1. Differential Semblance Optimization for Common Azimuth Migration
TRIP Annual meeting
Differential Semblance Optimization for Common Azimuth Migration
Alexandre KHOURY

2. Context of the project Prestack Wave Equation depth migration
Wavefield extrapolation method
Automating the velocity estimation loop (time-consuming)

3. Motivation of the project
Encouraging results in 2D for Shot-Record migration (Peng Shen, TRIP 2005)
Efficiency of the Common Azimuth Migration in 3D
enables sparse acquisition in one direction
very economic algorithm
Goal of the project:
Implement DSO for Common Azimuth Migration in 3D after a 2D validation

4. Common Azimuth Migration
Wavefield extrapolation in depth: “survey sinking” in the DSR equation h M
Subsurface offset
Variable used for Velocity Analysis : Subsurface offset

5. Subsurface Offset S M R S M R h’ M' M' R’=S’ R’ S’ For true velocity
For wrong velocity

6. Example: two reflectors data set

7. True velocity common image gather
Offset gather at x=1000 m

8. Example: two reflectors data set
One gather at midpoint x=1000m

9. Differential Semblance Optimization
From
we define the objective function :
For
Criteria for determining the true velocity !

10. Differential Semblance Optimization
Plot of the objective function with respect to the velocity
c=ctrue

11. Gradient calculation The objective function : Gradient calculation :
Adjoint-state calculation (Lions, 1971):
code operator

12. Migration: Structure of the Common Azimuth Migration
DSR equation:
Wavefield at depth z
Phase-Shift
in the Fourier domain H1 H2
Lens-Correction
in the space domain H3
General Screen Propagator or FFD
in the space domain
Imaging condition
Wavefield at depth z+Dz
Image at depth
z+Dz

13. Algorithm of the gradient calculation
Wavefield pz
Gradient at depth z+Dz
MIGRATION H
H-1
H*,B*
Wavefield pz+Dz
Gradient at depth z+2Dz H
H-1
H*,B*
Adjoint variables propagation Dp, Dc
Wavefield pz+2Dz

14. Algorithm of the gradient calculation
Velocity representation on a B-spline grid:
B-Spline transformation
Fine grid
B-Spline grid
LBFGS Optimizer
Adjoint B-Spline transformation
Gradient calculation respect to B-Spline grid
Gradient calculation respect to Fine Grid

15. Several critical points
- Avoid wrap-around in the subsurface offset domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity in a specified range
-Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

16. Several critical points
- Avoid wrap-around in the subsurface offset domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity in a specified range
-Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

17. Wrap-around in the subsurface offset domain
For wrong velocity
Image
Gather

18. Wrap-around in the subsurface offset domain
Effect of padding and split-spread for wrong velocity h
Image
Gather

19. Several critical points
- Avoid wrap-around in the subsurface offset domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity in a specified range
-Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

20. Artifacts propagation
Necessity to taper the data on both offset and midpoint axes and in time

21. Several critical points
- Avoid wrap-around in the subsurface offset domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity in a specified range
-Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

22. Several critical points
- Avoid wrap-around in the subsurface offset domain
-Avoid artifacts propagation by tapering the data
-Constrain the optimization to keep the velocity in a specified range
-Careful choice of migration parameters for the accuracy of the gradient (not necessarily for the migration)

23. Differential Semblance Optimization
Tests on different data sets:
-Test on flat reflectors with a constant background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model

24. Differential Semblance Optimization
Tests on different data sets:
-Test on flat reflectors with a constant background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model

25. Differential Semblance Optimization
Start of the optimization: V=2300
Image
Gather

26. Differential Semblance Optimization
10 iterations: Right position
Image
Gather

27. Differential Semblance Optimization
Tests on different data sets:
-Test on flat reflectors with a constant background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model

28. Differential Semblance Optimization
Top of salt : image
x=5000

29. Differential Semblance Optimization
Top of salt : one gather

30. Differential Semblance Optimization
Plot of : localization of the energy of the objective function

31. Differential Semblance Optimization
Tests on different data sets:
-Test on flat reflectors with a constant background velocity
-Test on the top of a salt model
-Test on a 4-Reflectors model

32. Differential Semblance Optimization
True velocity

33. Differential Semblance Optimization
Starting velocity

34. Differential Semblance Optimization
Starting image

35. Differential Semblance Optimization
Optimized image

36. Differential Semblance Optimization
True image

37. Differential Semblance Optimization
Optimized velocity

38. Conclusion Migration is critical and has to be artifacts free.
Is the DSR Migration precise enough for optimization of complex models ?
Can we deal with complex velocity model ?
Next: test on the Marmousi data set and on a 3D data set.

39. Acknowledgment Prof. William W. Symes Total E&P
Dr. Peng Shen, Dr Henri Calandra, Dr Paul Williamson