A theoretical study of correlation between scaled energy and earthquake magnitude based on two source displacement models

The correlation of the scaled energy, <i<ê</i< = <i<E</i<<sub<s</sub</<i<M</i<<sub<0</sub<, versus earthquake magnitude, <i<M</i<<sub<s</sub<, is studied based on two models: (1) Model 1 based on the use of the time function of the average displacements, with a <i<ω</i<<sup<-2</sup< source spectrum, across a fault plane; and (2) Model 2 based on the use of the time function of the average displacements, with a <i<ω</i<<sup<-3</sup< source spectrum, across a fault plane. For the second model, there are two cases: (a) As <i<τ</i< ≒ <i<T</i<, where <i<τ</i< is the rise time and <i<T</i< the rupture time, lg(<i<ê</i<) ~ -<i<M</i<<sub<s</sub<; and (b) As <i<τ</i< ≪ <i<T</i<, lg(<i<ê</i<) ~ -(1/2)<i<M</i<<sub<s</sub<. The second model leads to a negative value of <i<ê</i<. This means that Model 2 cannot work for studying the present problem. The results obtained from Model 1 suggest that the source model is a factor, yet not a unique one, in controlling the correlation of <i<ê</i< versus <i<M</i<<sub<s</sub<.