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3D gravity inversion incorporating prior information through an adaptive learning procedure

3D gravity inversion incorporating prior information through an adaptive learning procedure. Fernando J. S. Silva Dias Valéria C. F. Barbosa National Observatory João B. C. Silva Federal University of Pará. Content. Introduction and Objective. Methodology.

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3D gravity inversion incorporating prior information through an adaptive learning procedure

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  1. 3D gravity inversion incorporating prior information through an adaptive learning procedure Fernando J. S. Silva Dias Valéria C. F. Barbosa National Observatory João B. C. Silva Federal University of Pará

  2. Content • Introduction and Objective • Methodology • Synthetic Data Inversion Results • Real Data Inversion Result • Conclusions

  3. Introduction and Objective density contrast (g/cm3) Methods that estimate 3D density-contrast distributions: • Bear et al. (1995) • Li and Oldenburg (1998) • Portniaguine and Zhdanov (1999) • Zhdanov et al. (2004) Objective To estimate 3D source geometries that may give rise to interfering gravity anomalies.

  4. Methodology • Forward modeling of gravity anomalies • Inverse Problem • Adaptive Learning Procedure

  5. Forward modeling of gravity anomalies Gravity anomaly Source Region x y x y Depth 3D gravity sources z

  6. Forward modeling of gravity anomalies Source Region The source region is divided into an mx× my× mzgrid of M 3D vertical juxtaposed prisms dy dz x dx y Depth z

  7. Forward modeling of gravity anomalies Observed gravity anomaly Source Region To estimate the 3D density-contrast distribution x x y y Depth z

  8. Methodology - z ' z òòò = g r i r ( r ' ) dv ' i 3 - r ' r i - z ' z òòò = g i A ( r ) dv ' ij i 3 - V r ' r j i The vertical component of the gravity field produced by the density-contrast distribution r (r’): g ( ) V The discrete forward modeling operator for the gravity anomaly can be expressed by: g = A p (N x 1) (NxM) (M x 1) where Steiner (1978)

  9. o g The problem of obtaining a vector of parameter estimates, p , that minimizes this functional is an ill-posed problem. ^ Methodology The unconstrained Inverse Problem The linear inverse problem can be formulated by minimizing 2 1 g p - A = f N

  10. Methodology Source Region Concentrationof mass excess aboutNE specifiedgeometric elements (axes and points) x y Depth z

  11. Methodology Iterative inversion method that: • fits the gravity data • satisfies two constraints: • The density-contrast distribution must assume just two known density contrast values: or a nonnull value. zero • The estimated nonnulldensity contrast must be concentrated about a set of NE geometric elements (axes and points)

  12. Methodology 2 k k k 1/2 ( ( ( ) ) ) Δp W p 2 o = d (po+ Δp ) - g 1 A N 1/2 Prior reference vector 3 d 1/2 k k k ( ( ( ) ) ) w j Wp { } + = ( k ) ( k 1 ) ( k ) = + ˆ ˆ ≡ p p Δ p jj o ( ) k-1 + e ˆ p j The method estimates iteratively the constrained parameter correction Δp by Minimizing Subject to and updates the density-contrast estimates by

  13. Methodology d l j x d j y d j l xe ye ze ) ) , , l l l z 2 2 2 [ ] 1 / 2 - + - + - = = = d x xe ) ( y ye ) ( z ze ) 1 , , N , j 1 , , M ( l L L j j j E l l l j l = } d { min j £ £ 1 N l E The method defines dj as the distance from the center of the j th prism to the closest geometric element closest geometric element

  14. Methodology = d { d } min j j l x £ £ 1 N l E d j y xe ye ze ) ) , , l l l z target * = r = = p , arg min { d }, j 1 ,..., M l £ £ j j * 1 n l l l E The method assigns to the jth prism the target density contrast of the geometric element closest to the jth prism

  15. Methodology target x p d j j y point r = + 0.3 g/cm3 axis d j z axis r = + 0.2 g/cm3 Static Geologic Reference Model and The method assigns to each prism a pair of: j pjtarget = + 0.3 g/cm3 . At the first iteration: r = - 0.1 g/cm3 • Initial interpretation model • First-guess geometric elements • The corresponding target density contrasts

  16. Methodology + ( k ) ( k 1 ) ˆ p p o k ( ) ( k ) p target p ( ( ( ( ) ) ) ) k k k k ˆ ˆ ˆ ˆ p p p p o j j target target p p j j j j j j = 1/2 10+8 wp = jj ( k ) = + ˆ Δ p Penalization Algorithm: • For positive target density contrast 0 (g/cm3) • For negative target density contrast 0 (g/cm3) or 0 (g/cm3)

  17. Methodology k ( ) ( k ) ( k ) p target p p ( ( ( ( ( ( ) ) ) ) ) ) k k k k k k ˆ ˆ ˆ ˆ ˆ ˆ 3 p p p p p p d wp o o p j target j j j = target target p p j j j j j j j 2 jj ( ) k-1 + e ˆ j j p j = = p target j 2 1/2 + 0 (g/cm3) ( k ) ( k 1 ) ( k ) = + ˆ ˆ p p Δ p o Penalization Algorithm: • For positive target density contrast 0 (g/cm3) • For negative target density contrast 0 (g/cm3)

  18. The choice of the interpretation model

  19. The choice of the interpretation model True source noise-corrupted gravity anomaly geometric element rtarget = 0.4 g/cm3. r = 0.4 g/cm3.

  20. Rough interpretation model: 4×4×4 grid of 3D prisms density contrast (g/cm3) Rough interpretation model:5×5×5 grid of 3D prisms 0.00 0.10 0.20 0.30 0.40 Fitted anomaly True source 1 2 1 0 1 2 8 Fitted anomaly 1 0 6 y(km) 8 y(km) 6 4 4 2 2 0 0 - 2 - 2 - 4 - 6 - 4 -6 -4 -2 0 2 4 6 8 10 12 density contrast (g/cm3) - 6 -4 -2 0 2 4 6 8 10 12 -6 x(km) 0.40 0.00 0.10 0.20 0.30 x(km) True source First Second

  21. Refined interpretation model:12×12×12 grid of 3D prisms Refined interpretation model: 24×24×24 grid of 3D prisms Fitted anomaly Fitted anomaly True source Fourth density contrast (g/cm3) 0.00 0.10 0.20 0.30 0.40 True source Third

  22. Adaptive Learning Procedure • New interpretation model • New geometric elements • Associated target density contrasts

  23. Adaptive Learning Procedure Source Region x FirstIteration y z

  24. Adaptive Learning Procedure x Second Iteration y z Each 3D prism is divided

  25. Adaptive Learning Procedure New interpretation model x Iteration 1 y Iteration 2 Iteration 3 Iteration 4 z

  26. Adaptive Learning Procedure First density-contrast distribution estimate static geologic reference model New geometric elements (points) and associated target density contrasts New interpretation model First interpretation model and the static geologic reference model x First Iteration y Second Iteration Dynamic geologic reference model z

  27. Adaptive Learning Procedure Static geologic reference model rtarget = 0.4 g/cm3.

  28. Adaptive Learning Procedure True source density contrast (g/cm3) density contrast (g/cm3) 0.00 0.00 0.10 0.10 0.20 0.20 0.30 0.30 0.40 0.40 Without using the adaptive learning procedure Fourth iteration Both interpretation models consist of 24×24×24 grid of 3D prisms

  29. Inversions of Synthetic Data

  30. Large source surrounding a small source Granite ( 0.2 g/cm3 ) 4 3 2 x (km) 1 0 -1 -2 -2 -1 0 1 2 3 4 y (km) density contrast (g/cm3) density contrast (g/cm3) Anorthosite ( 0.4 g/m3 )

  31. Large source surrounding a small source The red dots are the first-guess skeletal outlines: static geologic reference model

  32. Large source surrounding a small source density contrast (g/cm3) First iteration Interpretation model: 4×3×3 grid of 3D prisms. Fitted anomaly

  33. Large source surrounding a small source density contrast (g/cm3) Fourth iteration interpretation model: 32×24×24 grid of 3D prisms. Fitted anomaly

  34. Multiple buried sources at different depths 0.4 g/cm3 0.15 g/cm3 0.3g/cm3 density contrast (g/cm3) density contrast (g/cm3) The axes are the first-guess skeletal outlines: static geologic reference model

  35. Multiple buried sources at different depths density contrast (g/cm3) Third iteration Interpretation model: 28×48×24 grid of 3D prisms. Fitted anomaly

  36. Real Gravity Data Redenção Granite (Brazil)

  37. Localization and Geological Setting The Amazon Craton in northern Brazil, within the Archean Greenstone unit, comprising a part of the Carajás metallogenic province. Brazil Oliveira et al. (2007)

  38. Geologic Map of the Redenção Area SRTM / Gamma Thorium Oliveira et al. (2007)

  39. Redenção Granite -0.09 g/cm3 -0.12 g/cm3 The red dots are the first-guess skeletal outlines static reference model The associated target density contrasts are: -0.09 g/cm3 or -0.12 g/cm3

  40. Redenção Granite Fourth iteration Interpretation model: 64×72×32 grid of 3D prisms. Dynamic Geometric Elements Fitted anomaly density contrast (g/cm3)

  41. Conclusions

  42. 3D gravity inversion incorporating prior information through an adaptive learning procedure New skeletal first-guess skeletal outlines density contrast (g/cm3) density contrast (g/cm3) density contrast (g/cm3) The proposed method: • Estimates 3D source geometries that may give rise to an interfering gravity anomaly • Concentrates the largest density contrast estimates about first-guess skeletal outlines • Creates new skeletal outlines and a new refined interpretation model • Makes it possible to reconstruct a sharp image of multiple and closely located gravity sources.

  43. Thank You I cordially invite you to attend the upcoming

  44. Extra Figures 1 CPU ATHLON with one core and 2.4 GHertz and 1 MB of  cache L22GB of  DDR1 memory

  45. r = 0.4 g/cm3. Isolated gravity anomalies

  46. Li and Oldenburg (1998) density contrast (g/cm3)

  47. Portniaguine and Zhdanov (2002) density contrast (g/cm3)

  48. Our gravity inversion density contrast (g/cm3)

  49. Interfering gravity anomalies

  50. Li and Oldenburg (1998) density contrast (g/cm3)

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