Tomographic velocity model estimation with data-derived ... · Tomographic velocity model...

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