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)=

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

d(x) = d(x) = x’x’

~~g(g(xx|x’)|x’)

A(xA(x,x’,x’))

xxx’x’iiee m(x’)m(x’)d(x) d(x) ~~m(x’)m(x’)

xx

dd(x, )(x, )xxx’x’FourierFourierTransformTransform

Forward Modeling (Forward Modeling (dd = L = Loo))ZO Depth Migration (ZO Depth Migration (mm L L dd))TT

m(x’)m(x’)

reflectivityreflectivity

xx

dd((xx, ), )A(xA(x,x’,x’))

xxx’x’==....

~~ 22

....

MigrationMigration

Migration Migration

IntuitiveIntuitiveLeast SquaresLeast Squares

Green’s TheoremGreen’s Theorem

Forward Problem: d=Forward Problem: d=LLmm

mm==LL d dTT

Japan Sea ExampleJapan Sea Example

LSM Image with a Ricker Wavelet (15 Hz)LSM Image with a Ricker Wavelet (15 Hz)

Actual Model 0

2D

epth

(km

)0 2

X (km)

LSM ImageKirchhoff Migration

Image

2D Poststack Data from Japan Sea2D Poststack Data from Japan SeaJAPEX 2D SSP marine data description:JAPEX 2D SSP marine data description:

Acquired in 1974, Acquired in 1974,

Dominant frequency of 15 Hz.Dominant frequency of 15 Hz.

00

55

TW

T (

s)T

WT

(s)

00 2020X (km)X (km)

LSM vs. Kirchhoff MigrationLSM vs. Kirchhoff MigrationLSM ImageLSM Image

0.70.7

1.91.9

Dep

th (

km)

Dep

th (

km)

2.42.4 4.94.9X (km)X (km)

0.70.7

1.91.9

Dep

th (

km)

Dep

th (

km)

2.42.4 4.94.9X (km)X (km)

Kirchhoff Migration Kirchhoff Migration ImageImage

MigrationMigration

Migration Migration

IntuitiveIntuitiveLeast SquaresLeast Squares

Green’s TheoremGreen’s Theorem

SummarySummary

m(x’)m(x’)

Approx. reflectivityApprox. reflectivity

xx

dd((xx, ), )A(xA(x,x’,x’))

xxx’x’==....

cos cos 1. ZO migration:1. ZO migration:

obliquityobliquity

x’x’

xx

2. ZO migration assumptions: Single scattering data2. ZO migration assumptions: Single scattering data

3. ZO migration matrix-vec: 3. ZO migration matrix-vec: mm=L =L ddTT~~

4. LSM ZO migration matrix-vec: 4. LSM ZO migration matrix-vec: mm=[L L] L =[L L] L ddTT TT-1-1

Compensates forCompensates forIllumination footprint and poorIllumination footprint and poorilluminationillumination

5. ZO migration smears an event along appropriate5. ZO migration smears an event along appropriate doughnutdoughnut

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

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*