Radial gravity inversion constrained by total anomalous mass excess for retrieving 3D bodies Vanderlei Coelho Oliveira Junior Valéria C. F. Barbosa Observatório

Radial gravity inversion constrained by total

anomalous mass excess for retrieving 3D bodies

Vanderlei Coelho Oliveira Junior

Valéria C. F. Barbosa

Observatório Nacionalwww.on.br

Contents• Objective

• Methodology

• Real Data Inversion Result

• Conclusions

• Synthetic Data Inversion Result

3D Gravity inversion method

Do the gravity data have resolution to retrieve the 3D source?

yxN E

zDep

thD

epth

3D source

ObjectiveEstimate from gravity data the geometry of an isolated 3D source

Gravity dataGravity data

Source’s top

Methodology

y

x

z

y

x Gravity observations

go NR

3D Source

Dep

thD

epth S

The 3D source has an unknown closed surface S.

Methodology

x

z

y


go NR

3D Source

Dep

thD

epth

Approximate the 3D source by a set of 3D juxtaposed prisms in the vertical direction .

S

y

Methodology

x

z

y


go NR

3D Source

Dep

thD

epth

We set the thicknesses and density contrasts of all prisms

y

Methodology

x

z

y


go NR

3D Source

Dep

thD

epth

y

MethodologyThe horizontal cross-section of each prism is described by

an unknown polygon The polygon sides approximately describe the edges of a

horizontal depth slice of the 3D source.

y


go NR

MethodologyThe polygon sides of an ensemble of vertically stacked prisms represent

a set of juxtaposed horizontal depth slices of the 3D source.

x

z3D Source

Dep

thD

epth S

y

y


go NR

MethodologyThe polygon sides of an ensemble of vertically stacked prisms represent

a set of juxtaposed horizontal depth slices of the 3D source.

x

z

Dep

thD

epth

y

x

z

Dep

thD

epth

y

Methodology

We expect that a set of juxtaposed estimated horizontal depth slices defines the geometry of a 3D source

x

z

The horizontal coordinates of the polygon vertices represent the

edges of horizontal depth slices of the 3D source.

MethodologyD

epth

Dep

th

y

jj yx ,

jj yx ,

The polygon vertices of each prism are described by polar coordinates

Methodology

x

z

Dep

thD

epth

y

with an arbitrary origin within the top of each prism.

Arbitrary origin

11 r ,( )22 r ,( )

33 r ,( )( 44 r , )

( 55 r , )

( 66 r , ) ( 77 r , ) ( 88 r , )

x

z

De

pth

De

pth

y

11 r ,( )22 r ,( )

33 r ,( )( 44 r , )

( 55 r , )

( 66 r , )( 77 r , ) ( 88 r , )

The vertical component of the gravity field produced by the kth prism at the

Methodology

),,,,,(k1kzdz

km,, iziy

ixf

k

1kz

i th observation point (xi, yi, zi) is given by (Plouff, 1976)

dz

T

M

k

M

kk

11][

TkkkM

kkyxrrm 001 ][

M )2(1

ig =

The gravity data produced by the set of L vertically stacked prisms at the i th observation

point (xi, yi, zi) is given by

Methodology

L

k

kf1

ig = ),,,,,( 1kzdz

km,, iziy

ix

k

x

z

Dep

thD

epth

y

MethodologyTHE INVERSE PROBLEM

Estimate the radii associated with polygon vertices

x

z

Dep

thD

epth

y

Arbitrary origin

1r2r3

r4r

5r6r

7r 8r

and the horizontal Cartesian coordinates of the origin.

( xo , yo )

y


MethodologyBy estimating the radii associated with polygon vertices and the

horizontal Cartesian coordinates of the arbitrary origin from gravity data,

x

z3D Source

Dep

thD

epth

y

(xo , yo)

we retrieve a set of vertically stacked prisms

The Inverse Problem

)(mParameter vector The data-misfit function

The constrained inversion obtains the geometry of 3D source by minimizing :

)(m

2

)( mggo

2

The constrained function

)(m

=

The constrained function (m) is defined as a sum of several

constraints:

Smoothness constraint on the adjacent radii defining the horizontal section of each vertical prism

The Inverse Problem

The first-order Tikhonov regularization on the radii of horizontally adjacent prisms

x

z

Dep

thD

epth

y2r3

r4r

5r6r

7r 8r1r

This constraint favors solutions composed by vertical prisms defined by

approximately circular cross-sections.

1r 2r

kjr

1kjr

Inverse ProblemSmoothness constraint on the adjacent radii of the

verticallyadjacent prisms

x

z

Dep

thD

epth

y

The first-order Tikhonov regularization on the radii of vertically adjacent prisms

This constraint favors solutions with a vertically cylindrical shape.

jr jrk k+1

Inverse ProblemSmoothness constraint on the horizontal position of the arbitrary origins of the vertically adjacent

prisms

It imposes smooth horizontal displacement between all vertically adjacent prisms.

x

z

Dep

thD

epth

y

),( 00kk yx

),( 00k +1k+1 yx

),( 00kk yx ),( 00

k +1k+1 yx

Inverse ProblemThe estimation of the depth of the bottom of the

geologic body

x

z

Dep

thD

epth

y

dz

zo

The interpretation model implicitly defines the maximum depth to the bottom (zmax) of the estimated body by

1

.

.

L

L . dzz max o z

zmax

How do we choose zmax ?Do the gravity data have resolution to retrieve the 3D source?

Inverse ProblemThe depth-to-the-bottom estimate of the geologic

body1) We assign a small value to zmax, setting up the first interpretation model.

gObserved gravity data

zo

zmax 1

z

x

2) We run our inversion method to estimate a stable solution

Fitted gravity data

s

mt

zmax 1

mt X s-curve

5) We repeat this procedure for increasingly larger

values of zmax of the interpretation model

zmax 2

g Observed gravity data

zo

zmax 2

z

x

Fitted gravity data


zo

zmax 3

z

x

Fitted gravity data

zmax 3


zo

zmax 4z

x

Fitted gravity data

4) We plot a point of the mt X s-curve

3) We compute the L1-norm of the data misfit (s) and the

estimated total-anomalous mass ( mt )

zmax 4

Optimum depth-to-bottom estimate

Inverse ProblemDo the gravity data have resolution to retrieve the

3D source?

Correct depth-to-bottom estimate

mt X s-curve

L1-norm of the data misfit

Est

imat

ed t

otal

-ano

mal

ous

mas

s

s (mGal)

mt


zo

zz

x

Fitted gravity data

The gravity data are able to resolve the source’s bottom.

z

Inverse ProblemDo the gravity data have resolution to retrieve the

3D source?

Minimum depth-to-bottom estimate

s (mGal)

mt

mt X s-curve

L1-norm of the data misfit

Est

imat

ed t

otal

-ano

mal

ous

mas

s


zo

z

z

x

Fitted gravity data

The gravity data are unable to resolve the source’s bottom.

z

INVERSION OF

SYNTHETIC GRAVITY DATA

Synthetic TestsTwo outcropping dipping bodies with density contrast of 0.5 g/cm³.

Simulated shallow-bottomed body

Simulated deep-bottomed body

Maximum bottom depth of 3 km

Maximum bottom depth of 9 km

Synthetic TestsShallow-bottomed dipping body (true depth to the bottom is 3 km)

The estimated total-anomalous mass ( mt ) x the L1-norm of the data misfit (s)

s (mGal) L1-norm of the data misfit

mt (

kg x

10

12 )

Est

ima

ted

tota

l-an

om

alo

us

ma

ss

mt X s - curve

0 0.2 0.40

5

10

zmax = 1.0 km

zmax = 11.0 km

zmax = 2.0 km zmax = 3.0 km

s (mGal) L1-norm of the data misfit m

t (kg

x 1

012)

Est

ima

ted

tot

al-a

nom

alou

s m

ass

0 0.2 0.4

zmax = 1.0 km

zmax = 11.0 km

zmax = 2.0 km zmax = 3.0 km

0

5

10


Dep

th (

km)

y(km) x(km)

Initial guessTrue Body

mt X s - curve


True Body Estimated body

Dep

th (

km)

x(km)y(km)

Synthetic TestsDeep-bottomed dipping body (true depth to the bottom is 9.0 km)



mt (

kg x

10

12 )

Est

ima

ted

tota

l-an

om

alo

us

ma

ss

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

5

10

15

20

25zmax = 6.0 km

mt X s - curve

True depth to the bottom

Lower bound estimate of the depth to the bottom

zmax = 9.0 km

De

pth

(k

m)

x(km)y(km)

True Body


0

4.9

9.9

y(km) x(km)

0

4.9

9.9

By assuming two interpretation models with maximum bottom depths of 6 km and 9 km

Initial guess (6 km) Initial guess (9 km)True Body

De

pth

(k

m)

Lower bound estimate of the depth to the bottom (6 km) True depth to the bottom (9 km)

6 km9 km


By assuming two interpretation models with maximum bottom depths of 6 km and 9 km

0

4.9

9.9

De

pth

(k

m)

y(km) x(km)

True Body

x(km)y(km)

0

4.9

9.9

True BodyEstimated bodyEstimated body

6 km6 km

Lower bound estimate of the depth to the bottom (6 km) True depth to the bottom (9 km)

9 km6 km

INVERSION OF

REAL GRAVITY DATA

Application to Real Data

Real gravity-data set over greenstone rocks in Matsitama, Botswana.

Study area

Simplified geologic map of greenstone rocks in Matsitama, Botswana.

(see Reeves, 1985)


20

20 40 60

40

60

80 100 120

80

100

120

140

160N

orth

ing

(km

)

Easting (km)

Gravity-data set over greenstone rocks in Matsitama (Botswana).


A

B




mt (

kg x

10

12 )

Est

ima

ted

tota

l-an

om

alo

us

ma

ss

mt X s - curve

zmax = 3.0 km

zmax = 10 km zmax = 8 km

Estimated greenstone rocks in Matsitama (Botswana).


Dep

th (

km)

Estimated BodyInitial guess

Northing (km)

N

Easting (km)


Northing (km) Easting (km)

Northing (km)Easting (km)

Estimated greenstone rocks

in Matsitama (Botswana).


Nor

thin

g (k

m)

Easting (km)

The fitted gravity anomaly produced by the estimated greenstone rock in Matisitama

Conclusions

Conclusions The proposed gravity-inversion method

• Estimates the 3D geometry of isolated source

• Introduces homogeneity and compactness constraints via the interpretation model

• To reduce the class of possible solutions, we use a criterion based on the curve of

the estimated total-anomalous mass (mt) versus data-misfit measure (s).

• The solution depends on the maximum depth to

the bottom assumed for the interpretation model.

•The correct depth-to-bottom estimate of the

source is obtained if the minimum of s on the

mt × s curve is well defined

Otherwise this criterion provides just a lower

bound estimate of the source’s depth to the

bottom.

Dep

th (

km)


mt X s - curve

mt (

kg x

1012

)

Est

ima

ted

to

tal-a

no

ma

lou

s m

ass

0

zmax = 3.0 km

0

5

10

0.2 0.4

0

4.9

9.9

Dep

th (

km)

mt (

kg x

1012

)

Est

imat

ed to

tal-a

nom

alou

s m

ass

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

5

10

15

20

25


zmax = 6 km Lower bound estimate of the

depth to the bottom

mt X s - curve

Thank you

for your attention

Documents

Radial gravity inversion constrained by total anomalous mass excess for retrieving 3D bodies Vanderlei Coelho Oliveira Junior Valéria C. F. Barbosa Observatório