Robust 3D gravity gradient inversion by planting anomalous densities

Transcription

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 g αβ

10 Observed g αβ g αβ

11 Observed g αβ g αβ Anomalous density

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

13 Interpretative model

14 Interpretative model Right rectangular prism Δ ρ= p j

15 Prisms with p j =0 not shown

16 Prisms with p j =0 not shown [] p1 p= p 2 pm

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

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

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

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 yz d zz d d

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

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

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

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

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

27 Inverse problem: p? g

28 Inverse problem

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

30 Minimize difference between Residual vector r= g d Data misfit function: ( ϕ( p)= r 2 = N (gi d i ) i=1 g ) and d

31 Minimize difference between Residual vector r= g d Data misfit function: ( ϕ( p)= r 2 = ℓ2 norm of r N (gi d i ) i=1 g ) and d

32 Minimize difference between Residual vector r= g d Data misfit function: ( ϕ( p)= r 2 = N (gi d i ) i=1 ℓ2 norm of r Least squares fit g ) and d

33 Minimize difference between Residual vector g and d r= g d Data misfit function: ( ϕ( p)= r 2 = N (gi d i ) i=1 ℓ2 norm of r Least squares fit ) N ϕ( p)= r 1= gi d i i=1 ℓ1 norm of r

34 Minimize difference between Residual vector g and d r= g d Data misfit function: ( ϕ( p)= r 2 = N (gi d i ) i=1 ℓ2 norm of r Least squares fit ) N ϕ( p)= r 1= gi d i i=1 ℓ1 norm of r Robust fit

35 ill posed problem non existent non unique non stable

36 constraints ill posed problem non existent non unique non stable

37 constraints ill posed problem well posed problem non existent exist non unique unique non stable 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 User specified prisms Given density contrasts ρs Any # of density contrasts

42 Constraints: 1. Compact no holes inside 2. Concentrated around seeds User specified 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 User specified 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 Well posed problem: Minimize goal function Γ( p)=ϕ( p)+μ θ( p)

45 Well posed problem: Minimize goal function Γ( p)=ϕ( p)+μ θ( p) Data misfit function

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

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

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

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

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

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

52 Constraints: 1. Compact 2. Concentrated around seeds 3. Only p j =0 or p j =ρs 4. p j =ρs of closest seed Regularization Algorithm 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 Include seeds d = predicted data Compute initial residuals (0) r = g d (0) Prisms with p j =0 not shown

60 Setup: g = observed data Define interpretative model All parameters zero N S seeds Include seeds d = predicted data Compute initial residuals (0) r = g d (0) 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 Find neighbors of seeds Neighbors 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 p j =ρs (New elements) j = chosen Update residuals r ( new) =r (old ) 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 p j =ρs (New elements) j = chosen Update residuals r ( new) =r (old ) 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 p j =ρs (New elements) j = chosen Update residuals r ( new) =r (old ) 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 p j =ρs (New elements) j = chosen Update residuals r ( new) =r (old ) 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 p j =ρs (New elements) j = chosen Update residuals r ( new) =r (old ) 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 p j =ρs (New elements) j = chosen Update residuals r ( new) =r (old ) pj aj None found = no accretion At least one seed grow? j Prisms with p j =0 not shown

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

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

74 Advantages: Compact & non smooth Any number of sources Any number of different density contrasts No large equation system Search limited to neighbors

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

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

77 Remember equations: Initial residual (0) ( r = g Update residual vector NS 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 (0) ( r = g Update residual vector NS 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 (0) ( r = g Update residual vector NS 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 (0) ( r = g Update residual vector NS 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 & non smooth Any number of sources Any number of different density contrasts No large equation system Search limited to neighbors

82 Advantages: Compact & non smooth 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 & non smooth 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 non targeted

101 Robust procedure: Seeds only for targets ℓ1 norm to ignore non targeted

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 Recover shape of targets Total time = 2.2 minutes (on laptop) 37,500 prisms 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 Predicted Observed Inversion: 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 Conclusions New 3D gravity gradient inversion Multiple sources Interfering gravitational effects Non targeted sources No matrix multiplications No linear systems Lazy evaluation of Jacobian matrix

138 Conclusions Estimates geometry Given density contrasts Ideal for: Sharp contacts Well constrained physical properties Ore bodies Intrusive rocks Salt domes

139 Thank you