Reverse Time Migration =

Reverse Time Migration Reverse Time Migration ==

Generalized Diffraction MigrationGeneralized Diffraction Migration

OutlineOutline1. RTM = GDM1. RTM = GDM2. Implications2. Implications

FilteringFilteringSuperresolutionSuperresolution

Target Oriented RTMTarget Oriented RTMFast LSMFast LSMDiffraction SelectiveDiffraction Selective

Perfect Migration OperatorsPerfect Migration Operators

d(r)d(r) = = m(x)m(x)G(G(ss|x)|x) G(x|G(x|rr))[[ ]]****

T=0T=0

Reverse Time MigrationReverse Time MigrationGeneralized Diff. MigrationGeneralized Diff. MigrationCalc. Green’s Func. By FD solvesCalc. Green’s Func. By FD solves

= dot product data with hyperbola= dot product data with hyperbolaGeneralized Kirchhoff kernelGeneralized Kirchhoff kernel

Convolution of G(Convolution of G(ss|x) with G(x||x) with G(x|rr))

QED:QED: RTM can now enjoy: RTM can now enjoy:

Anti-aliasing filterAnti-aliasing filter

Obliquity factorObliquity factor

Angle GathersAngle Gathers

UD SeparationUD Separation

Decomplexify back&forwardDecomplexify back&forward

felds according 2 tastefelds according 2 taste

Etc. etc.Etc. etc.

d(r)d(r) = = m(x)m(x)G(G(ss|x)|x) G(x|G(x|rr))[[ ]]****

T=0T=0

Reverse Time MigrationReverse Time MigrationGeneralized Kirch. MigrationGeneralized Kirch. MigrationCalc. Green’s Func. By FD solvesCalc. Green’s Func. By FD solves

= dot product data with hyperbola= dot product data with hyperbolaGeneralized Kirchhoff kernelGeneralized Kirchhoff kernel

Convolution of G(Convolution of G(ss|x) with G(x||x) with G(x|rr))

QED:QED: RTM can now enjoy: RTM can now enjoy:

Anti-aliasing filterAnti-aliasing filter

Obliquity factorObliquity factor

Angle GathersAngle Gathers

UD SeparationUD Separation

Decomplexify back&forwardDecomplexify back&forward

felds according 2 tastefelds according 2 taste

Etc. etc.Etc. etc.


FilteringFilteringSuperresolutionSuperresolution



Resolution of KM vs GDMResolution of KM vs GDM

timetime timetime

MultiplesMultiples

PrimaryPrimaryPrimaryPrimary

Kirchhoff Mig. vs GDMKirchhoff Mig. vs GDM

1. Low-Fold Stack vs Superstack 1. Low-Fold Stack vs Superstack

2. Poor Resolution vs Superresolution 2. Poor Resolution vs Superresolution

MultiplesMultiples

3. Caution: RTM sensitive to mig. vel. errors3. Caution: RTM sensitive to mig. vel. errors

Rayleigh ResolutionRayleigh Resolution

timetime

LL

migratemigrate

x = 0.25x = 0.25z/Lz/L

Is Superresolution by RTM Achievable?

Tucson, Arizona Test

T u n n el a t level 2(30 m from grou n d )

T u n n el a t leve l 3(45 m from grou n d )

Shaf

t G ro u n d S u rfa ce30 m

1 5 m

S h o t loca tio nR ec e iv e r lo ca tio n

S h a ft en tran ce

60 m

Poststack MigrationPoststack Migration

1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0X (m )

-0 .2

0

0 .2

0 .4

0 .6

0 .8

1

Nor

mal

ized

Am

plitu

de

D irec t - F u ll A p ra tu reD irec t - H a lf A p ra tu reS ca tte re rs - F u ll A p ra tu reS ca tte re rs - H a lf A p ra tu re

S u p er-reso lu tion T est - P o in t # 14 a t X = 2 4m

~Kirchhoff Mig.

~Scattered RTM

This is highest fruit on the tree..who dares pick it?This is highest fruit on the tree..who dares pick it?

(Hanafy et al., 2008)

Can Scatterers Beat the Resolution Limit?

Recorded Green’s functions G(s|x) divided into:

- Shot gathers with direct arrivals only

- Shot gathers with scattered arrivals only

0 20 4 0 60 80 1 0 0 1 2 0

D ista n ce (m )

0 .5

0 .4

0 .3

0 .2

0 .1

0

Time

(ms)

G reen 's F u n ctionA fter b an d -p ass filter

0 20 40 60 8 0 10 0 12 0

D ista n ce (m )

0 .5

0 .4

0 .3

0 .2

0 .1

0

Time

(ms)

G reen 's F u n ctio nA fter b a n d -p a ss filter

0 20 40 60 80 1 0 0 1 2 0

D ista n ce (m )

0 .5

0 .4

0 .3

0 .2

0 .1

0

Time

(ms)

G reen 's F u n ctionA fter b an d -p ass filter


Filtering: 1Filtering: 1stst Arrival Filtering Arrival FilteringSuperresolutionSuperresolution



[[GG((ss|x)G(x|g)]* d(s|g) |x)G(x|g)]* d(s|g) s,g

1. RTM: 1. RTM: [{[{ } {} { } ]* d(s|g) } ]* d(s|g)

GG((ss|x)|x) GG((ss|x)|x) GG(x|g) (x|g) GG(x|g)(x|g)++ ++s,g==

dd (x) = (x) =

GG((ss|x)|x) GG(x|g)(x|g){ }s,g~~ ** d(s|g) d(s|g)

Super-wide Angle PhaseSuper-wide Angle Phase

Shift MigrationShift Migration

First Arrival FilterFirst Arrival Filter

Single Arrival KirchhoffSingle Arrival Kirchhoffw/o high-freq. appoxw/o high-freq. appox

Early Arrival FilterEarly Arrival Filter

Multiple Arrival KirchhoffMultiple Arrival Kirchhoffw/o high-freq. appoxw/o high-freq. appox

(or Super beam migration)(or Super beam migration)

Frechet DerivativeFrechet Derivative

True RTMTrue RTM

Phase Shift, Beam, Kirchhoff Migrations Phase Shift, Beam, Kirchhoff Migrations

are Special Cases of True RTMare Special Cases of True RTM

dsds

First Arrival FilterFirst Arrival Filter

& U p+Down filter& U p+Down filter

Efficient RT Migration OperatorsEfficient RT Migration Operators

SALTSALTFD only in FD only in

expanding boxexpanding box

Example Example (Min Zhou, 2003)(Min Zhou, 2003)

Standard FD Wavefront G(s|x)Standard FD Wavefront G(s|x) Early Arrival FD Wavefront G(s|x)Early Arrival FD Wavefront G(s|x)

Standard RTM vs Early Arrival RTMStandard RTM vs Early Arrival RTM

Standard FDStandard FD

0 4.5 km0 4.5 km

00

1.5 km1.5 km

Wavefront FDWavefront FDEfficient RT Migration OperatorsEfficient RT Migration Operators

FD/ Wavefront FD CostFD/ Wavefront FD Cost

# Gridpts along side# Gridpts along side500 3000500 3000

4545

55

FD/ W

avef

ront

FD

Cos

tFD

/ Wav

efro

nt F

D C

ost

ModelModel

0 4.5 km0 4.5 km

00

1.5 km1.5 km

00

1.5 km1.5 km

Wavefront Migration ImageWavefront Migration Image

1.5 km/s1.5 km/s2.2 km/s2.2 km/s

1.8 km/s1.8 km/s

Wavefront Migration ImageWavefront Migration Image

Reverse Time MigrationReverse Time Migration

00

1.5 km1.5 km

0 4.5 km0 4.5 km

00

1.5 km1.5 km

1.5 km/s1.5 km/s2.2 km/s2.2 km/s

1.8 km/s1.8 km/s





SALTSALT

Filtering of Wave Equation Filtering of Wave Equation Migration OperatorsMigration Operators

SALTSALT

Truncation: anti-aliasingTruncation: anti-aliasing

SALTSALT

Slant stackSlant stack

Filtering of Wave Equation Filtering of Wave Equation Migration OperatorsMigration Operators

Filtering of Filtering of WaveWave Equation Migration Operators Equation Migration Operators

0 s0 s

1.0 s1.0 s

Tim

e (s

)T

ime

(s)

0 km 0 km 4.5 km 4.5 km X (km)X (km) 0 km 0 km 4.5 km 4.5 km X (km)X (km)

0 s0 s

1.0 s1.0 s

0 s0 s

1.0 s1.0 s

COG Mig. Op.COG Mig. Op. Filtered COG Mig. Op.Filtered COG Mig. Op.

Z=70 mZ=70 m

Z=270 mZ=270 m

Z=1190 mZ=1190 m





25

Standard Reverse Time RedatumingStandard Reverse Time Redatuming

Cost = 10 FD SolvesCost = 10 FD Solves

to get G(to get G(xx||xx))

Special case:Special case: 10 Shot Gathers at the Surface, 3 Receivers at Depth10 Shot Gathers at the Surface, 3 Receivers at Depth

Procedure:Procedure: Compute 10 FD Solves, one for each shot at z=0Compute 10 FD Solves, one for each shot at z=0

DatumDatum

Special case:Special case: 10 Shot Gathers at the Surface, 3 Receivers at Depth10 Shot Gathers at the Surface, 3 Receivers at Depth

Procedure:Procedure: Compute 3 FD Solves, one for each shot at z=datumCompute 3 FD Solves, one for each shot at z=datum

Cost = 3 FD SolvesCost = 3 FD Solves

to get to get GG((xx||xx))

Trick:Trick: By ReciprocityBy Reciprocity GG((xx||xx)=)=GG((xx||xx))

Target Oriented Reverse Time RedatumingTarget Oriented Reverse Time Redatuming

DatumDatum

Benefit: Several ordersBenefit: Several orders

magnitude less expensivemagnitude less expensive

27

Dep

th (K

m)

Dep

th (K

m)

00

WW EE

3.53.5Offset (km)Offset (km)00

22.0.0

A slice of 3D SEG/EAGE model at x=2.0 kmA slice of 3D SEG/EAGE model at x=2.0 km

1.241.24

3D Synthetic Data (Dong)3D Synthetic Data (Dong)

Kirchhoff MigrationKirchhoff Migration

Redatum + KMRedatum + KM

33.5.5Offset (km)Offset (km)00

Z (k

m)

Z (k

m)

00

8.08.0

y (km)y (km)

6.06.0

00

x (km)x (km)1212

00

Interval velocity model

3D Field Data Test3D Field Data Test

OBC geometry:

50,000 shots

180 receivers per shot

Datum depth:

1.5 km

RVSP Green’s functions:

5,000 shots

180 receivers per shot

km/s5.55.5

1.51.5

New Datum


y (km)y (km)00 4.54.5

Tim

e (s

)Ti

me

(s)

00

6.06.0

Original CSG

y (km)y (km)00 4.54.5

Tim

e (s

)Ti

me

(s)

00

6.06.0

Redatumed CSG

KM of RTD data

Z (k

m)

Z (k

m)

00

88

y (km)y (km)

55

00

x (km)x (km)1212

00

KM of redatumed data

Z (k

m)

Z (k

m)

00

88

y (km)y (km)

55

00

x (km)x (km)1212

00

KM of original data


( Inline No. 61 )( Inline No. 61 )

X (km)X (km)00 1212

Z (k

m)

Z (k

m)

00

8.08.0

KM of RTD data

X (km)X (km)00 1212

Z (k

m)

Z (k

m)

00

8.08.0


KM of original data

( Crossline No. 41 )( Crossline No. 41 )

Y (km)Y (km)00 5.05.0

Z (k

m)

Z (k

m)

00

8.08.0

KM of RTD data

Y (km)Y (km)00 5.05.0

Z (k

m)

Z (k

m)

00

8.08.0


KM of original data

( Crossline No. 61 )( Crossline No. 61 )

Y (km)Y (km)00 5.05.0

Z (k

m)

Z (k

m)

00

8.08.0

KM of RTD data

Y (km)Y (km)00 5.05.0

Z (k

m)

Z (k

m)

00

8.08.0


KM of original data

( Depth 2.0 km )( Depth 2.0 km )

X (km)X (km)00 1212

Y (k

m)

Y (k

m)

00

5.05.0

KM of RTD data

X (km)X (km)00 1212

Y (k

m)

Y (k

m)

00

5.05.0


KM of original data


X (km)X (km)00 1212

Y (k

m)

Y (k

m)

00

5.05.0

KM of RTD data

X (km)X (km)00 1212

Y (k

m)

Y (k

m)

00

5.05.0


KM of original data


X (km)X (km)00 1212

Y (k

m)

Y (k

m)

00

5.05.0

KM of RTD data

X (km)X (km)00 1212

Y (k

m)

Y (k

m)

00

5.05.0


KM of original data

RTM (CPU-hours)

RTD (CPU-hours)

Speed up

3D field data test 5,000,000 (estimated) 52,000 100

Computational CostsComputational Costs





Motivation (Ge Zhan)Motivation (Ge Zhan)• ProblemProblem

Conventional RTM suffers from imaging artifacts.Conventional RTM suffers from imaging artifacts.

• SolutionSolution

Wavelet compression of Green’s functions (10x or more).Wavelet compression of Green’s functions (10x or more).

Compressed generalized diffraction-stack migration (GDM) .Compressed generalized diffraction-stack migration (GDM) .

Kirchhoff (diffraction-stack) migration is efficient Kirchhoff (diffraction-stack) migration is efficient

but with a high-frequency approximation.but with a high-frequency approximation.

WEM method (RTM)WEM method (RTM) is accurate but computationallyis accurate but computationally

intensive compared to KM.intensive compared to KM.

Least squares algorithm.Least squares algorithm.

Migration OperatorMigration Operator

Size = nx*nz*ns*ng*nt = 645*150*323*176*1001*4 = 20 TB

Too big to store.

G G2D Wavelet Transform

rr

xx

ss

( | )G s x ( | )G x r

appropriate threshold

10x compression

TheoryTheory

G(s|x)G(x|g) (5 dimensions)

Green’s Function trace

Can Scatterers Beat the Resolution Limit ?

TheoryTheory

Numerical ResultsNumerical ResultsSEG/EAGE Salt Model

X (km)

Z (k

m)

0 15

0

3

X (km)

Z (k

m)

0 15

0

3

Zoom View

323 shots

176 geophones

peak freq = 13 Hz

dx = 24.4 m

dg = 24.4 m

ds = 48.8 m

nsamples = 1001

dt = 0.008 s

1.5

2.5

3.5

4.5

km/s

Calculated GF

Tim

e (s

)

Trace #

4

0

4011Ti

me

(s)

Trace#

4

1.5

4011

Trace Comparison

101 201 301

200 MB

Trace #4011

Reconstructed GF

20 MB

Wavelet Transform CompressionWavelet Transform Compression

Numerical ResultsNumerical Results

Early-arrivals

Tim

e (s

)

Trace#

4

0

4011 Trace# 4011

Multiples


X (km)

Z (k

m)

0 15

0

3

(a) GDM using Early-arrivals

X (km)0 15

(b) GDM using Full Wavefield

X (km)

Z (k

m)

0 15

0

3

(c) GDM using Multiples

X (km)0 15

(d) Optimal Stack of (a) and (c)






Exact Migration Operators from VSP Exact Migration Operators from VSP IMPLICATION #2IMPLICATION #2

SALTSALT

g(s|x)g(s|x)

SALTSALT

**Exact Migration Operators from VSP Exact Migration Operators from VSP

IMPLICATION #2IMPLICATION #2

g(r|x)g(r|x) g(s|x)g(s|x)**

Exxon RVSP DataExxon RVSP Data0 s0 s

0.5 s0.5 s

Z = .18 kmZ = .18 km

DirectDirectReflections Reflections

MultiplesMultiples

0 km 0 km 0.2 km 0.2 km XX

FocusingFocusing

OperatorOperatorg(x|r)g(x|r)g(s|x)g(s|x)

Exxon RVSP Data Exxon RVSP Data

0 km 0 km 0.2 km 0.2 km

0 km 0 km 0.2 km 0.2 km XX

XX

Prim Refl. Focusing OperatorPrim Refl. Focusing Operator 0.2 s0.2 s

0.28 s0.28 s

Interbed Multiple Refl. Focusing OperatorInterbed Multiple Refl. Focusing Operator 0.31 s0.31 s

0.37 s0.37 s

Documents

Reverse Time Migration =