SAPP (version 1.0.9-1)

respoi: The Residual Point Process of the ETAS Model

Description

Compute the residual of modified Omori Poisson process and display the cumulative curve and magnitude v.s. transformed time.

Usage

respoi(time, mag, param, zts, tstart, zte, threshold = 0.0, plot = TRUE)
respoi2(etas, param, zts, tstart, zte, threshold = 0.0, plot = TRUE)

Value

trans.time

transformed time \(F(t_i), i = 1,2,...,N.\)

cnum

cumulative number of events.

resData

a res-format dataset on 7 variables

(no., longitude, latitude, magnitude, time, depth and trans.time)

Arguments

time

the time measured from the main shock (t = 0).

mag

magnitude.

etas

an etas-format dataset on 9 variables

(no., longitude, latitude, magnitude, time, depth, year, month and days).

param

the four parameters \(B\), \(K\), \(c\) and \(p\).

zts

the start of the precursory period.

tstart

the start of the target period.

zte

the end of the target period.

threshold

threshold magnitude.

plot

logical. If TRUE (default) cumulative curve and magnitude v.s. transformed time \(F(t_i)\) are plotted.

Details

The function respoi and respoi2 compute the following output for displaying the goodness-of-fit of Omori-Utsu model to the data. 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.

respoi2 is equivalent to respoi except that input and output forms are different. When a etas-format dataset is given, respoi2 returns the dataset with the format as described below.

References

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

Examples

data(main2003JUL26)  # The aftershock data of 26th July 2003 earthquake of M6.2 

# output transformed times and cumulative numbers
x <- main2003JUL26
respoi(x$time, x$magnitude, param = c(0,0.96021e+02, 0.58563e-01, 0.96611e+00),
       zts = 0.0, tstart = 0.01, zte = 18.68, threshold = 2.5)

# output a res-format dataset
respoi2(main2003JUL26, param = c(0,0.96021e+02, 0.58563e-01, 0.96611e+00),
        zts = 0.0, tstart = 0.01, zte = 18.68, threshold = 2.5)