Abstract
We propose several parametric point process models to describe different types of earthquake activity, all of which are based upon ideas from fractional calculus’ application to rheology. For a single mainshock-aftershock sequence we study a novel Hawkes process model coined the “Seismic Fractional Hawkes Process” (SFHP). We compare this model to the gold standard “Epidemic Type Aftershock Sequence” (ETAS) model on four data sets: three from Southern California and one from New Zealand. The SFHP does not consistently outperform the ETAS model, however we find that on mainshock-aftershock sequences that exhibit stronger aftershock clustering is when the SFHP performs most favourably. Subsequently, we modify the SFHP to account for short term aftershock incompleteness using history dependent and independent methods. We find that the history dependent methods generally perform the best and may reduce bias in the parameter estimates caused by short term catalogue incompleteness. Following this, we shift our attention to modelling the long-term recurrence of earthquakes. To do so we develop two models, called the stress release fractional Hawkes process and the multidimensional fractional Hawkes process, that incorporate history dependence into the magnitude distribution in different ways. 2 We apply both models to a long-term Japanese data set and analyse their performance using information criteria, residual diagnostics, and simulation. We find that the stress release fractional Hawkes process may be able to describe the main features of the data, however further refinements of this model are necessary. Finally, we find that the multidimensional fractional Hawkes process captures the main features of the data sufficiently well, and from this we are able to infer characteristics of the data in a completely novel manner.