W I T
Tomographic velocity model estimationwith data-derived first and second
spatial traveltime derivatives
Eric Duveneck, Tilman Klüver, Jürgen Mann∗
Geophysical InstituteUniversity of Karlsruhe
Germany
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TOverview
Introduction
Velocity determination with CRS attributes
A synthetic data example
A real data example
Extension to 3D
Advantages/Limitations
Conclusions
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TIntroduction
Problem: Determination of velocity modelfor depth imaging
Tomographic approach based on CRS stack results
Smooth model description
Advantages:picking in simulated ZO section of high S/N ratiopick locations independent of each other⇒ very few picks required
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TIntroduction
Problem: Determination of velocity modelfor depth imaging
Tomographic approach based on CRS stack results
Smooth model description
Advantages:picking in simulated ZO section of high S/N ratiopick locations independent of each other⇒ very few picks required
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TIntroduction
Problem: Determination of velocity modelfor depth imaging
Tomographic approach based on CRS stack results
Smooth model description
Advantages:picking in simulated ZO section of high S/N ratiopick locations independent of each other⇒ very few picks required
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TIntroduction
Problem: Determination of velocity modelfor depth imaging
Tomographic approach based on CRS stack results
Smooth model description
Advantages:picking in simulated ZO section of high S/N ratiopick locations independent of each other⇒ very few picks required
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TCRS stack – 3D data example
4000 −
2000 −
6000 −
8000 −
0 −
4000 −
2000 −
6000 −
8000 −
0 −
PreSDM PostSDM of CRS stack
Data courtesy of ENI E&P Division
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TCRS stack and attributes
t2 (ξm,h) =(
t0 + 2 sinαv0
(ξm−ξ ))2
+2 t0 cos2 α
v0
((ξm−ξ )2
RN+ h2
RNIP
)
ξ ξ
α α
NIP NIP
NIP NRR
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TCRS attributes and velocities
NIP
α
ξNIPR In the vicinity of a ZO ray:CRP-response can beapproximately described byt0, ξ , RNIP, α
Velocity model is consistentif RNIP = 0 at t = 0 for all con-sidered data points
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TCRS attributes and velocities
NIP
α
ξNIPR In the vicinity of a ZO ray:CRP-response can beapproximately described byt0, ξ , RNIP, α
Velocity model is consistentif RNIP = 0 at t = 0 for all con-sidered data points
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TTomography with CRS attributes
Data and model components
θ
ξ
α
MT
(x,z)
v(x,z)
Data: (T , M, α , ξ )i
Model: (x, z, θ )i, v jk
M = 1/v0RNIP
T = t0/2
v jk: B-spline coefficients
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TTomography with CRS attributes
Data and model components
θ
ξ
α
MT
(x,z)
v(x,z)
Data: (T , M, α , ξ )i
Model: (x, z, θ )i, v jk
M = 1/v0RNIP
T = t0/2
v jk: B-spline coefficients
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TForward modeling
Kinematic ray-tracing
⇒ T , α , ξ
Dynamic ray-tracing
⇒ Ray propagator matrix ΠΠΠ =
(Q1 Q2P1 P2
)
⇒ M = P2/Q2
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TForward modeling
Kinematic ray-tracing
⇒ T , α , ξ
Dynamic ray-tracing
⇒ Ray propagator matrix ΠΠΠ =
(Q1 Q2P1 P2
)
⇒ M = P2/Q2
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TInversion procedure
nonlinear least-squares problem
⇒ iterative solution, linearize locally
model update ∆m: least-squares solution of
F∆m = ∆d
with ∆d : data misfitF : Fréchet derivatives
calculation of Fréchet derivatives:ray perturbation theory
regularization ⇒ F̂∆m = ∆d̂(minimization of second derivatives of velocity)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TInversion procedure
nonlinear least-squares problem
⇒ iterative solution, linearize locally
model update ∆m: least-squares solution of
F∆m = ∆d
with ∆d : data misfitF : Fréchet derivatives
calculation of Fréchet derivatives:ray perturbation theory
regularization ⇒ F̂∆m = ∆d̂(minimization of second derivatives of velocity)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TInversion procedure
nonlinear least-squares problem
⇒ iterative solution, linearize locally
model update ∆m: least-squares solution of
F∆m = ∆d
with ∆d : data misfitF : Fréchet derivatives
calculation of Fréchet derivatives:ray perturbation theory
regularization ⇒ F̂∆m = ∆d̂(minimization of second derivatives of velocity)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TInversion procedure
nonlinear least-squares problem
⇒ iterative solution, linearize locally
model update ∆m: least-squares solution of
F∆m = ∆d
with ∆d : data misfitF : Fréchet derivatives
calculation of Fréchet derivatives:ray perturbation theory
regularization ⇒ F̂∆m = ∆d̂(minimization of second derivatives of velocity)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TA synthetic data example
Original velocity model
-1500 500 2500 4500 6500x [m]
-3000
-2000
-1000
0
z [m
]
2000
3000
4000
5000
m/s
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TSynthetic data example
0
1
2
t [s]
0 2 4 6x [km]
0
1
2
t [s]
0 2 4 6x [km]
0
5
10
Rni
p [k
m]
0
1
2
t [s]
0 2 4 6x [km]
-20
0
20
angl
e [°
]
CRS stack RNIP section α section
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TSynthetic data example
Picked input data for the inversion
−2000 0 2000 4000 6000 8000
0
200
400
600
800
1000
1200
1400
Xo [m]
T [1
0−3
s]
−2000 0 2000 4000 6000 80000
200
400
600
800
1000
1200
1400
1600
1800
Xo [m]
M [1
0−9
s/m
2 ]
−2000 0 2000 4000 6000 8000−30
−20
−10
0
10
20
30
Xo [m]
alph
a [
o ]
T M α
Model parametrization:B-spline knot spacing ∆x = 500 m, ∆z = 300 m
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TSynthetic data example
Residual data error after 12 iterations
−2000 0 2000 4000 6000 8000−6
−4
−2
0
2
4
6
Xo [m]
Del
ta T
[10
−3 s
]
6
4
2
0
−2
−4−6
−2000 0 2000 4000 6000 8000−8
−6
−4
−2
0
2
4
6
8
10
Xo [m]
Del
ta M
[10
−9 s
/m2 ]
8
4
0
−4
−8−2000 0 2000 4000 6000 8000
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
Xo [m]
Del
ta a
lpha
[ o ]
0.02
0.01
0
−0.01
−0.02
∆T [10−3s] ∆M [10−9 s/m2] ∆α [◦]
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TSynthetic data example
Inversion result
-1500 500 2500 4500 6500x [m]
-3000
-2000
-1000
0
z [m
]
2000
3000
4000
5000
m/s
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TSynthetic data example
Inversion result
-1500 500 2500 4500 6500x [m]
-3000
-2000
-1000
0
z [m
]
2000
3000
4000
5000
m/s
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TReconstructed vs. original model
Reconstructed velocity and reflector elements
-1500 500 2500 4500 6500x [m]
-3000
-2000
-1000
0
z [m
]
2000
3000
4000
5000
m/s
Original velocity and reconstructed reflector elements
-1500 500 2500 4500 6500x [m]
-3000
-2000
-1000
0
z [m
]
2000
3000
4000
5000
m/s
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TPrestack migration results
1
2
3
z [k
m]
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0CIG location [km]
common-image gatherscommon-image gathers (maximum offset=2000m)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TIncluding additional constraints
. . . important in the case of data gaps!
v(x,z) values at arbitrary locations (x,z)
spatially dependent regularization(smoothness of velocity model)
force velocity structure to follow local reflectorstructure
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TIncluding additional constraints
. . . important in the case of data gaps!
v(x,z) values at arbitrary locations (x,z)
spatially dependent regularization(smoothness of velocity model)
force velocity structure to follow local reflectorstructure
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TIncluding additional constraints
. . . important in the case of data gaps!
v(x,z) values at arbitrary locations (x,z)
spatially dependent regularization(smoothness of velocity model)
force velocity structure to follow local reflectorstructure
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TReal data example
0
1
2
3
t [s]
0 1 2 3 4 5 6x [km]
CRS stack
0
1
2
3t [
s]
0 1 2 3 4 5 6x [km]
coherence
0
0.1
0.2
0.3
0.4
0.5
CRS stack section coherence section(semblance)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TReal data example
0
1
2
3
t [s]
0 1 2 3 4 5 6x [km]
Rnip
0
2
4
6
8
10
0
1
2
3t [
s]
0 1 2 3 4 5 6x [km]
angle
-40
-20
0
20
40
RNIP section [km] angle section [◦]
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TReal data example
Picked input data for the inversion
0 1000 2000 3000 4000 5000 6000
0
200
400
600
800
1000
1200
1400
x [m]
T [
10−3
s]
0 1000 2000 3000 4000 5000 60000
200
400
600
800
1000
1200
1400
1600
1800
x [m]
M [
10−9
s/m
2 ]
0 1000 2000 3000 4000 5000 6000−20
−15
−10
−5
0
5
x [m]
angl
e [
o ]
T M α
Model parametrization:B-spline knot spacing ∆x = 500 m, ∆z = 300 m
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TReal data example
Residual data error after 12 iterations
0 1000 2000 3000 4000 5000 6000−20
−15
−10
−5
0
5
10
15
x [m]
Del
ta T
[10
−3 s
]
0
10
−10
0 1000 2000 3000 4000 5000 6000−80
−60
−40
−20
0
20
40
60
x [m]
Del
ta M
[10
−9 s
/m2 ]
0
−40
40
0 1000 2000 3000 4000 5000 6000−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
x [m]
Del
ta a
ngle
[ o ]
0
−0.1
0.1
∆T [10−3s] ∆M [10−9 s/m2] ∆α [◦]
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TReal data example
Inversion result0
1000
2000
3000
4000
5000
z [m
]0 1000 2000 3000 4000 5000 6000
x [m]
inversion result
2000
3000
4000
5000
velo
city
[m/s
]
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TReal data example
Inversion result0
1000
2000
3000
4000
5000
z [m
]0 1000 2000 3000 4000 5000 6000
x [m]
inversion result
2000
3000
4000
5000
velo
city
[m/s
]
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TPoststack migration
0
1000
2000
3000
4000
5000
z [m
]
0 1000 2000 3000 4000 5000 6000x [m]
inversion result
2000
3000
4000
5000
velo
city
[m/s
]
1
2
3
4
5
z [k
m]
0 1 2 3 4 5 6x [km]
CRS poststack migration
Reconstructed velocity Poststack migrationmodel and dip bars of CRS stack result
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TPrestack migration results
1
2
3
4
5
z [k
m]
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0CIG location [km]
common-image gatherscommon-image gathers (maximum offset=2000m)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D CRS attributes
t2 (∆ξξξ ,h) =(
t0 + 2pξ ·∆ξξξ)2
+2t0
(∆ξξξ T Mξ ∆ξξξ + hT Mh h
)
pξ = 12∂ t/∂ξξξ = 1
v0(sinα cosψ ,sinα sinψ)T
Mh = 12∂ 2t/∂h2 = 1
v0DKNIPDT
Mξ = 12∂ 2t/∂ξξξ 2 = 1
v0DKNDT
Independent of near-surface velocity v0 !8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D CRS attributes
t2 (∆ξξξ ,h) =(
t0 + 2pξ ·∆ξξξ)2
+2t0
(∆ξξξ T Mξ ∆ξξξ + hT Mh h
)
For a tomographic inversion, only
one azimuth φ of Mh is required: Mφ !
⇒ Data: (T ,Mφ , pξx, pξy
,ξx,ξy)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D inversion with CRS attributes
Data and model components
(ξ ,ξ )
x
x y
y
yx φ
(x,y,z)
(e ,e )
M(p ,p )T
v(x,y,z)
Data:(T , Mφ , pξx
, pξy, ξx, ξy)i
Model:(x, y, z, ex, ey)i, v jkl
T = t0/2
v jkl: B-spline coefficients
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D inversion with CRS attributes
Data and model components
(ξ ,ξ )
x
x y
y
yx φ
(x,y,z)
(e ,e )
M(p ,p )T
v(x,y,z)
Data:(T , Mφ , pξx
, pξy, ξx, ξy)i
Model:(x, y, z, ex, ey)i, v jkl
T = t0/2
v jkl: B-spline coefficients
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D synthetic example
Cut through original and reconstructed 3D models
Model
v [k
m/s
]
5
43
2
34
5
x [km]
01
y [km]2
16
1.5
4.3
original model
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D synthetic example
Cut through original and reconstructed 3D models
Inversion result
v [k
m/s
]
5
43
2
34
5
x [km]
01
y [km]2
16
1.5
4.3
inversion result
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D synthetic example
Depth slice at z=1500 m
Model
0
1
2
3
4
5
1 2 3 4 5 6y [km]
4.3
z = 1500 m
x [km]
v [k
m/s
]
1.5
original model8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T3D synthetic example
Depth slice at z=1500 m
Inversion result
0
1
2
3
4
5
1 2 3 4 5 6y [km]
4.3
x [km]
v [k
m/s
]
z = 1500 m
1.5
inversion result8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAdvantages/Limitations
Input is a by-product of CRS stack
Very few picks are required
Picking in ZO section of increased S/N ratio
No assumptions about reflector continuity
Smooth model (ideal for ray-tracing)
Smooth velocity description must be valid
Limited lateral variation within CRS aperture(approximately hyperbolic traveltimes)
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TConclusions
CRS stack yields information useful fordetermination of smooth velocity modelsfor depth imaging
Tomographic inversion method based onCRS attributes
Implementation in 2D and 3D
Applied to 2D real data
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TConclusions
CRS stack yields information useful fordetermination of smooth velocity modelsfor depth imaging
Tomographic inversion method based onCRS attributes
Implementation in 2D and 3D
Applied to 2D real data
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TConclusions
CRS stack yields information useful fordetermination of smooth velocity modelsfor depth imaging
Tomographic inversion method based onCRS attributes
Implementation in 2D and 3D
Applied to 2D real data
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TConclusions
CRS stack yields information useful fordetermination of smooth velocity modelsfor depth imaging
Tomographic inversion method based onCRS attributes
Implementation in 2D and 3D
Applied to 2D real data
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I TAcknowledgment
This work has been supported by the sponsors of theWave Inversion Technology (WIT) consortium.
W I T
8th Int. Congress, Brasilian Geophysical Society, Rio 2003
W I T
8th Int. Congress, Brasilian Geophysical Society, Rio 2003