### Abstract:

Moderate to large shallow earthquakes generally give rise to aftershocks due to stress redistribution around the fault zone. For emergency management and for insurance purposes it would be of value to quantify the likelihood of the occurrence of an aftershock in a given time and magnitude interval. There is nothing measurable that sets an aftershock apart from any other earthquake. Therefore an aftershock has to be defined in a statistical way by its closeness in space and time to previous event.
Global data from earthquakes with depths shallower than 70 km were combined from the international Seismological Centre, the US National Earthquake Information Center, Blacknest, and Harvard. An extensive magnitude and catalogue completeness study defined a 'best' magnitude using the Harvard moment as a reference. The data were divided into six tectonic settings, and searched for related events using a simple window in space and time. An objective method was developed to define an elliptical aftershock area.
The database of aftershock sequences has about 28000 mainshocks of which about 2400 have a magnitude M ≥ 6.0, and these were followed by a total of about 7000 aftershocks. The database was analysed in space, time, magnitude, and number of aftershocks in a sequence, hereafter called abundance. The aftershock decay in time and the magnitude-frequency distribution follow well established empirical laws. The p-value in Omori's law for the aftershock decay in time can be assumed to be 1 for subduction and collision zones, and for regions of mixed tectonic character like New Zealand. For mid-ocean ridges the p-value of the present dataset is 1.19 ± 0.08 and for intracontinental zones 0.8 ± 0.14. The b-value of the magnitude-frequency relation is 1.0 for aftershock sequencues in all region. No variation of the b-value with time were observed. The abundance varies greatly from sequence to sequence, but has not been studied in detail before. The abundance for a given mainshock magnitude can be modelled by a geometric distribution, and the mean abundance grows exponentially with mainshock magnitude. The distribution parameters for time, magnitude and abundance were combined to probabilistically predict the number of aftershocks in a given time and magnitude interval from the mainshock.
As an example, the probability of observing one or more aftershocks in the magnitude range [6.0, 8.0] for an earthquake of M = 8.0 ranges between about 40% and 60% in the time interval 7 - 30 days for young subduction zones and intracontinental zones respectively. These results were gained in retrospect, i.e. from the analysis of aftershock sequences once the mainshock was determined. For real-time prediction there is an additional small probability that an earthquake may be followed by a larger event. The incorperation of foreshocks into a final model still needs to be done.