Rank-ordering Statistics of Extreme Events: Application to the Distribution of Large Earthquakes

Rank-ordering statistics provides a perspective on the rare, largest elements of a population, whereas the statistics of cumulative distributions are dominated by the more numerous small events. The exponent of a power law distribution can be determined with good accuracy by rank-ordering statistics from the observation of only a few tens of the largest events. Using analytical results and synthetic tests, we quantify the systematic and the random errors. We also study the case of a distribution defined by two branches, each having a power law distribution, one defined for the largest events and the other for smaller events, with application to the World-Wide (Harvard) and Southern California earthquake catalogs. In the case of the Harvard moment catalog, we make more precise earlier claims of the existence of a transition of the earthquake magnitude distribution between small and large earthquakes; the b-values are b2 = 2.3 ± 0.3 for large shallow earthquakes and b1 = 1.00 ± 0.02 for smaller shallow earthquakes. However, the cross-over magnitude between the two distributions is ill-defined. The data available at present do not provide a strong constraint on the cross-over which has a 50% probability of being between magnitudes 7.1 and 7.6 for shallow earthquakes; this interval may be too conservatively estimated. Thus, any influence of a universal geometry of rupture on the distribution of earthquakes world-wide is ill-defined at best. We caution that there is no direct evidence to confirm the hypothesis that the