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Signal Analysis and Imaging Group Department of Physics University of Alberta

Regularized Migration/Inversion Henning Kuehl (Shell Canada) Mauricio Sacchi (Physics, UofA) Juefu Wang (Physics, UofA) Carrie Youzwishen (Exxon/Mobil) This doc -> http://www.phys.ualberta.ca/~sacchi/pims03.ppt Email: sacchi@phys.ualberta.ca.

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Signal Analysis and Imaging Group Department of Physics University of Alberta

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  1. Regularized Migration/InversionHenning Kuehl (Shell Canada)Mauricio Sacchi (Physics, UofA) Juefu Wang (Physics, UofA)Carrie Youzwishen (Exxon/Mobil)This doc ->http://www.phys.ualberta.ca/~sacchi/pims03.pptEmail: sacchi@phys.ualberta.ca Signal Analysis and Imaging Group Department of PhysicsUniversity of Alberta

  2. Outline • Motivation and Goals • Migration/inversion - Evolution of ideas and concepts • Quadratic versus non-quadratic regularization • two examples • The migration problem • RLS migration with quadratic regularzation • Examples • RLS migration with non-quadratic regularization • Examples • Summary

  3. Motivation To go beyond the resolution provided by the data (aperture and band-width) by incorporating quadratic and non-quadratic regularization terms into migration/inversion algorithms This is not a new idea…

  4. Evolution of ideas and concepts Migration with Adjoint Operators [Current technology] RLS Migration (Quadratic Regularization) [Not in production yet] RLS Migration (Non-Quadratic Regularization) [??] Resolution

  5. Evolution of ideas and concepts Migration with Adjoint Operators 1 RLS Migration (Quadratic Regularization) 2 RLS Migration (Non-Quadratic Regularization) Resolution

  6. Evolution of ideas and concepts Two examples LS Deconvolution Sparse Spike Deconvolution LS Radon Transforms HR Radon Transforms Quadratic Regularization Stable and Fast Algorithms Low Resolution Non-quadratic Regularization Requires Sophisticated Optimization Enhanced Resolution

  7. Deconvolution Quadratic versus non-quadratic regularization

  8. Convolution / Cross-correlation / Deconv Convolution Cross-correlation Deconvolution

  9. Quadratic and Non-quadratic Regularization Cost Quadratic Non-quadratic

  10. Adj LS HR

  11. Radon Transform Quadratic versus non-quadratic regularization

  12. LS Radon q h

  13. HR Radon q h

  14. Modeling / Migration / Inversion Modeling Migration De-blurring problem

  15. Back to migration Inducing a sparse solution via non-quadratic regularization appears to be a good idea (at least for the two previous examples) Q. Is the same valid for Migration/Inversion?

  16. Back to migration Inducing a sparse solution via non-quadratic regularization appears to be a good idea (at least for the two previous examples) Q. In the same valid for Migration/Inversion? A. Not so fast… First, we need to define what we are inverting for…

  17. Migration as an inverse problem Forward Adjoint Data “Image” or Angle Dependent Reflectivity

  18. Migration as an inverse problem • There is No general agreement about what type of regularization should be used when inverting for m=m(x,z)… Geology is too complicated… • For angle dependent images, m(x,z,p), we attempt to impose horizontal smoothness along the “redundant” variable in the CIGs: Kuehl, 2002, PhD Thesis UofA url: cm-gw.phys.ualberta.ca~/sacchi/saig/index.html

  19. Quadratic Regularization Migration Algorithm Features: DSR/AVP Forward/Adjoint Operator (Prucha et. al, 1999) PSPI/Split Step Optimization via PCG 2D/3D (Common Azimuth) MPI/Open MP Kuehl and Sacchi, Geophysics, January 03 [Amplitudes in 2D]

  20. 2D - Synthetic data example

  21. Marmousi model (From H. Kuehl Thesis, UofA, 02) AVA target Velocity Density

  22. Migrated AVA CMP position [m] Ray parameter [mu s/m] 7000 7500 8000 0 200 400 600 800 0 0 0.5 0.5 1.0 1.0 ] ] m m k k [ [ h h t t 1.5 1.5 p p e e D D 2.0 2.0 2.5 2.5

  23. Regularized Migrated AVA CMP position [m] Ray parameter [mu s/m] 7000 7500 8000 0 200 400 600 800 0 0 0.5 0.5 1.0 1.0 ] ] m m k k [ [ h h t t 1.5 p 1.5 p e e D D 2.0 2.0 2.5 2.5

  24. Ray parameter [mu s/m] Ray parameter [mu s/m] 0 200 400 600 800 0 200 400 600 800 0.5 0.5 0.7 ] 0.7 ] m m k k [ [ h h t t p p 0.9 0.9 e e D D 1.1 1.1 0.2 0.2 0.18 0.18 0.16 0.16 0.14 0.14 0.12 0.12 Reflection coefficient Reflection coefficient 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 10 20 30 40 50 0 10 20 30 40 50 Angle of incidence (deg) Angle of incidence (deg) RLSM Migration

  25. Offset [m] Offset [m] 200 400 600 800 1000 1200 200 400 600 800 1000 1200 0 0 0.5 0.5 1.0 1.0 ] ] c c e e s s [ [ e e 1.5 1.5 m m i i T T 2.0 2.0 2.5 2.5 Incomplete CMP (30 %) Complete CMP

  26. Ray parameter [mu s/m] Ray parameter [mu s/m] 0 200 400 600 800 0 200 400 600 800 0.5 0.5 0.7 ] 0.7 ] m m k k [ [ h h t t p p 0.9 0.9 e e D D 1.1 1.1 0.2 0.2 0.18 0.18 0.16 0.16 0.14 0.14 0.12 0.12 Reflection coefficient Reflection coefficient 0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02 0 0 0 10 20 30 40 50 0 10 20 30 40 50 Angle of incidence (deg) Angle of incidence (deg) Migration RLSM

  27. CMP position [m] 3000 4000 5000 6000 7000 8000 9000 0 0.5 1.0 ] m k [ h 1.5 t p e D 2.0 2.5 Constant ray parameter image (incomplete)

  28. CMP position [m] 3000 4000 5000 6000 7000 8000 9000 0 0.5 1.0 ] m k [ h 1.5 t p e D 2.0 2.5 Constant ray parameter image (RLSM)

  29. 3D - Synthetic data example (J Wang) Parameters for the 3D synthetic data x-CDPs : 40 Y-CDPs : 301 Nominal Offset: 50 dx=5 m dy=10 m dh=10 m 90% traces are randomly removed to simulate a sparse 3D data.

  30. Incomplete Data 4 adjacent CDPs Reconstructed data (after 12 Iterations)

  31. B A C A: Iteration 1 B: Iteration 3 C: Iteration 12

  32. At y=950 m, x=130 m

  33. Misfit Curve Convergence – Misfit vs. CG iterations

  34. Stacked images comparison (x-line 30) A complete data B.Remove 90% traces C. least-squares

  35. Field data example ERSKINE (WBC) orthogonal 3-D sparse land data set dx=50.29m dy=33.5m Comment on Importance of W

  36. CIG at x-line #10, in-line #71 Iteration 1 Iteration 3 Iteration 7 Cost Optimization

  37. Stacked image, in-line #71 Migration RLS Migration

  38. Stacked image, in-line #71 Migration RLS Migration

  39. Image and Common Image Gather (detail)

  40. Image and Common Image Gather (detail)

  41. Stacked Image (x-line #10) comparison LS Migration Migration

  42. Non-quadratic regularization applied to imaging

  43. Non-quadratic regularization But first, a little about smoothing: Quadratic Smoothing Non-Quadratic Smoothing

  44. Quadratic regularization ->Linear filters Non-quadratic -> Non-linear filters

  45. Segmentation/Non-linear Smoothing

  46. Segmentation/Non-linear Smoothing 1/Variance at edge position

  47. 2D Segmentation/Non-linear Smoothing

  48. * * * * * * GRT Inversion - VSP(Carrie Youzwishen, MSc 2001)

  49. Dx Dz m

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