Nonlinear least-squares full waveform inversion: SVD analysis

Authors

  • K.G. Gadylshin
  • V.A. Tcheverda

Keywords:

Key words: Helmholtz equation
macrovelocity component
forward map
full waveform inversion
singular value decomposition
resolution analysis

Abstract

Nonlinear least-squares formulation for the inverse problem of seismic wave propagation is in the focus of computational geophysics community since 80th of the last century. Around the same time, the problem of trend component recovery becomes known. It is connected in the necessity to possess unrealistic low frequencies or extremely large source-receiver offsets in the data acquired to provide a reliable reconstruction of macrovelocity. At the same time, this component of the model determines the correct position of seismic images in space. Recently, thanks to the progress in the geophysical instrument industry, the substantive information becomes available for as low time frequencies as 5 Hz; as a rule, however, this is not sufficient for the proper reconstruction of macrovelocity. This paper deals with the numerical singular value decomposition of the linearized forward map to bring to the light the mathematical roots of the troubles. On this basis, a modification of the cost function is proposed, discussed and compared with the standard least-square approach. The high reliability of this modification in macrovelocity determination is confirmed numerically.

New computational technologies

Downloads

Published

2014-08-20

Issue

Vol. 15 (2014): Issue 3.

Section

Section 1. Numerical methods and applications

Author Biographies

K.G. Gadylshin

Novosibirsk State University,
Department of Mathematics and Mechanics
• PhD Student

V.A. Tcheverda

Trofimuk Institute of Petroleum Geology and Geophysics of SB RAS
• Head of Department

References

  1. Алексеев А.С., Костин В.И., Хайдуков В.Г., Чеверда В.А. Восстановление двумерных возмущений скорости вертикально-неоднородной акустической среды по данным многократного перекрытия (линеаризованная постановка) // Геология и геофизика. 1997. 38, № 12. 1980-1992.
  2. Ахиезер Н.И., Глазман И.М. Теория линейных операторов в гильбертовом пространстве. М.: Наука, 1966.
  3. Годунов С.К., Антонов А.Г., Кирилюк О.П., Костин В.И. Гарантированная точность решения систем линейных уравнений в евклидовых пространствах. Новосибирск: Наука, 1992.
  4. Бакушинский А.Б., Гончарский А.В. Итеративные методы решения некорректных задач. М.: Наука, 1989.
  5. Вайнберг Б.Р. Принципы излучения, предельного поглощения и предельной амплитуды в общей теории уравнений с частными производными // Успехи математических наук. 1966. 21, № 3. 115-194.
  6. Канторович Л.В., Акилов Г.П. Функциональный анализ. М.: Наука. 1977.
  7. Костин В.И., Хайдуков В.Г., Чеверда В.А. r-решения уравнений первого рода с компактным оператором в гильбертовых пространствах: существование и устойчивость // Доклады АН. 1997. 335, № 3. 308-312.
  8. Чеверда В.А., Костин В.И. r-псевдообратный для компактного оператора // Сибирские электронные математические известия. 2010. 7. 258-282.
  9. Протасов М.И., Чеверда В.А. Построение сейсмических изображений в истинных амплитудах // Доклады АН. 2006. 407, № 4. 528-532.
  10. Эйдус Д.М. О принципе предельного поглощения // Математический сборник. 1962. 57, № 1. 13-44.
  11. Bamberger A., Chavent G., Hemon Ch., Lailly P. Inversion of normal incidence seismograms // Geophysics. 1982. 47, N 5. 757-770.
  12. Bao G., Symes W.W. Regularity of an inverse problem in wave propagation // Lecture Notes in Physics. Vol. 486. Heidelberg: Springer, 1997. 226-235.
  13. Beylkin G. Imaging of discontinuities in the inverse scattering problem by inversion of causal generalized Radon transform // J. Math. Phys. 1985. 26, N 1. 99-108.
  14. Brenders A.J., Pratt R.G. Full waveform tomography for lithospheric imaging: results from a blind test in a realistic crustal model // Geophys. J. Int. 2007. 168, N 1. 133-151.
  15. Cheverda V.A., Kostin V.I. R-pseudoinverses for compact operators in Hilbert spaces: existence and stability // Journal of Inverse and Ill-Posed Problems. 1995. 3, N 2. 131-148.
  16. Clément F., Chavent G., G’omez S. Migration-based traveltime waveform inversion of 2-D simple structures: A synthetic example // Geophysics. 2001. 66, N 3. 845-860.
  17. Jannane M., Beydoun W., Crase E., el al. Wavelengths of earth structures that can be resolved from seismic reflection data // Geophysics. 1989. 54, N 7. 906-910.
  18. Gauthier O., Virieux J., Tarantola A. Two-dimensional nonlinear inversion of seismic waveforms: Numerical results // Geophysics. 1986. 51, N 7. 1387-1403.
  19. Lailly P. The seismic inverse problem as a sequence of before stack migrations // Proc. of the Conference on Inverse Scattering: Theory and Applications. Philadelphia: SIAM, 1983. 206-220.
  20. Paige C.C., Saunders M.A. LSQR: An algorithm for sparse linear equations and sparse least squares // Transactions on Mathematical Software. 1982. 8, N 1. 43-71.
  21. Protasov M.I., Tcheverda V.A. True amplitude imaging by inverse generalized Radon transform based on Gaussian beam decomposition of the acoustic Green’s function // Geophysical Prospecting. 2011. 59, N 2. 197-209.
  22. Silvestrov I., Neklyudov D., Kostov C., Tcheverda V. Full-waveform inversion for macro-velocity model reconstruction in look-ahead offset vertical seismic profile: numerical singular value decomposition-based analysis // Geophysical Propspecting. 2013. 61, N 6. 1099-1113.
  23. Tarantola A. Inversion of seismic reflection data in the acoustic approximation // Geophysics. 1984. 49, N 8. 1259-1266.
  24. Virieux J., Operto S. An overview of full-waveform inversion in exploration geophysics // Geophysics. 2009. 74, N 6. WCC1-WCC26.
  25. Yilmaz Ö. Seismic data analysis. Processing, inversion and interpretation of seismic data. Vol. 2. Tulsa: Society of Exploration Geophysicists, 2001.