Adaptive learning gravity inversion for 3D salt body imaging

1. Adaptive learning gravity inversion for 3D salt body imaging 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 Result • Real Data Inversion Result • Conclusions

3. Introduction Seismic and gravity data are combined to interpret salt bodies Brazilian sedimentary basin

4. Introduction It is much harder to “see” what lies beneath salt bodies. Where is the base of the salt body ? Top of the salt body

5. Objective Methods that reconstruct 3D (or 2D) salt bodies from gravity data Interactive gravity forward modeling: Starich et al. (1994) Yarger et al. (2001) Oezsen (2004) Huston et al. (2004) Gravity inversion methods Jorgensen and Kisabeth (2000) Bear et al. (1995) Moraes and Hansen (2001) Routh et al. (2001) Krahenbuhl and Li (2006) We adapted the 3D gravity inversion through an adaptive learning procedure (Silva Dias et al., 2007) to estimate the shape of salt bodies.

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

7. Forward modeling of gravity anomalies Gravity anomaly Source Region x y x y Depth 3D salt body z

8. 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

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

10. 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

11. o g Methodology The unconstrained Inverse Problem The linear inverse problem can be formulated by minimizing 2 1 g p - A = f N ill-posed problem

12. Methodology Source Region Concentrationof salt mass aboutspecifiedgeometric elements (axes and points) x y Depth 3D salt body z

13. r = - 0.3 g/cm3 homogeneous sediments Methodology Homogeneous salt body embedded in homogeneous sediments First-guess skeletal outline of the salt body Only one target density contrast 3D salt body Depth z

14. Methodology Homogeneous salt body embedded in a heterogeneous sedimentary pack A reversal 3D density-contrast distribution 3D salt body Depth Heterogeneous sedimentary pack z

15. r = + 0.3 g/cm3 r = + 0.2 g/cm3 r = - 0.1 g/cm3 r = - 0.2 g/cm3 Methodology Heterogeneous salt body embedded in homogeneous sediments First-guess skeletal outline of a particular homogeneous section of the salt body A reversal 3D density-contrast distribution Heterogeneous salt body Depth Homogeneous sediments z

16. Methodology x y x pjtarget = - 0.3 g/cm3 z x y y z z Iterative inversion method consists of two nested iterative loops: The outer loop: adaptive learning procedure • Coarse interpretation model • refined interpretation model • first-guess geometric elements (axes and points) • new geometric elements (points) • corresponding target density contrasts • corresponding target density contrasts The inner loop: Iterative inversion method • fits the gravity data • satisfies two constraints: • Density contrast values: zero or a nonnull value. • Concentration of the estimated nonnulldensity contrast about a set of geometric elements (axes and points)

17. 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 inversion method of the inner loopestimates iteratively the constrained parameter correction Δp by Minimizing Subject to and updates the density-contrast estimates by

18. 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 Inner loop = } 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

19. Adaptive Learning Procedure Outer Loop • Interpretation model • Geometric elements • Associated target density contrasts

20. Adaptive Learning Procedure INNER LOOP: First density-contrast distribution estimate static geologic reference model First interpretation model first-guess geometric elements and associated New interpretation model New geometric elements (points) and associated target density contrasts target density contrasts x OUTER LOOP: Second Iteration OUTER LOOP: First Iteration y Dynamic geologic reference model z Each 3D prism is divided

21. Inversion of Synthetic Data

22. 9 8 7 0.5 6 ) m 0.3 k 5 ( x 4 0.1 3 -0.1 mGal 2 1 -1 0 1 2 3 4 5 6 7 y (km) Synthetic example with a variable density contrast Noise-corrupted gravity anomaly

23. Synthetic example with a variable density contrast Homogeneous salt dome with density of 2.2 g/cm3 embedded in five sedimentary layers with density varying with depth from 1.95 to 2.39 g/cm3. 1.95 g/cm3 1.5 km Nil zone 2.39 g/cm3 Depth 3D salt body

24. Synthetic example with a variable density contrast The true reversal 3D density-contrast distribution Depth (km) below above Density contrast (g/cm3)

25. Synthetic example with a variable density contrast The blue axes are the first-guess skeletal outlines: static geologic reference model

26. Synthetic example with a variable density contrast Interpretation model at the fourth iteration: 80×72×40 grid of 3D prisms. True Salt Body Estimated Salt Body

27. Synthetic example with a variable density contrast Estimated Salt Body Fitted anomaly 9 8 7 6 ) m k 5 ( x 4 3 2 1 -1 0 1 2 3 4 5 6 7 y (km)

28. Real Gravity Data Galveston Island salt dome Texas

29. Localization of Galveston Island salt dome Study area

30. Localization of Galveston Island salt dome Study area Location map of the study area (after Fueg, 1995; Moraes and Hansen, 2001)

31. Galveston Island salt dome N N 3152 3150 3148 3146 3144 3142 3140 3138 3136 3134 km E 314 320 326 332 (UTM15) km E (UTM15) mGal Fueg’s (1995) density models 2.2 1 -0.2 -1.4 Bouguer anomaly maps

32. Galveston Island salt dome 0.08 0.08 0.00 (g/cm3) 0.00 (g/cm3) 0.15 0.15 0.20 (g/cm3) 0.20 (g/cm3) 0.5 0.5 0.10 (g/cm3) 0.10 (g/cm3) 0.8 0.8 0.06 (g/cm3) 0.06 (g/cm3) Depth (km) 1.2 1.2 Depth (km) 0.02 (g/cm3) 0.02 (g/cm3) 1.5 1.5 - 0.04 (g/cm3) - 0.04 (g/cm3) 2.0 2.0 - 0.08 (g/cm3) - 0.08 (g/cm3) 2.6 - 0.13 (g/cm3) 3.2 3.4 - 0.18 (g/cm3) - 0.13 (g/cm3) 3.8 - 0.23 (g/cm3) 3.9 The first geologic hypothesis about the salt dome First static geologic reference model based on Fueg’s (1995) density models

33. Galveston Island salt dome The first estimated reversal 3D density-contrast distribution

34. Galveston Island salt dome N 3152 3150 3148 3146 3144 3142 3140 3138 3136 3134 314 320 326 332 km E (UTM15) mGal 2.2 1 -0.2 -1.4 The second geologic hypothesis about the salt dome 0.04 0.00 (g/cm3) 0.31 0.19 (g/cm3) 0.35 0.08 (g/cm3) 1.2 Depth (km) - 0.04 (g/cm3) 2.0 - 0.13 (g/cm3) 2.2

35. Galveston Island salt dome Density contrast (g/cm3) -0.13 -0.042 0.045 0.22 0.13 The second estimated reversal 3D density-contrast distribution Overhang

36. Conclusions

37. Adaptive learning gravity inversion for 3D salt body imaging

38. Thank You We thank Dr. Roberto A. V. Moraes and Dr. Richard O. Hansen for providing the real gravity data

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

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

41. Large source surrounding a small source Fifth iteration interpretation model: 48×48×24 grid of 3D prisms.

42. Multiple buried sources at different depths 0.4 g/cm3 0.15 g/cm3 0.3g/cm3 density contrast (g/cm3) The points are the first-guess skeletal outlines: static geologic reference model Third iteration Interpretation model: 28×48×24 grid of 3D prisms.

43. 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)

44. 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)