Abstract. 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.

[1] We compare the aftershock decay rate in natural data with predictions from a stochastic analytical model based on a Markov process with stationary transition rates. These transition rates vary according to the magnitude of a scalar representing the state of stress and defined as the overload. Thus, the aftershock decay rate in the model is a sum of independent exponential decay functions with different characteristic times. From different shapes of the overload distribution and different expressions of the transition rates, we discuss the magnitude of the exponent of the power law aftershock decay rate and the time interval over which we can expect to observe this regime. Before and after this time interval, we show that the decay is linear and exponential, respectively. From our analytical solutions, we deduce a model of aftershock decay rate in which a power law scaling exponent and two characteristic rates emerge. One rate is a short-term linear decrease before the onset of the power law decay to account for a finite number of events at zero time, and the other one can be interpreted as an inverse correlation time, after which aftershocks no longer occur. Then, we interpret the empirical modified Omori law (MOL) and its parameters in the framework of our theoretical model. We suggest a technique to systematically estimate and interpret the temporal limits of the power law aftershock decay rate in real sequences. We approximate these temporal limits from data available from several well-known aftershock sequences and show from an Akaike Information Criteria (AIC) that, in almost all cases examined here, our model fits better the aftershock decay rate than the MOL despite a quantitative penalty for the extra parameter required. From this work, we conclude that the time delay before the onset of the power law decay may be related to the recurrence time of an earthquake. Finally, we suggest that the power law decay rates extend over longer times according to the concentration of the deformation along dominant major faults.INDEX TERMS: 3299 Mathematical Geophysics: General or miscellaneous; 5199 Physical Properties of Rocks: General or miscellaneous; 7230 Seismology: Seismicity and seismotectonics; 7260 Seismology: Theory and modeling Citation: Narteau, C., P. Shebalin, and M. Holschneider, Temporal limits of the power law aftershock decay rate,

Two of the long-standing relationships of statistical seismology are power laws: the Gutenberg-Richter relation describing the earthquake frequency-magnitude distribution, and the Omori-Utsu law characterizing the temporal decay of aftershock rate following a main shock. Recently, the effect of stress on the slope (the b value) of the earthquake frequency-magnitude distribution was determined by investigations of the faulting-style dependence of the b value. In a similar manner, we study here aftershock sequences according to the faulting style of their main shocks. We show that the time delay before the onset of the power-law aftershock decay rate (the c value) is on average shorter for thrust main shocks than for normal fault earthquakes, taking intermediate values for strike-slip events. These similar dependences on the faulting style indicate that both of the fundamental power laws are governed by the state of stress. Focal mechanisms are known for only 2 per cent of aftershocks. Therefore, c and b values are independent estimates and can be used as new tools to infer the stress field, which remains difficult to measure directly.

[1] In order to examine variations in aftershock decay rate, we propose a Bayesian framework to estimate the {K, c, p}-values of the modified Omori law (MOL), l(t) = K(c + t) Àp. The Bayesian setting allows not only to produce a point estimator of these three parameters but also to assess their uncertainties and posterior dependencies with respect to the observed aftershock sequences. Using a new parametrization of the MOL, we identify the trade-off between the c and p-value estimates and discuss its dependence on the number of aftershocks. Then, we analyze the influence of the catalog completeness interval [t start , t stop ] on the various estimates. To test this Bayesian approach on natural aftershock sequences, we use two independent and non-overlapping aftershock catalogs of the same earthquakes in Japan. Taking into account the posterior uncertainties, we show that both the handpicked (short times) and the instrumental (long times) catalogs predict the same ranges of parameter values. We therefore conclude that the same MOL may be valid over short and long times.

[1] In order to elucidate how structural heterogeneities affect the aftershock decay rate, we examine the aftershock sequences produced by a slider-block model of seismicity. In this model, the geometry of the seismic zone is the only free parameter and all aspects of the system are known. The power law aftershock decay rate holds only for smooth faults. A band-limited power law emerges at intermediate fault complexity. For rough faults, only a transient regime toward an exponential decay is observed. In all fault geometries examined, a band-limited power law model fits the synthetic aftershock decay rate better than the Modified Omori Law. Then, as the connected seismic elements form a simpler localised surface, we show that the power law aftershock decay rate extends over longer time, and that the power law exponent increases. These results support the inference that the correlation time of the power law aftershock decay rate increases as the deformation localises along dominant major faults.INDEX TERMS: 3299 Mathematical Geophysics: General or miscellaneous; 7230 Seismology: Seismicity and seismotectonics; 7260 Seismology: Theory and modeling. Citation: Narteau, C., P. Shebalin, S. Hainzl, G. Zöller, and M. Holschneider, Emergence of a band-limited power law in the aftershock decay rate of a slider-block model, Geophys.

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We describe an iterative method to combine seismicity forecasts. With this method, we produce the next generation of a starting forecast by incorporating predictive skill from one or more input forecasts. For a single iteration, we use the differential probability gain of an input forecast relative to the starting forecast. At each point in space and time, the rate in the next-generation forecast is the product of the starting rate and the local differential probability gain. The main advantage of this method is that it can produce high forecast rates using all types of numerical forecast models, even those that are not rate-based. Naturally, a limitation of this method is that the input forecast must have some information not already contained in the starting forecast. We illustrate this method using the Every Earthquake a Precursor According to Scale (EEPAS) and Early Aftershocks Statistics (EAST) models, which are currently being evaluated at the US testing center of the Collaboratory for the Study of Earthquake Predictability. During a testing period from July 2009 to December 2011 (with 19 target earthquakes), the combined model we produce has better predictive performance -in terms of Molchan diagrams and likelihood -than the starting model (EEPAS) and the input model (EAST). Many of the target earthquakes occur in regions where the combined model has high forecast rates. Most importantly, the rates in these regions are substantially higher than if we had simply averaged the models.

Abstract. We apply a simple linear transform, the alongtrack second derivative, to four years of scalar and vectorial data from the CHAMP satellite. This transform, reminiscent of techniques used in the interpretation of aeromagnetic surveys, is applied either to the geocentric spherical components of the field or to its intensity. After averaging in time and space, we first produce a map of the crustal field, then maps of the equatorial electrojet field at all local times and all universal times. The seasonal variation of the electrojet, its evolution with the solar cycle, and the effect of geomagnetic activity are discussed. The variation of the electrojet with longitude, an intriguing feature revealed by satellite data, is described in some detail, and it is shown that this longitude dependance is stable in time. The existence of a counterelectrojet in the morning, everywhere except over the Pacific Ocean, is established. The signatures of closure electric currents and of interhemispheric currents are also evidenced.