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

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

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

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: (1) Find: stationary point x* s.t. F(x*)=0 D Soln: Newton’s Method

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

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

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

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

VSP Example Model Src-Rec & Rays

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

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

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

LSM with Maximum Entropy Entropy Reg.

Outline

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