Inverse Problems for Maxwell's Equations [Reprint 2014 ed.] 9783110900101, 9783110354997

Table of contents :
Preface
Introduction
1 Cauchy problem for Maxwell’s equations
1.1 Maxwell’s equations as a hyperbolic symmetric system
1.2 Structure of the Cauchy problem solution in case of the current located on the media interface
2 One-Dimensional Inverse Problems
2.1 Structure of the Fourier-image of the Cauchy problem solution for one-dimensional medium in case of the current located at a point
2.2 The problem of determining the medium permittivity
2.3 The problem of determining the conductivity coefficient
2.4 The problem of determining all the coefficients of Maxwell’s equations
3 Multidimensional Inverse Problems
3.1 Linearization method applied to the inverse problems
3.2 Investigation of the linearized problem of determining the permittivity coefficient
3.3 Unique solvability theorem for a two-dimensional problem of determining the conductivity coefficient analytic in one variable
3.4 On the uniqueness of the solution of three-dimensional inverse problems
4 Inverse Problems in the Case of Source Periodic in Time
4.1 One-dimensional inverse problems
4.2 Linear one-dimensional inverse problem
4.3 Linearized three-dimensional inverse problem
5 Inverse Problems for Quasi-Stationary Maxwell’s Equations
5.1 On correspondence between the solutions of quasi-stationary and wave Maxwell’s equations
5.2 An one-dimensional inverse problem of determining the conductivity and permeability coefficients
5.3 The one-dimensional inverse problem for wave-quasistationary system of equations
6 The Inverse Problems for the Simplest Anisotropic Media
6.1 On the uniqueness of determination of permittivity and permeability in anisotropic media
6.2 On the problem of determining permittivity and conductivity tensors
7 Numerical Methods
7.1 Projection method for solving the Multidimensional Inverse Problems
7.2 One-dimensional problems
7.3 Two-dimensional direct problems
7.4 Finite-difference scheme inversion (FDSI)
7.5 Linearization method
7.6 Newton-Kantorovich method
7.7 Optimization methods
7.8 Dynamical version of the Gel’fand-Levitan method
8 Convergence Results
8.1 Definitions and examples
8.2 Local well-posedness and uiqueness on the whole
8.3 Well-posedness in a neighborhood of the exact solution
8.4 Convergence of the finite-difference scheme inversion
9 Examples
9.1 Inductive dielectric well-logging
9.2 Nearsurface radarlocation problem
References

Citation preview

INVERSE AND I I I POSED PROBLEMS SERIES

Inverse Problems for Maxwell's Equations

INVERSE AND ILL-POSED PROBLEMS SERIES

Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

///VSP///

Utrecht, The Netherlands, 1994

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

© VSP BV 1994 First published in 1994 ISBN 90-6764-172-3

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Romanov, V.G. Inverse problems for Maxwell's equations / V.G. Romanov and S.I. Kabanikhin. - Utrecht: VSP Withref. ISBN 90-6764-172-3 bound NUGI811 Subject headings: differential equations.

Printed in The Netherlands by Koninklijke Wöhrmann bv, Zutphen.

1

Contents Preface Introduction 1

Cauchy problem for Maxwell's equations 1.1 1.2

2

Maxwell's equations as a hyperbolic symmetric system Structure of the Cauchy problem solution in case of the current located on the media interface

iii 1 4 5 12

One-Dimensional Inverse Problems 2.1 Structure of the Fourier-image of the Cauchy problem solution for onedimensional medium in case of the current located at a point 2.2 The problem of determining the medium permittivity 2.3 The problem of determining the conductivity coefficient 2.4 The problem of determining all the coefficients of Maxwell's equations . .

21

Multidimensional Inverse Problems 3.1 Linearization method applied to the inverse problems 3.2 Investigation of the linearized problem of determining the permittivity coefficient 3.3 Unique solvability theorem for a two-dimensional problem of determining the conductivity coefficient analytic in one variable 3.4 On the uniqueness of the solution of three-dimensional inverse problems .

54 54

4

Inverse Problems in the Case of Source Periodic in T i m e 4.1 One-dimensional inverse problems 4.2 Linear one-dimensional inverse problem 4.3 Linearized three-dimensional inverse problem

83 84 92 96

5

Inverse Problems for Quasi-Stationary Maxwell's Equations 5.1 On correspondence between the solutions of quasi-stationary and wave Maxwell's equations 5.2 An one-dimensional inverse problem of determining the conductivity and permeability coefficients 5.3 The one-dimensional inverse problem for wave-quasistationary system of equations

3

6

22 27 39 43

58 62 74

101 102 104 114

The Inverse Problems for the Simplest Anisotropic Media 120 6.1 On the uniqueness of determination of permittivity and permeability in anisotropic media 121 6.2 On the problem of determining permittivity and conductivity tensors . . . 129

ii 7

Numerical Methods 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

8

Convergence Results 8.1 8.2 8.3 8.4

9

Definitions and examples Local well-posedness and uiqueness on the whole Well-posedness in a neighborhood of the exact solution Convergeiice of the finite-difference scheme inversion

Examples 9.1 9.2

146

Projection method for solving the Multidimensional Inverse Problems . . . 147 One-dimensional problems 151 Two-dimensional direct problems 162 Finite-difference scheme inversion ( F D S I ) 165 Linearization method 177 Newton-Kantorovich method 188 Optimization methods 197 Dynamical version of the Gel'fand-Levitan method 206

Inductive dielectric well-logging Nearsurface radarlocation problem

References

212 212 216 218 221

224 225 230

238

iii The book offers a simultaneous presentation of the theory and numerical treatment of inverse problems for Maxwell's equations. The inverse problems are central to many areas of science and technology such as geophysical exploration, remote sensing, nearsurface radarlocation, dielectric logging, medical imaging, etc. The basic idea of inverse methods is to extract from the evaluation of measured electromagnetic field the details of the medium considered. The inverse problems are investigated not only for Maxwell's equations but also for their guasistationary approximation and in the case of harmonic dependence in time. Starting with the simplest one-dimensional inverse problems, the book leads its readers to more complicated multidimensional ones studied for media of various kinds. The unique solvability of a number of the considered problems is shown as well as the stability of their solutions. The numerical analysis ranges from the finitedifference scheme inversion to the linearization method and finally the dynamic variant of the Gel'fand-Levitan method. Computational results are presented. The book is intended to provide graduate students in applied mathematics and geophysics, as well as the researches in the field, with an understanding of inverse problem theory. Although the main part of the book is rather theoretical in nature, it is also of practical value to experimentalists and engineers. Chapters 1-6 were written by V. G. Romanov and Chapters 7-9 by S. I. Kabanikhin. The book contains the results obtained by either the authors or their younger colleagues. So it does not pretend to display systematically the extensive general theory of geoelectric inverse problems whose origins date back to Tikhonov (1946, 1949, 1965). We mention also the books by Van'yan (1965), Dmitriev (1977), Berdichevskii and Zhdanov (1981) highlighting different aspects of interpretation of the electromagnetic field observed. We express sincere gratitude to our colleagues V. I. Priimenko, T . P. Pukhnacheva, K. S. Abdiev, S. Sh. Bimuratov, Κ. T . Iskakov, S. V. Martakov, T. Yu. Morgunova, and A. D. Satybaev for permission to use their results. We are indebted to I. G. Belinskaya for editing and translating the book into English. We are also thankful to S. P. Belinskii, A. L. Karchevskii, S. V. Martakov, and Ν. I. Martakova for their help in preparing the text to publication. The researches presented in the book were supported in part by the Russian Foundation of Fundamental Investigations under Grant 93-011-1739 and carried out. at the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. V. G. Romanov S. I. Kabanikhin

1

INTRODUCTION The book provides a study of the direct and inverse problems for Maxwell's equations curl Η =

+ σ Ε -f j ,

curl Ε = - μ η £ ·

(0.1)

Here Ε = (Ει,Ε^,Εζ)" and Η = (Hi, Η2, Η3)* are vector-functions depending on a point χ € R 3 , X = (χχ,χ2, .τ 3 ), and on the time variable t; the coefficients ε, μ, σ are functions of χ € R 3 ; and j = j ( x , t). The space R 3 is divided by the plane .T3 = 0 into two half-spaces R3_ = { x € R 3 I x3 < 0 } ,

R 3 = { x € R 3 I a-3 > 0}.

The coefficients ε, μ, σ are thought of as constant in R i and smooth in R 3 . On the common boundary X3 = 0 of R 3 , R 3 these coefficients may have a finite jump. Equations (0.1) are therefore understood in the following sense: they are assumed to hold in each of the domains R i , R 3 , while at the plane £3 = 0 the tangential components of Ε and Η should satisfy the continuity conditions Ei | X 3 = _ o = Ε{ |*3=+ο.

Hi | l 3 _ _ 0 = Η,· |r3=+o,

i = 1,2.

(0.2)

The vectors Ε , Η are also supposed to vanish identically up to the moment t = 0, whence ( Ε , H ) « o ξ 0, j | t ,·, «/>;, i — 1,2, can not be set arbitrarily since they are connected by two linear relations. Indeed, given ,·, we can solve the initial boundary value problem (0.1), (0.3) in the domain D~ with the boundary conditions E, |j-,=o= φί, i = 1,2. Thereby, we uniquely determine in D~ all the components of Ε , Η and, hence, calculate φί from the functions φί, i = 1,2. Thus, for the problem to be well-posed, only two of the four functions in (0.4) can be set independently. Any pair of (φι, ι ^ ) , (Φι, Φι), {ψ11 Φι), (ψ2, Φι) may be taken. For each of them one can define a well-posed initial boundary value problem in D~ and, having solved it, find the rest of Ε, Η components at the plane x3 = 0. But we shall not restrict ourselves to a special choice of independent functions, because for the general formulation of inverse problems it is more convenient to use the information on solution to (0.1)-(0.3) in the form (0.4). We shall specify the choice of data when considering particular cases. The presented inverse problems arise frequently in the mathematical modeling of various physical phenomena. In particular, they are closely connected with the practical needs of geophysical exploration. As an illustration, we take the following problem of geoelectrics. Let the Earth, with the flat surface { x | . t 3 = 0}, be modeled as the halfspace R + , and the air correspond to R l . The function j ( x , t) stands for an exterior current generating an electromagnetic field. The functions Ε , Η are treated as the electric and magnetic strength vectors of the electromagnetic field described by equations (0.1). The coefficients ε,μ,σ denote the permittivity, permeability, and conductivity of a medium, respectively. These coefficients are some known constants in the air R? and varying unknowns in the earth R+. They are of major interest in geophysical exploration. To determine them, an exterior current is generated and the strength components φ, ψ of the electromagnetic field are measured on the surface. The mathematical theory presented in the book affords ground for the interpretation of the electromagnetic field observed, i.e., for recovering the unknown medium parameters from the measurements. Let us make now a general terminological note. Usually system (0.1) is supplemented by the two equations div (μΗ) = 0, div ( i E ) = 4ττ ρ

(0.5)

Strictly speaking, it is the system (0.1), (0.5) which used to be called a system of Maxwell's equations. However, in the book we do not use equations (0.5) anywhere, regarding system (0.1) as an independent object. This treatment is based on the following reasoning. The first equation of (0.5) is a direct consequence of (0.1),(0.3); so it is fulfilled for any solution to problem (0.1)-(0.3). The second equation of (0.5) can be naturally considered as an independent equation for determining the charge density ρ which is beyond our interest here. At the same time, the electric strength vector Ε can be found from (0.1)—(0.3). Thus, equations (0.1) are the major and quite independent part of Maxwell's equations.

3 In the book the one-, two-, and three-dimensional direct and inverse problems for equations (0.1) are considered (the dimensionality of inverse problems is defined by the number of variables upon which the desired coefficients depend). We study the well-posedness of different problems of determining the conductivity, permittivity, and permeability. The simplest cases of anisotropy are considered, and linearized problems are investigated. Numerical methods for solving the direct and inverse problems for Maxwell's equations are constructed on the basis of the projection-difference approach. The comparative analysis of methods of finite difference scheme inversion, Newton-Kantorovich type methods, optimization methods is performed. The results of computational experiments are presented. Chapters 1-6, 8 should be regarded as an account of the inverse problem theory for Maxwell's equations; that is why we have striven for fullness, rigor, and brevity in presenting the material of these chapters, having in mind the maximal attainable generality. Chapters 7, 9 contain the description of numerical methods for solving the inverse problems for Maxwell's equations. Here the major objectives were rather simplicity and clearness than rigor, the attention being directed to algorithmic details and the main technical tools.

4

CHAPTER 1

Cauchy Problem for Maxwell's Equations

The topic of this book is a study of the inverse problems for Maxwell's equat ions. However, it is obvious that the investigation of an inverse problem necessarily includes a research of the corresponding direct problem and an analysis of its solution properties. The theory of Cauchy problem (0.1)-(0.3) is sufficiently elaborated ( see, for example. Courant, 1962; Sobolev, 1963; Bers et ai, 1964; Mizohata, 1973). So we shall use it here insofar as the inverse problems are concerned. It turned out to be convenient to consider Maxwell's equations as a symmetric hyperbolic system ( Section 1.1 ). Taking into account the special role of the variable x3, it is natural to isolate it and represent system (0.1) in terms of new variables x , where χ = ( a - ^ x j ) and 2 = s ( x ) is a solution to the oikonal equation |Vz|2 = εμ, ζ |r3=0= o, 2*3 > 0. It means that x3 is replaced by the variable 2 equal in modulus to the travel time of an electromagnetic signal ( its speed, as is known, is ) coming from a point χ to the plane x3 = 0. The plane 2 = 0 corresponds to the plane x3 = 0. After introducing some new functions, the principal part of the differential operator in (0.1) is reduced to a special canonical form with respect to the variables 2 , i (see (1.1.17)), which was convenient for an investigation of the main properties of the Cauchy problem solution. In geophysical exploration the exterior current is located, as a rule, on the earth-air interface and directed along it. In this case it is simulated by the function j ( x , / ) = +0 or ζ —* —0, respectively. Thus, the initial Cauchy problem is reduced to problem (1.1.17), (1.1.21). (1.1.23). In the following we shall consider, as a rule, the electromagnetic oscillations which are generated by an exterior current j located on the plane .T3 = 0 and directed along it. j = j°(x,/)«(i3),

j° = (j?,j?6). Denoting W1 =

W

W

=

=

« = 1,2,

(1.1.30)

we can write, for every vector-function W ' , Li

W

W = F",

=

ζ φ 0.

(1.1.31)

Here A' = diag ( - 1 , 1 , 0 ) , F' =

(Fj.

F, + 2,

t = 1.2,

F+λ),

1

the matrices C , C'l are formed by the elements of B\, B 2 with both subscripts odd. and C2, C'l are made of elements of the same matrices B\. B2 but with both subscripts even. In particular, / pi/c 0 1l(cy/2) \ & = \ . C2

0

-Pl/c

l/(cV2)

- 1 /(cy/2)

(

p,/c

0

0

= y/εμ

-ρ,/c

1l(cs/2)

- 1 l(cy/2)

,

0

)

l/(cy/2) 11

\

(cy/2) 0

\I(cy/2)

c = y j \ + p i r*

-

r