1. LSM with Sparsity Constraints
Additive noisy data, minimize influence of wild
Outlier noise by choosing p=1
1 < p
Minimize S S |Lijmj – di| + l P(m) p
i j or
Minimize S |mj | subject to Lm=d p
Undetermined problems, where sparsest soln
Is desired so choose something close to p=0. Choose
A domain where model signal and noise separate

2. Motivation Problem: Noise (inconsistent physics) in model
Correct velocity
e=||Lm-d||2
e=||Lm-d||2 + l||m||2
e=||Lm-d||2 + l||𝛻2m||2
Solution: Sparsity Constraint

3. Motivation Problem: Noise (inconsistent physics) in model
Correct velocity
e=||Lm-d||2
e=||Lm-d||2 + l||m||2
e=||Lm-d||2 + l||𝛻2m||2
Solution: Sparsity Constraint
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2

4. Motivation Problem: Noise (inconsistent physics) in model
Incorrect velocity (5.5 km/s instead of 5.0 km/s)
e=||Lm-d||2
e=||Lm-d||2 + l||m||2
e=||Lm-d||2 + l||𝛻2m||2
Solution: Sparsity Constraint
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2

5. Motivation Problem: Noise (inconsistent physics) in model
e=||Lm-d||2 + l||m||2
Solution: Sparsity Constraint
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2

6. Motivation Problem: Noise (inconsistent physics) in model
e=||Lm-d||2 + l||m||2
Solution: Sparsity Constraint
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2

7. Entropy Regularization
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2
Entropy:
Property: Entropy minimum when Si clumped
Spike Example (s_1=1):
Uniform Example:

8. Entropy Regularization
dS/dsi= -[s’i + 1]ds’i/dsi
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2

9. Entropy Regularization
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2
Use previous migration so ds’_i/ds_j=0
Step 1:
Step 2:
dS/dsi
Will this lead to a symmetric
SPD Hessian?

10. Outline LSM with Cauchy Constraint: Part Admundsen
Reweighted Least Squares
LSM with Entropy Regularization

11. Newton -> Steepest Descent Method
Given:
(1)
Find: stationary point x* s.t. F(x*)=0 D
Soln: Newton’s Method

12. Gradients of L2 vs Cauchy Norms
Frechet derivative: dPi/dsj

13. Gradients of L2 vs Cauchy Norms
Key Benefit: Large Residual DP are
Downweighted. l is like standard dev.

14. Adjust l to insure SPD diagonal

15. Adjust l to insure SPD diagonal

16. Numerical Tests

17. Simulated Data

18. Outline LSM with Cauchy Constraint: Saachi Reweighted Least Squares
LSM with Entropy Regularization

19. LSM with Sparsity Constraint
Given:
Solve: m subject to
Iterative Rewighted
Least Squares:
Note: Large values of Dm downweighted

20. Multichannel Decon

21. Standard LSM vs LSM with Sparsity
Migration CIG Precon. LSM CIG Sparsity :LSM CIG

22. Standard LSM vs LSM with Sparsity
Migration Precon. LSM Sparsity :LSM

23. References

24. Outline LSM with Cauchy Constraint Reweighted Least Squares
LSM with Entropy Regularization

25. Solve: m subject to (your Choice)
LSM with Sparsity Constraint
Given:
Solve: m subject to (your Choice)
Iterative Rewighted
Least Squares:
Rii = (|ri|)
p-2
For small r replace with cutoff or waterr level

26. Solve: m subject to (your Choice)
LSM with Sparsity Constraint
Given:
Solve: m subject to (your Choice)
Iterative Rewighted
Least Squares:
Rii = (|ri|)
p-2
For small r replace with cutoff or water level

27. VSP Example
Model
Src-Rec & Rays

28. Outline LSM with Cauchy Constraint Reweighted Least Squares
LSM with Entropy Regularization

29. Solve: m subject to (your Choice)
LSM with Maximum Entropy
Given:
Solve: m subject to (your Choice)
Entropy Reg.

30. Solve: m subject to (your Choice)
LSM with Maximum Entropy
Given:
Solve: m subject to (your Choice)
Entropy Reg.

31. LSM with Maximum Entropy
Entropy Reg.

38. Outline

39. Minimum Entropy Inversion:
Given:
Solve:
Minimum Entropy Inversion:
Normal eqs.