normalized so that sum(STF) = 1.0 holds. The algorithm has been modified after Hirano(2022; 2023). This code generates STFs with arbitrary lengths by using Bessel bridges, while the original model has probabilistic lengths following the Gutenberg-Richter law.

References

• Hirano, S. (2022), "Source time functions of earthquakes based on a stochastic differential equation", Scientific Reports, 12:3936, https://doi.org/10.1038/s41598-022-07873-2
• Hirano, S. (2023), "Stochastic source time functions with double corner frequencies", AGU23 Fall Meeting, S13F-0407, https://agu.confex.com/agu/fm23/meetingapp.cgi/Paper/1299761

Fortran90+: Usage and Compilation

Simply after compiling m_stochasticSTF.f90, the function StochasticSTF(n,r) will be callable.

For example,

gfortran m_stochasticSTF.f90 main.f90
./a.out

results in an ascii file STF.txt, and the column therein is the amplitude of a stochastic source time function. See lines 2-7 in main.f90 for usage.

To specity precision, see the comments in m_stochasticSTF.f90.

Julia and Matlab: Usage

Running main.jl or main.m calculates and plots a stochastic STF.

LinearAlgebra, DSP, and Plots are required for julia.