Canonical representation – Chapter 10

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 10
ISBN 9781560801481
Store SEG Online Store

the canonical representation of a seismic trace states that a nonminimum-delay wavelet can be represented uniquely by the convolution of an all-pass filter (which has a flat magnitude spectrum) and a minimum-delay wavelet. Both wavelets have the same magnitude spectrum (i.e., the same color). This representation is called the canonical representation (Figure 4). Let us call white anything that has a flat magnitude spectrum, whereas we call colored anything that has a curved, or nonwhite, magnitude spectrum. The canonical representation then states that a nonminimum-delay wavelet is equal to the convolution of a minimum-delay wavelet of the same color as the given nonminimum-delay wavelet with an all-pass system, which is white.

Figure 4.  A minimum-delay wavelet convolved with an all-pass filter gives a mixed-delay wavelet. Both wavelets have the same autocorrelation.

1) The trace model is: seismic trace = mixed-delay wavelet * reflectivity.

2) A mixed-delay wavelet has both a colored and a white component: mixed-delay wavelet = minimum-delay wavelet * all-pass wavelet.

3) Thus, we can write our seismic trace as: seismic trace = minimum-delay wavelet * (all-pass filter * reflectivity).

On the right-hand side of the equation in (3), two members are expressed in parentheses. These parentheses contain the white components (i.e., the all-pass filter and the reflectivity function). Because the convolution of two white waveforms is itself white, the parentheses enclose a white component.

How does deconvolution work in such a case? The process of predictive deconvolution separates on the basis of two fundamental criteria: “minimum-delay” and “white.” Thus, the method of predictive convolution attempts to remove the minimum-delay component to yield an estimate of the white component. That is, if we deconvolve the seismic trace according to the above model we obtain as the output the white component: (all-pass filter * reflectivity function).

Thus, we do not obtain the desired reflectivity function; instead, we obtain the reflectivity function passed through an unknown all-pass filter. If the source wavelet was close to a minimum-delay wavelet, the action of the all-pass filter will not be severe, and the above result will be acceptable. Otherwise, the desired reflectivity function can be distorted appreciably by the presence of this unknown and unwanted all-pass filter.

Over the years, hundreds of papers have been written in geophysics on deconvolution, and many more have been published in other fields of science. Much research has been devoted to extensions of the simple convolutional model by assuming that the source wavelet is mixed delay and/or that the reflectivity is not white but, say, blue. A blue reflectivity is one that is still random but is richer in the higher, or “blue,” frequencies than in the lower, or “red,” frequencies. Jensen et al. (1988)[1] made a blueness compensation in deconvolution for the reflectivity sequence. Marschall and Knecht (1986)[2] devised a reflectivity-corrected deconvolution and described its influence on inversion. Saggaf and Toksöz (1999)[3] presented an analysis of deconvolution in which they modeled reflectivity by fractionally integrated noise. Saggaf and Robinson (2000)[4] provided a unified framework for the deconvolution of traces with nonwhite reflectivity. Porsani and Ursin (1998)[5] devised methods of mixed-phase deconvolution. Carrion (1986)[6] gave a layer-stripping technique to suppress multiples, and Carrion and de Pinto Braga (1990)[7] developed methods of iterative trace deconvolution. Mendel (1983)[8] used an estimation-based approach for optimal seismic deconvolution. Mendel (1990)[9] also pioneered the effective and advantageous method called maximum-likelihood deconvolution.

Fokkema and Ziolkowski (1987)[10] gave a proposition that they called the critical-reflection theorem, which can be used for deconvolution. Kelamis and Chiburis (1988)[11] originated the method of postcritical deconvolution, which emphasizes reliance on postcritical reflection data. Ulrych (1971)[12] and Buttkus (1975)[13] gave an outstanding development of homomorphic deconvolution in seismology. Ulrych and Bishop (1975)[14] treated maximum-entropy spectral analysis and autoregressive decomposition. Ooe and Ulrych (1979)[15] extended the efficacy of minimum entropy deconvolution by use of an exponential transformation. Ulrych and Matsuoka (1991)[16] gave insights on the character of the output of predictive deconvolution. Robinson and Treitel (1980)[17] discussed maximum entropy and the relationship of the partial autocorrelation to the reflection coefficients of a layered system. Carrion and Foster (1985)[18] treated the inversion of seismic data using precritical reflection and refraction data. Ulrych and Walker (1982)[19] discussed analytic minimum-entropy deconvolution.

Many important papers on deconvolution are found in the SEG reprint volumes: Webster (1978)[20], Osman and Robinson (1996)[21], and Robinson and Osman (1996). Excellent examples of deconvolution are given in Yilmaz (2001)[22].


  1. Jensen, O. G., T. J. Ulrych, J. P. Todeschuk, W. S. Leaney, and C. Walker, 1988, Bluenesscompensation in deconvolution for the reflectivity sequence: 58th Annual International Conference, SEG, Expanded Abstracts, 939-942.
  2. Marschall, R., and M. Knecht, 1986, Reflectivity-corrected deconvolution and its influence on inversion: Presented at the Research Workshop on Deconvolution and Inversion, EAGE.
  3. Saggaf, M. M., and N. Toksöz, 1999, An analysis of deconvolution, modeling reflectivity by fractionally integrated noise: Geophysics, 64, 1093-1107.
  4. Saggaf, M. M., and E. A. Robinson, 2000, A unified framework for the deconvolution of traces of nonwhite reflectivity: Geophysics, 65, 1660-1676.
  5. Porsani, M. J., and B. Ursin, 1998, Mixed-phase deconvolution: Geophysics, 63, 637-647.
  6. Carrion, P. M., 1986, A layer-stripping technique for the suppression of water-bottom multiple reflections: Geophysical Prospecting, 34, 330-342.
  7. Carrion, P. M., and A. de Pinto Braga, 1990, Iterative trace deconvolution and noncausal transform for processing band-limited data: Geophysics, 55, 1549-1557.
  8. Mendel, J. M., 1983, Optimal seismic deconvolution, an estimation based approach: Academic Press Inc.
  9. Mendel, J. M., 1990, Maximum-likelihood deconvolution, a journey into model-based signal processing: Springer-Verlag.
  10. Fokkema, J. T., and A. Ziolkowski, 1987, The critical reflection theorem: Geophysics, 52, 965-972.
  11. Kelamis, P. G., and E. F. Chiburis, 1988, Post-critical wavelet estimation and deconvolution: Geophysical Prospecting, 36, 504-522.
  12. Ulrych, T. J., 1971, Application of homomorphic deconvolution to seismology: Geophysics, 36, 650–660.
  13. Buttkus, B., 1975, Homomorphic filtering, theory and practice: Geophysical Prospecting, 23, 712-748.
  14. Ulrych, T. J., and T. N. Bishop, 1975, Maximum entropy spectral analysis and autoregressive decomposition: Reviews of Geophysics and Space Physics, 13, 183-00.
  15. Ooe, M., and T. J. Ulrych. 1979, Minimum entropy deconvolution with an exponential transformation: Geophysical Prospecting, 27, 458-473.
  16. Ulrych, T. J., and T. Matsuoka, 1991, The output of predictive deconvolution: Geophysics, 56, 371-377.
  17. Robinson, E. A., and S. Treitel, 1980, Maximum entropy and the relationship of the partial autocorrelation to the reflection coefficients of a layered system: IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-28, 224-235.
  18. Carrion, P. M., and D. J. Foster, 1985, Inversion of seismic data using the precritical reflection and refraction data: Geophysics, 50, 759-765.
  19. Ulrych, T. J., and C. Walker, 1982, Analytic minimum entropy deconvolution: Geophysics, 47, 1295-1302.
  20. Webster, G. M., 1978, Deconvolution: SEG Geophysics Reprint Series No. 1.
  21. Osman, O. M., and E. A. Robinson 1996, Seismic source signature estimation and measurement: SEG Geophysics Reprint Series No. 18.
  22. Yilmaz, Ö., 2001, Seismic data analysis: Processing, inversion, and interpretation of seismic data, 2 v.: SEG.

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