simulateETAS {bayesianETAS}

R Documentation

Simulates synthetic data from the ETAS model

Description

This function simulates sample data from the ETAS model over a particular interval [0,T]. The Epidemic Type Aftershock Sequence (ETAS) model is widely used to quantify the degree of seismic activity in a geographical region, and to forecast the occurrence of future mainshocks and aftershocks (Ross 2016). The temporal ETAS model is a point process where the probability of an earthquake occurring at time t depends on the previous seismicity H_t, and is defined by the conditional intensity function:

\lambda(t|H_t) = \mu + \sum_{t[i] < t} \kappa(m[i]|K,\alpha) h(t[i]|c,p)

where

\kappa(m_i|K,\alpha) = Ke^{\alpha \left( m_i-M_0 \right)}

and

h(t_i|c,p) = \frac{(p-1)c^{p-1}}{(t-t_i+c)^p}

where the summation is over all previous earthquakes that occurred in the region, with the i'th such earthquake occurring at time t_i and having magnitude m_i. The quantity M_0 denotes the magnitude of completeness of the catalog, so that m_i \geq M_0 for all i. The temporal ETAS model has 5 parameters: \mu controls the background rate of seismicity, K and \alpha determine the productivity (average number of aftershocks) of an earthquake with magnitude m, and c and p are the parameters of the Modified Omori Law (which has here been normalized to integrate to 1) and represent the speed at which the aftershock rate decays over time. Each earthquake is assumed to have a magnitude which is an independent draw from the Gutenberg-Richter law p(m_i) = \beta e^{\beta(m_i-M_0)}. This function simulates sample data from the ETAS model over a particular interval [0,T].

Usage

simulateETAS(mu, K, alpha, c, p, beta, M0, T, displayOutput = TRUE)

Arguments

mu	Parameter of the ETAS model as described above.
K	Parameter of the ETAS model as described above.
alpha	Parameter of the ETAS model as described above.
c	Parameter of the ETAS model as described above.
p	Parameter of the ETAS model as described above.
beta	Parameter of the Gutenberg-Richter law used to generate earthquake magnitudes.
M0	Magnitude of completeness.
T	Length of the time window [0,T] to simulate the catalog over.
displayOutput	If TRUE then prints the number of earthquakes simulated so far.

Value

A list consisting of

ts	The simulated earthquake times
magnitudes	The simulated earthquake magnitudes
branching	The simulated branching structure, where branching[i] is the index of the earthquake that triggered earthquake i, or 0 if earthquake i is a background event

Author(s)

Gordon J Ross

References

Gordon J. Ross - Bayesian Estimation of the ETAS Model for Earthquake Occurrences (2016), available from http://www.gordonjross.co.uk/bayesianetas.pdf

Examples

## Not run: 
beta <- 2.4; M0 <- 3
simulateETAS(0.2, 0.2, 1.5, 0.5, 2, beta, M0, T=500, displayOutput=FALSE)

## End(Not run)

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