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Earthquakes and solar flares are phenomena involving huge and rapid releases of energy characterized by complex temporal occurrence. By analyzing available experimental catalogs, we show that the stochastic processes underlying these apparently different phenomena have universal properties. Namely, both problems exhibit the same distributions of sizes, interoccurrence times, and the same temporal clustering: We find after flare sequences with power law temporal correlations as the Omori law for seismic sequences. The observed universality suggests a common approach to the interpretation of both phenomena in terms of the same driving physical mechanism.

A crucial point in the debate on the feasibility of earthquake predictions is the dependence of an earthquake magnitude from past seismicity. Indeed, while clustering in time and space is widely accepted, much more questionable is the existence of magnitude correlations. The standard approach generally assumes that magnitudes are independent and therefore in principle unpredictable. Here we show the existence of clustering in magnitude: earthquakes occur with higher probability close in time, space, and magnitude to previous events. More precisely, the next earthquake tends to have a magnitude similar but smaller than the previous one. A dynamical scaling relation between magnitude, time, and space distances reproduces the complex pattern of magnitude, spatial, and temporal correlations observed in experimental seismic catalogs.

We propose a branching process based on a dynamical scaling hypothesis relating time and mass. In the context of earthquake occurrence, we show that experimental power laws in size and time distribution naturally originate solely from this scaling hypothesis. We present a numerical protocol able to generate a synthetic catalog with an arbitrary large number of events. The numerical data reproduce the hierarchical organization in time and magnitude of experimental interevent time distribution.

We show that, under a suitable assumption on the pressure tensor, the mass and momentum balance equations of hydrodynamical theory, introduced in the early 1980s by many authors to describe matrix separation, yield the equations of the Madelung fluid that are equivalent to the Schrödinger-like equation with logarithmic nonlinearity. This equation has solitary-waves solutions as required by many experimental volcanic models.

An increase in the number of smaller magnitude events, retrospectively named foreshocks, is often observed before large earthquakes. We show that the linear density probability of earthquakes occurring before and after small or intermediate mainshocks displays a symmetrical behavior, indicating that the size of the area fractured during the mainshock is encoded in the foreshock spatial organization. This observation can be used to discriminate spatial clustering due to foreshocks from the one induced by aftershocks and is implemented in an alarm-based model to forecast m > 6 earthquakes. A retrospective study of the last 19 years Southern California catalog shows that the daily occurrence probability presents isolated peaks closely located in time and space to the epicenters of five of the six m > 6 earthquakes. We find daily probabilities as high as 25% (in cells of size 0.04 × 0.04deg2), with significant probability gains with respect to standard models.

We investigate the spatial distribution of aftershocks, and we find that aftershock linear density exhibits a maximum that depends on the main shock magnitude, followed by a power law decay. The exponent controlling the asymptotic decay and the fractal dimensionality of epicenters clearly indicate triggering by static stress. The nonmonotonic behavior of the linear density and its dependence on the main shock magnitude can be interpreted in terms of diffusion of static stress. This is supported by the power law growth with exponent H approximately 0.5 of the average main-aftershock distance. Implementing static stress diffusion within a stochastic model for aftershock occurrence, we are able to reproduce aftershock linear density spatial decay, its dependence on the main shock magnitude, and its evolution in time.

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