Inversão robusta do gradiente da gravidade através da plantação de anomalias de densidade

84d34651c3931a54310a57484a109821?s=47 Leonardo Uieda
September 21, 2011

Inversão robusta do gradiente da gravidade através da plantação de anomalias de densidade

Seminário anual (2011) da pós-graduação em geofísica do Observatório Nacional.

84d34651c3931a54310a57484a109821?s=128

Leonardo Uieda

September 21, 2011
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  1. Robust 3D gravity gradient inversion by planting anomalous densities Leonardo

    Uieda Valéria C. F. Barbosa 2011 Observatório Nacional
  2. Outline

  3. Forward Problem Outline

  4. Inverse Problem Forward Problem Outline

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

    Outline
  6. Inverse Problem Planting Algorithm Synthetic Data Forward Problem Inspired by

    René (1986) Outline
  7. Inverse Problem Planting Algorithm Synthetic Data Real Data Forward Problem

    Inspired by René (1986) Outline Quadrilátero Ferrífero
  8. Forward problem

  9. Observed g αβ gαβ

  10. Observed g αβ gαβ

  11. Observed g αβ gαβ Anomalous density

  12. Observed g αβ gαβ Anomalous density Want to model this

  13. Interpretative model

  14. Interpretative model Right rectangular prism Δρ=p j

  15. Prisms with not shown p j =0

  16. p= [p 1 p 2 ⋮ p M ] Prisms

    with not shown p j =0
  17. Predicted g αβ dαβ p= [p 1 p 2 ⋮

    p M ] Prisms with not shown p j =0
  18. Predicted g αβ dαβ p= [p 1 p 2 ⋮

    p M ] dαβ=∑ j=1 M p j a j αβ Prisms with not shown p j =0
  19. Predicted g αβ dαβ p= [p 1 p 2 ⋮

    p M ] dαβ=∑ j=1 M p j a j αβ Contribution of jth prism Prisms with not shown p j =0
  20. dxx dxy dxz dyy dyz dzz More components:

  21. d dxx dxy dxz dyy dyz dzz More components:

  22. d dxx dxy dxz dyy dyz dzz =∑ j=1 M

    p j a j More components:
  23. d dxx dxy dxz dyy dyz dzz =∑ j=1 M

    p j a j = A p More components:
  24. d dxx dxy dxz dyy dyz dzz =∑ j=1 M

    p j a j = A p Jacobian (sensitivity) matrix More components:
  25. d dxx dxy dxz dyy dyz dzz =∑ j=1 M

    p j a j = A p Column vector of More components: A
  26. Forward problem: p d d=∑ j=1 M p j a

    j
  27. ̂ p g ? Inverse problem:

  28. Inverse problem

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

  30. 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
  31. 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
  32. 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
  33. 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 ϕ( p)=∥r∥1 =∑ i=1 N ∣g i −d i ∣ ℓ1­norm of r
  34. 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 ϕ( p)=∥r∥1 =∑ i=1 N ∣g i −d i ∣ ℓ1­norm of r Robust fit
  35. ill­posed problem non­existent non­unique non­stable

  36. ill­posed problem non­existent non­unique non­stable constraints

  37. ill­posed problem non­existent non­unique non­stable well­posed problem exist unique stable

    constraints
  38. Constraints: 1. Compact

  39. Constraints: 1. Compact no holes inside

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

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

    • User­specified prisms • Given density contrasts • Any # of ≠ density contrasts ρs
  42. Constraints: 1. Compact no holes inside 2. Concentrated around “seeds”

    • User­specified prisms • Given density contrasts 3. Only • Any # of ≠ density contrasts or p j =0 p j =ρs ρs
  43. Constraints: 1. Compact no holes inside 2. Concentrated around “seeds”

    • User­specified prisms • Given density contrasts 3. Only • Any # of ≠ density contrasts or p j =0 p j =ρs ρs 4. of closest seed p j =ρs
  44. Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μθ( p)

  45. Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μθ( p) Data­misfit

    function
  46. Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μθ( p) (Tradeoff

    between fit and regularization) Regularizing parameter
  47. Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μθ( p) Regularizing

    function θ( p)=∑ j=1 M p j p j +ϵ l j β
  48. Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μθ( p) Regularizing

    function θ( p)=∑ j=1 M p j p j +ϵ l j β Similar to Silva Dias et al. (2009)
  49. Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μθ( p) Regularizing

    function θ( p)=∑ j=1 M p j p j +ϵ l j β Similar to Silva Dias et al. (2009) Distance between jth prism and seed
  50. Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μθ( p) Regularizing

    function θ( p)=∑ j=1 M p j p j +ϵ l j β Similar to Silva Dias et al. (2009) Distance between jth prism and seed Imposes: • Compactness • Concentration around seeds
  51. Constraints: 1. Compact 2. Concentrated around “seeds” Regularization 3. Only

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

    Only or p j =0 p j =ρs 4. of closest seed p j =ρs Based on René (1986)
  53. Planting Algorithm

  54. Setup: g = observed data

  55. Setup: Define interpretative model Interpretative model g = observed data

  56. Setup: Define interpretative model All parameters zero Interpretative model g

    = observed data
  57. Setup: seeds N S Define interpretative model All parameters zero

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

    Include seeds Prisms with not shown p j =0 g = observed data
  59. Setup: 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 g = observed data d = predicted data
  60. Setup: 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 g = observed data Predicted by seeds d = predicted data
  61. Setup: seeds N S Define interpretative model All parameters zero

    Include seeds Compute initial residuals Prisms with not shown p j =0 g = observed data d = predicted data r(0)=g− (∑ s=1 N S ρ s a j S )
  62. 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 Neighbors Find neighbors of seeds p j =0 Setup:
  63. Prisms with not shown Try accretion to sth seed: p

    j =0 Growth:
  64. Prisms with not shown Try accretion to sth seed: 1.

    Reduce data misfit 2. Smallest goal function Choose neighbor: p j =0 Growth:
  65. Prisms with not shown Try accretion to sth seed: 1.

    Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Choose neighbor: p j =0 Growth: (New elements) j
  66. Prisms with not shown Try accretion to sth seed: 1.

    Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Choose neighbor: p j =0 Growth: (New elements) j Update residuals r(new)=r(old )− p j a j
  67. Prisms with not shown Try accretion to sth seed: 1.

    Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Choose neighbor: p j =0 Growth: (New elements) j Update residuals r(new)=r(old )− p j a j Contribution of j
  68. Prisms with not shown Try accretion to sth seed: 1.

    Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Choose neighbor: p j =0 Growth: (New elements) j Update residuals r(new)=r(old )− p j a j Variable sizes None found = no accretion
  69. Prisms with not shown None found = no accretion N

    S Try accretion to sth seed: 1. Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Update residuals r(new)=r(old )− p j a j Choose neighbor: p j =0 Growth: (New elements)
  70. Prisms with not shown None found = no accretion N

    S Try accretion to sth seed: 1. Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Update residuals r(new)=r(old )− p j a j Choose neighbor: p j =0 Growth: j (New elements)
  71. Prisms with not shown None found = no accretion N

    S Try accretion to sth seed: 1. Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Update residuals r(new)=r(old )− p j a j Choose neighbor: At least one seed grow? p j =0 Growth: j (New elements)
  72. Prisms with not shown None found = no accretion N

    S Try accretion to sth seed: 1. Reduce data misfit 2. Smallest goal function p j =ρ s j = chosen Update residuals r(new)=r(old )− p j a j Choose neighbor: At least one seed grow? Yes p j =0 Growth: j (New elements)
  73. Prisms with not shown None found = no accretion N

    S Try accretion to sth seed: 1. Reduce data misfit 2. Smallest goal function p j =ρ 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 Growth: Done! j (New elements)
  74. Advantages: Compact & non­smooth Any number of sources Any number

    of different density contrasts No large equation system Search limited to neighbors
  75. 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
  76. 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 +)
  77. 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
  78. 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
  79. 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
  80. 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
  81. Advantages: Compact & non­smooth Any number of sources Any number

    of different density contrasts No large equation system Search limited to neighbors
  82. 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
  83. 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
  84. Synthetic Data

  85. Data set: • 3 components • 51 x 51 points

    • 2601 points/component • 7803 measurements • 5 Eötvös noise
  86. Model:

  87. Model: • 11 prisms

  88. Model: • 11 prisms • 4 outcropping

  89. Model: • 11 prisms • 4 outcropping

  90. Model: • 11 prisms • 4 outcropping

  91. • Strongly interfering effects

  92. • Strongly interfering effects • What if only interested in

    these?
  93. • Common scenario

  94. • Common scenario • May not have prior information •

    Density contrast • Approximate depth
  95. • Common scenario • May not have prior information •

    Density contrast • Approximate depth • No way to provide seeds
  96. • Common scenario • May not have prior information •

    Density contrast • Approximate depth • No way to provide seeds • Difficult to isolate effect of targets
  97. Robust procedure:

  98. Robust procedure: • Seeds only for targets

  99. Robust procedure: • Seeds only for targets

  100. Robust procedure: • Seeds only for targets • ℓ1 ­norm

    to “ignore” non­targeted
  101. Robust procedure: • Seeds only for targets • ℓ1 ­norm

    to “ignore” non­targeted
  102. Inversion: • 13 seeds • 7,803 data

  103. Inversion: • 13 seeds • 7,803 data

  104. Inversion: • 13 seeds • 7,803 data

  105. Inversion: • 13 seeds • 7,803 data

  106. Inversion: • 13 seeds • 7,803 data

  107. Inversion: • 13 seeds • 7,803 data • 37,500 prisms

  108. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    Only prisms with zero density contrast not shown
  109. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    Only prisms with zero density contrast not shown
  110. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    Only prisms with zero density contrast not shown
  111. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    Only prisms with zero density contrast not shown
  112. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    Only prisms with zero density contrast not shown
  113. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    • Recover shape of targets Only prisms with zero density contrast not shown
  114. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    • Recover shape of targets • Total time = 2.2 minutes (on laptop) Only prisms with zero density contrast not shown
  115. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    Predicted data in contours
  116. Inversion: • 7,803 data • 37,500 prisms • 13 seeds

    Predicted data in contours Effect of true targeted sources
  117. Real Data

  118. Data: • 3 components • FTG survey • Quadrilátero Ferrífero,

    Brazil
  119. Data: • 3 components • FTG survey • Quadrilátero Ferrífero,

    Brazil Targets: • Iron ore bodies • BIFs of Cauê Formation
  120. Data: • 3 components • FTG survey • Quadrilátero Ferrífero,

    Brazil Targets: • Iron ore bodies • BIFs of Cauê Formation
  121. Data: Seeds for iron ore: • Density contrast 1.0 g/cm3

    • Depth 200 m
  122. Inversion: Observed Predicted • 46 seeds • 13,746 data

  123. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  124. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  125. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  126. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  127. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  128. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  129. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

    Only prisms with zero density contrast not shown
  130. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

    Only prisms with zero density contrast not shown
  131. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

    Only prisms with zero density contrast not shown
  132. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  133. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

  134. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

    • Agree with previous interpretations (Martinez et al., 2010)
  135. Inversion: • 46 seeds • 13,746 data • 164,892 prisms

    • Agree with previous interpretations (Martinez et al., 2010) • Total time = 14 minutes (on laptop)
  136. Conclusions

  137. • New 3D gravity gradient inversion • Multiple sources •

    Interfering gravitational effects • Non­targeted sources • No matrix multiplications • No linear systems • Lazy evaluation of Jacobian matrix Conclusions
  138. • Estimates geometry • Given density contrasts • Ideal for:

    • Sharp contacts • Well­constrained physical properties – Ore bodies – Intrusive rocks – Salt domes Conclusions
  139. Mestrado • 2010: Cumprir disciplinas • 2010­2011: 7 trabalhos em

    congresso (5 primeiro autor) • 10/2011: Submeter artigo para Geophysics • 11/2011: Defesa da dissertação de mestrado Continuação • Adaptar para gravimetria e magnetometria • Disponibilizar software Open Source Cronograma