Truncated exponential frequency-magnitude relationship in earthquake statistics

The following doubly truncated exponential probability density distribution f ( M ) = β exp ⁡ { − β ( M − M 0 ) } / { 1 − exp ⁡ { − β ( M p − M 0 ) } } for M 0 ≦ M ≦ M p f ( M ) = 0 for M ≧ M p , ( a ) where M0 is the threshold magnitude value, and β is a parameter, is proposed for the earthquake occurrence. The relation (a) has been obtained carrying out a simple model based on a number of assumptions, among which the more characterizing is the existence of a maximum regional finite magnitude value Mp. This assumption, derived by an evidence recognized by most seismologists, allows a simple explanation of the known behavior of the experimental cumulative frequency-magnitude graphs. In order to estimate β and Mp the moments method is suggested, which also represents a maximum likelihood method for β estimation. Finally, some results of application of the model to six seismic regions are presented.