A three-parameter formula for magnetude distribution of earthquakes.

A formula for magnitude distribution of earthquakeslog n(M)=a-bM+log(c-M)has an advantage over the well-known Gutenberg-Richter formula for it can represent a sharp decrease of frequency near the upper limit of magnitude c, which is recognized in some earthquake groups. In the range where M<<c, the above formula is practically the same as the G-R formula. A method for obtaining the maximum likelihood estimates of b and c in the above formula has been developed and the accuracy of the estimates has been investigated. The a-, b-, and c-values have been calculated for earthquakes in various regions of Japan and in the world. The c-value seems to exhibit regional variations. It is suggested that in some cases the spatial or temporal variations in b-value for the G-R formula or in the average magnitude M are interpretable as due to the variation in the upper limit of magnitude c.