1887
Volume 58 Number 6
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

We review the multifocusing method for traveltime moveout approximation of multicoverage seismic data. Multifocusing constructs the moveout based on two notional spherical waves at each source and receiver point, respectively. These two waves are mutually related by a focusing quantity. We clarify the role of this focusing quantity and emphasize that it is a function of the source and receiver location, rather than a fixed parameter for a given multicoverage gather. The focusing function can be designed to make the traveltime moveout exact in certain generic cases that have practical importance in seismic processing and interpretation. The case of a plane dipping reflector (planar multifocusing) has been the subject of all publications so far. We show that the focusing function can be generalized to other surfaces, most importantly to the spherical reflector (spherical multifocusing). At the same time, the generalization implies a simplification of the multifocusing method. The exact traveltime moveout on spherical surfaces is a very versatile and robust formula, which is valid for a wide range of offsets and locations of source and receiver, even on rugged topography. In two‐dimensional surveys, it depends on the same three parameters that are commonly used in planar multifocusing and the common‐reflection surface (CRS) stack method: the radii of curvature of the normal and normal‐incidence‐point waves and the emergence angle. In three dimensions the exact traveltime moveout on spherical surfaces depends on only one additional parameter, the inclination of the plane containing the source, receiver and reflection point. Comparison of the planar and spherical multifocusing with the CRS moveout expression for a range of reflectors with increasing curvature shows that the planar multifocusing can be remarkably accurate but the CRS becomes increasingly inaccurate. This can be attributed to the fact that the CRS formula is based on a Taylor expansion, whereas the multifocusing formulae are double‐square root formulae. As a result, planar and spherical multifocusing are better suited to model the moveout of diffracted waves.

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2010-02-09
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References

  1. AlkhalifahT. and TsvankinI.1995. Velocity analysis for tranversely isotropic media. Geophysics60, 1550–1566.
    [Google Scholar]
  2. De BazelaireE.1988. Normal moveout correction revisited: inhomogeneous media and curved interfaces. Geophysics53, 143–157.
    [Google Scholar]
  3. BerglerS., HubralP., MarchettiP., CristiniA. and CardoneG.2002. 3D common‐reflection‐surface stack and kinematic wavefield attributes. The Leading Edge21, 1010.
    [Google Scholar]
  4. BerkovichA., BelferI. and LandaE.2008, Multifocusing as a method of improving subsurface imaging. The Leading Edge27, 250–257.
    [Google Scholar]
  5. BurnsideW.S. and PantonA.W.1960. The Theory of Equations With an Introduction To The Theory of Binary Algebraic Forms . Dover , New York.
    [Google Scholar]
  6. CastleR.1994. A theory of normal moveout. Geophysics59, 983–999.
    [Google Scholar]
  7. CausseE., HangenG.U. and RommelB.2000. Large‐offset approximation to seismic reflection traveltimes. Geophysical Prospecting48, 763–778.
    [Google Scholar]
  8. CausseE.2002. Seismic traveltime approximations with high accuracy at all offsets. 64th EAGE meeting, Florence, Italy, Extended Abstracts.
  9. ČervenýV.2001. Seismic Ray Theory . Cambridge University Press.
    [Google Scholar]
  10. ChoiY.K., WangW. and KimM.S.2003. Exact collision detection of two moving ellipsoids under rational motions. Proceedings IEEE International Conference on Robotics and Automation1, 349–354.
    [Google Scholar]
  11. DörrieH.1965. 100 Great Problems of Elementary Mathematics: Their History and Solutions . Dover , New York.
    [Google Scholar]
  12. DrexlerM. and GanderM.J.1998. Circular billiard. SIAM Review40, 315–323.
    [Google Scholar]
  13. Eisenberg‐KleinG., PrüssmannJ., GierseG. and TrappeH.2008. Noise reduction in 2D and 3D seismic imaging by the CRS method. The Leading Edge27, 258–265.
    [Google Scholar]
  14. FomelS.2003. Theory of differential offset continuation. Geophysics68, 718–732.
    [Google Scholar]
  15. FomelS. and KazinnikR.2009. Non‐hyperbolic common reflection surface. 79th SEG meeting, Houston, Texas, USA, Extended Abstracts.
  16. GelchinskyB., BerkovitchA. and KeydarS.1999a. Multifocusing homeomorphic imaging – Part 1. Basic concepts and formulae. Journal of Applied Geophysics42, 229–242.
    [Google Scholar]
  17. GelchinskyB., BerkovitchA. and KeydarS.1999b. Multifocusing homeomorphic imaging – Part 2. Multifold data set and multifocusing. Journal of Applied Geophysics42, 243–260.
    [Google Scholar]
  18. GurevichB., KeydarS. and LandaE.2002. Multifocusing imaging over an irregular topography. Geophysics67, 639–643.
    [Google Scholar]
  19. HöchtG., De BazelaireE., MajerP. and HubralP.1999. Seismics and optics: hyperbolae and curvatures. Journal of Applied Geophysics42, 261–281.
    [Google Scholar]
  20. HubralP.1983. Computing true amplitude reflections in a laterally inhomogeneous earth. Geophysics48, 1051–1062.
    [Google Scholar]
  21. HubralP.1999. Macro model independent seismic reflection imaging. Journal of Applied Geophysics42, 3–4.
    [Google Scholar]
  22. JägerR., MannJ., HöchtG. and HubralP.2001. Common‐reflecting‐surface stack: Image and attributes. Geophysics66, 97–109.
    [Google Scholar]
  23. KhaidukovV., LandaE. and MoserT.J.2004. Diffraction imaging by focusing‐defocusing: An outlook on seismic superresolution. Geophysics69, 1478–1490.
    [Google Scholar]
  24. LandaE.2007. Beyond Conventional Seismic Imaging . EAGE.
    [Google Scholar]
  25. LandaE., GurevichB., KeydarS. and TrachtmanP.1999. Application of multifocusing method for subsurface imaging. Journal of Applied Geophysics42, 283–300.
    [Google Scholar]
  26. MalovichkoA.A.1978. A new representation of the travel time curve of reflected waves in horizontally layered media. Applied Geophysics91, 47–53 (in Russian).
    [Google Scholar]
  27. MannJ., JägerR., MüllerT., HöchtG. and HubralP.1999. Common‐reflection‐surface stack a real data example. Journal of Applied Geophysics42, 301–318.
    [Google Scholar]
  28. MayB.T. and StraleyD.K.1979. Higher‐order moveout spectra. Geophysics44, 1193–1207.
    [Google Scholar]
  29. MoserT.J. and ČervenýV.2007. Paraxial ray methods in anisotropic inhomogeneous media. Geophysical Prospecting55, 21–37.
    [Google Scholar]
  30. NeumannP.M.1998. Reflections on reflection in a spherical mirror. American Mathematical Monthly105, 523–528.
    [Google Scholar]
  31. SalmonG.1960. Conic Sections , 6th edn. Chelsea , New York.
    [Google Scholar]
  32. ScrutonR.2001. Short History of Modern Philosophy . Routledge.
    [Google Scholar]
  33. SwordC.H.1987. A Soviet look at datum shift. SEP‐51, Stanford Exploration Project, 313–316.
    [Google Scholar]
  34. TanerM.T. and KoehlerF.1969. Velocity spectra digital computer derivation and applications of velocity functions. Geophysics34, 859–881.
    [Google Scholar]
  35. TanerM.T., TreitelS., Al‐ChalabiM. and FomelS.2007. An offset dependent NMO velocity model. 69th EAGE meeting, London, UK, Extended Abstracts.
  36. TygelM., SantosL.T. and SchleicherJ.1999. Multifocus moveout revisited: derivations and alternative expressions. Journal of Applied Geophysics42, 319–331.
    [Google Scholar]
  37. TygelM. and SantosL.T.2007. Quadratic normal moveouts of symmetric reflections in elastic media: A quick tutorial. Studia Geophysica et Geodætica51, 185–206.
    [Google Scholar]
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  • Article Type: Research Article
Keyword(s): Moveout formulae , Multifocusing method and Subsurface imaging
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