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Introduction The Ddirect Problem Inverse Problem Numerical Examples

Complex Electrical Impedance Tomography in Geoelectrics

Andreas Helfrich-Schkarbanenko July 17, 2009 Institute for Algebra and Geometry University of Karlsruhe Germany

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples

Complex Electrical Impedance Tomography in Geoelectrics

1

Introduction Problem Setting

2

The Ddirect Problem Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

3

Inverse Problem Operators Implementation

4

Numerical Examples Synthetic Examples Real Example

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Problem Setting

Model

Ω ⊂ Rd. d ∈ {2, 3};

R

• R

R

X1 X2

• Figure: Problem setting

u : Ω → C is electrical potential which is induced by current pattern f and solves −∇ · (γ∇u) = f in Ω (1a) ν · ∇u = 0 on Γ1 (1b) ν · ∇u + αu = 0 on Γ2. (1c)

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

Weak Formulation of the Problem

Let Bγ,α : H1(Ω) × H1(Ω) → C defined by Bγ,α(u, v) := Z

Ω

γ∇u, ∇vdx + Z

Γ2

γαuvds, u, v ∈ H1(Ω) (2) and the linear form Lf : H1(Ω) → C defined by Lf (v) := Z

Ω

f vdx, v ∈ H1(Ω). (3)

Problem

For given f ∈ H−1(Ω), α ∈ L∞(Γ2, R+) and γ ∈ L∞(Ω, R+) find u ∈ H1(Ω) with: Bγ,α(u, v) = Lf (v), ∀v ∈ H1(Ω). (4) Is this problem uniquely solvable?

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

Theorem (Existence and uniqueness of weak solution)

Let Ω be a Lipschitz domain, f ∈ H−1(Ω), α ∈ L∞(Γ2, R+) and γ(x, ω) = σ(x) + iωǫ(x) ∈ L∞(Ω × R+, C+), where C+ := {x ∈ C : Re(x) > 0}. Then the variational problem (4) has a unique solution in the Sobolev space H1(Ω).

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

Theorem (Existence and uniqueness of weak solution)

Let Ω be a Lipschitz domain, f ∈ H−1(Ω), α ∈ L∞(Γ2, R+) and γ(x, ω) = σ(x) + iωǫ(x) ∈ L∞(Ω × R+, C+), where C+ := {x ∈ C : Re(x) > 0}. Then the variational problem (4) has a unique solution in the Sobolev space H1(Ω).

Proof idea.

Apply the

Lax-Milgram theorem

• n Bγ,α, Lf and use

lemma

in P.Monk Finite Element Methods for Maxwell’s Equations, 2003.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

Theorem (Existence and uniqueness of weak solution)

Let Ω be a Lipschitz domain, f ∈ H−1(Ω), α ∈ L∞(Γ2, R+) and γ(x, ω) = σ(x) + iωǫ(x) ∈ L∞(Ω × R+, C+), where C+ := {x ∈ C : Re(x) > 0}. Then the variational problem (4) has a unique solution in the Sobolev space H1(Ω).

Proof idea.

Apply the

Lax-Milgram theorem

• n Bγ,α, Lf and use

lemma

in P.Monk Finite Element Methods for Maxwell’s Equations, 2003. FE method: Find u ∈ Hh = span{ψk}Np

k=1 ⊂ H1(Ω).

Does the unique solution exist in Hh, too?

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

Theorem (Existence and uniqueness of weak solution)

Let Ω be a Lipschitz domain, f ∈ H−1(Ω), α ∈ L∞(Γ2, R+) and γ(x, ω) = σ(x) + iωǫ(x) ∈ L∞(Ω × R+, C+), where C+ := {x ∈ C : Re(x) > 0}. Then the variational problem (4) has a unique solution in the Sobolev space H1(Ω).

Proof idea.

Apply the

Lax-Milgram theorem

• n Bγ,α, Lf and use

lemma

in P.Monk Finite Element Methods for Maxwell’s Equations, 2003. FE method: Find u ∈ Hh = span{ψk}Np

k=1 ⊂ H1(Ω).

Does the unique solution exist in Hh, too? Yes! By means of

C´ ea lemma

.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

Further results

For the case of ν · ∇u = g on Γ1 the problem has a unique solution as well (provided some properties on f , g and u|Γ1). Prove it as above. Stability: uH1(Ω) ≤ C(f H−1(Ω)) + gH−1/2(Γ1)), C > 0. u depends continuously on γ = γ(x, t), if γ ∈ C 1(Ω, R).

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

−5 5 −6 −4 −2 2 Im{u} für f1

−4 −2 2 x 10

−3

−5 5 −6 −4 −2 2 Im{u} für f2

−6 −4 −2 2 x 10

−3

−5 5 −6 −4 −2 2 Im{u} für f3

−8 −6 −4 −2 2 x 10

−3

−5 5 −6 −4 −2 2 Im{u} für f4

−10 −5 x 10

−3

−5 5 −6 −4 −2 2 Im{u} für f5

−15 −10 −5 x 10

−3

−5 5 −6 −4 −2 2 Im{u} für f6

−10 −5 x 10

−3

−5 5 −6 −4 −2 2 Im{u} für f7

−8 −6 −4 −2 2 4 x 10

−3

−5 5 −6 −4 −2 2 Im{u} für f8

−6 −4 −2 2 x 10

−3

−5 5 1 2 3 x Re{uΓ

1

} −5 5 −15 −10 −5 5 x 10

−3

x Im{uΓ

1

} −2 −1 1 2 1 2 3 x Re{uE

k

} −2 −1 1 2 −15 −10 −5 5 x 10

−3

x Im{uE

k

}

Figure: Solution of the forward problem, Im{u} and the corresponding measurements

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

Solving 2.5D and 2D direct problems

1 Solving 3D problem is time- and memory-consuming 2 Assume γ = γ(x1, x2) and apply the Fourier-transform with respect to x3, [DM76] 3 We get 2D Helmholtz-equation with a wave number 4 Solve the obtained problem in image space and transform it back

2

X1 X2

• X3
• 3

+ +

• Andreas Helfrich-Schkarbanenko

Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Weak Formulation of the Problem Existence and uniqueness Numerical Solution Solving 2.5D and 2D direct problems

3D 2.5D 2D Ω3 ⊂ R3 γ = γ(x1, x2, x3)

❄

BVP in Ω3

❄

FEM 3D u3

✲

assume

• n Ω2

=

if γ = γ(x1, x2)

Ω3 ⊂ R3 γ = γ(x1, x2)

❄

BVP in Ω3

❄

Fcos Helmholtz BVP

❄

FEM 2D {e uω}ω∈R

❄

F−1

cos for x3 = 0

u3|Ω2 Ω2 ⊂ R2 γ = γ(x1, x2)

❄

BVP in Ω2

❄

FEM 2D u2 (a) (b) (c)

Figure: Comparison of 3D, 2.5D and 2D forward problem.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Operators Implementation

The inverse problem consists of the reconstruction of γ in Ω from current and voltage measurement data on Γ1 ⊂ ∂Ω, i.e. we have to find γ for the given Neumann-to-Dirichlet operator Λγ : f → u|Γ1. In addition, we consider a nonlinear operator with respect to γ Ff :=  L∞(Ω) → H1(Ω) γ → u = Λγf (5) so the inverse problem is nonlinear, too. For the numerical implementation of Newton-like algorithms the differentiation of Ff is essential.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Operators Implementation

Fr´ echet differentiability of Ff

Theorem

The map Ff : L∞(Ω) → H1(Ω), γ → u is Fr´ echet differentiable, i.e. it can be approximated locally by a linear operator. For h ∈ A, with γ + h ∈ A and u0 := Ff (γ) the derivative F′

f (γ)h =: w solves the equation

Bγ,α(w, v) = − „Z

Ω

h∇u0, ∇vdx + Z

Γ2

hαu0vds « , ∀v ∈ H.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Operators Implementation

Newton-like reconstruction method

In application instead of Λγ we have un = Λγfn, n = 1, ..., p. minγk∈A V − F(γk), V : measurement data, F(γk) model response The idea is to linearize the direct operator F at a reference distridution γ0. This leads to the Newton method: hN,k = −F ′(γk)−1(F(γk ) − F(γ)), γk+1 = γk + hN,k.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Operators Implementation

Newton-like reconstruction method

In application instead of Λγ we have un = Λγfn, n = 1, ..., p. minγk∈A V − F(γk), V : measurement data, F(γk) model response The idea is to linearize the direct operator F at a reference distridution γ0. This leads to the Newton method: hN,k = −F ′(γk)−1(F(γk ) − F(γ)), γk+1 = γk + hN,k. The EIT problem is ill-posed → regularisation techniques. The ansatz for minimizing functional Jθ(hT ) := F ′(γ)hT − (F(γ) − V )2

2 + θWhT 2 2 leads to

the generalised Tikhonov reconstruction method: hT,k = ` F ′(γk)∗F ′(γk) + θkW ∗W ´−1 F ′(γk)∗(F(γk ) − V ), γk+1 = γk + hT,k.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

1st Example (layer model)

−5 5 −4 −2 2 Vorwärts−Gitter (2294 FE) −5 5 −4 −2 2 Re(γ)

1 1.2 1.4

−5 5 −4 −2 2 Im(γ)

0.005 0.01

Figure: Parameter mesh TΩ of the domain Ω; Re{γ} and Im{γ} valued on TΩ; Number of electrodes: 13

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−005

−0.4 −0.2 0.2

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−006

−0.4 −0.2 0.2

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−007

−0.4 −0.2 0.2

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−008

−0.4 −0.2 0.2

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−009

−0.4 −0.2 0.2 0.4

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−010

−0.4 −0.2 0.2 0.4

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−011

−0.5 0.5

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−012

−0.5 0.5

−5 5 −4 −2 2 Im(h

k) für θ k=4.4e−005

−1 1 2 3 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−006

2 4 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−007

2 4 6 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−008

2 4 6 8 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−009

−2 2 4 6 8 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−010

5 10 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−011

−2 2 4 6 8 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−012

−2 2 4 6 8 x 10

−3

Figure: Re{hT,0} and Im{hT,0} for different regularisation parameters θ0n; γ0 = 1.1

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

2nd example (ring model)

−5 5 −4 −2 2 Parameter−Gitter (593 FE) −5 5 −4 −2 2 Re(γ0)

1 1.5

−5 5 −4 −2 2 Im(γ0)

0.005 0.01

Figure: Parameter mesh Tγ (593 nodes) of the domain Ω; Re{γ} and Im{γ} valued on Tγ; 13 electrodes; forward mesh: 2255 nodes

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−007

−0.5 0.5

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−008

−0.5 0.5

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−009

−0.5 0.5

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−010

−0.5 0.5

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−011

−0.5 0.5

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−012

−0.5 0.5 1

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−013

−0.5 0.5 1

−5 5 −4 −2 2 Re(h

k) für θ k=4.4e−014

−0.5 0.5 1

−5 5 −4 −2 2 Im(h

k) für θ k=4.4e−007

10 20 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−008

10 20 x 10

−3 −5 5 −4 −2 2 Im(h

k) für θ k=4.4e−009

−0.01 0.01

−5 5 −4 −2 2 Im(h

k) für θ k=4.4e−010

−0.01 0.01 0.02

−5 5 −4 −2 2 Im(h

k) für θ k=4.4e−011

−0.01 0.01 0.02

−5 5 −4 −2 2 Im(h

k) für θ k=4.4e−012

−0.01 0.01 0.02

−5 5 −4 −2 2 Im(h

k) für θ k=4.4e−013

−0.01 0.01 0.02

−5 5 −4 −2 2 Im(h

k) für θ k=4.4e−014

−0.01 0.01 0.02

Figure: Re{hT,0} and Im{hT,0} for different regularisation parameters θ0n; γ0 = 1.1

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

Figure: Result (resistivity and phase shift) for a geophysical measurement

data ; 50 electrodes Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

Thank you! ⋄ ⋄ ⋄ This work was supported by Deutsche Forschungsgemeinschaft DFG, (WE 1557/12-1,3).

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

A.Dey, H.F.Morrison: Resistivity Modelling for Arbitrary Shaped Two Dimensional Structures, Part I: Theoretical Formulation, Energy and Environment Division, Lawrence Berkeley Laboratory, University of California/Berkeley, Oktober (1976). J.P.Kaipio, V.Kohlemainen, E.Somersalo, M.Vauhkonen: Statistical inversion and Monte Carlo sampling nethods in electrical impedance theory, Inverse Problems 16 (2000), (1487-1522). P.Monk: Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford Science Publication (2003).

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

Theorem (Lax-Milgram Theorem)

Let B : H × H → C be a sesquilinear form on a Hilbert space H with following properties

1 continuity:

∃M > 0 : |B(u, v)| ≤ MuHvH, ∀u, v ∈ H,

2 H-coercivity:

∃C > 0 : |B(u, u)| ≥ Cu2

H,

∀u ∈ H, and let L : H → C be a bounded linear functional on H. Then there exists u0 ∈ H uniquely such that B(u0, v) = L(v), ∀v ∈ H. In addition the estimate u0 ≤ M

C LH∗ holds, where H∗ is the dual space of H.

[Mo03]

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

Lemma

There exists a constant C > 0 such that uH1(Ω) ≤ C „ ∇uL2(Ω) + | Z

∂Ω

uds| « for all u ∈ H1(Ω). This result is also valid, if ∂Ω is replaced by a subset of ∂Ω with a positive measure. Furthermore the integral on the right hand side can be replaced by an integral over Ω. [Mo03, Lemma 3.13]

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

Lemma (C´ ea)

Let Hh ⊂ H, h > 0 be a family of finite-dimensional subspaces of a Hilbert space H. Let B : H × H → C be a bounded, coercitive sesqui linear form and f ∈ H′. Then the problem of determining uh ∈ Hh, with B(uh, φh) = f (φh) for all φh ∈ Hh has a unique solution. If u ∈ H is exact solution of B(u, φ) = f (φ) for all φ ∈ H, then there exists a constant C which is independent on u, uh and h, such that the following estimate holds u − uhH ≤ C inf

vh∈Hh

u − uhH.

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics

Introduction The Ddirect Problem Inverse Problem Numerical Examples Synthetic Examples Real Example

A B M N ρ (Ωm) (mrad) x y x y x y x y . . . . . . . . . . . . . . . . . . . . 6.698 78.991 15.404 76.884 9.574 78.073 12.454 77.323 53.60 0.00 7.649 78.666 16.399 76.739 10.525 77.808 13.421 77.128 50.85 0.00 8.618 78.374 17.420 76.630 11.463 77.577 14.413 77.020 58.30 0.00 9.574 78.073 18.402 76.594 12.454 77.323 15.404 76.884 62.97 0.00 10.525 77.808 19.387 76.532 13.421 77.128 16.399 76.739 68.76 0.00 11.463 77.577 20.373 76.518 14.413 77.020 17.420 76.630 71.79 0.00 12.454 77.323 21.407 76.522 15.404 76.884 18.402 76.594 74.56 0.00 13.421 77.128 22.396 76.516 16.399 76.739 19.387 76.532 75.07 0.00 14.413 77.020 23.380 76.500 17.420 76.630 20.373 76.518 74.04 0.00

• 24.428

76.349

• 12.434

77.113

• 20.388

76.485

• 16.465

76.776 66.72 0.00

• 23.439

76.348

• 11.469

77.349

• 19.414

76.583

• 15.415

76.842 76.42 0.00

• 22.461

76.392

• 10.507

77.601

• 18.476

76.668

• 14.480

76.884 82.61 0.00

• 21.442

76.462

• 9.524

77.887

• 17.472

76.620

• 13.464

76.947 85.20 0.00

• 20.388

76.485

• 8.582

78.183

• 16.465

76.776

• 12.434

77.113 82.86 0.00 . . . . . . . . . . . . . . . . . . . .

Table: A fragment of a geoelectrical measurement data on a dike; dipole-dipole electrode configuration; a total of 387 measurements (of specific resistivity); A,B: active (current) electrodes; M,N: passive (measuring) electrodes

Andreas Helfrich-Schkarbanenko Complex Electrical Impedance Tomography in Geoelectrics