Abstract To measure the size of an earthquake, we can use two physically different quantities: the radiated seismic energy $$E_{{\text{R}}}$$ E R , from which seismic magnitude scales are defined, and the seismic moment $$M_{0}$$ 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 $$E_{{\text{R}}}$$ E R and $$M_{0}$$ M 0 , we can define a magnitude scale derived from the seismic moment. The widely accepted moment magnitude scale, $$M_{{\text{w}}} = {{(\log M_{0} - 9.1)} \mathord{\left/ {\vphantom {{(\log M_{0} - 9.1)} {1.5}}} \right. \kern-0pt} {1.5}}$$ M w = ( log M 0 - 9.1 ) ( log M 0 - 9.1 ) 1.5 1.5 in MKS units, was established by substituting the scaling relation of $$E_{{\text{R}}} = 5 \times 10^{ - 5} M_{0}$$ E R = 5 × 10 - 5 M 0 into the Gutenberg-Richter empirical energy–magnitude relation, $$\log E_{{\text{R}}} = 1.5M_{{\text{s}}} + 4.8$$ log E R = 1.5 M s + 4.8 . The above scaling relation comes from the energy–moment relation, $$E_{{\text{R}}} = ({{\Delta \tau } \mathord{\left/ {\vphantom {{\Delta \tau } {2\mu }}} \right. \kern-0pt} {2\mu }})M_{0}$$ E R = ( Δ τ Δ τ 2 μ 2 μ ) M 0 , based on a simplified energy balance equation in earthquake faulting with a uniform stress drop $$\Delta \tau$$ Δ τ . 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 $$E_{{\text{R}}}$$ E R is expressed as a function of cumulative moment $$M_{0} (t)$$ M 0 ( t ) . In the case of a self-similar circular crack expanding at a constant rupture velocity $$v_{{\text{r}}}$$ v r with a uniform stress drop $$\Delta \tau$$ Δ τ , the cumulative moment is calculated as $$M_{0} (t) = ({{16} \mathord{\left/ {\vphantom {{16} 7}} \right. \kern-0pt} 7})\Delta \tau v_{{\text{r}}}^{3} t^{3}$$ M 0 ( t ) = ( 16 167 7 ) Δ τ v r 3 t 3 . Substituting this expression into the theoretical formula of $$E_{{\text{R}}}$$ E R , we obtain a simple energy–moment relation, $$E_{{\text{R}}} = ({{v_{{\text{r}}} } \mathord{\left/ {\vphantom {{v_{{\text{r}}} } {V_{{\text{S}}} }}} \right. \kern-0pt} {V_{{\text{S}}} }})^{3} ({{\Delta \tau } \mathord{\left/ {\vphantom {{\Delta \tau } \mu }} \right. \kern-0pt} \mu })M_{0}$$ E R = ( v r v r V S V S ) 3 ( Δ τ Δ τ μ μ ) M 0 , which leads to the correction of the original moment magnitude scale $$M_{{\text{w}}}$$ M w as $$M^{\prime}_{{\text{w}}} = M_{{\text{w}}} + 2\log ({{v_{{\text{r}}} } \mathord{\left/ {\vphantom {{v_{{\text{r}}} } {V_{{\text{S}}} }}} \right. \kern-0pt} {V_{{\text{S}}} }}) + 0.2$$ M w ′ = M w + 2 log ( v r v r V S V S ) + 0.2 . Graphical Abstract