momori: Maximum Likelihood Estimates of Parameters in The Omori-Utsu (modified Omori) Formula

Description

Compute the maximum likelihood estimates (MLEs) of parameters in the Omori-Utsu (modified Omori) formula representing for the decay of occurrence rate of aftershocks with time.

Usage

momori(data,mag=NULL,threshold=0.0,tstart,tend,parami,tmpfile=NULL, nlmax=1000)

Arguments

data

point process data.

mag

magnitude.

threshold

threshold magnitude.

tstart

the start of the target period.

tend

the end of the target period.

parami

the initial estimates of the four parameters $B$, $K$, $c$ and $p$.

tmpfile

write the process of minimizing to $tmpfile$. If "" print the process to the standard output and if NULL (default) no report.

nlmax

the maximum number of steps in the process of minimizing.

Value

paramthe final estimates of the four parameters $B$, $K$, $c$ and $p$.
ngmlenegative max likelihood.
aicAIC = -2$LL$ + 2*(number of variables), and the number=4 in this case.
plistlist of parameters $t_i$, $K$, $c$, $p$ and $cls$.

Details

The modified Omori formula represent the delay law of aftershock activity in time. In this equation, $f(t)$ represents the rate of aftershock occurrence at time $t$, where $t$ is the time measured from the origin time of the main shock. $B$, $K$, $c$ and $p$ are non-negative constants. $B$ represents constant-rate background seismicity which may included in the aftershock data. $$f(t) = B + K/(t+c)^p$$ In this function the negative log-likelihood function is minimized by the Davidon-Fletcher-Powell algorithm. Starting from a given set of initial guess of the parameters parai, momori() repeats calculations of function values and its gradients at each step of parameter vector. At each cycle of iteration, the linearly searched step ($lambda$), negative log-likelihood value ($-LL$), and two estimates of square sum of gradients are shown ($process=1$). The cumulative number of earthquakes at time $t$ since $t_0$ is given by the integration of $f(t)$ with respect to the time $t$, $$F(t) = B(t-t_0) + K{c^{1-p}-(t-t_i+c)^{1-p}} / (p-1)$$ where the summation of $i$ is taken for all data event.

References

Y.Ogata (2006) Computer Science Monographs, No.33, Statistical Analysis of Seismicity - updated version (SASeies2006). The Institute of Statistical Mathematics.

Examples

Run this code

data(main2003JUL26)  # The aftershock data of 26th July 2003 earthquake of M6.2 
  x <- main2003JUL26
  momori(x$time, x$magnitude, 2.5, 0.01, 18.68,
         c(0,0.96021E+02,0.58563E-01,0.96611E+00))

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