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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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Fernando J. S. Silva Dias
Valéria C. F. Barbosa
National Observatory
João B. C. Silva
Federal University of Pará
Seismic and gravity data are combined to interpret salt bodies
Brazilian sedimentary basin
It is much harder to “see” what lies beneath salt bodies.
Where is the base of the salt body ?
Top of the salt body
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.
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
Observed gravity anomaly
Source Region
To estimate the 3D densitycontrast distribution
x
x
y
y
Depth
z

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 densitycontrast 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
g
Methodology
The unconstrained Inverse Problem
The linear inverse problem can be formulated by minimizing
2
1
g
p

A
=
f
N
illposed problem
Source Region
Concentrationof salt mass aboutspecifiedgeometric elements (axes and points)
x
y
Depth
3D salt body
z
homogeneous sediments
Methodology
Homogeneous salt body embedded in homogeneous sediments
Firstguess skeletal outline of the salt body
Only one target density contrast
3D salt body
Depth
z
Homogeneous salt body embedded in a heterogeneous sedimentary pack
A reversal 3D densitycontrast distribution
3D salt body
Depth
Heterogeneous sedimentary pack
z
r = + 0.2 g/cm3
r =  0.1 g/cm3
r =  0.2 g/cm3
Methodology
Heterogeneous salt body embedded in homogeneous sediments
Firstguess skeletal outline of a particular homogeneous section of the salt body
A reversal 3D densitycontrast distribution
Heterogeneous salt body
Depth
Homogeneous sediments
z
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
The inner loop: Iterative inversion method
zero
or a nonnull value.
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
(
)
k1
+
e
ˆ
p
j
The inversion method of the inner loopestimates iteratively the constrained parameter correction Δp by
Minimizing
Subject to
and updates the densitycontrast estimates by
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
Outer Loop
INNER LOOP:
First densitycontrast distribution estimate
static geologic reference model
First interpretation model
firstguess 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
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
Noisecorrupted gravity anomaly
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
The true reversal 3D densitycontrast distribution
Depth (km)
below
above
Density contrast (g/cm3)
The blue axes are the firstguess skeletal outlines: static geologic reference model
Interpretation model at the fourth iteration: 80×72×40 grid of 3D prisms.
True Salt Body
Estimated Salt Body
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)
Galveston Island salt dome Texas
Study area
Location map of the study area (after Fueg, 1995; Moraes and Hansen, 2001)
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
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
The first estimated reversal 3D densitycontrast distribution
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
Density contrast (g/cm3)
0.13
0.042
0.045
0.22
0.13
The second estimated reversal 3D densitycontrast distribution
Overhang
We thank Dr. Roberto A. V. Moraes and Dr. Richard O. Hansen for providing the real gravity data
The red dots are the firstguess skeletal outlines: static geologic reference model
Fifth iteration
interpretation model: 48×48×24 grid of 3D prisms.
0.4 g/cm3
0.15 g/cm3
0.3g/cm3
density contrast (g/cm3)
The points are the firstguess skeletal outlines:
static geologic reference model
Third iteration
Interpretation model: 28×48×24 grid of 3D prisms.
+
(
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:
0 (g/cm3)
0 (g/cm3)
or
0 (g/cm3)
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
(
)
k1
+
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:
0 (g/cm3)
0 (g/cm3)