Tpd, a damped predominant period function with improvements for magnitude estimation

Quantifying the Magnitude of an earthquake has been a fundamental endeavour in seismology since its beginnings as a science. Nevertheless, it can sometimes be hours before traditional methods of calculating magnitudes reach a robust value. It is now recognised, whether for early warning measures and planning civil response, or simply for public interest, that it is useful to have the earliest possible measures of earthquake magnitudes after the occurrence of large earthquakes. Clearly, such methods must concentrate on information in the first P-wave arrivals, rather than waiting for later seismic phases. In recent years, a great deal of interest has been stimulated by attempts to use a time-domain estimate of the predominant period of a waveform, to estimate seismic magnitude from the very first P-wave arrivals in seismograms. We use an aftershock dataset of over 1500 events (ML = 0.7-5.8), to study the relationship between magnitude, and the predominant period calculated from the initial P-wave arrival. We calculate TpMax (Nakamura, 1988; Allen and Kanamori, 2003), and find that there is a trend between TpMax and magnitude, as reported by previous authors. However, the trend is weaker than expected. We calculate an alternative predominant period function, τc (Kanamori, 2005), and find virtually no relationship to magnitude for this data. We therefore implement a modified, damped version of the Tp function, which we term Tpd. The Tpd function introduces an additional term, Ds, aimed at stabilising the predominant period function in the transition between noise and signal. We show that TpdMax has an improved relationship to magnitude, with the average coefficient of determination (R2) increasing from 0.15 for TpMax to 0.5 for TpdMax. This improvement is consistent for all stations. We then apply the Tpd function to the displacement waveforms, calling the associated function Tpd_D. The trend in the Tpd_DMax vs. magnitude relationship is superior to that of τc. Analysing the Tpd function, we conclude that improvements result from damping large values in the noise region, or reducing spikes during the noise-to-signal transition, thus preventing incorrect maxima being selected. We conclude that the performance of TpdMax is superior to TpMax and τc, and should be considered for real-time magnitude estimation.