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Seismological outliers: L₁ or adaptive L_p norm application

ISS International Ltd., P.O. Box 2083, Welkom 9460 Republic of South Africa

Abstract

In the early 1930s, Jeffreys recognized that the arrival time errors do not follow the Gaussian distribution required by least-squares (L₂ norm) location techniques. Large errors in the arrival times can displace the least-squares estimate of the hypocenter far away from the true one. As an alternative to the least-squares procedure, the minimization of the sum of absolute residuals (L₁ norm), as protection against the effects of outliers, is often used. In this note, a essential modification of inversion procedures is proposed which incorporates an adaptive algorithm for L_p norm estimation. A detailed study by numerical simulation demonstrates that in the presence of outlying observations, an L_p norm procedure can select the proper value of p, not necessarily equal to 1. The advantage of this procedure is that no a priori decision to use L₁ norm is to be made, but instead the data prescribes the most appropriate value of p.

In on-line seismic networks where the system is fully automated, and where there is no chance for manual intervention, it is suggested that the adaptive L_p norm offers the most reliable method for data processing.

Footnotes

^* Correspondence to: Western Deep Level Mine, Seismic Systems, P.O. Box 8044, Western Levels 2501, Republic of South Africa.