1. Attenuation of Diffracted Multiples
Gabriel Alvarez
Biondo Biondi
Antoine Guitton
Good morning:
My talk today will be on the attenuation of diffracted multiples. This is work that I have done with Biondo and Antoine and the corresponding paper is in the present report.
Stanford University

2. Alvarez-Biondi-Guitton
Goal
Introduce a method to attenuate diffracted multiples
on 2D data, based on an apex-shifted Radon transform.
The method is applied in image space, in particular to Angle Domain Common Image Gathers (ADCIG).
Before presenting the apex-shifted Radon transform, and at the risk of stating the obvious let me pose the following question: Can SRME attenuate diffracted multiples?
Well, the answer is yes, but SRME requires dense, regular, large aperture data which is hard to come by, specially with 3-D marine streamer acquisition.
It is here, then, that the apex-shifted Radon transform can become a suitable alternative.
Alvarez-Biondi-Guitton

3. Alvarez-Biondi-Guitton
Motivation
ADCIG
Depth (m)
Aperture angle 60
-60
1000
4000
Primaries
Aperture angle 60
-60
Multiples
Aperture angle 60
-60
Last year, Paul and Antoine showed that there were advantages in attenuating the multiples in the image space, that is, in common offset or angle image gathers. In particular, the showed, that, if we have an ADCIG like the
one shown here, applying a tangent-squared Radon tarnsform it was possible to attenuate the multiples while
preserving the primaries. The method works well as indicated in this example.
Notice that we used reciprocity to generate positive and negative aperture angles.
Alvarez-Biondi-Guitton

4. Alvarez-Biondi-Guitton
Motivation
Depth (m)
1000
4000
ADCIG
Aperture angle 60
-60 60
-60
Aperture angle
Primaries
Multiples
There are, however, situations in which the apex of the multiples is not located at zero aperture-angle and in that case the method does not perform very well. Since the basic assumption of the standard Radon transform is that the apex should be at zero-aperture angle, the transform gets confused and the apex-shifted multiples end up with the primaries.
Clearly, in this case something else is needed and this is what I will show here: that by explicitely accounting for the apex-shift distance during the Radon transform the apex-shifted multiples can be well-attenuated.
Alvarez-Biondi-Guitton

5. Apex-shifted Multiple
Horizontal Coordinate (m)
Depth (m)
500
3000
-1000
Receiver-side peg-leg multiples
Aperture angle (degrees)
Time (s) 20 40
1.35
1.6
Arrival times
Apex-shifted multiples can happen for a variety of reasons, the simplest of which is a dipping reflector as illustrated here. The different rays were traced from one Huygen’s source on the reflector to the surface. Here is the corresponding traveltimes plotted as a function of the aperture angle. Clearly, the apex of the hyperbola corresponds to an aperture angle for which the peg-leg multiple is simply vertical.
That peg-legs split and therefore do not have their apex at zero offset in CMPs was recognized long ago by Levin and Shah.
Alvarez-Biondi-Guitton

6. Alvarez-Biondi-Guitton
A Note on Terminology
Diffracted multiples:
Multiples with apex-shifted residual moveout
on ADCIGs migrated with primary velocity.
Specularly-reflected multiples:
Multiples whose residual moveout have their
apex at zero aperture angle on ADCIGs
migrated with primary velocity.
Before I go any further let me take a license with words and call diffracted multiples any multiple whos apex is shifted away from the zero aperture angle and specularly reflected multiple any multiples whose apex is at zero aperture angle.
Alvarez-Biondi-Guitton

7. Alvarez-Biondi-Guitton
What about SRME?
Can Surface-Related Multiple Elimination (SRME) attenuate the apex-shifted multiples?
Yes, but SRME requires dense, regular, large-aperture acquisition which is not generally available with today’s 3D streamer geometries.
An alternative: apex-shifted Radon transform of ADCIGs.
Before presenting the apex-shifted Radon transform, and at the risk of stating the obvious let me pose the following question: Can SRME attenuate diffracted multiples?
Well, the answer is yes, but SRME requires dense, regular, large aperture data which is hard to come by, specially with 3-D marine streamer acquisition.
It is here, then, that the apex-shifted Radon transform can become a suitable alternative.
Alvarez-Biondi-Guitton

8. Alvarez-Biondi-Guitton
Standard Radon Transform
Depth
ADCIG
Aperture angle
(γ)
Primaries
Multiple
Curvature (q)
Depth
We all know that the standard Radon transform will simply map an ADCIG to a 2-D space in which the horizontal coordiante is some measure of curvature and such that the primaries will then be be mapped near the zero curvature line.
Alvarez-Biondi-Guitton

9. Alvarez-Biondi-Guitton
Apex-shifted Radon Transform z’ q γa
Primaries
multiples γ z
ADCIG
S-R multiples
The apex-shifted Radon transform is similar, but instead of mapping the ADCIG to a 2-D plane it maps it to a 3-D cube in which the extra dimension is apex-shift distance. I am using here the letter h that should not be confused with half-offset.
In this cube, the perfectly-flat primaries would map to this
plane, the plane of zero-curvature, wereas events with curvature will map to similar planes according to the value of their curvature. Interestingly, the specularly reflected mutliples in particular would map to the zero apex-shift plane, since that is my definition of specularly reflected multiples. So, ideally, the primaries and the SR multiples will map to two orthogonal planes whereas the diffracted multiples will map elsewhere in the cube according to their apex-shift distance and their curvature.
Alvarez-Biondi-Guitton

10. Alvarez-Biondi-Guitton
Separating Primaries and Multiples
τ,z’ q hγ
Zero out the primary region
in the (z’,q,h) cube.
Map the multiples back to
the (z,γ) plane (ADCIG).
Subtract the multiples
from the original ADCIG
to get the primaries.
The process of separating the primaries and the multiples should look pretty straightforward now, since all we need to do is to zero-out, with an appropriate taper, the primary region which will leave only the multiples, then inverse transforn the multiples and subtract them from the original data to get the primaries. Pretty much as with the standard approach.
Alvarez-Biondi-Guitton

11. Alvarez-Biondi-Guitton
Image space vs. data space
In complex geology the primaries are more likely to be
horizontal in ADCIGs than in NMO-corrected CMPs.
However
In data space the multiple moveout is more intuitive
(predictable)and easy to visualize.
I hope that it is clear that there is nothing in the previous description that is specific to the image space, so we could just as well apply the same procedure in data space, that is, to NMO-corrected CMPs. There is, therefore, the question of which domain is better. There are arguments in favor of both choices, but we have preferred to use the image space because, in complex media, the priamries are much more likely to be flat after prestack migration than after NMO. But again, this choice is not without consequence and we could make a case in favor of the data space based on the fact that the multiples are more predictable and more important, because attenuating the multiples in data space would help the choice of the migration velocities for the primaries. As most of the things we do, this is a judgment call which needs to be considered on a case-by-case basis.
Delaying the multiple removal after imaging may
compromise the choice of the migration velocities
for the primaries.
Alvarez-Biondi-Guitton

12. Apex-shifted Radon Transform
From model space to data space (ADCIGs):
γ: aperture angle
γa: apex-shift distance
q: moveout curvature
z, z’: depth
O.K., I will show you now the specifics of our Radon transform implementation. We use a modified version of the tangent-square transform following the work by Biondi and Symes.
The modeling equation is this where gamma is the aperture angle and as usual q is the curvature measure.
The adjoint transform is this, which gives a cube as shown previously.
The transform is implemented as a least-squares inverse
with a sparsity constraint.
Alvarez-Biondi-Guitton

13. Alvarez-Biondi-Guitton
2D GOM seismic line
CMP position
4000
24000
1000
Depth (m)
To illustrate attenuation of the diffracted multiples, we applied the apex-shifted Radon transform to a part of this
2D seismic line from the Gulf of Mexico. I will illustrate the
prestack results with two ADCIGs from each edge of the salt body as indicated by the arrows. Of particular interest will be the attenuation of these criscrossing noises which we associate with the presence of the diffracted multiples.
5000
Stack of migrated angle gathers
Alvarez-Biondi-Guitton

14. Alvarez-Biondi-Guitton
ADCIGs
Sediments
Depth (m)
1000
5000 60
-60
Aperture-angle (degrees)
Aperture-angle (degrees)
-60 60
1000
5000
Depth (m)
Below salt
Just to show that these events don’t happen everywhere in the line, here is an ADCIG below the sediments. Notice how the apex correspond very closely to zero aperture-angle. Similar situation here for an ADCIG below the salt body. This shows that the shift in the apex of the moveout is not due to a systematic migration error, for example.
Alvarez-Biondi-Guitton

15. Alvarez-Biondi-Guitton
2D GOM seismic line
Stack of migrated angle gathers
Depth (m)
1000
5000
CMP position
4000
24000
To illustrate attenuation of the diffracted multiples, we applied the apex-shifted Radon transform to a part of this
2D seismic line from the Gulf of Mexico. I will illustrate the
prestack results with two ADCIGs from each edge of the salt body as indicated by the arrows. Of particular interest will be the attenuation of these criscrossing noises which we associate with the presence of the diffracted multiples.
Alvarez-Biondi-Guitton

16. Alvarez-Biondi-Guitton
Salt-edge ADCIGs
ADCIG1: Left edge
Depth (m)
1000
5000 60
-60
Aperture-angle (degrees)
Aperture-angle (degrees)
-60 60
1000
5000
Depth (m)
ADCIG2: Right edge
This is the first ADCIG, taken at the left edge of the salt. The arrows point to a couple of diffracted multiples.
This is the ADCIG on the right edge of the salt. Although no so clearly visible, some of the multiples are actually diffracted multiples as I will show later.
Alvarez-Biondi-Guitton

17. Comparison of Radon Transforms
2D transform
Depth (m)
1000
5000
2000
-400
Curvature (m)
1000
5000
Depth (m)
γa=0 plane from 3D transform
Curvature (m)
2000
-400
Primaries
Let me show you very quickly a comparison of the 2D and 3D Radon transforms. In the standard Radon transform the priamries, the specularly-reflected multiples and the diffracted multiples are all mapped to a single plane. The primaries are clelarly distinguished here in the shallow part but it is not so clear which are diffracted multiples and which are specularly-reflected multiples. Furthermore, the diffracted multiples are not really focused anywhere, but rather they are mapped as background noise, specially at the highest and smallest curvature values. By contrast, if we take the zero apex-shift plane from the cube, we get a similar representation, but without the diffracted multiples, since by definition they are mapped else where. Note that the primaries look weak because they are mapped to all the other h planes as well.
Alvarez-Biondi-Guitton

18. A look at the 3-D Radon cube
plane q=0
Depth (m)
1000
4000 30
-30
Apex-shift (degrees)
plane q=1200 30
-30
Apex-shift (degrees)
-400
2000
Curvature (m)
plane γa=10
At the risk of over-stting the obvious, let me show you some other planes from the 3-D cube. Here is the zero-curvature plane, that is, the plane to which the primaries map. The primaries look like horizontal lines because they have zero curvature irrespective of the apex-shift distance. In this particular case, the primaries are only visible at the shallow parts of the data. For comparison, here is a similar plane but at a higher curvature value. The energy near zero-apex shift corresponds to the specularly reflected multiples whereas the rest corresponds to diffracted multiples. Finally, let me show you the plane for an apex-shift of 10 degrees (remember that in an ADCIG the horizontal axis is angle). Here we see the diffracted mu.tiples and in particular the intersection with the previous plane.
Alvarez-Biondi-Guitton

19. Results of Multiple Attenuation
ADCIG1: Left edge
Depth (m)
1000
5000 60
-60
Aperture-angle (degrees)
Aperture-angle (degrees)
-60 60
1000
5000
Depth (m)
ADCIG2: Right edge
I will now show the results of the multiple attenuation on the two ADCIGs below the salt edges. I will do so for a close-up region indicated by the green rectangles.
Alvarez-Biondi-Guitton

20. Alvarez-Biondi-Guitton
Primaries ADCIG 1
standard transform
Depth (m)
3200
5200 40
-40
Aperture-angle (degrees)
Aperture-angle (degrees)
-40 40
3200
5200
Depth (m)
apex-shifted transform
Here is the primary result using the 2D RT. The specularly reflected multiples were satisfactorily attenuated but no so the diffracted multiples. Here is the result with the 3D RT. The diffracted multiples were not completely removed but were attenuated. Unfortunately, it doesn’t seem as if there are primaries here that could be uncovered by the attenuation of the multiples.
Alvarez-Biondi-Guitton

21. Alvarez-Biondi-Guitton
Multiples ADCIG 1
standard transform
Depth (m)
3200
5200 40
-40
Aperture-angle (degrees)
Aperture-angle (degrees)
-40 40
3200
5200
Depth (m)
apex-shifted transform
To further stress this point, here is the multiples panel for the 2D RT and for the 3D RT. Notice that the diffracted multiples were now correctly identified as multiples.
Alvarez-Biondi-Guitton

22. Discrimination of Multiples ADCIG 1
Diffracted multiples
Depth (m)
3200
5200 40
-40
Aperture-angle (degrees)
Aperture-angle (degrees)
-40 40
3200
5200
Depth (m)
S-R multiples
Just for fun, I tried further segregating the diffracted mutiples from the specularly reflected multiples for the 3D result.
Alvarez-Biondi-Guitton

23. Alvarez-Biondi-Guitton
Primaries ADCIG 2
Standard transform
Depth (m)
3200
5200 40
-40
Aperture-angle (degrees)
Aperture-angle (degrees)
-40 40
3200
5200
Depth (m)
Apex-shifted transform
Here is the comparison of the primaries for the second ADCIG, that is, below the right edge of the salt. Again, with the 2D transform the attenuation of the specularly reflected multiples is acceptable, but not so for the diffracted multiples. In contrast, the 3D result shows good attenuation of the diffracted multiples.
Alvarez-Biondi-Guitton

24. Alvarez-Biondi-Guitton
Multiples ADCIG 2
Standard transform
Depth (m)
3200
5200 40
-40
Aperture-angle (degrees)
Aperture-angle (degrees)
-40 40
3200
5200
Depth (m)
Apex-shifted transform
Here is the comparison of the multiples. Again, notice how the diffracted multiples do not appear on the result with the 2D transform. They do appear, however, in the result with the 3D transform.
Alvarez-Biondi-Guitton

25. Alvarez-Biondi-Guitton
2D GOM seismic line
Stack of migrated angle gathers
Depth (m)
1000
5000
CMP position
4000
24000
I wil finish by showing the results of the multiple attenuation on the stack of the ADCIGs. I will show the results only on this subset of the data.
Alvarez-Biondi-Guitton

26. Primaries on Angle Stacks
CMP position (m)
4000
10000
Depth (m)
3000
5000
2D-RT
3000
5000
Depth (m)
3D-RT
3000
5000
Depth (m)
Diff
Here is the stack of the primaries with the 2D transform. Above the green arrow, it is primaries only and below the green arrow it is essentially residual multiples. Here is the result with the 3D transform. In the primary region, above the arrow, the results are similar to the 2D case but below, much more multiple energy, mostly from the diffracted multiples have been removed, specially in the oval region.
The difference panel shows that not much difference exists in the primary region but significant difference exists in the multiple region.
Alvarez-Biondi-Guitton

27. Multiples on Angle Stacks
CMP position (m)
4000
10000
Depth (m)
3000
5000
2D-RT
3000
5000
Depth (m)
3D-RT
3000
5000
Depth (m)
Diff
Here is a similar comparison for the multiples. Again, the main difference is in the attenuation of the diffracted multiples as expected.
Alvarez-Biondi-Guitton

28. Alvarez-Biondi-Guitton
Conclusions
The apex-shifted Radon transform is an effective
way to attenuate diffracted multiples in the image
space.
The usual trade-off between multiple attenuation
and primary preservation remains.
Alvarez-Biondi-Guitton

29. Alvarez-Biondi-Guitton
Opportunities and challenges
In 3D data, the apex-shift is a function of azimuth
as well as aperture angle. We will takeadvantage
of the development of 3D ADCIGs.
The challenge is to handle the increased
dimensionality of the problem.
Alvarez-Biondi-Guitton

30. Alvarez-Biondi-Guitton
Thank you for your attention.
I will be happy to entertain your
questions.
Thank you very much for your attention.
Alvarez-Biondi-Guitton

31. Apex-shifted Radon Transform
From model space to data space (ADCIGs):
From data space to model space:
γ: aperture angle
h: apex-shift distance
q: moveout curvature
z, z’: depth
γ: aperture angle
h: apex-shift distance
q: moveout curvature
z, z’: depth
O.K., I will show you now the specifics of our Radon transform implementation. We use a modified version of the tangent-square transform following the work by Biondi and Symes.
The modeling equation is this where gamma is the aperture angle and as usual q is the curvature measure.
The adjoint transform is this, which gives a cube as shown previously.
Alvarez-Biondi-Guitton

32. Constrained LS inversion
Apex-shifted Radon transform:
Objective Function:
ε: controls level of sparsity in model space
b: controls minimum value below which to zero out model space
Cauchy regularization
If we consider the apex-shifted RT as a linear operator, then we can say that d=Lm and we choose m as the LS solution of this objective function. Here obviously, the fiorst term is the data fitting term and the second one is a Cauchy regularization to enforce sparseness in the Radon domain. Here epsilon and b are two constants chosen a priori, epsilon balances the relative contribution of the data-fitting goal with the regularization and b is a threshold parameter to set the noise level in the RT domain.
Alvarez-Biondi-Guitton