Imaging of diffraction objects
using post-stack
reverse-time migration
I. Silvestrov* (OPERA, IPGG SB RAS), R. Baina (OPERA)
and E. Landa (OPERA)
Outline
•
•
•
•
Motivation
Description of the proposed algorithm
Synthetic example based on Sigsbee model
Real-data example
2
Diffraction imaging algorithm in dip-angle domain
Landa, E., Fomel S., and Reshef M. 2008. 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
3
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)
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Post-stack reverse-time migration
For given data d(x; t) we solve wave-equation with half-velocity V/2 in reverse-time:
1
 2u
 Δu =  d ξ,t δ  x  ξ δ  z , t < T
2
2
V x, z  / 2 t
x
u  x, z,t  t T = u / t
t T
(1)
= 0,
Image is simply the wavefield at zero time:
I x, z  = ux, z, t  t 0
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.
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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):
p  2 sin( ) / V
p0

p  0.0012s / m
Model
Zero-offset data
Plane-wave components of ZO data
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Common image gathers in surface dip-angle domain
1 2
p=-0.0012 s/m
p=0
p=0.0012 s/m
Migration of plane-wave data components
1
2
p
p
CIGs with respect to
“surface” dip
Reflection is a focused event.
Diffraction is a horizontal line at the correct diffraction position.
How to separate them?
Diffraction separation using Kurtosis measure
Inverse Kurtosis measure:
2
 N 2  N 4
1
K =   xi  /  N  xi 
 i=1   i=1 
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
Plane-wave decomposition using
sparse local Radon transform
Local Radon transform is defined as:
And its adjoint as:
mτ, p,x0  =
~
d x,t =
x  x0  w
 d x,τ + px  x 
x  x0  w
p  p0 x0  x  w
  mt  px  x , p,x 
p  p0 x0  x  w
To find the model m( , p, x0 )
squares misfit:
0
0
0
we use greedy approach to minimize the least-
2
J m = d  Lm l
2
The plane-wave data section is obtained by summation over all local windows:
d p x,t =
x0  x  w
 mt  px  x , p,x .
x0  x  w
0
0
Wang, J., Ng, M., and Perz, M. 2010. Seismic data interpolation by greedy local Radon
transform. Geophysics 75(6), WB225-WB234.
Giboli, M., Baina, R., and Landa, E., 2013. Depth migration in the offset-aperture domain:
Optimal summation. SEG Technical Program Expanded Abstracts, 3866-3871
The proposed algorithm for diffraction separation
based on post-stack RTM
1. Plane-wave decomposition of zero-offset stack
•
Sparse local Radon transform based on greedy approach
2. Depth migration of each plane-wave seismogram
• RTM with zero-time imaging condition
3. Resorting of images into CIGs with respect to dip in
data domain
4. Diffraction/reflection separation based on defocusing
criteria
• Sparse local Radon transform based on greedy approach
• Inverse kurtosis as a measure of defocusing
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Sigsbee model (post-stack RTM result)
Simple part
Complex part
Two parts of the model will be considered in diffraction imaging
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Zero-offset section
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Plane-wave data component of zero-offset section
Horizontal slowness p=-0.00014 s/m
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Plane-wave data component of zero-offset section
Horizontal slowness p=0
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Common image gathers in simple part
X=6000
Before separation
X=6000
After separation
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Diffraction separation result in simple part
Before separation
After separation
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Diffraction separation result in complex part
Before separation
After separation
CIG at 15200m
before and after
separation
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Snapshots for diffraction and reflection below salt body
Reflection
Diffraction
Exploding reflector modeling
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Snapshots for diffraction and reflection below salt body
Reflection
Diffraction
Exploding reflector modeling
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Snapshots for diffraction and reflection below salt body
Diffraction’s and reflection’s responses are similar at the surface
Diffraction
Reflection
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
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Zero-offset section after redatuming
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Diffraction separation result in complex part
Initial image
Diffraction image
for initial data
Diffraction image
for redatumed data
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CIGs before and after redatuming
X=15200
Before redatuming
After redatuming
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Real-data example. Oseberg oil field in the North Sea.
Zero-offset stack obtained using path-integral summation approach
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Common image gathers
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Diffraction image
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Diffraction wavefield
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Full wavefield
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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
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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.
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