Scaling law of seismic spectrum

The dependence of the amplitude spectrum of seismic waves on source size is investigated on the basis of two dislocation models of an earthquake source. One of the models (by N. Haskell) is called the ω³ model, and the other, called the ω² model, is constructed by fitting an exponentially decaying function to the autocorrelation function of the dislocation velocity. The number of source parameters is reduced to one by the assumption of similarity. We found that the most convenient parameter for our purpose is the magnitude Ms, defined for surface waves with period of 20 sec. Spectral density curves are determined for given Ms. Comparison of the theoretical curves with observations is made in two different ways. The observed ratios of the spectra of seismic waves with the same propagation path but from earthquakes of different sizes are compared with the corresponding theoretical ratios, thereby eliminating the effect of propagation on the spectrum. The other method is to check the theory with the empirical relation between different magnitude scales defined for different waves at different periods. The ω² model gives a satisfactory agreement with such observations on the assumption of similarity, but the ω³ model does not. We find, however, some indications of departure from similarity. The efficiency of seismic radiation seems to increase with decreasing magnitude if the Gutenberg-Richter magnitude-energy relation is valid. The assumption of similarity implies a constant stress drop independent of source size. A preliminary study of Love waves from the Parkfield earthquake of June 28, 1966, shows that the stress drop at the source of this earthquake is lower than the normal value (around 100 bars) by about 2 orders of magnitude.