momori | R Documentation |

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.

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

`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 |

`tmpfile` |
a character string naming the file to write the process of minimizing.
If "" print the process to the standard output and if |

`nlmax` |
the maximum number of steps in the process of minimizing. |

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 be 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.

`param` |
the final estimates of the four parameters |

`ngmle` |
negative max likelihood. |

`aic` |
AIC = -2 |

`plist` |
list of parameters |

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

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

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