3D gravity inversion by planting anomalous densities

Transcript

1. 3D gravity inversion by planting anomalous densities Leonardo Uieda and Valéria C. F. Barbosa August, 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
8. Forward problem
9. Surface of the Earth
10. Observations of g z Surface of the Earth
11. Observations of g z Group in a vector: g= [g 1 g 2 ⋮ g N ] N×1 = observed data g Surface of the Earth
12. = observed data g
13. = observed data g Assume caused by anomalous sources
14. = observed data g Assume caused by anomalous sources Δρ Density contrast =
15. Parametrize the gravitational effect
16. Parametrize the gravitational effect Linearize
17. Parametrize the gravitational effect Discretize into M elements Linearize Interpretative model
18. 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
19. 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
20. Discretize into M elements Parametrize the gravitational effect p j =Δρ Prisms with not shown p j =0
21. Discretize into M elements Parametrize the gravitational effect g≈d Prisms with not shown p j =0 p j =Δρ
22. Discretize into M elements Parametrize the gravitational effect g≈d Predicted data Prisms with not shown p j =0 p j =Δρ
23. 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
24. 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
25. 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
26. 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
27. 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
28. 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
29. 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
30. Solved forward problem: p d d=∑ j=1 M p j a j
31. ̂ p g ? How to do the inverse?
32. Inverse problem
33. Minimize difference between
34. Minimize difference between g Observed data
35. Minimize difference between and g d Predicted data
36. Minimize difference between and g d r=g−d
37. Minimize difference between and g d r=g−d Residual vector
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
39. 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
40. ill­posed problem non­existent non­unique non­stable
41. ill­posed problem non­existent non­unique non­stable constraints
42. ill­posed problem non­existent non­unique non­stable well­posed problem exist unique stable constraints
43. Constraints: 1. Compact
44. Constraints: 1. Compact no holes inside
45. Constraints: 1. Compact no holes inside 2. Concentrated around “seeds”
46. Constraints: 1. Compact no holes inside 2. Concentrated around “seeds” Similar to René (1986)
47. Constraints: 1. Compact no holes inside 2. Concentrated around “seeds” • User­specified prisms Similar to René (1986)
48. Constraints: 1. Compact no holes inside 2. Concentrated around “seeds” • User­specified prisms • Given density contrasts ρs Similar to René (1986)
49. 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)
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 Similar to René (1986) Not like René (1986)
51. 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)
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)
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 parameter
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
55. 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
56. Regularization: θ( p)=∑ j=1 M p j p j +ϵ l j β Γ( p)=ϕ( p)+μθ( p)
57. Regularization: θ( p)=∑ j=1 M p j p j +ϵ l j β Γ( p)=ϕ( p)+μθ( p) Similar to Silva Dias et al. (2009)
58. Regularization: θ( p)=∑ j=1 M p j p j +ϵ 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)
59. Regularization: θ( p)=∑ j=1 M p j p j +ϵ l j β Γ( p)=ϕ( p)+μθ( p) For p j ≠0:
60. Regularization: θ( p)=∑ j=1 M p j p j +ϵ l j β Γ( p)=ϕ( p)+μθ( p) distance from seeds For p j ≠0:
61. Regularization: θ( p)=∑ j=1 M p j p j +ϵ l j β Γ( p)=ϕ( p)+μθ( p) distance from seeds regularizing function For p j ≠0:
62. Regularization: θ( p)=∑ j=1 M p j p j +ϵ l j β Γ( p)=ϕ( p)+μθ( p) distance from seeds regularizing function Imposes: • Compactness • Concentration around seeds For p j ≠0:
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
64. 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
65. Planting Algorithm
66. Based on René (1986) Overview:
67. Based on René (1986) Start with seeds Overview:
68. Based on René (1986) Start with seeds Overview:
69. Based on René (1986) Start with seeds known density contrast & position Overview:
70. Based on René (1986) Start with seeds known density contrast & position All other parameters set to 0 Overview:
71. Based on René (1986) Start with seeds All other parameters set to 0 Iteratively grow Overview: known density contrast & position
72. Based on René (1986) Start with seeds All other parameters set to 0 Iteratively grow add neighbor of seed Overview: known density contrast & position
73. 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
74. 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
75. Algorithm:
76. Algorithm: Define interpretative model Interpretative model
77. Algorithm: Define interpretative model g = observed data Interpretative model
78. Algorithm: Define interpretative model All parameters zero g = observed data Interpretative model
79. Algorithm: seeds N S Define interpretative model All parameters zero g = observed data Interpretative model
80. Algorithm: seeds N S Define interpretative model All parameters zero Include seeds Prisms with not shown p j =0 Seeds
81. 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
82. 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
83. 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
84. 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
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=∑ j=0 M p j a j d = predicted data Prisms with not shown p j =0
86. 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
87. 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
88. 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
89. 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
90. 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
91. 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
92. Prisms with not shown Growth: p j =0
93. Prisms with not shown Growth: Try accretion to sth seed: p j =0
94. Prisms with not shown Growth: Try accretion to sth seed: Choose neighbor: p j =0
95. Prisms with not shown Growth: Try accretion to sth seed: Choose neighbor: 1. Reduce data misfit ϕ( p)=∥r∥ 2 p j =0
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 =0
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) j = chosen j p j =0
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 =ρ s j = chosen j p j =0 (New elements)
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 =ρ s j = chosen j p j =0 (New elements) new predicted data
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 =ρ s j = chosen Update residuals r(new)=g−d(new) p j =0 (New elements) new predicted data j
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 =ρ s j = chosen Update residuals r(new)=g−d(new) p j =0 (New elements) new predicted data j d(old)+ effect of j
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 =ρ 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 +
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 =ρ 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 +
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 =ρ 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
105. 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 =ρ 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)
106. 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 =ρ s j = chosen Update residuals p j =0 (New elements) new predicted data j r(new)=r(old )− p j a j
107. Prisms with not shown Growth: None found = no accretion 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
108. Prisms with not shown Growth: None found = no accretion 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: Variable sizes p j =0
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 =ρ s j = chosen Update residuals r(new)=r(old )− p j a j Choose neighbor: p j =0
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 =ρ s j = chosen Update residuals r(new)=r(old )− p j a j Choose neighbor: p j =0 (New elements) j
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 =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
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 =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
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 =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 (New elements) j
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 =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 (New elements) j
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 =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 j
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 =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 j
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 =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 j
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 =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 j
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 =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 j
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 =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 j
121. 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 =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
122. 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 =ρ 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
123. Advantages: Compact & non­smooth Any number of sources Any number of different density contrasts No large equation system Search limited to neighbors
124. 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
125. 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 +)
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
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
128. 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
129. 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
130. Advantages: Compact & non­smooth Any number of sources Any number of different density contrasts No large equation system Search limited to neighbors
131. 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
132. 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
133. Synthetic Data
135. • Sources = 1 km X 1 km X 1 km
136. Δρ=0.5 g/cm3 • Sources = 1 km X 1 km X 1 km
137. Δρ=1.0 g/cm3 Δρ=0.5 g/cm3 • Sources = 1 km X 1 km X 1 km
138. • Sources = 1 km X 1 km X 1 km Depth=0.8 km
139. • Sources = 1 km X 1 km X 1 km Depth=1.6 km Depth=0.8 km
140. • Sources = 1 km X 1 km X 1 km • Data set = 375 observations Depth=1.6 km Depth=0.8 km
141. • 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
142. • 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
143. • Interpretative model = 151,875 prisms • Prisms = 66.7 m X 66.7 m X 66.7 m
144. • Used 2 seeds
145. • Used 2 seeds • Placed in center of sources
146. • Used 2 seeds • With corresponding density contrasts • Placed in center of sources
147. Δρ=0.5 g/cm3 • Used 2 seeds • With corresponding density contrasts • Placed in center of sources
148. Δρ=1.0 g/cm3 Δρ=0.5 g/cm3 • Used 2 seeds • With corresponding density contrasts • Placed in center of sources
149. Inversion result
150. Inversion result
151. Inversion result • compact • concentrated around seeds • recover correct geometry of sources
152. Predicted data Inversion result • compact • concentrated around seeds • recover correct geometry of sources
153. Predicted data Observed data Inversion result • compact • concentrated around seeds • recover correct geometry of sources
154. Predicted data Observed data Inversion result • compact • concentrated around seeds • fits observations • recover correct geometry of sources
155. Predicted data Observed data On laptop with 2.0 GHz • 375 data • 151,875 prisms • Total time≈4.4 min
156. Real Data
157. After Carminatti et al. (2003)
158. Cana Brava complex (CBC) After Carminatti et al. (2003)
159. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS) After Carminatti et al. (2003)
160. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS) • Outcropping • North of Goiás • Tocantins Province • Amazonian & São Francisco cratons After Carminatti et al. (2003)
161. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS) Gravimetric data: • 132 observations • Residual Bouguer • Max 45 mGal After Carminatti et al. (2003)
162. 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
163. 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
164. 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
165. Interpretative model Size: 120 km X 50 km X 11 km 480,000 prisms Prism size: 500 m X 500 m X 575 m
166. Inversion result
167. Inversion result
168. Inversion result Predicted data Observed data
169. Inversion result
170. Inversion result
171. Inversion result
172. Inversion result
173. Inversion result
174. Inversion result
175. Inversion result ≈6km Max depth Agree with previous interpretation Compact Fits observations
176. Inversion result On laptop with 2.0 GHz • 132 data • 480,00 prisms • Total time≈3.75 min
177. Conclusions
178. • 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
179. • 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
180. Thank you