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MigrationMigration
Migration Migration
IntuitiveIntuitiveLeast SquaresLeast Squares
Green’s TheoremGreen’s Theorem
2-w
ay ti
me
2-w
ay ti
me
((xx--x x ) + ) + yy 2222
c/2c/2xxxx ==
xxxx + + TToo
ZO MigrationZO Migration Smear Reflections along Fat CirclesSmear Reflections along Fat Circles
xx
xx
dd((xx , ) , )xxxx
Thickness = c*T /2Thickness = c*T /2oo
Where did reflectionsWhere did reflectionscome from?come from?
1-w
ay ti
me
1-w
ay ti
me
ZO MigrationZO MigrationSmear Reflections along Fat CirclesSmear Reflections along Fat Circles
xx
& Sum& Sum
Hey, that’s ourHey, that’s ourZO migration formulaZO migration formula dd((xx , ) , )xxxx
1-w
ay ti
me
1-w
ay ti
me
ZO MigrationZO MigrationSmear Reflections along CirclesSmear Reflections along Circles
xx
& Sum& Sum
In-PhaseIn-Phase
Out-of--PhaseOut-of--Phase
dd((xx , ) , )xxxxm(x)=m(x)=
MigrationMigration
Migration Migration
IntuitiveIntuitiveLeast SquaresLeast Squares
Green’s TheoremGreen’s Theorem
Forward Problem: d=Forward Problem: d=LLmm
mm==LL d dTT
Born Forward ModelingBorn Forward Modeling
d(x) = d(x) = x’x’
~~
dd = = LL m miijj jjjjii
g(g(xx|x’)|x’)A(xA(x,x’,x’))
xxx’x’iiee m(x’)m(x’)
reflectivityreflectivity
Given: Given: dd = L= LmmSeismic Inverse ProblemSeismic Inverse Problem
Find: Find: m(x,y,z)m(x,y,z)
Soln: min || LSoln: min || Lmm--dd || ||22
mm = [L L] L = [L L] L ddTT TT-1-1
‘‘
L L ddTT
migrationmigration
waveformwaveforminversioninversion
Forward Modeling (Forward Modeling (dd = L = Loo))
d(x) = d(x) = x’x’
~~g(g(xx|x’)|x’)
A(xA(x,x’,x’))
xxx’x’iiee o(x’)o(x’)d(x) d(x) ~~m(x’)m(x’)
xx
dd(x, )(x, )xxx’x’FourierFourierTransformTransform
ZO Depth Migration (ZO Depth Migration (oo L L dd))TT
M(x’)M(x’)
reflectivityreflectivity
xx
dd((xx, ), )A(xA(x,x’,x’))
xxx’x’==....
ReviewReview• Inverse Acoustic Problem Inverse Acoustic Problem
Find:Find: m( r ) (c - c )/(c + c )m( r ) (c - c )/(c + c )11 1122 22
Given:Given: d(r )d(r )
rm)(Soln:Soln: L L dd
TTSoln:Soln:
o rdrr )()(grm
)(
dr**
M(x’)M(x’)
reflectivityreflectivity
xx
dd((xx, ), )A(xA(x,x’,x’))
xxx’x’== SmearSmear& Sum& SumDataData
ZO DataZO DataZO DataZO DataMigrationMigration
0 km 7 km0 km 7 km
0 km0 km
3 km3 km
Least SquaresLeast Squares
Recall: Lm=dRecall: Lm=d
jj
iijj
Find: m that minimizes sum of squaredFind: m that minimizes sum of squared residuals r = L m - dresiduals r = L m - d
jj iiii
(r ,r) = ([Lm-d],[Lm-d])(r ,r) = ([Lm-d],[Lm-d]) = m L Lm -2m Ld-d d= m L Lm -2m Ld-d d
L Lm = LdL Lm = Ld Normal equationsNormal equations
For all For all ii(r ,r)(r ,r)dddmdmii
= 2 m L Lm -2 m Ld= 2 m L Lm -2 m Lddddmdmii
dddmdmii
= 0= 0
Prestack MigrationPrestack Migration
W(W( ) )~~A(xA(x,x’,x’))
xxx’x’iiee
x’x’
x’x’x’’x’’iiee
A(x’’A(x’’,x’,x’))m(x) =m(x) = d(x’)d(x’)dd dd~~
d(x’d(x’, , + + ))xxx’x’ x’x’x’’x’’
A(xA(x,x’,x’))
x’x’ A(x’’A(x’’,x’,x’))==
**-- --
Broadband caseBroadband caseW(W( )=1 )=1~~
Narrow band case: direct wave correlated with dataNarrow band case: direct wave correlated with data
....
115.
Diffraction Stack Migration: PrestackDiffraction Stack Migration: Prestack
Exploding ReflectorExploding Reflector½ Velocity½ Velocity
Dep
thD
epth
Tim
eT
imeV/2V/2
oo rorr )()(grd
)(
dr
o
oo rorr )()(grd
)(
ro
d = L d = L oo
rorr )()(grd
)(
drExplodingExploding
ReflectorReflector
Forward Acoustic Problem Forward Acoustic Problem
Given:Given: dd + k + k d d = 0= 022 22
Find:Find: d(r)
Soln:Soln:
rmrr )()(grd)(
d(r )dr
inc
rmrr )()(grd
)(
rd = d = LL mm
ReflectivityReflectivity
Dot Products and Adjoint OperatorsDot Products and Adjoint Operators
Recall: (u,u) = u* uRecall: (u,u) = u* u ii
ii ii
Recall: (v,Lu) = v* ( L u )Recall: (v,Lu) = v* ( L u ) jj
ii iijj ii
jj
[ L v* ]u [ L v* ]u jj
ii
jjiijj ii==
[ L* v ]* u [ L* v ]* u jj
ii
jjiijj ii==
So adjoint of L is LSo adjoint of L is L ii
iijjL*L*
Acoustic ZO MigrationAcoustic ZO MigrationD
epth
Dep
th
cc22
cc11
Find:Find: m( r ) (c - c )/(c + c )m( r ) (c - c )/(c + c )11 1122 22
Given:Given: d(r )=Ld(r )=Lmm
ro)(Soln:Soln: L L dd
TTSoln:Soln:
o rdrr )()(grm
)(
dr**
