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The Omori formula n(t)=K(t+c)-1 and its modified form n(t)=K(t+c)-P have been successfully applied to many aftershock sequences since the former was proposed just 100 years ago. This paper summarizes studies using these formulae. The problems of fitting these formulae and related point process models to observational data are discussed mainly. Studies published during the last 1/3 century confirmed that the modified Omori formula generally provides an appropriate representation of the temporal variation of aftershock activity. Although no systematic dependence of the index p has been found on the magnitude of the main shock and on the lowest limit of magnitude above which aftershocks are counted, this index (usually p = 0.9-1.5) differs from sequence to. sequence. This variability may be related to the tectonic condition of the region such as structural heterogeneity, stress, and temperature, but it is not clear which factor is most significant in controlling the p value. The constant c is a controversial quantity. It is strongly influenced by incomplete detection of small aftershocks in the early stage of sequence. Careful analyses indicate that c is positive at least for some sequences. Point process models for the temporal pattern of shallow seismicity must include the existence of aftershocks, most suitably expressed by the modified Omori law. Among such models, the ETAS model seems to best represent the main features of seismicity with only five parameters. An anomalous decrease in aftershock activity below the level predicted by the modified Omori formula sometimes precedes a large aftershock. An anomalous decrease in seismic activity of a region below the level predicted by the ETAS model is sometimes followed by a large earthquake in the same or in a neighboring region.

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When dealing with classical spike train analysis, the practitioner often performs goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of probabilistic model (Yana et al. in Biophys.

[1] On the basis of the epidemic-type aftershock sequence (ETAS) model and the thinning procedure, this paper gives the method about how to classify the earthquakes in a given catalogue into different clusters stochastically. The key points of this method are the probabilities of one event being triggered by another previous event and being a background event. Making use of these probabilities, we can reconstruct the functions associated with the characteristics of earthquake clusters to test a number of important hypotheses about the earthquake clustering phenomena. We applied this reconstruction method to the shallow seismic data in Japan and also to a simulated catalogue. The results show the following assertions: (1) The functions for each component in the formulation of the space-time ETAS model are good enough as a first-order approximation for describing earthquake clusters; (2) a background event triggers less offspring in expectation than a triggered event of the same magnitude; (3) the magnitude distribution of the triggered event depends on the magnitude of its direct ancestor; (4) the diffusion of the aftershock sequence is mainly caused by cascades of individual triggering processes, while no evidence shows that each individual triggering process is diffusive; and (5) the scale of the triggering region is still an exponential law, as formulated in the model but not the same one for the expected number of offspring.

Statistical models are introduced for the simultaneous estimation of the changes of the b value and detection rate of earthquakes in a catalogue which develops in time or varies in space. The three characteristic parameters, including b, for the magnitude frequency distribution of detected earthquakes, are represented by respective three B-spline functions of time or location in a space. An objective Bayesian method is adopted for the optimal estimate of such functions with many unknown coefficients. The present procedure is applied to earthquake catalogues for Japanese seismic activity.