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Robust 3D gravity gradient inversion by planting anomalous densities

Leonardo Uieda, profile picture
Leonardo Uieda

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.

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Robust 3D gravity gradient inversion by planting anomalous densities Leonardo Uieda Valéria C. F. Barbosa Observatório Nacional September, 2011    
Outline    
Outline Forward Problem    
Outline Forward Problem Inverse Problem    
Outline Forward Problem Inverse Problem Planting Algorithm Inspired by René (1986)    
Outline Forward Problem Inverse Problem Planting Algorithm Inspired by René (1986) Synthetic Data    
Outline Forward Problem Inverse Problem Planting Algorithm Inspired by René (1986) Synthetic Data Real Data Quadrilátero Ferrífero, Brazil    
Forward problem    
Observed g αβ αβ g    
Observed g αβ αβ g    
Observed g αβ αβ g Anomalous density    
Observed g αβ αβ g Want to model this Anomalous density    
Interpretative model    
Interpretative model Right rectangular prism Δ ρ= p j    
  Prisms with  p j =0   not shown
[] p1 p= p 2 ⋮ pM   Prisms with  p j =0   not shown
Predicted g αβ αβ d [] p1 p= p 2 ⋮ pM   Prisms with  p j =0   not shown
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
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
More components: xx d xy d xz d yy d yz d zz d    
More components: xx d xy d xz d yy d d yz d zz d    
More components: xx d xy d M xz d yy d =∑ p j a j d j=1 yz d zz d    
More components: xx d xy d M xz d yy d =∑ p j a j = A p d j=1 yz d zz d    
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    
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    
Forward problem: M p d=∑ p j a j d j=1    
Inverse problem: p ̂ ? g    
Inverse problem    
Minimize difference between g and d Residual vector r= g−d    
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 )    
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    
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    
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    
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    
ill­posed problem non­existent non­unique non­stable    
constraints ill­posed problem non­existent non­unique non­stable    
constraints ill­posed problem well­posed problem non­existent exist non­unique unique non­stable stable    
Constraints: 1. Compact    
Constraints: 1. Compact no holes inside    
Constraints: 1. Compact no holes inside 2. Concentrated around “seeds”    
Constraints: 1. Compact no holes inside 2. Concentrated around “seeds” ● User­specified prisms ●  Given density contrasts ρs ●  Any # of ≠ density contrasts    
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    
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     
Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μ θ( p)    
Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μ θ( p) Data­misfit function    
Well­posed problem: Minimize goal function Γ( p)=ϕ( p)+μ θ( p) Regularizing parameter (Tradeoff between fit and regularization)    
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    
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    
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    
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 ●    
Constraints: 1. Compact Regularization 2. Concentrated around “seeds” 3. Only  p j =0 or p j =ρs 4.  p j =ρs of closest seed     
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)    
Planting Algorithm    
Setup: g = observed data    
Setup: g = observed data Define interpretative model Interpretative model    
Setup: g = observed data Define interpretative model All parameters zero Interpretative model    
Setup: g = observed data Define interpretative model All parameters zero N S seeds Interpretative model    
Setup: g = observed data Define interpretative model All parameters zero N S seeds Include seeds     Prisms with  p j =0 not shown
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
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
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
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
Growth: Try accretion to sth seed:     Prisms with  p j =0 not shown
Growth: Try accretion to sth seed: Choose neighbor: 1. Reduce data misfit 2. Smallest goal function     Prisms with  p j =0 not shown
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
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
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
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
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
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
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
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
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
Advantages: Compact & non­smooth Any number of sources Any number of different density contrasts No large equation system Search limited to neighbors    
Remember equations: Initial residual Update residual vector NS (0) r = g− ( ∑ ρs a j s=1 S ) r (new) =r (old ) − pj aj    
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 +)    
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    
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    
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    
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    
Advantages: Compact & non­smooth Any number of sources Any number of different density contrasts No large equation system Search limited to neighbors    
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    
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    
Synthetic Data    
Data set:  3 components ●  51 x 51 points ●  2601 points/component ●  7803 measurements ●  5 Eötvös noise ●    
Model:    
Model:  11 prisms ●    
Model:  11 prisms ●  4 outcropping ●    
Model:  11 prisms ●  4 outcropping ●    
Model:  11 prisms ●  4 outcropping ●    
 Strongly interfering effects ●    
 Strongly interfering effects ●  What if only interested in these? ●    
 Common scenario ●    
 Common scenario ●  May not have prior information ●  Density contrast ●  Approximate depth ●    
 Common scenario ●  May not have prior information ●  Density contrast ●  Approximate depth ●  No way to provide seeds ●    
 Common scenario ●  May not have prior information ●  Density contrast ●  Approximate depth ●  No way to provide seeds ●  Difficult to isolate effect of targets ●    
Robust procedure:    
Robust procedure: ● Seeds only for targets    
Robust procedure: ● Seeds only for targets    
Robust procedure: ● Seeds only for targets ●  ℓ1­norm to “ignore” non­targeted    
Robust procedure: ● Seeds only for targets ●  ℓ1­norm to “ignore” non­targeted    
Inversion: ● 13 seeds ● 7,803 data    
Inversion: ● 13 seeds ● 7,803 data    
Inversion: ● 13 seeds ● 7,803 data    
Inversion: ● 13 seeds ● 7,803 data    
Inversion: ● 13 seeds ● 7,803 data    
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ●    
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ● Only prisms with zero     density contrast not shown
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ● Only prisms with zero     density contrast not shown
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ● Only prisms with zero     density contrast not shown
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ● Only prisms with zero     density contrast not shown
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ● Only prisms with zero     density contrast not shown
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ● ●  Recover shape of targets Only prisms with zero     density contrast not shown
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
Inversion: ● 13 seeds ● 7,803 data  37,500 prisms ● Predicted data in contours    
Inversion: ● 13 seeds ● 7,803 data ● 37,500 prisms Predicted data in contours Effect of true targeted sources    
Real Data    
Data:  3 components ●  FTG survey ●  Quadrilátero Ferrífero, Brazil ●    
Data:  3 components ● Targets:  FTG survey ●  Iron ore bodies ●  Quadrilátero Ferrífero, Brazil ●  BIFs of Cauê Formation ●    
Data:  3 components ● Targets:  FTG survey ●  Iron ore bodies ●  Quadrilátero Ferrífero, Brazil ●  BIFs of Cauê Formation ●    
Data: Seeds for iron ore:  Density contrast 1.0 g/cm3 ●  Depth 200 m ●    
Inversion:  46 seeds  13,746 data Observed Predicted ● ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ● Only prisms with zero     density contrast not shown
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ● Only prisms with zero     density contrast not shown
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ● Only prisms with zero     density contrast not shown
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ●  164,892 prisms ●    
Inversion:  46 seeds  13,746 data ● ● ● 164,892 prisms ●  Agree with previous interpretations (Martinez et al., 2010)    
Inversion:  46 seeds  13,746 data ● ● ● 164,892 prisms ●  Agree with previous interpretations (Martinez et al., 2010) ●  Total time = 14 minutes (on laptop)    
Conclusions    
Conclusions ● New 3D gravity gradient inversion ● Multiple sources ● Interfering gravitational effects ● Non­targeted sources ● No matrix multiplications ● No linear systems ● Lazy evaluation of Jacobian matrix    
Conclusions ● Estimates geometry ● Given density contrasts ● Ideal for: ● Sharp contacts ● Well­constrained physical properties – Ore bodies – Intrusive rocks – Salt domes    
Thank you    

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Robust 3D gravity gradient inversion by planting anomalous densities

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