simulateETAS: Simulates synthetic data from the ETAS model

In bayesianETAS: Bayesian Estimation of the Temporal and Spatio-Temporal ETAS Models for Earthquake Occurrences

Simulates synthetic data from the ETAS model

Description

The 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 2019). 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-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}

and s() is a spatial triggering kernel. The public spatio-temporal routines currently use a Gaussian spatial triggering kernel with a homogeneous spatial background over the supplied area and no spatial edge correction. s() can also be omitted to fit the temporal ETAS model without a spatial component. The summation is over all previous earthquakes that occurred in the region, with the i'th such earthquake occurring at time t_i at spatial location ( x_i,, y_i) aand 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)}.

Usage

simulateETAS(
  mu,
  K,
  alpha,
  c,
  p,
  beta,
  M0,
  maxTime,
  sigma2x = NULL,
  sigma2y = NULL,
  xmin = NULL,
  xmax = NULL,
  ymin = NULL,
  ymax = NULL
)

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.
maxTime	Length of the time window [0,maxTime] the catalog was observed over. If not specified, will be taken as the time of the last earthquake.
sigma2x	(For the spatio-temporal ETAS model only) Parameter of the spatio-temporal model as described above.
sigma2y	(For the spatio-temporal ETAS model only) Parameter of the spatio-temporal model as described above.
xmin	(For the spatio-temporal ETAS model only) Minimum x value of the spatial region
xmax	(For the spatio-temporal ETAS model only) Maximum x value of the spatial region
ymin	(For the spatio-temporal ETAS model only) Minimum y value of the spatial region
ymax	(For the spatio-temporal ETAS model only) Maximum y value of the spatial region

Details

In the spatio-temporal version of the model, s() is the multivariate Gaussian with diagonal covariance parameters \sigma^2_x and \sigma^2_y. Spatial simulation uses a bounded-window discard convention: proposed spatial offspring outside the requested window are discarded and discarded offspring do not generate descendants. This is not an edge-corrected simulation of an infinite-plane ETAS process observed through a bounded window.

This function simulates sample data from the ETAS model (either temporal or spatio-temporal) over a particular time interval [0,T], For the spatio-temporal model we also specify the range of the x axis region [xmin,xmax] and the y axis region [ymin,ymax].

Value

A list consisting of

ts	The simulated earthquake times
ms	The simulated earthquake magnitudes
locs	(For the spatio-temporal ETAS model only) A two column matrix wher the i'th row is the (x,y) location of earthquake i

Author(s)

Gordon J Ross

References

Gordon J. Ross - Bayesian Estimation of the ETAS Model for Earthquake Occurrences (2019)

Examples

set.seed(1)

#temporal model
beta <- 2.4; M0 <- 3; maxTime <- 20
catalog <- simulateETAS(0.1, 0.05, 1.0, 0.5, 2, beta, M0, maxTime)

#spatio-temporal model
sigma2x <- 0.2; sigma2y <- 0.2
xmin <- 0; xmax <- 10; ymin <- 0; ymax <- 10
catalog <- simulateETAS(0.1, 0.05, 1.0, 0.5, 2, beta, M0, maxTime,
   sigma2x, sigma2y, xmin, xmax, ymin, ymax)

bayesianETAS documentation built on May 26, 2026, 1:06 a.m.