Seismic imaging in the curvelet domain: achievements and perspectives

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Seismic imaging in the curvelet domain
achievements and perspectives

• Hervé Chauris(1) Jianwei Ma(1,2)
• EAGE 2009 - Amsterdam

• Centre de Géosciences, Mines ParisTech, France
• Institute of Seismic Exploration, School of
  Aerospace, Tsinghua University, Beijing, China

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Locally coherent events
Time common-offset section
Same section after depth migration
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Locally coherent events

• Context seismic modeling / migration

Diffraction point
Globally coherent event
Locally coherent event
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Curvelets and other lets
Taking into account band-limited data ?
Curvelet
Ridgelet
Contourlet
Shearlet
Wavelet
EPW
Surfacelet
Seislet
Bandlet,
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Potential of curvelets
Single curvelet
Seismic data
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Introduction Curvelets
Curvelets for different coefficients cµ(x,z)
shift
stretch
rotation
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Content

• Introduction what are curvelets
• Curvelets and seismic processing tasks
• pre-/post-processing
• seismic migration
• seismic demigration/migration velocity
  estimation
• Conclusions

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Curvelets and seismic applications

• Review
• Ma and Plonka, 2009
• Workshop EAGE 2007 (London), Chauris and Douma
• Data denoising, interpolation and compression
• Hennenfent and Herrmann, 2006 Droujinine et al.,
  2007
• Herrmann et al., 2008a Sacchi et al., 2007
• Herrmann et al., 2008b Fomel, 2007
• Lin and Herrmann, 2007
• Neelamani et al, 2008
• Seismic modeling and migration
• Douma and de Hoop, 2007
• Chauris, 2006
• Velocity model estimation
• Chauris and Nguyen, 2007
• Chauris and Nguyen, 2008

redundant transform
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Seismic propagation/migration

• Theoretical results
• Candes and Donoho, 2003
• Practical results
• Douma and de Hoop, 2006
• Douma and de Hoop, 2007
• Chauris, 2006

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Use of digital curvelets
Fast curvelet transform
Migrated section in initial model
Curvelet processing
Fast inverse curvelet transform
Perturbed section
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Migration in heterogeneous models
Smooth heterogeneous 2-D model
Kirchhoff migration
First-order curvelet migration
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Migration in heterogeneous models
First-order curvelet migration
Kirchhoff migration
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Migration in heterogeneous models
First-order approximation not good enough
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Demigration/migration
Sensitivity of the migrated result with respect
to the velocity model
Initial velocity model
Triplicated ray field
Velocity perturbation (up to 200 m/s)
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Demigration/migration

• Sensitivity of a migrated image with respect to
  the background velocity model

Given velocity model
Migration
?
Local velocity perturbation
Migration
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Input data
RayBorn 2-D synthetic data set (offsets from 0
to 2 km)
Offset 600 m
Initial image
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Demigration/migration
Sensitivity of the migrated result with respect
to the velocity model
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Demigration/migration
Depth difference reduced from 60 m to less than 2
m
Initial / exact
Predicted / exact
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Demigration/migration
Sensitivity of the migrated result with respect
to the velocity model
Unperturbed part
Modified part
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Common Image Gathers
The prediction takes into account the lateral
velocity variations
Prediction with curvelets
Exact
Reference image
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Demigration/migration

• Sensitivity of a migrated image with respect to
  the background velocity model

Given velocity model
Migration
?
Local velocity perturbation
Migration
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Velocity estimation

• Sensitivity of a migrated image with respect to
  the background velocity model

Given velocity model
Migration
Improved seismic section
?
Optimal velocity perturbation?
Migration
Cost function
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DSO in curvelet domain
Feasibility study 2-D synthetic rayBorn data set
Initial
After 1 iteration
Exact
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DSO in curvelet domain
Stack of offsets between 100 and 800 m
Initial
After 1 iteration
Exact
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Conclusions perspectives

• Seismic imaging operators
• Curvelets more suited for demigration/migration
  than for migration or demigration (modeling)
  alone
• Applications limited to smooth background
  velocity models
• A similar analysis should be conducted for
  (non-smooth) general background velocity models
  (without the use of geometrical optics)
• Perspectives new transform (e.g. with explicit
  curvature)?

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Acknowledgements
We would like to thank F. ten Kroode (Shell
EP) for fruitful discussions and support M.
Noble and P. Podvin (Mines ParisTech) Shell
EP for partly funding the project