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W I T

Tomographic velocity model estimation with data-derived first and second

spatial traveltime derivatives

Eric Duveneck, Tilman Klüver, Jürgen Mann∗

Geophysical Institute University of Karlsruhe

Germany

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Overview

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 T Introduction

Problem: Determination of velocity model for depth imaging

Tomographic approach based on CRS stack results

Smooth model description

Advantages: picking in simulated ZO section of high S/N ratio pick locations independent of each other ⇒ very few picks required

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Introduction

Problem: Determination of velocity model for depth imaging

Tomographic approach based on CRS stack results

Smooth model description

Advantages: picking in simulated ZO section of high S/N ratio pick locations independent of each other ⇒ very few picks required

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Introduction

Problem: Determination of velocity model for depth imaging

Tomographic approach based on CRS stack results

Smooth model description

Advantages: picking in simulated ZO section of high S/N ratio pick locations independent of each other ⇒ very few picks required

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Introduction

Problem: Determination of velocity model for depth imaging

Tomographic approach based on CRS stack results

Smooth model description

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T CRS 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 T CRS stack and attributes

t2 (ξm,h) = (

t0 + 2 sinα

v0 (ξm−ξ )

)2

+ 2 t0 cos

2 α v0

( (ξm−ξ )2

RN + h

2

RNIP

)

ξ ξ

α α

NIP NIP

NIP NRR

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T CRS attributes and velocities

NIP

α

ξNIPR In the vicinity of a ZO ray: CRP-response can be approximately described by t0, ξ , RNIP, α

Velocity model is consistent if RNIP = 0 at t = 0 for all con- sidered data points

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T CRS attributes and velocities

NIP

α

ξNIPR In the vicinity of a ZO ray: CRP-response can be approximately described by t0, ξ , RNIP, α

Velocity model is consistent if RNIP = 0 at t = 0 for all con- sidered data points

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Tomography with CRS attributes

Data and model components

θ

ξ

α

M T

(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 T Tomography with CRS attributes

Data and model components

θ

ξ

α

M T

(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 T Forward modeling

Kinematic ray-tracing

⇒ T , α , ξ

Dynamic ray-tracing

⇒ Ray propagator matrix ΠΠΠ = (

Q1 Q2 P1 P2

)

⇒ M = P2/Q2

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Forward modeling

Kinematic ray-tracing

⇒ T , α , ξ

Dynamic ray-tracing

⇒ Ray propagator matrix ΠΠΠ = (

Q1 Q2 P1 P2

)

⇒ M = P2/Q2

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Inversion procedure

nonlinear least-squares problem

⇒ iterative solution, linearize locally

model update ∆m: least-squares solution of F∆m = ∆d

with ∆d : data misfit F : 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 T Inversion procedure

nonlinear least-squares problem

⇒ iterative solution, linearize locally model update ∆m: least-squares solution of

F∆m = ∆d with ∆d : data misfit

F : 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 T Inversion procedure

nonlinear least-squares problem

⇒ iterative solution, linearize locally model update ∆m: least-squares solution of

F∆m = ∆d with ∆d : data misfit

F : 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 T Inversion procedure

nonlinear least-squares problem

⇒ iterative solution, linearize locally model update ∆m: least-squares solution of

F∆m = ∆d with ∆d : data misfit

F : 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 T A synthetic data example

Original velocity model

-1500 500 2500 4500 6500 x [m]

-3000

-2000

-1000

0

z [m

]

2000

3000

4000

5000

m /s

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Synthetic data example

0

1

2

t [ s]

0 2 4 6 x [km]

0

1

2

t [ s]

0 2 4 6 x [km]

0

5

10

R ni

p [k

m ]

0

1

2

t [ s]

0 2 4 6 x [km]

-20

0

20

an gl

e [°

]

CRS stack RNIP section α section

8th Int. Congress, Brasilian Geophysical Society, Rio 2003

W I T Synthetic 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 [

10 −3

s ]

−2000 0 2000 4000 6000 8000 0

200

400

600

800

1000

1200

1400

1600

1800

Xo [m]

M [

10 −9

s /m

2 ]

−2000 0 2000 4000 6000 8000 −30

−20

−10

0

10

20

30

Xo [m]

al p

h 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 T Synthetic data example

Residual data error after 12 iterations

−2000 0 2000 4000 6000 8000 −6

−4

−2

0

2

4

6

Xo [m]

D el

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]

D el

ta M

[ 10

−9 s

/m 2 ]

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]

D el

ta a

lp h

a [

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 T Synthetic data example

Inversion result

-1500 500 2500 4500 6500 x [m]

-3000

-2000

-1000

0

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