robust 3d gravity gradient inversion by planting anomalous densities
DESCRIPTION
Paper presented at the 2011 SEG Annual MeetingTRANSCRIPT
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Robust 3D gravity gradient inversionby planting anomalous densities
Leonardo Uieda
Valéria C. F. Barbosa
September, 2011
Observatório Nacional
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Outline
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Forward Problem
Outline
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Inverse ProblemForward Problem
Outline
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Inverse Problem Planting AlgorithmForward Problem
Inspired by René (1986)
Outline
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Inverse Problem Planting Algorithm
Synthetic Data
Forward Problem
Inspired by René (1986)
Outline
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Inverse Problem Planting Algorithm
Synthetic Data Real Data
Forward Problem
Inspired by René (1986)
Outline
Quadrilátero Ferrífero, Brazil
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Forward problem
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Observed gαβ
gαβ
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Observed gαβ
gαβ
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Observed gαβ
gαβ
Anomalous density
![Page 12: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/12.jpg)
Observed gαβ
gαβ
Anomalous densityWant to model this
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Interpretative model
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Interpretative model
Right rectangular prism
Δρ=p j
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Prisms with not shown
p j=0
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p=[p1
p2
⋮
pM]
Prisms with not shown
p j=0
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Predicted gαβ
dαβ
p=[p1
p2
⋮
pM]
Prisms with not shown
p j=0
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Predicted gαβ
dαβ
p=[p1
p2
⋮
pM]
dαβ=∑j=1
M
p j a jαβ
Prisms with not shown
p j=0
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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
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d xx
d xy
d xz
d yy
d yz
d zz
More components:
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d
d xx
d xy
d xz
d yy
d yz
d zz
More components:
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d
d xx
d xy
d xz
d yy
d yz
d zz
=∑j=1
M
p j a j
More components:
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d
d xx
d xy
d xz
d yy
d yz
d zz
=∑j=1
M
p j a j =A p
More components:
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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:
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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
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Forward problem:
p dd=∑j=1
M
p j a j
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p̂ g?Inverse problem:
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Inverse problem
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Minimize difference between andg d
r=g−dResidual vector
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Minimize difference between andg d
r=g−dResidual vector
Datamisfit function:
ϕ( p)=∥r∥2=(∑i=1
N
(gi−d i)2)
12
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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
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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
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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
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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
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illposed problem
nonexistent
nonunique
nonstable
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illposed problem
nonexistent
nonunique
nonstable
constraints
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illposed problem
nonexistent
nonunique
nonstable
wellposed problem
exist
unique
stable
constraints
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Constraints:
1. Compact
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Constraints:
1. Compact no holes inside
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Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
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Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● Userspecified prisms● Given density contrasts● Any # of ≠ density contrasts
ρs
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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
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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
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Wellposed problem: Minimize goal function
Γ( p)=ϕ( p)+μθ( p)
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Wellposed problem: Minimize goal function
Γ( p)=ϕ( p)+μθ( p)
Datamisfit function
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Wellposed problem: Minimize goal function
Γ( p)=ϕ( p)+μθ( p)
(Tradeoff between fit and regularization)Regularizing parameter
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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
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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)
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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
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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
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Constraints:
1. Compact
2. Concentrated around “seeds”Regularization
3. Only orp j=0 p j=ρs
4. of closest seed p j=ρs
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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)
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Planting Algorithm
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Setup: g = observed data
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Setup:Define interpretative model
Interpretative model
g = observed data
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Setup:Define interpretative model
All parameters zero
Interpretative model
g = observed data
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Setup:
seedsN S
Define interpretative model
All parameters zero
Interpretative model
g = observed data
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Setup:
seedsN S
Define interpretative model
All parameters zero
Include seeds
Prisms with not shown
p j=0
g = observed data
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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
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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
![Page 61: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/61.jpg)
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)
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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:
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Prisms with not shown
Try accretion to sth seed:
p j=0
Growth:
![Page 64: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/64.jpg)
Prisms with not shown
Try accretion to sth seed:
1. Reduce data misfit
2. Smallest goal function
Choose neighbor:
p j=0
Growth:
![Page 65: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/65.jpg)
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
![Page 66: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/66.jpg)
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
![Page 67: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/67.jpg)
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
![Page 68: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/68.jpg)
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
![Page 69: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/69.jpg)
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)
![Page 70: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/70.jpg)
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)
![Page 71: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/71.jpg)
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)
![Page 72: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/72.jpg)
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)
![Page 73: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/73.jpg)
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)
![Page 74: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/74.jpg)
Advantages:
Compact & nonsmooth
Any number of sources
Any number of different density contrasts
No large equation system
Search limited to neighbors
![Page 75: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/75.jpg)
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
![Page 76: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/76.jpg)
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 +)
![Page 77: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/77.jpg)
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
![Page 78: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/78.jpg)
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
![Page 79: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/79.jpg)
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
![Page 80: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/80.jpg)
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
![Page 81: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/81.jpg)
Advantages:
Compact & nonsmooth
Any number of sources
Any number of different density contrasts
No large equation system
Search limited to neighbors
![Page 82: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/82.jpg)
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
![Page 83: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/83.jpg)
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
![Page 84: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/84.jpg)
Synthetic Data
![Page 85: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/85.jpg)
Data set:
● 3 components
● 51 x 51 points
● 2601 points/component
● 7803 measurements
● 5 Eötvös noise
![Page 86: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/86.jpg)
Model:
![Page 87: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/87.jpg)
Model: ● 11 prisms
![Page 88: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/88.jpg)
Model: ● 11 prisms ● 4 outcropping
![Page 89: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/89.jpg)
Model: ● 11 prisms ● 4 outcropping
![Page 90: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/90.jpg)
Model: ● 11 prisms ● 4 outcropping
![Page 91: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/91.jpg)
● Strongly interfering effects
![Page 92: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/92.jpg)
● Strongly interfering effects
● What if only interested in these?
![Page 93: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/93.jpg)
● Common scenario
![Page 94: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/94.jpg)
● Common scenario
● May not have prior information
● Density contrast
● Approximate depth
![Page 95: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/95.jpg)
● Common scenario
● May not have prior information
● Density contrast
● Approximate depth
● No way to provide seeds
![Page 96: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/96.jpg)
● Common scenario
● May not have prior information
● Density contrast
● Approximate depth
● No way to provide seeds
● Difficult to isolate effect of targets
![Page 97: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/97.jpg)
Robust procedure:
![Page 98: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/98.jpg)
Robust procedure:● Seeds only for targets
![Page 99: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/99.jpg)
Robust procedure:● Seeds only for targets
![Page 100: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/100.jpg)
Robust procedure:● Seeds only for targets
● ℓ1norm to “ignore” nontargeted
![Page 101: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/101.jpg)
Robust procedure:● Seeds only for targets
● ℓ1norm to “ignore” nontargeted
![Page 102: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/102.jpg)
Inversion: ● 13 seeds ● 7,803 data
![Page 103: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/103.jpg)
Inversion: ● 13 seeds ● 7,803 data
![Page 104: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/104.jpg)
Inversion: ● 13 seeds ● 7,803 data
![Page 105: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/105.jpg)
Inversion: ● 13 seeds ● 7,803 data
![Page 106: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/106.jpg)
Inversion: ● 13 seeds ● 7,803 data
![Page 107: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/107.jpg)
Inversion: ● 13 seeds ● 7,803 data ● 37,500 prisms
![Page 108: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/108.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
Only prisms with zerodensity contrast not shown
![Page 109: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/109.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
Only prisms with zerodensity contrast not shown
![Page 110: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/110.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
Only prisms with zerodensity contrast not shown
![Page 111: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/111.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
Only prisms with zerodensity contrast not shown
![Page 112: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/112.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
Only prisms with zerodensity contrast not shown
![Page 113: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/113.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
● Recover shape of targets
Only prisms with zerodensity contrast not shown
![Page 114: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/114.jpg)
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
![Page 115: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/115.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
Predicted data in contours
![Page 116: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/116.jpg)
Inversion: ● 7,803 data ● 37,500 prisms● 13 seeds
Predicted data in contours
Effect of true targeted sources
![Page 117: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/117.jpg)
Real Data
![Page 118: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/118.jpg)
Data:
● 3 components
● FTG survey
● Quadrilátero Ferrífero, Brazil
![Page 119: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/119.jpg)
Data:
● 3 components
● FTG survey
● Quadrilátero Ferrífero, Brazil
Targets:
● Iron ore bodies
● BIFs of Cauê Formation
![Page 120: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/120.jpg)
Data:
● 3 components
● FTG survey
● Quadrilátero Ferrífero, Brazil
Targets:
● Iron ore bodies
● BIFs of Cauê Formation
![Page 121: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/121.jpg)
Data:
Seeds for iron ore:
● Density contrast 1.0 g/cm3
● Depth 200 m
![Page 122: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/122.jpg)
Inversion:O
bser
ved
Pre
dict
ed● 46 seeds ● 13,746 data
![Page 123: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/123.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 124: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/124.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 125: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/125.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 126: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/126.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 127: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/127.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 128: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/128.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 129: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/129.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
Only prisms with zerodensity contrast not shown
![Page 130: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/130.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
Only prisms with zerodensity contrast not shown
![Page 131: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/131.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
Only prisms with zerodensity contrast not shown
![Page 132: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/132.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 133: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/133.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
![Page 134: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/134.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
● Agree with previous interpretations
(Martinez et al., 2010)
![Page 135: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/135.jpg)
Inversion: ● 46 seeds ● 13,746 data ● 164,892 prisms
● Agree with previous interpretations
(Martinez et al., 2010)
● Total time = 14 minutes
(on laptop)
![Page 136: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/136.jpg)
Conclusions
![Page 137: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/137.jpg)
● New 3D gravity gradient inversion● Multiple sources● Interfering gravitational effects● Nontargeted sources● No matrix multiplications● No linear systems● Lazy evaluation of Jacobian matrix
Conclusions
![Page 138: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/138.jpg)
● Estimates geometry● Given density contrasts● Ideal for:
● Sharp contacts● Wellconstrained physical properties
– Ore bodies– Intrusive rocks– Salt domes
Conclusions
![Page 139: Robust 3D gravity gradient inversion by planting anomalous densities](https://reader033.vdocument.in/reader033/viewer/2022052412/558382b8d8b42ac6268b54d7/html5/thumbnails/139.jpg)
Thank you