Scaling relations for strong ground motion prediction models with M2 terms

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.