3D gravity inversion by planting anomalous densities
-
Upload
leonardo-uieda -
Category
Technology
-
view
556 -
download
0
description
Transcript of 3D gravity inversion by planting anomalous densities
![Page 1: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/1.jpg)
3D gravity inversion by planting anomalous densities
Leonardo Uieda and Valéria C. F. Barbosa
August, 2011
Observatório Nacional
![Page 2: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/2.jpg)
Outline
![Page 3: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/3.jpg)
Forward Problem
Outline
![Page 4: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/4.jpg)
Inverse ProblemForward Problem
Outline
![Page 5: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/5.jpg)
Inverse Problem Planting AlgorithmForward Problem
Inspired by René (1986)
Outline
![Page 6: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/6.jpg)
Inverse Problem Planting Algorithm
Synthetic Data
Forward Problem
Inspired by René (1986)
Outline
![Page 7: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/7.jpg)
Inverse Problem Planting Algorithm
Synthetic Data Real Data
Forward Problem
Inspired by René (1986)
Outline
![Page 8: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/8.jpg)
Forward problem
![Page 9: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/9.jpg)
Surface of the Earth
![Page 10: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/10.jpg)
Observations of gz
Surface of the Earth
![Page 11: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/11.jpg)
Observations of gz
Group in a vector:
g=[g1
g2
⋮gN
]N×1
= observed datag
Surface of the Earth
![Page 12: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/12.jpg)
= observed datag
![Page 13: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/13.jpg)
= observed datag
Assume caused by anomalous sources
![Page 14: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/14.jpg)
= observed datag
Assume caused by anomalous sources
ΔρDensity contrast =
![Page 15: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/15.jpg)
Parametrize the gravitational effect
![Page 16: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/16.jpg)
Parametrize the gravitational effect
Linearize
![Page 17: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/17.jpg)
Parametrize the gravitational effect
Discretize intoM elements
Linearize
Interpretative model
![Page 18: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/18.jpg)
Right rectangular prisms
Parametrize the gravitational effect
Discretize intoM elements
Homogeneous density contrast
jth element
Linearize
(Nagy et al., 2000)
Interpretative model
p j
![Page 19: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/19.jpg)
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
![Page 20: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/20.jpg)
Discretize intoM elements
Parametrize the gravitational effect
p j=Δρ
Prisms with not shown
p j=0
![Page 21: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/21.jpg)
Discretize intoM elements
Parametrize the gravitational effect
g≈d
Prisms with not shown
p j=0
p j=Δρ
![Page 22: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/22.jpg)
Discretize intoM elements
Parametrize the gravitational effect
g≈dPredicted data
Prisms with not shown
p j=0
p j=Δρ
![Page 23: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/23.jpg)
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
![Page 24: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/24.jpg)
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
![Page 25: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/25.jpg)
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
![Page 26: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/26.jpg)
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
![Page 27: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/27.jpg)
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
![Page 28: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/28.jpg)
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
![Page 29: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/29.jpg)
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
![Page 30: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/30.jpg)
Solved forward problem:
p dd=∑j=1
M
p j a j
![Page 31: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/31.jpg)
p̂ g?How to do the inverse?
![Page 32: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/32.jpg)
Inverse problem
![Page 33: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/33.jpg)
Minimize difference between
![Page 34: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/34.jpg)
Minimize difference between g
Observed data
![Page 35: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/35.jpg)
Minimize difference between andg d
Predicted data
![Page 36: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/36.jpg)
Minimize difference between andg d
r=g−d
![Page 37: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/37.jpg)
Minimize difference between andg d
r=g−d
Residual vector
![Page 38: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/38.jpg)
Minimize difference between andg d
r=g−d
Residual vector
Datamisfit function: ϕ( p)=∥r∥2=(∑i=1
N
(gi−d i)2)
12
![Page 39: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/39.jpg)
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
![Page 40: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/40.jpg)
illposed problem
nonexistent
nonunique
nonstable
![Page 41: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/41.jpg)
illposed problem
nonexistent
nonunique
nonstable
constraints
![Page 42: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/42.jpg)
illposed problem
nonexistent
nonunique
nonstable
wellposed problem
exist
unique
stable
constraints
![Page 43: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/43.jpg)
Constraints:
1. Compact
![Page 44: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/44.jpg)
Constraints:
1. Compact no holes inside
![Page 45: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/45.jpg)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
![Page 46: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/46.jpg)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
Similar to René (1986)
![Page 47: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/47.jpg)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● Userspecified prisms
Similar to René (1986)
![Page 48: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/48.jpg)
Constraints:
1. Compact no holes inside
2. Concentrated around “seeds”
● Userspecified prisms● Given density contrasts ρs
Similar to René (1986)
![Page 49: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/49.jpg)
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)
![Page 50: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/50.jpg)
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)
![Page 51: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/51.jpg)
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)
![Page 52: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/52.jpg)
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)
![Page 53: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/53.jpg)
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
![Page 54: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/54.jpg)
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
![Page 55: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/55.jpg)
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
![Page 56: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/56.jpg)
Regularization:
θ( p)=∑j=1
M p j
p j+ϵl j
β
Γ( p)=ϕ( p)+μθ( p)
![Page 57: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/57.jpg)
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)
![Page 58: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/58.jpg)
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)
![Page 59: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/59.jpg)
Regularization:
θ( p)=∑j=1
M p j
p j+ϵl jβ
Γ( p)=ϕ( p)+μθ( p)
For p j≠0 :
![Page 60: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/60.jpg)
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 :
![Page 61: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/61.jpg)
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 :
![Page 62: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/62.jpg)
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 :
![Page 63: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/63.jpg)
Constraints:
1. Compact
2. Concentrated around “seeds”
3. Only orp j=0 p j=Δρs
4. of closest seed p j=Δρs
Regularization
![Page 64: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/64.jpg)
Constraints:
1. Compact
2. Concentrated around “seeds”
3. Only orp j=0 p j=Δρs
4. of closest seed p j=Δρs
Regularization
Algorithm
![Page 65: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/65.jpg)
Planting Algorithm
![Page 66: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/66.jpg)
Based on René (1986)
Overview:
![Page 67: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/67.jpg)
Based on René (1986)
Start with seeds
Overview:
![Page 68: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/68.jpg)
Based on René (1986)
Start with seeds
Overview:
![Page 69: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/69.jpg)
Based on René (1986)
Start with seeds known density contrast & position
Overview:
![Page 70: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/70.jpg)
Based on René (1986)
Start with seeds known density contrast & position
All other parameters set to 0
Overview:
![Page 71: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/71.jpg)
Based on René (1986)
Start with seeds
All other parameters set to 0
Iteratively grow
Overview:
known density contrast & position
![Page 72: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/72.jpg)
Based on René (1986)
Start with seeds
All other parameters set to 0
Iteratively grow add neighbor of seed
Overview:
known density contrast & position
![Page 73: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/73.jpg)
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
![Page 74: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/74.jpg)
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
![Page 75: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/75.jpg)
Algorithm:
![Page 76: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/76.jpg)
Algorithm:Define interpretative model
Interpretative model
![Page 77: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/77.jpg)
Algorithm:Define interpretative model
g = observed data
Interpretative model
![Page 78: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/78.jpg)
Algorithm:Define interpretative model
All parameters zero
g = observed data
Interpretative model
![Page 79: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/79.jpg)
Algorithm:
seedsN S
Define interpretative model
All parameters zero
g = observed data
Interpretative model
![Page 80: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/80.jpg)
Algorithm:
seedsN S
Define interpretative model
All parameters zero
Include seeds
Prisms with not shown
p j=0
Seeds
![Page 81: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/81.jpg)
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
![Page 82: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/82.jpg)
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
![Page 83: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/83.jpg)
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
![Page 84: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/84.jpg)
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
![Page 85: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/85.jpg)
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
![Page 86: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/86.jpg)
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
![Page 87: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/87.jpg)
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
![Page 88: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/88.jpg)
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
![Page 89: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/89.jpg)
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
![Page 90: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/90.jpg)
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
![Page 91: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/91.jpg)
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
![Page 92: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/92.jpg)
Prisms with not shown
Growth:
p j=0
![Page 93: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/93.jpg)
Prisms with not shown
Growth:
Try accretion to sth seed:
p j=0
![Page 94: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/94.jpg)
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
p j=0
![Page 95: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/95.jpg)
Prisms with not shown
Growth:
Try accretion to sth seed:
Choose neighbor:
1. Reduce data misfit
ϕ( p)=∥r∥2
p j=0
![Page 96: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/96.jpg)
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
![Page 97: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/97.jpg)
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
![Page 98: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/98.jpg)
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)
![Page 99: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/99.jpg)
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
![Page 100: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/100.jpg)
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
![Page 101: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/101.jpg)
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
![Page 102: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/102.jpg)
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+
![Page 103: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/103.jpg)
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+
![Page 104: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/104.jpg)
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
![Page 105: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/105.jpg)
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)
![Page 106: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/106.jpg)
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
![Page 107: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/107.jpg)
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
![Page 108: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/108.jpg)
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
![Page 109: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/109.jpg)
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
![Page 110: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/110.jpg)
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
![Page 111: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/111.jpg)
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
![Page 112: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/112.jpg)
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
![Page 113: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/113.jpg)
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
![Page 114: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/114.jpg)
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
![Page 115: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/115.jpg)
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
![Page 116: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/116.jpg)
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
![Page 117: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/117.jpg)
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
![Page 118: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/118.jpg)
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
![Page 119: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/119.jpg)
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
![Page 120: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/120.jpg)
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
![Page 121: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/121.jpg)
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
![Page 122: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/122.jpg)
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
![Page 123: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/123.jpg)
Advantages:
Compact & nonsmooth
Any number of sources
Any number of different density contrasts
No large equation system
Search limited to neighbors
![Page 124: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/124.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 125: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/125.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 126: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/126.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 127: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/127.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 128: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/128.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 129: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/129.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 130: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/130.jpg)
Advantages:
Compact & nonsmooth
Any number of sources
Any number of different density contrasts
No large equation system
Search limited to neighbors
![Page 131: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/131.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 132: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/132.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 133: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/133.jpg)
Synthetic Data
![Page 134: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/134.jpg)
![Page 135: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/135.jpg)
● Sources = 1 km X 1 km X 1 km
![Page 136: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/136.jpg)
Δρ=0.5 g/ cm3
● Sources = 1 km X 1 km X 1 km
![Page 137: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/137.jpg)
Δρ=1.0 g/ cm3
Δρ=0.5 g/ cm3
● Sources = 1 km X 1 km X 1 km
![Page 138: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/138.jpg)
● Sources = 1 km X 1 km X 1 km
Depth=0.8 km
![Page 139: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/139.jpg)
● Sources = 1 km X 1 km X 1 km
Depth=1.6 km
Depth=0.8 km
![Page 140: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/140.jpg)
● Sources = 1 km X 1 km X 1 km ● Data set = 375 observations
Depth=1.6 km
Depth=0.8 km
![Page 141: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/141.jpg)
● 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
![Page 142: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/142.jpg)
● 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
![Page 143: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/143.jpg)
● Interpretative model = 151,875 prisms
● Prisms = 66.7 m X 66.7 m X 66.7 m
![Page 144: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/144.jpg)
● Used 2 seeds
![Page 145: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/145.jpg)
● Used 2 seeds ● Placed in center of sources
![Page 146: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/146.jpg)
● Used 2 seeds
● With corresponding density contrasts
● Placed in center of sources
![Page 147: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/147.jpg)
Δρ=0.5 g/ cm3
● Used 2 seeds
● With corresponding density contrasts
● Placed in center of sources
![Page 148: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/148.jpg)
Δρ=1.0 g/cm3
Δρ=0.5 g/ cm3
● Used 2 seeds
● With corresponding density contrasts
● Placed in center of sources
![Page 149: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/149.jpg)
Inversion result
![Page 150: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/150.jpg)
Inversion result
![Page 151: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/151.jpg)
Inversion result
● compact● concentrated around seeds● recover correct geometry of sources
![Page 152: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/152.jpg)
Predicted data
Inversion result
● compact● concentrated around seeds● recover correct geometry of sources
![Page 153: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/153.jpg)
Predicted dataObserved data
Inversion result
● compact● concentrated around seeds● recover correct geometry of sources
![Page 154: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/154.jpg)
Predicted dataObserved data
Inversion result
● compact● concentrated around seeds
● fits observations
● recover correct geometry of sources
![Page 155: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/155.jpg)
Predicted dataObserved data
On laptop with 2.0 GHz
● 375 data● 151,875 prisms● Total time≈4.4 min
![Page 156: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/156.jpg)
Real Data
![Page 157: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/157.jpg)
After Carminatti et al. (2003)
![Page 158: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/158.jpg)
Cana Brava complex (CBC)
After Carminatti et al. (2003)
![Page 159: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/159.jpg)
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
After Carminatti et al. (2003)
![Page 160: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/160.jpg)
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
● Outcropping
● North of Goiás
● Tocantins Province
● Amazonian & São Francisco cratons
After Carminatti et al. (2003)
![Page 161: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/161.jpg)
Cana Brava complex (CBC) & Palmeirópolis sequence (PVSS)
Gravimetric data:
● 132 observations
● Residual Bouguer
● Max 45 mGal
After Carminatti et al. (2003)
![Page 162: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/162.jpg)
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
![Page 163: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/163.jpg)
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
![Page 164: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/164.jpg)
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
![Page 165: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/165.jpg)
Interpretative model
Size: 120 km X 50 km X 11 km
480,000 prisms
Prism size: 500 m X 500 m X 575 m
![Page 166: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/166.jpg)
Inversion result
![Page 167: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/167.jpg)
Inversion result
![Page 168: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/168.jpg)
Inversion result
Predicted dataObserved data
![Page 169: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/169.jpg)
Inversion result
![Page 170: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/170.jpg)
Inversion result
![Page 171: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/171.jpg)
Inversion result
![Page 172: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/172.jpg)
Inversion result
![Page 173: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/173.jpg)
Inversion result
![Page 174: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/174.jpg)
Inversion result
![Page 175: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/175.jpg)
Inversion result
≈6 kmMax depth
Agree with previous interpretation
Compact
Fits observations
![Page 176: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/176.jpg)
Inversion result
On laptop with 2.0 GHz
● 132 data● 480,00 prisms● Total time≈3.75 min
![Page 177: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/177.jpg)
Conclusions
![Page 178: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/178.jpg)
● 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
![Page 179: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/179.jpg)
● 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
![Page 180: 3D gravity inversion by planting anomalous densities](https://reader033.fdocuments.in/reader033/viewer/2022052311/558383eed8b42aef748b4f9b/html5/thumbnails/180.jpg)
Thank you