Abstract
The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l’Aquila region, Italy, with the devastating event on 6 April 2009, is presented.
Highlights
Point process models are often used in statistical seismology for describing the occurrence of earthquakes in a spatio-temporal setting
The epidemic type aftershock sequence (ETAS) model is characterized by its conditional intensity function, that is, the rate of arriving events conditioned on the history of previous events
This time-dependent conditional intensity function itself consists of two parts, (i) a background intensity μ of a Poisson process, which models the arrival of spontaneous events, and (ii) a timedependent triggering function φ which encodes the form of self-excitation by adding a positive impulse response for each event, that is, an instantaneous jump which decays gradually at time progresses
Summary
Point process models are often used in statistical seismology for describing the occurrence of earthquakes (point data) in a spatio-temporal setting. The ETAS model as a variant of a Hawkes process model assigns the earthquake magnitude as a mark to each event, and it usually employs a specific mark depending excitation kernel (Ogata 1988, 1998). The ETAS model (Ogata 1998), describes a stochastic process, which generates point pattern over some domain X × T × M, where T × X is the time-space window and M the mark space of the process Realisations of this point process are denoted by D = {(ti , xi , mi )}iN=D1, which in seismology can be interpreted as an earthquake catalog consisting of ND observed events. One way to define the ETAS model is by a conditional intensity function, which models the infinitesimal rate of expected arrivals around (t, x) given the history Ht = {(ti , xi , mi ) : ti < t} of the process until time t. The minimal bandwidth is commonly chosen as dmin ∈ [0.02, 0.05] degrees, which is in the range of the localisation error (Zhuang et al 2002)
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