1. Seismic imaging in the curvelet domain: achievements and perspectives Hervé Chauris (1) & Jianwei Ma (1,2) EAGE 2009 - Amsterdam (1)Centre de Géosciences, Mines ParisTech, France (2)Institute of Seismic Exploration, School of Aerospace, Tsinghua University, Beijing, China

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

3. Locally coherent events Context : seismic modeling / migration Diffraction point Globally coherent event Locally coherent event

4. Curvelets and other …lets EPW Ridgelet SurfaceletSeislet ContourletShearlet Wavelet Bandlet, … Taking into account band-limited data:  Curvelet

5. Potential of curvelets Single curveletSeismic data

6. Introduction – Curvelets Curvelets for different coefficients c μ (x,z) shift stretchrotation

7. Content Introduction – what are curvelets Curvelets and seismic processing tasks pre-/post-processing seismic migration seismic demigration/migration – velocity estimation Conclusions

8. Curvelets and seismic applications Review: Ma and Plonka, 2009 Workshop EAGE 2007 (London), Chauris and Douma Data denoising, interpolation and compression Hennenfent and Herrmann, 2006Droujinine et al., 2007 Herrmann et al., 2008aSacchi et al., 2007 Herrmann et al., 2008bFomel, 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

9. Seismic propagation/migration Theoretical results Candes and Donoho, 2003 Practical results Douma and de Hoop, 2006 Douma and de Hoop, 2007 Chauris, 2006

10. Curvelet processing Migrated section in initial model Perturbed section Fast curvelet transform Fast inverse curvelet transform Use of digital curvelets

11. Migration in heterogeneous models Kirchhoff migrationFirst-order curvelet migration Smooth heterogeneous 2-D model

12. Migration in heterogeneous models Kirchhoff migration First-order curvelet migration

13. Migration in heterogeneous models Kirchhoff migration of a few curvelets First-order approximation not good enough

14. Demigration/migration Sensitivity of the migrated result with respect to the velocity model Initial velocity model Velocity perturbation (up to 200 m/s) Triplicated ray field

15. Demigration/migration Given velocity model Local velocity perturbation Sensitivity of a migrated image with respect to the background velocity model Migration ?

16. Input data Ray+Born 2-D synthetic data set (offsets from 0 to 2 km) Initial image Offset 600 m

17. Demigration/migration Sensitivity of the migrated result with respect to the velocity model Initial imageExact imagePredicted image

18. Demigration/migration Initial / exact Predicted / exact Depth difference reduced from 60 m to less than 2 m

19. Demigration/migration Sensitivity of the migrated result with respect to the velocity model Unperturbed partModified part

20. Common Image Gathers Reference image Prediction with curvelets Exact The prediction takes into account the lateral velocity variations

21. Demigration/migration Given velocity model Local velocity perturbation Sensitivity of a migrated image with respect to the background velocity model Migration ?

22. Velocity estimation Given velocity model Optimal velocity perturbation? Sensitivity of a migrated image with respect to the background velocity model Migration ? Cost function Improved seismic section

23. DSO in curvelet domain Feasibility study 2-D synthetic ray+Born data set InitialAfter 1 iterationExact

24. DSO in curvelet domain Initial After 1 iteration Exact Stack of offsets between 100 and 800 m

25. 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)?

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