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Paper presented at the 2011 SEG Annual Meeting
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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
