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 forumla (Nakamura, 1988; Allen and Kanamori, 2003) and find that there is a trend between forumla 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 these 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 stabilizing the predominant period function in the transition between noise and signal. We show that forumla has an improved relationship to magnitude, with the average coefficient of determination (R2) increasing from 0.15 for forumla to 0.5 for forumla. 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 forumla versus magnitude relationship is superior to that of τc. Analyzing 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 from being selected. We attempt to optimize the forumla and τc results, and find that although the results improve, they are still significantly worse than for forumla. The forumla performance is shown to be robust and less dependent on the choice of parameters than forumla. We then apply forumla and forumla to estimating magnitudes. Average errors are 20% smaller for forumla estimates compared with optimized forumla results, with greater improvement for unoptimized parameters. We conclude that the performance of forumla is superior to forumla and τc and should be considered for real-time magnitude estimation.

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