Maximum-likelihood event magnitude estimation with bootstrapping for uncertainty estimation

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Abstract
The bootstrap procedure of Efron may be used with maximum-likelihood magnitude estimation to estimate the standard deviation of the event magnitude and the distribution standard deviation. The procedure resamples the observations randomly with replacement. The event magnitude, m, and station magnitude distribution, σ, are then estimated for each random sampling of the observations. This generates a sequence of event magnitude estimations that are used to estimate the event magnitude standard error, σm, and the uncertainty in the distribution, σσ. Maximum-likelihood mb event magnitudes with uncertainties are provided for events at NTS, Amchitka, Tuamotu, Novaya Zemlya, and Eastern Kazakh. These magnitudes based on WWSSN film chip readings illustrate the importance of nondetection for events with magnitudes below mb < 5 and of clipping for events with magnitudes above mb > 6.5. The uncertainties in the event magnitudes are found to be close to the uncertainty in the mean of the observed signals. The introduction of the maximum-likelihood procedure does not significantly improve the precision of the event magnitude estimate, and furthermore it may actually increase the estimated uncertainty with introduction of censoring information. However, the maximum-likelihood bootstrap estimate is a more accurate estimate of the total uncertainty in the event magnitude.

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