Advertisement

Plane- Wave Decomposition: A Tool for Deconvolution

  • M. Tygel
  • P. Hubral
  • F. Wenzel
Part of the Modern Approaches in Geophysics book series (MAGE, volume 8)

Abstract

Most (prestack) deconvolution methods for seismic shot records are based on simple stochastic models for the traces. They do not exploit the fact that these traces have to satisfy the wave equation. A plane-wave decomposition of a point-source seismogram recorded over a vertically inhomogeneous layered (acoustic) medium is a process based on the wave equation. It offers the possibility to extract both (a) the causal source pulse of arbitrary unknown shape and (b) the unknown broad-band plane-wave response (reflectivity) for any incidence angle or ray parameter. This was hitherto considered to be impossible but has become a reality if one exploits the wave theory that permits us to compose and decompose point-source responses from vertically inhomogeneous media in terms of plane waves. As starting point for constructing the seismogram by a plane-wave composition serves the Weyl Integral in either its time-harmonic or transient form. The central concept, on the other hand, upon which the plane-wave decomposition is based is the slant stack. The theory of plane-wave composition is briefly reviewed in order to establish the background for the new deconvolution procedure which essentially uses the properties that the reflectivity function (i.e. the reflection response of a layered medium for an incident plane wave) has for above-critical incidence angles.

Keywords

Source Pulse Reflectivity Function Deconvolution Procedure Source Wavelet Reflection Response 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bube, K.P. and Burridge, R., 1983, The one-dimensional inverse problem of reflection seismology, SIAM Rev. 25, 497-559.CrossRefGoogle Scholar
  2. Brekhovskikh, L.M., 1980, Waves in Layered Media, Academic Press, New York.Google Scholar
  3. Fokkema, J. and Ziolkowski, A., 1987, The critical reflection theorem, Geophysics 52, 965-972.CrossRefGoogle Scholar
  4. Fuchs, K., 1986, The reflection of spherical waves from transition zones with arbitrary depth-dependent elastic moduli and density, J. Phys. Earth 16, 27-41.CrossRefGoogle Scholar
  5. Kelamis, P. G., Chiburis, E. F., and Friedmann, V., 1987, Post-critical wavelet estimation and deconvolution, in Technical Programme of the EAEG Meeting at Belgrade, Yugoslavia, 9-12 June 1987. Google Scholar
  6. Müller, G., 1971, Direct inversion of seismic observations, J. Geophys. 37, 225-235Google Scholar
  7. Poritzky, H., 1951, Extension of Weyl's integral for harmonic spherical waves to arbitrary wave shapes, Comm. Pure Appl. Math. 4, 33-42.CrossRefGoogle Scholar
  8. Robinson, E.A., 1954, Predictive decomposition of time series with applications to seismic exploration, PhD Thesis, MIT, Cambridge, Mass. Also in Geophysics 32, 418-484 (1967).Google Scholar
  9. Sonnevend, G. Y., 1987, Sequential and stable methods for the solution of mass recovery problems (estimation of the spectrum and of the impedance function). Paper presented at the 5th International Seminar on Model Optimization in Exploration Geophysics, Berlin.Google Scholar
  10. Symes, W.W., 1983, Impedance profile inversion via the finite transport equation, J. Math. Anal. Appl., 94, 435-453.CrossRefGoogle Scholar
  11. Tygel, M. and Hubral, P., 1987, Transient Waves in Layered Media, Elsevier, Amsterdam.Google Scholar
  12. Ursin, B. and Berteussen, K.A., 1986, Comparison of some inverse methods for wave propagation in layered media, Proceedings of the IEEE, 74, 389-400.CrossRefGoogle Scholar
  13. Weyl, H., 1919, Ausbreitung elektromagnetischer Wellen über einen ebenen Leiter, Ann. Phys. 60, 481-500.CrossRefGoogle Scholar
  14. Yagle, A.E. and Levy, B.C., 1984, Application of the Schur algorithm to the inverse problem for a layered acoustic medium, J. Acoust. Soc. Am. 76, 301-308.CrossRefGoogle Scholar
  15. Ziolkowski, A.M., Fokkema, J.T., Baeten, G.J.M., and Ras, P.A.W., 1987, Extraction of the dynamite wavelet on real data using the critical reflection theorem, in Technical Programme of the EAEG Meeting at Belgrade, Yugoslavia, 9-12 June 1987 Google Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • M. Tygel
    • 1
  • P. Hubral
    • 2
  • F. Wenzel
    • 2
  1. 1.Department of Applied MathematicsIMECC/UNICAMPCampinasBrazil
  2. 2.Geophysical InstituteUniversity of KarlsruheKarlsruheWest Germany

Personalised recommendations