Tpd, a Damped Predominant Period Function with Improvements for Magnitude Estimation

Abstract We use an aftershock dataset of over 1500 events ( M L 0.7–5.8) to study the relationship between magnitude and the predominant period calculated from the initial P -wave arrival. We calculate (Nakamura, 1988; Allen and Kanamori, 2003) and find that there is a trend between and magnitude, as reported by previous authors. However, the trend is weaker than expected. We calculate an alternative predominant period function, τ c (Kanamori, 2005), and find virtually no relationship to magnitude for these data. We therefore implement a modified, damped version of the T p function, which we term T pd . The T pd function introduces an additional term, D s , aimed at stabilizing the predominant period function in the transition between noise and signal. We show that has an improved relationship to magnitude, with the average coefficient of determination ( R 2 ) increasing from 0.15 for to 0.5 for . This improvement is consistent for all stations. We then apply the T pd function to the displacement waveforms, calling the associated function T pd _ D . The trend in the versus magnitude relationship is superior to that of τ c . Analyzing the T pd 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 from being selected. We attempt to optimize the and τ c results, and find that although the results improve, they are still significantly worse than for . The performance is shown to be robust and less dependent on the choice of parameters than . We then apply and to estimating magnitudes. Average errors are 20% smaller for estimates compared with optimized results, with greater improvement for unoptimized parameters. We conclude that the performance of is superior to and τ c and should be considered for real-time magnitude estimation.