Complexity, aftershock sequences, and uncertainty in earthquake statistics

Earthquake statistics is a growing field of research with direct application to probabilistic seismic hazard evaluation. The earthquake process is a complex spatio-temporal phenomenon, and has been thought to be an example of the self-organised criticality (SOC) paradigm, in which events occur as cascades on a wide range of sizes, each determined by fine details of the rupture process. As a consequence, deterministic prediction of specific event sizes, locations, and times may well continue to remain elusive. However, probabilistic forecasting, based on statistical patterns of occurrence, is a much more realistic goal at present, and is being actively explored and tested in global initiatives. This thesis focuses on the temporal statistics of earthquake populations, exploring the uncertainties in various commonly-used procedures for characterising seismicity and explaining the origins of these uncertainties. Unlike many other SOC systems, earthquakes cluster in time and space through aftershock triggering. A key point in the thesis is to show that the earthquake inter-event time distribution is fundamentally bimodal: it is a superposition of a gamma component from correlated (co-triggered) events and an exponential component from independent events. Volcano-tectonic earthquakes at Italian and Hawaiian volcanoes exhibit a similar bimodality, which in this case, may arise as the sum of contributions from accelerating and decelerating rates of events preceding and succeeding volcanic activity. Many authors, motivated by universality in the scaling laws of critical point systems, have sought to demonstrate a universal data collapse in the form of a gamma distribution, but I show how this gamma form is instead an emergent property of the crossover between the two components. The relative size of these two components depends on how the data is selected, so there is no universal form. The mean earthquake rate—or, equivalently, inter-event time—for a given region takes time to converge to an accurate value, and it is important to characterise this sampling uncertainty. As a result of temporal clustering and non-independence of events, the convergence is found to be much slower than the Gaussian rate of the central limit theorem. The rate of this convergence varies systematically with the spatial extent of the region under consideration: the larger the region, the closer to Gaussian convergence. This can be understood in terms of the increasing independence of the inter-event times with increasing region size as aftershock sequences overlap in time to a greater extent. On the other hand, within this high-overlap regime, a maximum likelihood inversion of parameters for an epidemic-type statistical model suffers from lower accuracy and a systematic bias; specifically, the background rate is overestimated. This is because the effect of temporal overlapping is to mask the correlations and make the time series look more like a Poisson process of independent events. This is an important result with practical relevance to studies using inversions, for example, to infer temporal variations in background rate for time-dependent hazard estimation.