Robust inversion using biweight norm and its application to seismic inversion

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Contents Vol 43(2)

doi:10.1071/EG12014

The Bulletin of the Australian Society of Exploration Geophysicists

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Robust inversion using biweight norm and its application to seismic inversion

Jun Ji

Hansung University, 389 Samsun-dong 2-ga, Seongbuk-gu, Seoul 136-792, Korea. Email: junji64@gmail.com

Exploration Geophysics 43(2) 70-76 http://dx.doi.org/10.1071/EG12014
Submitted: 15 February 2012 Accepted: 23 February 2012 Published: 28 March 2012





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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.

References

Beaton, A. E., and Tukey, J. W., 1974, The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data: Technometrics, 16, 147–185
| CrossRef |

Bube, K. P., and Langan, R. T., 1997, Hybrid l ₁/l ₂ minimization with applications to tomography: Geophysics, 62, 1183–1195
| CrossRef |

Chapman, N. R., and Barrodale, I., 1983, Deconvolution of marine seismic data using the l ₁ norm: Geophysical Journal of the Royal Astronomical Society, 72, 93–100
| CrossRef |

Claerbout, J. F., 1992, Earth soundings analysis, processing versus inversion: Blackwell Scientific Publication.

Claerbout, J. F., 2009, Blocky models via the l ₁/l ₂ hybrid norm: SEP-Report, 139, 1–10

Claerbout, J. F., and Muir, F., 1973, Robust modeling with erratic data: Geophysics, 38, 826–844
| CrossRef |

Daubechies, I., Devore, R., Fornasier, M., and Gunturk, C. S., 2010, Iteratively reweighted least squares minimization for sparse recovery: Communications on Pure and Applied Mathematics, 63, 1–38
| CrossRef |

Fomel, S., and Claerbout, J. F., 1995, Searching the Sea of Galilee: the splendors and miseries of iteratively reweighted least squares: SEP-Report, 84, 259–270

Guitton, A., and Claerbout, J., 2004, Interpolation of bathymetry data from the Sea of Galilee: a noise attenuation problem: Geophysics, 69, 608–616
| CrossRef |

Guitton, A., and Symes, W., 2003, Robust inversion of seismic data using the Huber norm: Geophysics, 68, 1310–1319
| CrossRef |

Holland, P. W., and Welsch, R. E., 1977, Robust regression using iteratively reweighted least-squares: Communications in Statistics Theory and Methods, 6, 813–827
| CrossRef |

Huber, P. J., 1981, Robust statistics: John Wiley & Sons.

Ji, J., 2006, CG method for robust inversion and its application to velocity stack inversion: Geophysics, 71, R59–R67
| CrossRef |

Li, Y., Zhang, Y., and Claerbout, J., 2010, Geophysical applications of a novel and robust l ₁ solver: SEG Technical Program Expanded Abstracts, 29, 3519–3523
| CrossRef |

Sacchi, M. D., and Ulrych, T. J., 1995, High-resolution velocity gathers and offset space reconstruction: Geophysics, 60, 1169–1177
| CrossRef |

Scales, J. A., and Gersztenkorn, A., 1988, Robust methods in inverse theory: Inverse Problems, 4, 1071–1091
| CrossRef |

Taylor, H. L., Banks, S. C., and McCoy, J. F., 1979, Deconvolution with the L-one norm: Geophysics, 44, 39–52
| CrossRef |

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