Leonardo Uieda
September 21, 2011
39

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

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. Observed g
αβ
gαβ

10. Observed g
αβ
gαβ

11. Observed g
αβ
gαβ
Anomalous density

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

13. Interpretative model

14. Interpretative model
Right rectangular prism
Δρ=p
j

15. Prisms with
not shown
p
j
=0

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

20. dxx
dxy
dxz
dyy
dyz
dzz
More components:

21. d
dxx
dxy
dxz
dyy
dyz
dzz
More components:

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

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

27. ̂
p g
?
Inverse problem:

28. Inverse problem

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

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

35. ill­posed problem
non­existent
non­unique
non­stable

36. ill­posed problem
non­existent
non­unique
non­stable
constraints

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

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

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

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

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

46. Well­posed problem: Minimize goal function
Γ( p)=ϕ( p)+μθ( p)
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

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

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

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

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)

53. Planting Algorithm

54. Setup: g = observed data

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

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:

63. Prisms with
not shown
Try accretion to sth seed:
p
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

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

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

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

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

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

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

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

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

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)

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

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

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

ℓ1
­norm to “ignore” non­targeted

101. Robust procedure:

Seeds only for targets

ℓ1
­norm 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: ●
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

117. Real Data

118. Data:

3 components

FTG survey

119. Data:

3 components

FTG survey

Targets:

Iron ore bodies

BIFs of Cauê Formation

120. Data:

3 components

FTG survey

Targets:

Iron ore bodies

BIFs of Cauê Formation

121. Data:
Seeds for iron ore:

Density contrast 1.0 g/cm3

Depth 200 m

122. Inversion:
Observed
Predicted

46 seeds ●
13,746 data

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. 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

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