Measuring and modelling the occupation probability to characterize the temporal statistics of seismic sequences

SUMMARY: The probability that any given time interval of duration $\tau$ is occupied by one earthquake, or more, characterizes the temporal statistics of seismic sequences and, therefore, the temporal clustering of events. The occupation probability reveals the fractal behaviour of seismic sequences, $\Phi (\tau ) \sim \tau ^{1 - D_\tau }$, at short times, defining a temporal fractal dimension, $D_\tau$, for seismic events. We introduce an empirical model of the occupation probability, parametrized by the fractal dimension and two other parameters. We use the mathematical relationship between the occupation probability and the interevent time probability density to develop intuition about the model parameters and to compare the proposed model with the conventional gamma model for interevent times. Our model captures the statistical properties of a wide range of seismic sequences where the gamma model fails, from slow-slip driven swarms to burst-like episodes at the bottom of the seismogenic zone and low-frequency earthquakes. Using real and synthetic catalogues, we find that the model parameters are related to the degree of intermittency of the seismic activity, the characteristics of the bursts and the timescale of the quiescent periods. Measuring and modelling the occupation probability constitutes a valuable tool to categorize seismic sequences over a wide spectrum of seismic occurrence patterns. When applied to the new generation of earthquake catalogues, this empirical method highlights intermittency and fractal bursts, challenging conventional seismicity models and emphasizing the need to refine them to better capture these key features.