We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction. Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems. Both these results have implications for the study and prediction of natural hazards and their human impacts.

Mechanisms of stress transfer and probabilistic models have been widely investigated to explain earthquake clustering features. However, these approaches are still far from being able to link individual events and to determine the number of earthquakes caused by a single event. An alternative approach based on proximity functions allows us to generate hierarchical clustering trees and to identify pairs of nearest-neighbors between consecutive levels of hierarchy. Then, the productivity of an earthquake is the number of events of the next level to which it is linked. Using a relative magnitude threshold ΔM to account for scale invariance in the triggering process, we show that the ΔM-productivity attached to each event is a random variable that follows an exponential distribution. The exponential rate of this distribution does not depend on the magnitude of triggering events and systematically decreases with depth. These results could now be used to characterize active fault systems and improve epidemic models of seismicity.

Characterising the state of stress in the brittle upper-crust is essential in mechanics of faulting, industrial production processes, and operational earthquake forecasting. Nevertheless, unresolved questions concern the variation of pore-fluid with depth and the absolute strength on tectonically active faults. Here we show that, along the San Andreas fault system, the time-delay before the onset of the power-law aftershock decay rate (the c-value) varies by three orders of magnitude in the first 20 km below the surface. Despite the influence of the lithostatic stress, there is no continuous change in c-value with depth. Instead, two decay phases are separated by an abrupt increase at an intermediate depth range of 2–5 km. This transitional regime is the only one observed in fluid-injection-induced seismic areas. This provides strong evidence for the role of fluid and a porosity reduction mechanism at depth of few kilometres in active fault zones. Aftershock statistics can then be used to predict changes in differential shear stress with depth until the brittle-ductile transition is reached.