For a general use of the frequency-magnitude (FM) relation in seismic risk assessment, we formulate a multi-scale approach that relies on the hypothesis that only the ensemble of events that are geometrically small, compared with the elements of the seismotectonic regionalization, can be described by a log-linear FM relation. It follows that the seismic zonation must be performed at several scales, depending upon the self-similarity conditions of the seismic events and the linearity of the log FM relation, in the magnitude range of interest. The analysis of worldwide seismicity, using the Harvard catalog, where the seismic moment is recorded as the earthquake size, corroborates the idea that a single FM relation is not universally applicable. The multi-scale model of the FM relation is tested in the Italian region.

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