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

The correlation of the scaled energy, ê = E_s/M₀, versus earthquake magnitude, M_s, is studied based on two models: (1) Model 1 based on the use of the time function of the average displacements, with a ω^-2 source spectrum, across a fault plane; and (2) Model 2 based on the use of the time function of the average displacements, with a ω^-3 source spectrum, across a fault plane. For the second model, there are two cases: (a) As τ ≒ T, where τ is the rise time and T the rupture time, lg(ê) ~ -M_s; and (b) As τ ≪ T, lg(ê) ~ -(1/2)M_s. The second model leads to a negative value of ê. 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 ê versus M_s.