Leonardo Uieda
August 15, 2011
360

# 3D gravity inversion by planting anomalous densities

August 15, 2011

## 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

l
j
β
Γ( p)=ϕ( p)+μθ( p)

57. Regularization:
θ( p)=∑
j=1
M p
j
p
j

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

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

l
j
β
Γ( p)=ϕ( p)+μθ( p)
For p
j
≠0:

60. Regularization:
θ( p)=∑
j=1
M p
j
p
j

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

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

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)
Overview:

68. Based on René (1986)
Overview:

69. Based on René (1986)
Overview:

70. Based on René (1986)
All other parameters set to 0
Overview:

71. Based on René (1986)
All other parameters set to 0
Iteratively grow
Overview:
known density contrast & position

72. Based on René (1986)
All other parameters set to 0
Iteratively grow add neighbor of seed
Overview:
known density contrast & position

73. Based on René (1986)
All other parameters set to 0
Iteratively grow add neighbor of seed
accretion
Overview:
known density contrast & position

74. Based on René (1986)
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Compact & non­smooth
Any number of sources
Any number of different density contrasts
No large equation system
Search limited to neighbors

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

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

134. Sources = 1 km X 1 km X 1 km

135. Δρ=0.5 g/cm3

Sources = 1 km X 1 km X 1 km

136. Δρ=1.0 g/cm3
Δρ=0.5 g/cm3

Sources = 1 km X 1 km X 1 km

137. Sources = 1 km X 1 km X 1 km
Depth=0.8 km

138. Sources = 1 km X 1 km X 1 km
Depth=1.6 km
Depth=0.8 km

139. Sources = 1 km X 1 km X 1 km ●
Data set = 375 observations
Depth=1.6 km
Depth=0.8 km

140. 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

141. 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

142. Interpretative model = 151,875 prisms

Prisms = 66.7 m X 66.7 m X 66.7 m

143. Used 2 seeds

144. Used 2 seeds ●
Placed in center of sources

145. Used 2 seeds

With corresponding density contrasts

Placed in center of sources

146. Δρ=0.5 g/cm3

Used 2 seeds

With corresponding density contrasts

Placed in center of sources

147. Δρ=1.0 g/cm3
Δρ=0.5 g/cm3

Used 2 seeds

With corresponding density contrasts

Placed in center of sources

148. Inversion result

149. Inversion result

150. Inversion result

compact

concentrated around seeds

recover correct geometry of sources

151. Predicted data
Inversion result

compact

concentrated around seeds

recover correct geometry of sources

152. Predicted data
Observed data
Inversion result

compact

concentrated around seeds

recover correct geometry of sources

153. Predicted data
Observed data
Inversion result

compact

concentrated around seeds

fits observations

recover correct geometry of sources

154. Predicted data
Observed data
On laptop with 2.0 GHz

375 data

151,875 prisms

Total time≈4.4 min

155. Real Data

156. After Carminatti et al. (2003)

157. Cana Brava complex (CBC)
After Carminatti et al. (2003)

158. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
After Carminatti et al. (2003)

159. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)

Outcropping

North of Goiás

Tocantins Province

Amazonian & São Francisco cratons
After Carminatti et al. (2003)

160. Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
Gravimetric data:

132 observations

Residual Bouguer

Max 45 mGal
After Carminatti et al. (2003)

161. 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

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
Test this hypothesis

163. 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

164. Interpretative model
Size: 120 km X 50 km X 11 km
480,000 prisms
Prism size: 500 m X 500 m X 575 m

165. Inversion result

166. Inversion result

167. Inversion result
Predicted data
Observed data

168. Inversion result

169. Inversion result

170. Inversion result

171. Inversion result

172. Inversion result

173. Inversion result

174. Inversion result
≈6km
Max depth
Agree with previous interpretation
Compact
Fits observations

175. Inversion result
On laptop with 2.0 GHz

132 data

480,00 prisms

Total time≈3.75 min

176. Conclusions

177. 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