1. Imaging of diffraction objects using post-stack reverse-time migration
I. Silvestrov* (OPERA, IPGG SB RAS), R. Baina (OPERA)
and E. Landa (OPERA)

2. Outline
Motivation
Description of the proposed algorithm
Synthetic example based on Sigsbee model
Real-data example

3. Diffraction imaging algorithm in dip-angle domain
Landa, E., Fomel S., and Reshef M Separation, imaging, and velocity analysis of seismic diffractions using migrated dip-angle gathers. 78th Annual International Meeting, SEG, Expanded Abstracts, 2176–2180
Diffractions and reflections have different shapes in migrated dip-angle domain

4. Post-stack Reverse-time migration (RTM)
Motivation
Diffraction imaging in areas where Kirchhoff migration fails (e.g. subsalt)
Numerical efficiency of the diffraction imaging algorithm
Our choice:
Post-stack Reverse-time migration (RTM)

5. Post-stack reverse-time migration
For given data d(x; t) we solve wave-equation with half-velocity V/2 in reverse-time:
(1)
Image is simply the wavefield at zero time:
Why we can not use previous approach for diffraction separation?
Due to summation over receivers in (1) we do not have extra dimension for straightforward construction of CIGs
Analyzing the wavefield at zero time is equivalent to analyzing the image itself. However, we want to analyze the data and not the image.

6. Common image gathers in surface dip-angle domain
As an extra dimension in CIGs we propose to use dip of event in data domain
(horizontal slowness):
Model
Zero-offset data
Plane-wave components of ZO data

7. Common image gathers in surface dip-angle domain1 2 1 2 p p
p= s/m
p=0
p= s/m
CIGs with respect to
“surface” dip
Migration of plane-wave data components
Reflection is a focused event.
Diffraction is a horizontal line at the correct diffraction position.
How to separate them?

8. Diffraction separation using Kurtosis measure
Inverse Kurtosis measure:
Kurtosis is a measure of peakedness of a probability distribution.
Inverse Kurtosis is low for focused events.
At the same time inverse Kurtosis is large for coherent events as a correlation of a squared signal with a constant.
The events above a predefined threshold level are considered as diffractions

9. Plane-wave decomposition using sparse local Radon transform
Local Radon transform is defined as:
And its adjoint as:
To find the model we use greedy approach to minimize the least-squares misfit:
The plane-wave data section is obtained by summation over all local windows:
Wang, J., Ng, M., and Perz, M Seismic data interpolation by greedy local Radon transform. Geophysics 75(6), WB225-WB234.
Giboli, M., Baina, R., and Landa, E., Depth migration in the offset-aperture domain: Optimal summation. SEG Technical Program Expanded Abstracts,

10. Plane-wave decomposition of zero-offset stack
The proposed algorithm for diffraction separation based on post-stack RTM
Plane-wave decomposition of zero-offset stack
Sparse local Radon transform based on greedy approach
Depth migration of each plane-wave seismogram
RTM with zero-time imaging condition
Resorting of images into CIGs with respect to dip in data domain
Diffraction/reflection separation based on defocusing criteria
Inverse kurtosis as a measure of defocusing

11. Sigsbee model (post-stack RTM result)
Simple part
Complex part
Two parts of the model will be considered in diffraction imaging

12. Zero-offset section

13. Plane-wave data component of zero-offset section
Horizontal slowness p= s/m

14. Plane-wave data component of zero-offset section
Horizontal slowness p=0

15. Common image gathers in simple part
X=6000
X=6000
Before separation
After separation

16. Diffraction separation result in simple part
Before separation
After separation

17. Diffraction separation result in complex part
CIG at 15200m
before and after separation
Before separation
After separation
After separation

18. Snapshots for diffraction and reflection below salt body
Exploding reflector modeling

19. Snapshots for diffraction and reflection below salt body
Exploding reflector modeling

20. Snapshots for diffraction and reflection below salt body
Diffraction’s and reflection’s responses are similar at the surface
Reflection
Diffraction
Redatuming level
Redatuming may be used to simplify the wavefield
Reshef M., Lipzer N., Dafni R. and Landa E., 3D post-stack interval velocity analysis with effective use of datuming, Geophysical Prospecting 1(60), 18–28, January 2012

21. Zero-offset section after redatuming

22. Diffraction separation result in complex part
Diffraction image
for redatumed data
Initial image
Diffraction image
for initial data

23. CIGs before and after redatuming
X=15200
Before redatuming
After redatuming

24. Real-data example. Oseberg oil field in the North Sea.
Zero-offset stack obtained using path-integral summation approach

25. Common image gathers

26. Diffraction image

27. Diffraction wavefield

28. Full wavefield

29. Conclusion
We propose a method for imaging small scale diffraction objects based on post-stack Reverse-time migration
The method is based on separation between specular reflection and diffraction components of the total wavefield in the migrated domain. We used continuity of diffractions in the surface dip-angle CIGs as a criterion for separating reflections from diffractions
Synthetic and real data examples illustrate efficient application of the method

30. Acknowledgements
The authors thank TOTAL for supporting this research. OPERA is a private organization funded by TOTAL and supported by Pau University whose main objective is to carry out applied research in petroleum geophysics.