1. Prestack depth migration in angle-domain using beamlet decomposition:
Local image matrix and local AVA
Ru-Shan Wu and Ling Chen
Modeling and Imaging Laboratory/IGPP
University of California, Santa Cruz
†Presently at Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China

2. Beamlet decomposition: Wave field in angle-domain
Local image matrix and local scattering matrix
Effect of acquisition aperture
Local AVA: Preliminary tests
Conclusion

3. True-reflection imaging in angle-domain
Preserving the relative amplitudes of scattered waves w.r.t. incident waves.
Benefits:
Improve image (total strength image) quality, especially for steep reflectors.
Reduce artifacts (angle-domain filtering).
Provide basis for local AVA and local inversion

4. True-amplitude imaging in angle-domain
Amplitude corrections (for ray theory see Hubral et al., Bleistein et al., Xu et al., Audebert et al., ……):
Transmission loss (boundary reflection and scattering)
Geometric spreading (for ray method)
Nonuniform information distribution: Jacobian (Beylkin determinant)
Acquisition aperture effects (hit-count for ray method)
Intrinsic attenuation (Anelasticity)
For Ray theory: Peter Hubral’s group, Norm Bleistein, Martin Tygel, S. Xu and others.

5. True-reflection imaging in angle-domain for wave-equation based methods
Preserving the relative amplitudes of scattered waves w.r.t. incident waves:
Nonuniform information distribution: Jacobian
Acquisition aperture effects (in angle-domain) (including the geometric spreading and hit-count for ray method)
Transmission and anelastic losses are less important, especially for small-angle reflections

7. (each beamlet is a windowed plane wave)
Windowed plane waves
G-D frame atoms
is a Windowed Plane wave
(each beamlet is a windowed plane wave)

8. Local plane waves
Local plane wave: a superposition of windowed plane waves of the same local wavenumber from all neighboring windows:
Partial reconstruction of wavefield (mixed domain wavefield: local phase–space):
The corresponding propagating angle:

9. (includes aperture and propagation effects)
Source
Receiver * *
High-velocity
body
Target area
Local Image Matrix
(includes aperture and propagation effects)
Local Scattering
Matrix

10. Local image matrix in homogeneous medium
Point scatterer
Planar reflector
shallow
deep
Local image matrix in homogeneous medium
(total 201 shots with 176 left-hand receivers )

11. Local image matrix: image condition in
beamlet domain and mixed domain
Forward-propagated source field:
Backward-propagated scattered field:

12. Local image matrix: Serves as the Jacobian Where
Ws and WRs are the wave fields in angle-domain by beamlet
decomposition

13. Stacking over frequency to get the final image
In the local angle-domain:
The final image in space domain:

14. Local Reflection Analysis (AVA):
For planar reflectors: the local image matrix can be represented as:
with

15. CDAI (common dip-angle image) gathers
Sum up all reflections for a common dip-angle: CDAI gather
Obtain the dip-angle of the local reflectors from CDAI.

16. CRAI (common reflection-angle image) gathers
Sum up reflected energy for a common reflection-angle for all possible dip-angles: CRAI.
Performing local AVA from CRAI gathers.
The calculation of local reflection coefficients:

17. Local AVA for an oblique interface in homogeneous background

18. Local image matrices at a point on the middle of dipping interface 14°
obtained from 80 shots with a two-side receiver array (513 receivers).
The dotted line corresponds to the theoretical prediction without aperture effect.

19. CDAI gathers for a local reflector at its central location
Obtain the dip-angle of local reflectors from
the CDAI gathers

20. Calculated reflection coefficients from CRAI gathers
. Angle-dependent reflection coefficients at the interface using 256 shots
with 513 two-side detectors for the horizontal layered model
with different velocity contrasts: (a) 10%; (b) 25%; (c) 50%; (d)150%

21. Dotted: synthetic; Red: 513 points two-sides
Blue: 257 points one-side; Green: 129 points two-sides
Angle-dependent reflection coefficients at the interface obtained from LIM in case of 10% velocity contrast for the horizontal layered model

22. Local image matrix and the local scattering matrix
The local image matrix has the acquisition-aperture and propagation effects included. The purpose of the imaging/inversion is to recover the real local scattering matrix and obtain the local reflection coefficients. To achieve the true-reflection imaging, we need to estimate the acquisition-aperture effect and apply the correction.

23. Acquisition-Aperture Efficacy (Effect of the source-receiver configuration)
Acquisition-Aperture Efficacy (AAE) Matrix
Acquisition-Aperture Dip-response function
Aperture corrections

24. (includes propagation effects)
Source
Receiver * *
Overburden
structures
Assume scattering
Coefficients as 1
Target area
Acquisition-Aperture Efficacy:
(includes propagation effects)

25. Acquisition-aperture efficacy matrix
With unit impulses at both source and receivers, the local acquisition-aperture efficacy matrix is obtained as:
Where G’s are the Green’s functions in beamlet domain

26. dip-response function
Acquisition-aperture
dip-response function
Acquisition-aperture dip-response as a function of dip-angle of local interface (reflector), which reduce the AAE matrix into a vector:
with

27. Acquisition-Aperture Dip-Response
(Acquisition Configuration Response) * S1 S2 * * S3

28. image by common-shot prestack G-D migration
Acquisition-Dip-Response (horizontal reflector) from all the 325 shots
Acquisition-Dip-Response (45 down from horizontal) from all the 325 shots

29. Directional illumination
G-D beamlet prestack
migration image
Acquisition-Dip
Response for 45o
reflectors
Improved image after
Directional illumination
correction

30. Conclusion
Local image matrix can be obtained from the local incident and scattered plane waves based on beamlet decomposition
The goal of true-reflection imaging in angle-domain is to remove the acquisition aperture effect and propagation effect through directional illumination analysis and the corresponding corrections

31. Conclusion (continued)
CDAI and CRAI gathers can be deduced from local image matrices (after corrections)
CDAI gathers can be used to determine the dip-angles of local reflectors
CRAI gathers can be used for local AVA analysis (and further for local inversion)

32. Acknowlegement
We thank the support from WTOPI Research Consortium at UCSC
We thank the support from DOE Project at UCSC
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