6. Seismic Anisotropy
A.Stovas, NTNU
2005
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
isotropic
VTI 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 2
v v
3 3 1 21 1 2 2
1 22 2
1 1 2 2
h hh v h v
v vS
h v h v
2S 1 8
2
1 2 1 22
1 2 1 2
h h v v10
2 v v h h
22 21 2 1 2
2
1 2 1 1 2 2
h h v v10
8 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
u
t x
Equation of motion
Stress-strain relation (Hook’s law)
stress
strain
The elasticity tensor
2 2i m
ijmn2n j
u u
t 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 C
C C C C C C
C C C C C C
C C C C C C
C C C C C C
C C C C C C
ij ji 11 22 33 23 13 12
1 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 0
2 0 0 0
2 0 0 0
0 0 0 0 0
0 0 0 0 0
0 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 C
C 2C C C
C C C
C
C
C
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 C
C C C
C C C
C
C
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
2
33 44
2 213 44 33 44 11 33 44
2 2 411 33 44 13 44
1v C C C C sin D
21
v 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 2
13 33 33 44
33 33 44
66 44
44
C C
2C
C C C C
2C C C
C C
2C
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 cos
v
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, [
km/s
]
Phase angle, [degrees]
0 10 20 30 40 50 60 70 80
3,7
3,8
3,9
4,0
4,1
4,2
4,3
4,4
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 sin
v 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 norm
al
Wav
efro
nt ta
ngen
t
2
phase2 2group phase
phasegroup phase
phase
phase
2 22group
vV v
dv1V cos v cos 1 tan
v d
vk
Vk k
3P P
2 2 2P P
pv vtan
v 1 p v
P PP
P
P P
2 2 2P P
1 1
V 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 t
v
2
2 2NMO x 0
1 t
v x
2 2P NMO P0
22 2 P0SV NMO S0 2
S0
2 2SH NMO S0
v v 1 2
vv v 1 2
v
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
sin
pv
2
P 0 P0 P 0 P02 2
0
1 S Hx t tan p t
1 p 1 S
2 20 P0 0
P P02 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 1p S
v v 2 j
j 0j 0
S a p
dSp
H2 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 1
1
24a 1
1 1
The P-wave
The S-wave
The vertical Vp-Vs ratio
22 P00 2
S0
v
v
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 Hpt1
xV 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 a
5 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 4
2 0 0 0 0
S , k 2,3,...
4aS 1
1 a
4a 8aS 1
1 a 1 a
a24 a 32 a 16 a 64S 1
5 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
2
3 2 2 3
3 2
4 2 2 2
2 3 3 4
t x t 1 x c x c x c x ...
xx
v t
1c S 1
41
c 2 S 1 3 S 1 S 181
c 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 1
S
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 STsvankin T homsen (1994)
v v 2 4B
c 2S S SUr sin Stovas (2004)
c 2 S 1
Shifted hyperbola
Continued fraction
The continued fraction approximations
2
22xP
xlim t x
v 2
22 2
23 2
S S 1, Layered isotropic mediumS S 1
, Single VTI layer2
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 1
11 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
tP
P [
ms]
Offset [km]
0,0 0,2 0,4 0,6 0,8 1,0-4
-3
-2
-1
0
1
2
3
4
tP
SV [
ms]
Offset [km]
0,0 0,2 0,4 0,6 0,8 1,0-4
-3
-2
-1
0
1
2
3
4
tS
VS
V [
ms]
Offset [km]
0,0 0,2 0,4 0,6 0,8 1,0-4
-3
-2
-1
0
1
2
3
4
tS
HS
H [
ms]
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
S3
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
S3
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
S3
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
S3
S2
VTI
VNMO for dipping reflector
2
2
NMO
1 v1
vvv
tan vcos1
v
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/kmp
1=-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 22PP PP2 2 20
0 1 1PP
T 0 v1 1
T 0 2
2 40 PP PP PP PP
222
0PP PP
1 T 0 v S T 01
T 0 v
0 PPT 0z
2
00
0
2PP PP
20 PP
T 0 v1 11
2 T 0
2 2PP PP SS SS
2 20 PP SS 0
T 0 v T 0 v1 1 11
2 T 0 T 0
Ursin and Stovas, 2004
2 2SS SS PP 02 2
1 02SS PP PP
T 0 v T 0 1 2
T 0 1 2T 0 v
SS 00
PP 0
T 0
T 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 [k
m/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
0d
0dz
Abb
B
Trzrz U,S,S,U b
133
21311
213313
13313
133
cccppcc
pccc
A
1
44cp
pB
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
id
idz
u q 0 F G u
d 0 q G F d
diag q ,q q
q and q are verical slownesses for P- and S-wave
d
uLb
With linear transformation
The transformation matrix
T T2 1 2 1 L L L L I
1 1
2 2
i i1
2
L LL
L L
with the symmetries
Scattering matrices
0f
f0F
2212
1211
gg
ggG
TFF TGG
dz
d
dz
d
2
1 2T1
1T2
LL
LLF
dz
d
dz
d
2
1 2T1
1T2
LL
LLG
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
1
D 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 2p
c 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 2p
4 4 c c2q 2q q q
2 22 20 44
22 02 2 2 2440 0 0 0
p c1 1 1g 1 2p
2 2 c2 q 4 q
02 2 2 2440 0 0 0
44 0 00 0
qcpf 2 q q p 2p
c 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 sin
2 Z 2 V 2 V
1 1R 2 sin
2 1
41 1 1sin
4 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
Horizontal slowness [ms/m]
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
Horizontal slowness [ms/m]
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
Horizontal slowness [ms/m]
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 g
2 21 1
f 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 g
21
g 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
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
Horizontal slowness [ms/m]
Visco-elastic parameters
44
33
13
11
2
1211
1211
1211
44
33
13
11
c
c
c
c
M000
MM2
1M0
2
1M
MM22
1M1
2
1M
MM2
1M0
2
1M
c
c
c
c ij ij
0c 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
Horizontal slowness [ms/m]
0,00 0,05 0,10 0,15 0,20 0,25 0,30
-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
Horizontal slowness [ms/m]
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
Horizontal slowness [ms/m]
0,00 0,05 0,10 0,15 0,20 0,25 0,30
-0,0010
-0,0005
0,0000
Im rPS
Exact TIV (Q01
=100, Q02
=50): f=10 Hz f=40 Hz f=100 Hz
Horizontal slowness [ms/m]
Transmission fot the stack of the layers
0,00 0,02 0,04 0,06
0,0
0,2
0,4
0,6
0,8
1,0
0 20 40 60 80 100 120
0,0
0,2
0,4
0,6
0,8
1,0
0,00 0,02 0,04 0,06
0,0
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