LSM Theory: Overdetermined vs Underdetermined Workflow: Diffraction Stack vs RTM Sensitivity: LSM sensitivity to Dv(x,z) Dot Product Test: (Lm,d)=(m,LT d) Examples Summary

Iterative Least Squares Migration Step 1: Step 2: Step 3: Step 4:

Motivation LSM

LSM=Antialiasing

Iterative Least Squares Migration Overdetermined Wrong v(x,z) will misposition reflector for src on left vs src on right. Inconsistent Set of equations so reflector will be blurred rthat expalins all CSGs Liability: wrong velocity model smears reflectivity Migration image

Iterative Least Squares Migration Invert each shot gather Underdetermined i Misfit+regularization Invert each shot gather separately Advantage: correct migration images prior to stacking Penalize CSG images if they are different

LSM Theory: Overdetermined vs Underdetermined Workflow: Diffraction Stack vs RTM Sensitivity: LSM sensitivity to Dv(x,z) Dot Product Test: (Lm,d)=(m,LT d) Examples Summary

MATLAB SD Least Squares Diffraction Stack Migration Unlike RTM, Kirchhoff not bothered by rabbit ears Note: no update to smooth background c, only hi-wavenumber m p=p0 % Data without direct wave m=adjoint(p,c) % Initial reflectivity model c % Velocity model for i=1:niter p=forward(m,c) % Kirchhoff predicted data alpha=step(p,p0,c,m) % step length dP=p-p0 % data residual dm =adjoint(dP,c) % migrate residual m = m –alpha*dm % Update model end - =

Iterative Least Squares Migration We are now using RTM so Both rabbit ears+ellipses - Note: no update to smooth background so, only hi-wavenumber ds LD D U

LSM Theory: Overdetermined vs Underdetermined Workflow: Diffraction Stack vs RTM Sensitivity: LSM sensitivity to Dv(x,z) Dot Product Test: (Lm,d)=(m,LT d) Examples Summary

Sensitivity to V(x,z) Error

LSM Theory: Overdetermined vs Underdetermined Workflow: Diffraction Stack vs RTM Sensitivity: LSM sensitivity to Dv(x,z) Dot Product Test: (Lm,d)=(m,LT d) Examples Summary

Dot Product Test with CG code Actual model Predicted model Actual data Predicted data d Lm =(d,Lm) = (Lm,d) = m L d T T T d=forward(m,c) m=adjoint(d,c) d d = T m T m All migration codes should pass the dot product test

LSM Theory: Overdetermined vs Underdetermined Workflow: Diffraction Stack vs RTM Sensitivity: LSM sensitivity to Dv(x,z) Dot Product Test: (Lm,d)=(m,LT d) Examples Summary

2D Poststack Data from Japan Sea JAPEX 2D SSP marine data description: Acquired in 1974, Dominant frequency of 15 Hz. 5 TWT (s) 20 X (km) 16

Poststack LSM vs. Kirchhoff Migration LSM Image 0.7 1.9 Depth (km) 2.4 4.9 X (km) 0.7 1.9 Depth (km) 2.4 4.9 X (km) Kirchhoff Migration Image

Multi-scale LSM Applied to JAPEX Data Multi-scale (MS) LSM vs. Standard LSM Convergence Curves MS LSM Image 0.7 1.9 Depth (km) 2.4 4.9 X (km) Standard LSM Image 0.7 1.9 2.4 4.9 X (km) X10 5 3.0 0.5 Residual 40 Iteration Multi-scale LSM Standard LSM 20　Hz 25 I apply band-pass filter to data. The frequency band increase with iteration 30 32 34 36 38 40 18

GOM Poststack Data Poststack LSM somewhat insensitive to Dv(x,z)

LSM=Antialiasing

Prestack KM vs LSM Prestack LSM sensitive to Dv(x,z)

Attenuation RTM vs LSM

Attenuation RTM vs LSM

Nonlinear LSRTM

Nonlinear LSRTM

NL Means Filter for Trim Statics Problem: velocity error CIGs misaligned  poor stacking Washed out Two prestack image patches out of phase: Solution: Xcorr patches; recursive stacking needs no pilot 1 A 2 B 3 4 C shift

NL Means Filter for Trim Statics Advantage: drastic improvement in feature coherency Examples: Disadvantage: strong migration artifacts may mislead Simple Stacking Future Work: 3D trim statics Trim Statics

Statics+ LSRTM

LSM Summary 1. Prefer undetermined LSM if large Dv(xz)>>0 2. Biggest challenge: LSM sensitive Dv(x,z) 3. Q-LSM much better than LSM or RTM if Q<30 4. LSRTM cost O(20x) more than RTM 5. Multisource encoded O(LSRTM) cost = O(RTM) 6. Iterative LSM+DSO+FWI is future.