Statistical Detection and Characterization of a Deviation from the Gutenberg-Richter Distribution above Magnitude 8

— We present a quantitative statistical test for the presence of a crossover c₀ in the Gutenberg-Richter distribution of earthquake seismic moments, separating the usual power-law regime for seismic moments less than c₀ from another faster decaying regime beyond c₀. Our method is based on the transformation of the ordered sample of seismic moments into a series with uniform distribution under condition of no crossover. A simulation method allows us to estimate the statistical significance of the null hypothesis H₀ of an absence of crossover (c₀=infinity). When H₀ is rejected, we estimate the crossover c₀ using two different competing models for the second regime beyond c₀ and the simulation method. For the catalog obtained by aggregating 14 subduction zones of the Circum-Pacific Seismic Belt, our estimate of the crossover point is log(c₀)=28.14 ± 0.40 (c₀ in dyne-cm), corresponding to a crossover magnitude m_W=8.1 ± 0.3. For separate subduction zones, the corresponding estimates are substantially more uncertain, so that the null hypothesis of an identical crossover for all subduction zones cannot be rejected. Such a large value of the crossover magnitude makes it difficult to associate it directly with a seismogenic thickness as proposed by many different authors. Our measure of c₀ may substantiate the concept that the localization of strong shear deformation could propagate significantly in the lower crust and upper mantle, thus increasing the effective size beyond which one should expect a change of regime.     

Power law distributions of seismic rates

We report an empirical determination of the probability density functions P_data(r) of the number r of earthquakes in finite space–time windows for the California catalog for different space (5 × 5 to 50 × 50 km²) and time intervals (0.1 to 1000 days). The data can be represented by asymptotic power law tails together with several cross-overs reasonably explained by one of the most used reference model in seismology (ETAS), which assumes that each earthquake can trigger other earthquakes according to complex cascades. These results are useful to constrain the physics of earthquakes and to estimate the performance of forecasting models of seismicity.     

Power-law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering

We consider a general stochastic branching process,which is relevant to earthquakes as well as to many other systems, and we study the distributions of the total number of offsprings (direct and indirect aftershocks in seismicity) and of the total number of generations before extinction. We apply our results to a branching model of triggered seismicity, the ETAS (epidemic-type aftershock sequence) model. The ETAS model assumes that each earthquake can trigger other earthquakes (aftershocks). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. Due to the large fluctuations of the number of aftershocks triggered directly by any earthquake (fertility), there is a large variability of the total number of aftershocks from one sequence to another, for the same mainshock magnitude. We study the regime in which the distribution of fertilities is characterized by a power law ~1/¹⁺. For earthquakes we expect such a power-distribution of fertilities with =b/ based on the Gutenberg-Richter magnitude distribution ~ 10^–bm and on the increase ~ 10^–m of the number of aftershocks with the mainshock magnitude m. We derive the asymptotic distributions p_r(r) and p_g(g) of the total number r of offsprings and of the total number g of generations until extinction following a mainshock. In the regime < 2 for which the distribution of fertilities has an infinite variance, we find This should be compared with the distributions obtained for standard branching processes with finite variance. These predictions are checked by numerical simulations. Our results apply directly to the ETAS model whose preferred values =0.8–1 and b=1 puts it in the regime where the distribution of fertilities has an infinite variance. More generally, our results apply to any stochastic branching process with a power-law distribution of offsprings per mother     

Characterization of the Frequency of Extreme Earthquake Events by the Generalized Pareto Distribution

Recent results in extreme value theory suggest a new technique for statistical estimation of distribution tails (Embrechts et al., 1997), based on a limit theorem known as the Gnedenko-Pickands-Balkema-de Haan theorem. This theorem gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD), which is a family of distributions with two parameters. The GPD has been successfully applied in a number of statistical problems related to finance, insurance, hydrology, and other domains. Here, we apply the GPD approach to the well-known seismological problem of earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law. We analyze shallow earthquakes (depth h<70 km) in the Harvard catalog over the period 1977–2000 in 12 seismic zones. The GPD is found to approximate the tails of the seismic moment distributions quite well over the lower threshold approximately M 10²⁴ dyne-cm, or somewhat above (i.e., moment-magnitudes larger than m _W =5.3). We confirm that the b-value is very different (b=2.06 ± 0.30) in mid-ocean ridges compared to other zones (b=1.00 ± 0.04) with a very high statistical confidence and propose a physical mechanism contrasting crack-type rupture with dislocation-type behavior. The GPD can as well be applied in many problems of seismic hazard assessment on a regional scale. However, in certain cases, deviations from the GPD at the very end of the tail may occur, in particular for large samples signaling a novel regime.     

Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory

The present work is a continuation and improvement of the method suggested in Pisarenko et al. (Pure Appl Geophys 165:1–42, 2008) for the statistical estimation of the tail of the distribution of earthquake sizes. The chief innovation is to combine the two main limit theorems of Extreme Value Theory (EVT) that allow us to derive the distribution of T-maxima (maximum magnitude occurring in sequential time intervals of duration T) for arbitrary T. This distribution enables one to derive any desired statistical characteristic of the future T-maximum. We propose a method for the estimation of the unknown parameters involved in the two limit theorems corresponding to the Generalized Extreme Value distribution (GEV) and to the Generalized Pareto Distribution (GPD). We establish the direct relations between the parameters of these distributions, which permit to evaluate the distribution of the T-maxima for arbitrary T. The duality between the GEV and GPD provides a new way to check the consistency of the estimation of the tail characteristics of the distribution of earthquake magnitudes for earthquake occurring over an arbitrary time interval. We develop several procedures and check points to decrease the scatter of the estimates and to verify their consistency. We test our full procedure on the global Harvard catalog (1977–2006) and on the Fennoscandia catalog (1900–2005). For the global catalog, we obtain the following estimates: \( \hat{M}_{{\rm max} } \)  = 9.53 ± 0.52 and \( \hat{Q}_{10} (0.97) \)  = 9.21 ± 0.20. For Fennoscandia, we obtain \( \hat{M}_{{\rm max} } \)  = 5.76 ± 0.165 and \( \hat{Q}_{10} (0.97) \)  = 5.44 ± 0.073. The estimates of all related parameters for the GEV and GPD, including the most important form parameter, are also provided. We demonstrate again the absence of robustness of the generally accepted parameter characterizing the tail of the magnitude-frequency law, the maximum possible magnitude M _max, and study the more stable parameter Q _T (q), defined as the q-quantile of the distribution of T-maxima on a future interval of duration T.     

Statistical physics of fault patterns self-organized by repeated earthquakes

This work presents at attempt to model brittle ruptures and slips in a continental plate and its spontaneous organization by repeated earthquakes in terms of coarse-grained properties of the mechanical plate. A statistical physics model, which simulates anti-plane shear deformation of a thin plate with inhomogeneous elastic properties, is thus analyzed theoretically and numerically in order to study the spatio-temporal evolution of rupture patterns in response to a constant applied strain rate at its borders, mimicking the effect of neighboring plates. Rupture occurs when the local stress reaches a threshold value. Broken elements are instantaneously healed and retain the original material properties, enabling the occurrence of recurrent earthquakes. Extending previous works (Cowie et al., 1993;Miltenberger et al., 1993), we present a study of the most startling feature of this model which is that ruptures become strongly correlated in space and time leading to the spontaneous development of multifractal structures and gradually accumulate large displacements. The formation of the structures and the temporal variation of rupture activity is due to a complex interplay between the random structure, long-range elastic interactions and the threshold nature of rupture physics. The spontaneous formation of fractal fault structures by repeated earthquakes is mirrored at short times by the spatio-temporal chaotic dynamics of earthquakes, well-described by a Gutenberg-Richter power law. We also show that the fault structures can be understood as pure geometrical objects, namely minimal manifolds, which in two dimensions correspond to the random directed polymer (RDP) problem. This mapping allows us to use the results of many studies on the RDP in the field of statistical physics, where it is an exact result that the minimal random manifolds in 2D systems are self-affine with a roughness exponent 2/3. We also present results pertaining to the influence of the degree of stress release per earthquake on the competition between faults. Our results provide a rigorous framework from which to initiate rationalization of many, reported fractal fault studies.