27

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

#### Leonardo Uieda

September 21, 2011

## Transcript

1. ### Robust 3D gravity gradient inversion by planting anomalous densities Leonardo

Uieda Valéria C. F. Barbosa 2011 Observatório Nacional

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

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

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

j

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

constraints

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

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)

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:

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

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

• 2601 points/component • 7803 measurements • 5 Eötvös noise

these?

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

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

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

to “ignore” non­targeted

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

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

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

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)

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