Adaptive learning gravity inversion for 3D salt body imaging
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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. Content. Introduction and Objective. Methodology. Synthetic Data Inversion Result.

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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


Content

  • Introduction and Objective

  • Methodology

  • Synthetic Data Inversion Result

  • Real Data Inversion Result

  • Conclusions


Introduction

Seismic and gravity data are combined to interpret salt bodies

Brazilian sedimentary basin


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


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.


Methodology

  • Forward modeling of gravity anomalies

  • Inverse Problem

  • Adaptive Learning Procedure


Forward modeling of gravity anomalies

Gravity anomaly

Source Region

x

y

x

y

Depth

3D salt body

z


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


Forward modeling of gravity anomalies

Observed gravity anomaly

Source Region

To estimate the 3D density-contrast distribution

x

x

y

y

Depth

z


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


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


Methodology

Source Region

Concentrationof salt mass aboutspecifiedgeometric elements (axes and points)

x

y

Depth

3D salt body

z


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


Methodology

Homogeneous salt body embedded in a heterogeneous sedimentary pack

A reversal 3D density-contrast distribution

3D salt body

Depth

Heterogeneous sedimentary pack

z


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


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)


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


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


Adaptive Learning Procedure

Outer Loop

  • Interpretation model

  • Geometric elements

  • Associated target density contrasts


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



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


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


Synthetic example with a variable density contrast

The true reversal 3D density-contrast distribution

Depth (km)

below

above

Density contrast (g/cm3)


Synthetic example with a variable density contrast

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


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


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)


Real Gravity Data

Galveston Island salt dome Texas



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

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


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


Galveston Island salt dome

The first estimated reversal 3D density-contrast distribution


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


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




Thank You

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


Extra figures
Extra Figures

1 CPU ATHLON with one core and

2.4 GHertz and 1 MB of  cache L22GB 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

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.


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)


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)


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