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Characterization and `Source-Receiver' Continuation of Seismic Reflection Data

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Abstract

In reflection seismology one places sources and receivers on the Earth's surface. The source generates elastic waves in the subsurface, that are reflected where the medium properties, stiffness and density, vary discontinuously. In the field, often, there are obstructions to collect seismic data for all source-receiver pairs desirable or needed for data processing and application of inverse scattering methods. Typically, data are measured on the Earth's surface. We employ the term data continuation to describe the act of computing data that have not been collected in the field. Seismic data are commonly modeled by a scattering operator developed in a high-frequency, single scattering approximation. We initially focus on the determination of the range of the forward scattering operator that models the singular part of the data in the mentioned approximation. This encompasses the analysis of the properties of, and the construction of, a minimal elliptic projector that projects a space of distributions on the data acquisition manifold to the range of the mentioned scattering operator. This projector can be directly used for the purpose of seismic data continuation, and is derived from the global parametrix of a homogeneous pseudodifferential equation the solution of which coincides with the range of the scattering operator. We illustrate the data continuation by a numerical example.

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References

  1. Beylkin, G.: The inversion problem and applications of the generalized Radon transform. Comm. Pure Appl. Math. XXXVII, 579–599 (1984)

    Google Scholar 

  2. Biondi, B., Fomel, S., Chemingui, N.: Azimuth moveout for 3-D prestack imaging. Geophysics 63, 574–588 (1998)

    Article  Google Scholar 

  3. Bolondi, G., Loinger, E., Rocca, F.: Offset continuation of seismic sections. Geoph. Prosp. 30, 813–828 (1982)

    Article  Google Scholar 

  4. De Hoop, M.V., Bleistein, N.: Generalized radon transform inversions for reflectivity in anisotropic elastic media. Inverse Problems 13, 669–690 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. De Hoop, M.V., Brandsberg-Dahl, S.: Maslov asymptotic extension of generalized Radon transform inversion in anisotropic elastic media: A least-squares approach. Inverse Problems 16, 519–562 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. De Hoop, M.V., Malcolm, A.E., Le Rousseau, J.H.: Seismic wavefield `continuation' in the single scattering approximation: A framework for Dip and Azimuth MoveOut. Can. Appl. Math. Q. 10, 199–238 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Deregowski, S.G., Rocca, F.: Geometrical optics and wave theory of constant offset sections in layered media. Geoph. Prosp. 29, 374–406 (1981)

    Article  Google Scholar 

  8. Duistermaat, J.J.: Fourier integral operators. Boston: Birkhäuser, 1996

  9. Fomel, S.: Theory of differential offset continuation. Geophysics 68, 718–732 (2003)

    Article  Google Scholar 

  10. Gel'fand, I.M., Graev, M.I.: Complexes of straight lines in the space . Funct. Anal. Appl. 2, 39–52 (1968)

    Article  MATH  Google Scholar 

  11. Goldin, S.: Superposition and continuation of transformations used in seismic migration. Russ. Geol. and Geophys. 35, 131–145 (1994)

    MathSciNet  Google Scholar 

  12. Grechka, V., Tsvankin, I., Cohen, J.K.: Generalized Dix equation and analytic treatment of normal-moveout velocity for anisotropic media. Geoph. Prosp. 47, 117–148 (1999)

    Article  Google Scholar 

  13. Guillemin, V.: In: Pseudodifferential operators and applications (Notre Dame, Ind., 1984), Chapter ``On some results of Gel'fand in integral geometry'', Providence, RI: Amer. Math. Soc., 1985, pp. 149–155

  14. Guillemin, V., Sternberg, S.: Some problems in integral geometry and some related problems in microlocal analysis. Amer. J. of Math. 101, 915–955 (1979)

    MATH  MathSciNet  Google Scholar 

  15. Guillemin, V., Uhlmann, G.: Oscillatory integrals with singular symbols. Duke Math. J. 48, 251–267 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hansen, S.: Solution of a hyperbolic inverse problem by linearization. Commun. Par. Differ. Eqs. 16, 291–309 (1991)

    MATH  Google Scholar 

  17. Hörmander, L.: The analysis of linear partial differential operators. Volume IV. Berlin: Springer-Verlag, 1985

  18. Hörmander, L.: The analysis of linear partial differential operators. Volume III. Berlin: Springer-Verlag, 1985

  19. John, F.: The ultrahyperbolic differential equation with four independent variables. Duke Math. J. 4, 300–322 (1938)

    Article  MATH  MathSciNet  Google Scholar 

  20. Malcolm, A.E., De Hoop, M.V., Le Rousseau, J.H.: The applicability of DMO/AMO in the presence of caustics. Geophysics 70, 51 (2005)

    Article  Google Scholar 

  21. Maslov, V.P., Fedoriuk, M.V.: Semi-classical approximation in quantum mechanics. Dordrecht: Reidel Publishing Company, 1981

  22. Patch, S.K.: Computation of unmeasured third-generation VCT views from measured views. IEEE Trans. Med. Imaging 21, 801–813 (2002)

    Article  Google Scholar 

  23. Stolk, C.C., De Hoop, M.V.: Microlocal analysis of seismic inverse scattering in anisotropic, elastic media. Comm. Pure Appl. Math. 55, 261–301 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Taylor, M.E.: Reflection of singularities of solutions to systems of differential equations. Comm. Pure Appl. Math. 28, 457–478 (1975)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Center for Computational and Applied Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    Maarten V. de Hoop

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    Gunther Uhlmann

Authors
  1. Maarten V. de Hoop
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  2. Gunther Uhlmann
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Corresponding author

Correspondence to Maarten V. de Hoop.

Additional information

Communicated by P. Constantin

This research was supported in part under NSF CMG grant EAR-0417891.

Partly supported by a John Simon Guggenheim fellowship.

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Hoop, M., Uhlmann, G. Characterization and `Source-Receiver' Continuation of Seismic Reflection Data. Commun. Math. Phys. 263, 1–19 (2006). https://doi.org/10.1007/s00220-005-1491-6

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  • DOI: https://doi.org/10.1007/s00220-005-1491-6

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