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

The Epidemic-Type Aftershock Sequence (ETAS) model is widely used to describe the occurrence of earthquakes in space and time, but there has been little discussion dedicated to the limits of, and influences on, its estimation. Among the possible influences we emphasize in this article the effect of the cut-off magnitude, M_cut, above which parameters are estimated; the finite length of earthquake catalogs; and missing data (e.g., during lively aftershock sequences). We analyze catalogs from southern California and Italy and find that some parameters vary as a function of M_cut due to changing sample size (which affects e.g. Omori’s c constant) or an intrinsic dependence on M_cut (as M_cut increases, absolute productivity and background rate decrease). We also explore the influence of another form of truncation—the finite catalog length—that can bias estimators of the branching ratio. Being also a function of Omori’s p-value, the true branching ratio is underestimated by 45% to 5% for 1.05<p<1.2. Finite sample size affects the variation of the branching ratio estimates. Moreover, we investigate the effect of missing aftershocks and find that the ETAS productivity parameters (α and K₀) and the Omori’s c- and p-values are significantly changed for M_cut<3.5. We further find that conventional estimation errors for these parameters, inferred from simulations that do not account for aftershock incompleteness, are underestimated by, on average, a factor of eight.