To measure the size of an earthquake, we can use two physically different quantities: the radiated seismic energy \(E_{{\text{R}}}\) , from which seismic magnitude scales are defined, and the seismic moment \(M_{0}\) . The former depends on the dynamic process of earthquake faulting, whereas the latter does not. Nevertheless, if there exists a scaling relation between \(E_{{\text{R}}}\) and \(M_{0}\) , we can define a magnitude scale derived from the seismic moment. The widely accepted moment magnitude scale, \(M_{{\text{w}}} = {{(\log M_{0} - 9.1)} \mathord{\left/ {\vphantom {{(\log M_{0} - 9.1)} {1.5}}} \right. \kern-0pt} {1.5}}\) in MKS units, was established by substituting the scaling relation of \(E_{{\text{R}}} = 5 \times 10^{ - 5} M_{0}\) into the Gutenberg-Richter empirical energy–magnitude relation, \(\log E_{{\text{R}}} = 1.5M_{{\text{s}}} + 4.8\) . The above scaling relation comes from the energy–moment relation, \(E_{{\text{R}}} = ({{\Delta \tau } \mathord{\left/ {\vphantom {{\Delta \tau } {2\mu }}} \right. \kern-0pt} {2\mu }})M_{0}\) , based on a simplified energy balance equation in earthquake faulting with a uniform stress drop \(\Delta \tau\) . However, this energy–moment relation is a bit strange, because the right-hand side depends exclusively on the final static state of earthquake faulting, while the left-hand side must depend on the whole dynamic process of earthquake faulting. Theoretically, the radiated seismic energy \(E_{{\text{R}}}\) is expressed as a function of cumulative moment \(M_{0} (t)\) . In the case of a self-similar circular crack expanding at a constant rupture velocity \(v_{{\text{r}}}\) with a uniform stress drop \(\Delta \tau\) , the cumulative moment is calculated as \(M_{0} (t) = ({{16} \mathord{\left/ {\vphantom {{16} 7}} \right. \kern-0pt} 7})\Delta \tau v_{{\text{r}}}^{3} t^{3}\) . Substituting this expression into the theoretical formula of \(E_{{\text{R}}}\) , we obtain a simple energy–moment relation, \(E_{{\text{R}}} = ({{v_{{\text{r}}} } \mathord{\left/ {\vphantom {{v_{{\text{r}}} } {V_{{\text{S}}} }}} \right. \kern-0pt} {V_{{\text{S}}} }})^{3} ({{\Delta \tau } \mathord{\left/ {\vphantom {{\Delta \tau } \mu }} \right. \kern-0pt} \mu })M_{0}\) , which leads to the correction of the original moment magnitude scale \(M_{{\text{w}}}\) as \(M^{\prime}_{{\text{w}}} = M_{{\text{w}}} + 2\log ({{v_{{\text{r}}} } \mathord{\left/ {\vphantom {{v_{{\text{r}}} } {V_{{\text{S}}} }}} \right. \kern-0pt} {V_{{\text{S}}} }}) + 0.2\) .
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