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3D gravity inversion by planting anomalous densities

Leonardo Uieda and Valéria C. F. Barbosa

August, 2011

Observatório Nacional

Outline

Forward Problem

Outline

Inverse ProblemForward Problem

Outline

Inverse Problem Planting AlgorithmForward Problem

Inspired by René (1986)

Outline

Inverse Problem Planting Algorithm

Synthetic Data

Forward Problem


Outline

Inverse Problem Planting Algorithm

Synthetic Data Real Data

Forward Problem


Outline

Forward problem

Surface of the Earth

Observations of gz


Observations of gz

Group in a vector:

g=[g1

g2

⋮gN

]N×1

= observed datag


= observed datag

= observed datag

Assume caused by anomalous sources

= observed datag

Assume caused by anomalous sources

ΔρDensity contrast =

Parametrize the gravitational effect


Linearize


Discretize intoM elements

Linearize

Interpretative model

Right rectangular prisms



Homogeneous density contrast

jth element

Linearize

(Nagy et al., 2000)


p j

Arrange M density contrasts in a vector:



p=[p1

p2

⋮pM

]M×1

Parameter vector

Linearize




p j=Δρ

Prisms with not shown

p j=0



g≈d


p j=0

p j=Δρ



g≈dPredicted data


p j=0

p j=Δρ



Gravitational effect is linear

d=∑j=1

M

p j a j

g≈dPredicted data


p j=0

p j=Δρ




d=∑j=1

M

p j a j

Density contrast of jth prism

g≈dPredicted data


p j=0

p j=Δρ




d=∑j=1

M

p j a j

g≈dPredicted data

Effect of prism with unit density Prisms with not shown

p j=0

p j=Δρ




g≈dPredicted data

d=∑j=1

M

p j a j=A p


p j=0

p j=Δρ




g≈dPredicted data

Parameter vector

d=∑j=1

M

p j a j=A p


p j=0

p j=Δρ




g≈dPredicted data

Jacobian (sensitivity) matrix

d=∑j=1

M

p j a j=A p


p j=0

p j=Δρ




g≈dPredicted data

Column vector of A

d=∑j=1

M

p j a j=A p


p j=0

p j=Δρ

Solved forward problem:

p dd=∑j=1

M

p j a j

p̂ g?How to do the inverse?

Inverse problem

Minimize difference between

Minimize difference between g

Observed data

Minimize difference between andg d

Predicted data


r=g−d


r=g−d

Residual vector


r=g−d

Residual vector

Datamisfit function: ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12


r=g−d

Residual vector

Datamisfit function: ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12

ℓ2norm of r

Leastsquares fit

illposed problem

nonexistent

nonunique

nonstable

illposed problem

nonexistent

nonunique

nonstable

constraints

illposed problem

nonexistent

nonunique

nonstable

wellposed problem

exist

unique

stable

constraints

Constraints:

1. Compact

Constraints:

1. Compact no holes inside

Constraints:


2. Concentrated around “seeds”

Constraints:



Similar to René (1986)

Constraints:



● Userspecified prisms


Constraints:



● Userspecified prisms● Given density contrasts ρs


Constraints:



● Userspecified prisms● Given density contrasts● Any n° of ≠ density contrasts

ρs


Not like René (1986)

Constraints:



● Userspecified prisms● Given density contrasts

3. Only

● Any n° of ≠ density contrasts

orp j=0 p j=ρs

ρs



Constraints:



● Userspecified prisms● Given density contrasts

3. Only

● Any n° of ≠ density contrasts

orp j=0 p j=ρs

ρs

4. of closest seed p j=ρs



illposed problem wellposed problemconstraints

ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12

Minimize data misfit Minimize goal function

Γ( p)=ϕ( p)+μθ( p)


ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12


Γ( p)=ϕ( p)+μθ( p)

Regularizing parameter


ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12


Γ( p)=ϕ( p)+μθ( p)

Regularizing function


ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12


Γ( p)=ϕ( p)+μθ( p)

Regularizing function

μ = tradeoff between fit and regularization

Regularization:

θ( p)=∑j=1

M p j

p j+ϵl j

β

Γ( p)=ϕ( p)+μθ( p)

Regularization:

θ( p)=∑j=1

M p j

p j+ϵl j

β

Γ( p)=ϕ( p)+μθ( p)Similar to

Silva Dias et al. (2009)

Regularization:

θ( p)=∑j=1

M p j

p j+ϵl jβ

Γ( p)=ϕ( p)+μθ( p)

ϵ = avoid singularity

l j = distance between jth prism and seed

β = how much compactness (3 to 7)

Similar to Silva Dias et al. (2009)

Regularization:

θ( p)=∑j=1

M p j

p j+ϵl jβ

Γ( p)=ϕ( p)+μθ( p)

For p j≠0 :

Regularization:

θ( p)=∑j=1

M p j

p j+ϵl j

β

Γ( p)=ϕ( p)+μθ( p)

distance from seeds

For p j≠0 :

Regularization:

θ( p)=∑j=1

M p j

p j+ϵl j

β

Γ( p)=ϕ( p)+μθ( p)

distance from seeds regularizing function

For p j≠0 :

Regularization:

θ( p)=∑j=1

M p j

p j+ϵl j

β

Γ( p)=ϕ( p)+μθ( p)

distance from seeds regularizing function

Imposes:

● Compactness ● Concentration around seeds

For p j≠0 :

Constraints:

1. Compact


3. Only orp j=0 p j=Δρs

4. of closest seed p j=Δρs

Regularization

Constraints:

1. Compact


3. Only orp j=0 p j=Δρs

4. of closest seed p j=Δρs

Regularization

Algorithm

Planting Algorithm

Based on René (1986)

Overview:


Start with seeds

Overview:


Start with seeds

Overview:


Start with seeds known density contrast & position

Overview:


Start with seeds known density contrast & position

All other parameters set to 0

Overview:


Start with seeds


Iteratively grow

Overview:

known density contrast & position


Start with seeds


Iteratively grow add neighbor of seed

Overview:



Start with seeds



accretion

Overview:



Start with seeds



accretion

Controlled by goal function and data misfit function

Overview:

Γ( p)=ϕ( p)+μθ( p) ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12


Algorithm:

Algorithm:Define interpretative model



g = observed data



All parameters zero

g = observed data


Algorithm:

seedsN S

Define interpretative model

All parameters zero

g = observed data


Algorithm:

seedsN S


All parameters zero

Include seeds


p j=0

Seeds

Algorithm:

seedsN S


All parameters zero

Include seeds

Compute initial residuals

r(0)=g−d(0)


p j=0

Algorithm:

Residual vector

seedsN S


All parameters zero

Include seeds


r(0)=g−d(0)


p j=0

Algorithm:

Observed data

seedsN S


All parameters zero

Include seeds


r(0)=g−d(0)

g = observed data


p j=0

Algorithm:

Predicted by seeds

seedsN S


All parameters zero

Include seeds


r(0)=g−d(0)

g = observed data

d = predicted data


p j=0

Algorithm:

seedsN S


All parameters zero

Include seeds


r(0)=g−d(0)

g = observed data

d=∑j=0

M

p j a j

d = predicted data


p j=0

Algorithm:

seedsN S


All parameters zero

Include seeds


r(0)=g−d(0)

g = observed data

d=∑j=0

M

p j a j

Many=0

d = predicted data


p j=0

Algorithm:

seedsN S


All parameters zero

Include seeds


r(0)=g−d(0)

g = observed data

d=∑s=1

N S

ρs a jS

d = predicted data


p j=0

Algorithm:

seedsN S


All parameters zero

Include seeds


r(0)=g−(∑s=1

N S

ρs a jS)


g = observed data

d = predicted data

p j=0

Algorithm:

Density contrastof sth seed

seedsN S


All parameters zero

Include seeds


r(0)=g−(∑s=1

N S

ρs a jS)


g = observed data

d = predicted data

p j=0

seedsN S


All parameters zero

Include seeds


r(0)=g−(∑s=1

N S

ρs a jS)Column vector

of APrisms with not shown

g = observed data

d = predicted data

p j=0

seedsN S


All parameters zero

Include seeds


r(0)=g−(∑s=1

N S

ρs a jS)


g = observed data

d = predicted data

NeighborsFind neighbors of seeds

p j=0


Growth:

p j=0


Growth:

Try accretion to sth seed:

p j=0


Growth:


Choose neighbor:

p j=0


Growth:


Choose neighbor:

1. Reduce data misfit

ϕ( p)=∥r∥2

p j=0


Growth:


Choose neighbor:


2. Smallest goal function

ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=0


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

j = chosen

j

p j=0


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

j

p j=0

(New elements)


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

j

p j=0

(New elements)new predicted data


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

Update residuals

r(new)=g−d(new)

p j=0


j


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

Update residuals

r(new)=g−d(new)

p j=0


j

d(old )+ effect of j


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

Update residuals

r(new)=g−d(new)

p j=0


j


∑s=1

N S

ρs a j Sp j a j+


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

Update residuals

r(new)=g−d(new)

p j=0


j


∑s=1

N S

ρs a j Sp j a j+


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

Update residuals

r(new)=g−∑

s=1

N S

ρs a j S− p j a j

p j=0


j


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

Update residuals

r(new)=g−∑

s=1

N S

ρs a j S− p j a j

p j=0


j{

r(0)


Growth:


Choose neighbor:



ϕ( p)=∥r∥2

Γ( p)=ϕ( p)+μθ( p)

p j=ρsj = chosen

Update residuals

p j=0


jr(new)

=r(old )− p j a j


Growth:

None found = no accretion




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

p j=0


Growth:





p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

Variable sizes

p j=0


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

p j=0


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

p j=0

(New elements)

j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

At least one seed grow?

Yes No

p j=0


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0

(New elements)

j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0

(New elements)

j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0

j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0

j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0

j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0

j


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

p j=0


Growth:


N S




p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:


Yes No

Done!p j=0

Advantages:

Compact & nonsmooth

Any number of sources

Any number of different density contrasts

No large equation system

Search limited to neighbors

Remember equations:

r(0)=g−(∑

s=1

N S

ρs a jS) r(new)=r(old )

− p j a j

Initial residual Update residual vector

Remember equations:

r(0)=g−(∑

s=1

N S


− p j a j


No matrix multiplication (only vector +)


Remember equations:

r(0)=g−(∑

s=1

N S


− p j a j


Only need some columns of A


Remember equations:

r(0)=g−(∑

s=1

N S


− p j a j



Calculate only when needed


Remember equations:

r(0)=g−(∑

s=1

N S


− p j a j



Calculate only when needed & delete after update


Remember equations:

r(0)=g−(∑

s=1

N S


− p j a j



Calculate only when needed

Lazy evaluation

& delete after update

Advantages:

Compact & nonsmooth





Advantages:

Compact & nonsmooth






Lazy evaluation of Jacobian

Advantages:

Compact & nonsmooth






Lazy evaluation of Jacobian

Fast inversion + low memory usage

Synthetic Data

● Sources = 1 km X 1 km X 1 km

Δρ=0.5 g/ cm3


Δρ=1.0 g/ cm3

Δρ=0.5 g/ cm3



Depth=0.8 km


Depth=1.6 km

Depth=0.8 km

● Sources = 1 km X 1 km X 1 km ● Data set = 375 observations

Depth=1.6 km

Depth=0.8 km


● Area = 5 km X 3 km

● Data set = 375 observations

Depth=1.6 km

Depth=0.8 km

● 0.05 mGal Gaussian noise


● Area = 5 km X 3 km

● Data set = 375 observations

Depth=1.6 km

Depth=0.8 km

● Interpretative model = 151,875 prisms

● Prisms = 66.7 m X 66.7 m X 66.7 m

● Used 2 seeds

● Used 2 seeds ● Placed in center of sources

● Used 2 seeds

● With corresponding density contrasts

● Placed in center of sources

Δρ=0.5 g/ cm3

● Used 2 seeds



Δρ=1.0 g/cm3

Δρ=0.5 g/ cm3

● Used 2 seeds



Inversion result

Inversion result

Inversion result

● compact● concentrated around seeds● recover correct geometry of sources

Predicted data

Inversion result


Predicted dataObserved data

Inversion result



Inversion result

● compact● concentrated around seeds

● fits observations

● recover correct geometry of sources


On laptop with 2.0 GHz

● 375 data● 151,875 prisms● Total time≈4.4 min

Real Data

After Carminatti et al. (2003)

Cana Brava complex (CBC)


Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)



● Outcropping

● North of Goiás

● Tocantins Province

● Amazonian & São Francisco cratons



Gravimetric data:

● 132 observations

● Residual Bouguer

● Max 45 mGal



Previous interpretation:


● Carminatti et al. (2003)

● PVSS

● CVC● Max depth

Δρ=0.27 g/cm3

Δρ=0.39 g /cm3

≈6 km


Previous interpretation:


● Carminatti et al. (2003)

● PVSS

● CVC● Max depth

Δρ=0.27 g/cm3

Δρ=0.39 g /cm3

≈6 kmTest this hypothesis

Assign seeds

Green:z=0 km

Δρ=0.27 g /cm3

Blue:z=2 km

Δρ=0.27g /cm3

Red:z=0 km

Δρ=0.39 g /cm3

Total = 269

Assign seeds


Size: 120 km X 50 km X 11 km

480,000 prisms

Prism size: 500 m X 500 m X 575 m

Inversion result

Inversion result

Inversion result


Inversion result

Inversion result

Inversion result

Inversion result

Inversion result

Inversion result

Inversion result

≈6 kmMax depth

Agree with previous interpretation

Compact

Fits observations

Inversion result

On laptop with 2.0 GHz

● 132 data● 480,00 prisms● Total time≈3.75 min

Conclusions

● New 3D gravity inversion● Multiple sources● Interfering gravitational effects● Abrupt densitycontrast distribution● No matrix multiplication● No need to solve large linear systems● Ideal for: ore bodies, intrusions, salt domes, etc

Conclusions

● Developed for gravity gradients

● Presented at EAGE 2011 preliminary results

● To be presented at SEG 2011:

● Final results

● Robust method to handle nontargeted sources

Previous and future work

Thank you