A Poisson model for identifying characteristic size effects in frequency data: Application to frequency-size distributions for global earthquakes, "starquakes", and fault lengths

The standard Gaussian distribution for incremental frequency data requires a constant variance which is independent of the mean. We develop a more gen eral and appropriate method based on the Poisson distribution, which assume s different unknown variances for the frequencies, equal to the means. We e xplicitly include "empty bins", and our method is quite insensitive to the choice of bin width. We develop a maximum likelihood technique that minimiz es bias in the curve fits, and penalizes additional free parameters by obje ctive information criteria. Various data sets are used to test three differ ent physical models that have been suggested for the density distribution: the power law; the double power law; and the "gamma" distribution. For the CMT catalog of global earthquakes, two peaks in the posterior distribution are observed at moment magnitudes m* = 6.4 and 6.9 implying a bimodal distr ibution of seismogenic depth at around 15 and 30 km, respectively. A simila r break at a characteristic length of 60 km or so is observed in moment-len gth data, but this does not outperform the simpler power law model. For the earthquake frequency-moment data the gamma distribution provides the best overall fit to the data, implying a finite correlation length and a system near but below the critical point. In contrast, data from soft gamma ray re peaters show that the power law is the best fit, implying infinite correlat ion length and a system that is precisely critical. For the fault break dat a a significant break of slope is found instead at characteristic scale of 44 km, implying a typical seismogenic thickness of up to 22 km or so in wes t central Nevada. The exponent changes from 1.5 to -2.1, too large to be ac counted for by changes in sampling for an ideal, isotropic fractal set.