1. The Rice Inversion Project
An automatic wave equation migration velocity analysis by differential semblance optimization
The Rice Inversion Project

2. Objective Simultaneous optimization for velocity and image
Shot-record wave-equation migration.

3. Theory Nonlinear Local Optimization Remark: Objective function
Gradient of the objective function
Remark:
Objective function requires to be smooth .
Differential semblance objective function is smooth.

4. Differential semblance criteriaz x
offset image
angle image z z h h

5. Objective function I : offset domain image c : velocity
h : offset parameter
P : differential semblance operator
|| ||: L2 norm
M : set of smooth velocity functions

7. Gradient calculation Definitions:
Downward continuation and upward continuation S0 R0
gradient
derivative cross correlate*
down
down SZ RZ
DS*
DR*
cross correlate up up
S*z
R*z
image
cross correlate reference field

8. Gradient smoothing using spline evaluation
Vimage I
gimage
spline
spline*
Vmodel
gmodel
migration
differential migration*
M : set of smooth velocity functions

9. Optimization cout Iout BFGS algorithm for nonlinear iteration
Objective function evaluation
Gradient calculation
loop
Update search direction
cout Iout

10. Synthetic Examples
Flat reflector, constant velocity
Marmousi data set

11. Experiment of flat reflector at constant velocity
Ccorrect = 2km/sec z

12. Initial iterate: Image (v0 = 1.8km/sec) Offset image Angle image
Image space: 401 by 80
Model space: 4 by 4
Offset image
Angle image

13. Iteration 5:
Image
Offset image
Angle image

14. Iterations
v5: Output velocity at
iteration 5
vbest - v5

15. Marmousi data set

16. Marmousi data set

17. V

18. Initial iterate: Image (v0=1.8km/sec) Offset image Angle image
Image space: 921 by 60
Model space: 6 by 6
Offset image
Angle image

19. Iterate 5:
Image
Offset image
Angle image

20. v5 - vbest iterations v5: output velocity at iteration 5
vbest: best spline interpolated velocity
v5 - vbest
iterations

21. Low velocity lense + constant velocity background
Vbackground = 2 km/sec

22. Seismogram Shot gathers far away from the low velocity lense
Shot gathers near the low velocity lense

23. Iteration 1
Start with v0 = 2km/sec
Iteration 2
Iteration 3
Iteration 4

25. Conclusions
Offset domain DSO is a good substitute for angle domain DSO.
Image domain gradient needs to be properly smoothed.
DSO is sensitive to the quality of the image.
Differential semblance optimization by wave equation migration is promising.