6. Seismic Anisotropy

6. Seismic Anisotropy

A.Stovas, NTNU2005

Definition (1)

• Seismic anisotropy is the dependence of seismic velocity upon angle

• This definition yields both P- and S-waves

Definition (2)

• Saying ”velocity” we mean

• ray veocity and wavefront velocity• group velocity and phase velocity• interval velocity and average velocity• NMO velocity and RMS velocity

Definition (3)

• We have to distinguish between anisotropy and heterogeneity

• Heterogeneity is the dependence of physical properties upon position

• Heterogeneity on the small scale can appear as seismic anisotropy on the large scale

Notationshi – layer thicknessvi – layer velocityt0 – vertical traveltimevP0 – vertical velocityvNMO – normal moveout velocity, – anisotropy parametersS2 – heterogeneity coefficient

Simple example of anisotropy (two isotropic layers model)

1 2P0

1 2

1 2

h hv h hv v

1h

2h

1v

2v

isotropic

isotropicVTI

P0v

2 1 1 2 2NMO

1 2

1 2

h v h vv h hv v

2 2NMO P0v v 1 2

1 20

1 2

h ht 2v v

3 3 1 21 1 2 2

1 22 2

1 1 2 2

h hh v h vv v

Sh v h v

2S 1 8

21 2 1 2

21 2 1 2

h h v v1 02 v v h h

22 21 2 1 2

21 2 1 1 2 2

h h v v1 08 v v h v h v

hi – layer thicknessvi – layer velocityt0 – vertical traveltimevP0 – vertical velocityvNMO – normal moveout velocity, – anisotropy parametersS2 – heterogeneity coefficient

Elasticity tensor

τ CE

2iji

2j

ut x

Equation of motion

Stress-strain relation (Hook’s law)

stress

strain

The elasticity tensor

2 2i m

ijmn2n j

u ut x x

C

Symmetry

11 12 13 14 15 16

12 22 23 24 25 26

13 23 33 34 35 36

14 24 34 44 45 46

15 25 35 45 55 56

16 26 36 46 56 66

C C C C C CC C C C C CC C C C C CC C C C C CC C C C C CC C C C C C

ij ji 11 22 33 23 13 121 2 3 4 5 6

We convert stiffness tensor Cijmn to the stiffness matrix C

The best case:Isotropic symmetry2 different elements

The worst case:Triclinic symmetry21 different elements

2 0 0 02 0 0 0

2 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

Lame parameters: and

Seismic anisotropy symmetries

11 11 66 13

11 66 11 13

13 13 33

44

44

66

C C 2C CC 2C C C

C C CC

CC

Orthorombicic symmetry9 different elements(shales, thin-bed sequenceswith vertical crack-sets)

Trasverse isotropy symmetry5 different elements(shales, thin-bed sequences

11 12 13

12 22 23

13 23 33

44

55

66

C C CC C CC C C

CC

C

VTI and HTI anisotropy

VTI

HTI

symmetry axis

symmetry plane

The phase velocities (velocities of plane waves)

2 2P 33 44 11 33

2 2SV 33 44 11 33

2 2 2SH 44 66

233 44

2 213 44 33 44 11 33 44

2 2 411 33 44 13 44

1v C C C C sin D21v C C C C sin D2

v C cos C sin

D C C

2 2 C C C C C C 2C sin

C C 2C 4 C C sin

Cij – stiffness coefficientsvi – phase velocity– phase angle

Parametrization (Thomsen, 1984)

33 44P0 S0

C Cv v

11 33

33

2 213 33 33 44

33 33 44

66 44

44

C C2C

C C C C2C C C

C C2C

Vertical velocities

Anisotropy parameters

Interpretation of anisotropy parameters

P0 P0v P wave velocity v S wave velocity

0

Isotropy reduction

2 2P P0

2 2SV S0

2 2SH S0

v v 1 22

v v2

v v 1 22

Horizontal propagation

Weak anisotropy approximation

, , are small numbers

2 2 4P P0

2

2 2P0SV S0

SV0

2SH S0

v v 1 sin cos sin

vv v 1 sin cosv

v v 1 sin

Weak anisotropy for laminated siltstone

0 10 20 30 40 50 60 70 80 901600

1800

2000

2200

2400

2600 Exact Thomsen (T,T) Stovas&Ursin ()

Phase angle, degrees

0 10 20 30 40 50 60 70 80 904400

4600

4800

5000

5200

v SV,

m/s

v P, m

/s

Exact Thomsen (T,T) Stovas&Ursin (

T,)

Mesaverde shale/sandstone

0 10 20 30 40 50 60 70 802,65

2,70

2,75

2,80

2,85

Mesaverde sandstone

Mesaverde shale

VS

V, [k

m/s

]

Phase angle, [degrees]

0 10 20 30 40 50 60 70 80

3,73,83,94,04,14,24,34,4

Mesaverde sandstone

Mesaverde shale

VP,

[km

/s]

Nonellipticity

2 4P P0

2 2

2 4P0 P0SV S0

SV0 SV0

v v 1 sin sin

v vv v 1 sin sinv v

20

2 20

The anellipticity definitions(T homsen)

(Alkhalifah)1 2

21 (Stovas and Ur sin)11 2

Wave propagation in homogeneous anisotropic medium

Wavefront normal

Wavefr

ont ta

ngen

t

2phase2 2

group phase

phasegroup phase

phase

phase

2 22group

vV v

dv1V cos v cos 1 tanv d

vk

Vk k

3P P

2 2 2P P

pv vtan

v 1 p v

P PP

P

P P2 2 2P P

1 1V cos dv1v cos 1 tan

v d

v pv

v 1 p v

k – wavenumberVgroup – group velocity – angular frequencyvphase – phase velocityp – horizontal slowness– group and phase angles

The anisotropic moveout

2

2 20 2

NMO

xt x tv

2

2 2NMO x 0

1 tv x

2 2P NMO P0

22 2 P0SV NMO S0 2

S0

2 2SH NMO S0

v v 1 2

vv v 1 2v

v v 1 2

The hyperbolic moveout

The Taylor series coefficient

The moveout velocity

(x – offset ot source-receiver separation)

The anisotropic qP-traveltime in p-domain

sinpv

2

P 0 P0 P 0 P0 2 20

1 S Hx t tan p t

1 p 1 S

2 20 P0 0

P P0 2 2P P 0

t 1 p Ht t

V cos 1 p 1 S

The horizontal slowness

The offset

The traveltime

22 2P P0

1 1 p Sv v 2 j

j 0j 0

S a p

dSpH

2 dp

S – deviation of the slowness squared between VTI and isotropic casesaj – coefficients for expansion in order of slowness

The anisotropic traveltime parameters

P0

20

P1 20

220

P2 02 20 0

a 2

2a 2 11

24a 11 1

The P-wave

The S-wave

The vertical Vp-Vs ratio2

2 P00 2

S0

vv

2S0 0a 2 2 j

Sj 0 Pja a

The moveout velocity

2 2 2 2 2P P 0P2 22 2 2

PP 0P P

d t d t d p 1 p Hpt1xV 1 S Hd x d x d p

22 2c 0 0 0 1p 1 1 S 1 a 1 a a

The critical slowness

2 2 2 21 2P c 0 0 0 P NMO

0

a S 3V p p 1 S 1 a V

1 a 4

The velocity moments

0

2 22 0 0 NMO

244 0 0 1

366 0 0 0 1 2

4 28 28 0 0 0 1 0 2 1 3

1

v 1 a v

v 1 a 4a

v 1 a 4 1 a a 8a

24 32 16 64v 1 a 1 a a 1 a a a a5 5 5 5

v0 = 0

The heterogeneity coefficients

2kk k

2

4 12 2 2

2 0

6 1 23 3 2 3

2 0 0

28 31 2 1

4 4 2 3 4 42 0 0 0 0

S , k 2,3,...

4aS 11 a

4a 8aS 11 a 1 a

a24 a 32 a 16 a 64S 15 5 5 51 a 1 a 1 a 1 a

The Taylor series

2 2 2 4 6 80 2 3 4

22

2 2NMO 0

2 2

23 2 2 3

3 24 2 2 2

2 3 3 4

t x t 1 x c x c x c x ...

xxv t

1c S 14

1c 2 S 1 3 S 1 S 181c 24 S 1 63 S 1 30 S 164

24 S 1 S 1 20 S 1 5 S 1

The normalized offset squared

The traveltime approximations

20 2

2

1t x t 1 1 x S 1S

422 2 2

0 2

S 1 xt x t 1 x

4 1 Bx

22 P 2

2 2P NMO P

23 2 2 3

2 2

c v 2 3 S Tsvankin T homsen (1994)v v 2 4

Bc 2S S S Ur sin Stovas (2004)c 2 S 1

Shifted hyperbola

Continued fraction

The continued fraction approximations

2

22xP

xlim t xv 2

22 2

23 2

S S 1, Layered isotropic mediumS S 1, Single VTI layer

2

Tsvankin-Thomsen

Ursin-Stovas Correct 3c

The heterogeneity coefficient S2

2

8S 1 8 1

1 2

Alkhalifah and Tsvankin:

Ursin and Stovas:

201

2 2 2 200

8 24aS 1 1 111 a 1 2

S2 (Ursin and Stovas) reduces to

S2 (Alkhalifah and Tsvankin) if is large

(acoustic approximation)

The traveltime approximations(single VTI layer)

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8

0

2

4

6

8 Equation (38) Equation (37) Equation (39) Equation (36)

PS

Offset [km]

0,0 0,2 0,4 0,6 0,8 1,0 1,2

0

2

4

6

8

Equation (30) Equation (29) Equation (31) Equation (26)

Equation (20) Equation (19) Equation (21) Equation (14)

SS

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

0

2

4

6

8

[ms]

[ms]

[ms]

PP

P0 S0v 2.0km s, v 1.0km s,H 1km, 0.1, 0.05

Bold – two terms TaylorEmpty circles – shifted hyperbolaFilled circles – Tsvankin-ThomsenEmpty stars – Stovas-Ursin

The traveltime approximations(stack of VTI layers)

0,0 0,2 0,4 0,6 0,8 1,0-4

-3

-2

-1

0

1

2

3

4

tPP

[ms]

Offset [km]

0,0 0,2 0,4 0,6 0,8 1,0-4

-3

-2

-1

0

1

2

3

4

tPS

V [m

s]

Offset [km]

0,0 0,2 0,4 0,6 0,8 1,0-4

-3

-2

-1

0

1

2

3

4

tSV

SV [m

s]

Offset [km]

0,0 0,2 0,4 0,6 0,8 1,0-4

-3

-2

-1

0

1

2

3

4

tSH

SH [m

s]

Offset [km]

Layering against anisotropy

0,9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,80,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4 PP

S 3

S2

VTI ISO

-0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6-6

-4

-2

0

2

4

SVSV

S 3

S2

VTI ISO

1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4 3,6

2

4

6

8

10

12

14

16 PSV

S 3

S2

VTI ISO

1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4

1

2

3

4

5

6

7

8

9

SHSH

S 3

S2

VTI

VNMO for dipping reflector

2

2

NMO

1 v1vv

vtan vcos 1v

Tsvankin, 1995

is the angle for dipping reflector

VTI DMO operator

1,0

0,8

0,6

0,4

0,2

0,0-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6

a)

h=0.4 km, VNMO

(0)=2.0 km/s

t0=1.0 s

t0=0.1 s

p1=-0.26 s/km

p2=-0.145 s/km

p2=-0.22 s/km

p1=-0.42 s/km

Midpoint, km

Tim

e, s

1,2

1,0

0,8

0,6

0,4

0,2

0,0-0,4 -0,2 0,0 0,2 0,4

b)

h=0.4 km, VNMO

(0)=2.0 km/s

t0=1.0 s

t0=0.1 s

p3=p

4=-0.11 s/kmp

2=-0.195 s/kmp1=-0.46 s/km

p2=-0.195 s/km

p4=-0.085 s/km

p3=-0.175 s/km

p1=-0.44 s/km

Midpoint, km

Tim

e, s

Stovas, 2002

Operator shape depends on the sign of

If we ignore anisotropy in post-stack time migration

Dipping reflectors are mispositioned laterally. Mislocation is a function of:

- magnitude of the average foroverburden - dip of the reflector - thickness of anisotropic overburdenDiffractions are not completely collapsed,

leaving diffraction tails, etc.

Alkhalifah and Larner, 1994

Determining

• Use VP-NMO from well-log

• The residual moveout gives

Dix-type equations (1)

2 2 2PP PP2 2 200 1 1

PP

T 0 v1 1

T 0 2

2 40 PP PP PP PP

22 20 PP PP

1 T 0 v S T 01

T 0 v

0 PPT 0z

2

00

0

2PP PP

20 PP

T 0 v1 1 12 T 0

2 2PP PP SS SS

2 20 PP SS 0

T 0 v T 0 v1 1 112 T 0 T 0

Ursin and Stovas, 2004

2 2SS SS PP 02 2

1 02SS PP PP

T 0 v T 0 1 2T 0 1 2T 0 v

SS 00

PP 0

T 0T 0

Dix-type equations (2)(error in parameters due to error in

S2

2,00 2,02 2,04 2,06 2,08 2,10 2,12 2,14 2,16 2,18 2,202,30

2,35

2,40

2,45

2,50

2,55

2,60

2,65

2,70

Estimated values of 0

Correct value

0 [km

/s]

SPP

2,00 2,02 2,04 2,06 2,08 2,10 2,12 2,14 2,16 2,18 2,200,94

0,96

0,98

1,00

1,02

1,04

1,06 Estimated values of z Correct value

z, [

km]

SPP

2,00 2,02 2,04 2,06 2,08 2,10 2,12 2,14 2,16 2,18 2,200,10

0,15

0,20

0,25

0,30

Estimated values of Correct value

epsi

lon

SPP

2,00 2,02 2,04 2,06 2,08 2,10 2,12 2,14 2,16 2,18 2,20-0,02

0,00

0,02

0,04

0,06

0,08

0,10

0,12

Estimated values of Correct value

delta

SPP

Wave propagation in VTI medium

0d0dz

Ab bB

Trzrz U,S,S,U b

133

21311

213313

13313

133

cccppccpccc

A

1

44cpp

B

Uz and Ur are transformed verical and horizontal displacement components;

Sz and Sr are transformed vertical and horizontal stress components

Stovas and Ursin, 2003

Up- and down-wave decomposition

ididz

u q 0 F G ud 0 q G F d

diag q ,q q

q and q are verical slownesses for P- and S-wave

du

Lb

With linear transformation

The transformation matrix

T T2 1 2 1 L L L L I

1 1

2 2

i i12

L LL

L L

with the symmetries

Scattering matrices

0ff0

F

2212

1211

gggg

G

TFF TGG

dzd

dzd

21 2T

11T

2LLLLF

dzd

dzd

21 2T

11T

2LLLLG

Symmetry relations

The vertical slowness

4 2q Tr q det 0 H H

*H A B

The vertical slownesses squared are the eigenvalues of the matrix

and are found by solving the characteristic equation

The R/T coefficients

T1 22 1

T1 21 2

C L L

D L L

1D 2 T C D

where superscripts (1) and (2) denote the upper and lower medium, respectively

with

1D

DCDCR

ICD T

The weak-contrast R/T coefficients

02 2 2 24412 0 0 0 0

44 0 00 0

qcpg 2 q q p 2pc q q2 q q

4 2 2 2 22 20 33 44

11 02 2 2 233 440 0 0 0

p c cp 1 p 1 pg 1 1 2p4 4 c c2q 2q q q

2 22 20 44

22 02 2 2 2440 0 0 0

p c1 1 1g 1 2p2 2 c2 q 4 q

02 2 2 2440 0 0 0

44 0 00 0

qcpf 2 q q p 2pc q q2 q q

q is the vertical slowness

The weak-contrast R/T coefficients(Rueger, 1996)

2 2 4P0 P0PP 0

P0 P0

PS 00

230 0

0 00 0

V V1 Z 1 1R 4 sin sin2 Z 2 V 2 V

1 1R 2 sin2 1

41 1 1 sin4 2 4 1 1

is shear wave bulk modulus:2

S0V

Parametrization

• Stiffness coefficients• Velocities• Impedances• Mixed

Effect of anisotropy

0,00 0,05 0,10 0,15 0,20 0,25 0,300,0

0,1

0,2

0,3

Exact TIV Weak TIV Isotropic

Re rPP

0,00 0,05 0,10 0,15 0,20 0,25 0,30

-0,2

-0,1

0,0

0,1

Re rPS

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,0010

-0,0005

0,0000

0,0005

0,0010

0,0015

0,0020

Im rPP

Horizontal slowness [ms/m]

0,00 0,05 0,10 0,15 0,20 0,25 0,30

-0,0020

-0,0016

-0,0012

-0,0008

-0,0004

0,0000

Im rPS


Different parametrizations

0,00 0,05 0,10 0,15 0,20 0,25 0,30

0,0

0,1

0,2

0,3

Re rPP

0,00 0,05 0,10 0,15 0,20 0,25 0,30

0,0

0,1

Re rPS

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,0010

-0,0005

0,0000

0,0005

0,0010

0,0015

0,0020

Isotropic background Exact TIV

Linear: (c

ij,)

(Iij,)

(,,Iii,)

(T,

T,I

ii,)

Im rPP

Horizontal slowness [ms/km]0,00 0,05 0,10 0,15 0,20 0,25 0,30

-0,0020

-0,0016

-0,0012

-0,0008

-0,0004

0,0000

Im rPS


Contribution from the contrasts

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,2

0,0

0,2

0,4 TotalContribution from:

c11

c13

c33

c44

Re rPP

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,1

0,0

0,1

Re rPS

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,001

0,000

0,001

0,002

0,003

Im rPP

Horizontal slowness [ms/m]0,00 0,05 0,10 0,15 0,20 0,25 0,30

-0,0012

-0,0008

-0,0004

0,0000

Im rPS


Second-order R/T

2 2 211 12 12 11 22

2D

2 2 212 11 22 12 22

1 11 g g f f g g g2 2

1 1f g g g 1 g g f2 2

T

11 12 12 11 222

D

12 11 22 22 12

1g g f g f g g2

1g f g g g g f2

R

Effect of second-order R/T

0,00 0,05 0,10 0,15 0,20 0,25 0,30

0,0

0,1

0,2

0,3

Exact TIVTIV background(c

ij,)-parametrization

Linear Quadratic

Re rPP

0,00 0,05 0,10 0,15 0,20 0,25 0,30

0,0

0,1

Re rPS

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,0010

-0,0005

0,0000

0,0005

0,0010

0,0015

0,0020

Im rPP


-0,0020

-0,0016

-0,0012

-0,0008

-0,0004

0,0000

Im rPS


Visco-elastic parameters

44

33

13

11

2

1211

1211

1211

44

33

13

11

cccc

M000

MM2

1M0

21M

MM22

1M1

21M

MM2

1M0

21M

cccc

ij ij0

c lim c

j

j j

1 iM

1 i

11Q

Q2

j0j0

0j

11Q

Q2

j0j0

0j

Linear solid model (Carcione, 1997) The real coefficients

The modified comples Zener moduliThe relaxation times

0 50 100 150 200 250

0.000

0.005

0.010

0.015

0.020

Frequency, Hz

0 50 100 150 200 250

0.96

0.97

0.98

0.99

c11 c13 c33 c44

Frequency, Hz

Complex stiffness coefficients versus frequency

Clay shale (real part is to the top, imaginary part is to the bottom)

The effect of viscoelasticity (1)

0,00 0,05 0,10 0,15 0,20 0,25 0,300,0

0,1

0,2

0,3

0,4

0,5

Re rPP

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,2

-0,1

0,0

0,1

Re rPS

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,008

-0,006

-0,004

-0,002

0,000Im r

PP


-0,005

-0,004

-0,003

-0,002

-0,001

0,000

Im rPS

Exact TIV (f=40 Hz): Q

01=100, Q

02=50

Q01

=50, Q02

=20 Q

01=20, Q

02=10


The effect of viscoelasticity (2)

0,00 0,05 0,10 0,15 0,20 0,25 0,300,0

0,1

0,2

0,3

0,4

0,5

Re rPP

0,00 0,05 0,10 0,15 0,20 0,25 0,30

-0,2

-0,1

0,0

0,1

Re rPS

0,00 0,05 0,10 0,15 0,20 0,25 0,30-0,0020

-0,0015

-0,0010

-0,0005

0,0000

0,0005

Im rPP


-0,0010

-0,0005

0,0000

Im rPS

Exact TIV (Q01

=100, Q02

=50): f=10 Hz f=40 Hz f=100 Hz


Transmission fot the stack of the layers

0,00 0,02 0,04 0,060,0

0,2

0,4

0,6

0,8

1,0

0 20 40 60 80 100 1200,0

0,2

0,4

0,6

0,8

1,0

0,00 0,02 0,04 0,060,0

0,2

0,4

0,6

0,8

1,0

Isotropic model

Elastic model Visco-elastic model

f=40 Hz

Horizontal wavenumber, m-10 20 40 60 80 100 120

0,0

0,2

0,4

0,6

0,8

1,0Transversely isotropic model

k=0.06 m-1

Frequency, Hz

Conclusion

• In practice the weak-anisotropy approximation is very useful

6. Seismic Anisotropy

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