This document outlines an algorithm for robust 3D gravity gradient inversion by planting anomalous densities. It describes the forward problem of modeling gravity gradients from anomalous densities, as well as the inverse problem of estimating densities from observed gradients. The algorithm formulates the inverse problem as a regularized optimization that minimizes data misfit while imposing constraints like compactness and concentration around seed densities. Neighboring prisms are iteratively accreted to seeds in a manner that reduces misfit and regularization cost. The algorithm is inspired by previous work and aims to robustly estimate densities from real geophysical data.

1. Robust 3D gravity gradient inversion
by planting anomalous densities
Leonardo Uieda
Valéria C. F. Barbosa
Observatório Nacional
September, 2011

2. Outline

3. Outline
Forward Problem

4. Outline
Forward Problem Inverse Problem

5. Outline
Forward Problem Inverse Problem Planting Algorithm
Inspired by René (1986)

6. Outline
Forward Problem Inverse Problem Planting Algorithm
Inspired by René (1986)
Synthetic Data

7. Outline
Forward Problem Inverse Problem Planting Algorithm
Inspired by René (1986)
Synthetic Data Real Data
Quadrilátero Ferrífero, Brazil

8. Forward problem

9. Observed g αβ
αβ
g

10. Observed g αβ
αβ
g

11. Observed g αβ
αβ
g
Anomalous density

12. Observed g αβ
αβ
g
Want to model this Anomalous density

13. Interpretative model

14. Interpretative model
Right rectangular prism
Δ ρ= p j

15.
Prisms with p j =0
not shown

16. []
p1
p= p 2
⋮
pM
Prisms with p j =0
not shown

17. Predicted g αβ
αβ
d
[]
p1
p= p 2
⋮
pM
Prisms with p j =0
not shown

18. Predicted g αβ
αβ
d
M
d =∑ p j a
αβ αβ
j
j=1
[]
p1
p= p 2
⋮
pM
Prisms with p j =0
not shown

19. Predicted g αβ
αβ
d
M
d =∑ p j a
αβ αβ
j
j=1
Contribution of jth prism
[]
p1
p= p 2
⋮
pM
Prisms with p j =0
not shown

20. More components:
xx
d
xy
d
xz
d
yy
d
yz
d
zz
d

21. More components:
xx
d
xy
d
xz
d
yy
d
d
yz
d
zz
d

22. More components:
xx
d
xy
d
M
xz
d
yy
d =∑ p j a j
d j=1
yz
d
zz
d

23. More components:
xx
d
xy
d
M
xz
d
yy
d =∑ p j a j = A p
d j=1
yz
d
zz
d

24. More components:
xx
d
xy
d
M
xz
d
yy
d =∑ p j a j = A p
d j=1
yz
d Jacobian (sensitivity) matrix
zz
d

25. More components:
xx
d
xy
d
M
xz
d
yy
d =∑ p j a j = A p
d j=1
yz
d Column vector of A
zz
d

26. Forward problem:
M
p d=∑ p j a j d
j=1

27. Inverse problem:
p
̂ ? g

28. Inverse problem

29. Minimize difference between g and d
Residual vector r= g−d

30. Minimize difference between g and d
Residual vector r= g−d
Data­misfit function:
N 1
ϕ( p)=∥r∥2 =
( 2 2
∑ (gi−d i )
i=1
)

31. Minimize difference between g and d
Residual vector r= g−d
Data­misfit function:
N 1
ϕ( p)=∥r∥2 =
( 2 2
∑ (gi−d i )
i=1
)
ℓ2­norm of r

32. Minimize difference between g and d
Residual vector r= g−d
Data­misfit function:
N 1
ϕ( p)=∥r∥2 =
( 2 2
∑ (gi−d i )
i=1
)
ℓ2­norm of r
Least­squares fit

33. Minimize difference between g and d
Residual vector r= g−d
Data­misfit function:
N 1 N
ϕ( p)=∥r∥2 =
( 2 2
∑ (gi−d i )
i=1
) ϕ( p)=∥r∥1=∑ ∣gi −d i∣
i=1
ℓ2­norm of r ℓ1­norm of r
Least­squares fit

34. Minimize difference between g and d
Residual vector r= g−d
Data­misfit function:
N 1 N
ϕ( p)=∥r∥2 =
( 2 2
∑ (gi−d i )
i=1
) ϕ( p)=∥r∥1=∑ ∣gi −d i∣
i=1
ℓ2­norm of r ℓ1­norm of r
Least­squares fit Robust fit

35. ill­posed problem
non­existent
non­unique
non­stable

36. constraints
ill­posed problem
non­existent
non­unique
non­stable

37. constraints
ill­posed problem well­posed problem
non­existent exist
non­unique unique
non­stable stable

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 ρs
● Any # of ≠ density contrasts

42. Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● User­specified prisms
● Given density contrasts ρs
● Any # of ≠ density contrasts
3. Only p j =0 or p j =ρs

43. Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● User­specified prisms
● Given density contrasts ρs
● Any # of ≠ density contrasts
3. Only p j =0 or p j =ρs
4. p j =ρs of closest seed

44. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μ θ( p)

45. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μ θ( p)
Data­misfit function

46. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μ θ( p)
Regularizing parameter
(Tradeoff between fit and regularization)

47. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μ θ( p)
Regularizing function
M
pj
θ( p)=∑ l β
j
j=1 p j +ϵ

48. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μ θ( p)
Regularizing function
M
pj
Similar to θ( p)=∑ l β
j
Silva Dias et al. (2009) j=1 p j +ϵ

49. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μ θ( p)
Regularizing function
M
pj
Similar to θ( p)=∑ l β
j
Silva Dias et al. (2009) j=1 p j +ϵ
Distance between
jth prism and seed

50. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μ θ( p)
Regularizing function
M
pj
Similar to θ( p)=∑ l β
j
Silva Dias et al. (2009) j=1 p j +ϵ
Distance between
jth prism and seed
Imposes:
● Compactness Concentration around seeds
●

51. Constraints:
1. Compact
Regularization
2. Concentrated around “seeds”
3. Only p j =0 or p j =ρs
4. p j =ρs of closest seed

52. Constraints:
1. Compact
Regularization
2. Concentrated around “seeds”
3. Only p j =0 or p j =ρs
Algorithm
4. p j =ρs of closest seed Based on René (1986)

53. Planting Algorithm

54. Setup: g = observed data

55. Setup: g = observed data
Define interpretative model
Interpretative model

56. Setup: g = observed data
Define interpretative model
All parameters zero
Interpretative model

57. Setup: g = observed data
Define interpretative model
All parameters zero
N S seeds
Interpretative model

58. Setup: g = observed data
Define interpretative model
All parameters zero
N S seeds
Include seeds
Prisms with p j =0
not shown

59. Setup: g = observed data
Define interpretative model
All parameters zero
N S seeds
d = predicted data
Include seeds
Compute initial residuals
(0) (0)
r = g− d
Prisms with p j =0
not shown

60. Setup: g = observed data
Define interpretative model
All parameters zero
N S seeds
d = predicted data
Include seeds
Compute initial residuals
(0) (0)
r = g− d
Predicted by seeds
Prisms with p j =0
not shown

61. Setup: g = observed data
Define interpretative model
All parameters zero
N S seeds
d = predicted data
Include seeds
Compute initial residuals
NS
(∑ )
r (0)= g−
s=1
ρs a j S
Prisms with p j =0
not shown

62. Setup: g = observed data
Define interpretative model
All parameters zero
N S seeds
d = predicted data
Include seeds
Compute initial residuals
NS
(∑ )
r (0)= g−
s=1
ρs a j S
Neighbors
Find neighbors of seeds
Prisms with p j =0
not shown

63. Growth:
Try accretion to sth seed:
Prisms with p j =0
not shown

64. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
Prisms with p j =0
not shown

65. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
j = chosen p j =ρs (New elements)
j
Prisms with p j =0
not shown

66. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
j
Prisms with p j =0
not shown

67. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
j
Contribution of j
Prisms with p j =0
not shown

68. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
None found = no accretion j
Variable sizes
Prisms with p j =0
not shown

69. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
NS 2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
None found = no accretion
Prisms with p j =0
not shown

70. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
NS 2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
None found = no accretion
j
Prisms with p j =0
not shown

71. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
NS 2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
None found = no accretion
j
At least one seed grow?
Prisms with p j =0
not shown

72. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
NS 2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
None found = no accretion
j
At least one seed grow?
Yes
Prisms with p j =0
not shown

73. Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
NS 2. Smallest goal function
j = chosen p j =ρs (New elements)
Update residuals
( new) (old )
r =r − pj aj
None found = no accretion
j
At least one seed grow?
Yes No
Prisms with p j =0
Done! not shown

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:
Initial residual Update residual vector
NS
(0)
r = g− ( ∑ ρs a j
s=1
S ) r (new)
=r (old )
− pj aj

76. Remember equations:
Initial residual Update residual vector
NS
(0)
r = g−
( ∑ ρs a j
s=1
S ) r
(new)
=r
(old )
− pj aj
No matrix multiplication (only vector +)

77. Remember equations:
Initial residual Update residual vector
NS
(0)
r = g−
( ∑ ρs a j
s=1
S ) r
(new)
=r
(old )
− pj aj
No matrix multiplication (only vector +)
Only need some columns of A

78. Remember equations:
Initial residual Update residual vector
NS
(0)
r = g−
( ∑ ρs a j
s=1
S ) r
(new)
=r
(old )
− pj aj
No matrix multiplication (only vector +)
Only need some columns of A
Calculate only when needed

79. Remember equations:
Initial residual Update residual vector
NS
(0)
r = g−
( ∑ ρs a j
s=1
S ) r
(new)
=r
(old )
− pj aj
No matrix multiplication (only vector +)
Only need some columns of A
Calculate only when needed & delete after update

80. Remember equations:
Initial residual Update residual vector
NS
(0)
r = g−
( ∑ ρs a j
s=1
S ) r
(new)
=r
(old )
− pj aj
No matrix multiplication (only vector +)
Only need some columns of A
Calculate only when needed & delete after update
Lazy evaluation

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
● ℓ1­norm to “ignore” non­targeted

101. Robust procedure:
● Seeds only for targets
● ℓ1­norm 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: ● 13 seeds ● 7,803 data 37,500 prisms
●
Only prisms with zero
density contrast not shown

109. Inversion: ● 13 seeds ● 7,803 data 37,500 prisms
●
Only prisms with zero
density contrast not shown

110. Inversion: ● 13 seeds ● 7,803 data 37,500 prisms
●
Only prisms with zero
density contrast not shown

111. Inversion: ● 13 seeds ● 7,803 data 37,500 prisms
●
Only prisms with zero
density contrast not shown

112. Inversion: ● 13 seeds ● 7,803 data 37,500 prisms
●
Only prisms with zero
density contrast not shown

113. Inversion: ● 13 seeds ● 7,803 data 37,500 prisms
●
● Recover shape of targets
Only prisms with zero
density contrast not shown

114. Inversion: ● 13 seeds ● 7,803 data 37,500 prisms
●
● Recover shape of targets
● Total time = 2.2 minutes (on laptop) Only prisms with zero
density contrast not shown

115. Inversion: ● 13 seeds ● 7,803 data 37,500 prisms
●
Predicted data in contours

116. Inversion: ● 13 seeds ● 7,803 data ● 37,500 prisms
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
● Targets:
FTG survey
● Iron ore bodies
●
Quadrilátero Ferrífero, Brazil
● BIFs of Cauê Formation
●

120. Data:
3 components
● Targets:
FTG survey
● Iron ore bodies
●
Quadrilátero Ferrífero, Brazil
● BIFs of Cauê Formation
●

121. Data:
Seeds for iron ore:
Density contrast 1.0 g/cm3
●
Depth 200 m
●

122. Inversion: 46 seeds 13,746 data
Observed
Predicted ● ●

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. Conclusions
● New 3D gravity gradient inversion
● Multiple sources
● Interfering gravitational effects
● Non­targeted sources
● No matrix multiplications
● No linear systems
● Lazy evaluation of Jacobian matrix

138. Conclusions
● Estimates geometry
● Given density contrasts
● Ideal for:
● Sharp contacts
● Well­constrained physical properties
– Ore bodies
– Intrusive rocks
– Salt domes

139. Thank you