In most previous studies on the attenuation relation of strong ground motion, the increase in amplitude with earthquake magnitude was expressed by a linear relation. However, if the ω−2 source model is assumed, the corner period of the spectrum varies with the magnitude, and the scaling law of the source spectrum amplitude becomes a complicated function of magnitude. The scaling relation of the spectrum amplitude with MW can be simply approximated as a quadratic function, and the coefficient of the MW2 term should be negative. On the contrary, positive coefficients of the ML2, mbLg2, and MJ2 (MJ: magnitude of Japan Meteorological Agency) terms have been derived by regression analyses of strong ground motions.
Based on the ω−2 model, semi-empirical relations between Mo and the magnitudes were derived. Further, the seismological scaling law of the source spectrum obtained from the relation between Mo and the magnitudes was approximated as a quadratic function of the magnitude, and positive coefficients of the squared magnitude terms were obtained. These results stem from the definitions of magnitude derived from amplitudes of seismograms around a specific period of 1.0 sec for ML and mbLg and of 5.0 sec for MJ.