On the saturation of earthquake magnitudes

As MS, mb, or unified magnitude approaches 8, there is an abrupt change in the rate at which frequency of occurrence decreases with magnitude. This change is not observed for moment-magnitude, MW. Data on past earthquakes are insufficient to prove whether there is a finite upper limit on MS, mb, or unified magnitude or whether earthquake probability is better described by an unlimited logarithmic relation of the form M = E + F ( I n ( G − I n ( − I n probability ) ) ) . This empirical equation, however, fits the world data best. If there is an upper limit on size, it is approximately 8.1 for mb, 8.7 for MS, and 8.9 for unified magnitude. For regions such as the contiguous United States and Alaska, there are too few large earthquakes for saturation to be clearly observed, but the data at large magnitudes fit an empirical equation of shape M = A + ( M max − A ) tanh ( C X − D ) better than either the Gumbel I or III theory. This equation provides for a more sudden saturation than Gumbel's theory allows.