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Pseudo-Hessian matrix for the logarithmic objective function in full waveform inversion

WANSOO HA1 ,  WOOKEEN CHUNG2 ,  CHANGSOO SHIN1
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1 Department of Energy Systems Engineering, Seoul National University, Seoul 151-742, South Korea.,
2 Department of Energy and Resources Engineering, Korea Maritime University, Busan 606-791, South Korea. wkchung@hhu.ac.kr,
JSE 2012, 21(3), 201–214;
Submitted: 27 December 2011 | Accepted: 29 March 2012 | Published: 1 August 2012
© 2012 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

Ha, W.S., Chung, W.K. and Shin, C.S., 2012. Pseudo-Hessian matrix for the logarithmic objective function in full waveform inversion. Journal of Seismic Exploration, 21: 201-214. The pseudo-Hessian matrix of the J, objective function has been used in previous inversion studies because it was observed to yield satisfactory results in both the frequency and Laplace domains. However, there are fundamental differences between the pseudo-Hessian of the logarithmic objective function and that of the J, objective function. In this study, we first derive the pseudo-Hessian matrix for full waveform inversion using the logarithmic objective function. We then compare the two pseudo-Hessian matrices in both Laplace-and frequency-domain waveform inversions using the logarithmic objective function. A Laplace-domain inversion using the pseudo-Hessian of the logarithmic objective function yields a better result than that obtained using the pseudo-Hessian of the /, objective function. On the other hand, frequency domain inversion results using the two pseudo-Hessians yield similar results. The differences originate from the different behaviour of wavefields in the Laplace and frequency domains.

Keywords
pseudo-Hessian matrix
logarithmic objective function
frequency domain
waveform inversion
References
  1. Amundsen, L., 1991. Comparison of the least-squares criterion and the Cauchy criterion infrequency-wavenumber inversion. Geophysics, 56: 2027-2035.
  2. Bae, H.S., Shin, C., Cha, Y.H., Choi, Y. and Min, D.J., 2010. 2D acoustic-elastic coupledwaveform inversion in the Laplace domain. Geophys. Prosp., 58: 997-1010.
  3. Ben-Hadj-Ali, H., Operto, S. and Virieux, J., 2008. Velocity model-building by 3D frequencydomain, full-waveform inversion of wide-aperture seismic data. Geophysics, 73: VE101-VE117.
  4. Brenders, A. and Pratt, R., 2007. Efficient waveform tomography for lithospheric imaging:
  5. Implications for realistic, two-dimensional acquisition geometries and low-frequency data.Geophys. J. Internat., 168: 152-170.
  6. Bube, K.P. and Langan, R.T., 1997. Hybrid 1(1)/1(2) minimization with applications to tomography.Geophysics, 62: 1183-1195.214 HA, CHUNG & SHIN
  7. Chung, W., Shin, C. Pyun, S. and Calandra, H., 2010. 2D elastic waveform inversion in the
  8. Laplace domain. Expanded Abstr., 80th Ann. Internat. SEG Mtg., Denver: 1059-1064.
  9. Claerbout, J. and Muir, F., 1973. Robust modeling with erratic data. Geophysics, 38: 826-844.
  10. Gauthier, O., Virieux, J. and Tarantola, A., 1986. Two-dimensional nonlinear inversion of seismicwaveforms: numerical results. Geophysics, 51: 1387-1403.
  11. Ha, W., Pyun, S., Yoo, J. and Shin, C., 2010. Acoustic full waveform inversion of synthetic landand marine data in the Laplace domain. Geophys. Prosp., 58: 1033-1047.
  12. Levenberg, K., 1944. A method for the solution of certain nonlinear problems in least squares.Quart. J. Appl. Mathemat., 2: 164-168.
  13. Marquardt, D., 1963. An algorithm for least-squares estimation of nonlinear parameters. J. Soc.Industr. Appl. Mathemat., 11: 431-441.
  14. Mora, P., 1987. Nonlinear two-dimensional elastic inversion of multi-offset seismic data.Geophysics, 52: 1211-1228.
  15. Operto, S., Ravaut, C., Improta, L., Virieux, J., Herrero, A. and Dellapos-Aversana, P., 2004.
  16. Quantitative imaging of complex structures from dense wide-aperture seismic data bymultiscale traveltime and waveform inversions: A case study. Geophys. Prosp., 52: 625-51.
  17. Pratt, R., Shin, C. and Hicks, G., 1998. Gauss-Newton and full Newton methods in frequencyspaceseismic waveform inversion. Geophys. J. Internat., 133: 341-62.
  18. Shin, C., Jang, S. and Min, D., 2001. Improved amplitude preservation for prestack depth migrationby inverse scattering theory. Geophys. Prosp., 49: 592-606.
  19. Shin, C. and Min, D., 2006. Waveform inversion using a logarithmic wavefield. Geophysics, 71:R31-R42.
  20. Shin, C. and Cha, T., 2008. Waveform inversion in the Laplace domain. Geophys. J. Internat., 173:922-931.
  21. Shin, C. and Cha, T., 2009. Waveform inversion in the Laplace-Fourier domain. Geophys. J.Internat., 177: 1067-1079.
  22. Shin, C., Pyun, S. and Bednar, J.B., 2007. Comparison of waveform inversion, Part 1:
  23. Conventional wavefield vs. logarithmic wavefield. Geophys. Prosp., 55: 449-464.
  24. Symes, W. and Carazzone, J., 1991. Velocity inversion by differential semblance optimization.Geophysics, 56: 654-63.
  25. Tarantola, A., 1984. Inversion of seismic-reflection data in the acoustic approximation. Geophysics,49: 1259-1266.
  26. Versteeg, R., 1994. The Marmousi experience: Velocity model determination on a synthetic complexdata set. The Leading Edge, 13: 927-936.
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Journal of Seismic Exploration, Print ISSN: 0963-0651, Published by AccScience Publishing
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