Rank-ordering statistics of extreme events: Application to the distribution of large earthquakes

Rank-ordering statistics provide 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 worldwide (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 b(2) = 2.3 +/- 0.3 for large shallow earthquakes and b(1) = 1.00 +/- 0.02 for smaller shallow earthquakes. However, the crossover magnitude between the two distributions is ill defined. The data available at present do not provide a strong constraint on the crossover 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 worldwide is ill defined at best. We caution that there is no direct evidence to confirm the hypothesis that the large-moment branch is indeed a power law. In fact, a gamma distribution fits the entire suite of earthquake moments from the smallest to the largest satisfactorily. There is no evidence that the earthquakes of the southern California catalog have a distribution with two branches or that a rolloff in the distribution is needed; for this catalog, b = 1.00 +/- 0.02 up to the largest magnitude observed, M(W) similar or equal to 7.5; hence we conclude that the thickness of the seismogenic layer has no observable influence whatsoever on the frequency distribution in this region.