Physical basis of earthquake magnitudes: an extreme value of seismic amplitudes from incoherent fracture of random fault patches

A relation between surface-wave magnitude M_s and fault area S for great earthquakes has been proposed

log S 2M_{s ? 11.5 (}M_{s > 7.5)}

A similar formula has been also derived for body-wave magnitude m_b^* redetermined from maximum amplitudes of short-period P-wavetrains

log S ∝ 1.7 m_b^*

These are quite different from a theoretical relation expected on the basis of long-wave approximation. Because wavelengths of seismic waves used for the above magnitudes are very short compared to the size of earthquake sources, these relations represent the short-period nature of the earthquake process. The statistical theory of extreme values has been applied to understand the relations considering that the component waves which constitute the wavetrains for m_b^* and M_s determinations originate from the random fracture of fault heterogeneities.