Wave-Equation Migration in Wave-Equation Migration in Anisotropic MediaAnisotropic Media

Jianhua YuJianhua Yu

University of UtahUniversity of Utah

Contents Contents

Motivation

Anisotropic Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE model

3-D SEG/EAGE model

Contents Contents

Motivation

Anisotropic Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE model

3-D SEG/EAGE model

What Blurs Seismic Images? What Blurs Seismic Images?

Irregular acquisition geometry

Bandwidth source wavelet

Velocity errors

Higher order phenomenon: Anisotropy

Anisotropic ImagingAnisotropic Imaging

Ray-based anisotropic migration: Anisotropic velocity model

Anisotropic wave-equation migration:

---Ristow et al, 1998

---Han et al. 2003

Objective: Objective:

High efficiency

Improve image accuracy

Develop 3-D anisotropic wave-equation migration method in orthorhombic model

>78 wave propagatoro

Contents Contents

Motivation

Anisotropy Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE model

3-D SEG/EAGE model

General Wave EquationGeneral Wave EquationWave equation in displacement

il

kijkl

j

i Fx

uC

xt

u

)(2

2

Ui : displacement component

Cijkl : 4th-order stiffness tensor

3 3

3,2,1,,k l

jiklijklij ec

Eigensystem EquationEigensystem Equation

0

3

2

1

2333231

232

2221

13122

11

U

U

U

V

V

V

Polarization components of P-P, SV, and SH waves

Orthorhombic AnisotropicOrthorhombic Anisotropic2355

2266

211111 ncncncΓ

21661112 n)nc(cΓ

31551313 n)nc(cΓ

2344

2222

216622 ncncncΓ

2333

2244

215533 ncncncΓ

Orthorhombic AnisotropicOrthorhombic Anisotropic

21662121 n)nc(cΓ

32442323 n)nc(cΓ

31553131 n)nc(cΓ

32443232 n)nc(cΓ

Decoupled P plane Wave Motion Decoupled P plane Wave Motion Equations in (x,z) and (y,z) planesEquations in (x,z) and (y,z) planes

0)(

)(

2

1

2233

2555513

551322

552

11

U

U

KcKcKKcc

KKccKcKc

zxzx

zxzx

0)(

)(

3

2

2233

2444423

442322

442

22

U

U

KcKcKKcc

KKccKcKc

zxzx

zxzx

and

Decoupled P plane Wave Motion Decoupled P plane Wave Motion Equations in (x,z) and (y,z) planesEquations in (x,z) and (y,z) planes

0)(

)(

2

1

2233

2555513

551322

552

11

U

U

KcKcKKcc

KKccKcKc

zxzx

zxzx

0)(

)(

3

2

2233

2444423

442322

442

22

U

U

KcKcKKcc

KKccKcKc

zxzx

zxzx

and

det

det

Dispersion EquationsDispersion Equations

24)1()1(22

2)1(2242

)(2

)1(

x

xz K

KK

(x,z) plane

24)2()2(22

2)2(2242

)(2

)1(

y

yz K

KK

(y,z) plane

Thomsen’s Parameters

33c

33

3322)1(

2c

cc

)(2

)()(

443333

24433

24423)1(

ccc

cccc

Thomsen’s Parameters

33

3311)2(

2c

cc

)(2

)()(

553333

25533

25513)2(

ccc

cccc

VTI:

)2()1(

)2()1(

5544 cc

20

20

2

0 )](1[

)(

x

xm

A

KBB

KA

24000

20

2

200

242

)(2

)21(

x

xz K

KK

)11

(0

FFD algorithm

FFD Anisotropy Migration

)21( )1(00 aA

)21( )1(aA

)1(0

)1(00 )(2)2(2 bababB

)1()1( )(2)2(2 bababB

How to Set Velocity and Anisotropy Parameters

a & b : Optimization coefficients of Pade approximation for FD

d 0Velocity:

Anisotropy:

)1()1(0

)1( d

)1()1(0

)1( d

0

5

Err

or %

0 90

Pade Approximation Comparison

Angle

0

0.05

Error %

0 78

Pade Approximation Comparison

Angle

Beyond 78 within 0.02 %

Contents Contents

Motivation

Anisotropy Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE model

3-D SEG/EAGE model

0.6

0

Kz

Kx -0.3 0.3 Kx -0.3 0.3

Weak Anisotropy Strong Anisotropy

Exact Exact

** Approximation ** Approximation

2.01.0 00 4.05.0

00 00 0015.005.0

0.3

0

Kz

Kx -0.3 0.3

Dispersion Equation Approximation

Strong anisotropy

0

2.0

Dep

th (

km

)

V/V0=3

V/V0=3

iso

iso

New

Sta

nd

ard

0

2.0

Dep

th (

km

)

V/V0=3

V/V0=3

Weak Aniso

Strong Aniso

2.01.0 00 4.05.0

00 00 0015.005.0

Contents Contents

Motivation

Anisotropy Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE model

3-D SEG/EAGE model

00

1

Dep

th (

km

)1.5X (km)

Velocity (2.0-3.0 km/s)Velocity (2.0-3.0 km/s)

00

1

Tim

e (s

)

1.5X (km)

Velocity (2.0-3.0 km/s)

0 1.5

Anisotropic data (SUSYNLVFTI)

0

1.2

Tim

e (s

)

X (km)

Isotropic data (SUSYNLY)

1.004.0 00 00

00

1

Dep

th (

km

)

1.5

X (km)

Isotropic data Isotropic mig (su)

0 1.5

Anisotropic data Isotropic mig

0 1.5

Anisotropic data Anisotropic mig

Contents Contents

Motivation

Anisotropy Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE model

3-D SEG/EAGE model

00

4

Dep

th (

km

)5X (km)

Salt Model (VTI)

1.0045.0 00 00

00

4

Dep

th (

km

)5X (km)

Iso-mig

00

4

Dep

th (

km

)5X (km)

VTI Aniso-mig

0 1.5

Anisotropy Error 40 %

X (km)

0

4

Dep

th (

km

)

0 1.5

Anisotropy Error 10 %

X (km)0 1.5

Anisotropy Error 20 %

X (km)

Inaccurate Thomsen’s Parameters (VTI)

5 10

Anisotropy Error 40 %

X (km)

3

4

Dep

th (

km

)

5 10

Anisotropy Error 10 %

X (km)5 10

Anisotropy Error 20 %

X (km)

Inaccurate Thomsen’s Parameters

Contents Contents

Motivation

Anisotropy Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE VTI model

3-D SEG/EAGE VTI model

0

4

Dep

th (

km

)0 5X (km) 0 5X (km)

VTI Aniso (y=1.5 km)Iso (y=1.5 km)

1.0045.0 00 00

0

4

Dep

th (

km

)0 5Y (km) 0 5Y (km)

VTI Aniso (x=1.5 km)Iso (x=1.5 km)

0

4

Dep

th (

km

)0 5Y (km) 0 5Y (km)

VTI Aniso (x=3 km)Iso (x=3 km)

00

5

Y (

km

)

5X (km) 0 5X (km)

VTI Aniso (z=0.5 km)Iso (z=0.5 km)

00

5

Y (

km

)

5X (km) 0 5X (km)

VTI Aniso (z=2.5 km)Iso (z=2.5 km)

Contents Contents

Motivation

Anisotropy Wave-Equation Migration

Numerical Examples:

Cusp model

Conclusions

2-D SEG/EAGE model

3-D SEG/EAGE model

Conclusions Conclusions

Works for 2-D and 3-D media

New > 78 Anisotropic wave propagator:

Improves spatial resolution

Valid for VTI and TI

o

78 Propagator Cost = Cost of Standard 45^o propagator

o

Thanks To Thanks To

2003 UTAM Sponsors

CHPC