Leonardo Uieda
September 15, 2011
270

Robust 3D gravity gradient inversion by planting anomalous densities

Leonardo Uieda

September 15, 2011

Transcript

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

Uieda Valéria C. F. Barbosa September, 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, Brazil

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