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

Introduction and Objective Methodology Real Data Inversion Result Conclusions Synthetic Data Inversion Result Content

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

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

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

Methodology Forward modeling of gravity anomalies Inverse Problem Adaptive Learning Procedure

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

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

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

The vertical component of the gravity field produced by the density-contrast distribution  ( r’ ): )(g i r )'(r  V i ' ' ' 3   i dv zz rr   Methodology The discrete forward modeling operator for the gravity anomaly can be expressed by: g  A p ' ' ' )( 3     j V i i iij dv zz A rr r  where (N x 1)(M x 1)(NxM)

Methodology 2 A g o  1  N g  The unconstrained Inverse Problem The linear inverse problem can be formulated by minimizing ill-posed problem p

x y z Source Region Depth Methodology Concentration of salt mass about specified geometric elements (axes and points) 3D salt body

z Depth 3D salt body Homogeneous salt body embedded in homogeneous sediments Methodology First-guess skeletal outline of the salt body Only one target density contrast  g/cm 3 homogeneous sediments

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

z Depth Heterogeneous salt body Methodology Homogeneous sediments  g/cm 3  g/cm 3  g/cm 3  g/cm 3 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

Methodology Iterative inversion method consists of two nested iterative loops: The outer loop : adaptive learning procedure The inner loop: Iterative inversion method  fits the gravity data  satisfies two constraints: Density contrast values: zeroor a nonnull value. Concentration of the estimated nonnull density contrast about a set of geometric elements (axes and points) Coarse interpretation model first-guess geometric elements (axes and points) corresponding target density contrasts x y z x y z p j target  g/cm 3 x z y refined interpretation model new geometric elements (points) corresponding target density contrasts

The inversion method of the inner loop estimates iteratively the constrained parameter correction Δp by Minimizing Subject to Methodology Δp 2 )( k W )( k 1/2 p and updates the density-contrast estimates by 2 A g o  1 N   Δp ) (p o + )( k )( k )( )( )1( ˆˆ k k k pΔpp o    ≡ )( 3 ˆ k-1 j j jj p d w WpWp )( k 1/2 )( k = {} Prior reference vector

} { min 1N j d E d j     MjNzezyeyxd E jjj,,1,,,1)()( ( 2/1 222  xe) j   Methodology z y x xe ) ye,, ze ) j d The method defines d j as the distance from the center of the j th prism to the closest geometric element closest geometric element d j Inner loop

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

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

Inversion of Synthetic Data

Noise-corrupted gravity anomaly Synthetic example with a variable density contrast

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

Synthetic example with a variable density contrast Density contrast (g/cm 3 ) Depth (km) The true reversal 3D density-contrast distribution abovebelow

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

True Salt Body Estimated Salt Body Interpretation model at the fourth iteration: 80×72×40 grid of 3D prisms.

Synthetic example with a variable density contrast Estimated Salt Body Fitted anomaly y (km) x ( k m )

Real Gravity Data Galveston Island salt dome Texas

Localization of Galveston Island salt dome Study area

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

Galveston Island salt dome (UTM15) km E N Bouguer anomaly maps (UTM15) km E N mGal Fueg’s (1995) density models

Galveston Island salt dome Depth (km) (g/cm 3 ) 0.20 (g/cm 3 ) 0.10 (g/cm 3 ) 0.06 (g/cm 3 ) 0.02 (g/cm 3 ) (g/cm 3 ) (g/cm 3 ) (g/cm 3 ) Depth (km) (g/cm 3 ) 0.20 (g/cm 3 ) 0.10 (g/cm 3 ) 0.06 (g/cm 3 ) 0.02 (g/cm 3 ) (g/cm 3 ) (g/cm 3 ) (g/cm 3 ) (g/cm 3 ) (g/cm 3 ) First static geologic reference model based on Fueg’s (1995) density models The first geologic hypothesis about the salt dome

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

Depth (km) (g/cm 3 ) 0.19 (g/cm 3 ) 0.08 (g/cm 3 ) (g/cm 3 ) (g/cm 3 ) Galveston Island salt dome (UTM15) km E N mGal The second geologic hypothesis about the salt dome

Galveston Island salt dome The second estimated reversal 3D density-contrast distribution Density contrast (g/cm 3 ) Overhang

Conclusions

Adaptive learning gravity inversion for 3D salt body imaging

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

Extra Figures 1 CPU ATHLON with one core and 2.4 GHertz and 1 MB of cache L2 2GB of DDR1 memory

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

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

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

Methodology Penalization Algorithm: )( ˆ k j p j p target 0 (g/cm 3 ) j p target 0 (g/cm 3 ) For positive target density contrast For negative target density contrast )( ˆ k j p )( ˆ k j p )( ˆ k j p jj wpwp )( k 1/2 =   target j p or 0 (g/cm 3 ) )( ˆ k pΔ )(k p o  )1( ˆ k p  ( k ) o p j 

Methodology Penalization Algorithm: j p target 0 (g/cm 3 ) j p target 0 (g/cm 3 ) For positive target density contrast For negative target density contrast )( ˆ k j p )( )( )1( ˆˆ k k k pΔpp o   p j target 2 p j 2 )( ˆ k j p )( ˆ k j p )( ˆ k j p j p ( k ) o p j  o p j  0 (g/cm 3 )  )( 3 ˆ k-1 j j jj p d wpwp )( k 1/2 = )( ˆ k j p )( ˆ k j p