Finite-Horizon Triggering Cores of Earthquake Sequences

Abstract: Models of triggered seismicity commonly aggregate the contributions of all past earthquakes into a total predicted rate. This aggregation does not distinguish a forecast dominated by one recent earthquake from an equal-rate forecast supported by many events distributed across a longer history, even though these configurations can have different persistence and sensitivity to catalog and parameter uncertainties. We introduce the \emph{finite-horizon triggering core size} (N_{p,T}(t)), defined as the minimum number of past earthquakes whose expected contributions account for a fraction (p) of the total triggering potential over the future interval ([t,t+T]). The definition applies to any triggering model that assigns nonnegative event-specific expected contributions over a specified future window. We develop its theory for marked Hawkes and epidemic-type aftershock sequence processes, where the weights are the expected numbers of direct offspring produced by individual past earthquakes. The finite horizon makes the observable well defined for every normalized Omori kernel with exponent (θ>0), including the long-memory regime in which the corresponding infinite-horizon quantity diverges. We derive continuum and marked-Poisson benchmarks, an exact representation of the marked-Poisson distribution, and high-(p) distributional laws. For finite-horizon Omori memory, the triggering core grows as ((1-p)^{-1/θ}) with a realization-dependent amplitude controlled by the total triggering weight; for exponential memory, the growth is logarithmic.[...]