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The l 1 Norm in Seismic Data Processing

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Developments in Geophysical Exploration Methods

Part of the book series: The Developments Series ((TDS))

Summary

For solving a linear model by minimising the residuals ri = bi − ΣaijXjthe l 1 norm uses the sum of absolute values of the residuals rather than the sum of the squares of the residuals, which is used in the least-squares procedures associated with the l 2 norm. The l 1 norm defines a robust procedure which is useful in handling certain types of model errors and data containing a few wild data points. The l 1 norm solution x to r = b − Ax is also the maximum likelihood estimate of the system Ax + e = b where the errors e have a Laplace distribution. In seismic data processing, the l 1 norm has possible applications in earthquake centre location and in numerous reflection seismic prospecting steps, including residual statics, velocity analysis, stacking, filter design and deconvolution. The l 1 deconvolution of a seismic trace is of special interest since the resulting spike train contains a sparse spike representation of the reflectivity train of the earth rather than a smooth band-limited representation. The sparse spike representation can be useful for wavelet extraction, production of a stacked section and correlation with well log data.

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References

  1. PLACKETT, R. L., Studies in the history of probability and statistics; Chapter 29, The discovery of the method of least squares, Biometrika 59 p. 239, 1972.

    Google Scholar 

  2. HARDY, G. H., LITTLEWOOD, J. E. and POLYA, G., Inequalities, Cambridge University Press, Cambridge, 1934.

    Google Scholar 

  3. BARRODALE, I., L 1 approximation and the analysis of data, Appl. Statist., 17, p. 51, 1968.

    Article  Google Scholar 

  4. KARST, O. J., Linear curve fitting using least deviations, J. Am. Statist. Ass., 53, p. 118, 1958.

    Article  Google Scholar 

  5. WAGNER, H. M., Linear programming techniques for regression analysis, J. Am. Statist. Ass., 54, p. 206, 1959.

    Article  Google Scholar 

  6. RICE, J. R., The approximation of functions, Addison-Wesley, Reading, Mass., 1964.

    Google Scholar 

  7. CHENEY, E. W., Introduction to approximation theory, McGraw-Hill, New York, 1966.

    Google Scholar 

  8. CHARNES, A., COOPER, W. W. and FERGUSON, R., Optimal estimation of executive compensation by linear programming, Manag. Sci., 1, p. 138, 1955.

    Article  Google Scholar 

  9. BARRODALE, I. and YOUNG, A., Algorithm for best L1 and L∞ linear approximation on a discrete set, Numer. Math., 8, p. 295, 1966.

    Article  Google Scholar 

  10. RABINOWITZ, P., Applications of linear programming to numerical analysis, SIAM Rev., 10, p. 121, 1968.

    Article  Google Scholar 

  11. DOUGHERTY, E. L. and SMITH, S. T., The use of linear programming to filter digitized map data,Geophysics, 31, p. 253, 1966.

    Article  Google Scholar 

  12. CLAERBOUT, J. F. and MUIR, F., Robust modeling with erratic data,Geophysics, 38, p. 826, 1973.

    Article  Google Scholar 

  13. CLAERBOUT, J. F., Fundamentals of geophysical data processing with application to petroleum prospecting, McGraw-Hill, New York, 1976.

    Google Scholar 

  14. GASS, S. I., Linear programming methods and applications, McGraw-Hill, New York, 1958.

    Google Scholar 

  15. RUST, B. W. and BURRUS, W. R., Mathematical programming and the numerical solution of linear equations, Elsevier, New York, 1972.

    Google Scholar 

  16. SAFON, C., VASSUER, G. and CUER, M., Some applications of linear programming to the inverse gravity problem, Geophysics, 42, p. 1215, 1977.

    Article  Google Scholar 

  17. TAYLOR, H. L. and WOOD, M., Wavelet contraction and shaping using minimax criteria, presented Soc. Explor. Geophys. 49th Ann. Mtg., New Orleans, 1979 (submitted to Geophysics, 1980).

    Google Scholar 

  18. HADLEY, G., Linear programming, Addison-Wesley, Reading, Mass, 1962.

    Google Scholar 

  19. WATSON, G. A., The calculation of best linear one-sided L p approximation, Math. Comput., 27, p. 607, 1973.

    Google Scholar 

  20. BARRODALE, I. and ROBERTS, F D. K, An improved algorithm for discrete l 1 linear approximation, SIAM J. Numer. Anal., 10, p. 839, 1973.

    Article  Google Scholar 

  21. BARRODALE, I. and ROBERTS, F. D. K., Algorithm 478, solution of an overdetermined system of equations in the l 1 norm, Commun. ACM , 17, p. 319, 1974.

    Article  Google Scholar 

  22. FRASER, D. A. S., Statistics, an introduction, John Wiley, New York, 1958.

    Google Scholar 

  23. RICE, J. R. and WHITE, J. S., Norms for smoothing and estimation, SIAM Rev., 6, p. 243, 1964.

    Article  Google Scholar 

  24. GEARY, R. C., Testing for normality, Biometrika, 34, p. 209, 1947.

    Google Scholar 

  25. ANDREWS, D. F., A robust method for multiple linear regression, Technometries, 16, p. 523, 1974.

    Article  Google Scholar 

  26. HARVEY, A. C., A comparison of preliminary estimators for robust regression, J. Am. Statist. Ass., 72, p. 910, 1977.

    Article  Google Scholar 

  27. HOGG, R. V., ‘Adaptive robust procedures: a partial review and some suggestions for future applications and theory’, J. Am. Statist. Assoc., 69, p.909, 1974.

    Article  Google Scholar 

  28. EKBLOM, E. and HENDRIKSSON, S., L p —Criteria for the estimation of location parameters,SIAM J. Appl. Math., 17, p. 1130, 1969.

    Article  Google Scholar 

  29. HUBER, P. J., Robust estimation of a location parameter, Ann. Math. Statist., 35, p. 73, 1964.

    Article  Google Scholar 

  30. SHANNON, C. E. and WEAVER, W., The mathematical theory of communication, University of Illinois Press, Urbana, Illinois, 1963.

    Google Scholar 

  31. WIGGINS, R. A., LARNER, K. L. and WISECUP, R. D., Residual statics analysis as a general linear inverse problem, Geophysics, 41, p. 922, 1976.

    Article  Google Scholar 

  32. DONOHO, D., Robust estimation of residual statics corrections, presented Soc. Explor. Geophys. 49th Ann. Mtg., New Orleans, 1979.

    Google Scholar 

  33. WEBSTER, G. M., Deconvolution (vols. I and II.), Geophysics Reprint Series, Society of Exploration Geophysicists, Tulsa, Okla., 1978.

    Google Scholar 

  34. Carter, R. D., Kemp, L. F., Jr., Pierce, A. C. and Williams, D, L., Performance matching with constraints, Soc. Pet. Engrs. Preprint. No. 4260, 1973.

    Google Scholar 

  35. CARTER, R. C., KEMP, L. F., JR., PIERCE, A. C. and WILLIAMS, D. L., Performance matching with constraints, Soc. Pet. Engrs J., p. 187, 1974.

    Google Scholar 

  36. GODFREY, R., MUIR, F. and FABBIO, R., ‘Modeling seismic impedance with Markov chains’, Geophysics, 45, p. 1351, 1980.

    Article  Google Scholar 

  37. TAYLOR, H. L., BANKS, S. C. and MCCOY, J. F., Deconvolution in the l 1 norm, Geophysics, 44, p. 39, 1979.

    Article  Google Scholar 

  38. BANKS, S. C. and TAYLOR, H. L., A modification to the discrete l 1 linear approximation algorithm of Barrodale and Roberts, SIAM J. Sci. Statist. Comput., 1, pp. 187–98, June 1980.

    Article  Google Scholar 

  39. BAMBERGER, A., CHAVENT, G., HEMON, CH., and LAILLY, P., Inversion of normal incidence seismograms, presented Soc. Explor. Geophys.48th Ann. Mtg, San Francisco, 1978.

    Google Scholar 

  40. TAYLOR, H. L., SIRAKI, E. S. and WHITSETT, R., The poor man’s wavelet extractor, presented Soc. Explor. Geophys. 48th Ann. Mtg, San Francisco, 1978.

    Google Scholar 

  41. GODFREY, R. J., A stochastic model for seismogram analysis, 1979 Ph.D. Dissertation, Stanford University, Calif., 1979.

    Google Scholar 

  42. GRAY, W., Variable norm deconvolution, Ph.D. Dissertation, Stanford University, Stanford, Calif., 1979.

    Google Scholar 

  43. WIGGINS, R., Minimum entropy deconvolution, p. 7, Proc. IEEE Computer Society International Symposium on Computer-Aided Seismic Analysis and Discrimination, Falmouth, Mass., June 9–10, 1977.

    Google Scholar 

  44. WIGGINS, R., LARNER, K. and JOHNSTON, D., Minimum entropy decon- volution, presented Soc. Explor. Geophys. 47th Ann. Mtg, Calgary, 1977.

    Google Scholar 

  45. OOE, M. and ULRYCH, T. J., Minimum entropy deconvolution with an exponential transformation, Geophys. Prospecting, 27, p. 458, 1979.

    Article  Google Scholar 

  46. STONE, D. G., Robust wavelet estimation by structural deconvolution, presented Soc. Explor. Geophys. 46th Ann. Mtg, Houston, Texas, 1976.

    Google Scholar 

  47. STONE, D. G., Estimation of reflection coefficients from seismic data, presented Soc. Explor. Geophys. 47th Ann. Mtg, Calgary, 1977.

    Google Scholar 

  48. STONE, D. G. and ATHERTON, J. L., Decomposition of seismic traces, presented Soc. Explor. Geophys. 49th Ann. Mtg, New Orleans, 1979.

    Google Scholar 

  49. KORMYLO, J. and MENDEL, J. M., On maximum likelihood detection and estimation of reflection coefficients,presented Soc. Explor. Geophys. 48th Ann. Mtg, San Francisco, 1978.

    Google Scholar 

  50. SHAH, P. C., GAVALAS, G. R. and SEINFELD, J. H., Error analysis in history matching: the optimum level of parameterization,Soc. Pet. Engrs J., p. 219, 1978.

    Google Scholar 

  51. SIRAKI, E. S. and TAYLOR, H. L., An application of sparse spike train concepts to seismic data processing, presented Soc. Explor. Geophys. 48th Ann. Mtg, San Francisco, 1978.

    Google Scholar 

  52. TAYLOR, H. L., Spikes and waveforms in well log and seismic trace comparisons, presented Houston Geophys. Soc. Symp. on Seismically Derived Pseudo Acoustic Impedance Logs, Houston, 1978.

    Google Scholar 

  53. BENVENISTE, A., GOURSAT, M. and ROGET, G. A robust adaptive procedure for solving a non-Gaussian identification problem. Optimization techniques—Part l, ed. J. Stoer, Springer-Verlag, Berlin, Heidelberg, New York, 1978.

    Google Scholar 

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Authors and Affiliations

  1. P.O. Box 354, Richardson, Texas, 75080, USA

    H. L. Taylor

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  1. H. L. Taylor
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Editors and Affiliations

  1. Seismograph Service (England) Limited, Keston, Kent, UK

    A. A. Fitch (Consultant) (Consultant)

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© 1981 Applied Science Publishers Ltd

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Taylor, H.L. (1981). The l 1 Norm in Seismic Data Processing. In: Fitch, A.A. (eds) Developments in Geophysical Exploration Methods. The Developments Series. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8105-8_3

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  • DOI: https://doi.org/10.1007/978-94-009-8105-8_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-8107-2

  • Online ISBN: 978-94-009-8105-8

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