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3D gravity inversion by planting anomalous densities

3D gravity inversion by planting anomalous densities

Leonardo Uieda

August 15, 2011
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  1. 3D gravity inversion by planting
    anomalous densities
    Leonardo Uieda and Valéria C. F. Barbosa
    August, 2011
    Observatório Nacional

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  2. Forward Problem
    Outline

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  3. Inverse Problem
    Forward Problem
    Outline

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  4. Inverse Problem Planting Algorithm
    Forward Problem
    Inspired by René (1986)
    Outline

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  5. Inverse Problem Planting Algorithm
    Synthetic Data
    Forward Problem
    Inspired by René (1986)
    Outline

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  6. Inverse Problem Planting Algorithm
    Synthetic Data Real Data
    Forward Problem
    Inspired by René (1986)
    Outline

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  7. Forward problem

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  8. Surface of the Earth

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  9. Observations
    of g
    z
    Surface of the Earth

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  10. Observations
    of g
    z
    Group in a vector:
    g=
    [g
    1
    g
    2

    g
    N
    ]
    N×1
    = observed data
    g
    Surface of the Earth

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  11. = observed data
    g

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  12. = observed data
    g
    Assume caused by
    anomalous sources

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  13. = observed data
    g
    Assume caused by
    anomalous sources
    Δρ
    Density contrast =

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  14. Parametrize the gravitational effect

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  15. Parametrize the gravitational effect
    Linearize

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  16. Parametrize the gravitational effect
    Discretize into
    M elements
    Linearize
    Interpretative model

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  17. Right rectangular prisms
    Parametrize the gravitational effect
    Discretize into
    M elements
    Homogeneous density contrast
    jth element
    Linearize
    (Nagy et al., 2000)
    Interpretative model
    p
    j

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  18. Arrange M density contrasts in a vector:
    Parametrize the gravitational effect
    Discretize into
    M elements
    p=
    [p
    1
    p
    2

    p
    M
    ]
    M×1
    Parameter vector
    Linearize
    Interpretative model

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  19. Discretize into
    M elements
    Parametrize the gravitational effect
    p
    j
    =Δρ
    Prisms with
    not shown
    p
    j
    =0

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  20. Discretize into
    M elements
    Parametrize the gravitational effect
    g≈d
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  21. Discretize into
    M elements
    Parametrize the gravitational effect
    g≈d
    Predicted data
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  22. Discretize into
    M elements
    Parametrize the gravitational effect
    Gravitational effect is linear
    d=∑
    j=1
    M
    p
    j
    a
    j
    g≈d
    Predicted data
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  23. Discretize into
    M elements
    Parametrize the gravitational effect
    Gravitational effect is linear
    d=∑
    j=1
    M
    p
    j
    a
    j
    Density contrast of jth prism
    g≈d
    Predicted data
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  24. Discretize into
    M elements
    Parametrize the gravitational effect
    Gravitational effect is linear
    d=∑
    j=1
    M
    p
    j
    a
    j
    g≈d
    Predicted data
    Effect of prism with unit density Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  25. Discretize into
    M elements
    Parametrize the gravitational effect
    Gravitational effect is linear
    g≈d
    Predicted data
    d=∑
    j=1
    M
    p
    j
    a
    j
    =A p
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  26. Discretize into
    M elements
    Parametrize the gravitational effect
    Gravitational effect is linear
    g≈d
    Predicted data
    Parameter vector
    d=∑
    j=1
    M
    p
    j
    a
    j
    =A p
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  27. Discretize into
    M elements
    Parametrize the gravitational effect
    Gravitational effect is linear
    g≈d
    Predicted data
    Jacobian (sensitivity) matrix
    d=∑
    j=1
    M
    p
    j
    a
    j
    =A p
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  28. Discretize into
    M elements
    Parametrize the gravitational effect
    Gravitational effect is linear
    g≈d
    Predicted data
    Column vector of A
    d=∑
    j=1
    M
    p
    j
    a
    j
    =A p
    Prisms with
    not shown
    p
    j
    =0
    p
    j
    =Δρ

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  29. Solved forward problem:
    p d
    d=∑
    j=1
    M
    p
    j
    a
    j

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  30. ̂
    p g
    ?
    How to do the inverse?

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  31. Inverse problem

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  32. Minimize difference between

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  33. Minimize difference between g
    Observed data

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  34. Minimize difference between and
    g d
    Predicted data

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  35. Minimize difference between and
    g d
    r=g−d

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  36. Minimize difference between and
    g d
    r=g−d
    Residual vector

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  37. Minimize difference between and
    g d
    r=g−d
    Residual vector
    Data­misfit function: ϕ( p)=∥r∥2
    =
    (∑
    i=1
    N
    (g
    i
    −d
    i
    )2
    )1
    2

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  38. Minimize difference between and
    g d
    r=g−d
    Residual vector
    Data­misfit function: ϕ( p)=∥r∥2
    =
    (∑
    i=1
    N
    (g
    i
    −d
    i
    )2
    )1
    2
    ℓ2­norm of r
    Least­squares fit

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  39. ill­posed problem
    non­existent
    non­unique
    non­stable

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  40. ill­posed problem
    non­existent
    non­unique
    non­stable
    constraints

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  41. ill­posed problem
    non­existent
    non­unique
    non­stable
    well­posed problem
    exist
    unique
    stable
    constraints

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  42. Constraints:
    1. Compact

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  43. Constraints:
    1. Compact no holes inside

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  44. Constraints:
    1. Compact no holes inside
    2. Concentrated around “seeds”

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  45. Constraints:
    1. Compact no holes inside
    2. Concentrated around “seeds”
    Similar to René (1986)

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  46. Constraints:
    1. Compact no holes inside
    2. Concentrated around “seeds”

    User­specified prisms
    Similar to René (1986)

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  47. Constraints:
    1. Compact no holes inside
    2. Concentrated around “seeds”

    User­specified prisms

    Given density contrasts ρs
    Similar to René (1986)

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  48. Constraints:
    1. Compact no holes inside
    2. Concentrated around “seeds”

    User­specified prisms

    Given density contrasts

    Any n° of ≠ density contrasts
    ρs
    Similar to René (1986)
    Not like René (1986)

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  49. Constraints:
    1. Compact no holes inside
    2. Concentrated around “seeds”

    User­specified prisms

    Given density contrasts
    3. Only

    Any n° of ≠ density contrasts
    or
    p
    j
    =0 p
    j
    =ρs
    ρs
    Similar to René (1986)
    Not like René (1986)

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  50. Constraints:
    1. Compact no holes inside
    2. Concentrated around “seeds”

    User­specified prisms

    Given density contrasts
    3. Only

    Any n° of ≠ density contrasts
    or
    p
    j
    =0 p
    j
    =ρs
    ρs
    4. of closest seed
    p
    j
    =ρs
    Similar to René (1986)
    Not like René (1986)

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  51. ill­posed problem well­posed problem
    constraints
    ϕ( p)=∥r∥2
    =
    (∑
    i=1
    N
    (g
    i
    −d
    i
    )2
    )1
    2
    Minimize data misfit Minimize goal function
    Γ( p)=ϕ( p)+μθ( p)

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  52. ill­posed problem well­posed problem
    constraints
    ϕ( p)=∥r∥2
    =
    (∑
    i=1
    N
    (g
    i
    −d
    i
    )2
    )1
    2
    Minimize data misfit Minimize goal function
    Γ( p)=ϕ( p)+μθ( p)
    Regularizing parameter

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  53. ill­posed problem well­posed problem
    constraints
    ϕ( p)=∥r∥2
    =
    (∑
    i=1
    N
    (g
    i
    −d
    i
    )2
    )1
    2
    Minimize data misfit Minimize goal function
    Γ( p)=ϕ( p)+μθ( p)
    Regularizing function

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  54. ill­posed problem well­posed problem
    constraints
    ϕ( p)=∥r∥2
    =
    (∑
    i=1
    N
    (g
    i
    −d
    i
    )2
    )1
    2
    Minimize data misfit Minimize goal function
    Γ( p)=ϕ( p)+μθ( p)
    Regularizing function
    μ = tradeoff between fit and regularization

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  55. Regularization:
    θ( p)=∑
    j=1
    M p
    j
    p
    j

    l
    j
    β
    Γ( p)=ϕ( p)+μθ( p)

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  56. Regularization:
    θ( p)=∑
    j=1
    M p
    j
    p
    j

    l
    j
    β
    Γ( p)=ϕ( p)+μθ( p)
    Similar to
    Silva Dias et al. (2009)

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  57. Regularization:
    θ( p)=∑
    j=1
    M p
    j
    p
    j

    l
    j
    β
    Γ( p)=ϕ( p)+μθ( p)
    ϵ = avoid singularity
    l
    j
    = distance between jth prism and seed
    β = how much compactness (3 to 7)
    Similar to
    Silva Dias et al. (2009)

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  58. Regularization:
    θ( p)=∑
    j=1
    M p
    j
    p
    j

    l
    j
    β
    Γ( p)=ϕ( p)+μθ( p)
    For p
    j
    ≠0:

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  59. Regularization:
    θ( p)=∑
    j=1
    M p
    j
    p
    j

    l
    j
    β
    Γ( p)=ϕ( p)+μθ( p)
    distance from seeds
    For p
    j
    ≠0:

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  60. Regularization:
    θ( p)=∑
    j=1
    M p
    j
    p
    j

    l
    j
    β
    Γ( p)=ϕ( p)+μθ( p)
    distance from seeds regularizing function
    For p
    j
    ≠0:

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  61. Regularization:
    θ( p)=∑
    j=1
    M p
    j
    p
    j

    l
    j
    β
    Γ( p)=ϕ( p)+μθ( p)
    distance from seeds regularizing function
    Imposes:

    Compactness ●
    Concentration around seeds
    For p
    j
    ≠0:

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  62. Constraints:
    1. Compact
    2. Concentrated around “seeds”
    3. Only or
    p
    j
    =0 p
    j
    =Δρs
    4. of closest seed
    p
    j
    =Δρs
    Regularization

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  63. Constraints:
    1. Compact
    2. Concentrated around “seeds”
    3. Only or
    p
    j
    =0 p
    j
    =Δρs
    4. of closest seed
    p
    j
    =Δρs
    Regularization
    Algorithm

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  64. Planting Algorithm

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  65. Based on René (1986)
    Overview:

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  66. Based on René (1986)
    Start with seeds
    Overview:

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  67. Based on René (1986)
    Start with seeds
    Overview:

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  68. Based on René (1986)
    Start with seeds known density contrast & position
    Overview:

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  69. Based on René (1986)
    Start with seeds known density contrast & position
    All other parameters set to 0
    Overview:

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  70. Based on René (1986)
    Start with seeds
    All other parameters set to 0
    Iteratively grow
    Overview:
    known density contrast & position

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  71. Based on René (1986)
    Start with seeds
    All other parameters set to 0
    Iteratively grow add neighbor of seed
    Overview:
    known density contrast & position

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  72. Based on René (1986)
    Start with seeds
    All other parameters set to 0
    Iteratively grow add neighbor of seed
    accretion
    Overview:
    known density contrast & position

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  73. Based on René (1986)
    Start with seeds
    All other parameters set to 0
    Iteratively grow add neighbor of seed
    accretion
    Controlled by goal function and data misfit function
    Overview:
    Γ( p)=ϕ( p)+μθ( p) ϕ( p)=∥r∥2
    =
    (∑
    i=1
    N
    (g
    i
    −d
    i
    )2
    )1
    2
    known density contrast & position

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  74. Algorithm:
    Define interpretative model
    Interpretative model

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  75. Algorithm:
    Define interpretative model
    g = observed data
    Interpretative model

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  76. Algorithm:
    Define interpretative model
    All parameters zero
    g = observed data
    Interpretative model

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  77. Algorithm:
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    g = observed data
    Interpretative model

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  78. Algorithm:
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Prisms with
    not shown
    p
    j
    =0
    Seeds

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  79. Algorithm:
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−d(0)
    Prisms with
    not shown
    p
    j
    =0

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  80. Algorithm:
    Residual vector
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−d(0)
    Prisms with
    not shown
    p
    j
    =0

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  81. Algorithm:
    Observed data
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−d(0)
    g = observed data
    Prisms with
    not shown
    p
    j
    =0

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  82. Algorithm:
    Predicted by seeds
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−d(0)
    g = observed data
    d = predicted data
    Prisms with
    not shown
    p
    j
    =0

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  83. Algorithm:
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−d(0)
    g = observed data
    d=∑
    j=0
    M
    p
    j
    a
    j
    d = predicted data
    Prisms with
    not shown
    p
    j
    =0

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  84. Algorithm:
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−d(0)
    g = observed data
    d=∑
    j=0
    M
    p
    j
    a
    j
    Many=0
    d = predicted data
    Prisms with
    not shown
    p
    j
    =0

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  85. Algorithm:
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−d(0)
    g = observed data
    d=∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    d = predicted data
    Prisms with
    not shown
    p
    j
    =0

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  86. Algorithm:
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    )
    Prisms with
    not shown
    g = observed data
    d = predicted data
    p
    j
    =0

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  87. Algorithm:
    Density contrast
    of sth seed
    seeds
    N
    S
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    )
    Prisms with
    not shown
    g = observed data
    d = predicted data
    p
    j
    =0

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  88. seeds
    N
    S
    Algorithm:
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    )
    Column vector
    of A
    Prisms with
    not shown
    g = observed data
    d = predicted data
    p
    j
    =0

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  89. seeds
    N
    S
    Algorithm:
    Define interpretative model
    All parameters zero
    Include seeds
    Compute initial residuals
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    )
    Prisms with
    not shown
    g = observed data
    d = predicted data
    Neighbors
    Find neighbors of seeds
    p
    j
    =0

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  90. Prisms with
    not shown
    Growth:
    p
    j
    =0

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  91. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    p
    j
    =0

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  92. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    p
    j
    =0

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  93. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    ϕ( p)=∥r∥
    2
    p
    j
    =0

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  94. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j
    =0

    View full-size slide

  95. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    j = chosen
    j
    p
    j
    =0

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  96. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    j
    p
    j
    =0
    (New elements)

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  97. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    j
    p
    j
    =0
    (New elements)
    new predicted data

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  98. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=g−d(new)
    p
    j
    =0
    (New elements)
    new predicted data
    j

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  99. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=g−d(new)
    p
    j
    =0
    (New elements)
    new predicted data
    j
    d(old)+ effect of j

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  100. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=g−d(new)
    p
    j
    =0
    (New elements)
    new predicted data
    j
    d(old)+ effect of j

    s=1
    N
    S
    ρs
    a
    j
    S
    p
    j
    a
    j
    +

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  101. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=g−d(new)
    p
    j
    =0
    (New elements)
    new predicted data
    j
    d(old)+ effect of j

    s=1
    N
    S
    ρs
    a
    j
    S
    p
    j
    a
    j
    +

    View full-size slide

  102. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=g−∑
    s=1
    N
    S
    ρs
    a
    j
    S
    − p
    j
    a
    j
    p
    j
    =0
    (New elements)
    new predicted data
    j

    View full-size slide

  103. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=g−∑
    s=1
    N
    S
    ρs
    a
    j
    S
    − p
    j
    a
    j
    p
    j
    =0
    (New elements)
    new predicted data
    j
    {
    r(0)

    View full-size slide

  104. Prisms with
    not shown
    Growth:
    Try accretion to sth seed:
    Choose neighbor:
    1. Reduce data misfit
    2. Smallest goal function
    ϕ( p)=∥r∥
    2
    Γ( p)=ϕ( p)+μθ( p)
    p
    j

    s
    j = chosen
    Update residuals
    p
    j
    =0
    (New elements)
    new predicted data
    j
    r(new)=r(old )− p
    j
    a
    j

    View full-size slide

  105. Prisms with
    not shown
    Growth:
    None found = no accretion
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    p
    j
    =0

    View full-size slide

  106. Prisms with
    not shown
    Growth:
    None found = no accretion
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    Variable sizes
    p
    j
    =0

    View full-size slide

  107. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    p
    j
    =0

    View full-size slide

  108. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    p
    j
    =0
    (New elements)
    j

    View full-size slide

  109. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0

    View full-size slide

  110. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0

    View full-size slide

  111. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    (New elements)
    j

    View full-size slide

  112. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    (New elements)
    j

    View full-size slide

  113. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    j

    View full-size slide

  114. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    j

    View full-size slide

  115. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    j

    View full-size slide

  116. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    j

    View full-size slide

  117. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    j

    View full-size slide

  118. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0
    j

    View full-size slide

  119. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    p
    j
    =0

    View full-size slide

  120. Prisms with
    not shown
    Growth:
    None found = no accretion
    N
    S
    Try accretion to sth seed:
    1. Reduce data misfit
    2. Smallest goal function
    p
    j

    s
    j = chosen
    Update residuals
    r(new)=r(old )− p
    j
    a
    j
    Choose neighbor:
    At least one seed grow?
    Yes No
    Done!
    p
    j
    =0

    View full-size slide

  121. Advantages:
    Compact & non­smooth
    Any number of sources
    Any number of different density contrasts
    No large equation system
    Search limited to neighbors

    View full-size slide

  122. Remember equations:
    r(0)=g−
    (∑
    s=1
    N
    S
    ρs
    a
    j
    S
    ) r(new)=r(old)− p
    j
    a
    j
    Initial residual Update residual vector

    View full-size slide

  123. Remember equations:
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    ) r(new)=r(old)− p
    j
    a
    j
    Initial residual Update residual vector
    No matrix multiplication (only vector +)

    View full-size slide

  124. No matrix multiplication (only vector +)
    Remember equations:
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    ) r(new)=r(old)− p
    j
    a
    j
    Initial residual Update residual vector
    Only need some columns of A

    View full-size slide

  125. No matrix multiplication (only vector +)
    Remember equations:
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    ) r(new)=r(old)− p
    j
    a
    j
    Initial residual Update residual vector
    Only need some columns of A
    Calculate only when needed

    View full-size slide

  126. No matrix multiplication (only vector +)
    Remember equations:
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    ) r(new)=r(old)− p
    j
    a
    j
    Initial residual Update residual vector
    Only need some columns of A
    Calculate only when needed & delete after update

    View full-size slide

  127. No matrix multiplication (only vector +)
    Remember equations:
    r(0)=g−
    (∑
    s=1
    N
    S
    ρ
    s
    a
    j
    S
    ) r(new)=r(old)− p
    j
    a
    j
    Initial residual Update residual vector
    Only need some columns of A
    Calculate only when needed
    Lazy evaluation
    & delete after update

    View full-size slide

  128. Advantages:
    Compact & non­smooth
    Any number of sources
    Any number of different density contrasts
    No large equation system
    Search limited to neighbors

    View full-size slide

  129. Advantages:
    Compact & non­smooth
    Any number of sources
    Any number of different density contrasts
    No large equation system
    Search limited to neighbors
    No matrix multiplication (only vector +)
    Lazy evaluation of Jacobian

    View full-size slide

  130. Advantages:
    Compact & non­smooth
    Any number of sources
    Any number of different density contrasts
    No large equation system
    Search limited to neighbors
    No matrix multiplication (only vector +)
    Lazy evaluation of Jacobian
    Fast inversion + low memory usage

    View full-size slide

  131. Synthetic Data

    View full-size slide


  132. Sources = 1 km X 1 km X 1 km

    View full-size slide

  133. Δρ=0.5 g/cm3

    Sources = 1 km X 1 km X 1 km

    View full-size slide

  134. Δρ=1.0 g/cm3
    Δρ=0.5 g/cm3

    Sources = 1 km X 1 km X 1 km

    View full-size slide


  135. Sources = 1 km X 1 km X 1 km
    Depth=0.8 km

    View full-size slide


  136. Sources = 1 km X 1 km X 1 km
    Depth=1.6 km
    Depth=0.8 km

    View full-size slide


  137. Sources = 1 km X 1 km X 1 km ●
    Data set = 375 observations
    Depth=1.6 km
    Depth=0.8 km

    View full-size slide


  138. Sources = 1 km X 1 km X 1 km

    Area = 5 km X 3 km

    Data set = 375 observations
    Depth=1.6 km
    Depth=0.8 km

    View full-size slide


  139. 0.05 mGal Gaussian noise

    Sources = 1 km X 1 km X 1 km

    Area = 5 km X 3 km

    Data set = 375 observations
    Depth=1.6 km
    Depth=0.8 km

    View full-size slide


  140. Interpretative model = 151,875 prisms

    Prisms = 66.7 m X 66.7 m X 66.7 m

    View full-size slide


  141. Used 2 seeds

    View full-size slide


  142. Used 2 seeds ●
    Placed in center of sources

    View full-size slide


  143. Used 2 seeds

    With corresponding density contrasts

    Placed in center of sources

    View full-size slide

  144. Δρ=0.5 g/cm3

    Used 2 seeds

    With corresponding density contrasts

    Placed in center of sources

    View full-size slide

  145. Δρ=1.0 g/cm3
    Δρ=0.5 g/cm3

    Used 2 seeds

    With corresponding density contrasts

    Placed in center of sources

    View full-size slide

  146. Inversion result

    View full-size slide

  147. Inversion result

    View full-size slide

  148. Inversion result

    compact

    concentrated around seeds

    recover correct geometry of sources

    View full-size slide

  149. Predicted data
    Inversion result

    compact

    concentrated around seeds

    recover correct geometry of sources

    View full-size slide

  150. Predicted data
    Observed data
    Inversion result

    compact

    concentrated around seeds

    recover correct geometry of sources

    View full-size slide

  151. Predicted data
    Observed data
    Inversion result

    compact

    concentrated around seeds

    fits observations

    recover correct geometry of sources

    View full-size slide

  152. Predicted data
    Observed data
    On laptop with 2.0 GHz

    375 data

    151,875 prisms

    Total time≈4.4 min

    View full-size slide

  153. After Carminatti et al. (2003)

    View full-size slide

  154. Cana Brava complex (CBC)
    After Carminatti et al. (2003)

    View full-size slide

  155. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
    After Carminatti et al. (2003)

    View full-size slide

  156. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)

    Outcropping

    North of Goiás

    Tocantins Province

    Amazonian & São Francisco cratons
    After Carminatti et al. (2003)

    View full-size slide

  157. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
    Gravimetric data:

    132 observations

    Residual Bouguer

    Max 45 mGal
    After Carminatti et al. (2003)

    View full-size slide

  158. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
    Previous interpretation:
    After Carminatti et al. (2003)

    Carminatti et al. (2003)

    PVSS

    CVC

    Max depth
    Δρ=0.27 g/cm3
    Δρ=0.39g/cm3
    ≈6 km

    View full-size slide

  159. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
    Previous interpretation:
    After Carminatti et al. (2003)

    Carminatti et al. (2003)

    PVSS

    CVC

    Max depth
    Δρ=0.27 g/cm3
    Δρ=0.39g/cm3
    ≈6 km
    Test this hypothesis

    View full-size slide

  160. Assign seeds
    Green:
    z=0 km
    Δρ=0.27g/cm3
    Blue:
    z=2 km
    Δρ=0.27g/cm3
    Red:
    z=0 km
    Δρ=0.39 g/cm3
    Total = 269
    Assign seeds

    View full-size slide

  161. Interpretative model
    Size: 120 km X 50 km X 11 km
    480,000 prisms
    Prism size: 500 m X 500 m X 575 m

    View full-size slide

  162. Inversion result

    View full-size slide

  163. Inversion result

    View full-size slide

  164. Inversion result
    Predicted data
    Observed data

    View full-size slide

  165. Inversion result

    View full-size slide

  166. Inversion result

    View full-size slide

  167. Inversion result

    View full-size slide

  168. Inversion result

    View full-size slide

  169. Inversion result

    View full-size slide

  170. Inversion result

    View full-size slide

  171. Inversion result
    ≈6km
    Max depth
    Agree with previous interpretation
    Compact
    Fits observations

    View full-size slide

  172. Inversion result
    On laptop with 2.0 GHz

    132 data

    480,00 prisms

    Total time≈3.75 min

    View full-size slide


  173. New 3D gravity inversion

    Multiple sources

    Interfering gravitational effects

    Abrupt density­contrast distribution

    No matrix multiplication

    No need to solve large linear systems

    Ideal for: ore bodies, intrusions, salt domes, etc
    Conclusions

    View full-size slide


  174. Developed for gravity gradients

    Presented at EAGE 2011 preliminary results

    To be presented at SEG 2011:

    Final results

    Robust method to handle non­targeted sources
    Previous and future work

    View full-size slide