Advertisement

From Radon to Kirchhoff Migration

  • M. Novotný
Part of the Theory and Practice of Applied Geophysics book series (THPAG, volume 5)

Abstract

The interrelation between the Radon wave field extrapolation and integral solutions of boundary value problems posed for scalar wave equation is studied. Using the stationary phase method it is found that the Radon extrapolator degenerates into the Rayleigh-Sommerfeld solution of the Dirichlet problem in the far-field zone. The derivations are done for the 3-D case, but the 2-D analogues are also mentioned. The practical consequences and advantages of slant-stack representation of space-under sampled noisy data are discussed.

Keywords

Green Function Dirichlet Problem Integral Solution Scalar Wave Equation Stationary Phase Method 
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. Berkhout, A.J., 1980: Seismic migration — Imaging of acoustic energy by wave field extrapolation. Amsterdam/New York, Elsevier (north Holland Publ.Co.)Google Scholar
  2. Bessonova, E.N., V.M. Fishman, V.Z. Ryaboi and G.A. Sitnikova, 1974: The Tau method for inversion of traveltimes — I. Deep sounding data. Geophys. J.R. Astr. Soc., 36, 377–398.CrossRefGoogle Scholar
  3. Born, M. and E. Wolf, 1975: Principles of optics, 5th ed., New York, Pergamon Press.Google Scholar
  4. Drysk, H. and D.W. McGowan, 1986: A slant-stack procedure for point-source data, Geophysics 51, 1370–1386.CrossRefGoogle Scholar
  5. Chapman, C.H., 1978: A new method for computing synthetic seismograms, Geophys. J.R. Astr. Soc. 54, 481–518.CrossRefGoogle Scholar
  6. Chapman, C. II., 1981: Generalized Radon transform and slant stacks, Geophys. J.R. Astr. Soc. 66, 445–453.CrossRefGoogle Scholar
  7. Claerbout, J.F., 1985: Imaging the Earth’s Interior. Blackwell scientific publications.Google Scholar
  8. Clayton, R.W. and A. McMechan, 1981: Inversion of reflection data by wave-field continuation. Geophysics 46, 860–868.CrossRefGoogle Scholar
  9. Diebold, J.B. and P.L. Stoffa, 1981: The traveltime equation, tau-p mapping, and inversion of common midpoint data. Geophysics 46, 238–254.CrossRefGoogle Scholar
  10. Estevez, R., 1977: Slant stacks and interval of optimum stacking. SEP Report No.11, Stanford University,Stanford,California.Google Scholar
  11. Hubral, P., 1981: Slant-stack migration. In Festschrift Theodor Krey, Prakla-Seismos, Hannover, 72–78.Google Scholar
  12. Kuhn, M.J. and K.A. Alhilali, 1977: Weighting factors in the construction and reconstruction of acoustical wave fields. Geophysics 42, 6, 1183–1198.CrossRefGoogle Scholar
  13. Levin, S., 1980: A frequency-dip formulation of wave-theoretic migration in stratified media. Acoustical Imaging 9, Plenum Press, 681–697.Google Scholar
  14. McMechan, G.A., 1983: P-x imaging by localized slant-stacks of T-x data, Geophys. J.R. Astr. Soc. 72, 213–221.CrossRefGoogle Scholar
  15. Milkerit, B., 1987: description and inversion of seismic data — an instantaneous slowness approach. Geophysical prospecting 35, 875–894.Google Scholar
  16. Morgan, T.R., 1983: Foundations of wave theory for seismic exploration. Boston, International Human Resources Development Corporation.Google Scholar
  17. Morse, P.M. and H. Feshbach, 1953: Methods of Theoretical Physics, New York. McGraw-Hill Book company.Google Scholar
  18. Müller, G., 1971: Direct inversion of seismic observations. Zeits. Geophys., 37, 225–235.Google Scholar
  19. Novotny, M., 1990: Trace interpolation by slant-stack migration. Geophysical Prospecting, to be published.Google Scholar
  20. Papoulis, A., 1968: Systems and transforms with applications in optics. New York, McGraw-Hill Book Company.Google Scholar
  21. Radon, J., 1917: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Bei. Vehr. Akad. Wiss., 69, 262–277.Google Scholar
  22. Rayleigh, 1897: On the passage of waves through apertures in plane screens and allied problems. Phil. Mag. 43, 259.CrossRefGoogle Scholar
  23. Sommerfeld, A., 1912: Die Greensche Funktion der Schwingungsgleichungen. Jahresber.Deut.Math.Ver., 21, 309.Google Scholar
  24. Schultz, P.S. and J.F. Claerbout, 1978: Velocity estimation and downward continuation by wave front synthesis, Geophysics, 43, 691–714.CrossRefGoogle Scholar
  25. Stoffa, P.L., P. Buhl, J.B. Diebold and F. Wenzel, 1981: Direct mapping of seismic data to the domain of intercept time and ray parameter — A plane wave decomposition. Geophysics 46, 255–267.CrossRefGoogle Scholar
  26. Treitel, S., P.R. Gutowski and D.E. Wagner, 1972: Plane-wave decomposition of seismograms. Geophysics 47, 1375–1401.CrossRefGoogle Scholar
  27. Tygel, M. and P. Hubral, 1989: Constant velocity migration in the various guises of plane-wave theory, to be published in Surveys of Geophysics, Kluwer Academic Publishers.Google Scholar

Copyright information

© Springer Fachmedien Wiesbaden 1992

Authors and Affiliations

  • M. Novotný
    • 1
  1. 1.Geofyzika BrnoBrnoCzechoslovakia

Personalised recommendations