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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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slide1
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

slide2
Content
  • Introduction and Objective
  • Methodology
  • Synthetic Data Inversion Result
  • Real Data Inversion Result
  • Conclusions
slide3
Introduction

Seismic and gravity data are combined to interpret salt bodies

Brazilian sedimentary basin

slide4
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

slide5
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.

slide6
Methodology
  • Forward modeling of gravity anomalies
  • Inverse Problem
  • Adaptive Learning Procedure
slide7
Forward modeling of gravity anomalies

Gravity anomaly

Source Region

x

y

x

y

Depth

3D salt body

z

slide8
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

slide9
Forward modeling of gravity anomalies

Observed gravity anomaly

Source Region

To estimate the 3D density-contrast distribution

x

x

y

y

Depth

z

slide10
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

slide11
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

slide12
Methodology

Source Region

Concentrationof salt mass aboutspecifiedgeometric elements (axes and points)

x

y

Depth

3D salt body

z

slide13
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

slide14
Methodology

Homogeneous salt body embedded in a heterogeneous sedimentary pack

A reversal 3D density-contrast distribution

3D salt body

Depth

Heterogeneous sedimentary pack

z

slide15
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

slide16
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)
slide17
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

slide18
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

slide19
Adaptive Learning Procedure

Outer Loop

  • Interpretation model
  • Geometric elements
  • Associated target density contrasts
slide20
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

slide22
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

slide23
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

slide24
Synthetic example with a variable density contrast

The true reversal 3D density-contrast distribution

Depth (km)

below

above

Density contrast (g/cm3)

slide25
Synthetic example with a variable density contrast

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

slide26
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

slide27
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)

slide28
Real Gravity Data

Galveston Island salt dome Texas

slide30
Localization of Galveston Island salt dome

Study area

Location map of the study area (after Fueg, 1995; Moraes and Hansen, 2001)

slide31
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

slide32
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

slide33
Galveston Island salt dome

The first estimated reversal 3D density-contrast distribution

slide34
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

slide35
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

slide38
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

slide40
Large source surrounding a small source

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

slide41
Large source surrounding a small source

Fifth iteration

interpretation model: 48×48×24 grid of 3D prisms.

slide42
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.

slide43
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)

slide44
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)

ad