1887
Volume 38 Number 4
  • E-ISSN: 1365-2478

Abstract

A

The purpose of deconvolution is to retrieve the reflectivity from seismic data. To do this requires an estimate of the seismic wavelet, which in some techniques is estimated simultaneously with the reflectivity, and in others is assumed known. The most popular deconvolution technique is inverse filtering. It has the property that the deconvolved reflectivity is band‐limited. Band‐limitation implies that reflectors are not sharply resolved, which can lead to serious interpretation problems in detailed delineation.

To overcome the adverse effects of band‐limitation, various alternatives for inverse filtering have been proposed. One class of alternatives is L‐norm deconvolution, Lnorm deconvolution being the best‐known of this class.

We show that for an exact convolutional forward model and statistically independent reflectivity and additive noise, the maximum likelihood estimate of the reflectivity can be obtained by L‐norm deconvolution for a range of multivariate probability density functions of the reflectivity and the noise. The L‐norm corresponds to a uniform distribution, the L‐norm to a Gaussian distribution, the L‐norm to an exponential distribution and the L‐norm to a variable that is sparsely distributed. For instance, if we assume sparse and spiky reflectivity and Gaussian noise with zero mean, the L‐norm deconvolution problem is solved best by minimizing the L‐norm of the reflectivity and the L‐norm of the noise. However, the L‐norm is difficult to implement in an algorithm. From a practical point of view, the frequency‐domain mixed‐norm method that minimizes the Lnorm of the reflectivity and the L‐norm of the noise is the best alternative.

L‐norm deconvolution can be stated in both time and frequency‐domain. We show that both approaches are only equivalent for the case when the noise is minimized with the L‐norm.

Finally, some L‐norm deconvolution methods are compared on synthetic and field data. For the practical examples, the wide range of possible L‐norm deconvolution methods is narrowed down to three methods with = 1 and/or 2. Given the assumptions of sparsely distributed reflectivity and Gaussian noise, we conclude that the mixed Lnorm (reflectivity) L‐norm (noise) performs best. However, the problems inherent to single‐trace deconvolution techniques, for example the problem of generating spurious events, remain. For practical application, a greater problem is that only the main, well‐separated events are properly resolved.

Loading

Article metrics loading...

/content/journals/10.1111/j.1365-2478.1990.tb01852.x
2006-04-27
2020-03-30
Loading full text...

Full text loading...

References

  1. Debeye, H.W.J.1987. Deconvolution with L p ‐norms. Report. Delft University of Technology, Faculty of Applied Physics, Laboratory of Seismics and Acoustics, The Netherlands .
    [Google Scholar]
  2. Fullagar, P.K.1985. Spike recovery deconvolution. In: Developments in Geophysical Exploration Methods, A. A.Fitch (ed.), Vol. 6. Elsevier Applied Science Publishers.
    [Google Scholar]
  3. Gill, P.E., Murray, W. and Wright, M.H.1981. Practical Optimization. Academic Press, Inc.
    [Google Scholar]
  4. Jackson, D.D.1979. The use of a priori data to resolve non‐uniqueness in linear inversion. Geophysical Journal of the Royal Astronomical Society57, 137–157.
    [Google Scholar]
  5. Johnson, N.L. and Kotz, S.1972. Continuous Multivariate Distributions, Part IV. John Wiley and Sons, Inc.
    [Google Scholar]
  6. Levy, S. and Fullagar, P.K.1981. Reconstruction of a sparse spike‐train from a portion of its spectrum and application to high‐resolution deconvolution. Geophysics46, 1235–1243.
    [Google Scholar]
  7. Oldenburg, D.W., Levy, S. and Sttnson, K.J.1986. Inversion of band‐limited reflection seismograms: theory and practice. Proceedings of the IEEE74, 487–497.
    [Google Scholar]
  8. Oldenburg, D.W., Scheuer, T. and Levy, S.1983. Recovery of the acoustic impedance from reflection seismograms. Geophysics48, 1318–1337.
    [Google Scholar]
  9. Santosa, F. and Symes, W.W.1984. Linear inversion of band limited reflection seismograms. Report, Department of Theoretic and Applied Mechanics, Cornell University, New York .
    [Google Scholar]
  10. Tarantola, A.1987. Inverse Problem Theory, Methods for Data Fitting and Model Estimation. Elsevier Science Publishers.
    [Google Scholar]
  11. Tarantola, A. and Valette, B.1982. Inverse problems = quest for information. Journal of Geophysics50, 159–170.
    [Google Scholar]
  12. Taylor, H.L., Banks, S.C. and McCoy, J.F.1979. Deconvolution with the Lrnorm. Geophysics44, 39–52.
    [Google Scholar]
  13. Van de Panne, C.1975. Methods for Linear and Quadratic Programming. North‐Holland Publishing Company.
    [Google Scholar]
  14. Van Riel, P. and Berkhout, A.J.1985. Resolution in seismic trace inversion by parameter estimation. Geophysics50, 1140–1455.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1111/j.1365-2478.1990.tb01852.x
Loading
  • Article Type: Research Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error