Introduction to 3D MT inversion code x3Di - DIAS – DIAS 2020...Anna Avdeeva ([email protected]), Dmitry Avdeev and Marion Jegen 31 March 2011 Introduction to 3D MT inversion

Introductionto 3D MT

inversion codex3Di

A. Avdeeva,D. Avdeev,M. Jegen

X3DI

OptimizationMethod

Calculation ofthe Gradients

Parametrization

Regularization

Salt OverhangDetectability

Conclusions

References

Introduction to 3D MT inversion code x3Di

Anna Avdeeva ([email protected]), Dmitry Avdeevand Marion Jegen

31 March 2011


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Outline

1 Essential Parts of 3D MT Inversion Code X3DIOptimization MethodCalculation of the GradientsParametrizationRegularization

2 Salt Dome Overhang Detectability Study with x3Di

3 Conclusions


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Outline



3 Conclusions


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

How 3D inverse problem is commonly solved

Traditionally solution is sought as a stationary point of a penaltyfunction

ϕ(m, λ) = ϕd(m) + λϕs(m) −→m

min (1)

ϕd(m) data misfit

ϕs(m) Tikhonov-type stabilizer

λ regularization parameter

m vector of model parameters (conductivities)

ϕd(m) =1

2

∥∥dobs − F (m)∥∥2 (2)

F (m) is a forward problem mapping

ϕs(m) =1

2

∥∥W(m−mref )∥∥2 (3)

mref vector of model parameters, for some reference model,which usually include some a priori information.


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Where are the differences between 3D inversioncodes?

1 model parameters (σ, ρ, log ρ, log σ etc.)

2 forward problem solver (FD, FE or IE)

- X3D

3 optimization method (GN, QN, LMQN, NLCG etc.)

4 form of the data misfit ϕd

5 form of the stabilizer ϕs


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References



2 forward problem solver (FD, FE or IE)

- X3D





inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References



2 forward problem solver (FD, FE or IE) - X3D





inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References



2 forward problem solver (FD, FE or IE) - X3D





inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Optimization method

All iterative methods work at the same manner. At each iteration lthey find:

1 a search direction vector p(l)

2 a step length α(l) using an inexact line search along p(l)

3 the next, improved model given by m(l+1) = m(l) + α(l)p(l)

The difference is in how a specific method finds the search directionand in what price is paid for this.

NLCG p(l) = −g(l) + γ(l)p(l−1), where γ(l) = g(l)·g(l)g(l−1)·g(l−1)

Newton, GN H(l)p(l) = −g(l)

QN m(i), g(i) : i = 1, ..., l → p(l)

LMQN m(i), g(i) : i = l − ncp, ..., l → p(l)

H(l)ij = ∂2ϕ

∂mi∂mj(m(l)), g

(l)i = ∂ϕ

∂mi(m(l))


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Optimization method

All iterative methods work at the same manner. At each iteration lthey find:

1 a search direction vector p(l)

2 a step length α(l) using an inexact line search along p(l)

3 the next, improved model given by m(l+1) = m(l) + α(l)p(l)

The difference is in how a specific method finds the search directionand in what price is paid for this.

NLCG p(l) = −g(l) + γ(l)p(l−1), where γ(l) = g(l)·g(l)g(l−1)·g(l−1)

Newton, GN H(l)p(l) = −g(l)

QN m(i), g(i) : i = 1, ..., l → p(l)

LMQN m(i), g(i) : i = l − ncp, ..., l → p(l)

H(l)ij = ∂2ϕ

∂mi∂mj(m(l)), g

(l)i = ∂ϕ

∂mi(m(l))


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Adjoint approach

Calculation of gradients requires 2 forward modellings ateach frequency

Important: Straightforward calculation of the gradientswould require N + 1 forward modellings, i.e. in ≈ N/2times more

Example: number of model parameters = 3000

Single forward modelling requires ≈ 4 min on PC⇒ Straightforward calculation of the single gradient requires 8 daysUsually ≈ 200 iterations(gradients) is needed⇒ Total inversion time is ≈ 4 years.


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

3D data misfit

ϕd(σ) =1

2

NS∑i=1

NT∑j=1

βij tr[A

Tij (σ)Aij(σ)

](4)

σ = (σ1, ..., σN)T the vector of the electrical conductivities ofthe cells, N number of cells

NS number of MT sites ri = (xi , yi , z = 0)

NT number of frequencies ωj

Aij = Zij −Dij 2× 2 matrices

Zij complex-valued predicted Z(ri , ωj) impedance

Dij complex-valued observed D(ri , ωj) impedance

βij some positive weights

tr[A

TA]

= A11A11 + A12A12 + A21A21 + A22A22


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Gradients: Adjoint approach

∂ϕd

∂σk= Re

NT∑j=1

2∑p=1

∫Vk

(u(p)x E(p)

x + u(p)y E(p)y + u(p)z E(p)

z

)dV

(5)

Maxwell’s equation

∇×∇× E(p)j −

√−1ωjµσ(r)E

(p)j =

√−1ωjµJ

(p)j (6)

Adjoint Maxwell’s equation

∇×∇× u(p)j −

√−1ωjµσ(r)u

(p)j =

√−1ωjµ

(j(p)j +∇× h

(p)j

)(7)

j(p)j and h

(p)j - horizontal electric and magnetic dipoles at the MT sites

p = 1, 2 - polarization

The forward modelling is performed with X3D by (Avdeev et al.,

2002).


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Grid and MT sites

N= 20× 20× 9 cellsdx = dy = 4 kmNS= 80 MT sitesNT=3 periods:100, 300, 1000 snoise - 1%initial guess model:50 Ωm halfspace


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Logarithmic parametrization vs conductivities:inversion results


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Logarithmic parametrization vs conductivities:convergence curves

0 100 200 300 400 500 600

−2

−1

0

1

2

3

Mis

fit

nfg

−2−10123456789

log 10

(λ)

without additional regularization

m= σ m=log

10 σ

Target misfit

0 100 200 300 400 500 600

−2

−1

0

1

2

3

Mis

fit

nfg

−2−10123456789

log 10

(λ)

with additional regularization

m= σ m=log

10 σ

Target misfit


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Regularization

Tikhonov-type stabilizers:

Laplace

ϕs = dxdy∑αβγ

[∂2m

∂x2+∂2m

∂y2+∂2m

∂z2

]2αβγ

dzγ (8)

Gradient

ϕs = dxdy∑αβγ

[(∂m

∂x

)2

+

(∂m

∂y

)2

+

(∂m

∂z

)2]αβγ

dzγ (9)


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Gradient vs Laplace: inversion results


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Gradient vs Laplace: inversion results

Laplace

Gradient

True model


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Gradient vs Laplace: convergence curves

0 100 200 300 400 50010

−2

10−1

100

101

102

103

104

Mis

fit

nfg

0 100 200 300 400 500

−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

3

4

log 10

(λ)

Laplace regul.Gradient regul.Target misfit


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Outline



3 Conclusions


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

3D Model of a salt wall


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

1995 MT sites

N= 129× 69× 13 cellsdx = dy = 0.25 kmNS=1995 MT sitesNT=5 frequencies:10−3 - 101 Hznoise - 5%initial guess model:11 Ωm halfspace


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Inversion result for model 2 with overhang.1995 MT sites. Horizontal slices

0.002−0.05 km

y (

km

)

−30

−20

−10 y

(k

m)

x (km)−15−10 −5 0

−30

−20

−10

0.05−0.275 km

x (km)−15−10 −5 0

0.95−1.2 km

x (km)−15−10 −5 0

1.7−2.2 km

x (km)−15−10 −5 0

2.7−3.7 km

x (km)

Model 2: with overhang

−15−10 −5 0

Resistivity (Ωm)10

−110

010

110

2

z :

Inversion result

0.002−0.05 km 0.05−0.275 km 0.95−1.2 km 1.7−2.2 km 2.7−3.7 kmz :

−15−10 −5 0 −15−10 −5 0 −15−10 −5 0 −15−10 −5 0 −15−10 −5 0


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Inversion result: model 1 vs model 2.1995 MT sites. Vertical slices


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

105 MT sites along three profiles

N= 129× 69× 13 cellsdx = dy = 0.25 kmNS=105 MT sitesNT=5 frequencies:10−3 - 101 Hznoise - 5%initial guess model:11 Ωm halfspace


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Inversion result for model 2 with overhang.105 MT sites. Horizontal slices

y (

km

)

−30

−20

−10 y

(k

m)

x (km)

−30

−20

−10

x (km) x (km) x (km)

C

B

A

x (km)

C

B

A

Resistivity (Ωm)10

−110

010

110

2

0.002−0.05 km

−15−10 −5 0

0.05−0.275 km

−15−10 −5 0

0.95−1.2 km

−15−10 −5 0

1.7−2.2 km

−15−10 −5 0

2.7−3.7 km

Model 2: with overhang

−15−10 −5 0

z :

Inversion result

0.002−0.05 km 0.05−0.275 km 0.95−1.2 km 1.7−2.2 km 2.7−3.7 kmz :

−15−10 −5 0 −15−10 −5 0 −15−10 −5 0 −15−10 −5 0 −15−10 −5 0


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Inversion result: model 1 vs model 2.105 MT sites. Vertical slices


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Outline



3 Conclusions


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Conclusions

Logarithmic parametrization is beneficial in terms ofinversion result and computational time

Regularization based on Gradient suppresses spatialresistivity gradients

The x3Di code produces encouraging results for salt domeoverhang detectability


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Acknowledgements

We would like to thank

Wintershall Holding AG, who funded the inversion studyand 20 ocean bottom MT instruments


inversion codex3Di


X3DI

OptimizationMethod


Parametrization

Regularization


Conclusions

References

Thanks for your attention

Avdeev, D. B. and Avdeeva, A. D. (2009). Three-dimensional magnetotelluricinversion using a limited-memory QN optimization. Geophysics, 74(3),F45–F57.

Avdeev, D. B., Kuvshinov, A. V., Pankratov, O. V., and Newman, G. A. (2002).Three-dimensional induction logging problems, Part I: An integral equationsolution and model comparisons. Geophysics, 67, 413–426.

Avdeeva, A. D. and Avdeev, D. B. (2006). A limited-memory quasi-newtoninversion for 1D magnetotellurics. Geophysics, 71, G191–G196.

Introduction to 3D MT inversion code x3Di - DIAS – DIAS 2020...Anna Avdeeva ([email protected]), Dmitry Avdeev and Marion Jegen 31 March 2011 Introduction to 3D MT inversion

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