3d gravity inversion by planting anomalous densities
DESCRIPTION
Paper presented at the 2011 SBGf International Congress in Rio de Janeiro, Brazil. Abstract: This paper presents a novel gravity inversion method for estimating a 3D density-contrast distribution defined on a grid of prisms. Our method consists of an iterative algorithm that does not require the solution of a large equation system. Instead, the solution grows systematically around user-specified prismatic elements called “seeds”. Each seed can have a different density contrast, allowing the interpretation of multiple bodies with different density contrasts and interfering gravitational effects. The compactness of the solution around the seeds is imposed by means of a regularizing function. The solution grows by the accretion of neighboring prisms of the current solution. The prisms for the accretion are chosen by systematically searching the set of current neighboring prisms. Therefore, this approach allows that the columns of the Jacobian matrix be calculated on demand. This is a known technique from computer science called “lazy evaluation”, which greatly reduces the demand of computer memory and processing time. Test on synthetic data and on real data collected over the ultramafic Cana Brava complex, central Brazil, confirmed the ability of our method in detecting sharp and compact bodies.TRANSCRIPT
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
Inspired by René (1986)
Outline
Inverse Problem Planting Algorithm
Synthetic Data Real Data
Forward Problem
Inspired by René (1986)
Outline
Forward problem
Surface of the Earth
Observations of gz
Surface of the Earth
Observations of gz
Group in a vector:
g=[g1
g2
⋮gN
]N×1
= observed datag
Surface of the Earth
= observed datag
= observed datag
Assume caused by anomalous sources
= observed datag
Assume caused by anomalous sources
ΔρDensity contrast =
Parametrize the gravitational effect
Parametrize the gravitational effect
Linearize
Parametrize the gravitational effect
Discretize intoM elements
Linearize
Interpretative model
Right rectangular prisms
Parametrize the gravitational effect
Discretize intoM elements
Homogeneous density contrast
jth element
Linearize
(Nagy et al., 2000)
Interpretative model
p j
Arrange M density contrasts in a vector:
Parametrize the gravitational effect
Discretize intoM elements
p=[p1
p2
⋮pM
]M×1
Parameter vector
Linearize
Interpretative model
Discretize intoM elements
Parametrize the gravitational effect
p j=Δρ
Prisms with not shown
p j=0
Discretize intoM elements
Parametrize the gravitational effect
g≈d
Prisms with not shown
p j=0
p j=Δρ
Discretize intoM elements
Parametrize the gravitational effect
g≈dPredicted data
Prisms with not shown
p j=0
p j=Δρ
Discretize intoM elements
Parametrize the gravitational effect
Gravitational effect is linear
d=∑j=1
M
p j a j
g≈dPredicted data
Prisms with not shown
p j=0
p j=Δρ
Discretize intoM elements
Parametrize the gravitational effect
Gravitational effect is linear
d=∑j=1
M
p j a j
Density contrast of jth prism
g≈dPredicted data
Prisms with not shown
p j=0
p j=Δρ
Discretize intoM elements
Parametrize the gravitational effect
Gravitational effect is linear
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
Discretize intoM elements
Parametrize the gravitational effect
Gravitational effect is linear
g≈dPredicted data
d=∑j=1
M
p j a j=A p
Prisms with not shown
p j=0
p j=Δρ
Discretize intoM elements
Parametrize the gravitational effect
Gravitational effect is linear
g≈dPredicted data
Parameter vector
d=∑j=1
M
p j a j=A p
Prisms with not shown
p j=0
p j=Δρ
Discretize intoM elements
Parametrize the gravitational effect
Gravitational effect is linear
g≈dPredicted data
Jacobian (sensitivity) matrix
d=∑j=1
M
p j a j=A p
Prisms with not shown
p j=0
p j=Δρ
Discretize intoM elements
Parametrize the gravitational effect
Gravitational effect is linear
g≈dPredicted data
Column vector of A
d=∑j=1
M
p j a j=A p
Prisms with not shown
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
Minimize difference between andg d
r=g−d
Minimize difference between andg d
r=g−d
Residual vector
Minimize difference between andg d
r=g−d
Residual vector
Datamisfit function: ϕ( p)=∥r∥2=(∑i=1
N
(gi−d i)2)
12
Minimize difference between andg d
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:
1. Compact no holes inside
2. Concentrated around “seeds”
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
Similar to René (1986)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● Userspecified prisms
Similar to René (1986)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● Userspecified prisms● Given density contrasts ρs
Similar to René (1986)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● Userspecified prisms● Given density contrasts● Any n° of ≠ density contrasts
ρs
Similar to René (1986)
Not like René (1986)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● Userspecified prisms● Given density contrasts
3. Only
● Any n° of ≠ density contrasts
orp j=0 p j=ρs
ρs
Similar to René (1986)
Not like René (1986)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● 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
Similar to René (1986)
Not like René (1986)
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)
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)
Regularizing parameter
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)
Regularizing function
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)
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
2. Concentrated around “seeds”
3. Only orp j=0 p j=Δρs
4. of closest seed p j=Δρs
Regularization
Constraints:
1. Compact
2. Concentrated around “seeds”
3. Only orp j=0 p j=Δρs
4. of closest seed p j=Δρs
Regularization
Algorithm
Planting Algorithm
Based on René (1986)
Overview:
Based on René (1986)
Start with seeds
Overview:
Based on René (1986)
Start with seeds
Overview:
Based on René (1986)
Start with seeds known density contrast & position
Overview:
Based on René (1986)
Start with seeds known density contrast & position
All other parameters set to 0
Overview:
Based on René (1986)
Start with seeds
All other parameters set to 0
Iteratively grow
Overview:
known density contrast & position
Based on René (1986)
Start with seeds
All other parameters set to 0
Iteratively grow add neighbor of seed
Overview:
known density contrast & position
Based on René (1986)
Start with seeds
All other parameters set to 0
Iteratively grow add neighbor of seed
accretion
Overview:
known density contrast & position
Based on René (1986)
Start with seeds
All other parameters set to 0
Iteratively grow add neighbor of seed
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
known density contrast & position
Algorithm:
Algorithm:Define interpretative model
Interpretative model
Algorithm:Define interpretative model
g = observed data
Interpretative model
Algorithm:Define interpretative model
All parameters zero
g = observed data
Interpretative model
Algorithm:
seedsN S
Define interpretative model
All parameters zero
g = observed data
Interpretative model
Algorithm:
seedsN S
Define interpretative model
All parameters zero
Include seeds
Prisms with not shown
p j=0
Seeds
Algorithm:
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
Algorithm:
Residual vector
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
Algorithm:
Observed data
seedsN S
Define interpretative model
All parameters zero
Include seeds
Compute initial residuals
r(0)=g−d(0)
g = observed data
Prisms with not shown
p j=0
Algorithm:
Predicted by seeds
seedsN S
Define interpretative model
All parameters zero
Include seeds
Compute initial residuals
r(0)=g−d(0)
g = observed data
d = predicted data
Prisms with not shown
p j=0
Algorithm:
seedsN S
Define interpretative model
All parameters zero
Include seeds
Compute initial residuals
r(0)=g−d(0)
g = observed data
d=∑j=0
M
p j a j
d = predicted data
Prisms with not shown
p j=0
Algorithm:
seedsN S
Define interpretative model
All parameters zero
Include seeds
Compute initial residuals
r(0)=g−d(0)
g = observed data
d=∑j=0
M
p j a j
Many=0
d = predicted data
Prisms with not shown
p j=0
Algorithm:
seedsN S
Define interpretative model
All parameters zero
Include seeds
Compute initial residuals
r(0)=g−d(0)
g = observed data
d=∑s=1
N S
ρs a jS
d = predicted data
Prisms with not shown
p j=0
Algorithm:
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
p j=0
Algorithm:
Density contrastof sth seed
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
p j=0
seedsN S
Algorithm:Define interpretative model
All parameters zero
Include seeds
Compute initial residuals
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
Algorithm: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
Prisms with not shown
Growth:
p j=0
Prisms with not shown
Growth:
Try accretion to sth seed:
p j=0
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
p j=0
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
ϕ( p)=∥r∥2
p j=0
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=0
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
j = chosen
j
p j=0
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=ρsj = chosen
j
p j=0
(New elements)
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=ρsj = chosen
j
p j=0
(New elements)new predicted data
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=ρsj = chosen
Update residuals
r(new)=g−d(new)
p j=0
(New elements)new predicted data
j
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=ρsj = chosen
Update residuals
r(new)=g−d(new)
p j=0
(New elements)new predicted data
j
d(old )+ effect of j
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=ρsj = chosen
Update residuals
r(new)=g−d(new)
p j=0
(New elements)new predicted data
j
d(old )+ effect of j
∑s=1
N S
ρs a j Sp j a j+
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=ρsj = chosen
Update residuals
r(new)=g−d(new)
p j=0
(New elements)new predicted data
j
d(old )+ effect of j
∑s=1
N S
ρs a j Sp j a j+
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
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
(New elements)new predicted data
j
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
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
(New elements)new predicted data
j{
r(0)
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
2. Smallest goal function
ϕ( p)=∥r∥2
Γ( p)=ϕ( p)+μθ( p)
p j=ρsj = chosen
Update residuals
p j=0
(New elements)new predicted data
jr(new)
=r(old )− p j a j
Prisms with not shown
Growth:
None found = no accretion
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
Prisms with not shown
Growth:
None found = no accretion
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:
Variable sizes
p j=0
Prisms with not shown
Growth:
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
Prisms with not shown
Growth:
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
(New elements)
j
Prisms with not shown
Growth:
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
Prisms with not shown
Growth:
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
Prisms with not shown
Growth:
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
(New elements)
j
Prisms with not shown
Growth:
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
(New elements)
j
Prisms with not shown
Growth:
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
j
Prisms with not shown
Growth:
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=0j
Prisms with not shown
Growth:
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
j
Prisms with not shown
Growth:
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=0j
Prisms with not shown
Growth:
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
j
Prisms with not shown
Growth:
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
j
Prisms with not shown
Growth:
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
Prisms with not shown
Growth:
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
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
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
● Sources = 1 km X 1 km X 1 km
Δρ=0.5 g/ cm3
● Sources = 1 km X 1 km X 1 km
Δρ=1.0 g/ cm3
Δρ=0.5 g/ cm3
● Sources = 1 km X 1 km X 1 km
● Sources = 1 km X 1 km X 1 km
Depth=0.8 km
● Sources = 1 km X 1 km X 1 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
● Sources = 1 km X 1 km X 1 km
● Area = 5 km X 3 km
● Data set = 375 observations
Depth=1.6 km
Depth=0.8 km
● 0.05 mGal Gaussian noise
● Sources = 1 km X 1 km X 1 km
● 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
● With corresponding density contrasts
● Placed in center of sources
Δρ=1.0 g/cm3
Δρ=0.5 g/ cm3
● Used 2 seeds
● With corresponding density contrasts
● Placed in center of sources
Inversion result
Inversion result
Inversion result
● compact● concentrated around seeds● recover correct geometry of sources
Predicted data
Inversion result
● compact● concentrated around seeds● recover correct geometry of sources
Predicted dataObserved data
Inversion result
● compact● concentrated around seeds● recover correct geometry of sources
Predicted dataObserved data
Inversion result
● compact● concentrated around seeds
● fits observations
● recover correct geometry of sources
Predicted dataObserved data
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)
After Carminatti et al. (2003)
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
After Carminatti et al. (2003)
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
● Outcropping
● North of Goiás
● Tocantins Province
● Amazonian & São Francisco cratons
After Carminatti et al. (2003)
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
Gravimetric data:
● 132 observations
● Residual Bouguer
● Max 45 mGal
After Carminatti et al. (2003)
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
Previous interpretation:
After Carminatti et al. (2003)
● Carminatti et al. (2003)
● PVSS
● CVC● Max depth
Δρ=0.27 g/cm3
Δρ=0.39 g /cm3
≈6 km
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
Previous interpretation:
After Carminatti et al. (2003)
● 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
Interpretative model
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
Predicted dataObserved data
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