Geophysics and Geophysical Exploration (지구물리와물리탐사)

Korean Society of Earth and Exploration Geophysicists (한국지구물리·물리탐사학회)
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An Efficient Implementation of Hybrid $l^1/l^2$ Norm IRLS Method as a Robust Inversion

강인한 역산으로서의 하이브리드 $l^1/l^2$ norm IRLS 방법의 효율적 구현기법

  • Ji, Jun (Department of Information System Engineering, Hansung University)
  • 지준 (한성대학교 정보시스템공학과)
  • Published : 2007.05.31
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Abstract

Least squares ($l^2$ norm) solutions of seismic inversion tend to be very sensitive to data points with large errors. The $l^1$ norm minimization gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) method gives efficient approximate solutions of these $l^1$ norm problems. I propose an efficient implementation of the IRLS method for a hybrid $l^1/l^2$ minimization problem that behaves as $l^2$ norm fit for small residual and $l^1$ norm fit for large residuals. The proposed algorithm shows more robust characteristics to the decision of the threshold value than the l1 norm IRLS inversion does with respect to the threshold value to avoid singularity.

탄성파 역산에 있어서 가장 널리 사용되는 최소자승($l^2$ norm)해는 이상치(outlier)에 매우 민감하게 반응하는 경향이 있다. 이에 반해서 $l^1$ norm을 최소화하는 해는 이상치에 강인한 면을 보이나 일반적으로 좀 더 많은 계산이 필요하다. 반복적가중의 최소자승법(Iteratively reweighted least squares [IRLS] method)을 이용하면 이러한 $l^1$ norm 문제의 근사해(approximate solution)를 효율적으로 구할 수 있다. 본 논문에서는 작은 크기의 잔여분은 $l^2$ norm으로 처리하며, 큰 크기의 잔여분은 $l^1$ norm으로 처리하는 하이브리드 $l^1/l^2$ norm 최소화를 IRLS 방법에 쉽게 적용하는 구현 기법을 소개한다. 소개된 알고리즘은 특이치(singularity)처리를 위한 임계값의 결정에 민감하게 반응하는 기존의 $l^1$ norm IRLS 방법과는 달리 임계값 결정에 상관없이 늘 강인한 역산의 특성을 보여준다.

Keywords

References

  1. Bube, K. P., and Langan, R. T., 1997, Hybrid $l^{1}/l^{2}$ minimization with applications to tomography, Geophysics, 62, 1183-1195 https://doi.org/10.1190/1.1444219
  2. Claerbout, J., and Muir, F., 1973, Robust modeling with erratic data, Geophysics, 38, 826-844 https://doi.org/10.1190/1.1440378
  3. Claerbout, J., 1992, Earth sounding analysis: Processing versus inversion, Blackwell Scientific Publ. Inc
  4. Guitton, A., and Symes, W. S., 2003, Robust inversion of seismic data using the Huber norm, Geophysics, 68, 1310-1319 https://doi.org/10.1190/1.1598124
  5. Hampson, D., 1986, Inverse velocity stacking for multiple elimination, J. Can. Soc. Expl. Geophysics., 22, 44-45
  6. Ji, J., 2006, Conjugate guided gradient (CGG) method for robust inversion and its application to velocity-stack inversion, Geophysics, 72, R59-R67
  7. Sacchi, M., and Ulrych, T., 1995, High-resolution velocity gathers and offset space reconstruction, Geophysics, 60, 1169-1177 https://doi.org/10.1190/1.1443845
  8. Scales, J. A., and Gersztenkorn, A., 1987, Robust methods in inverse theory, Scales, J. A. Ed., Geophysical imaging, Symposium of geophysical society of Tulsa, SEG, 25-50
  9. Scales, J. A., Gersztenkorn, A., Treitel, S., and Lines, L. R., 1988, Robust optimization methods in geophysical inverse theory, SEG 58th Ann. Internat. Mtg., Session:S7.1
  10. Taner, M. T., and Koehler, F., 1969, Velocity spectra - Digital computer derivation and application of velocity functions, Geophysics, 34, 859-881 https://doi.org/10.1190/1.1440058
  11. Thorson, J. R., and Claerbout, J. F., 1985, Velocity stack and slant stochastic inversion, Geophysics, 50, 2727-2741 https://doi.org/10.1190/1.1441893
  12. Tred, D., Ulrych, T., and Sacchi M., 2003, Latest views of the sparse Radon transform, Geophysics, 68, 386-399 https://doi.org/10.1190/1.1543224