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

Outline ObjectiveObjective Numerical TestsNumerical Tests 3D U model3D U model ConclusionConclusion Preconditioned Conjugate GradientPreconditioned Conjugate Gradient IntroductionIntroduction Theory of Deblurring filterTheory of Deblurring filter

Introduction Standard migration: Least-squares migration: Standard migrationLSM ProsFast, robustHigh resolution images ConsImages of low qualityHigh computation cost Forward modeling:

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 Gradient: Step length: Conjugate direction: Update:

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

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

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

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

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

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

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

Numerical Tests model: 3D U model grid: grid interval: 10m background velocity: 1500 m/s 300 shots and 300 receivers on the surface Recording geometry X (m)01500 Y (m) 0 Fig. 1. Recording geometry. Red stars indicate sources and blue triangles indicate receivers.

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

3D U model Fig. 3. One horizontal and one vertical slices of the U model. X (m)01500 Y (m) 0 X (m) Z (m) 0

Fig. 4. Standard migration result. The same slices as previous figure are shown here. Standard migration image X (m) Y (m) X (m) Z (m)

Fig. 5. Vertical slice of the reference model and its corresponding standard migration image. Reference model X (m) Z (m) 0 X (m)01500 Z (m) 0

Fig. 6. Standard migration image and deblurred image for the reference model (vertical slices). X (m)01500 Z (m) 0 X (m)01500 Z (m) Standard migration image vs Deblurred image

X (m)01500 Y (m) X (m) Y (m) Standard migration image vs Deblurred image Fig. 7. Standard migration image and deblurred image for the 3D U model (horizontal slice along 2 nd reflectivity layer).

Fig. 8. Standard migration image and deblurred image for the 3D U model (vertical slice y=500m) X (m) Z (m) X (m)01500 Z (m) Standard migration image vs Deblurred image

Fig. 9. Standard migration image and deblurred image for the 3D U model (vertical slice x=550m) Y (m) Z (m) Y (m)01500 Z (m) Standard migration image vs Deblurred image

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

Fig. 11. Image after 3 iterations of PCG X (m)01500 Y (m) 0 X (m)01500 Z (m) Three iterations result

Fig. 12. Image after 5 iterations of PCG. 0 X (m)01500 Y (m) 0 X (m) Z (m) Five iterations result

0 X (m)01500 Y (m) 0 X (m) Z (m) Fig. 13. Image after 10 iterations of PCG. Ten iterations result

0 X (m) Y (m) 0 X (m) Z (m) Fig iterations results of PCG and CG (horizontal slices). PCG vs CG

1 X (m) Z (m) 0 X (m) Z (m) Fig iterations result of PCG and CG (vertical slices). PCG vs CG

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.