A modified formula for frequency-magnitude distribution

A modification of the well known Gutenberg-Richter formula leads to the relation $$N(M) = - A_1 + A_2 exp ( - A_3 M)$$ whereN(M) is the cumulative exceedence frequency of an event of magnitudeM. Therefore, from the above generalized formula is concluded that the log cumulative frequency-magnitude relation is not linear. This conclusion is valid for the ranges of magnitudes in contradiction with the classical Gutenberg-Richter relationship. The proposed distribution is bounded to the right. The upper bound is probably related to the maximum magnitude of a certain region or to the maximum magnitude of a given earthquake sample. The values of the upper bound, estimated for numerous samples, are in good agreement with the corresponding values estimated using the third type asymptotic distribution of Gumbel's extreme value statistics. The modified formula of the cumulative frequency-magnitude relation was tested in numerous samples of seismological data from the major area of Greece. The modified relation fits very well at low and high magnitudes range from all earthquake samples where it has been tested.