LSM with Sparsity
Constraints
Additive noisy data, minimize influence of wild
Outlier noise by choosing p=1
Minimize S
i
1< p
S |Lijmj – di|
p
+l
P(m)
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
Motivation
Problem: Noise (inconsistent physics) in model
e=||Lm-d||2
Correct velocity
e=||Lm-d||2 + l||m||2
e=||Lm-d||2 + l||𝛻2m||2
Solution: Sparsity Constraint
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
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
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
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
Entropy Regularization
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2
Entropy:
Property: Entropy minimum when Si clumped
Spike Example (s_1=1):
Uniform Example:
Entropy Regularization
e = ||Lm-d||2 + l||𝑒𝑛𝑡𝑟𝑜𝑝𝑦(𝑚)||2
dS/dsi= -[s’i + 1]ds’i/dsi
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?
Outline
• LSM with Cauchy Constraint: Part Admundsen
• Reweighted Least Squares
• LSM with Entropy Regularization
Newton -> Steepest Descent Method
Given:
Soln: Newton’s Method
D
Find: stationary point x* s.t.
(1)
F(x*)=0
L2 vs Cauchy Norms
Gradients of L2 vs Cauchy Norms
Frechet derivative: dPi/dsj
L2 vs Cauchy Norms
Gradients of L2 vs Cauchy Norms
Key Benefit: Large Residual DP are
Downweighted. l is like standard dev.
Adjust l to insure SPD diagonal
Adjust l to insure SPD diagonal
Numerical Tests
Simulated Data
Outline
• LSM with Cauchy Constraint: Saachi
• Reweighted Least Squares
• LSM with Entropy Regularization
LSM with Sparsity Constraint
Given:
:
Solve m subject to
Iterative
Rewighted
Least Squares:
Note: Large values of Dm downweighted
Multichannel Decon
Standard LSM vs LSM with Sparsity
Migration CIG
Precon. LSM CIG
Sparsity :LSM CIG
Standard LSM vs LSM with Sparsity
Migration
Precon. LSM
Sparsity :LSM
References
Outline
• LSM with Cauchy Constraint
• Reweighted Least Squares
• LSM with Entropy Regularization
LSM with Sparsity Constraint
Given:
:
Solve m subject to (your Choice)
Iterative
Rewighted
Least Squares:
p-2
Rii = (|ri|)
For small r replace with cutoff or waterr level
LSM with Sparsity Constraint
Given:
:
Solve m subject to (your Choice)
Iterative
Rewighted
Least Squares:
p-2
Rii = (|ri|)
For small r replace with cutoff or water level
VSP Example
Model
Src-Rec & Rays
Outline
• LSM with Cauchy Constraint
• Reweighted Least Squares
• LSM with Entropy Regularization
LSM with Maximum Entropy
Given:
:
Solve m subject to (your Choice)
Entropy Reg.
LSM with Maximum Entropy
Given:
:
Solve m subject to (your Choice)
Entropy Reg.
LSM with Maximum Entropy
Entropy Reg.
Outline
Given
Solve
:
:
Minimum
Entropy
Inversion:
Normal eqs.