Statistical estimation of the duration of aftershock sequences

Abstract

S U M M A R YIt is well known that large earthquakes generally trigger aftershock sequences. However, the duration of those sequences is unclear due to the gradual power-law decay with time. The triggering time is assumed to be infinite in the epidemic type aftershock sequence (ETAS) model, a widely used statistical model to describe clustering phenomena in observed earthquake catalogues. This assumption leads to the constraint that the power-law exponent p of the Omori-Utsu decay has to be larger than one to avoid supercritical conditions with accelerating seismic activity on long timescales. In contrast, seismicity models based on rate-and statedependent friction observed in laboratory experiments predict p ≤ 1 and a finite triggering time scaling inversely to the tectonic stressing rate. To investigate this conflict, we analyse an ETAS model with finite triggering times, which allow smaller values of p. We use synthetic earthquake sequences to show that the assumption of infinite triggering times can lead to a significant bias in the maximum likelihood estimates of the ETAS parameters. Furthermore, it is shown that the triggering time can be reasonably estimated using real earthquake catalogue data, although the uncertainties are large. The analysis of real earthquake catalogues indicates mainly finite triggering times in the order of 100 days to 10 years with a weak negative correlation to the background rate, in agreement with expectations of the rate-and state-friction model. The triggering time is not the same as the apparent duration, which is the time period in which aftershocks dominate the seismicity. The apparent duration is shown to be strongly dependent on the mainshock magnitude and the level of background activity. It can be much shorter than the triggering time. Finally, we perform forward simulations to estimate the effective forecasting period, which is the time period following a mainshock, in which ETAS simulations can improve rate estimates after the occurrence of a mainshock. We find that this effective forecasting period is only in the order of 100 days for moderate mainshocks and in the order of a few years for large events, even if the underlying triggering process lasts much longer.

Estimating ETAS: The effects of truncation, missing data, and model assumptions

“…Consequently, declustering methods, which aim to recover the triggering structure, would always suffer from the bias in if they showed perfect performance. We described the temporal decay with the standard modified Omori formulation; for different formulations, e.g., a truncated power law decay [ Hainzl et al ., ] or stretched exponential [ Mignan , , , ], we expect a different size of the bias.…”

Exploring probabilistic seismic risk assessment accounting for seismicity clustering and damage accumulation: Part I. Hazard analysis

“…First of all, it is important to stress that although our findings suggest that ETAS-type models can, at the very least, serve as a legitimate alternative view of seismicity, there are still some unresolved open issues and certainly room for further refinement. Several authors have reported sources of bias related to the lower magnitude cut-off (Harte, 2016; Schoenberg et al, 2010; Seif et al, 2017; Wang et al, 2010), the finite catalog length and the duration of sequences (Hainzl et al, 2016; van der Elst, 2017; Wang et al, 2010), aftershock incompleteness (e.g. Hainzl et al, 2013; Seif et al, 2017; Zhuang et al, 2017), potential time-dependence of the background rate (Hainzl et al, 2013), the use of an isotropic kernel for the spatial distribution of direct aftershocks (Bach and Hainzl, 2012; Guo et al, 2015; Hainzl et al, 2008), sample size (Harte, 2018) and potential spatial, temporal, and intersequence variability of parameters (e.g.…”

Universal avalanche statistics and triggering close to failure in a mean-field model of rheological fracture

“…aftershock) within the same cluster, only when h > 0. Hence, mean field models are unable to generate the superposition of complex triggering trees identified in natural phenomena [72,73,99,100] and modeled in spatio-temporal stochastic point processes [71,87,101]. Instead, all aftershocks triggered due to the breaking of a given fiber S k are correlative in time and occur in the same cluster.…”