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Authors:  Lombardi, A. M.^{*} 
Title:  Probabilistic interpretationof «Bath's Law» 
Issue Date:  2002 
Series/Report no.:  45 (34) 
URI:  http://hdl.handle.net/2122/704 
Keywords:  magnitude distribution cluster size bvalue order statistics 
Abstract:  Assuming that, in a catalog, all the earthquakes with magnitude larger than or equal to a cutoff magnitude M c follow the GutenbergRichter Law, the compatibility of this hypothesis with «Bath’s Law» is examined. Considering the mainshock M 0 and the largest aftershock M 1 of a sequence respectively as the first and the second largest order statistic of a sample of independent and identically distributed exponential random variables, the distribution of M 0 , M 1 and of their difference D 1 is evaluated. In particular, it is analyzed as the distribution of D 1 changes when only the sequences with the magnitude of the mainshock above a second threshold M c*•M c are considered. It results that the distributions of M 0 , M 1 and D 1 depend on the difference M c*•M c and on the number of events in the sequence. Moreover, the expected value of D 1 increases with increasing of M c*•M c for every value of N. Then it is shown that «Bath’s Law» could be ascribed to selection of data caused by the two thresholds M c and M c* and that it has a qualitative agreement with the model proposed.
Key words Assuming that, in a catalog, all the earthquakes with magnitude larger than or equal to a cutoff magnitude M c follow the GutenbergRichter Law, the compatibility of this hypothesis with «Bath’s Law» is examined. Considering the mainshock M 0 and the largest aftershock M 1 of a sequence respectively as the first and the second largest order statistic of a sample of independent and identically distributed exponential random variables, the distribution of M 0 , M 1 and of their difference D 1 is evaluated. In particular, it is analyzed as the distribution of D 1 changes when only the sequences with the magnitude of the mainshock above a second threshold M c*•M c are considered. It results that the distributions of M 0 , M 1 and D 1 depend on the difference M c*•M c and on the number of events in the sequence. Moreover, the expected value of D 1 increases with increasing of M c*•M c for every value of N. Then it is shown that «Bath’s Law» could be ascribed to selection of data caused by the two thresholds M c and M c* and that it has a qualitative agreement with the model propose. 
Appears in Collections:  04.05.08. Instruments and techniques 05.05.99. General or miscellaneous Annals of Geophysics

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