Robust earthquake location using M-estimates

Abstract

In the 1930's, Jeffreys recognized that large errors among the arrival times used to locate an earthquake can move the least-squares estimate of the epicenter far from the true epicenter. This has often been remedied by using M-estimates of epicentral location that iteratively weight each equation of condition by a function of its residual size. Six estimators, least absolute value, least squares, bisquare, Huber, Jeffreys, and sine, are compared for robustness when applied to the earthquake location problem. Although M-estimates are more robust than least squares, they are sensitive to the initial location estimate, the scale of the residuals, and the geometry of the recording network. A detailed study of one earthquake located by both least squares and the bisquare method is reported. This shows that when using least squares, error at one station can inflate the residuals at other stations so that a true outlier may be hidden. Also, error at influential stations can go completely unnoticed by either method.