Right rectangular prisms Parametrize the gravitational effect Discretize into M 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 into M elements p= [p 1 p 2 ⋮ p M ] M×1 Parameter vector Linearize Interpretative model

Discretize into M elements Parametrize the gravitational effect Gravitational effect is linear d=∑ j=1 M p j a j g≈d Predicted data Prisms with not shown p j =0 p j =Δρ

Discretize into M elements Parametrize the gravitational effect Gravitational effect is linear d=∑ j=1 M p j a j Density contrast of jth prism g≈d Predicted data Prisms with not shown p j =0 p j =Δρ

Discretize into M elements Parametrize the gravitational effect Gravitational effect is linear d=∑ j=1 M p j a j g≈d Predicted data Effect of prism with unit density Prisms with not shown p j =0 p j =Δρ

Discretize into M elements Parametrize the gravitational effect Gravitational effect is linear g≈d Predicted data d=∑ j=1 M p j a j =A p Prisms with not shown p j =0 p j =Δρ

Discretize into M elements Parametrize the gravitational effect Gravitational effect is linear g≈d Predicted data Parameter vector d=∑ j=1 M p j a j =A p Prisms with not shown p j =0 p j =Δρ

Discretize into M elements Parametrize the gravitational effect Gravitational effect is linear g≈d Predicted 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 into M elements Parametrize the gravitational effect Gravitational effect is linear g≈d Predicted 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

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 or p 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 or p j =0 p j =ρs ρs 4. of closest seed p j =ρs Similar to René (1986) Not like René (1986)

illposed problem wellposed problem constraints ϕ( p)=∥r∥2 = (∑ i=1 N (g i −d i )2 )1 2 Minimize data misfit Minimize goal function Γ( p)=ϕ( p)+μθ( p)

illposed problem wellposed problem constraints ϕ( p)=∥r∥2 = (∑ i=1 N (g i −d i )2 )1 2 Minimize data misfit Minimize goal function Γ( p)=ϕ( p)+μθ( p) Regularizing parameter

illposed problem wellposed problem constraints ϕ( p)=∥r∥2 = (∑ i=1 N (g i −d i )2 )1 2 Minimize data misfit Minimize goal function Γ( p)=ϕ( p)+μθ( p) Regularizing function

illposed problem wellposed problem constraints ϕ( p)=∥r∥2 = (∑ i=1 N (g i −d i )2 )1 2 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 +ϵ l j β 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 +ϵ l j β Γ( p)=ϕ( p)+μθ( p) distance from seeds regularizing function Imposes: ● Compactness ● Concentration around seeds For p j ≠0:

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 (g i −d i )2 )1 2 known density contrast & position

Algorithm: Residual vector seeds N 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 seeds N 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 seeds N 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: seeds N 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: seeds N 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: seeds N 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 ρ s a j S d = predicted data Prisms with not shown p j =0

Algorithm: seeds N S Define interpretative model All parameters zero Include seeds Compute initial residuals r(0)=g− (∑ s=1 N S ρ s a j S ) Prisms with not shown g = observed data d = predicted data p j =0

Algorithm: Density contrast of sth seed seeds N S Define interpretative model All parameters zero Include seeds Compute initial residuals r(0)=g− (∑ s=1 N S ρ s a j S ) Prisms with not shown g = observed data d = predicted data p j =0

seeds N S Algorithm: Define interpretative model All parameters zero Include seeds Compute initial residuals r(0)=g− (∑ s=1 N S ρ s a j S ) Column vector of A Prisms with not shown g = observed data d = predicted data p j =0

seeds N S Algorithm: Define interpretative model All parameters zero Include seeds Compute initial residuals r(0)=g− (∑ s=1 N S ρ s a j S ) Prisms with not shown g = observed data d = predicted data Neighbors Find neighbors of seeds 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) p j =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 S p 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 =ρ s j = 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 S p 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = chosen Update residuals p j =0 (New elements) new predicted data j r(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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = 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 =ρ s j = chosen Update residuals r(new)=r(old )− p j a j Choose neighbor: At least one seed grow? Yes No Done! p j =0

Remember equations: r(0)=g− (∑ s=1 N S ρ s a j S ) 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 ρ s a j S ) 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 ρ s a j S ) 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 ρ s a j S ) 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 ρ s a j S ) 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 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

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) Previous interpretation: After Carminatti et al. (2003) ● Carminatti et al. (2003) ● PVSS ● CVC ● Max depth Δρ=0.27 g/cm3 Δρ=0.39g/cm3 ≈6 km Test this hypothesis

● 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