Fast 3 D Leastsquares Migration with a Deblurring

Fast 3 D Least-squares Migration with a Deblurring Filter Wei Dai

Outline • • • Introduction Objective Preconditioned Conjugate Gradient Theory of Deblurring filter Numerical Tests • 3 D U model • Conclusion

Introduction Forward modeling: Standard migration: Least-squares migration: Standard migration LSM Pros Fast, robust High resolution images Cons Images of low quality High computation cost

Conjugate Gradient Misfit functional: Normal equation: Direct solver: Need to invert huge matrix. Iterative solver: Iterative conjugate gradient method:

Conjugate Gradient vs Steepest Descent

Conjugate Gradient: Conjugate direction: Step length: Update:

Objective: Reduce the iteration numbers required for LSM Proposal: A good preconditioner to accelerate the convergence.

Preconditioned Conjugate Gradient To improve the condition number: Problem: is not symmetric. Solution: to decompose and solve: By change of variables:

Preconditioned Conjugate Gradient: Conjugate direction: Step length: Update: Problem: Need to calculate

Preconditioned Conjugate Gradient By change of variables: Gradient: Conjugate direction:

Preconditioned Conjugate Gradient Step length: Update: Advantage: only need M. Requirement: M to be SPD.

Theory of the Deblurring Reference model : grid model with evenly distributed point scatterers. Calculate its standard migration image:

Construct an image, which is an approximation of Rewrite in matrix notation so,

Numerical Tests grid: 1500 model: 3 D U model Recording geometry 300 shots and 300 receivers on the surface 0 background velocity: 1500 m/s Y (m) grid interval: 10 m 0 X (m) 1500 Fig. 1. Recording geometry. Red stars indicate sources and blue triangles indicate receivers.

3 D U model Fig. 2. 3 D view of the U model. Courtesy of Naoshi Aoki

Z (m) 0 0 Y (m) 1500 3 D U model 0 X (m) 1500 0 X (m) Fig. 3. One horizontal and one vertical slices of the U model. 1500

1500 Standard migration image 0. 02 Y (m) Z (m) 0. 2 -0. 1 0 0 -0. 12 0 X (m) 1500 0 X (m) Fig. 4. Standard migration result. The same slices as previous figure are shown here. 1500

0 0 Z (m) 1500 Reference model 0 X (m) 1500 0 X (m) Fig. 5. Vertical slice of the reference model and its corresponding standard migration image. 1500

1500 Z (m) 0 0. 5 -0. 3 0 1500 0 Z (m) Standard migration image vs Deblurred image X (m) 1500 0 X (m) 1500 Fig. 6. Standard migration image and deblurred image for the reference model (vertical slices).

1500 Standard migration image vs Deblurred image 0. 02 Y (m) 0. 1 0 0 -0. 12 0 X (m) 1500 Fig. 7. Standard migration image and deblurred image for the 3 D U model (horizontal slice along 2 nd reflectivity layer). -0. 6

0. 6 Z (m) 0. 2 1500 Standard migration image vs Deblurred image 0 -0. 4 0 0 -0. 1 X (m) 1500 0 X (m) 1500 Fig. 8. Standard migration image and deblurred image for the 3 D U model (vertical slice y=500 m)

1500 Standard migration image vs Deblurred image 0. 5 Z (m) 0. 25 0 Y (m) 1500 0 0 -0. 15 0 Y (m) 1500 Fig. 9. Standard migration image and deblurred image for the 3 D U model (vertical slice x=550 m) -0. 5

0 Data residual 0. 7 Residual curves 30 0 Iteration number Fig. 10. Residual curves for SD, PSD, CG and PCG.

0. 8 0 -0. 8 0 X (m) 1500 -0. 6 0 Y (m) Z (m) 0. 1 1500 Three iterations result 0 X (m) Fig. 11. Image after 3 iterations of PCG. 1500

1500 Five iterations result 1 0 -1 0 X (m) 1500 -0. 8 0 Y (m) Z (m) 0 0 X (m) Fig. 12. Image after 5 iterations of PCG. 1500

1500 Ten iterations result 1 0 -1 0 X (m) 1500 -0. 8 0 Y (m) Z (m) 0 0 X (m) Fig. 13. Image after 10 iterations of PCG. 1500

1500 PCG vs CG 0 Y (m) Z (m) 0 0 X (m) 1500 -0. 9 0 0 -1 0 X (m) Fig. 14. 30 iterations results of PCG and CG (horizontal slices). 1500

0. 8 Z (m) 1 1500 PCG vs CG 0 X (m) 1500 -0. 8 0 0 -1 0 X (m) Fig. 15. 30 iterations result of PCG and CG (vertical slices). 1500

Conclusions • Our deblurring filter is a good approximation to the Hessian inverse. • It can improve the standard migration image and reduce its data residual by about 50%. • Our deblurring filter as a preconditioner in LSM can speed up convergence rate by several times • 3 iterations of PCG are equivalent to 10 iterations of CG.