Adaptive learning gravity inversion for

3D salt body imagingFernando J. S. Silva Dias

Valéria C. F. Barbosa National Observatory

João B. C. SilvaFederal 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

IntroductionWhere 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

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

Forward modeling of gravity anomalies

x

Observed gravity anomaly

y

z

Dep

th

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

Methodology

2

Ago 1N

g

The unconstrained Inverse Problem

The linear inverse problem can be formulated by

minimizing

ill-posed problem

p

x

y

z Source Region

Dep

thMethodology

Concentration of salt mass about specified

geometric elements (axes and points)

3D salt body

z

Dep

th

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/cm3

homogeneous sediments

Homogeneous salt body embedded in a heterogeneous sedimentary pack

zHeterogeneous

sedimentary pack

Dep

th

3D salt body

Methodology

A reversal 3D density-contrast distribution

z

Dep

th Heterogeneous salt body

Methodology

Homogeneous sediments

g/cm3g/cm3

g/cm3

g/cm3

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

MethodologyIterative 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: zero or 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

pjtargetg/

cm3

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

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

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 model first-guess geometric elements and associated

target density contrasts

Inversion of Synthetic Data

Noise-corrupted gravity anomaly

Synthetic example with a variable density contrast

-1 0 1 2 3 4 5 6 7

y (km)

1

2

3

4

5

6

7

8

9x

(km

)

-0.1

0.1

0.3

0.5

mGal

Homogeneous salt dome with density of 2.2 g/cm3 embedded in five sedimentary layers

Synthetic example with a variable density contrast

with density varying with depth from 1.95 to 2.39 g/cm3.D

epth

3D salt body

1.5 km Nil zone

1.95 g/cm3

2.39 g/cm3

Synthetic example with a variable density contrast

Density contrast (g/cm3)

Dep

th (

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

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 BodyFitted anomaly

-1 0 1 2 3 4 5 6 7y (km)

1

2

3

4

5

6

7

8

9

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

NBouguer anomaly maps

(UTM15) km E N

314 320 326 332

3134

3136

3138

3140

3142

3144

3146

3148

3150

3152

-1.4-0.212.2mGalFueg’s (1995)

density models

Galveston Island salt domeD

epth

(km

)

0.08 0.00 (g/cm3)

0.20 (g/cm3)

0.10 (g/cm3)

0.06 (g/cm3)

0.02 (g/cm3)

- 0.04 (g/cm3)

- 0.08 (g/cm3)

- 0.13 (g/cm3)

0.15

0.5

0.8

1.2

1.5

2.0

3.4

Dep

th (

km)

0.08 0.00 (g/cm3)

0.20 (g/cm3)

0.10 (g/cm3)

0.06 (g/cm3)

0.02 (g/cm3)

- 0.04 (g/cm3)

- 0.08 (g/cm3)

- 0.18 (g/cm3)

0.15

0.5

0.8

1.2

1.5

2.0

3.2

2.6

3.83.9 - 0.23 (g/cm3)

- 0.13 (g/cm3)

First static geologic reference model based on Fueg’s (1995) density models

The first geologic hypothesis about the salt dome

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

Dep

th (

km)

0.04 0.00 (g/cm3)

0.19 (g/cm3)

0.08 (g/cm3)

- 0.04 (g/cm3)

0.31

0.35

1.2

2.0

2.2 - 0.13 (g/cm3)

Galveston Island salt dome

(UTM15) km E N

314 320 326 332

3134

3136

3138

3140

3142

3144

3146

3148

3150

3152

-1.4-0.212.2mGal

The second geologic hypothesis about the salt dome

Galveston Island salt domeThe second estimated reversal 3D density-contrast distribution

Density contrast (g/cm3)

-0.13 -0.042 0.045 0.13 0.22

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 L22GB of  DDR1 memory

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

static geologic reference model

(a)

(b)

Silva Dias et al. Fig. 8

Estimated density contrast (g/cm3)0.1 0.2 0.3 0.4 0.5

(a)

(b)

Silva Dias et al. Fig. 8

Estimated density contrast (g/cm3)0.1 0.2 0.3 0.4 0.5

Large source surrounding a small sourceFifth 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/cm3)

0.15 g/cm3

0.3g/cm3

0.4 g/cm3

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

Silva Dias et al. Fig. 8

(d)

(e)

0.15 0.2 0.3 0.4Estimated density contrast (g/cm3)

Silva Dias et al. Fig. 8

(d)

(e)

0.15 0.2 0.3 0.4Estimated density contrast (g/cm3)

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