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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

    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 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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