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Maximum entropy and the magnitude distribution

DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF CANTERBURY, CHRISTCHURCH, New Zealand

Abstract

The principle of maximum entropy is used to determine the distribution of earthquake magnitudes. Given no upper bound to magnitude, the maximum entropy principle generates the Gutenberg and Richter recurrence relation. When an upper bound m₁ is assumed, the principle leads to the truncated exponential distribution proposed by Cornell and Vanmarcke (1969),

in which N denotes the number of earthquakes with magnitude equal to or greater than M, and T is the total number of earthquakes with nonnegative magnitudes. This relationship agrees with observed data both at small magnitudes, where it coincides with the Gutenberg and Richter relation, and at large magnitudes, where it fits the data remarkably well. Further, given only the mean rate of occurrence, the principle leads to a Poisson temporal distribution of earthquakes.

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