Electromagnetic inverse profiling of stratified media

ELECTROMAGNETIC

INVERSE PROFILING OF

STRATIFIED MEDIA

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan

de Technische Universiteit Delft, op gezag van

de Rector Magnificus, prof. drs. P.A. Schenck,

in het openbaar te verdedigen ten overstaan van

een commissie aangewezen door het College van Dekanen

op donderdag 21 september 1989 te 16.00 uur

door

Michael Cosmas Silvester Zeijlmans

geboren te 's-Gravenhage

natuurkundig doctorandus

TR diss

1752

FIELD DERIVATIVES WITH RESPECT TO PROFILE PARAMETERS: A DERIVATION BASED UPON RECIPROCITY

2.1 I n t r o d u c t i o n 7 2.2 P r o p e r t i e s of t h e e l e c t r o m a g n e t i c f i e l d s i n the

s c a t t e r i n g c o n f i g u r a t i o n 9 2 . 3 Formulation of the problem 11 2.4 E x c i t a t i o n by an e l e c t r i c - or m a g n e t i c - c u r r e n t source and o b s e r v a t i o n at a p o i n t 17 2.5 E x c i t a t i o n by an e l e c t r i c - or m a g n e t i c - c u r r e n t source and o b s e r v a t i o n i n a s p e c i f i c d i r e c t i o n i n t h e f a r - f i e l d r e g i o n 22 2.6 E x c i t a t i o n by an e l e c t r o m a g n e t i c p l a n e wave and o b s e r v a t i o n a t a p o i n t 24 2.7 E x c i t a t i o n by an e l e c t r o m a g n e t i c p l a n e wave and o b s e r v a t i o n i n a s p e c i f i c d i r e c t i o n i n t h e f a r - f i e l d r e g i o n 26 2.8 F i e l d d e r i v a t i v e s i n p l a n e - s t r a t i f i e d and c i r c u l a r l y c y l i n d r i c a l l y s t r a t i f i e d , d i e l e c t r i c media 29 2 . 8 . 1 P l a n e - s t r a t i f i e d media 29 2 . 8 . 2 C i r c u l a r l y c y l i n d r i c a l l y s t r a t i f i e d media 34

INVERSE PROFILING

3.1 I n t r o d u c t i o n 39 3.2 Formulation of the d a t a f i t t i n g problem 42

3.3 Levenberg-Marquardt methods for n o n - l i n e a r l e a s t

s q u a r e s problems 44 3.4 The s o l u t i o n of a system of l i n e a r e q u a t i o n s i n t h e

Levenberg-Marquardt a l g o r i t h m , u s i n g the method of

s i n g u l a r v a l u e decomposition 50

3 . 4 . 1 S c a l i n g 51 3.5 E r r o r a n a l y s i s 53 3.6 The i n v e r s i o n a l g o r i t h m 55

3.7 The r e l a t i o n between a Levenberg-Marquardt scheme and

a Born-type i t e r a t i v e scheme 60

ELECTROMAGNETIC RADIATION OF A MAGNETIC DIPOLE IN A PLANE-STRATIFIED, DIELECTRIC MEDIUM

4.1 I n t r o d u c t i o n 65 4 . 2 F o r m u l a t i o n of the problem 68 4 . 3 Method of s o l u t i o n 69 4 . 3 - 1 S o l u t i o n i n t h e Hankel-transform domain 69 4 . 3 . 2 E x p r e s s i o n s f o r t h e e l e c t r o m a g n e t i c f i e l d components E ( r , z ) , H ( r , z ) and H ( r , z ) 76 Ü i Z 4.4 The f i e l d d e r i v a t i v e with r e s p e c t t o a p r o f i l e parameter 77 4.5 Numerical implementation and r e s u l t s 82

4 . 5 . 1 The system of f i r s t - o r d e r d i f f e r e n t i a l

e q u a t i o n s i n t h e Hankel-transform domain 82

4 . 5 . 2 The f i e l d d e r i v a t i v e s 84 4 . 5 . 3 Numerical r e s u l t s 85

5 . ELECTROMAGNETIC RADIATION OF AN ELECTRIC-CURRENT LOOP IN A CIRCULARLY CYLINDRICALLY STRATIFIED, DIELECTRIC MEDIUM

5.1 I n t r o d u c t i o n 93 5.2 Formulation of the problem 96

5 . 3 Method of s o l u t i o n 97 5 . 3 . 1 S o l u t i o n i n t h e F o u r i e r - t r a n s f o r m domain 97 5 . 3 . 2 Expressions f o r t h e e l e c t r o m a g n e t i c f i e l d components E ( r , z ) , H ( r , z ) and H ( r , z ) 103 5.4 The f i e l d d e r i v a t i v e with r e s p e c t t o a p r o f i l e parameter 104 5.5 Numerical implementation and r e s u l t s 108

5 . 5 . 1 The system of f i r s t - o r d e r d i f f e r e n t i a l

e q u a t i o n s i n t h e F o u r i e r - t r a n s f o r m domain 108

5 . 5 . 2 The f i e l d d e r i v a t i v e s 109 5 . 5 . 3 Numerical r e s u l t s 110

THE INVERSE PROFILING PROBLEM FOR A PLANE-STRATIFIED, DIELECTRIC MEDIUM

6.1 I n t r o d u c t i o n 117 6.2 Formulation of t h e problem 120

6.3 Numerical r e s u l t s 121 6 . 3 . 1 R e t r i e v a l of both p e r m i t t i v i t y and c o n d u c t i v i t y

p r o f i l e s 124 6 . 3 . 2 The impact of e r r o r s i n t h e measurements

on t h e r e t r i e v a l of t h e p r o f i l e s 131 6 . 3 . 3 The impact of t h e l o c a t i o n and frequency of the

THE INVERSE PROFILING PROBLEM FOR A CIRCULARLY CYLINDRICALLY STRATIFIED, DIELECTRIC MEDIUM

7.1 I n t r o d u c t i o n 1^3 7.2 F o r m u l a t i o n of the problem 116 7 . 3 Numerical r e s u l t s 150 7 . 3 . 1 R e t r i e v a l of both p e r m i t t i v i t y and c o n d u c t i v i t y p r o f i l e s 150 Appendices A. The L a p l a c e - t r a n s f o r m domain e l e c t r o m a g n e t i c r e c i p r o c i t y theorem 159 B. P r o p e r t i e s of the e l e c t r o m a g n e t i c f i e l d i n t h e f a r - f i e l d r e g i o n 161 C. T r a n s m i t t i n g and r e c e i v i n g a n t e n n a s 163

D. Asymptotic behavior of U(k , z ) as' k -►<» 165 / s i 1 / s

E. Asymptotic behavior of E„(r,k ) and H (r,k.) 6 z z z

a s Ik U » 169 1 z '

F. The Gauss-Newton method 175 G. The s i n g u l a r v a l u e decomposition 176

References 179

Summary 185

The theory of electromagnetic inverse profiling is concerned with the development of methods to determine the material constituents of an object by exciting it with electromagnetic radiation. Most of the time, use can be made of measurements of the resulting field at a number of locations. It is characteristic that the source of the radiation (the transmitter) as well as the field detectors (receivers) are placed outside the object (see Figure 1.1).

Applications of electromagnetic inverse profiling theory include the detection of buried objects beneath the earth's surface in civil engineering, the imaging of biological tissue in medical science, the non-destructive evaluation of conducting materials in material science, and the investigation of the earth's structure between boreholes in geophysics. The method of inversion that is discussed in this thesis can, however, also be fruitfully used in acoustics. Several surveys covering the many fields in which inversion methods may be applied are available to the interested reader (Langenberg [1989,1986], Daniels [1988], Boerner et al. [1985], Devaney [1985], Boerner, Jordan and Kay [1981], Baltes [1980,1978]).

RECEIVERS

Fig. 1.1 The inverse profiling configuration.

Most methods of s o l u t i o n for e l e c t r o m a g n e t i c i n v e r s e profiling problems belong t o one of t h r e e t y p e s , i r r e s p e c t i v e of whether the a n a l y s i s t a k e s p l a c e in the time domain or in the frequency domain. A d e t a i l e d d i s c u s s i o n of t h e s e t h r e e types of methods, for t h e o n e -d i m e n s i o n a l c a s e of a plane-wave i n c i -d e n t on a s l a b , i s given by Tijhuis [1987].

The methods of the f i r s t type of s o l u t i o n l i n k up with work by Kay [1955]. Through a Liouville transformation, he converted the inverse problem of a l o s s l e s s slab to a problem that had already been solved in

quantum-mechanical s c a t t e r i n g t h e o r y ( t h e Gel'fand-^Levitan p r o b l e m ) . The

application of these methods in p r a c t i c e , however, i s r e s t r i c t e d to one-dimensional problems.

The methods of the second type make use of an i n t e g r a l expression for the d i f f e r e n c e of the measured f i e l d a t the r e c e i v e r and t h e ( c a l c u l a t e d ) f i e l d at the receiver location in a background medium. The contrast function, being the d i f f e r e n c e of t h e unknown p r o f i l e and a known background p r o f i l e , i s expanded i n t o a number of base functions. Employing a Born-type approximation, one t h e n t r i e s t o recover the unknown p r o f i l e in an i t e r a t i v e process, while updating the background a t every i t e r a t i o n (see for example T i j h u i s [1987] and Habashy e t a l . [1986]).

With the t h i r d type, optimization techniques are employed, such as Newton-type methods and conjugate-gradient methods. I t i s c h a r a c t e r i s t i c of such methods that a model object i s introduced with a (known) p r o f i l e t h a t depends on a number of p r o f i l e parameters. These adjustable p r o f i l e parameters are t o be chosen such t h a t t h e ( c a l c u l a t e d ) f i e l d v a l u e s o u t s i d e the model object best f i t the measured data. In t h i s process an error function i s minimized (see Eaton [ 1 9 8 9 ] , Santosa e t a l . [ 1 9 8 8 ] , T i j h u i s [ 1 9 8 7 ] , Lines and T r e i t e l [ 1 9 8 4 ] , Roger et a l . [1986] and Lesselier [1982]).

With t h e Born-type i t e r a t i v e methods t h e following d i f f i c u l t i e s a r i s e , among o t h e r s : the convergence of the i t e r a t i v e scheme i s not guaranteed; the number of unknown parameters i s determined by the number of expansion functions necessary t o describe the contrast f u n c t i o n ; the i l l - p o s e d n e s s of the o r i g i n a l problem causes the r e s u l t i n g numerical problem to be i l l - c o n d i t i o n e d . To remedy the l a s t mentioned, one r e s o r t s t o r e g u l a r i z a t i o n methods. C r i t e r i a f o r choosing the r e g u l a r i z a t i o n parameters are seldom given; the numerical schemes a r e u s u a l l y d r a f t e d by t r i a l and e r r o r . An important advantage of the Born-type i t e r a t i v e methods i s the fast convergence in the region near the s o l u t i o n .

The o p t i m i z a t i o n methods on the other hand, which a l s o experience d i f f i c u l t i e s with the r e g u l a r i z a t i o n , show convergence t o a s t a t i o n a r y

p o i n t o r , in some c a s e s , to a (local) minimum. However, t h e i r rates of convergence can be very slow, l e n g t h e n i n g t h e r e q u i r e d computation times.

In t h i s t h e s i s the Levenberg-Marquardt algorithm for the minimization of a c u m u l a t i v e s q u a r e d e r r o r f u n c t i o n i s chosen (Chapter 3 ) . T h i s o p t i m i z a t i o n a l g o r i t h m combines t h e a d v a n t a g e of a g u a r a n t e e d c o n v e r g e n c e t o a s t a t i o n a r y p o i n t with r a p i d convergence near t h e solution (Fletcher [1980], Marquardt [1963] and Levenberg [ 1 9 4 4 ] ) . In f a c t , t h e L e v e n b e r g - M a r q u a r d t scheme combines the Gauss-Newton optimization method and the steepest-descent optimization method. In the i n i t i a l i t e r a t i o n s t e p s , the rapid convergence of the steepest descent method i s used a d v a n t a g e o u s l y . When a r r i v e d i n a r e g i o n near t h e s o l u t i o n , the good convergence properties of the Gauss-Newton method are

apparent. The what is called Levenberg-Marquardt parameter r e g u l a t e s which of the two methods dominates. In every i t e r a t i o n s t e p of the Levenberg-Marquardt a l g o r i t h m a system of l i n e a r e q u a t i o n s must be s o l v e d , t h e s i z e of which depends on t h e chosen number of p r o f i l e parameters of the model. The s o l u t i o n of t h i s system of equations i s obtained by making use of the method of singular value decomposition.

The Levenberg-Marquardt algorithm employs the f i e l d d e r i v a t i v e s w i t h r e s p e c t t o the p r o f i l e parameters of the model o b j e c t . These derivatives cannot be obtained by d i r e c t d i f f e r e n t i a t i o n , because no e x p l i c i t e x p r e s s i o n i s a v a i l a b l e for t h e f i e l d as a function of the p r o f i l e p a r a m e t e r s . T h i s problem i s c i r c u m v e n t e d by u s i n g t h e e l e c t r o m a g n e t i c r e c i p r o c i t y theorem in a s u i t a b l e manner (Chapter 2 ) . The a n a l y s i s i s carried out in the time-Laplace transform domain. F i e l d d e r i v a t i v e s are o b t a i n e d for s e v e r a l important t r a n s m i t t e r - r e c e i v e r s i t u a t i o n s , such as excitation by an e l e c t r i c or magnetic d i p o l e with o b s e r v a t i o n at a point, and e x c i t a t i o n by an electromagnetic plane wave with observation in a s p e c i f i c direction in the f a r - f i e l d region.

I t i s assumed t h a t in the measured data, which are the input data for the Levenberg-Marquardt a l g o r i t h m , t h e r e i s a c e r t a i n e r r o r . Of

course t h i s e r r o r in the data c a r r i e s over i n t o an error of the f i n a l , optimal, parameters. An e r r o r a n a l y s i s of t h i s p r o c e s s i s g i v e n . I t leads t o , among other t h i n g s , two c r i t e r i a for terminating the i t e r a t i v e scheme. A combination of the Levenberg-Marquardt scheme and the r e s u l t s of the e r r o r a n a l y s i s y i e l d s the complete (3-D) inversion algorithm (Subsection 3 . 6 ) . In a p r a c t i c a l s i t u a t i o n , the number of data r e q u i r e d t o determine a c e r t a i n number of p r o f i l e parameters of the model i s of importance. Insight into t h i s can be obtained from a c l o s e examination of t h e L e v e n b e r g - M a r q u a r d t scheme. The method of s i n g u l a r value decomposition i s herewith an important t o o l .

Use of t h e Levenberg-Marquardt method sheds new l i g h t on t h e complementary behavior of i n v e r s i o n schemes based on o p t i m i z a t i o n t e c h n i q u e s and t h o s e based on Born-type i t e r a t i v e methods (Tijhuis

[1987], page 316). In f a c t , i t brings them together, because i t i s shown t h a t the Born-type i t e r a t i v e algorithm can be considered as a p a r t i a l l y c a r r i e d out Levenberg-Marquardt a l g o r i t h m ( S u b s e c t i o n 3 . 7 ) . The mechanism, in the l a t t e r method, t o a d j u s t the Levenberg-Marquardt parameter i s essential to obtain convergence. In the Born-type i t e r a t i v e m e t h o d , h o w e v e r , such a mechanism t o a d j u s t the r e g u l a r i z a t i o n parameter, as i t i s called in t h i s context, does not e x i s t .

Numerical r e s u l t s with the inversion algorithm are obtained for two inverse problems in c i v i l engineering and geophysics. The f i r s t problem concerns the r e t r i e v a l of the unknown p e r m i t t i v i t y and conductivity profiles of a p l a n e - s t r a t i f i e d medium, e x c i t e d by a magnetic d i p o l e a n t e n n a , from measurements of the p e r p e n d i c u l a r component of the magnetic f i e l d s t r e n g t h (Chapter 6 ) . The second one c o n c e r n s t h e r e t r i e v a l of the permittivity and conductivity profiles of a c i r c u l a r l y c y l i n d r i c a l l y s t r a t i f i e d medium, excited by an e l e c t r i c - c u r r e n t l o o p , from measurements of the azimuthal component of the e l e c t r i c f i e l d strength (Chapter 7). In both cases the measurements are c a r r i e d out at s e v e r a l f r e q u e n c i e s and a t several l o c a t i o n s . The model object in the i n v e r s i o n scheme i s taken t o be p l a n e - s t r a t i f i e d , r e s p e c t i v e l y c i r c u l a r l y c y l i n d r i c a l l y s t r a t i f i e d , with piecewise l i n e a r p e r m i t t i v i t y

and conductivity p r o f i l e s . Special a t t e n t i o n i s paid t o the impact of e r r o r s in the measurements on the r e t r i e v a l of the p r o f i l e s , as well as on t h e impact of the l o c a t i o n of the r e c e i v e r s and f r e q u e n c y of o p e r a t i o n . The i n v e r s i o n schemes have been presented in such a manner t h a t they can be programmed s t r a i g h t away.

In s o l v i n g t h e above-mentioned i n v e r s i o n problems, d i r e c t - f i e l d computations have to be carried out (Chapters 4 and 5 ) , not only because t h e y form p a r t of t h e i n v e r s i o n a l g o r i t h m , but a l s o t o g e n e r a t e s y n t h e t i c d a t a . To c a l c u l a t e t h e f i e l d q u a n t i t i e s a H a n k e l t r a n s f o r m a t i o n i s p e r f o r m e d i n the case of the p l a n e - s t r a t i f i e d configuration, and a Fourier transformation i s performed in t h e case of the c i r c u l a r l y c y l i n d r i c a l l y s t r a t i f i e d configuration. In both cases, in the transform domain a s e t of coupled d i f f e r e n t i a l e q u a t i o n s r e s u l t s , which i s s o l v e d n u m e r i c a l l y . This i s d i f f e r e n t from the s t a n d a r d approach, in which the f i e l d in an inhomogeneous medium i s approximated by t h e f i e l d of a piecewise homogeneous medium. Expressions for the f i e l d derivatives with respect to the p r o f i l e parameters a r e given in Chapters 4 and 5 as w e l l . Their numerical evaluation i s discussed and elucidated with some examples.

CHAPTER 2

FIELD DERIVATIVES WITH RESPECT TO PROFILE PARAMETERS: A DERIVATION BASED UPON RECIPROCITY

2 . 1 . Introduction

In electromagnetic inverse profiling t h e o r y , one t r i e s t o r e c o n s t r u c t the unknown s p a t i a l d i s t r i b u t i o n of the c o n s t i t u t i v e parameters of an object that i s irradiated by a known e l e c t r o m a g n e t i c s o u r c e from the measured f i e l d values outside the object.

In t h i s c h a p t e r a s y s t e m a t i c d e r i v a t i o n i s p r e s e n t e d of f i e l d d e r i v a t i v e s with respect to p r o f i l e parameters of a model object. These field derivatives are used in the process of solving the above-mentioned t y p e of i n v e r s e p r o f i l i n g problems. The d e r i v a t i o n i s based upon reciprocity and i s carried out in the time-Laplace transform domain. I t i s assumed t h a t the media a r e t i m e - i n v a r i a n t and l i n e a r in t h e i r electromagnetic behavior.

A r e c i p r o c i t y theorem i n t e r r e l a t e s , in a specific way, the field quantities associated with two n o n - i d e n t i c a l p h y s i c a l s t a t e s t h a t can occur in one and the same space-time or space-frequency domain (De Hoop

[1987]). Such a reciprocity theorem applies to electromagnetic f i e l d s in t i m e - i n v a r i a n t c o n f i g u r a t i o n s t h a t a r e l i n e a r and l o c a l l y r e a c t i n g .

Because of t h i s time i n v a r i a n c e and l i n e a r i t y , r e c i p r o c i t y r e l a t i o n s a r e most e a s i l y formulated in t h e t i m e - L a p l a c e domain, t h e s-domain. In the s-domain one d i s t i n g u i s h e s between a f i e l d r e c i p r o c i t y t h e o r e m (named a f t e r H.A. Lorentz) and a power r e c i p r o c i t y theorem. Their c o u n t e r p a r t s i n t h e time domain, being of t h e t i m e - c o n v o l u t i o n and t i m e - c o r r e l a t i o n t y p e , r e s p e c t i v e l y , f o l l o w by a p p l y i n g the s t a n d a r d r u l e s for i n v e r s i o n from t h e s - t o t h e t i m e d o m a i n . For a d i s c u s s i o n of ( t i m e d o m a i n ) r e c i p r o c i t y theorems, t h e r e a d e r i s r e f e r r e d t o De Hoop [1987]» where a l i s t of r e f e r e n c e s to e a r l i e r papers can a l s o be found, and t o t h e paper by B o j a r s k i [ 1 9 8 3 ] . The Lorentz r e c i p r o c i t y theorem i n the s-domain i s given in Appendix A.

To d e t e r m i n e t h e p r o f i l e of the unknown o b j e c t , a model o b j e c t i s i n t r o d u c e d . The (known) p r o f i l e of t h i s model o b j e c t depends on a number of p r o f i l e p a r a m e t e r s . The m e a s u r e d v a l u e s of t h e f i e l d due t o the p r e s e n c e of the unknown o b j e c t a r e compared w i t h t h e c a l c u l a t e d v a l u e s of t h e f i e l d of t h e model o b j e c t a t t h e p o i n t s of measurement. In t h e i n v e r s i o n scheme the p r o f i l e of t h e model i s d e t e r m i n e d i n s u c h , away t h a t t h e c a l c u l a t e d f i e l d v a l u e s b e s t f i t the measured v a l u e s , according t o a c r i t e r i o n given i n advance. The i n v e r s i o n scheme makes u s e of t h e L e v e n b e r g - M a r q u a r d t m i n i m i z a t i o n a l g o r i t h m . This a l g o r i t h m employs t h e f i e l d d e r i v a t i v e s with r e s p e c t t o t h e p r o f i l e p a r a m e t e r s of t h e model o b j e c t . T h e s e f i e l d d e r i v a t i v e s a r e d e t e r m i n e d f o r t h e f o l l o w i n g t r a n s m i t t e r - r e c e i v e r s i t u a t i o n s , i m p o r t a n t i n i n v e r s e p r o f i l i n g e x p e r i m e n t s : ( i ) e x c i t a t i o n by an e l e c t r i c - or a m a g n e t i c - c u r r e n t s o u r c e and o b s e r v a t i o n a t a p o i n t ( S u b s e c t i o n 2 . 4 ) , ( i i ) e x c i t a t i o n by an e l e c t r i c - or a m a g n e t i c - c u r r e n t p o i n t and o b s e r v a t i o n in a s p e c i f i c d i r e c t i o n i n the f a r - f i e l d r e g i o n ( S u b s e c t i o n 2 . 5 ) , ( i i i ) e x c i t a t i o n by a n e l e c t r o m a g n e t i c p l a n e w a v e a n d o b s e r v a t i o n a t a p o i n t ( S u b s e c t i o n 2 . 6 ) , and ( i v ) e x c i t a t i o n by an e l e c t r o m a g n e t i c p l a n e wave a n d o b s e r v a t i o n i n a s p e c i f i c d i r e c t i o n i n t h e f a r - f i e l d r e g i o n ( S u b s e c t i o n 2 . 7 ) . As s p e c i f i c examples t h e c a s e of a p l a n e - s t r a t i f i e d medium ( s e e e . g . T i j h u i s [ 1 9 8 7 ] , Vidberg and Riska [ 1 9 8 5 ] , and Chew and Chuang [ 1 9 8 4 ] ) , which i s a c a n o n i c a l problem i n e l e c t r o m a g n e t i c l o g g i n g ,

and t h e c a s e of a c y l i n d r i c a l l y s t r a t i f i e d medium ( s e e e . g . Habashy e t a l . [ 1 9 8 6 ] ) , a b a s i s for r a d i a l - p r o f i l e r e c o n s t r u c t i o n i n w e l l l o g g i n g , a r e d i s c u s s e d in d e t a i l ( S u b s e c t i o n 2 . 8 ) .

Taking a n o t h e r approach, t h e i n v e r s e p r o f i l i n g problem i s sometimes f o r m u l a t e d , using t h e r e c i p r c i t y theorem i n i t s g l o b a l form, i n terms of domain i n t e g r a l r e l a t i o n s ( s i m i l a r t o r e l a t i o n s ( 2 . 4 . 4 ) and ( 2 . 4 . 9 ) i n S u b s e c t i o n 2 . 4 ) . They r e l a t e t h e e l e c t r o m a g n e t i c f i e l d i n a c o n f i g u r a t i o n w i t h t h e a c t u a l o b j e c t t o b e p r o b e d a n d t h e e l e c t r o m a g n e t i c f i e l d i n a c o n f i g u r a t i o n w i t h a known background o b j e c t ( s e e Blok and Zeylmans [ 1 9 8 7 ] , where a s c a l a r f o r m u l a t i o n h a s b e e n g i v e n ) . The i n v e r s i o n p r o b l e m t h e n i n v o l v e s t h e d e t e r m i n a t i o n of a c o n t r a s t f u n c t i o n , being t h e d i f f e r e n c e of t h e unknown a c t u a l p r o f i l e and the known background p r o f i l e .

2 . 2 . P r o p e r t i e s of t h e e l e c t r o m a g n e t i c f i e l d s i n t h e s c a t t e r i n g c o n f i g u r a t i o n

The sdomain r e c i p r o c i t y theorems f o r e l e c t r o m a g n e t i c f i e l d s i n t i m e -i n v a r -i a n t c o n f -i g u r a t -i o n s t h a t a r e l -i n e a r and l o c a l l y r e a c t -i n g -in t h e -i r e l e c t r o m a g n e t i c b e h a v i o r a p p l y t o a b o u n d e d s p a t i a l domain D. The b o u n d a r y of D i s denoted by 3D. F u r t h e r i s D* t h e complement of DU3D in

3 R . n i s t h e u n i t v e c t o r p o i n t i n g away from D, a l o n g t h e n o r m a l t o 3D. 3 The p o s i t i o n of o b s e r v a t i o n in R i s s p e c i f i e d by t h e v e c t o r r = xi + y i + z i w i t h r e s p e c t t o a f i x e d o r t h o g o n a l C a r t e s i a n — —x —y —z r

r e f e r e n c e frame; {± ,i_ ,± } a r e the t h r e e mutually perpendicular u n i t v e c t o r s . In the indicated order they form a r i g h t - h a n d e d system. Since i n p h y s i c a l r e a l i t y only c a u s a l phenomena are of i n t e r e s t , the time-Laplace transform for a space-time function f = f ( £ , t ) i s taken, defined by

f"

where tf i i s t h e i n s t a n t at which the s o u r c e s t h a t g e n e r a t e t h e f i e l d a r e s w i t c h e d o n . F o r R e ( s ) > 0 t h e i n t e g r a l i s c o n v e r g e n t f o r bounded q u a n t i t i e s . C a u s a l i t y of t h e f i e l d i s t a k e n i n t o a c c o u n t by t a k i n g R e ( s ) > 0 , and r e q u i r i n g t h a t f o r a l l r a l l c a u s a l f i e l d q u a n t i t i e s be a n a l y t i c f u n c t i o n s i n the r i g h t h a l f 0<Re(s)<°° of t h e complex s - p l a n e . F r e q u e n c y - d o m a i n r e l a t i o n s a r e o b t a i n e d by t a k i n g s->— iw, w i t h w r e a l , v i a Re(s)>0 i n t h e t i m e - L a p l a c e transformed r e l a t i o n s . In t h i s way t h e c a u s a l i t y c o n d i t i o n s remain s a t i s f i e d .

In each subdomain of t h e c o n f i g u r a t i o n t h e e l e c t r o m a g n e t i c f i e l d v e c t o r s s a t i s f y Maxwell's e q u a t i o n s i n the s-domain:

V_ x H ( r , s ) - s D ( r , s ) = J ( r , s ) ,

V x E ( r , s ) + s B ( r , s ) = - K ( r , s ) ,

( 2 . 2 . 2 )

where

E_ = s-domain e l e c t r i c f i e l d s t r e n g t h (Vs/m), H = s-domain magnetic f i e l d s t r e n g t h (As/m),

2 I) = s-domain e l e c t r i c f l u x d e n s i t y (Cs/m ) , B = s-domain magnetic f l u x d e n s i t y ( T s ) ,

2 <J = s-domain volume s o u r c e d e n s i t y of e l e c t r i c c u r r e n t (As/m ) ,

2 K = s-domain volume s o u r c e d e n s i t y of magnetic c u r r e n t (Vs/m ) ,

and V = i 9 + i 8 + i 3 . Equations ( 2 . 2 . 2 ) a r e supplemented by t h e - -x x - y y - z z H c o n s t i t u t i v e r e l a t i o n s . For a l i n e a r , t i m e - i n v a r i a n t , l o c a l l y and i n s t a n t a n e o u s l y r e a c t i n g , i s o t r o p i c medium t h e s e a r e D ( r , s ) = e ( r ) E ( r , s ) , ( 2 . 2 . 3 ) B ( r , s ) = y ( r ) H ( r , s ) , where

e(r) = permittivity (F/m), y(r) = permeability (H/m).

In vacuum y = u_ = 4TT X 10~' H/m and e = e = 1 /vQol with c = 299792*458

m / s . In the case of l o s s y media, the e l e c t r i c properties in the media a r e c h a r a c t e r i z e d by t h e c o n d u c t i v i t y o =-n(rO [ S / m ] and t h e p e r m i t t i v i t y e = e(r_). Then the s-dependent p e r m i t t i v i t y E ( £ , S ) i s introduced as

E( r , s ) = e ( r ) + o ( r ) / s . (2.2.4)

In a similar way an s-dependent permeability can be i n t r o d u c e d denoted by u(r_,s).

It i s assumed that e and u are piecewise continuous functions of r_. Note that across an i n t e r f a c e , the tangential components of the e l e c t r i c and the magnetic field strengths are continuous.

2 . 3 . Formulation of the problem

By making use of the Lorentz r e c i p r o c i t y theorem, f i e l d d e r i v a t i v e s are derived with respect to the p r o f i l e parameters of a model object. These f i e l d derivatives are used in solving inverse p r o f i l i n g problems. Four b a s i c c o n f i g u r a t i o n s are used in the derivation. Two of them are model c o n f i g u r a t o n s t h a t c o n t a i n a model o b j e c t and t h e a c t u a l s o u r c e d i s t r i b u t i o n . The other two are auxiliary configurations. They contain a model object and an e l e c t r i c - and m a g n e t i c - c u r r e n t p o i n t s o u r c e , r e s p e c t i v e l y . They a r e used t o i n t r o d u c e t h e e l e c t r i c and magnetic Green's s t a t e vectors. Using r e c i p r o c i t y , p r o f i l e d e r i v a t i v e s of the f i e l d s are determined at the locations at which measurements are carried o u t . To do so, the point sources in the auxiliary configurations should be chosen at t h e s e l o c a t i o n s of measurement. Therefore, the domain in

which t h e s o u r c e s of t h e a u x i l i a r y c o n f i g u r a t i o n s a r e l o c a t e d i s sometimes c a l l e d t h e domain of o b s e r v a t i o n .

The e l e c t r o m a g n e t i c f i e l d s i n the v a r i o u s c o n f i g u r a t i o n s s a t i s f y M a x w e l l ' s e q u a t i o n s i n the t i m e - L a p l a c e transform domain. Let in one of t h e model c o n f i g u r a t i o n s the p e r m i t t i v i t y of t h e medium be g i v e n by e , t h e p e r m e a b i l i t y of t h e medium by y , and t h e s o u r c e d e n s i t i e s by J and

3 3 K . The electric and magnetic field strengths are denoted by E and H ,

—a a a respectively. With (2.2.2) - (2.2.3) one has

V x H - s e E = J , — —a a -a -a

V x E + s y H = - K . — —a a -a -a

(2.3.D

The sources J and K are located in the bounded domain D . Consider the

—a —a a scattering object as a contrast with respect to a given embedding. Then,

the incident fields E and H are introduced that satisfy —a —a

V x H1 - s e„ E* = J , — —a e —a —a

(2.3-2) V x E1 + s y H1 = -K ,

- —a e -a -a

in which e = e (r,s) and u = u (r,s) are the permittivity and the e e e e

-permeability of the embedding, respectively. It is assumed that e and s s u are constant outside some sphere S . The scattered fields E„ and H , *e o —a —a defined by Es = E - E1, -a —a —a H3 = H - H1. -a —a -a (2.3.3)

V x HS - s e ES = JS, - -a e -a -a V x ES + s y HS = -KS. — -a e —a —a (2.3.4) s s

In (2.3.4), the contrast source densities J and K , introduced as

~""3. 3 (2.3.5) js = s[e - e ] E , -a a e -a KS = s[y - y ]H , -a a e -a g

differ from zero in the scattering domain D . a

Let in the other model configurations the permittivity of the medium be given by e , the permeability of the medium by y , and the source densities again by J and K . The electric and magnetic fields E.

—a —a —b and H satisfy V x H - s e. E. = J , — —b b —b —a V x E + s y.H. = -K . - -b b -b -a (2.3.6)

Again, a description is used in terms of contrast sources, with respect to the same embedding as in (2.3.2). Therefore, one writes

h=£

+

£

H = H* + H°, —b —b —b

(2.3.7)

in which the incident fields E. and H. satisfy

V X H ^ - S E E* = J ,

— -b e -b -a

(2.3.8) V x E* + s p H* = -K .

Then, t h e s c a t t e r e d f i e l d s E_ and H a r e given by V x at - s z . El = J3, — —b e —b —b V x r* + s u H3 = -K* — - b e —b - b s s

The contrast source densities J. and K. , introduced as —b —b (2.3.9) J. = s[e, - e ] E , — D b e —b

^b

= sCw

b "

M

e

]

h'

(2.3.10) g

differ from zero in the scattering domain D. . b

Introduce the difference fields E . and H, by E. = E - E. and —d —d —d —a —b H_, = H - H , i n w h i c h e x p r e s s i o n s E and H s a t i s f y ( 2 . 3 . 1 ) and —d —a —b —a —a

E and H. s a t i s f y ( 2 . 3 . 6 ) . Then E and H^ s a t i s f y

(2.3.11)

(2.3.12)

1 x Md " 3 eb Ed = Jd,

1 * h

+ S

^b «d

=

'h'

in which J , and K . are given by

—d —d

^ " s [ ea " eb] la»

Kc, = 3Cua " pb] Ha.

S 3 J . and K. a r e z e r o o u t s i d e t h e domain D UD. .

a GE GE

I n t r o d u c e t h e e l e c t r i c G r e e n ' s s t a t e v e c t o r s E. and H. as _{—D —b}

1 * t E ^ ' W3 ) " s eb(^ 's )^ bE (^ ^ '3 ) ■ j j f < D i y8 ) = ib 6 ( r - rT) ,

SOURCE DOMAIN DOMAIN OF OBSERVATION SCATTERING DOMAIN

GM GM

and t h e magnetic G r e e n ' s s t a t e v e c t o r s E and H as t h e v e c t o r s t h a t

s a t i s f y

V x H °M( r ; rn, s ) - s eb( r , s ) E ^M( r ; rQ, s ) = j J f V j r ^ . s ) = 0, ( 2 . 3 . U )

V x E ^M( r ; rf i, s ) + s ub( r , s ) H £M( r ; rn, s ) = - K J f V ^ . s ) = -k_b 6 ( r - rQ) .

The e l e c t r i c - and m a g n e t i c - c u r r e n t p o i n t s o u r c e s j _ 6(r_ - r_ ) and k,_ 6 ( r - r ) are l o c a t e d i n t h e domain of o b s e r v a t i o n D^. The v e c t o r s

- b — — ft b j _h = Ah(s) a n d ]Sh = !ih^s^ r e p r e s e n t t h e s-domain s p e c t r a of t h e s i g n a l s EE EM ME of t h e p o i n t s o u r c e s . The e l e c t r o m a g n e t i c G r e e n ' s t e n s o r s G. , G. , G. MM and G. a r e defined by T-.GE, . . « E E , ,

I

b

(!l;r

?

) = ±

b

' % (ryr),

( 2 . 3 . 1 5 ) Eb ( r ; rQ) = - ^ • gb ( r ^ r ) , „ G M , , , . M M , x Hb (L;£Q) = Hb • 2b ( rn; r ) .

Henceforth, the model objects will always be chosen such that

S S 3 3

D = D = D and such t h a t D c o n t a i n s t h e unknown, a c t u a l o b j e c t . In an s i

i n v e r s e p r o f i l i n g p r o b l e m t h e i n t e r s e c t i o n of D and D i s empty; t h e

3. 3.

transmitters are located outside the domain to be probed. Also, the s i

intersection of D and D. is empty, for the measurements are carried out i i

outside the unknown domain. D and D. may, however, overlap. The general

3. D c o n f i g u r a t i o n f o r i n v e r s e p r o f i l i n g p r o b l e m s i s d e p i c t e d i n F i g u r e 2 . 3 . 1 . I n t h e s u b s e q u e n t s u b s e c t i o n s , f i e l d d e r i v a t i v e s with r e s p e c t t o t h e p r o f i l e parameters of t h e model o b j e c t a r e d e t e r m i n e d f o r s e v e r a l t r a n s m i t t e r - r e c e i v e r s i t u a t i o n s t h a t a r e i m p o r t a n t i n i n v e r s e - p r o f i l i n g e x p e r i m e n t s : ( i ) e x c i t a t i o n by an e l e c t r i c - or a m a g n e t i c - c u r r e n t s o u r c e

and o b s e r v a t i o n a t a p o i n t ( S u b s e c t i o n 2 . 4 ) , ( i i ) e x c i t a t i o n by an e l e c t r i c - or a m a g n e t i c - c u r r e n t s o u r c e and o b s e r v a t i o n i n a s p e c i f i c d i r e c t i o n i n the f a r - f i e l d r e g i o n ( S u b s e c t i o n 2 . 5 ) , ( i i i ) e x c i t a t i o n by an e l e c t r o m a g n e t i c p l a n e w a v e a n d o b s e r v a t i o n a t a p o i n t ( S u b s e c t i o n 2 . 6 ) , and ( i v ) e x c i t a t i o n by an e l e c t r o m a g n e t i c plane wave and o b s e r v a t i o n i n a s p e c i f i c d i r e c t i o n i n t h e f a r - f i e l d r e g i o n ( S u b s e c t i o n 2 . 7 ) .

2 . 4 . E x c i t a t i o n by an e l e c t r i c - or m a g n e t i c - c u r r e n t s o u r c e and o b s e r v a t i o n a t a p o i n t

Let s t a t e A i n the r e c i p r o c i t y theorem (A.2) be t h e s t a t e given by

^ A ' » A ' 4 ' ^ ■ t l d ^ d ' ^ d ' ^1' { 2 A A )

and l e t s t a t e B be a s t a t e of t h e a u x i l i a r y c o n f i g u r a t i o n , given by

| EB, HB, £B, KB} = { E °E, H ^E, ib6 ( r - r¥) , 0 } . ( 2 . 4 . 2 )

The f i e l d s EJ and H. s a t i s f y ( 2 . 3 . 1 1 ) . The f i e l d s E " and H " s a t i s f y

—d —d — b — b ( 2 . 3 . 1 3 ) .

Next, apply t h e r e c i p r o c i t y theorem (A.2) t o t h e domain D i n t e r i o r t o a s p h e r e 3D = SD, w i t h r a d i u s R and c e n t e r a t t h e o r i g i n , t h a t

i i s

e n t i r e l y surrounds t h e domains D , D and D . The s u r f a c e i n t e g r a l i n

3 D

(A.2), in Appendix A, over the sphere 3D = S vanishes as R-*», because

CV PF

the f i e l d s EJ t H., E. and H. , b e i n g r e l a t e d t o c a u s a l t i m e - d o m a i n

—d —d —o — D

f i e l d s , decay e x p o n e n t i a l l y on SR as R-*-°°. Then, one f i n d s

{ i idM • l f ( r ) - i b • Ed( r ) 6 ( r - ry) _{( 2 . 4 . 3 )} K . ( r ) • H™(r)} dV(r) = 0.

Fig. 2.4.1 Excitation by an electric- or magnetic-current source. Computation of the field derivatives with respect to the profile parameters of the model at r = r,„ and r = r

- -¥

_-a

Application in (2.4.3) of the expressions for the Green's tensors given in (2.3.15) yields the relation between the electric field strengths E

cl and E_ , given by E (r,„) = Ew(r,„) + s Ae(r) C£"(r,„;r) ^E (r) dV(r) JDs -a -'T -b -¥ ,EE, + s ! s Ay(r) G"",(r,„;r)-H^(r) dV(r), 1 D ,EM. =b ^ y - ' -av-' (2.4.4) in which Ae(r) = e (r_) - eb(r), Au(r) = u (r) - uh(r) (2.4.5)

Ae and AM a r e t h e c o n t r a s t p e r m i t t i v i t y f u n c t i o n and t h e c o n t r a s t -p e r m e a b i l i t y f u n c t i o n of one model o b j e c t with r e s -p e c t t o t h e o t h e r .

Let, n e x t , s t a t e A in t h e r e c i p r o c i t y t h e o r e m ( A . 2 ) be t h e s t a t e given by

^ A ' 4 ' ^ - ^d, Hd, Jd, Kd} , ( 2 . 4 . 6 )

and let state B be a state of the auxiliary configuration, given by

{EB,HB,JB,KB} = {Ejf,Hj;M,0,kb6(r - r^)}. (2.4.7)

The fields E . and H. satisfy (2.3.11). E °M and H P1 satisfy (2.3-14). —a —a —b —b

Application of the reciprocity theorem (A.2) in a similar way as above yields

| i

i

d

(£) • i J V ) + k

b

• H

d

(r)6(r - r^)

,GM, (2.4.8) K ^ r ) • Hj"(r)} dV(r) = 0. U s i n g i n ( 2 . 4 . 8 ) t h e e x p r e s s i o n s for t h e G r e e n ' s t e n s o r s , one o b t a i n s t h e r e l a t i o n between t h e magnetic f i e l d s t r e n g t h s H and H , g i v e n by

—a —b

M O "

H

J

r

n>

+ s

L

Ae(r) G

w

E

(

r

n ;

r

)

, E

(

r

)

d V

^

r

)

—a —P. — o — Q J s — =b — ii — — a — — f MM + s Au(r) G' ( rn; r ) - H ( r ) d V ( r ) , ( 2 . 4 . 9 ) JDs ~ =b 0 a -with Ae and Au a s g i v e n i n ( 2 . 4 . 5 ) . I n t e g r a l e x p r e s s i o n s s i m i l a r t o ( 2 . 4 . 4 ) and ( 2 . 4 . 9 ) can be o b t a i n e d f o r E ( r ) and H ( r ) , a p p e a r i n g i n

—a a t h e i n t e g r a n d s of ( 2 . 4 . 4 ) and ( 2 . 4 . 9 ) .

Assume t h a t t h e p e r m i t t i v i t y f u n c t i o n s and t h e p e r m e a b i l i t y f u n c t i o n s can be expanded as

N N £a(^ - J1 ea , n V ^ ' ^ = ^ " afn * n(^ « ( 2 . 4 . 1 0 ) N N n=1 n=1 w h e r e {I)J (r_)} a r e s - i n d e p e n d e n t e x p a n s i o n f u n c t i o n s , d e f i n e d on D. I n t r o d u c e Ae and An , with n = 1 , . , , N , as Ae = e - E. , n a,n b , n ' % = ya , n ~ V n ' ( 2 . 4 . 1 1 )

Now, take two model configurations such that u = p. for all n,

8 1 n D i n

and e = e. f o r a l l n e x c e p t n = k , f o r which e , = e. , + Ae, . a , n b , n a,k b,k k Then, ( 2 . 4 . 4 ) can be w r i t t e n as

- a(- reb , 1 ", , Eb , k + A ek " -, eb , N 'l Ib , 1 ",'yb , N)

- I b(^; eb , 1 ' - - 'eb , N 'yb , 1 " "; Jb , N) ( 2 . 4 . 1 2 )

= Ae I_, + o(Ae. ) as Ae. ■* 0,

with

h

= S

| s V ^ S b ^ V ^ ' V ^

an

-

)

' (2.4.13)

In ( 2 . 4 . 1 2 ) , the dependence of the f i e l d s on t h e p r o f i l e p a r a m e t e r s has b e e n d e n o t e d e x p l i c i t l y . I t f o l l o w s from ( 2 . 4 . 1 2 ) t h a t t h e d e r i v a t i v e

dE

— with r e s p e c t t o t h e p r o f i l e parameter e. , of the model i s g i v e n ,

9 eb , k b'k

<*£>/-¥) f FF

a ¥ - s * . ( r ) G ^ ( rr a; r ) . E . ( r ) d V ( r ) . ( 2 . 4 . 1 4 ) 9 eb , k JDs k =b H b

-I f one t a k e s two model c o n f i g u r a t i o n s such t h a t e = e. for a l l n , a , n b,n

and u = p. for a l l n e x c e p t n = k, f o r w h i c h u . = u. . + Ay. ,

3. j n D j n 3. j K D | K K 9

ü

b

then i t follows t h a t the d e r i v a t i v e — , f o r k = 1 , . . , N , i s given by 3 Mb,k

^ * - s ! i k ( r ) G ^M( rm; r ) - HK( r ) d V ( r ) . ( 2 . 4 . 1 5 ) 3 ^ — = a JD S V£ _ , ub v 4 i L r V u

9MD 9ÜD

The d e r i v a t i v e s r— and r— a r e o b t a i n e d i n a s i m i l a r way as above, b , k b , k s t a r t i n g from r e l a t i o n ( 2 . 4 . 9 ) . They a r e g i v e n by 9H ( r ) e

a^f" -

s

J

D

s V ^ C ^ ^ ' V ^

dV(

^'

( 2 . 4 . 1 6 ) 9ü h(L0) f MM a - s ! *. ( r ) G ™ ( rn; r ) - H . ( r ) d V ( r ) . 3 yb , k JDS vk =b n b -The c o n f i g u r a t i o n and t h e t r a n s m i t t e r - r e c e i v e r s i t u a t i o n t h a t a r e d i s c u s s e d i n t h i s s u b s e c t i o n a r e d e p i c t e d i n F i g . 2 . 4 . 1 . P r o f i l e d e r i v a t i v e s with r e s p e c t t o t h e p r o f i l e p a r a m e t e r s of t h e model a r e c o m p u t e d a t £ = r_ a n d r = £ , f o r t h e c a s e of e x c i t a t i o n by an e l e c t r i c - or m a g n e t i c - c u r r e n t s o u r c e .

2 . 5 . E x c i t a t i o n by a m a g n e t i c - or e l e c t r i c - c u r r e n t s o u r c e and o b s e r v a t i o n i n a s p e c i f i c d i r e c t i o n i n t h e fai—field r e g i o n

Let s t a t e A i n t h e r e c i p r o c i t y theorem (A.2) be t h e s t a t e given by

^ A ' » A ' 4 ' ^ " Ud'ïïd'iïci'Sdï'

( 2

-

5

'

1 ) and l e t s t a t e B be a s t a t e of the a u x i l i a r y c o n f i g u r a t i o n , given by

i EB, HB, JB, KB} - {E^E,H^E,j_b 6(r - rf) . o } . ( 2 . 5 . 2 )

The f i e l d s E , and H . s a t i s f y ( 2 . 3 . 1 1 ) . The f i e l d s E" and H. s a t i s f y —d —d - b —b ( 2 . 3 . 1 3 ) .

P r o c e e d a s i n S u b s e c t i o n 2 . 4 , s t a r t i n g from r e l a t i o n s ( 2 . 4 . 4 ) and ( 2 . 4 . 9 ) . Apply t h e p r o p e r t i e s of t h e f i e l d i n t h e f a r - f i e l d r e g i o n , d i s c u s s e d i n A p p e n d i x B, a n d , i n a s i m i l a r way a s i n S u b s e c t i o n 2.4,

EE,°° EM,"" ME,00 make use of t h e e x p r e s s i o n s for t h e G r e e n ' s t e n s o r s G. , G. , Gb ,

MM °°

and G. ' given i n ( B . 6 ) , i n Appendix B. F i n a l l y , t h e d e r i v a t i v e s of the

00 00

a n g u l a r l y dependent s p h e r i c a l - w a v e amplitudes E and H are f o u n d , w i t h r e s p e c t t o t h e p r o f i l e p a r a m e t e r s e. . and u. . of t h e m o d e l . For k = 1 , . . , N , t h e y a r e given by

a^f- "

s

J

D

s W

G

=b ' %&-h<0

d V (

^

( 2 . 5 . 3 ) 3Mb(£u;) f EM »

3^7" =

S

J

D

s W C' %->rJ%^ dV(r).

w i t h r = | £ | and t h e u n i t v e c t o r £ = r_ /r i n t h e d i r e c t i o n of o b s e r v a t i o n , and

£*.£n

Fig. 2.5.1 Excitation by an e l e c t r i c - or magnetic-current source. Computation of the model p r o f i l e derivatives of the spherical-wave amplitudes in t h e directions of observation _B and JJ . 3 M 3 J ,

S^f" -

S

J

D

s V ^ C'"%-<D%(rJ dV(r),

(2.5.1) ^ ( J L J f MM c=

3^7" -

S

J

D

s \ ^ t ' (IfliD'Vr) dV(r).

w i t h rf i = | £n| and t h e u n i t vector jS = £ o/ ro i n t n e d i r e c t i o n of observation.

The c o n f i g u r a t i o n and the transmittei—receiver s i t u a t i o n that are discussed in t h i s subsection a r e depicted in F i g . 2 . 5 . 1 .

2 . 6 . E x c i t a t i o n by an e l e c t r o m a g n e t i c p l a n e wave and o b s e r v a t i o n at a p o i n t

Let t h e s o u r c e f u n c t i o n s J and K in ( 2 . 3 . 1 ) be given by

3. ~~3.

J ( r , s ) = 0 ,

K ( r , s ) = 0 ,

3

( 2 . 6 . 1 )

and l e t t h e s t a t e s "A" and "B" i n the r e c i p r o c i t y theorem (A.2) be given by t h e r e l a t i o n s ( 2 . 4 . 1 ) and ( 2 . 4 . 2 ) . The s c a t t e r i n g o b j e c t i n t h e

g

domain D i s now i r r a d i a t e d by an i n c i d e n t e l e c t r o m a g n e t i c plane wave p r o p a g a t i n g i n t h e d i r e c t i o n a ( s e e F i g . 2 . 6 . 1 ) ; i t i s given by E1 K( r ) = e* exp(-Y o e r ) , —a;b — —a;b e H= w( r> = ÏL h e x p ( - Y a - r ) , —a; b — —a; b e ( 2 . 6 . 2 )

i n which oca = 1, aQü, and Y = s ( e u ) . {e . ,h . } a r e c o n s t a n t

- e epe - a ; b ' - a - , b complex v e c t o r s t h a t s a t i s f y the r e l a t i o n s Z a x h . + e . = 0 , oce . = 0, e - - a ; b - a ; b - a;b Y a x e1 . - h1 . = 0, a-h1 . = 0, e - - a ; b - a ; b — a ; b ( 2 . 6 . 3 ) i n which Z = Y 1 = (M / e )1 / 2. e e e e A p p l i c a t i o n of t h e r e c i p r o c i t y t h e o r e m i n a s i m i l a r way a s i n S u b s e c t i o n 2.4 y i e l d s

Ë

a

(ïy«) = Ib

(

v«>

+ s

jis

Ae(!:)

2b

E(

-*

;

-

)

*-a

(

-

;

-

)dV(

-

) + s [ Ay(r) G^M(r,„;r)-H ( r ; a ) d V ( r ) ,

F i g . 2 . 6 . 1 E x c i t a t i o n by a p l a n e wave p r o p a g a t i n g i n t h e d i r e c t i o n a. Computation of t h e f i e l d d e r i v a t i v e s with r e s p e c t t o the p r o f i l e p a r a m e t e r s of t h e model a t t h e p o i n t s of o b s e r v a t i o n r = r,„ and r = r _ . — —T — — W MLva) - MJrn;a) + s A e ( r ) G "E( rn; r ) - E ( r ; a ) d V ( r ) ( 2 . 6 . 4 ) —a —si — —o — si — Jns — =b — Si — — a — — —

f MM

+ s A u ( r ) G ™ ( r ; r ) - H ( r ; a ) d V ( r ) , J s — =b —Ji — —a —

in which

Ae and Au a r e t h e c o n t r a s t p e r m i t t i v i t y f u n c t i o n and t h e c o n t r a s t -p e r m e a b i l i t y f u n c t i o n of one model o b j e c t with r e s -p e c t t o the o t h e r . I n

( 2 . 6 . 4 ) t h e a a f t e r t h e s e m i c o l o n denotes t h a t t h e f i e l d i s due t o an i n c i d e n t wave p r o p a g a t i n g i n t h e d i r e c t i o n a.

Proceeding i n t h e same way as in the former s u b s e c t i o n , one f i n a l l y f i n d s t h e f i e l d d e r i v a t i v e s with r e s p e c t t o t h e p r o f i l e p a r a m e t e r s E. and \i of the model. For k = 1 , . . , N , they a r e given by

3

iyr_u,;«) r

EE ^ W ' ^ f EM

a

V k

=

s

j

D

s \^ Sf^'£)-Hb^;«> dv(r),

9 H . ( r ; a ) <• Zb~k " = S JDs *k«r) G jE( rn; r ) . Eb( r;a ) d V ( r ) . 8 H ( r ; a ) r ° = s 4 ( r ) G™(r ; r ) . H ( r ; a ) d V ( r ) . 3 yb > k JDs k =b o b -' b , k ( 2 . 6 . 6 ) 2 . 7 . E x c i t a t i o n by an e l e c t r o m a g n e t i c p l a n e wave and o b s e r v a t i o n i n a s p e c i f i c d i r e c t i o n in t h e f a r - f i e l d r e g i o n

Let the s o u r c e f u n c t i o n s J and K in ( 2 . 3 . 1 ) be given by

—a —a

J ( r , s ) = 0, —a — —

K ( r , s ) = 0,

£f£n

Fig. 2 . 7 . 1 E x c i t a t i o n by a p l a n e wave p r o p a g a t i n g i n t h e d i r e c t i o n oi. Computation of the model p r o f i l e d e r i v a t i v e s of the s p h e r i c a l - w a v e amplitudes i n t h e d i r e c t i o n s of

o b s e r v a t i o n 0,„ and B„.

and l e t t h e s t a t e s "A" and "B" i n t h e r e c i p r o c i t y theorem (A.2) be given by t h e r e l a t i o n s ( 2 . 4 . 1 ) a n d ( 2 . 4 . 2 ) . The s c a t t e r i n g o b j e c t i n t h e domain D i s i r r a d i a t e d by an i n c i d e n t e l e c t r o m a g n e t i c p l a n e wave p r o p a g a t i n g in t h e d i r e c t i o n a ( s e e F i g u r e 2 . 7 . 1 ) ; i t i s given by

—a; b — e . exp(-Y a « r ) , —a; b e

ü a ; b(^ = na.be xP( - Yea . r ) ,

( 2 . 7 . 2 )

i n which wa = 1, aen. {e . ,h . } a r e constant complex vectors that _{- - a ; b - a ; b} s a t i s f y the r e l a t i o n s given in ( 2 . 6 . 3 ) .

Proceed as i n S u b s e c t i o n 2 . 6 . , s t a r t i n g from r e l a t i o n s ( 2 . 6 . 4 ) . Apply the properties of the f i e l d in the f a r - f i e l d region, d i s c u s s e d i n

Appendix B, and, i n a s i m i l a r way as in S u b s e c t i o n 2 . 6 , make u s e of t h e EE," EM,00 ME,00 MM.» e x p r e s s i o n s for t h e G r e e n ' s t e n s o r s G. , G. , G , and G. given i n (B.6) i n Appendix B. F i n a l l y , one f i n d s t h e f i e l d d e r i v a t i v e s w i t h r e s p e c t t o t h e p r o f i l e p a r a m e t e r s e and v . of t h e m o d e l . For D i K D | K k = 1 , . . , N , they a r e given by

9 W « > f

EE

.

1 ^ 7 -

= s

j

D

s v ^ i'

^ D - V ^

dv(

^-(2.7.3) ^ A u , : » ) f EM »

"4^—

= s

J

D

8 +k

(

n> 2

bEM

' <*,'£>%<*■> - v ( r ) .

with r = | r I and j$ = r v / r in t h e d i r e c t i o n of o b s e r v a t i o n , and

3H. (B„;ct) r „

~jt^ -

S

J

D

s W C %&%<**>

dV(

^'

(2.7.4)

3Hb(io!«) f MM »

l^f— =

S

J

D

s W C' %^-^L^ 0V(r).

2 . 8 . F i e l d d e r i v a t i v e s i n p l a n e - s t r a t i f i e d and c i r c u l a r l y c y l i n d r i c a l l y s t r a t i f i e d , d i e l e c t r i c media

2 . 8 . 1 . P l a n e - s t r a t i f i e d media

Assume t h a t t h e m e d i a i n t h e model and a u x i l i a r y c o n f i g u r a t i o n s a r e s t r a t i f i e d . I n t r o d u c e c i r c u l a r l y c y l i n d r i c a l c o o r d i n a t e s ( r , 6 , z ) , w i t h x = r c o s 6 , y = r s i n e , and z = z , where O S r O , 0S6<2TT, and -°°<z<°°. Let t h e z - a x i s be p e r p e n d i c u l a r t o t h e p l a n e s of s t r a t i f i c a t i o n . Then

N

a;b — a;b , a ; b , n n

where {i|> ( z ) } a r e e x p a n s i o n f u n c t i o n s , d e f i n e d f o r - » < z < ° ° . The p e r m e a b i l i t i e s of t h e media a r e chosen t o be equal t o t h e p e r m e a b i l i t y of vacuum (y ) :

Vi = u, = y . ( 2 . 8 . 2 ) a b o

Let the source terms i n ( 2 . 3 . 1 ) , ( 2 . 3 - 6 ) and ( 2 . 3 . 1 4 ) be given by

i n v \, I \ 6 ( r ,Z) .

J = 0 , K = k ( s ) —r—-— i ,

- a — - a z , a 2irr - z ( 2 . 8 . 3 )

J°M = 0 K®1 = k . ( s ) 6(r - ro) i , -b — —b z,b — — ü —z

i n which k ( s ) and k ,_(s) a r e t h e s - d o m a i n s p e c t r a of t h e source z , a z ,b

s i g n a l s i n t h e z - d i r e c t i o n . The t r a n s m i t t e r s i n t h e model c o n f i g u r a t i o n s a r e magnetic d i p o l e s l o c a t e d a t t h e o r i g i n of the c o o r d i n a t e s y s t e m . Due t o the choice of the source terms in t h e a u x i l i a r y c o n f i g u r a t i o n , f i e l d d e r i v a t i v e s a r e o b t a i n e d a t r = r . The e l e c t r o m a g n e t i c f i e l d s i n t h e model c o n f i g u r a t i o n s "a" and "b" a r e r o t a t i o n a l l y s y m m e t r i c a r o u n d t h e

z - a x i s , and t h e f i e l d q u a n t i t i e s a r e i n d e p e n d e n t of 9 . From M a x w e l l ' s e q u a t i o n s ( 2 . 3 . 1 ) i t then follows t h a t

HQ . = 0, E . = 0 and E . = 0. ( 2 . 8 . 4 )

9 , a ; b r , a ; b z , a ; b

The field components E , H and H satisfy the equations

tj|3] 0 r* j 3. J D Z j 3. j D

-3 E. . + s un H . = 0 ,

z9,a;b 0 r,a;b

3zHr,a;b " 3rHz,a;b " s ea;T>(a)Ee.a;b = °' ( 2"8'5 )

r_1

V

'

W

+S

*0

H

z,a;b - ^ a ^ l F '

Using the results obtained in Subsection 2.4, one finds that

(2.8.6) Hz , a (^ " « z . b ^ ' 2 )

= - — j dz Ae(z) j d9 dr r E ^ C r s O ) E ^ t r ; ^ ) ,

Z f D Z.. U U

z, b

and that the derivative with respect to the profile parameters 3 eb,k

e of the model i s given by

D f K

8H Ar

_z,b^fl

n;0) s rz0 r2Tr ;

^

3

f

Z

2 „ , , , f

Zir rffl

f „

^

=

~ ^ b J

Z l d Z

*

k ( Z )

J o

d 9

i o

d r r

(2.8.7)

In (2.8.6) and (2.8.7) the interval of the z-integration i s given by z < z < z„, because the contrast is assumed to be restricted to this i n t e r v a l . Not always, for example with a p p l i c a t i o n s in i n v e r s e p r o f i l i n g , the properties of the s t r a t i f i e d medium are known beyond

z = z?. Consequently, t h e z - i n t e r v a l of i n t e g r a t i o n s h o u l d e x t e n d t o i n f i n i t y . However, i n t h e s e , c a s e s the i n t e g r a t i o n i n t e r v a l i s u s u a l l y s p l i t up in an i n t e r v a l ( z . . , z?) and an i n t e r v a l ( z2, « > ) . The i n t e g r a l over t h e l a t t e r i n t e r v a l i s then approximated, s o t h a t only t h e i n t e g r a l over the f i r s t i n t e r v a l r e m a i n s .

CM

In the s p e c i a l case t h a t K. in ( 2 . 8 . 3 ) i s given by

PM 6 ( r , z - z ) C1 = k . ( s ) = il- i , ( 2 . 8 . 8 ) - b z , b 2irr - z ' f i e l d d e r i v a t i v e s a r e o b t a i n e d along t h e z - a x i s ( s e e F i g . 2 . 8 . 1 ) . The e l e c t r o m a g n e t i c f i e l d s i n t h e a u x i l i a r y c o n f i g u r a t i o n a r e t h e n r o t a t i o n a l l y s y m m e t r i c a r o u n d t h e z - a x i s , t o o . As a consequence, t h e e q u a t i o n s ( 2 . 8 . 6 ) and ( 2 . 8 . 7 ) s i m p l i f y c o n s i d e r a b l y . The p o s i t i o n v e c t o r i s d e n o t e d by r_ = ( r , z ) . From Maxwell's e q u a t i o n s ( 2 . 3 - 1 ^ ) i t follows t h a t

H™ = 0, EGMh = 0 and EGMh - 0. ( 2 . 8 . 9 )

8,b r , b z,b GM GM GM

The f i e l d components E„ . , H . and H . s a t i s f y

y 9,b r , b z , b z 9,b 0 r , b a HGMh - 3 H f f l - s ch(z)EG M h = 0, ( 2 . 8 . 1 0 ) z r , b r z,b b e,b 1 ™ ™ 6 ( r , z - z ) -1 „ ,. _GM > , „ G M . J2 r 8 ( r E„ . ) + s p„ H . = - k , . r 6,b 0 z , b z,b 2irr

D

ozn

(

0

D'

D

■ > r r ' z Z2

Fig. 2.8.1 Dipole transmitter at the origin of the coordinate system in a plane-stratified medium. Computation of the field derivatives with respect to the profile parameters of the model along the z-axis at z = z .

H

z , a

( 0

' V

0

'

0 )

"

H

z , b

( 0

' V ° '

0 ) ( 2 . 8 . 1 1 ) 2 i r s fz? f00 rM - - — d z A E ( Z ) d r r E ( r , z ; 0 , 0 ) Eg ( r , z ; 0 , z ) , z , b Jz J 0 ' ' 9 Hz b and t h e d e r i v a t i v e -—'— with r e s p e c t t o t h e p r o f i l e p a r a m e t e r s e. . of 9 eb , k b , k t h e model i s given by 9H , ( 0 , zn; 0 , 0 ) 2TTS rz

z , b ^ '

z

n

; u

'

u ;

<™ n «■, , , t

b,k z,b ' z . ' 0 dz i|;u(z) | dr r ( 2 . 8 . 1 2 ) x Ee > b( r , z ; 0 , 0 ) E ^ ( r ,z ; 0 , zf i) . A l t e r n a t i v e l y , t h e t r a n s m i t t e r can be d e s c r i b e d a s an e l e c t r i c -c u r r e n t loop i n t h e p l a n e z = 0. Then one s h o u l d -choose t h e sour-ce terms i n ( 2 . 3 . 1 ) and ( 2 . 3 . 6 ) as

6 ( r - r ,z)

J ( r , s ) = j (3) i , K ( r . s ) = 0. ( 2 . 8 . 1 3 ) - a - 6,a 2 i r r 6 a

-Let the s o u r c e terms of the a u x i l i a r y c o n f i g u r a t i o n i n ( 2 . 3 . 1 1 ) be given by

4G E< r , s ) = J6 > b( 3 ) ^ f ^ ie, Kb G E(r,s) = 0, ( 2 . 8 . 1 4 ) ' 2irr

i n which j ( s ) and j K( s ) a r e t h e s - d o m a i n s p e c t r a of the source

6, a b, D

s i g n a l s . As a r e s u l t of t h i s c h o i c e , f i e l d d e r i v a t i v e s a r e o b t a i n e d i n t h e p l a n e z = z . The e l e c t r o m a g n e t i c f i e l d s i n t h e model and a u x i l i a r y c o n f i g u r a t i o n s a r e r o t a t i o n a l l y s y m m e t r i c a r o u n d t h e z - a x i s , and t h e f i e l d q u a n t i t i e s a r e independent of 6. The p o s i t i o n v e c t o r i s denoted by

r = ( r , z ) . I n a way s i m i l a r t o t h e one d e s c r i b e d a b o v e f o r t h e c a s e of the p o i n t s o u r c e , one f i n a l l y a r r i v e s a t t h e f o l l o w i n g r e l a t i o n s : E

e,a

(

VW

0)

- " W W V

0

'

(2

-

8

-

15) f 2 f GE j dz Ae(z) i dr r EQ > a( r , z ; r . ,0) E ^ t r . z j r ^ . z ^ ) , and je , b Z1 ° 3 Ee > b( r ,z ; r 0) 2 lrs fz f

—TI— —

dz

V

z )

J

n dr r b'k J9 , b Z1 ° GE x Ee , b( r'z ;V0 ) Ee , b( r'z ; rfz^ (2.8.16)

E ( r , z ; r ' , z ' ) denotes the field at position r , generated by an

electric-ö — current loop with radius r' around the z-axis in the plane z = z ' .

2.8.2 Circularly cylindrically stratified media

Relations similar to (2.8.11), (2.8.12), (2.8.15) and (2.8.16) will be derived for a circularly cylindrically stratified medium. Such relations are used in the field of radial profile reconstruction, for example in a borehole configuration (see Figure 2 . 8 . 2 ) . For a discussion of this problem see Habashy et a l . [1986].

In the circularly cylindrical coordinates (r,6,z) l e t the z-axis be the axis of rotational symmetry of the configuration. The stratification is now in the r-direction. Then,

N

e, Jr) = e. h(r> " z E= K * J r ) . ( 2 . 8 . 1 7 ) a;b - a ; b a ; b , n n

w h e r e {ty ( r ) } a r e e x p a n s i o n f u n c t i o n s , d e f i n e d for 0 < r O . Again, the d i s c u s s i o n i s r e s t r i c t e d t o t h e case of magnetic d i p o l e s w i t h m a g n e t i c moments i n t h e z - d i r e c t i o n , s u c h t h a t ( 2 . 8 . 3 ) h o l d s . Because of the r o t a t i o n a l symmetry a r o u n d t h e z - a x i s , t h e f i e l d q u a n t i t i e s a r e independent of 9, and t h e p o s i t i o n v e c t o r i s denoted by r = ( r , z ) . Thus,

GM ( 2 . 8 . 4 ) and ( 2 . 8 . 9 ) a p p l y . Making use of t h e e x p r e s s i o n s f o r J , J. and

GM - a

K in ( 2 . 8 . 1 ) , and the expression for K in (2.8.8) together with the r e s u l t s obtained in Subsection 2.4, one finds that

(2.8.18) Hz , a( 0' V ° '0 ) -Hz , b( 0'zf i; 0'0 ) 2 l r S fr? f °° CM = - r — dr r Ae(r) dz E a( r , z ; 0 , 0 ) E ^1. ( r , z ; 0 , zo) , kz , b Jr1 J-<=° 6 , a 6 , b n 9Hz b

and that the derivative r—'— with r e s p e c t t o t h e p r o f i l e parameters

3 eb , k

e. of the model i s given by

3 H

z , b

( 0

' V ° -

0 ) 2lrS

f2

A

. , . .

ÏT- —

h

J

dr {r

* k

( r )

b,k z,b 'r.

(2.8.19) j dz E9 ) b( r , z ; 0 , 0 ) E™b(r ,z ; 0 , zQ) } .

In (2.8.18) and (2.8.19) the i n t e r v a l of the r - i n t e g r a t i o n i s given by

r < r < r „ , because the c o n t r a s t i s known t o be r e s t r i c t e d t o t h i s

i n t e r v a l . When t h i s r e s t r i c t i o n cannot be met in p r a c t i c a l inverse p r o f i l i n g e x p e r i m e n t s , s i m i l a r remarks apply as in the case of the p l a n e - s t r a t i f i e d medium (see text after r e l a t i o n ( 2 . 8 . 7 ) ) .

If the transmitter i s d e s c r i b e d in terms of an e l e c t r i c - c u r r e n t l o o p in t h e p l a n e z = 0, then the s o u r c e terms J and K should be

— a ""'a. chosen as in (2.8.13). To obtain the field derivatives in a plane

PR PR

z = z,„, the source terms J. and K,_ should be chosen as in (2.8.14). ¥ — b —b

O

@-— + @-—

<i>zn

Fig. 2.8.2 Dipole transmitter at the origin of the coordinate system in a c i r c u l a r l y c y l i n d r i c a l l y s t r a t i f i e d medium. Computation of the f i e l d derivatives with respect to the p r o f i l e parameters of the model along the z-axis at

The e l e c t r o m a g n e t i c f i e l d s in t h e model and a u x i l i a r y c o n f i g u r a t i o n s a r e r o t a t i o n a l l y s y m m e t r i c a r o u n d t h e z - a x i s , and t h e r e f o r e t h e - f i e l d q u a n t i t i e s a r e i n d e p e n d e n t of 6. The p o s i t i o n v e c t o r i s d e n o t e d . b y r_ = ( r , z ) . R e l a t i o n s ( 2 . 8 . 4 ) and ( 2 . 8 . 9 ) a p p l y . The f i e l d components E. . , H . and H . s a t i s f y t h e e q u a t i o n s 9 , a ; b ' r , a ; b z , a ; b J -9 E„ . + s p . H . = 0 , z 6 , a ; b 0 r , a ; b 6 ( r - r , z ) 9 H w - 3 H x - s e K( r ) EQ . = j D ~ , ( 2 . 8 . 2 0 ) z r , a ; b r z , a ; b a ; b e , a ; b J9 , a 2irr r~1 9 ( r EQ . ) + s uA H . = 0 . r 6 , a ; b 0 z , a ; b

The f i e l d components E„ . , H . and H , s a t i s f y

9,b r , b z,b J

'■'

V C ' * ■ "o C ■ »•

Then, for circularly cylindrically stratified media, the formulas comparable to (2.8.15) and (2.8.16) are

E

e , a

(

V W

0 )

-

E

9 , b

(

W V

0 ) (2

'

8

'

22)

= j dr r Ae(r) ! dz EQ a(r,z;ri>0) EQ fe(r . z ^ . z ^ ) ,

and 8 Ee , b(Wri '0 ) 2 i r s fr2 - ui n n>. \.i y ( 2 . 8 . 2 3 ) _ = d r {r ^ ( r ) 8S k i Jr k D*K J6 , b P1 j dz Ee ( b( r , z ; r . , 0 ) E ^ r , z ; r ^ . z ^ ) } . E ( r , z ; r ' , z ' ) d e n o t e s the f i e l d a t l o c a t i o n r , g e n e r a t e d by an e l e c t r i c -b —

CHAPTER 3

INVERSE PROFILING

3.1 Introduction

In i n v e r s e p r o f i l i n g t h e o r y , one t r i e s t o c o n s t r u c t the unknown d i s t r i b u t i o n of the c o n s t i t u t i v e parameters of an o b j e c t . T y p i c a l l y , some f i e l d measurements are carried out outside the object at different locations and at different frequencies. As a s t a r t i n g point for p r o f i l e r e c o n s t r u c t i o n , one o f t e n t a k e s an i n t e g r a l e x p r e s s i o n for the d i f f e r e n c e of t h e measured f i e l d and the ( c a l c u l a t e d ) f i e l d i n a background medium ( T i j h u i s [ 1 9 8 7 ] , Habashy e t a l . [ 1 9 8 6 ] , Chew and Chuang [1984], Tijhuis and Van der Worm [1984], Coen et a l . [1981]). The c o n t r a s t f u n c t i o n , being the difference of the unknown p r o f i l e and the known background p r o f i l e , i s expanded i n t o a number of base f u n c t i o n s . Employing a Born-type a p p r o x i m a t i o n , one then t r i e s t o recover the unknown p r o f i l e in an i t e r a t i v e process, while updating t h e background a t every i t e r a t i o n (Subsection 3 . 7 ) . Methods of t h i s type are therefore known as B o r n - t y p e i t e r a t i v e m e t h o d s . H e r e w i t h t h e f o l l o w i n g d i f f i c u l t i e s a r i s e , among o t h e r s : the convergence of the i t e r a t i v e scheme i s not guaranteed; the number of unknown parameters i s determined by the number of expansion functions necessary t o describe the contrast function; the ill-posedness of the original problem causes the r e s u l t i n g

numerical problem t o be i l l - c o n d i t i o n e d . To remedy t h i s l a s t d i f f i c u l t y , one r e s o r t s t o r e g u l a r i z a t i o n m e t h o d s . C r i t e r i a f o r c h o o s i n g t h e r e g u l a r i z a t i o n p a r a m e t e r s a r e seldom g i v e n . The numerical schemes are u s u a l l y d r a f t e d by t r i a l and e r r o r .

Some of t h e a b o v e - m e n t i o n e d d i f f i c u l t i e s can be overcome i f the i n v e r s e p r o f i l i n g problem i s c o n s i d e r e d as f o l l o w s . I t i s looked upon as a d a t a f i t t i n g p r o b l e m ( S u b s e c t i o n 3 . 2 ) . I n t h i s a p p r o a c h , a model o b j e c t i s i n t r o d u c e d with a (known) p r o f i l e t h a t depends on a number of p r o f i l e p a r a m e t e r s . The a d j u s t a b l e p r o f i l e p a r a m e t e r s are t o be chosen such t h a t t h e ( c a l c u l a t e d ) f i e l d v a l u e s o u t s i d e t h e model o b j e c t b e s t f i t t h e m e a s u r e d d a t a . I n t h i s p r o c e s s a c u m u l a t i v e s q u a r e d e r r o r f u n c t i o n i s minimized, u s i n g t h e Levenberg-Marquardt a l g o r i t h m f o r n o n ­ l i n e a r l e a s t s q u a r e s p r o b l e m s ( S u b s e c t i o n 3 . 3 ) . In Chapters 6 and 7, i n v e r s e p r o f i l i n g problems a r e c o n s i d e r e d i n which t h e d a t a a r e formed by t h e r e a l and i m a g i n a r y p a r t s of the f i e l d v a l u e s . The parameters t o be found a r e t h e r e a l - v a l u e d p e r m i t t i v i t y and c o n d u c t i v i t y parameters of t h e model.

The Levenberg-Marquardt a l g o r i t h m i s an o p t i m i z a t i o n a l g o r i t h m t h a t combines t h e advantage of a g u a r a n t e e d convergence t o a s t a t i o n a r y point ( t h e s t e e p e s t descent method) with f a s t convergence i n a r e g i o n near the s o l u t i o n ( t h e Gauss-Newton m e t h o d ) . I n e v e r y i t e r a t i o n s t e p of the L e v e n b e r g - M a r q u a r d t a l g o r i t h m a s y s t e m of l i n e a r e q u a t i o n s must be s o l v e d , t h e s i z e of which depends on t h e number of p r o f i l e parameters of the model. The s o l u t i o n of t h i s s y s t e m of e q u a t i o n s makes u s e of t h e method of s i n g u l a r v a l u e decomposition, and i s d i s c u s s e d i n Subsection

3 . 4 .

I t i s assumed t h a t i n the measured d a t a , which a r e i n p u t d a t a for t h e Levenberg-Marquardt a l g o r i t h m , t h e r e i s a c e r t a i n e r r o r . Of c o u r s e t h i s e r r o r c a r r i e s over i n t o an e r r o r of the f i n a l p a r a m e t e r s , found as a r e s u l t of the o p t i m i z a t i o n p r o c e s s . A f i r s t - o r d e r e r r o r a n a l y s i s of t h i s p r o c e s s i s given i n S u b s e c t i o n 3 . 5 . A combination of the Levenberg-M a r q u a r d t scheme and t h e r e s u l t s of t h e e r r o r a n a l y s i s y i e l d s t h e