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Attenuation of Diffracted Multiples

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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

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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

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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

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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

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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

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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

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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

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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

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Separating Primaries and Multiples τ,z’ q 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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


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