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. SilvaFederal 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/cm3)

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

xD

epth

y

z

Dep

th

x

Source Region

dy

dzdx

The source region is divided into an mx × my× mz grid

of M 3D vertical juxtaposed prisms


x

Observed gravity anomaly

y

z

Dep

th

Source Region

To estimate the 3D density-contrast distribution

y

x


The vertical component of the gravity field produced by the

density-contrast distribution (r’):

)(g ir )'(rV

i

''

'3

i dvzz

rr

Methodology

The discrete forward modeling operator for the gravity anomaly can be expressed by:

g A p

''

')( 3

jVi

iiij dv

zzA

rrr

where

Steiner (1978)

(N x 1) (M x 1)(NxM)

Methodology

2 Ago 1

Ng

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

Dep

thMethodology

Concentration of mass excess about NE specified

geometric elements (axes and points)

MethodologyIterative 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 NE 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

Δp2 )( k

W )( k1/2

p

and updates the density-contrast estimates by

2 Ago 1

NΔp )(po +

)( k )( k

)()()1( ˆˆ kkk pΔpp o

≡

)(

3

ˆ k-1j

jjj

p

dwWp

)( k1/2 )( k1/2

={ }Prior reference vector

}{min1 N

jdE

d j

MjNzezyeyxd Ejjj ,,1,,,1)()((2/1222

xe )j

Methodology

z

y

x

xe

)

ye, , ze)

jd

The method defines dj as the

distance from the center of the

j th prism to the

closest geometric elementclosest geometric element

d j

Mjdptargetj ,...,1},{min arg, j

**

En1

}{min1

jN

j ddE

z

y

x

xe

)

ye, , ze )

MethodologyThe method assigns to the jth prism the target density contrast

of the geometric element closest to the jth prism

d j

Methodology

z

x

axisaxis

point

jy

jd

g/cm3 g/cm3

g/cm3

At the first iteration:

• Initial interpretation model

• First-guess geometric elements• The corresponding target density

contrasts

Static Geologic Reference Model

pj targetg/cm3

jd andj

p targetThe method assigns to each prism a pair of:

.

Methodology

Penalization Algorithm:

)(ˆ kjp

jp target

0 (g/cm3)

jp target 0 (g/cm3)

• For positive target density contrast

• For negative target density contrast

)(ˆ kjp

)(ˆ kjp )(ˆ k

jp

jjwp

)( k1/2

=

target

jp or 0 (g/cm3)

)(ˆ kpΔ)(kp o )1(ˆ kp

( k )

op

j

Methodology

Penalization Algorithm:

jp target

0 (g/cm3)

jp target

0 (g/cm3)

• For positive target density contrast

• For negative target density contrast

)(ˆ kjp

)()()1( ˆˆ kkk pΔpp o

pjtarget

2

pjtarget

2

)(ˆ kjp)(ˆ k

jp

)(ˆ kjp

target

jp( k )

op

j

( k )

op

j

0 (g/cm3)

)(

3

ˆ k-1

j

j

jj p

dwp

)( k1/2

=)(ˆ k

jp

)(ˆ kjp

The choice of the interpretation model

noise-corrupted gravity anomaly

geometric element

The choice of the interpretation model

True source

target = 0.4 g/cm3. = 0.4 g/cm3.

x(km)-6 -4 -2 0 2 4 6 8 10 12

-6

-4

-2

0

2

4

6

8

10

12

y(k

m)

True source

Fitted anomaly

Rough interpretation model: 4×4×4 grid of 3D prisms

First

-6 -4 -2 0 2 4 6 8 10 12-6

-4

-2

0

2

4

6

8

10

12

x(km)

y(k

m)

True source

0.200.00 0.10 0.30 0.40density contrast (g/cm3)

density contrast (g/cm3)0.200.00 0.10 0.30 0.40

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

0.200.00 0.10 0.30 0.40density contrast (g/cm3)

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

zSource Region

First Iteration


x

y

z Each 3D prism is divided

Second Iteration


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


Static geologic reference model

target = 0.4 g/cm3.

Adaptive Learning Procedure Fourth iteration

0.200.00 0.10 0.30density contrast (g/cm3)

0.40

Without using the adaptive learning procedure

Both interpretation models consist of 24×24×24 grid of 3D prisms

0.200.00 0.10 0.30density contrast (g/cm3)

0.40

True source

Inversions of Synthetic Data

Large source surrounding a small source

-2 -1 0 1 2 3 4

y (km)

-2

-1

1

2

3

4

0

x (

km)

density contrast (g/cm3)density contrast (g/cm3)

Anorthosite ( 0.4 g/m3 )

Granite ( 0.2 g/cm3 )

Large source surrounding a small sourceThe red dots are the first-guess skeletal outlines:

static geologic reference model

Large source surrounding a small sourceFirst iteration

Interpretation model: 4×3×3 grid of 3D prisms.

density contrast (g/cm3)

-2 -1 0 1 2 3 4

Y (km)

-2

-1

0

1

2

3

4

X (

km)

Fitted anomaly

Large source surrounding a small sourceFourth iteration

interpretation model: 32×24×24 grid of 3D prisms.

-2 -1 0 1 2 3 4

Y (km)

-2

-1

0

1

2

3

4

X (

km)

Fitted anomaly density contrast (g/cm3)


Multiple buried sources at different depths

0.15 g/cm3

0.3g/cm3

0.4 g/cm3

The axes are the first-guess skeletal outlines: static geologic reference model


Multiple buried sources at different depths

Third iteration Interpretation model: 28×48×24 grid of 3D prisms.

density contrast (g/cm3)Fitted

anomaly

Real Gravity DataRedenção 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 outlinesstatic reference model

Redenção Granite

The associated target density contrasts are: -0.09 g/cm3 or -0.12 g/cm3

-0.09 g/cm3

-0.12 g/cm3

Redenção Granite


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:


first-guess skeletal outlines



New skeletal

Thank You I cordially invite

you to attend the upcoming

Extra Figures

1 CPU ATHLON with one core and 2.4 GHertz and 1 MB of cache L22GB of DDR1 memory

= 0.4 g/cm3.

Isolated gravity

anomalies


Li and Oldenburg (1998)


Portniaguine and Zhdanov (2002)


Our gravity inversion

Interfering gravity

anomalies


Li and Oldenburg (1998)


Portniaguine and Zhdanov (2002)


Our gravity inversion

MethodologyThe Iterative Constrained Inverse Problem

Starting from the minimum-norm solution

ogIAAAp 1o ) (ˆ TT

we look, at the kth iteration, for a constrained parameter correction

ˆˆˆ )(

o pΔpp ) 1( k k )( k

and update the density-contrast distribution estimate by

)(11)( ˆ)(ˆ k

o

T

)(k

T

)(k

kpAgIAA WAWpΔ

o 1

,

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

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