Epidemic-type aftershock sequences

Modeling the temporal occurrence of earthquakes with self-exciting point processes was first proposed by Ogata [11] who later extended his model to the spatio-temporal domain [12]. These epidemic-type aftershock sequence (ETAS) models are parametric models based on well-studied phenomenon of the temporal decay of aftershocks, via Omori-Utsu [19, 20], and the magnitude distributions of earthquakes, via Gutenberg-Richter [21].

ETAS models consider the triggering function ν to be composed of three separable functions g(t), h(x, y), and k(m) pertaining to time, space, and magnitude, respectively. That is, the conditional intensity function is defined as

The functions g(t) and h(x, y) then model how the conditional rate of events decays over time and space, respectively, while the function k(m) describes the productivity of previous events based on their marks. In seismology, for example, where marks are taken to be the magnitude, or amount of energy released during an earthquake, events with a larger magnitude will be more productive at producing offspring than an event of smaller magnitude.