Distribution of Maximum Earthquake Magnitudes in Future Time Intervals, Application to the Seismicity of Japan (1923-2007)

We modify the new method for the statistical estimation of the tail distribution of earthquake seismic moments introduced by Pisarenko et al. [2009] and apply it to the earthquake catalog of Japan (1923-2007). The method is based on the two main limit theorems of the theory of extreme values and on the derived duality between the Generalized Pareto Distribution (GPD) and Generalized Extreme Value distribution (GEV). We obtain the distribution of maximum earthquake magnitudes in future time intervals of arbitrary duration tau. This distribution can be characterized by its quantile Qq(tau) at any desirable statistical level q. The quantile Qq(tau) provides a much more stable and robust characteristic than the traditional absolute maximum magnitude Mmax (Mmax can be obtained as the limit of Qq(tau) as q tends to 1, and tau tends to infinity). The best estimates of the parameters governing the distribution of Qq(tay) for Japan (1923-2007) are the following: Form parameter for GEV = -0.1901 +- 0.0717; position parameter GEV(tau=200)= 6.3387 +- 0.0380; spread parameter for GEV(tau=200)= 0.5995 +- 0.0223; Q_0.90,GEV(tau=10)= 8.34 +- 0.32. We also estimate Qq(tau) for a set of q-values and future time periods in the range for tau between 1 and 50 years from 2007. For comparison, the absolute maximum estimate Mmax from GEV, which is equal to 9.57 +- 0.86, has a scatter more than twice that of the 90 percent quantile Q_{0.90,GEV}(tau=10) of the maximum magnitude over the next 10 years counted from 2007.Comment: 15 pages + 10 figure

On the Correct Use of Statistical Tests: Reply to "Lies, damned lies and statistics (in Geology)"

In a Forum published in EOS Transactions AGU (2009) entitled "Lies, damned lies and statistics (in Geology)", Vermeesch (2009) claims that "statistical significant is not the same as geological significant", in other words, statistical tests may be misleading. In complete contradiction, we affirm that statistical tests are always informative. We detail the several mistakes of Vermeesch in his initial paper and in his comments to our reply. The present text is developed in the hope that it can serve as an illuminating pedagogical exercise for students and lecturers to learn more about the subtleties, richness and power of the science of statistics.Comment: 7 pages and 1 figure: This text expands considerably the short text published under the same title in Eos Transactions AGU, Vol. 92, No. 8, page 64, 22 February 201

Rigorous statistical detection and characterization of a deviation from the Gutenberg-Richter distribution above magnitude 8 in subduction zones

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. The bootstrap 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 bootstrap 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 mW=8.1 +- 0.3. For separate subduction zones, the corresponding estimates are much 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 in the past. 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.Comment: pdf document of 40 pages including 5 tables and 19 figure

Characterization of the frequency of extreme events by the Generalized Pareto Distribution

Based on recent results in extreme value theory, we use a new technique for the statistical estimation of distribution tails. Specifically, we use the Gnedenko-Pickands-Balkema-de Haan theorem, which gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD). Useful in finance, insurance, hydrology, we investigate here the earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law and analyze shallow earthquakes (depth h < 70 km) in the Harvard catalog over the period 1977-2000 in 18 seismic zones. The whole GPD is found to approximate the tails of the seismic moment distributions quite well above moment-magnitudes larger than mW=5.3 and no statistically significant regional difference is found for subduction and transform seismic zones. We confirm that the b-value is very different in mid-ocean ridges compared to other zones (b=1.50=B10.09 versus b=1.00=B10.05 corresponding to a power law exponent close to 1 versus 2/3) with a very high statistical confidence. We propose a physical mechanism for this, contrasting slow healing ruptures in mid-ocean ridges with fast healing ruptures in other zones. Deviations from the GPD at the very end of the tail are detected in the sample containing earthquakes from all major subduction zones (sample size of 4985 events). We propose a new statistical test of significance of such deviations based on the bootstrap method. The number of events deviating from the tails of GPD in the studied data sets (15-20 at most) is not sufficient for determining the functional form of those deviations. Thus, it is practically impossible to give preference to one of the previously suggested parametric families describing the ends of tails of seismic moment distributions.Comment: pdf document of 21 pages + 2 tables + 20 figures (ps format) + one file giving the regionalizatio

Robust statistical tests of Dragon-Kings beyond power law distributions

We ask the question whether it is possible to diagnose the existence of "Dragon-Kings” (DK), namely anomalous observations compared to a power law background distribution of event sizes. We present two new statistical tests, the U-test and the DK-test, aimed at identifying the existence of even a single anomalous event in the tail of the distribution of just a few tens of observations. The DK-test in particular is derived such that the p-value of its statistic is independent of the exponent characterizing the null hypothesis, which can use an exponential or power law distribution. We demonstrate how to apply these two tests on the distributions of cities and of agglomerations in a number of countries. We find the following evidence for Dragon-Kings: London in the distribution of city sizes of Great Britain; Moscow and St-Petersburg in the distribution of city sizes in the Russian Federation; and Paris in the distribution of agglomeration sizes in France. True negatives are also reported, for instance the absence of Dragon-Kings in the distribution of cities in German

Vere-Jones' Self-Similar Branching Model

Motivated by its potential application to earthquake statistics, we study the exactly self-similar branching process introduced recently by Vere-Jones, which extends the ETAS class of conditional branching point-processes of triggered seismicity. One of the main ingredient of Vere-Jones' model is that the power law distribution of magnitudes m' of daughters of first-generation of a mother of magnitude m has two branches m'm with exponent beta+d, where beta and d are two positive parameters. We predict that the distribution of magnitudes of events triggered by a mother of magnitude $m$ over all generations has also two branches m'm with exponent beta+h, with h= d \sqrt{1-s}, where s is the fraction of triggered events. This corresponds to a renormalization of the exponent d into h by the hierarchy of successive generations of triggered events. The empirical absence of such two-branched distributions implies, if this model is seriously considered, that the earth is close to criticality (s close to 1) so that beta - h \approx \beta + h \approx \beta. We also find that, for a significant part of the parameter space, the distribution of magnitudes over a full catalog summed over an average steady flow of spontaneous sources (immigrants) reproduces the distribution of the spontaneous sources and is blind to the exponents beta, d of the distribution of triggered events.Comment: 13 page + 3 eps figure

Investigation of cosmic radiation on the AMS ''Luna-10''

Flux of primary cosmic radiation from Luna-1