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Volume 36 Number 4
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

The classical aim of non‐linear inversion of seismograms is to obtain the earth model which, for null initial conditions and given sources, best predicts the observed seismograms. This problem is currently solved by an iterative method: each iteration involves the resolution of the wave equation with the actual sources in the current medium, the resolution of the wave equation, backwards in time, with the current residuals as sources; and the correlation, at each point of space, of the two wavefields thus obtained.

Our view of inversion is more general: we want to obtain a whole set of earth model, initial conditions, source functions, and predicted seismograms, which are the closest to some values, and which are related through the wave equation. It allows us to justify the previous method, but it also allows us to set the same inverse problem in a different way: what is now searched for is the best fit between calculated and initial conditions, for given sources and observed surface displacements. This leads to a completely different iterative method, in which each iteration involves the downward extrapolation of given surface displacements and tractions, down to a given depth (the‘bottom’), the upward extrapolation of null displacements and tractions at the bottom, using as sources the initial time conditions of the previous field, and a correlation, at each point of the space, of the two wavefields thus obtained. Besides the theoretical interest of the result, it opens the way to alternative numerical methods of resolution of the inverse problem. If the non‐linear inversion using forward‐backward time propagations now works, this non‐linear inversion using downward‐upward extrapolations will give the same results but more economically, because of some tricks which may be used in depth extrapolation (calculation frequency by frequency, inversion of the top layers before the bottom layers, etc.).

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References

  1. AKI, K. and RICHaRDS, P.G.1980. Quantitative Seismology (2 volumes).Freeman and Co.
    [Google Scholar]
  2. ALTERMaN, Z.S. and KaRaL, F.C., JR.1968. Propagation of elastic waves in layered media by finite difference methods. Bulletin of the Seismological Society of America58, 367–398.
    [Google Scholar]
  3. BaYSaL, E., KOSLOFF, D.D. and SHERWOOD, J.W.C.1983. Reverse time migration. Geophysics48, 1514–1524.
    [Google Scholar]
  4. BEN‐MENaHEM, A. and SINGH, S.J.1981. Seismic Waves and Sources.Springer‐Verlag, Inc.
    [Google Scholar]
  5. CHaNG, W.F. and MCMECHaN, G.A.1986. Reverse‐time migration of offset VSP data using the excitation time imaging condition. Geophysics51, 67–84.
    [Google Scholar]
  6. CLaERBOUT, J.F.1971. Toward a unified theory of reflector mapping. Geophysics36, 467–481.
    [Google Scholar]
  7. DaUTRaY, R. and LIONS, J.L.1984. Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Vol. I. Masson , Paris.
    [Google Scholar]
  8. GaUTHIER, O., VIRIEUX, J. and TaRaNTOLa, A.1986. Two‐dimensional inversion of seismic waveforms: numerical results. Geophysics51, 1387–1403.
    [Google Scholar]
  9. JOBERT, G.1975. Propagator and Green matrices for body force and dislocation. Geophysical Journal of the Royal Astronomical Society42, 755–762.
    [Google Scholar]
  10. JOBERT, G.1977. On equivalent models of seismic sources. Journal of Geophysics43, 329–339.
    [Google Scholar]
  11. KOLB, P., COIXINO, F. and LaILLY, P.1986. Pre‐stack inversion of a ID medium. Proceedings of the IEEE74 (3), 498–508.
    [Google Scholar]
  12. Kuo, J.T. and Dai, T.F.1984. Kirchhoff elastic wave migration for the case of noncoincident source and receiver. Geophysics49, 1223–1238.
    [Google Scholar]
  13. LaILLY, P.1984. Migration methods: partial but efficient solutions to the seismic inverse problem. In: Inverse Problems of Acoustic and Elastic Waves. F.Santosa , Y.‐H.Pao , W.Symes , and Ch.Holland (eds), Society of Industrial and Applied Mathematics, Philadelphia .
    [Google Scholar]
  14. LEVIN, S.A.1984. Principle of reverse time migration. Geophysics49, 581–583.
    [Google Scholar]
  15. MORa, P.1987. Elastic inversion of field data. Stanford Exploration Project, report and Equipe de Tomographie Géophysique (Paris) report no. 7.
  16. MORa, P.1988. Elastic wavefield inversion and the adjoint operation for the elastic wave equation. In: Mathematical Geophysics, ed. by Vlaar, N.J. , Nolet, G. , Wortel, M.R.J , and Cloetingh, S.A.P.L. Reidel Pub. Co.
    [Google Scholar]
  17. MORSE, P.M. and FESHBaCH, H.1953. Methods of Theoretical Physics.McGraw‐Hill Book Co.
    [Google Scholar]
  18. PICa, A., DIET, J.P. and TaRaNTOLa, A.1987. Nonlinear inversion of real seismic data (preprint).
  19. RESHEF, M. and KOSLOFF, D.1986. Migration of common‐shot gathers. Geophysics51, 324–331.
    [Google Scholar]
  20. SUN, R. and MCMECHaN, G.1986. Prestack reverse time migration for elastic waves with application to synthetic offset vertical seismic profiles. Proceedings IEEE, 74 (3), 457–465.
    [Google Scholar]
  21. TaRaNTOLa, A.1984a. Inversion of seismic reflection data in the acoustic approximation. Geophysics49, 1259–1266.
    [Google Scholar]
  22. TaRaNTOLa, A.1984b. The seismic reflection inverse problem. In: Inverse Problems of Acoustic and Elastic Waves. F.Santosa , Y.‐H.Pao , W.Symes , and Ch.Holland (eds), Society of Industrial and Applied Mathematics, Philadelphia .
    [Google Scholar]
  23. TaRaNTOLa, A.1986. A strategy for nonlinear elastic inversion of seismic reflection data. Geophysics51, 1893–1903.
    [Google Scholar]
  24. TaRaNTOLa, A.1987. Inverse Problem Theory: methods for data fitting and model parameter estimation.Elsevier Science Publishing Co.
    [Google Scholar]
  25. TaYLOR, A.E. and LaY, D.C.1980. Introduction to Functional Analysis.John Wiley and Sons, Inc.
    [Google Scholar]
  26. VIRIEUX, J.1986. P‐SV wave propagation in heterogeneous media; velocity‐stress finite‐difference method. Geophysics51, 889–901.
    [Google Scholar]
  27. WaPENaaR, C.P.A. and BERKHOUT, A.J.1986. Wavefield extrapolation techniques for inhomogeneous media which include critical angle events, Parts II and III. Geophysical Prospecting34, 147–207.
    [Google Scholar]
  28. WaPENaaR, C.P.A., KINNEGING, N.A. and BERKHOUT, A.J.1987. Principle of prestack migration based on the full elastic two way wave equation. Geophysics52, 151–173.
    [Google Scholar]
  29. WHITMORE, N.D.1983. Iterative depth‐migration by backward time propagation. 53rd SEG meeting, Las Vegas , Expanded Abstracts, 382–385.
    [Google Scholar]
  30. WOODHOUSE, J.H.1974. Surface waves in a laterally varying layered structure. Geophysical Journal of the Royal Astronomical Society37, 461–490.
    [Google Scholar]
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