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 Real Data Inversion Result Conclusions Synthetic Data Inversion Results

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

Methodology Forward modeling of gravity anomalies Inverse Problem Adaptive Learning Procedure

Gravity anomaly x y z 3D gravity sources 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 Steiner (1978) (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 The problem of obtaining a vector of parameter estimates, p, that minimizes this functional is an ill-posed problem. ^ p

x y z Source Region Depth Methodology Concentration of mass excess about N E specified geometric elements (axes and points)

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

The method 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

M jdp target j,...,1},{min arg, j * *  E n1   }{ min 1 j N j dd E   z y x xe ) ye,, ze ) Methodology The method assigns to the j th prism the target density contrast of the geometric element closest to the j th prism d j

Methodology z x axis point j y j d  g/cm 3  g/cm 3  g/cm 3 At the first iteration: Initial interpretation model First-guess geometric elements The corresponding target density contrasts Static Geologic Reference Model p j target  g/cm 3 j d and j p target The method assigns to each prism a pair of:.

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

The choice of the interpretation model

noise-corrupted gravity anomaly geometric element The choice of the interpretation model True source  target = 0.4 g/cm 3.  = 0.4 g/cm 3.

True source Fitted anomaly Rough interpretation model: 4×4×4 grid of 3D prisms First x(km) y(km) True source density contrast (g/cm 3 ) Rough interpretation model : 5×5×5 grid of 3D prisms Fitted anomaly Second

True source Refined interpretation model: 24×24×24 grid of 3D prisms Fourth density contrast (g/cm 3 ) Fitted anomaly True source Refined interpretation model: 12×12×12 grid of 3D prisms Third Fitted anomaly

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

Adaptive Learning Procedure x y z Source Region First Iteration

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

Adaptive Learning Procedure x y z Iteration  Iteration  Iteration  Iteration  New interpretation model

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

Adaptive Learning Procedure Static geologic reference model  target = 0.4 g/cm 3.

Adaptive Learning Procedure Fourth iteration density contrast (g/cm 3 ) 0.40 Without using the adaptive learning procedure Both interpretation models consist of 24×24×24 grid of 3D prisms density contrast (g/cm 3 ) 0.40 True source

Inversions of Synthetic Data

Large source surrounding a small source y (km) x (km) density contrast (g/cm 3 ) Anorthosite ( 0.4 g/m 3 ) Granite ( 0.2 g/cm 3 )

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 First iteration Interpretation model: 4×3×3 grid of 3D prisms. density contrast (g/cm 3 ) Fitted anomaly

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

Multiple buried sources at different depths 0.15 g/cm 3 0.3g/cm g/cm 3 The axes are the first-guess skeletal outlines: static geologic reference model density contrast (g/cm 3 )

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

Real Gravity Data Redenção Granite (Brazil)

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

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

The red dots are the first-guess skeletal outlines static reference model Redenção Granite The associated target density contrasts are: g/cm 3 or g/cm g/cm g/cm 3

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

Conclusions

3D gravity inversion incorporating prior information through an adaptive learning procedure  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. The proposed method: density contrast (g/cm 3 ) first-guess skeletal outlines density contrast (g/cm 3 ) New skeletal

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

 = 0.4 g/cm 3. Isolated gravity anomalies

density contrast (g/cm 3 ) Li and Oldenburg (1998)

density contrast (g/cm 3 ) Portniaguine and Zhdanov (2002)

density contrast (g/cm 3 ) Our gravity inversion

Interfering gravity anomalies

density contrast (g/cm 3 ) Li and Oldenburg (1998)

density contrast (g/cm 3 ) Portniaguine and Zhdanov (2002)

density contrast (g/cm 3 ) Our gravity inversion

Methodology The Iterative Constrained Inverse Problem Starting from the minimum-norm solution o gIAAA p 1o ) ( ˆ   TT we look, at the kth iteration, for a constrained parameter correction ˆˆ ˆ )( o pΔp p   ) 1 ( k  k )( k and update the density-contrast distribution estimate by  )(11)( ˆ )( ˆ k o T )(k T )(k k pA g IAA WAWpΔ o  1    ,