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

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

Transcript

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