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Iterative network magnitude estimation and uncertainty assessment with noisy and clipped data

TELEDYNE GEOTECH ALEXANDRIA LABORATORIES, 314 MONTGOMERY STREET, ALEXANDRIA, VIRGINIA 22341-1581
DIVISION OF STATISTICS UNIVERSITY OF CALIFORNIA, DAVIS, CALIFORNIA 95616

Abstract

We briefly discuss the similarities and differences between two iterative estimators that are suitable for the network m_b estimation problem, namely a modification of the Iterative Least-Squares method (ILS) developed by Schmee and Hahn (1979) and the Maximum-Likelihood Estimator (MLE). Both methods reduce to the usual Least-Squares Multiple Factors (LSMF) method when the censored data are deleted from the network observational data. For censored case, the standard deviation () of the obscuring noise has to be solved through iteration along with the event magnitudes and the station corrections. An extra constraint on is necessary to determine which optimal estimation scheme is of interest. The final value of for each iterative scheme can be used as a good approximation to the unbiased estimate of the standard deviation of the perturbing noise. By scaling this value by the square root of the number of observations associated with each unknown parameter, the uncertainty in each estimated parameter can be approximated efficiently. These error estimates seem to differ from the unbiased standard errors only by a common multiplying constant across all stations and all event m_b s.

The bootstrap method is reviewed and adapted to the case of multivariate estimation with doubly censored data. The Monte Carlo resampling is carried out among the collection of residuals instead of the observational data. The pool of residuals is enlarged to include all censored residuals for random drawing. The bootstrap result confirms the aforementioned scaling relationship between the individual error estimates and the global of the perturbing noise. As a result, the bootstrap/jackknife technique might not be really worth the considerable computational effort they require on this specific application.

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