1. Imaging conditions in depth migration algorithms
Bogdan G. Nita
Assistant Professor, Dept. of Mathematical Sciences
Montclair State University
May 11, 2006

2. Main results presented in this talk
Summation over frequencies with the data in wavenumber-frequency domain is a imaging condition
Summation over frequencies with the data in space-frequency domain is a imaging condition

3. Outline Motivation Spherical and plane waves
Depth migration algorithms and imaging conditions
Analytic examples
Conclusions

4. Outline Motivation Spherical and plane waves
Depth migration algorithms and imaging conditions
Analytic examples
Conclusions

5. Motivation
Provide a better understanding of today’s most used migration algorithms
Differentiate between their efficiency and capability
Develop better seismic processing algorithms (for imaging, internal multiple elimination etc).

6. Motivation (cont’d)
M-OSRP 2004 report Nita & Weglein discussed the inverse scattering internal multiple attenuation algorithm. They showed
monotonicity in pseudo-depth is equivalent to vertical time monotonicity
The algorithm attenuates multiples with headwaves subevents.
Also discussed: casting the algorithm in other domains

7. Outline Spherical and plane waves Motivation
Depth migration algorithms and imaging conditions
Analytic examples
Conclusions

8. Spherical and plane waves
+BC FT FT
+BC
Point-source/point-receiver response is the sum of all planewave responses.

9. The geometry of planewaves
Equation of the wave front in the xz-plane

10. The geometry of planewaves
Vertical intercept time
Def.:

11. The geometry of planewaves
Moving planewave
Compare with a Fourier planewave
kzz is important in phase-shift migration algorithms
hints towards the type of imaging when summing over frequencies in wavenumbers domain.

12. Outline Depth migration algorithms and imaging conditions Motivation
Spherical and plane waves
Depth migration algorithms and imaging conditions
Analytic examples
Conclusions

13. Depth migration algorithms
Wavenumber-frequency
Gazdag’s phase-shift, Stolt’s f-k, etc.
Space-frequency
Kirchoff, etc.
Criteria for this classification: the domain in which the integration over frequency (imaging) takes place.

14. Wavenumber-frequency algorithm
Apply shift term

15. Wavenumber-frequency algorithm
Source and receiver horizontal coordinates
Corresponding horizontal wavenumbers
Temporal frequency

16. Imaging step/condition
Each planewave event in the data is completely described by horizontal wavenumbers and depth z.
Downward extrapolation results in a change of z only
Depth z, vertical wavenumber kz and frequency are related to the vertical intercept time
Integration over frequency amounts to a imaging condition

17. Space-frequency algorithm
Apply shift term
Apply Diffraction term

18. Space-frequency algorithm
Source and receiver horizontal coordinates
Corresponding horizontal wavenumbers
Temporal frequency

19. Wavenumber-frequency algorithm
Source and receiver horizontal coordinates
Corresponding horizontal wavenumbers
Temporal frequency

20. Space-frequency algorithm
Source and receiver horizontal coordinates
Corresponding horizontal wavenumbers
Temporal frequency

21. Imaging step/condition
Each event in the data is described by horizontal and vertical positions which relate to the total traveltime
Integration over frequency amounts to a t=0 imaging condition

22. Variables dependence for wavenumber-frequency algorithm
the set of independent variables are
integration over frequency is performed while keeping the horizontal wavenumbers constant.
only the time component related to vertical wavenumber is taken to zero.

23. Variables dependence for space-frequency algorithm
the extrapolation term is both a downward and a lateral extrapolation
approximating the inner integrals puts ray-paths conditions and imposes relationships between wavenumbers
integration over frequency is taking all time components and hence the total traveltime to zero

24. Outline Analytic examples Motivation Spherical and plane waves
Depth migration algorithms and imaging conditions
Analytic examples
Conclusions

25. Analytic example: Stolt migration
data includes reflection and headwave for large offsets
prime indicates time derivative of the data

26. Analytic example: Stolt migration
Downward extrapolation
Imaging
Change of variable and integrate

27. Analytic example: Kirchhoff migration
Stationary Phase Approx.
Reflection
Headwave

28. Analytic example: Kirchhoff migration
Events are approximated by ray-paths diagrams

29. Analytic example: Kirchhoff migration
t=0 imaging
headwave is imaged at the wrong depth

30. Outline Conclusions Motivation Spherical and plane waves
Depth migration algorithms and imaging conditions
Analytic examples
Conclusions

31. Conclusions
Summation over frequencies with the data in wavenumber-frequency domain is a imaging condition
Summation over frequencies with the data in space-frequency domain is a imaging condition
The two types of algorithms lead to different results even for simple 1.5D media.

32. Acknowledgements Art Weglein for stimulating discussions
Sam Kaplan for careful reading of the paper
M-OSRP sponsors for the support