robust 3d gravity gradient inversion by planting anomalous densities

Robust 3D gravity gradient inversionby planting anomalous densities

Leonardo Uieda

Valéria C. F. Barbosa

September, 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

Inspired by René (1986)

Outline

Inverse Problem Planting Algorithm

Synthetic Data Real Data

Forward Problem

Inspired by René (1986)

Outline

Quadrilátero Ferrífero, Brazil

Forward problem

Observed gαβ

gαβ

Observed gαβ

gαβ

Observed gαβ

gαβ

Anomalous density

Observed gαβ

gαβ

Anomalous densityWant to model this

Interpretative model

Interpretative model

Right rectangular prism

Δρ=p j

   Prisms with not shown

p j=0

p=[p1

p2

⋮

pM]

Prisms with not shown

p j=0

Predicted gαβ

dαβ

p=[p1

p2

⋮

pM]

Prisms with not shown

p j=0

Predicted gαβ

dαβ

p=[p1

p2

⋮

pM]

dαβ=∑j=1

M

p j a jαβ

Prisms with not shown

p j=0

Predicted gαβ

dαβ

p=[p1

p2

⋮

pM]

dαβ=∑j=1

M

p j a jαβ

Contribution of jth prism

Prisms with not shown

p j=0

d xx

d xy

d xz

d yy

d yz

d zz

More components:

d

d xx

d xy

d xz

d yy

d yz

d zz

More components:

d

d xx

d xy

d xz

d yy

d yz

d zz

=∑j=1

M

p j a j

More components:

d

d xx

d xy

d xz

d yy

d yz

d zz

=∑j=1

M

p j a j =A p

More components:

d

d xx

d xy

d xz

d yy

d yz

d zz

=∑j=1

M

p j a j =A p

Jacobian (sensitivity) matrix

More components:

d

d xx

d xy

d xz

d yy

d yz

d zz

=∑j=1

M

p j a j =A p

Column vector of

More components:

A

Forward problem:

p dd=∑j=1

M

p j a j

p̂ g?Inverse problem:

Inverse problem

Minimize difference between andg d

r=g−dResidual vector

Minimize difference between andg d

r=g−dResidual vector

Data­misfit function:

ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12

Minimize difference between andg d

r=g−dResidual vector

Data­misfit function:

ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12

 ℓ2­norm of r

Minimize difference between andg d

r=g−dResidual vector

Data­misfit function:

ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12

 ℓ2­norm of r

Least­squares fit

Minimize difference between andg d

r=g−dResidual vector

Data­misfit function:

ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12

 ℓ2­norm of r

Least­squares fit

ϕ( p)=∥r∥1=∑i=1

N

∣gi−d i∣

 ℓ1­norm of r

Minimize difference between andg d

r=g−dResidual vector

Data­misfit function:

ϕ( p)=∥r∥2=(∑i=1

N

(gi−d i)2)

12

 ℓ2­norm of r

Least­squares fit

ϕ( p)=∥r∥1=∑i=1

N

∣gi−d i∣

 ℓ1­norm of r

Robust 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:

1. Compact no holes inside

2. Concentrated around “seeds”

Constraints:

1. Compact no holes inside

2. Concentrated around “seeds”

● User­specified prisms● Given density contrasts● Any # of ≠ density contrasts

ρs

Constraints:

1. Compact no holes inside

2. Concentrated around “seeds”

● User­specified prisms● Given density contrasts

3. Only 

● Any # of ≠ density contrasts

orp j=0 p j=ρs

ρs

Constraints:

1. Compact no holes inside

2. Concentrated around “seeds”

● User­specified prisms● Given density contrasts

3. Only 

● Any # of ≠ density contrasts

orp j=0 p j=ρs

ρs

4.  of closest seed p j=ρs

Well­posed problem: Minimize goal function

Γ( p)=ϕ( p)+μθ( p)

Well­posed problem: Minimize goal function

Γ( p)=ϕ( p)+μθ( p)

Data­misfit function

Well­posed problem: Minimize goal function

Γ( p)=ϕ( p)+μθ( p)

(Tradeoff between fit and regularization)Regularizing parameter

Well­posed problem: Minimize goal function

Γ( p)=ϕ( p)+μθ( p)

Regularizing function

θ( p)=∑j=1

M p j

p j+ϵl j

β

Well­posed problem: Minimize goal function

Γ( p)=ϕ( p)+μθ( p)

Regularizing function

θ( p)=∑j=1

M p j

p j+ϵl j

βSimilar to Silva Dias et al. (2009)

Well­posed problem: Minimize goal function

Γ( p)=ϕ( p)+μθ( p)

Regularizing function

θ( p)=∑j=1

M p j

p j+ϵl j

βSimilar to Silva Dias et al. (2009)

Distance between jth prism and seed

Well­posed problem: Minimize goal function

Γ( p)=ϕ( p)+μθ( p)

Regularizing function

θ( p)=∑j=1

M p j

p j+ϵl j

βSimilar to Silva Dias et al. (2009)

Distance between jth prism and seed

Imposes:

● Compactness ● Concentration around seeds

Constraints:

1. Compact

2. Concentrated around “seeds”Regularization

3. Only  orp j=0 p j=ρs

4.  of closest seed p j=ρs

Constraints:

1. Compact

2. Concentrated around “seeds”Regularization

Algorithm3. Only  orp j=0 p j=ρs

4.  of closest seed p j=ρs Based on René (1986)

Planting Algorithm

Setup: g = observed data

Setup:Define interpretative model

Interpretative model

g = observed data

Setup:Define interpretative model

All parameters zero

Interpretative model

g = observed data

Setup:

seedsN S

Define interpretative model

All parameters zero

Interpretative model

g = observed data

Setup:

seedsN S

Define interpretative model

All parameters zero

Include seeds

Prisms with not shown

p j=0

g = observed data

Setup:

seedsN S

Define interpretative model

All parameters zero

Include seeds

Compute initial residuals

r(0)=g−d(0)

Prisms with not shown

p j=0

g = observed data

d = predicted data

Setup:

seedsN S

Define interpretative model

All parameters zero

Include seeds

Compute initial residuals

r(0)=g−d(0)

Prisms with not shown

p j=0

g = observed data

Predicted by seeds

d = predicted data

Setup:

seedsN S

Define interpretative model

All parameters zero

Include seeds

Compute initial residuals

Prisms with not shown

p j=0

g = observed data

d = predicted data

r(0)=g−(∑s=1

N S

ρs a jS)

seedsN S

Define interpretative model

All parameters zero

Include seeds

Compute initial residuals

r(0)=g−(∑s=1

N S

ρs a jS)

Prisms with not shown

g = observed data

d = predicted data

NeighborsFind neighbors of seeds

p j=0

Setup:

    Prisms with not shown

Try accretion to sth seed:

p j=0

Growth:

    Prisms with not shown

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

Choose neighbor:

p j=0

Growth:

    Prisms with not shown

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Choose neighbor:

p j=0

Growth:

(New elements)

j

    Prisms with not shown

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Choose neighbor:

p j=0

Growth:

(New elements)

j

Update residuals

r(new)=r(old )

− p j a j

    Prisms with not shown

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Choose neighbor:

p j=0

Growth:

(New elements)

j

Update residuals

r(new)=r(old )

− p j a j

Contribution of j

    Prisms with not shown

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Choose neighbor:

p j=0

Growth:

(New elements)

j

Update residuals

r(new)=r(old )

− p j a j

Variable sizes

None found = no accretion

    Prisms with not shown

None found = no accretion

N S

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

p j=0

Growth:

(New elements)

    Prisms with not shown

None found = no accretion

N S

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

p j=0

Growth:

j

(New elements)

    Prisms with not shown

None found = no accretion

N S

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

At least one seed grow?

p j=0

Growth:

j

(New elements)

    Prisms with not shown

None found = no accretion

N S

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

p j=ρsj = chosen

Update residuals

r(new)=r(old )

− p j a j

Choose neighbor:

At least one seed grow?

Yes

p j=0

Growth:

j

(New elements)

    Prisms with not shown

None found = no accretion

N S

Try accretion to sth seed:

1. Reduce data misfit

2. Smallest goal function

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:

Done!

j

(New elements)

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

ρs a jS) r(new)=r(old )

− p j a j

Initial residual Update residual vector

No matrix multiplication (only vector +)

No matrix multiplication (only vector +)

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

Only need some columns of A

No matrix multiplication (only vector +)

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

Only need some columns of A

Calculate only when needed

No matrix multiplication (only vector +)

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

Only need some columns of A

Calculate only when needed & delete after update

No matrix multiplication (only vector +)

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

Only need some columns of A

Calculate only when needed

Lazy evaluation

& delete after update

Advantages:

Compact & non­smooth

Any number of sources

Any number of different density contrasts

No large equation system

Search limited to neighbors

Advantages:

Compact & non­smooth

Any number of sources

Any number of different density contrasts

No large equation system

Search limited to neighbors

No matrix multiplication (only vector +)

Lazy evaluation of Jacobian

Advantages:

Compact & non­smooth

Any number of sources

Any number of different density contrasts

No large equation system

Search limited to neighbors

No matrix multiplication (only vector +)

Lazy evaluation of Jacobian

Fast inversion + low memory usage

Synthetic Data

Data set:

● 3 components

● 51 x 51 points

● 2601 points/component

● 7803 measurements

● 5 Eötvös noise

Model:

Model: ● 11 prisms

Model: ● 11 prisms ● 4 outcropping

Model: ● 11 prisms ● 4 outcropping

Model: ● 11 prisms ● 4 outcropping

● Strongly interfering effects

● Strongly interfering effects

● What if only interested in these?

● Common scenario

● Common scenario

● May not have prior information

● Density contrast

● Approximate depth

● Common scenario

● May not have prior information

● Density contrast

● Approximate depth

● No way to provide seeds

● Common scenario

● May not have prior information

● Density contrast

● Approximate depth

● No way to provide seeds

● Difficult to isolate effect of targets

Robust procedure:

Robust procedure:● Seeds only for targets

Robust procedure:● Seeds only for targets

Robust procedure:● Seeds only for targets

●  ℓ1­norm to “ignore” non­targeted

Robust procedure:● Seeds only for targets

●  ℓ1­norm to “ignore” non­targeted

Inversion: ● 13 seeds ● 7,803 data

Inversion: ● 13 seeds ● 7,803 data

Inversion: ● 13 seeds ● 7,803 data

Inversion: ● 13 seeds ● 7,803 data

Inversion: ● 13 seeds ● 7,803 data

Inversion: ● 13 seeds ● 7,803 data ● 37,500 prisms

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

Only prisms with zerodensity contrast not shown

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

Only prisms with zerodensity contrast not shown

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

Only prisms with zerodensity contrast not shown

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

Only prisms with zerodensity contrast not shown

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

Only prisms with zerodensity contrast not shown

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

● Recover shape of targets

Only prisms with zerodensity contrast not shown

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

● Recover shape of targets

● Total time = 2.2 minutes (on laptop) Only prisms with zerodensity contrast not shown

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

Predicted data in contours

Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds

Predicted data in contours

Effect of true targeted sources

Real Data

Data:

● 3 components

● FTG survey

● Quadrilátero Ferrífero, Brazil

Data:

● 3 components

● FTG survey

● Quadrilátero Ferrífero, Brazil

Targets:

● Iron ore bodies

● BIFs of Cauê Formation

Data:

● 3 components

● FTG survey

● Quadrilátero Ferrífero, Brazil

Targets:

● Iron ore bodies

● BIFs of Cauê Formation

Data:

Seeds for iron ore:

● Density contrast 1.0 g/cm3

● Depth 200 m

Inversion:O

bser

ved

Pre

dict

ed● 46 seeds ● 13,746 data

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Only prisms with zerodensity contrast not shown

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Only prisms with zerodensity contrast not shown

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Only prisms with zerodensity contrast not shown

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

● Agree with previous interpretations

(Martinez et al., 2010)

Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms

● Agree with previous interpretations

(Martinez et al., 2010)

● Total time = 14 minutes

(on laptop)

Conclusions

● New 3D gravity gradient inversion● Multiple sources● Interfering gravitational effects● Non­targeted sources● No matrix multiplications● No linear systems● Lazy evaluation of Jacobian matrix

Conclusions

● Estimates geometry● Given density contrasts● Ideal for:

● Sharp contacts● Well­constrained physical properties

– Ore bodies– Intrusive rocks– Salt domes

Conclusions

Thank you