Robust inversion using biweight norm and its application to seismic inversion
Jun JiHansung University, 389 Samsun-dong 2-ga, Seongbuk-gu, Seoul 136-792, Korea. Email: junji64@gmail.com
Exploration Geophysics 43(2) 70-76 https://doi.org/10.1071/EG12014
Submitted: 15 February 2012 Accepted: 23 February 2012 Published: 28 March 2012
Abstract
In spite of some minor drawbacks such as nonuniqueness and higher computational cost, finding the least-absolute (l1 norm) error solution to solve an optimisation problem is mostly known to give a better answer than the classical least-squares (l2 norm) method. This is because the robust property of the median value is affected little by outlier values and the solution of the least l1 norm error corresponds to the solution of minimum median error. Several variants of the l1 norm such as the Huber norm and the Hybrid norm have the same robust properties as the l1 norm. The optimisation methods based on l1 norm obtain their robustness by reducing the influence of outliers significantly, although never ignoring it. Therefore, if the proportion of outliers increases, most of the methods based on l1 norm may begin to be affected by the outliers. In such a case, other types of robust measures such as Tukey’s Biweight (Bisquare weight) norm, which excludes outliers in computing the misfit measure, could perform better. This paper describes the application of the Biweight norm using the IRLS (iteratively reweighted least-squares) method as a robust inversion and shows its possible improvement in robustness when dealing with data having many outliers.
Key words: biweight norm, IRLS, l1 norm, robust inversion, robust norm, weighted least-squares.
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