 = 0.5  j  r j (  kk’ (  m kk’ /  z) 2  m ii’ =  j  r j  r j /  m ii’ + (  kk’  m kk’ /  m ii’  m kk’ /  z) (1) m 11 m 12 m 13 m 14 m 15 m 16 m 21 m 22 m 23 m 24 m 25 m 26 m 31 m 32 m 33 m 34 m 35 m 36 Discrete Model  m kk’ /  m ii’ =  ki  k’i’ (2)  m kk’ /  z = (m k+1k’  m kk’ )/  z (3)  kk’  m kk’ /  m ii’  m kk’ /  x =  kk’  ki  k’i’ (m k+1k’  m kk’ )/  z = (m i+1i’  m ii’ )/  z  m ii’ =  m ii’ Plug eqs. 2-3 into 1 we get Smoothing regularizer Misfit + I get nervous about this..will it lead to a SPD Hessian? Better to do 2 nd derivative which is symmetric?

 = 0.5  j  r j  j  h j 2 + penalty different misfit misfit + Different misfits such as combining EM and seismic data. Surface waves with body waves, traveltimes with waveforms, etc. One is inverting for velocity and other conductivity. How to combine them? Penalty =  m EM x m seismic (recall r x c=|r||c|sin  ) Almost same, not quite Penalty function is small if normalized gradients are parallel. Why normalized? Different units+scales of EM vs seismic parameters (e.g., velocity dv/dx vs conductivity d  /dx). Choosing :

Choosing  for the Penalty Function       

Migration Migration Intuitive: Modeling Poststack Mig Poststack Mig Prestack Prestack Least Squares Green’s Theorem

Zero-Offset Data (aka poststack data) Depth Time V/2 oo xoxr)()(  g  rd)( dx o

ZO Data: d 0 km 7 km 0 km 3 km Assume energy came directly from below Wrong assumption for lateral heterogeneities Bowties?

Given: d = Lm Seismic Inverse Problem Find: m(x,y,z) Find: m(x,y,z) Soln: min || Lm-d || Soln: min || Lm-d ||2 m = [L L] L d T T L d L dT Migration Waveform inversion (non-linear) (non-linear) Least squares migration (linear)

ZO Data Migration: m~L T d 0 km 7 km 0 km 3 km

d(x i ) =  xj xj xj xj ~ d =  L m ijijijijj j i g(x i |x j ) A(x i,x j )  xixjxixjxixjxixj -2i  e m(x j ) reflectivity Born Forward Modeling: d=Lm Data  Model Reflected wavefield=sum of weighted point source responses at reflector (Huygen’s Principle). All exploding at halfway propagation time  xx

d =  L m ijijijijj j i d(x i ) =  xj xj xj xj ~ g(x|x’) A(x i,x j )  xixjxixjxixjxixj -2i  e m(x j ) Born Migration: d=Lm  m=L T d xi xi xi xi d(x i ) ~ m(x j ) didididi mjmjmjmjL* ijijijij i ~  d(x i,2  ij ) xi xi xi xi

Migration Migration Intuitive: Modeling Poststack Mig Poststack Mig Prestack Prestack Least Squares Green’s Theorem

2-way time (x-x ) + y c xxxxxxxx =  xxxxxxxx + T o ZO Migration ZO Migration Smear Reflections along Fat Circles Smear Reflections along Fat Circlesx x d(x, )  xxxxxxxx Thickness = c*T /2 o Where did reflections come from?

2-way time ZO Migration ZO Migration Smear Reflections along Fat Circles  x & Sum Hey, that’s our ZO migration formula d(x, )  xxxxxxxx

2-way time ZO Migration ZO Migration Smear Reflections along Circles  x & Sum In-Phase Out-of--Phase d(x, )  xxxxxxxx m(x)=

for ixtrace=1:ntrace; for ixtrace=1:ntrace; for ixs=istart:iend; for ixs=istart:iend; for izs=1:nz; for izs=1:nz; r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2); r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2); time = 1 + round( r/c/dt ); time = 1 + round( r/c/dt ); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); end; end; end; end; Traveltime Loop over x in model Loop over z in model Smear over circle ZO Migration: Smear Trace Sample over Circle d (g, )  xgxgxgxg m(x) =  g Loop over data

for ixtrace=1:ntrace; for ixtrace=1:ntrace; for ixs=istart:iend; for ixs=istart:iend; for izs=1:nz; for izs=1:nz; r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2); r = sqrt((ixtrace*dx-ixs*dx)^2+(izs*dx)^2); time = 1 + round( r/c/dt ); time = 1 + round( r/c/dt ); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); end; end; end; end; ZO Migration: Sum Trace Samples along migration hyperbola into m(x) Sum samples along hyperbola (x,z) (x’,z’) Loop over x in model Loop over z in model Loop over data

Trial image pt x traces ZO Diffraction Stack Migration g d (g, ) xgxgxgxg m(x) =  g  x

Trial image pt x traces g d (g, ) xgxgxgxg m(x) =  g  x Migration Image ZO Diffraction Stack Migration 2D dot product of migration Operator and d(g,t)

Trial image pt x d (g, ) xgxgxgxg m(x) =  g  ZO Diffraction Stack Migration: C(x,z) Ray tracing

Trial image pt x d (g, ) xgxgxgxg m(x) =  g  3D ZO Diffraction Stack Migration Impulse Response of Mig. Op.

Migration Migration Intuitive: Modeling Poststack Poststack Prestack Prestack Least Squares Green’s Theorem

3D Prestack Diffraction Stack Migration gs x Motivation: ZO only good if no lateral vel change

Trial image pt x m(x) = 3D Prestack Diffraction Stack Migration gs x= d(x’,  +  ) sxsxsxsx s,g xgxgxgxg

Prestack Migration Question: Why Prestack when poststack migration seems good enough? migration seems good enough? Answer: Stacking to get stacked section assumes layered medium assumption. layered medium assumption. Solution: Migrate shot gathers so no layer assumption needed. This is prestack assumption needed. This is prestack migration. migration.

Narrow band case: direct wave correlated with data Diffraction Stack Migration: Prestack  sxsxsxsx xgxgxgxg s xg Where is scatterer? T(s,g) =  sxsxsxsx xgxgxgxg + d(s,g, )  sxsxsxsx xgxgxgxg +  s,g s,g Down time Up time

W(  ) ~ A(s,x)  sxsxsxsx i e x  xgxgxgxg  i e A(g,x) d(s,g) = m(x)~115. Diffraction Stack Modeling: Prestack ~ 2 d = L m m = L d T

W(  ) ~ A(s,x)  sxsxsxsx  i e s,g s,g  xgxgxgxg  i e A(g,x) m(x) = d(s,g) dddd  dddd  ~ d(s,g,  +  ) sxsxsxsx xgxgxgxg A(s,x)  s,g s,g A(x,g) = * - - Broadband case W(  )=1 ~ Narrow band case: direct wave correlated with data Diffraction Stack Migration: Prestack ~ m(x) 2 m = L d T

MATLAB Inefficient Prestack Migration for isx=1:nx % Loop over shot for isx=1:nx % Loop over shot for igx=1:nx % Loop over receivers for igx=1:nx % Loop over receivers for ix=1:nx % Loop over model x for ix=1:nx % Loop over model x for iz=1:nx % Loop over model z for iz=1:nx % Loop over model z t=timer(ix,iz,isx)+timer(ix,iz,igx) t=timer(ix,iz,isx)+timer(ix,iz,igx) sample=gather(isx,igx,t) % Shot gather has 2 time derivatives sample=gather(isx,igx,t) % Shot gather has 2 time derivatives mig(ix,iz)=mig(ix,iz)+sample mig(ix,iz)=mig(ix,iz)+sample end end end Model Loops Data Loops

MATLAB Prestack Migration

Poststack vs Prestack Migration

Prestack Migration 1. No 1D assumption about velocity model 2. More sensitive to velocity model errors compared to poststack migration compared to poststack migration 3. More than 100 times slower than poststack migration migration 4. More sensitive to velocity model than time migration. Assumes no multiples. migration. Assumes no multiples.

Summary Trial image pt x d (g, ) xgxgxgxg m(x) =  g  3D ZO Diffraction Stack Migration Trial image pt x m(x) = = d(x’,  +  ) sxsxsxsx s,g xgxgxgxg 3D Prestack Diffraction Stack Migration Migration Motivation: diffractions, dipping layers, conflicting dips, out-of-plane reflections conflicting dips, out-of-plane reflections

Migration Migration Intuitive: Modeling Poststack Poststack Prestack Prestack Least Squares Green’s Theorem

Raypath Traveltime Inversion vs Full Wave Equation Inversion Ray Tomography doesn’t give hi-wavenumber model.

Raypath Traveltime Inversion  = 0.5  j  j 2 Object. Fcn. Wave Equation Waveform Inversion 5. s(x) (k+1)  s(x) (k) -  s(x) (k) [d  /dx] 2 + [d  /dz] 2 = s(x) 2 [d  /dx] 2 + [d  /dz] 2 = s(x) 2  s(x i )=  j  j  j /  s(x i ) Gradient ij Fre ’ chet Deriv. Pred. Data  xjxj  P + k P = s(x) (k+1)  s(x) (k) -  s(x) (k)  =  i |  P i | 2 Object. Fcn.  s(x j )=   P i  P i /  s(x j ) Gradient Pred. Data i Fre ’ chet Deriv. i j j * We should have Re[ ], but its ok xjxj

 P + k P =  =   P 2  s(x)=   P  P/  s(x) Wave Equation Waveform Inversion Wave Equation Traveltime Inversion  =    2  s(x)=     /  s(x)

Step 1: Step 1: Step 2: Step 2: Step 3: Step 3: Step 4: Step 4: Iterative Least Squares Migration Iterative Least Squares Migration L T (Lm-d) Each iteration, migrate residual

MATLAB SD Least Squares Migration p=p0 % Data without direct wave p=p0 % Data without direct wave m=adjoint(p,c) % Initial reflectivity model c % Velocity model c % Velocity model for i=1:niter p=forward(m,c) % predicted data p=forward(m,c) % predicted data alpha=step(p,p0,c,m) % step length alpha=step(p,p0,c,m) % step length dP=p-p0 % data residual dP=p-p0 % data residual dm =adjoint(dP,c) % migrate residual dm =adjoint(dP,c) % migrate residual m = m –alpha*dm % Update model m = m –alpha*dm % Update modelend

Dot Products and Adjoint Operators Recall: (u,u) = u* u  i ii Recall: (v,Lu) = v* ( L u )  j i ijijijij i j [ L v* ]u  j i j ijijijiji = [ L* v ]* u  j i j ijijijiji = So adjoint of L is L   i ijijijij L*

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

Migration Migration Intuitive: Modeling Poststack Poststack Prestack Prestack Least Squares: Examples ExamplesFootprint Green’s Theorem r = L L LLTT Migration butterfly

m = L d T Migration = Blurred r but d = L r Migrated Section Data Modeling

m = L T but d = L r Migrated Section L rL rL rL r Migration Image m = True Reflectivity Model r Migration = Blurred r

Migration Deconvolution m r LLT Migration image Reflectivity = Migration Green’s function

Migration Deconvolution m r LLT  LL T][ LL T ][ 1 1

m r  LL T][ 1

Assume Local v(z) Approximation m r  LL T][ 1 Migration Deconvolution

Migration noise and artifacts Migration Noise Problems Depth (km) Weak illumination Footprint Note: Artifacts stronger near surface. Why?

X (km) 0 10 Depth (km) Wave Equation Migration Before LSM 020

X (km) 0 Wave Equation Migration after MD X (km) 0 10 Depth (km) 020

Acquisition Footprint (Geophone Aliasing) Actual Model 0 2 Depth (km) 0 2 X (km) 0 2 Depth (km) 0 2 X (km) LSM Image (15 Hz) Coarse Kirchhoff Migration Image (15 Hz) Note: Artifacts stronger near surface. Why?

Standard Kirchhoff Image vs LSM Image Actual Model 0 2 Depth (km) 0 2 X (km) 0 2 Depth (km) 0 2 X (km) LSM Image (15 Hz) Kirchhoff Migration Image (15 Hz)

Migration Migration Intuitive: Modeling Poststack Poststack Prestack Prestack Least Squares: Examples ExamplesFootprint Green’s Theorem

2D Poststack Data from Japan Sea JAPEX 2D SSP marine data description: Acquired in 1974, Dominant frequency of 15 Hz. 0 5 TWT (s) 020 X (km)

Multi-scale LSM Applied to JAPEX Data Multi-scale (MS) LSM vs. Standard LSM Convergence Curves MS LSM Image Depth (km) X (km) Standard LSM Image X (km) X Residual 0 40 Iteration Multi- scale LSM Standard LSM 20 Hz

LSM vs. Kirchhoff Migration LSM Image Depth (km) X (km) Depth (km) X (km) Kirchhoff Migration Image

Kirchhoff MD

Kirchhoff LSM

Migration Summary Trial image pt x d (g, ) xgxgxgxg m(x) =  g  3D ZO Diffraction Stack Migration 3D ZO Diffraction Stack Migration Trial image pt x m(x) = = d(x’,  +  ) sxsxsxsx s,g xgxgxgxg 3D Prestack Diffraction Stack Migration 3D Prestack Diffraction Stack Migration Migration Motivation: diffractions, dipping layers, Migration Motivation: diffractions, dipping layers, conflicting dips, out-of-plane reflections conflicting dips, out-of-plane reflections m = m -  L  m (k+1) (k) (k) T Least Squares Migration Least Squares Migration Full Waveform Inversion Full Waveform Inversion m = m -  L  m (k+1) (k) (k) T (k)

ZO Summary m(x’) Approx. reflectivity  x d(x, ) A(x,x’)  xx’ =..cos 1. ZO migration: obliquity x’x 2. ZO migration assumptions: Single scattering data 3. ZO migration matrix-vec: m=L d T~ 4. LSM ZO migration matrix-vec: m=[L L] L d TT Compensates for Illumination footprint and poor illumination 5. ZO migration smears an event along appropriate doughnut doughnut

Summary m(x’) Approx. reflectivity  x d(x, ) A(x,x’)  xx’ =..cos 1. ZO migration: obliquity x’x 2. ZO migration assumptions: Single scattering data 3. ZO migration matrix-vec: m=L d T~ 4. LSM ZO migration matrix-vec: m=[L L] L d TT Compensates for Illumination footprint and poor illumination 5. ZO migration smears an event along appropriate doughnut doughnut

Kirchhoff MD

Least Squares Recall: Lm=d  j ijijijij Find: m that minimizes sum of squared residuals r = L m - d residuals r = L m - djii (r,r) = ([Lm-d],[Lm-d]) = m L Lm -2m Ld-d d L Lm = Ld Normal equations For all i (r,r) ddm i = 2 m L Lm -2 m Ld d dm i d dm i = 0

Review Inverse Acoustic ProblemInverse Acoustic Problem Find: m( r ) (c - c )/(c + c ) m( r ) (c - c )/(c + c )1122 Given: d(r ) rm  )( Soln: L d L d T Soln:  o rdrr)()(  g rm  )(  dr * m(x’) reflectivity  x d(x, ) A(x,x’)  xx’ = Smear & Sum Data

Prestack Migration

W(  ) ~ A(x,x’)  xx’  i e x’ x’  x’x’’  i e A(x’’,x’) m(x) = d(x’) dddd  dddd  ~ d(x’,  +  ) xx’ x’x’’ A(x,x’)  x’ x’ A(x’’,x’) = * - - Broadband case W(  )=1 ~ Narrow band case: direct wave correlated with data Diffraction Stack Migration: Prestack

Exploding Reflector ½ Velocity Depth Time V/2  oo rorr)()(  g  rd  )(  dr o  oo rorr)()( g  rd  )(  r o d = L o

 rorr)()(  g  rd  )(  dr ExplodingReflector Forward Acoustic Problem Given: d + k d = 0 22 Find: d(r) Soln:  rmrr)()(  g  rd  )(  d(r )dr inc  rmrr)()( g  rd  )(  r d = L m Reflectivity

Acoustic ZO Migration Depth c 2c1 Find: m( r ) (c - c )/(c + c ) m( r ) (c - c )/(c + c )1122 Given: d(r )=Lm ro  )( Soln: L d L d T Soln:  o rdrr)()(  g rm  )(  dr *

ZO Data Migration 0 km 7 km 0 km 3 km