Observation of di usion processes in earthquake populations , and implications on the predictability of seismicity systems

Scale invariance, either in space or in time, has been shown in many papers to characterize earthquake distributions. Unfortunately, only little work has been dedicated to looking at the general space-time scaling invariance of seismicity systems, even though a better understanding of how the two domains (spatial and temporal) link together could help to the development of the stochastic dynamical modeling of earthquake populations. We report in this paper the observation of di usion processes of temporally correlated seismic activity for three di erent datasets: a mine (Creighton Mine, Canada), the Long Valley Caldera in eastern California, and a seven year period of recorded seismic activity in southern California. The observed sub-di usion processes are indicative of the general space-time scaling of the system, taking the form of a slow power-law growth R(t) t of the mean distance R(t) between the main event and the temporally correlated afterevents occuring after a delay t. H is found on average to be small (0:1 for Creighton Mine, 0:08 for the Long Valley Caldera, and 0:22 for the southern California main events with magnitude greater or equal to 1.5), but uctuates signi cantly from one main event to the other: the di usion is found to be intermittent (non-Gaussian) and multiscaling, and, except for the Long Valley Caldera, a systematic correlation between the sizes of the main event and subsequent afterevents and the growth exponent H is observed. While classical viscous relaxation models (e.g., elastic listhosphere plastic asthenosphere coupling, or uid ow triggered by sudden changes in pore pressure) have been proposed to characterize this relaxation by homogeneous (i.e., non-intermittent) normal (H = 0:5) di usion processes, the direct implication of the reported results is that seismicity systems, at spatial scales from metres to hundreds of kilometres and small (micro-earthquakes in a mine) to intermediate magnitudes, relax spatio-temporally in a non-elastic way revealing the stochastic space-time scaleinvariant nature of such systems. Since these di usion processes correspond to a loss of information with time on the location of the main event, they can be used to investigate the limits of predictability, at all spatial scales, of seismic systems.