Plane Wave Propagation Problems in Electrically ... · Plane Wave Propagation Problems in...

Plane Wave Propagation Problems in Electrically Anisotropic

and Inhomogeneous Media with Geophysical Applications

Glenn Andrew Wilson

BAppSc (Physics) CQU

School of Microelectronic Engineering

Faculty of Engineering and Information Technology

Submitted as fulfilment of the requirements for the degree of Doctor of Philosophy

December 2002

ii

Abstract

Boundary value problems required for modelling plane wave propagation in

electrically anisotropic and inhomogeneous media relevant to the surface impedance

methods in electromagnetic geophysics are formally posed and treated. For a

homogeneous TM-type wave propagating in a half space with both vertical and

horizontal inhomogeneities where the TM-type wave is aligned with one of the

elements of the conductivity tensor, it is shown using exact solutions that the shearing

term in the homogeneous Helmholtz equation for inclined anisotropic media:

.0cossin 11 2

cossinsincos

,,

2

,

2,

2

,

2

,

2

2,

2

,

2

,

2

=−∂∂

∂

−+

∂∂

++

∂∂

+

mxmx

mmmtn,m

mx

mt

m

mn

mmx

mt

m

mn

m

Hjzy

Hσ

zH

yH

ωµαασ

σα

σα

σα

σα

unequivocally vanishes and solutions need only be sought to the homogeneous

Helmholtz equation for biaxial media:

.0cossinsincos

,2,

2

,

2

,

2

2,

2

,

2

,

2

=−∂

∂

++

∂∂

+ mx

mx

mt

m

mn

mmx

mt

m

mn

m HjzH

yH

ωµσ

ασ

ασ

ασ

α

This implies that those problems posed with an inclined uniaxial conductivity

tensor can be identically stated with a fundamental biaxial conductivity tensor,

provided that the conductivity values are the reciprocal of the diagonal terms from the

Euler rotated resistivity tensor:

mxxmxxmx ,1

,, σρσ == − ,

mzz

mzymyzmyymyy

mn

m

mt

mmy

,

,,,

1,

1

,

2

,

2

,sincos

σσσ

σρσ

ασ

ασ −==

+= −

−

,

myy

myzmzymzzmzz

mn

m

mt

mmz

,

,,,

1,

1

,

2

,

2

,cossin

σσσ

σρσ

ασ

ασ −==

+= −

−

.

iii

The applications of this consequence for numerical methods of solving arbitrary

two-dimensional problems for a homogeneous TM-type wave is that they need only to

approximate the homogeneous Helmholtz equation and neglect the corresponding

shearing term. The self-consistent impedance method, a two-dimensional finite-

difference approximation based on a network analogy, is demonstrated to accurately

solve for problems with inclined uniaxial anisotropy using the fundamental biaxial

anisotropy equivalence.

The problem of a homogeneous plane wave at skew incidence upon an inclined

anisotropic half space is then formally treated. In the half space, both TM- and TE-type

waves are coupled and the linearly polarised incident TM- and TE-type waves reflect

TE- and TM-type components. Equations for all elements of the impedance tensor are

derived for both TM- and TE-type incidence. This offers potential as a method of

predicting the direction of anisotropic strike from tensor impedance measurements in

sedimentary environments.

iv

Statement of Originality

This work has not previously been submitted for any other degree or diploma at

this or any other University, College or Institution. To the best of my knowledge and

belief, this thesis contains no material previously published or written by any other

person, except where due reference and acknowledgment is made in the thesis itself.

___________________________________

Glenn Andrew Wilson

v

Run rabbit run!

Dig that hole forget the sun,

And when at last the work is done,

Don�t sit down. It�s time to start another one.

Breathe, by Pink Floyd, 1973.

vi

List of Contents

Abstract��..��.. ii Statement of Originality��.�.� iv

List of Contents��..�.. vi

List of Figures��..�� ix

Acknowledgements��..�� xiii

Introduction��...��...�� 1

0.1. Overview��.�� 1

0.2. Outline of thesis��.��. 3

0.3. Detailed outline of thesis��.��3

0.4. Publications��. 7

0.5. References��...� 7

Chapter 1. Introductory theory��...9

1.1. The Maxwell equations��9

1.1.1. The Maxwell equations��.9

1.1.2. Auxiliary potential functions��14

1.1.3. Boundary conditions�� 15

1.1.4. Frames of reference��.. 16

1.1.5. Further notation and definitions��... 19

1.2. Surface impedance��.. 21

1.3. Electrical anisotropy��23

1.4. References��...� 27

Chapter 2. Exact solutions for one-dimensional electrical anisotropy

problems��..��... 30

2.1. Introduction��. 30

2.2. The problem of a homogeneous half space��. 33

2.2.1. The Chetaev method of functional rotation�..��34

2.2.2. The method of tensor rotation��.. 38

2.2.3. Discussion�� 43

2.3. The problem of a horizontally layered half space��.�.�. 43

2.4. References��..��. 46

vii

Chapter 3. Exact solutions for two-dimensional electrical anisotropy

problems��.��... 47


3.2. Exact formulation: preliminary considerations��..��. 48

3.3. Exact formulation: perfectly insulating basement solution�.�52

3.3.1. Inclined anisotropic conductivity��...�.. 52

3.3.2. Fundamental anisotropic conductivity��.�� 62

3.4. Exact formulation: perfectly conducting basement solution...�65

3.4.1. Inclined anisotropic conductivity��...��.. 65

3.4.2. Fundamental anisotropic conductivity��.�...�. 75

3.5. Discussion��..��. 78

3.6. References��.��.. 79

Chapter 4. Approximate solutions for two-dimensional electrical

anisotropy problems��.��..��.. 81


4.2. Formulation: self-consistent impedance method��.�82

4.3. Results: inclined anisotropy in inhomogeneous media��..�. 88

4.3.1. Homogeneous layer above a perfect electric

conductor��.��.88

4.3.2. Horizontal homogeneous layers above a perfect

electric conductor��.�� 92

4.3.3. Vertical dyke embedded in a homogeneous layer

above a perfect electric conductor��... 92

4.4. Discussion��.��.. 93

4.5. References��..��. 94

Chapter 5. Exact solutions for three-dimensional electrical anisotropy

problems��...��...95


5.2. General solutions for the electromagnetic fields in an

anisotropic medium��. 96

5.3. General solutions for the electromagnetic fields in air.��.. 104

5.4. The problem of a TE-type incident field��.� 105

5.5. The problem of a TM-type incident field��.��..�.108

5.6. Discussion��...�� 110

5.7. Conclusions��..��...�� 121

viii

5.8. References��....��...�� 121

Chapter 6. Conclusions��..��.. 123

6.1. Contributions��.��.��.123

6.2. Future research��..��.. 125

6.2.1. Geophysical applications��..125

6.2.2. Complex media.��..��. 127

6.3. References��.��.. 127

Appendix 1. Co-ordinate rotations��..�129

Appendix 2. Tensor rotation��132

Appendix 3. Solutions to one-dimensional magnetic field coefficients� 133

Appendix 4. Solutions to the Helmholtz equation��..�� 135

Appendix 5. Solution to Fourier series coefficients��..��. 138

Appendix 6. Finite-difference operators��.�� 141

Appendix 7. Inhomogeneous plane waves��.��.143

ix

List of Figures

Figure 1.1. Fundamental co-ordinate system for the solution to

problems��..��17

Figure 1.2. Inclined co-ordinate system relative to the fundamental

co-ordinate system with Euler angles about the x-axis

indicated��...��.. 18







Figure 2.1. Effective conductivity for 0 calculated using o90≤≤ mα

equations (2.1) and (2.2) for a homogeneous half space

where 001.0=tσ S/m and 4=λ ��..31

Figure 2.2. Surface impedance and magnitude of a homogeneous

TM-type wave incident upon a homogeneous half space

with the anisotropic parameters 001.0=tσ S/m, 4=λ

and ��..��. 32 o30=α

Figure 2.3. Surface impedance and magnitude of a homogeneous

TM-type wave incident upon a two-layered half space

where the upper layer is isotropic with conductivity of 0.1

S/m and 10 km thick. The bottom layer, of infinite depth

extent, has the anisotropic parameters 001.0=tσ S/m,

4=λ and ��.33 o30=α

Figure 2.4. Geometry for the inclined anisotropic half space. The

source fields are incident parallel to the x-axis. The z-axis

is normal to the surface of the earth��.��.. 34

Figure 2.5. Geometry for a two-layer horizontally stratified half space

where each layer exhibits inclined uniaxial anisotropic

conductivity��.��.��..45

Figure 3.1. Geometry and parameters of the dyke embedded in a

homogeneous layer��.��48

x

Figure 3.2. Fourier series representations, satisfying the )(zgm

boundary conditions that for and

��.�.�.��.��.. 53

0)(, =zH amx 0=z

hz =

Figure 3.3. Fourier series representations, satisfying the )(zgm

boundary conditions that for and 0)(, =zH amx 0=z

0)(, =

∂∂

zhH a

mx at z = h��.�..��..67

Figure 4.1. Schematic diagram in the y-z plane showing the numbering

convention for the impedance mesh. The dotted line shows

the integration path in the x-z plane for the application of

Ampere�s Law��..��.. 84

Figure 4.2. Surface impedance magnitude and phase response for an

anisotropic layer of 1 000 m thickness with 001.01, =tσ

S/m and 01.01, =nσ S/m above a perfectly conducting

basement. The anisotropic conductivity tensor is rotated

through the range . The impedance method o900 ≤≤ α

solutions for 10 Hz (+), 100 Hz (x), 1 kHz (o) and 10 kHz

(*) are presented. The exact solutions for 10 Hz (dotted),

100 Hz (solid), 1 kHz (dash-dotted) and 10 kHz (double-

dashed) are also shown. Note that at 1 kHz and 10 kHz, the

surface impedance is equal to the intrinsic impedance of the

layer and the phase is equal to 45o��...��.. 89


anisotropic layer of 20 m thickness with 01, 15εε =n ,

01.01, =nσ S/m, 01, 5εε =t and 001.01, =nσ S/m above a

perfect electrically. The anisotropic complex conductivity

tensor is rotated through the range . The o900 ≤≤ α

impedance method solutions for 104 Hz (o), 105 Hz (*) and

106 Hz (∆) are presented. The exact solutions for 104 Hz

(solid), 105 Hz (dashed) and 106 Hz (double dashed) are

also shown. Note that at due to the effects of displacement

currents at 106 Hz, the surface impedance is equal to the

xi

intrinsic impedance of the layer and the phase is not equal

to 45o��..�� 90

Figure 4.4. Surface impedance magnitude and phase response at 100

Hz for two anisotropic layers overlying a perfectly

conducting basement using the self-consistent impedance

method (o) and corresponding exact solutions (solid line).

The upper layer thickness was varied and the total depth

was kept constant at 1 000 m. The upper layer has

σt,1 = 0.01 S/m, σn,1 = 0.001 S/m and α1 = 30o. The lower

layer has σt,2 = 0.05 S/m, σn,2 = 0.5 S/m and α2 = 60o�..... 91

Figure 4.5. Surface impedance magnitude and phase response at a

single boundary at 10 kHz for a 500 m wide vertical dyke

with 01.01, =tσ S/m and 001.01, =nσ S/m embedded in an

otherwise homogeneous layer with 001.01, =tσ S/m and

01.01, =nσ S/m, terminated with a perfectly conducting

basement. The depth of the dyke and layer is 200 m. The

impedance method response (dots) is shown above the

exact response (solid line)��.��. 93

Figure 5.1. Geometry for the inhomogeneous plane wave incident at a

skew angle θ to the anisotropic half space inclined at angle

α about the x-axis��.. 97

Figure 5.2. xyZ of a 10 kHz homogeneous TM-type plane wave as a

function of θ for a half space with 001.0=tσ S/m,

01.0=nσ S/m, and =α 45o��..�� 112

Figure 5.3. of a 10 kHz homogeneous TM-type plane wave as a xyZ∠


01.0=nσ S/m, and =α 45o��..�� 113

Figure 5.4. yyZ of a 10 kHz homogeneous TM-type plane wave as a


01.0=nσ S/m, and =α 45o��..�� 114

xii

Figure 5.5. of a 10 kHz homogeneous TM-type plane wave as a yyZ∠


01.0=nσ S/m, and =α 45o��..�� 115

Figure 5.6. Angle of orientation of the surface impedance meter ψ

with respect to the skew angle of incidence θ and strike

of the anisotropic half space in the xy plane��.. 116

Figure 5.7. Normalized observed (measured) magnetic field of a 10 kHz

homogeneous TM-type plane wave as a function of ψ for

a half space with 001.0=tσ S/m, 01.0=nσ S/m, =α 45o

and θ =60o��. 117

Figure 5.8. Normalized observed (measured) electric field of a 10 kHz

homogeneous TM-type plane wave as a function of ψ for

a half space with 001.0=tσ S/m, 01.0=nσ S/m, =α 45o

and θ =60o��.. 118

Figure 5.9. Normalized observed (measured) magnetic field of a 19.8

kHz VLF wave as a function of ψ measured at different

times of day at the Callide Mine, Trap Gully B Area in July

1997 from [5.13]�� 119

Figure 5.10. Normalized observed (measured) electric field of a 19.8

kHz VLF wave as a function of ψ measured at different

times of day at the Callide Mine, Trap Gully B Area in July

1997 from [5.13]�� 120

xiii

Acknowledgments

During this thesis, I�ve been fortunate to have the opportunity to meet and work

with many great and inspiring people.

Foremost, my grateful acknowledgments to Professor D. V. Thiel for his

ongoing professional and personal encouragement, guidance and commitment to

supervise this thesis as Principal Supervisor, and to Dr S. G. O�Keefe for his role as

Associate Supervisor. I am most appreciative to Dr P. G. Keleher for taking on the role

of External Associate Supervisor whilst I was employed in Central Queensland during

2000. During 2002, I visited the Department of Geology and Geophysics, University of

Utah, Salt Lake City, as an exchange graduate student. I am indebted to Professor M. S.

Zhdanov for his hospitality and guidance, as well as for accepting the role of External

Associate Supervisor. Special thanks are also extended to Mr W. J. F. Nichols of

Callide Coalfields Pty Ltd, Biloela, for his unwavering support during the course of this

candidature.

As a self-funded candidate, there is absolutely no way this thesis could have

been completed without the financial support of several organisations, to which I am all

very grateful and would like to acknowledge here. Callide Coalfields Pty Ltd have

financially supported this work and their funding was primarily responsible for

enabling the exchange to the University of Utah. The International Centre of Griffith

University provided a long stay scholarship as part of the International Experience

Incentive Scheme. In 2001, I was awarded an Australian Institute of Geoscientists

Postgraduate Bursary and I acknowledge that support here.

I am also indebted to a great number of teachers, colleagues, associates and

fellow students, from whom along the way, I have learnt from. I wish to firstly

acknowledge my gratitude to those organisations which provided me with employment

and study opportunities during this thesis; Central Queensland University, Callide

Coalfields Pty Ltd, Griffith University and the University of Utah; but importantly,

those people to whom I have had the pleasure to meet, know and work with: Drs N. G.

Golubev, D. A. James, and Messrs A. Chernyavskiy, G. J. Durnan, H. Ebersbach, B. R.

Hanna, Y. Hill, A. V. Gribenko, S. A. Mehanee, S. Mukherjee, E. Peksen, S. A. Saario,

R. W. Schlub and E. Tolstaya.

Finally, but certainly not least, to my family for their on-going support and

encouragement, which is always appreciated.

1

Introduction

0.1. Overview

The importance of studying electrically anisotropic and inhomogeneous media

is of direct relevance to a wide range of electrodynamic disciplines, such as photonic

crystals [0.9] and inductive prospecting methods [0.10]. A variety of methods for

solving the boundary value problems associated with the electrodynamics of

anisotropic and inhomogeneous media have been developed over the last half-century.

These methods can be broadly described as being one of either:

• exact (or analytical) solutions to partial differential equations with boundary

value conditions;

• approximate (or numerical) solutions to partial differential equations with

boundary value conditions;

• simulations using RC network analogies; or

• high frequency scale modelling using electrolytic troughs.

In this thesis, only exact and approximate methods for the solutions to boundary

value problems related to plane wave propagation in electrically anisotropic and

inhomogeneous media with specific applications to low frequency electromagnetic

geophysical methods will be considered.

The first comprehensive catalogue of exact solutions for plane wave

propagation problems in electrically anisotropic and inhomogeneous media was

detailed by Porstendorfer [0.11, 0.12]. These same catalogues of models also detail

some of the earlier works on approximate and scale modelling solutions. The most

comprehensive catalogue and study of approximate solutions for plane wave

propagation problems in electrically inhomogeneous media, including finite-difference,

finite-element and integral equation methods, was conducted as part of the Comparison

of Modelling Methods for Electromagnetic Induction Problems (COMMEMI) project

[0.23]. However, no studies of approximate solutions for problems including

electrically anisotropic media were studied as part of the COMMEMI project.

2

In the geophysical community, both academic and industrial, there is now

significant interest in the development of three-dimensional approximate solutions to

problems including both electrically anisotropic and inhomogeneous media. It may be

argued that studying exact solutions to the electrodynamics of plane wave propagation

in anisotropic and inhomogeneous media for a few ideal models will not yield

significant information about the more general induction problem of arbitrarily

anisotropic and inhomogeneous media. However, the viewpoint is taken that the

principles one needs to understand the more complicated induction problems can be

developed one by one by first handling the simpler class of induction problems.

Methods of exactly solving the boundary value problems for the

electrodynamics of isotropic and inhomogeneous media for geophysical applications

have existed circa 1950. Following these early works, the mathematical foundations for

solving the exact boundary value problems for the electrodynamics of anisotropic and

inhomogeneous media for geophysical applications were developed circa 1960.

Literally, hundreds of references may be cited and in view of the great diversity of

references available in the literature, it would be difficult to cite every author who has

contributed to the discipline over the last half century. However, it is important to

mention here some of the outstanding developments in the literature; namely:

• A. N. Tikhonov�s [0.17] exact solution of a horizontal electric dipole on the

surface of an anisotropic half space;

• the extensive works of D. N. Chetaev [0.2-0.8] devoted to the electrodynamics

of anisotropic media; and

• the tensor surface impedance concept for magnetotelluric prospecting by T.

Cantwell [0.1] and I. I. Rokityanski [0.14].

These particular works have had an especially strong influence on the studies of

wave propagation in anisotropic media for geophysical applications, and specifically,

this thesis relies heavily on the works of D. N. Chetaev for developing exact solutions

to plane wave propagation problems in electrically anisotropic and inhomogeneous

media. However, it is recognised that exact solutions do have limitations, particularly

when one is trying to simulate the propagation of fields in complicated models. It is for

this reason that interest in approximate solutions has increased significantly in recent

years. However, this thesis only contains part of the material that constitutes

approximate solutions to anisotropic electrodynamic problems. For investigations of

3

approximate solutions, this thesis will only consider a two-dimensional finite-difference

method. Discussions and comparisons to corresponding finite-element methods are

presented. The theories of integral equations are not considered in this thesis.

0.2. Outline of thesis

This thesis structurally consists of two parts:

(1) an exact and approximate study of inclined and coplanar electrical anisotropy in

one- and two-dimensional wave propagation problems via vector function

solutions to boundary value problems, with particular reference to homogeneous

plane wave excitation (Chapters 2, 3 and 4). Consideration is only given to

those problems where the source field is incident parallel to one of the axes of

the conductivity tensor, allowing the TE- and TM-type waves to propagate

independently; and

(2) an exact study of the problem of inclined and coplanar electrical anisotropy at a

skew incidence to an incident homogeneous plane wave, derived using potential

function solutions to boundary value problems (Chapter 5). Consideration is

given to both TE- and TM-type source fields and equations for the coupled

reflected and transmitted waves are derived. Results from very low frequency

surface impedance surveys are also presented to verify the formulations

obtained.

0.3. Detailed outline

This thesis consists of six chapters. In some detail, we now outline the relevant

material in each chapter.

Chapter 1.

Electromagnetic fields are defined by the Maxwell equations. There exist a wide

variety of definitions and nomenclature used in the different scientific literatures, which

may lead to some confusion for the reader of this thesis. Due to constant referral to the

Maxwell equations throughout this thesis, and those formulae and conditions derived

4

from them, this thesis commences with a formal treatment of the equations of

electrodynamics, followed by brief reviews of the surface impedance and electrical

anisotropy concepts. Thus, Chapter 1 is auxiliary to the later chapters of this thesis,

where the different classes of plane wave propagation problems are treated.

Chapter 2.

In this chapter, we consider the surface impedance problem of a homogeneous

plane wave incidence upon a homogeneous half space with inclined electrical

anisotropy. It is assumed that the source field is polarised such that it is aligned parallel

to one of the elements of the inclined conductivity tensor, meaning that the TE- and

TM-type waves propagate independently. First, Chetaev�s method of functional

rotation [0.2] is considered, where the Maxwell equations are considered in the inclined

co-ordinate system and rotated into the fundamental co-ordinate system. The method of

tensor rotation is then considered where the inclined conductivity tensor is rotated into

the fundamental co-ordinate system with an Euler rotation.

It is shown that the class of problem with an inclined conductivity tensor is

equivalent to a class of problem with a biaxial conductivity tensor in the fundamental

co-ordinates, where:

zz

zyyzyyyy

nty σ

σσσρ

σα

σασ −==

+= −

−1

122 sincos , (0.1)

and

yy

yzzyzzzz

ntz σ

σσσρ

σα

σασ −==

+= −

−1

122 cossin . (0.2)

It is then shown that for homogeneous TM-type incidence, the surface

impedance depends only upon the conductivity given by equation (0.1). For incidence

of the TE-type, the surface impedance is independent of the anisotropy parameters.

These formulations are extended to the problem of a horizontally stratified half space

using Wait�s [0.18] transmission line analogy. This chapter has been published as

[0.19].

5

Chapter 3.

In the previous chapter, it was shown that for homogeneous TM-type waves, the

problem of an inclined conductivity tensor is equivalent to a class of problem with a

biaxial conductivity tensor in the fundamental co-ordinates, provided that the medium

is horizontally homogeneous. In this chapter, Chetaev�s method of functional rotation is

extended to consider the fields of a homogeneous plane wave in a laterally

inhomogeneous medium. This exact solution is based on Rankin�s [0.13] Fourier series

solution for TM-type incidence. TE-type incidence is not considered in this chapter

since, as was shown in Chapter 2, TE-type waves propagate independently of the

inclined anisotropic conductivity. Of principle interest in this chapter are the properties

of the anisotropic Helmholtz equation:

.0cossin 11 2

cossinsincos

2

2

222

2

222

=−∂∂

∂

−+

∂∂

++

∂∂

+

xx

tn

x

tn

x

tn

Hjzy

Hσ

zH

yH

ωµαασ

σα

σα

σα

σα

(0.3)

Using separation of variables and by imposing known boundary conditions, it is

shown that in laterally inhomogeneous media, the shearing term containing the mixed

partial derivative in equation (0.3) vanishes and only solutions to the bianisotropic

Helmholtz equation need be found:

0cossinsincos2

222

2

222

=−∂

∂

++

∂∂

+ x

x

tn

x

tn

HjzH

yH ωµ

σα

σα

σα

σα . (0.4)

From equations (0.1) and (0.2), one observes that equation (0.4) can be cast in a

form where the inclined conductivity tensor is equivalent to a biaxial conductivity

tensor in fundamental co-ordinates. The exact solution for a horizontally

inhomogeneous layer is then derived. The result is similar to the formulation by Rankin

[0.13]. The work presented in this chapter has been published as [0.21].

6

Chapter 4.

Chapter 4 commences by formally introducing the self-consistent impedance

method; a two-dimensional finite-difference method for TM-type propagation problems

developed by Thiel & Mittra [0.15]. This approximate method is presented in such

form that it can be applied to a wide range of electrodynamic problems beyond the

scope of the specific geophysical ones presented in this thesis. The formulation

presented can include displacement currents through a complex conductivity.

When other authors developed approximate methods for solving two-

dimensional TM-type propagation problems in inclined anisotropic media, they

considered approximate solutions to equation (0.3). In the formulation of the self-

consistent impedance method presented in this thesis, one is able to observe that only

approximate solutions to equation (0.4) need be sought. In previous chapters, it has

been shown that for homogeneous TM-type waves, the class of problem with an

inclined conductivity tensor is equivalent to a class of problem with a biaxial

conductivity tensor in the fundamental co-ordinates.

It is therefore concluded that by approximating equation (0.4), which is

identified as the Helmholtz equation for bianisotropic media in the fundamental co-

ordinate system, one is equivalently approximating equation (0.3), which has been

identified as the Helmholtz equation for inclined anisotropic media. Results are

compared with the corresponding exact solutions for inclined anisotropic media as

presented in Chapters 2 and 3. Whilst in this thesis, we have only chosen the self-

consistent impedance method to demonstrate this formulation, it should be noted that

the principle of the inclined anistropic/bianisotropic equivalence would hold for any

approximate method. The work presented in this chapter has been presented in papers

[0.16, 0.20, 0.22].

Chapter 5.

In this chapter, Chetaev�s method of auxiliary potentials is implemented to

solve for the fields of homogeneous plane waves at skew incidence to an inclined

anisotropic half space. This class of problem is a logical extension of the class of

problems considered in Chapter 2, which can be considered as a special class of the

solutions presented in this chapter. When a TM/TE-type field is incident at a skew

angle upon an inclined anisotropic half space, both TM- and TE-type field components

7

exist in the lower half space. As a result, both TM/TE- and TE/TM-type fields are

reflected from the surface into the upper half space. In effect, this means that a linearly

polarised field is reflected as an elliptically polarised field from an anisotropic half

space. First, the general expressions for all electromagnetic field components that exist

in the anisotropic half space are considered. Then, the separate problems of

inhomogeneous TE- and TM-type incident fields are treated, and equations for the

surface impedance tensor elements at the surface of the anisotropic half space are

derived. The response of one-dimensional surface impedance measurements are then

analysed and discussions related to the practical applications of this technique for

identifying the direction of anisotropic strike are presented. This chapter has not yet

been prepared for publication.

Chapter 6.

In this chapter, a summary of the thesis is presented. Achievements and specific

contributions to the solution of plane wave propagation problems in electrically

anisotropic and inhomogeneous media are presented. The thesis is concluded with some

brief suggestions for future research in the electrodynamics of anisotropic media.

0.4. Publications

Papers [0.19-0.21] have been prepared by the author in collaboration with the

thesis supervisors, and submitted to international journals. Papers [0.16, 0.22] have

been prepared by the author in collaboration with the thesis supervisors, and submitted

to international conferences.

0.5 References

[0.1] T. Cantwell, PhD thesis, Massachusetts Institute of Technology, 1960.

[0.2] D. N. Chetaev, Bull. Acad. Sci. USSR Geophys. Ser. 4, 407 (1960).

[0.3] D. N. Chetaev, Radioteck. Elektron. 8, 64 (1963).

[0.4] D. N. Chetaev, Phys. Solid Earth 2, 233 (1966); 2, 651 (1966).

[0.5] D. N. Chetaev, Sov. Phys. Dokl. 12, 555 (1967).

[0.6] D. N. Chetaev, in Estestvennoye Electromagnitoe Pole i Issledovaniya

Vnutrennego Stroeniya (Nauka, Moscow, 1971), pp. 15-39.

8

[0.7] D. N. Chetaev & M. G. Savin, Sov. Phy. Dokl. 12, 564 (1967).

[0.8] D. N. Chetaev & B. N. Belen�kaya, Phys. Solid Earth 7, 212 (1971); 8, 535-538

(1972).

[0.9] J. D. Joannopoulos, R. D. Meade & J. N. Winn, Photonic Crystals: Modeling

the Flow of Light (Princeton Uni. Press, Princeton, 1995).

[0.10] J. G. Negi & P. D. Saraf, Anisotropy in Geoelectromagnetism (Elsevier,

Amsterdam, 1989).

[0.11] G. Porstendorfer, Principles of Magneto-telluric Prospecting (Geopublication

Associates, Berlin, 1975).

[0.12] G. Porstendorfer, in Geoelectric and Geothermal Studies (East-Central Europe,

Soviet Asia, edited by A. Adam (Akad. Kiado, Budapest, 1976), pp. 152-164.

[0.13] D. Rankin, Geophysics 27, 666, 1962.

[0.14] I. I. Rokityanski, Bull. Acad. Sci. USSR Geophys. Ser. 10, 1050 (1961).

[0.15] D. V. Thiel & R. Mittra, Radio Sci. 36, 31 (2001).

[0.16] D. V. Thiel & G. A. Wilson, presented at URSI Nat. Sci. Meet., Boulder, CO,

Jan. 2002.

[0.17] A. N. Tikhonov, Sov. Phys. Dokl. 4, 566 (1959).

[0.18] J. R. Wait, Electromagnetic Waves in Stratified Media, 2nd ed. (Permagon Press,

Oxford, 1970).

[0.19] G. A. Wilson & D. V. Thiel, Radio Sci. 37, 1029, 2001RS002535 (2002).

[0.20] G. A. Wilson & D. V. Thiel, IEEE Trans. Geosci. Remote Sensing, submitted.

[0.21] G. A. Wilson & D. V. Thiel, J. Electromagn. Waves Applic., submitted.

[0.22] G. A. Wilson & D. V. Thiel, presented at IEEE Int. Antennas Propagat. Symp.,

San Antonio, TX, Jun. 2002.

[0.23] M. S. Zhdanov, I. M. Varenstov, J. T. Weaver, N. G. Golubev & V. A. Krylov,

J. App. Geophys. 37, 133 (1997).

9

Chapter 1

Introductory Theory

The material discussed in this thesis requires the knowledge of fundamental

concepts from electromagnetic theory. As we will continually be referring to the

Maxwell equations throughout this thesis, this chapter starts by defining the Maxwell

equations in the most general cases. We then obtain forms of these equations that are

used regularly throughout this thesis. After this, the frames of reference, further

notations and other definitions are presented. An introductory discussion on surface

impedance and electrical anisotropy in electromagnetic geophysics is then presented.

1.1. The Maxwell equations

1.1.1. The Maxwell equations

The theory for electrical geophysics has been developed from the laws

governing the behaviour of electromagnetic fields in an inhomogeneous conducting

earth. The Maxwell equations, which are the fundamental, mathematical formulations

describing the behaviour of electromagnetic fields, are the basis of this theory, e.g., see

[1.33]. For the sake of a thorough exposition, this section follows the formal

mathematical treatment of Zhdanov [1.32]. In differential form, the Maxwell equations,

excluding extraneous currents and sources in a homogeneous medium are:

t∂∂+=×∇ DJH , (1.1)

t∂∂−=×∇ BE , (1.2)

0=⋅∇ B , (1.3)

qρ=⋅∇ D , (1.4)

10

where qρ is the density of all charges in the medium. Equation (1.1) is an expression of

Ampere�s Law which relates the magnetic field intensity H with the current density J

and time-dependent displacement field D; equation (1.2) is an expression of Faraday�s

Law which relates the electric field intensity E with the time-dependent magnetic flux

density B; equation (1.3) states the non-existence of magnetic monopoles; and equation

(1.4) is Gauss� or Coulomb�s Law stating the existence of electric monopoles.

Equations (1.3) and (1.4) represent the continuous properties of the electromagnetic

fields. Equations (1.1) to (1.4) can also be expressed in integral form as:

sDJlH dt

dSC

⋅

∂∂+=⋅ ∫∫

, (1.5)

td

C ∂∂−=⋅∫φ

lE , (1.6)

0

=⋅∫SdsB , (1.7)

qdS

=⋅∫ sD , (1.8)

where q is the total charge and φ is the total flux through the surface S. As a result of

equations (1.1) and (1.4), the Maxwell equations are conservative such that:

tq

∂∂−

=⋅∇ρ

J . (1.9)

In addition to the four fundamental Maxwell equations, two constitutive

equations exist to relate the magnetic flux density B and magnetic field intensity H, and

the displacement field D and electric field intensity E vectors:

HB µ= , (1.10)

ED ε= , (1.11)

11

where µ and ε represent the permeability and permittivity of the medium. The current

density J in an extended medium is linearly proportional to the electric field vector E.

This is Ohm�s Law, and is stated as:

EJ σ= , (1.12)

where σ is the conductivity of the medium, and is the reciprocal of the resistivity ρ of

the medium. In equations (1.10) to (1.12), the terms µ, ε and σ respectively can have

either scalar or tensor properties. It is observed that equations (1.3) and (1.4) follow

from equations (1.1) and (1.2) with continuity equation (1.9).

To seek both exact and approximate solutions, it is useful to separate the

Maxwell equations (1.1) to (1.4) into equations of the electric and magnetic fields. To

do this, one can first re-write equations (1.1) and (1.2) exclusively in terms of E and H

by substituting constitutive relations (1.10) and (1.11), as well as equation (1.12). By

applying the curl operator to both sides of equation (1.2) and substituting the result into

equation (1.1), one obtains:

02

2

=∂∂+

∂∂+×∇×∇

ttEEE µσµε , (1.13)

for the electric field. Similarly, by applying the curl operator to both sides of equation

(1.1) and substituting into equation (1.2), one obtains:

02

2

=∂∂+

∂∂+×∇×∇

ttHHH µσµε , (1.14)

for the magnetic field. Using the vector identity:

EEE ∆−⋅∇∇=×∇×∇ )( ,

where ∆ is the Laplacian operator (n.b., ∇ is also commonly used in scientific

literature), and with Maxwell equation (1.4), equation (1.13) can be re-written as:

2

12

qttρ

εµσµε ∇=

∂∂−

∂∂−∆ 1

2

2 EEE . (1.15)

Free charges cannot exist in a homogeneous medium without extraneous

electric currents and sources, so equation (1.15) reduces to:

02

2

=∂∂−

∂∂−∆

ttEEE µσµε . (1.16)

One obtains a similar relationship for the magnetic field in equation (1.14),

using Maxwell equation (1.3) to obtain:

02

2

=∂∂−

∂∂−∆

ttHHH µσµε . (1.17)

Equations (1.16) and (1.17) are called the homogeneous telegraph equations for

a homogeneous medium in the absence of extraneous currents and charge. In many

problems in electromagnetic geophysics, the fields vary with time sufficiently slowly

that the ∂ term can be ignored. Such quasi-static electromagnetic fields satisfy

the Maxwell equations:

22 / t∂

JH =×∇ , (1.18)

t∂∂−=×∇ BE , (1.19)

0=⋅∇ B , (1.20)

and the continuity condition ∇ for an inhomogeneous medium with conductivity

gradients:

0=⋅ J

σσ

∇⋅−=⋅∇ EE 1 , (1.21)

13

in the absence of extraneous currents and sources. Equations (1.18) to (1.21) lead to the

homogeneous diffusion equations:

0=∂∂−∆

tEE µσ , (1.22)

0=∂∂−∆

tHH µσ . (1.23)

If one considers the Maxwell equations for a perfect electric insulator (i.e.,

0=σ ), then equation (1.1) takes the form of:

t∂∂=×∇ DH . (1.24)

In a homogeneous medium, equations (1.16) and (1.17) then take the form of

the homogeneous wave equations:

02

2

=∂∂−∆

tEE µε , (1.25)

02

2

=∂∂−∆

tHH µε . (1.26)

It is now assumed that all fields are monochromatic and vary with a time

dependence of )exp( tjω type where 1−=j . With few exceptions, any field that is

an arbitrary function of time can be synthesised using a sum of cisoidal functions since

any reasonably well-behaved function of time can be replaced with either a Fourier

integral or Fourier series of its spectral components [1.32].

In a homogeneous medium without extraneous currents and charges, the

telegraph, diffusion and wave equations all satisfy the homogeneous Helmholtz

equations:

02 =+∆ EE k , (1.27)

14

02 =+∆ HH k , (1.28)

where k is the wave number, and has the most general expression:

ωµσεµω jk −= 22 , (1.29)

when magnetic conductivity is neglected. To prevent exponentially divergent solutions

in both E and H with increasing values of either time or distance, the values of k are

chosen such that . In a homogeneous medium, the wave number can also be

written as:

0Re >k

*2 ωµσjk −= , (1.30)

where ωεσσ j+=* is called the complex conductivity. Hence, it is possible to obtain

frequency-dependent telegraph equation solutions using the diffusion equation provided

the conductivity is complex.

1.1.2. Auxiliary potential functions

Since B is rotational in the full space, as expressed in equation (1.3), it may

therefore be defined unequivocally by specifying a vector potential function A [1.32]:

AB ×∇= . (1.31)

Since , then equation (1.3) is satisfied identically, and it is noted

that A may have both rotational and lamellar parts. The use of A requires the

specification of the rotational and lamellar parts for unique representation of the field

quantities. The rotational part is normally specified based on the properties of the field.

The lamellar part must then be specified to be consistent with the field equations.

Substituting equation (1.31) into equation (1.2), then:

0( =×∇⋅∇ A)

0)( =+×∇ AE ωj . (1.32)

Since for any scalar Φ, equation (1.32) can be integrated to obtain: 0=Φ∇×∇

15

Φ∇−−= AE ωj , (1.33)

where Φ is a scalar function of position and is called the scalar potential function.

Equation (1.15) satisfies equation (1.3) identically. Since Φ and ∇ are as yet

arbitrary, we can choose a relationship between them with a view to simplifying the

equations satisfied by A. One condition is:

A⋅

0=⋅∇ A , (1.34)

and this is known as the Coulomb gauge condition. The second condition is:

Φ−=⋅∇ ωµεjA , (1.35)

and this relationship is known as the Lorentz gauge condition. It should be noted

however that equation (1.35) is by no means the only unique solution for the Lorentz

gauge condition. Any function which satisfies the Lorentz gauge condition then

satisfies equations (1.2) and (1.4) identically as they lead to the continuity equation, and

hence conservative properties of the Maxwell equations [1.6].

1.1.3. Boundary conditions

As written in equations (1.1) to (1.8), the Maxwell equations are only valid in a

homogeneous medium where the properties µ, ε and σ vary slowly in space such that

E, D, B and H can be differentiated with respect to spatial and time co-ordinates only.

If the properties do vary discontinuously, the Maxwell equations must be augmented

with appropriate boundary conditions. At a boundary between different regions in the

earth, µ, ε and σ can change abruptly. Referring to equations (1.10) to (1.12), this

results in a discontinuous change in several of the fields. This is considered as a

physical discontinuity in material properties between two media, referred to as either

medium 1 or medium 2. It is therefore essential to have boundary conditions to define

the fields at points immediately adjacent to each other on either side of a boundary

across which the properties of the medium changes. E is governed by equation (1.4)

and (1.11). By re-writing this in terms of the electric field intensity alone, the interface

16

conditions for the tangential t and normal n components of the field can be written

respectively as:

tt EE 21 = , (1.36)

snn DD ρ=− 21 , (1.37)

where sρ is the surface charge density at the surface. Thus the tangential component of

E is continuous regardless of the material properties. The normal component of D is

discontinuous and the discontinuity is equal to the surface charge density at the

boundary. In the particular case when 0=sρ , the normal component of E is

continuous across the boundary. The boundary conditions for B are similarly obtained.

However, in this case, the surfaces between the two media may contain a surface

current density instead of a surface charge density such that:

tt BB 21 = , (1.38)

snn JBB =− 21 , (1.39)

where is the surface current density at the surface. Thus the tangential component of

B is continuous regardless of the material properties. The normal component of B is

discontinuous and the discontinuity is equal to the surface current density at the

boundary. In the particular case when , the normal component of B is continuous

across the boundary. Equations (1.36) to (1.39) are a complete representation of the

boundary conditions for the tangential and normal components of E, D, B and H.

sJ

0=sJ

1.1.4. Frames of reference

In this thesis, all problems are considered within the Cartesian co-ordinate

system. The fundamental co-ordinate system is defined as the right-hand positive {x, y,

z} co-ordinate system, as shown in Figure 1.1. The xy-plane is parallel to the surface of

the earth and the z-axis is perpendicular to the surface of, and is directed into the earth.

Another right-handed positive Cartesian co-ordinate system exists and can be

17

represented by {x�, y�, z�} where the y�z�-plane is inclined at an angle α about the x-

axis, the x�z�-plane is inclined at an angle β about the y-axis and the x�y�-plane is

inclined at an angle θ about the z-axis. This is defined as the inclined co-ordinate

system (Figures 1.2, 1.3 and 1.4). α, β and θ represent the three elementary Euler angles

about the x, y and z axes respectively, which can be used to rotate one frame of

reference to another (see Appendix 1).

x y z

Figure 1.1. Fundamental co-ordinate system for the solution to problems.

18

x= x� y α α z� y� z

Figure 1.2. Inclined co-ordinate system relative to the fundamental co-ordinate

system with the Euler angles about the x-axis indicated.

x x� β y = y� β z� z


system with the Euler angles about the y-axis indicated.

19

x x� θ y θ y� z = z�


system with the Euler angles about the z-axis indicated.

1.1.5. Further notation and definitions

All dimensional quantities are expressed explicitly in the SI units of

measurement. All definitions used in this thesis conform to the IEEE Standard

Definitions of Terms for Radio Wave Propagation, IEEE Std 211-1997TM. However,

in the interest of clarifying some of those definitions, we now define some of the

important concepts, and their respective mathematical formulations.

Homogeneous plane waves are those waves where the planes of constant

magnitude and constant phase are parallel. Time harmonic homogeneous plane waves

propagating in the z-direction are characterised by the notation:

)exp()exp( zjktj z−ω , (1.40)

where is the wave number in the z-direction. Inhomogeneous plane waves are those

waves where the planes of constant magnitude and constant phase are not parallel.

Time harmonic inhomogeneous plane waves propagating in the x- and z-directions are

characterised by the notation:

zk

20

)exp()exp()exp( zjkxktj zx −−ω , (1.41)

where is the wave number in the x-direction. In the scientific literature, there is

considerable ambiguity in the definitions of anisotropy and inhomogeneity as a

descriptor for media, e.g. [1.7], especially when discussing macro-anisotropy. Here, the

definitions for anisotropy and inhomogeneity are clarified as they relate to the problems

discussed in this thesis.

xk

The full space is considered as two horizontal half spaces. The upper half space

will be air and have properties such that 0=σ , 0εε = and 0µµ = . The lower half

space will be earth and have properties such that 0≠σ , 0εεε r= and 0µµµ r= .

The lower half space is defined as homogeneous when the half space consists

only of one medium. The lower half space is defined as inhomogeneous when the half

space can be considered a juxtaposition of two or more media in any spatial

dimension(s) and the boundaries between the different media may be unconformal.

A medium is defined as isotropic when all values of the conductivity,

permittivity and permeability are represented by the scalar constants σ, ε and µ

respectively, and are position independent everywhere inside that medium.

A medium is defined as coplanar anisotropic when any, or all of the values of

the conductivity, permittivity and permeability are represented by the second rank

diagonal tensors σ� , ε� and µ� respectively, where each element of the tensor is

coplanar with the fundamental co-ordinate system, and are position independent

everywhere inside that medium. When two elements of the diagonal tensors are equal,

and different to the third element, the medium is defined as uniaxial anisotropic. When

all three elements of the diagonal tensors are different, the medium is defined as biaxial

anisotropic, or bianisotropic.

A medium is defined as inclined anisotropic when any, or all of the values of

the conductivity, permittivity and permeability are represented by the second rank

diagonal tensors σ� , ε� and µ� respectively, inclined at an arbitrary angle to the

fundamental co-ordinate system, and are position independent everywhere inside that

medium. In fundamental co-ordinates, the tensor will contain non-diagonal elements.

Most earth media are non-magnetic so unless explicitly stated otherwise for the

particular problems considered in this thesis, it is assumed that I0� µµ = in the full

space, where I is the identity matrix. In addition to the definitions for inhomogeneous

21

that have been presented above, the term also extends to the description for the right-

hand side of a differential equation not equalling zero.

1.2. Surface impedance theory

By examining the relationship between two orthogonal field components of an

electromagnetic wave incident upon the surface of the earth, it is possible to gain

information about the subsurface conductivity profile beneath the point of measurement

[1.4, 1.23]. This method of mapping the subsurface was initially developed from the

telluric method of geophysical mapping where voltage probes were used to measure the

potential difference at points across the earth�s surface [1.22]. It was realised that the

source fields for these earth currents were geomagnetic pulsations and associated

ionospheric currents, where the fields can be considered to propagate as homogeneous

plane waves [see 1.1]. It should be noted that there has been considerable debate on the

appropriateness of the homogeneous plane wave model [1.9, 1.25, 1.27, 1.28]. As a

result, a method of normalising the measured electric field data was achieved by also

measuring the associated magnetic field components. The surface impedance Zmn is

defined as the ratio between the horizontal electric Em and magnetic Hn field

components:

n

mmn H

EZ = , (1.42)

where yxnm , , = . Originally, all interpretation was conducted for one-dimensional

models of horizontally stratified, isotropic media [1.30]. Such models are referred to as

Tikhonov-Cagniard models. Since the inception of this method, it has been applied

across the frequency range from 10-4 Hz to 100 kHz using both natural radiation

sources such as geomagnetic pulsations, which includes the magnetotelluric (MT)

[1.24, 1.26] and audiomagnetotelluric (AMT) methods, and atmospheric based

techniques [1.11]. Artificial sources have also been introduced, and include the

controlled-source audiomagnetotelluric (CSAMT) [1.13], very low frequency (VLF)

[1.15], and radio magnetotelluric (RMT) [1.10] methods. In geophysics, it is common

to express the surface impedance as an apparent resistivity ρa defined by:

22

0

2

ωµρ mn

a

Z= , (1.43)

It should be noted that this definition of apparent resistivity has been questioned

e.g., [1.21] but in this thesis, where used, the definition of equation (1.43) is implied.

The concept of the apparent resistivity is that it is equal to the resistivity of the medium

as if it is a homogeneous medium.

If the radiation source is not linearly polarised or the earth�s subsurface is

laterally anisotropic, hence not corresponding to the Tikhonov-Cagniard model, the

perpendicularity of E and H does not hold [1.5, 1.19]. In such a case, each electric field

component is the linear sum of both components of the horizontal magnetic field, and

can be expressed as the linear relationship:

yxyxxxx HZHZE += , (1.44)

yyyxyxy HZHZE += , (1.45)

where the second rank impedance tensor Z� has been introduced:

=

yyyx

xyxx

ZZZZ

Z� . (1.46)

The tensor elements are defined by equation (1.43). It is specifically noted that

equation (1.46) is defined in the xy-plane. The components and are called the

principal impedances, and the components Z and are called the additional

impedances [1.20]. However, the tensor components change as the co-ordinate system

is rotated. If a right-handed rotation about the positive z-axis is applied to equation

(1.46) (see Appendix 1), then:

xyZ yxZ

xx yyZ

θθθθ 2sin)(21sincos)( 22

'' yxxyyyxxxx ZZZZZ +−+= , (1.47)


'' yyxxyxxyyx ZZZZZ −+−= , (1.48)

23


'' yyxxxyyxxy ZZZZZ −+−= , (1.49)


'' yxxyxxyyyy ZZZZZ +−+= . (1.50)

There are three expressions obtained from the impedance tensor that are

independent of the rotation of the measurement axes. These are called invariants [1.20]

and are denoted by I:

2

1 effyxxyyyxx ZZZZZI =−= , (1.51)

yyxx ZZI +=2 , (1.52)

ZZZI xyxy 23 =−= , (1.53)

where and 2effZ Z are called the effective and mean impedance values, respectively. If

the radiation source is linearly polarised and/or the earth�s subsurface is laterally

isotropic, then and . 0== yyZxxZ yxxy ZZ −=

1.3. Electrical anisotropy

The concepts of homogeneity and isotropy play an important role in current

electromagnetic plane wave modelling and inversion. Usually, multi-dimensional

models are considered to be a juxtaposition of elements that are homogeneous and

isotropic, e.g., see [1.34]. It has been well documented that in layered or laminated

sedimentary rocks, electrical current circulates with less resistance parallel to the

bedding planes than perpendicularly to them [e.g., 1.14, 1.18]. Several mechanisms

account for this conductivity difference including sorting (i.e., graded deposition), rock

matrix grain geometry, pore geometry and the preferential orientation and grain

geometry of conductive clay minerals. For convenience, early studies concerning

electrical anisotropy assumed the anisotropic model based upon the uniaxial crystal

structure in optics, e.g. [1.3, 1.16], where, in inclined co-ordinates, the uniaxial

conductivity tensor can be represented by:

24

=

n

t

t

σσ

σσ

000000

� , (1.54)

where tσ is the conductivity tangential to the bedding plane, and nσ is the

conductivity normal to the bedding plane. The coefficient of anisotropy [1.18] is

defined as:

n

t

σσλ = . (1.55)

This assumption is based upon the view that depositional conditions in different

lateral directions, representing different points in space at the same instant in time, have

less variation than depositional conditions in the vertical direction, representing the

same points in space at different instants in time. Whilst a uniaxial anisotropic

conductivity is clearly a reasonable approximation based on a simple stratigraphic

model, the assumption is also mathematically convenient. The Maxwell equations can

be decoupled in certain frames of reference, leading to relatively simple exact solutions.

If a more general model were to be considered, it would be easy to assume depositional

environments where the conditions vary laterally [1.12] and it would be necessary to

consider the biaxial conductivity tensor:

=

n

yt

xt

σσ

σσ

000000

� ,

,

. (1.56)

In this thesis, only uniaxial anisotropic media is considered. However, the

following discussion about the properties of the conductivity tensor is equally

applicable to both uniaxial and biaxial anisotropic conductivity tensors. Consider a

conductivity tensor '�σ analogous to either equations (1.54) or (1.56) in an inclined co-

ordinate system {x�, y� z�}. The conductivity tensor can be rotated into the fundamental

co-ordinate system {x, y, z} using the three Euler rotations [1.2]:

),,( '� ),,(� T αβθσαβθσ RR= , (1.57)

25

where ),,( αβθR is the Euler co-ordinate rotation matrix (see Appendix 1):

)()()(),,( αβθαβθ RRRR = . (1.58)

The resultant conductivity tensor σ� takes the most general form of:

=

zzzyzx

yzyyyx

xzxyxx

σσσσσσσσσ

σ� , (1.59)

where ) , , , ,( θβασσσ ntijij f= . The corresponding resistivity tensor ρ� is defined as

the corresponding inverse of the conductivity tensor:

I='� '� ρσ , (1.60)

and it follows that:

IRIRRR == ),,( ),,(),,('� '� ),,( TT αβθαβθαβθρσαβθ , (1.61)

IRRRR =),,('� ),,( ),,( '� ),,( TT αβθραβθαβθσαβθ , (1.62)

I=∴ ρσ � � . (1.63)

Equation (1.63) holds for any co-ordinate rotation. Furthermore, to define the

resistivity tensor, it is demonstrated that:

11 � � � � −− = ρρρσ I , (1.64)

1� � −= ρσ II , (1.65)

1�� −=∴ ρσ . (1.66)

26

However, it should be noted that and except when 1,,

−≠ jiji ρσ jiji ,1

, ρσ ≠−

IR = ),,( αβθ . σ� has the specific properties of non-negativity and symmetry [1.31].

σ� is symmetric whenever the magnetic field does not influence the conduction process

[1.17]. Symmetry is thus granted for purely ohmic conduction. However, in the

presence of Hall currents, such as in plasma, σ� will be asymmetric [1.29]. To

demonstrate this symmetry, an example from Dekker & Hastie [1.8] is used. Let us

consider a laterally anisotropic surface where the ohmic functions:

θθθ sincos)( yx EEE += , (1.67)

θθσθθσθ sin)(cos)()( yx EEJ += , (1.68)

exist, where )(θσ is the effective conductivity in the direction of θ , and is defined

simply as:

)()()(

θθθσ

EJ= . (1.69)

Now, if , then 0=xE θθσ sin)()0( yEJ = and:

∫∫−−

===2

2

2

2

)( cossin)0( cos1)0(π

π

π

π

θθσθθθθσ ddJEE

J

yyxy . (1.70)

If , then 0=yE θθσπ cos)()2

( xEJ = and:

∫∫ ===ππ

θθσθθθπθ

π

σ00

)( cossin)2

( sin1)2

(ddJ

EE

J

xxyx . (1.71)

We can then observe that:

27

∫∫ −=−−

ππ

π

θθσθθθθσθθσσ0

2

2

)( cossin)( cossin ddyxxy . (1.72)

If )()( θσθσ −= , then 0=− yxxy σσ . If not, then there will be rectification of

the electrical current in the direction of θ . Now, consideration is given to the second

condition of non-negativity in σ� . Assuming no magnetic conductivity exists, any

losses in fields are exclusively ohmic (i.e., from electrical conductivity). From

Poynting�s theorem, the expression for the specific energy dissipation u due to ohmic

losses, known as Joule�s Law, can be written as [1.16]:

2 � EEJ σ=⋅=u , (1.73)

Since from thermodynamic considerations, we observe that the time averaged

specific energy dissipation

0>u

u can be written as:

0�21

21 >⋅⋅=⋅= E*EJ*E σu , (1.74)

which implies that σ� must be positive semi-definite. This means that only positive

conductivity values can exist in the diagonal elements of σ� .

1.4. References

[1.1] M. N. Berdichevskiy & M. S. Zhdanov, Advanced Theory of Deep

Geomagnetic Sounding (Elsevier, Amsterdam, 1984).

[1.2] F. W. Byron & R. W. Fuller, Mathematics of Classical and Quantum Physics

(Dover Publications, New York, 1992).

[1.3] M. Born & E. Wolf, Principles of Optics (Permagon Press, Oxford, 1965).

[1.4] L. Cagniard, Geophysics 18, 605 (1953).


[1.6] D. N. Chetaev, Phys. Solid Earth 2, 555 (1966).

[1.7] D. N. Chetaev & B. N. Belen�kaya, Phys. Solid Earth 7, 212 (1971); 8, 535

(1972).

28

[1.8] D. L. Dekker & L. M. Hastie, Geophys. J. R. astron. Soc. 61, 11 (1980).

[1.9] V. I. Dmitriev & M. N. Berdichevsky, Proc. IEEE 67, 1034-1044 (1979).

[1.10] A. Dupis, A. Choquier & G. Borruet, Bull. Soc. Geol. France 166, 231 (1995).

[1.11] S. J. Garner & D. V. Thiel, Explor. Geophys. 31, 173 (2000).

[1.12] S. Gianzero, D. Kennedy, L. Gao & L. SanMartin, Petrophys. 43, 172 (2002).

[1.13] M. A. Goldstein & D. W. Strangway, Geophysics 40, 669 (1975).

[1.14] G. V. Keller & F. C. Frischknecht, Electrical Methods in Geophysical

Prospecting (Permagon Press, Oxford, 1966).

[1.15] R. J. King, IEEE Trans. Antennas Propagat. 24, 115 (1976).

[1.16] L. D. Landau, E. M. Lifshitz & L. P. Pitaevskii, Electrodynamics of Continuous

Media, 2nd ed. (Butterworth-Heinemann, Oxford, 1982).

[1.17] L. Onsager, Phys. Rev. 37, 405 (1931).

[1.18] E. I. Parkhomenko, Electrical Properties of Rocks (Plenum Press, New York,

1967).


[1.20] I. I. Rokityanski, Geoelectromagnetic Investigation of Earth's Crust and Mantle

(Springer-Verlag, Berlin, 1982).

[1.21] B. R. Spies & D. E. Eggers, Geophysics 51, 1462 (1986).

[1.22] D. V. Thiel, IEEE Trans. Antennas Propagat. 48, 1517 (2000).

[1.23] A. N. Tikhonov, Dokl. Akad. Nauk SSSR 73, 295 (1950).

[1.24] A. N. Tikhonov & M. N. Berdichevsky, Phys. Solid Earth 2, 93 (1966).

[1.25] A. N. Tikhonov, D. N. Chetaev, V. A. Morgunov, I. K. Chantladze, S. V.

Shamanin & Ye. A. Gerasimovich, Dokl. Akad. Nauk SSSR Zemli Nauk 217,

28 (1974).

[1.26] K. Vozoff, Geophysics 37, 98 (1972).

[1.27] J. R. Wait, Geophysics 19, 281 (1954).

[1.28] J. R. Wait, J. Res. Nat. Bur. Stand. D. Radio Propagat. 66D, 509 (1962).

[1.29] J. R. Wait, Electromagnetics and Plasmas (Holt, Rinehart and Winston, New

York, 1968).


Oxford, 1970).

[1.31] P. Weidelt, in Three-Dimensional Electromagnetics, edited by M. Oristaglio &

B. Spies (Soc. Explor. Geophys., Tulsa, 1999), pp. 119-137.

[1.32] M. S. Zhdanov, Integral Transforms in Geophysics (Springer-Verlag, Berlin,

1988).

29

[1.33] M. S. Zhdanov and G. V. Keller, The Geoelectrical Methods in Geophysical

Exploration (Elsevier, Amsterdam, 1994).


J. App. Geophys. 37, 133 (1997).

30

Chapter 2

Exact solutions for one-dimensional anisotropy problems

2.1. Introduction

The effect of fundamental uniaxial anisotropic conductivity in a horizontally

layered half space excited by an inhomogeneous plane wave has been previously

studied [2.4, 2.5, 2.14-2.16]. Chetaev [2.2, 2.3] was the first to consider the problem of

a homogeneous half space with inclined uniaxial conductivity anisotropy excited by a

homogeneous plane wave. Chetaev�s method of solution (referred to as the Chetaev

method of functional rotation) was later extended to a two-layer model by Sinha [2.20],

where only the basement was assumed to have inclined anisotropic conductivity, and

later by Negi & Saraf [2.9] where the upper layer was assumed to have inclined

anisotropic conductivity. Reddy & Rankin [2.17] extended the formulation to a multiple

layered half space where each layer was assumed to have inclined anisotropic

conductivity. Further discussion on horizontally layered problems with inclined

anisotropy subsequently followed [2.1, 2.6-2.8, 2.10-2.13, 2.18, 2.23-2.24]. Singh

[2.19] presented an album of sounding curves investigating the effect of conductive and

resistive basements. Thiel [2.21] observed that from Singh�s [2.19] results using the

formulation derived by Sinha [2.20], the effective horizontal conductivity of an inclined

anisotropic layer m was given by the equation:

1

,

2

,

2 sincos−

+=

mn

m

mt

mm σ

ασ

ασ , (2.1)

where mα is the angle of inclination of the anisotropic conductivity, measured with

respect to the horizontal plane. mt ,σ and mn,σ are the tangential and normal

conductivities of the medium and were discussed in Chapter 1. Thiel [2.21] observed

that equation (2.1) is the yyρ element of a uniaxial resistivity tensor rotated into the

horizontal plane. Thiel [2.21] suggested that perhaps the effective horizontal

conductivity should be given by the yyσ element of the uniaxial conductivity tensor

rotated into the horizontal plane, i.e.:

31

mmnmmtm ασασσ 2,

2, sincos += , (2.2)

which would produce a different set of results for the surface impedance than those

presented by Singh [2.19] when 0 . Figure 2.1 presents the effective

horizontal conductivity for calculated using equations (2.1) and (2.2) for

a homogeneous half space where

o90<< mαo90

001.0=t

0 ≤≤ mα

σ S/m and 4=λ . Thiel [2.21] based his

argument on the belief that the conductivity tensor is a more fundamental quantity than

the resistivity tensor, since the conductivity describes the relationship between H and E

in the Maxwell equation (1.1). The discussion was dismissed (without scientific

argument) by Singh [2.19], and no further discussion on this problem has arisen in the

literature.

Figure 2.1. Effective conductivity for 0 calculated using equations

(2.1) and (2.2) for a homogeneous half space where

o90≤≤ mα

001.0=tσ S/m

and 4=λ .

32

The difference in the surface impedance magnitude of a homogeneous TM-type

wave across the magnetotelluric frequency band (10-4 Hz to 10 Hz) for a half space

with inclined conductivity anisotropy is shown in Figure 2.2 with effective

conductivities given by both equations (2.1) and (2.2). The homogeneous half space has

the anisotropic parameters 001.0=tσ S/m, 4=λ and . For a homogeneous

half-space, the phase is equal to and is constant for all frequencies. Figure 3

presents the response for the same frequency band for a two-layered half-space, where

the upper layer has isotropic conductivity of 0.1 S/m and is 10 km thick, overlaying the

homogeneous half-space with inclined anisotropic conductivity described above.

o30=αo45

Figure 2.2. Surface impedance and magnitude of a homogeneous TM-type wave

incident upon a homogeneous half space with the anisotropic

parameters 001.0=tσ S/m, 4=λ and . o30=α

Now, given that some formulations rely on the rotation of the resistivity tensor,

based on Chetaev [2.2, 2.3], and other formulations rely on the rotation of the

conductivity tensor, e.g., [2.25], it is important to clarify under what conditions the two

methods are equal. The observation that the two rotations yield different results for the

33

same conductivity and angle of inclination is one that appears to have been overlooked.

In this chapter, the Chetaev method of function rotation [2.2, 2.3] and the method of

tensor rotation [2.13] will be reviewed to derive expressions for the surface impedance

of homogeneous TE- and TM-type waves incident upon horizontally homogeneous

inclined anisotropic media.

Figure 2.3. Surface impedance and magnitude of a homogeneous TM-type wave

incident upon a two-layered half space where the upper layer is

isotropic with conductivity of 0.1 S/m and 10 km thick. The bottom

layer, of infinite depth extent, has the anisotropic parameters

001.0=tσ S/m, 4=λ and . o30=α

2.2. The problem of a homogeneous half space

We will consider the surface impedance of a homogeneous, monochromatic

TM-type wave with time variance of )exp( tjω incident upon a homogeneous half

space with inclined uniaxial anisotropic conductivity. It is assumed that the

component of the TM-type wave is co-incident with the

xH

xxσ element of the inclined

34

conductivity tensor, and so propagates independently of any TE-type waves (see Figure

2.4). It is assumed that the component of the TE-type wave is co-incident with the xE

xxσ element of the inclined conductivity tensor, and so propagates independently of

any TM-type waves.

'E y

'Ez

σt x

y

α

σt y�

z� σn

z

Figure 2.4. Geometry for the inclined anisotropic half space. The source fields

are incident parallel to the x-axis. The z-axis is normal to the surface

of the earth.

2.2.1. The Chetaev method of functional rotation

In this section, the formulation of Chetaev [2.2] is followed. First, from

equations (1.18) and (1.19), we write the forms of Maxwell�s equations for a

homogeneous TM-type wave in the inclined co-ordinate system:

'1 '

zH x

t ∂∂

=σ

, (2.3)

'1 '

yH x

n ∂∂−=

σ, (2.4)

35

0'' ''' =+

∂∂

−∂

∂x

yz Hjz

Ey

E ωµ . (2.5)

Differentiating equations (2.3) and (2.4) with respect to z� and y� respectively,

and substituting the results into equation (2.5) obtains:

0'

1'

1'2

'2

2'

2

=−∂

∂+

∂∂

xx

t

x

n

HjzH

yH ωµ

σσ, (2.6)

which is the homogeneous Helmholtz equation for anisotropic media. Now, since the

half space is assumed to be infinitely deep, a solution to equation (2.6) in the form of a

down-going homogeneous plane wave is considered:

)cos'sin'(exp)exp( 1111' αα zykCzkCHH xx +−=−== , (2.7)

which satisfies equation (2.6) provided the wave number is given by: 1k

122

1sincos

−

+=

nt

jkσ

ασ

αωµ , (2.8)

and provided that Re to prevent an exponentially divergent solution in ; and

where is an arbitrary constant, which can be complex. Following equations (2.3)

and (2.4), the electric fields in inclined co-ordinates can be expressed as:

01 >k xH

1C

'1

' cos xt

y HkE ασ−

= , (2.9)

'1

' sin xn

z HkE ασ

= . (2.10)

Using a co-ordinate rotation (see Appendix 1), the horizontal electric field in

fundamental co-ordinates can be written as:

36

αα sincos '' zyy EEE −= . (2.11)

After substituting equations (2.9) and (2.10) into equation (2.11), then:

xnt

y HkE sincos 22

1

+−=

σα

σα , (2.12)

is obtained, and it follows that the surface impedance of the TM-type wave can be

expressed as:

+−=

ntyx jZ

σα

σαωµ

22 sincos . (2.13)

We present here another method of solving this problem using rotation of the

Maxwell equations. Consider the x-axis co-ordinate rotation (see Appendix 1):

−=

−=

zyx

zyx

zyx

mm

mm )(cossin0sincos0

001

'''

ααααα R , (2.14)

such that all the partial derivatives of any function f can be re-written as:

mm zf

yf

yf αα sincos' ∂

∂+∂∂=

∂∂ , (2.15)

mmmm zf

zyf

yf

yf αααα 2

2

222

2

2

2

2

sincossin2cos' ∂

∂+∂∂

∂+∂∂=

∂∂ , (2.16)

mm zf

yf

zf αα cossin' ∂

∂+∂∂−=

∂∂ , (2.17)

mmmm zf

zyf

yf

zf αααα 2

2

222

2

2

2

2

coscossin2sin' ∂

∂+∂∂

∂−∂∂=

∂∂ . (2.18)

37

Now, equation (2.6) can be re-written in y, z co-ordinates as:

.0cossin 11 2

cossinsincos

2

2

222

2

222

=−∂∂

∂

−+

∂∂

++

∂∂

+

xx

tn

x

tn

x

tn

Hjzy

Hσ

zH

yH

ωµαασ

σα

σα

σα

σα

(2.19)

From equation (2.19), the term containing the mixed partial derivative:

zyH

σx

tn ∂∂∂

−

2

cossin 11 2 αασ

,

is commonly called the shearing term, from the mechanics analogy of forces acting on

a beam carrying a load, where the diagonal terms of the force tensor are pressures, or

tensions, and the non-diagonal terms are shearing forces.

A general solution for a homogeneous plane wave, like equation (2.7), is

satisfied provided that the wave number is given by equation (2.8) and for

the same reasons as stated before. For any medium, m, the intrinsic impedance is

defined as:

0Re 1 >k

mm k

jZ ωµ−= . (2.20)

By substituting equation (2.8) into equation (2.20), the surface impedance is

identically obtained to equation (2.13). Now, the same formulation as shown above will

be considered but with considerations to a down-going homogeneous plane wave of the

of TE-type. From equations (1.18) and (1.19):

zE

jH x

y ∂∂−=

ωµ1 , (2.21)

yE

jH x

z ∂∂

=ωµ1 , (2.22)

and the Helmholtz equation for Ex is simply:

38

xxx Ek

yE

zE 2

22

2

2

2

=∂

∂+

∂∂

, (2.23)

which has a solution in the form of a down-going homogeneous plane wave:

)exp( 22 zkCEx −= , (2.24)

since the half space is assumed to be infinitely deep. Following equation (1.18), the

wave number k is given by: 2

tjk ωµσ=2 , (2.25)

provided to prevent an exponentially divergent solution in ; and where

is an arbitrary constant which can be complex. Since E

0Re 2 >k xE

2C x is laterally homogeneous,

Hz vanishes and the solution for Hx is simply:

xy Ejk

Hωµ

2= , (2.26)

and the surface impedance can be written as:

txy

jZσωµ= , (2.27)

which is independent of the anisotropic parameters α and σn.

2.2.2 Method of tensor rotation

Let us now consider the quasi-static Maxwell curl equations (1.18) to (1.19) and

expand them respectively to obtain:

xyz J

zH

yH

=∂

∂−

∂∂ , (2.28)

39

yzx J

xH

zH

=∂

∂−

∂∂

, (2.29)

zxy J

yH

xH

=∂

∂−

∂∂

, (2.30)

xyz Hj

zE

yE ωµ−=

∂∂

−∂

∂ , (2.31)

yzx Hj

xE

zE ωµ−=

∂∂

−∂

∂, (2.32)

zxy Hj

yE

xE

ωµ−=∂

∂−

∂∂

. (2.33)

From Ohm�s Law, equation (1.12), and for a full conductivity tensor, equation

(1.59), one can write the current density components as:

zxzyxyxxxx EEEJ σσσ ++= , (2.34)

zyzyyyxyxy EEEJ σσσ ++= , (2.35)

zzzyzyxzxz EEEJ σσσ ++= . (2.36)

Since we are considering a homogeneous half space excited by a homogeneous

plane wave, all partial derivatives with respect to x and y are equal to zero. From

equations (2.28) to (2.33), then it follows that:

zxzyxyxxxy EEE

zH

σσσ ++=∂

∂− , (2.37)

zyzyyyxyxx EEE

zH σσσ ++=∂

∂, (2.38)

40

xy Hj

zE

ωµ=∂

∂, (2.39)

yx Hj

zE ωµ−=∂

∂, (2.40)

and by implication it is observed that:

0=zH , (2.41)

0=++ zzzyzyxzx EEE σσσ . (2.42)

Equations (2.41) and (2.42) simply state that the vertical magnetic field and

vertical current density in the half space are equal to zero, when the source field is a

homogeneous plane wave. By differentiating equations (2.39) and (2.40) with respect to

z and substituting the results into equations (2.37) and (2.38), one obtains:

( zxzyxyxxxx EEEj

zE σσσωµ ++−

∂∂

= 2

2

0 ), (2.43)

( zyzyyyxyxy EEEj

zE

σσσωµ ++−∂

∂= 2

2

0 ). (2.44)

From equation (2.36), it is possible to write:

yzz

zyx

zz

zxz EEE

σσ

σσ

−−= . (2.45)

After substituting equation (2.45) into equations (2.43) and (2.44), two coupled

second order differential equations for the horizontal electric fields are obtained [2.13]:

yzz

zyxzxyx

zz

zxxzxx

x EjEjzE

−−

−−

∂∂

=σ

σσσωµ

σσσσωµ2

2

0 , (2.46)

41

yzz

zyyzyyx

zz

zxyzyx

y EjEjzE

−−

−−

∂∂

=σ

σσσωµ

σσσ

σωµ2

2

0 . (2.47)

Let us now assume that problem is posed with the inclined conductivity tensor

only being rotated about the x-axis by α such that the conductivity tensor can be written

as:

=

zzyz

zyyy

xx

σσσσ

σσ

00

00� , (2.48)

where the elements of the conductivity tensor then have the form:

txx σσ = , (2.49)

ασασσ 22 sincos ntyy += , (2.50)

ασασσ 22 cossin ntzz += , (2.51)

αασσσσ cossin)( ntzyyz −== , (2.52)

where 0==== zxxzyxxy σσσσ . With such conditions on the conductivity tensor,

equations (2.46) and (2.47) take the forms of the uncoupled second order ordinary

differential equations:

xxxx Ej

zE ωµσ−

∂∂

= 2

2

0 , (2.53)

yzz

zyyzyy

y EjzE

−−

∂∂

=σ

σσσωµ2

2

0 . (2.54)

Now, if one considers a solution for equation (2.53) of the form of a down-

going homogeneous plane wave of type:

42

)exp( 33 zkCEx −= , (2.55)

where is an arbitrary constant which may be complex, then it is easily observed

that:

3C

tjk ωµσ=3 , (2.56)

provided that to prevent an exponentially divergent solution in . The

surface impedance can then be written as:

0Re 3 >k xE

txy

jZσωµ= , (2.57)

which corresponds identically to the surface impedance of the TE-type wave, derived

earlier as equation (2.27). If one now considers a solution for equation (2.54) of the

form of the down-going homogeneous plane wave:

)exp( 44 zkCEy −= , (2.58)

where C is an arbitrary constant which may be complex, then from equations (2.49) to

(2.52), it can be determined that:

4

122 sincos

−

+=−

ntzz

zyyzyy σ

ασ

ασ

σσσ , (2.59)

implying that the yyρ term of the rotated resistivity tensor can be written in terms of

yyσ , yzσ , zyσ and zzσ , and vice versa. It then follows simply that:

122

4sincos

−

+=

nt

jkσ

ασ

αωµ , (2.60)

43

provided to prevent an exponentially divergent solution in . The surface

impedance can then be written as:

0Re 4 >k yE

+−=

ntyx jZ

σα

σαωµ

22 sincos , (2.61)

which corresponds identically to equation (2.13).

2.2.3 Discussion

For TM-type incidence in sub-Sections 2.21 and 2.22, one can observe that the

inclined uniaxial anisotropic conductivity tensor can be written as a corresponding

fundamental bianisotropic conductivity tensor where:

zz

zyyzyyyy

nty σ

σσσρ

σα

σασ −==

+= −

−1

122 sincos ,

yy

yzzyzzzz

ntz σ

σσσρ

σα

σασ −==

+= −

−1

122 cossin .

This is a very important concept. We can therefore state that for a homogeneous

plane wave propagating in a homogeneous half space, the inclined conductivity tensor

is equivalent to a bianisotropic conductivity tensor in the fundamental co-ordinates.

2.3. The problem of a horizontally layered half space

Thiel [2.21] suggested that the Wait�s [2.22] surface impedance solution for a

horizontally stratified half space consisting of isotropic layers could be used to

calculate the surface impedance of a horizontally stratified half space consisting of

layers with inclined anisotropy, provided the conductivity used in Wait�s [2.22]

expressions were that of the effective horizontal conductivity. This argument is a valid

one, given that for a homogeneous TM-type plane wave, it has been shown in Section

2.2 that the surface impedance is a function of the yyρ term only, automatically

implying that non-diagonal conductivity tensor terms are considered. One can then

44

write the surface impedance Z1,s expression for a horizontally stratified half space (e.g.,

see Figure 2.5) consisting of M layers with inclined anisotropy using Wait�s [2.22]

transmission line analogy:

)tanh()tanh(

11,21

111,21,1 hkZZ

hkZZZZ

s

ss +

+= , (2.63)

)tanh()tanh(

22,32

222,32,2 hkZZ

hkZZZZ

s

ss +

+= , (2.64)

��

)tanh()tanh(

,1

,1

mmsmm

mmmsmmms hkZZ

hkZZZZ

+

+

++

= , (2.65)

��

)tanh()tanh(

111

11111

−−−

−−−−− +

+=

MMMM

MMMMMM hkZZ

hkZZZZ , (2.66)

where m = 1, 2, �.., M, indicating the layer number from the surface and σn,m, σt,m and

αm are the conductivity and anisotropy inclination values for the mth layer.

45

x y z σt,1 Zs,1 z = 0 α1 σt,1 Z1 σn,1 α1 σt,2 Zs,2 z = h1 α2 Z2 σn,2 α2 σt,2

Figure 2.5. Geometry for a two-layer horizontally stratified half space where

each layer exhibits inclined uniaxial anisotropic conductivity.

Equations (2.63) to (2.66) can be used to exactly reproduce the results of Singh

[2.19], Negi & Saraf [2.9] and Sinha [2.20]. Equations (2.63) to (2.66) are derived on

the basis of the continuity of the tangential field components at the boundaries, with the

conditions that:

mxmx HH ,1, =− , (2.67)

zH

zH mx

mn

m

mt

mmx

mn

m

mt

m

∂∂

+=

∂∂

+ −

−

−

−

− ,

,

2

,

21,

1,

12

1,

12 sincossincos

σα

σα

σα

σα

. (2.68)

Boundary conditions (2.67) and (2.68) then lead to 2(M-1) equations which are

linear in their plane wave amplitude coefficients, to solve for 2(M-1) unknowns in

terms of the known incident plane wave amplitude. For waves of the TE-type, the wave

number will simply contain the σt,m terms, identically satisfying all other conditions

required for equations (2.63) to (2.66).

46

2.4. References

[2.1] N. M. Al�tgauzen (Althausen), Phys. Solid Earth 5, 510 (1969).



[2.4] D. L. Dekker & L. M. Hastie, Geophys. J. R. astron. Soc. 61, 11 (1980).

[2.5] R. J. King, App. Phys. 5, 187 (1974).

[2.6] R. J. King & J. R. Wait, Symp. Math. 17, 107 (1976).

[2.7] V. T. Levadnyy & I. V. Pavlova, Geomag. Aeron. 21, 657 (1981); 22, 689

(1982).

[2.8] D. Loewenthal & M. Landisman, Geophys. J. R. astron. Soc. 35, 195 (1973).

[2.9] J. G. Negi & P. D. Saraf, Geophys. Prosp. 20, 785 (1972).

[2.10] J. G. Negi & P. D. Saraf, Radio Sci. 11, 787 (1976).

[2.11] J. G. Negi & P. D. Saraf, Phys. Earth Planet. Int. 44, 324 (1986).

[2.12] D. P. O�Brien & H. F. Morrison, Geophysics 32, 668 (1967).

[2.13] J. Pek & F. A. M. Santos, presented at 19th DGG Coll. Electromagn. Depth Inv.,

Burg Ludwigstein, Germany, Oct. 2001.

[2.14] V. Petr, Stud. Geophys. Geod. 11, 291 (1967).

[2.15] O. Praus & V. Petr, Can. J. Earth Sci. 6, 759 (1969).

[2.16] G. G. Pukhov, Geol. Geofiz. 7, 68 (1965).

[2.17] I. K. Reddy & D. Rankin, Geophys. Prosp. 19, 84 (1971).

[2.18] P. D. Saraf & J. G. Negi, Geophys. J. R. astron. Soc. 74, 809 (1983).

[2.19] R. P. Singh, Geophys. Prosp. 33, 369 (1985); 34, 925 (1986).

[2.20] A. K. Sinha, Geoexpl. 7, 9 (1969).

[2.21] D. V. Thiel, Geophys. Prosp. 34, 923 (1986).


Oxford, 1970).

[2.23] J. R. Wait, IEEE Trans. Antennas Propagat. 39, 268 (1991).

[2.24] J. R. Wait, J. Electromagn. Waves Applic. 10, 871 (1996).




47

Chapter 3

Exact solutions for two-dimensional inclined anisotropy problems

3.1. Introduction

As discussed in Chapter 2, the exact one-dimensional surface impedance

expression for homogeneous plane wave incidence above a horizontally stratified

(vertically inhomogeneous) earth has been well documented. Exact and approximate

solutions for the surface impedance anomaly above a horizontally inhomogeneous half

space with isotropic conductivity have been previously investigated for both TE- and

TM-type waves [3.2, 3.3, 3.7-3.9, 3.13-3.16, 3.18-3.23]. The issue of a sloping contact

was considered theoretically by Dmitrieyev & Zakharov [3.4] and Geyer [3.5], and

numerically by Reddy & Rankin [3.12]. However, with the exception of the

fundamental uniaxial anisotropy solution of Obukhov [3.10], and the review of

d�Erceville & Kunetz�s [3.3] solution for inclined uniaxial anisotropic conductivity by

Grubert [3.6], no significant attention has been given to the problem of exactly solving

for the surface impedance anomaly above a vertical contact between two conductive

media that have inclined anisotropic conductivity.

The exact inclined anisotropic problem has been investigated for vertically

inhomogeneous problems (see Chapter 2). It is the purpose of this chapter to present the

exact quasi-static solution for the surface impedance response of a conducting layer

with inclined anisotropic conductivity, with lateral inhomogeneities. In a similar way

that the solutions of Weaver et al. [3.22, 3.23] were developed as control models for the

COMMEMI project, it is suggested that the development of an exact solution for a two-

dimensional control model with inclined uniaxial anisotropy will serve as a benchmark

for other approximate methods of solution. We will do this using an extension of the

Fourier series method presented by previous authors for isotropic and coplanar

anisotropic media. The formulation by Rankin [3.11] is used as the basis for the

formulation presented here.

It is assumed that the xxσ element of the conductivity tensor is parallel to the

strike of the dyke and the linearly polarised field, as this then allows one to solve

for the linearly polarised TM-type homogeneous plane wave, as the TE-type

homogeneous plane wave is uncoupled and will be propagated independently. By

xH

48

assuming that the lateral inhomogeneities have an infinite strike length, this reduces the

problem to a two-dimensional one and makes it only necessary to solve the Maxwell

equations in the region . 0≥z

bσ

-

σt,2

σt,2

3.2. Exact formulation: preliminary considerations

Consider a homogeneous dyke with inclined conductivity anisotropy extending

infinitely into the x-direction, embedded in an otherwise homogeneous layer, which

also exhibits inclined anisotropic conductivity. The common depth of the dyke and the

layer is h and they are both underlain by a homogeneous half space, or basement, with

isotropic conductivity . Figure 3.1 shows the solution space and respective variables.

By considering a homogeneous plane wave as the source field, then all partial

derivatives with respect to x are equal to zero.

x l/2 +l/2 y z σt,1 σt,2 z = 0 α2 α1 α2 σt,1 σt,2

α2 α1 α2 σn,2 σn,1 σn,2 z = h

σb

Figure 3.1. Geometry and parameters of the dyke embedded in a homogeneous

layer.

In the local inclined co-ordinate system characterised by angle of inclination 1α

about the x-axis, the conductivity of the dyke is represented with the uniaxial

conductivity tensor:

49

=

1,

1,

1,

1

000000

�

n

t

t

σσ

σσ . (3.1)

Similarly, in the local inclined co-ordinate system characterised by angle of

inclination 2α about the x-axis, the conductivity of the host layer is represented with

the uniaxial conductivity tensor:

=

2,

2,

2,

2

000000

�

n

t

t

σσ

σσ . (3.2)

t subscripts denote the conductivity tangential to the inclined horizontal axis of

the medium, and n subscripts denote the conductivity normal to the inclined horizontal

axis of the medium. When general solutions to the Maxwell equations are considered in

this section, subscript m is introduced to designate the medium, where m = 1 for the

dyke and m = 2 for the host layer. The equations presented here can be considered to

satisfy an arbitrary inclined co-ordinate system {x�, y�, z�} rotated through an angle mα

about the x-axis. Firstly, the forms of the Maxwell equations in the inclined co-ordinate

system are considered. From the frequency domain statement of Maxwell equation

(1.2):

HE ωµj−=×∇ , (3.3)

it follows that for a linearly polarised TM-type homogeneous plane

wave where is parallel to the

0,, == mzmy HH

mx,H xxσ component of mσ� . From equations (3.3) and

(1.18), we then obtain the family of equations:

'',','

,' zE

yE

Hj mymzmx ∂

∂−

∂∂

=− ωµ , (3.4)

'1 ,'

,,' y

HE mx

mtmz ∂

∂−=σ

, (3.5)

50

'1 ,'

,,' z

HE mx

mnmy ∂

∂=

σ. (3.6)

By differentiating equations (3.5) and (3.6) with respect to y' and z' respectively,

and by substituting these results into equation (3.4), one obtains the homogeneous

Helmholtz equation for anisotropic media:

0'

1'

1,'2

,'2

,2

,'2

,

=−∂

∂+

∂∂

mxmx

mt

mx

mn

HjzH

yH

ωµσσ

. (3.7)

From the co-ordinate rotations for transforming {x', y', z'} co-ordinates to {x, y,

z} co-ordinates (see Appendix 1), the co-ordinate rotations for transforming {x, y, z}

co-ordinates to {x', y', z'} co-ordinates can be written as:

−=

−=

zyx

zyx

zyx

mm

mm )(cossin0sincos0

001

'''

ααααα R , (3.8)

such that the partial derivatives of any function f can be re-written as:

mm zf

yf

yf αα sincos' ∂

∂+∂∂=

∂∂ , (3.9)

mmmm zf

zyf

yf

yf αααα 2

2

222

2

2

2

2

sincossin2cos' ∂

∂+∂∂

∂+∂∂=

∂∂ , (3.10)

mm zf

yf

zf αα cossin' ∂

∂+∂∂−=

∂∂ , (3.11)

mmmm zf

zyf

yf

zf αααα 2

2

222

2

2

2

2

coscossin2sin' ∂

∂+∂∂

∂−∂∂=

∂∂ . (3.12)

Equation (3.7) can be written in {x, y, z} co-ordinates as:

51

.0cossin 11 2

cossinsincos

,,

2

,

2,

2

,

2

,

2

2,

2

,

2

,

2

=−∂∂

∂

−+

∂∂

++

∂∂

+

mxmx

mmmtn,m

mx

mt

m

mn

mmx

mt

m

mn

m

Hjzy

Hσ

zH

yH

ωµαασ

σα

σα

σα

σα

(3.13)

The presence of the lateral inhomogeneities will generate anomalous magnetic

fields across the strike of the dyke. At an infinite distance from the dyke, the anomalous

fields must reduce to zero and the solution for the magnetic field will be identical to

that of a horizontally homogeneous medium. Near the dyke, the general solution for the

total magnetic field can be written as the sum of the background (b) and

anomalous (a) fields:

),(, zyH mx

),()(),( ,,, zyHzHzyH amx

bmxmx += , (3.14)

where is the anomalous field that exists due to the lateral inhomogeneities.

Since satisfies equation (3.13), then and must also

satisfy a form of equation (3.13) as linear sums of solution. In this particular problem,

the anisotropic medium is bound by a perfectly insulating layer (i.e., air) at the upper

boundary ( ), and a basement with isotropic conductivity

),(, zyH amx

),(, zyH mx

0=z

)(, zH bmx ),(, zyH a

mx

bσ at the lower

boundary ( ). A general solution for the background magnetic field , is a

homogeneous plane wave with both up-going and down-going components of the form:

hz = )(, zbmH x

)exp()exp()(, zkBzkAzH mmmmb

mx +−= , (3.15)

which satisfies equation (3.13) provided that the wave number is given by:

1

,

2

,

2 cossin−

+=

mt

m

mn

mm jk

σα

σαωµ , (3.16)

and to prevent an exponentially divergent solution in . It should be

noted that and in equation (3.15) are independent of {x, y, z, t} and are not

0Re >mk

mA

)(, zH bmx

mB

52

related to components of the vector potential A or magnetic flux density B. The

solution for is easily identified as the solution for the horizontally

homogeneous problem, as presented in Chapter 2.

)(, zH bmx

))

hhk

m

m

))

hkhk

m

m−

sinh((sinhhkhk

m

m

0)(, =zm

0)0 H=

()( gy mm

3.3. Exact formulation: perfectly insulating basement solution

3.3.1. Inclined anisotropic conductivity

If 0=bσ , then at and this boundary condition is

equivalent to the top of the basement being a perfect magnetic conductor [3.19]. It

follows (see Appendix 3) that the coefficients for equation (3.15) are given by:

0),(, =zyH mx hz =

sinh(2exp(0

kH

Am = , (3.17)

sinh(2exp(0H

Bm−

= , (3.18)

where is the magnetic field magnitude at , and is a constant which may be

complex. Equation (3.15) can then be written as:

0H 0=z

))

)( 0,

zHzH b

mx−

= . (3.19)

It is easily observed from equation (3.19) that when .

Similarly, when . This also implies that when

since . By separation of variables, the anomalous magnetic field will be

written as the product of two independent functions, and :

0)(, =zH bmx

)(, =zH amx

)(zgm

hz =

0=zH ax

,(y

hz = 0

,H mx

)(yfm

)),(, zfzyH amx = , (3.20)

53

where can be expressed as a Fourier series of sine terms with an argument of )(zgm

hznπ :

∑∞

=

=

1, sin)(

nnmm h

znAzg π ,

where n is the mode number (1 ) and where are the Fourier series

coefficients, and should not be confused with in equation (3.17), a coefficient of the

magnetic field wave equations. For convenience, we will include in the values of

at each n. As shown in Figure 3.2, this series satisfies the boundary conditions

that the anomalous magnetic field vanishes at both and .

∞ ...., 3, 2, , nmA ,

z =

mA

nmA ,

h

)(yfm

0=z

z = 0 z = h n = 1 n = 2 n = 3 n = 4

Figure 3.2. Fourier series representations, satisfying the boundary

conditions that for and .

)(, zg nm

0)(, =zH amx 0=z hz =

Equation (3.20) can then be written as:

=∑

∞

= hznyfzyH

nnm

amx

π1

,, sin)(),( . (3.21)

54

For , each term of equation (3.21) must satisfy a form of

equation (3.13) as a linear sum of solutions. For each term from equation (3.21), we

have the partial derivatives:

∞= ...., 3, 2, ,1n

∂∂

=∂

∂h

zny

yfy

zyH nma

nmx πsin)(),(

2,

2

2,,

2

, (3.22)

−=

∂∂

hznyf

hn

zzyH

nm

anmx ππ sin)(

),(,2

22

2,,

2

, (3.23)

∂∂

=∂∂

∂h

zny

yfh

nzy

zyH nma

nmx ππ cos)(),( ,,,

2

. (3.24)

The form of equation (3.13) that the anomalous fields must satisfy is then

written as:

.0sin)(

cos)(

cossin 11 2

sin)(cossin

sin)(sincos

,

,

,

,2

22

,

2

,

2

2,

2

,

2

,

2

=

−

∂∂

−+

+−

∂∂

+

hznyfj

hzn

yyf

hn

σ

hznyf

hn

hzn

yyf

nm

nmmm

mtn,m

nmmt

m

mn

m

nm

mt

m

mn

m

πωµ

ππαασ

ππσ

ασ

α

πσ

ασ

α

(3.25)

At the and boundaries, 0=z hz = 0sin =

hznπ and the shearing term:

0cos)(

cossin 11 2 ,

,

=

∂∂

−

hzn

yyf

hn

σnm

mmmtn,m

ππαασ

. (3.26)

At , 0=z 1cos =

hznπ , so equation (3.26) can only vanish for three possible

cases:

55

(a) if:

011

,

=−mtn,mσ σ

, (3.27)

which is the special case for an isotropic solution; or

(b) if:

0cossin =mm αα , (3.28)

which is only the special case of either 0=mα or , corresponding to

fundamental anisotropic solutions; or else,

o90=mα

(c) if:

ny

yf nm 0)(, ∀=

∂∂

. (3.29)

As a general solution for the inclined anisotropic problem is sought, equations

(3.27) and (3.28) are invalid (as they are special conditions) implying that equation

(3.29) must hold true in all cases. This means that equation (3.25) can be reduced to:

.0sin)(

sin)(cossin

sin)(sincos

,

,2

22

,

2

,

2

2,

2

,

2

,

2

=

−

+−

∂∂

+

hznyfj

hznyf

hn

hzn

yyf

nm

nmmt

m

mn

m

nm

mt

m

mn

m

πωµ

ππσ

ασ

α

πσ

ασ

α

(3.30)

A solution for must satisfy both equations (3.29) and (3.30). To satisfy

equation (3.30), a solution to is of form:

)(, yf nm

f )(, ynm

56

+

−=

hyq

bh

yqayf nm

nmnm

nmnm,

,,

,, expexp)( , (3.31)

provided that:

++

+=mmnmmt

mmnmmtmznm nhkq

ασασασασ

π 2,

2,

2,

2,2222

,, sincoscossin

, (3.32)

where is the vertical wave number: mzk ,

1

,

2

,

2

,sincos

−

+=

mt

m

mn

mmz jk

σα

σαωµ .

This solution satisfies the condition that equation (3.21) vanish for ∞=y , if we only

take the a-type terms for positive y and b-type terms for negative y both in m = 2, whilst

both positive and negative exponentials can exist in the finite region of m = 1.

Symmetry conditions at the 2ly ±= boundaries then permit us to write:

nmnm ba ,, = , (3.33)

which will ensure that is an even function about y = 0; i.e.,

. Employing this boundary condition is equivalent to using

one of the boundaries for solving the continuity of the magnetic field components, with

the remaining boundary condition to be available for solving the remainder of the

coefficients [3.11]. If the

),(, zyH mx

),(),( ,, zyHzyH mxmx −=

2l=y boundary is considered, then following from equation

(3.33), the use of symmetry implies that:

∑∞

=

−=

−−

112

,2,2

,1,1 sin

2exp

2cosh2

n

nn

nn HH

hzn

hlq

ahlq

a π , (3.34)

57

where:

hkzhkH

HHm

mbmxm sinh

)(sinh0,

−=≡ ,

for m = 1, 2. The expansion of into a sine series of argument 12 HH −h

znπ is written

as:

=− ∑

∞

= hznCHH

nn

π1

12 sin , (3.35)

where is a complex constant yet to be determined. At the boundary, term-by-term

must be equated, so both equations:

nC

nn

nn

n Ch

lqa

hlq

a =

−−

2

exp2

cosh2 ,2,2

,1,1 , (3.36)

02

exp2

sinh2 ,22,,2,2

,11,,1,1 =

−+

h

lqqa

hlq

qa nzznn

nzznn ρρ , (3.37)

must be satisfied, where equation (3.37) is obtained from the Maxwell equation:

yH

E mxmzzmz ∂

∂−= ,

,, ρ , (3.38)

stating the continuity of the tangential electric field across the boundary 2ly ±= .

Solutions for the a-type coefficients are then:

+

=

hlq

qq

hlq

Ca

n

n

nzz

nzz

zznn

2sinh2

2cosh2 ,1

,2

,11,

,12,

2,,1

ρρ

ρ, (3.39)

58

+

−

=

hlq

hlq

hlq

hlq

Ca

nzz

nzz

nnzzn

n

2sinh

2cosh

2sinh

2exp

,11,

,12,

,1,21,

,2

ρρ

ρ. (3.40)

Following the expansion of into an odd Fourier series with argument 12 HH −

hznπ , one obtains:

∫

−=

h

n dzh

znHHh

C0

12 sin)(2 π , (3.41)

where:

)sinh()(sinh

)sinh()(sinh

1

10

2

2012 hk

zhkHhk

zhkHHH

−−

−=− . (3.42)

Following through with equation (3.41) using integration by parts (see

Appendix 5), the solution for is: nC

+

+

−−=

2

222

12

222

22

21

220 )(2

hnk

hnkh

kknHCn ππ

π. (3.43)

Substituting equation (3.43) into equations (3.39) and (3.40), solutions for the a-

type coefficients now take the form:

+

+

+

−−=

hlq

qq

hlq

hnk

hnkh

kknHa

n

n

nzz

nzz

zzn

2sinh

2cosh

)(

,1

,2

,11,

,12,2

222

12

222

22

21

222,0

,1

ρρππ

πρ,

(3.44)

59

+

+

+

−−

=

hlq

qq

hlq

hnk

hnkh

hlq

hlq

kknHa

n

n

nzz

nzz

nnzz

n

2cosh

2sinh

2sinh

2exp)(2

,1

,1

,22,

,11,2

222

12

222

22

,1,221

221,0

,2

ρρππ

πρ.

(3.45)

For 22lyl ≤≤− , the anomalous magnetic field is written as:

=∑

∞

= hzn

hyq

azyHn

nn

ax

π1

,1,11, sincosh2),( . (3.46)

The total magnetic field can then be written as:

+

−= ∑

∞

= hzn

hyq

ahk

zhkHzyH

n

nnx

π1

,1,1

1

101, sincosh2

)sinh()(sinh

),( . (3.47)

From Maxwell equations (3.5) and (3.6) rotated about the x-axis, we have:

zH

E xyyy ∂

∂= 1,

1,1, ρ , (3.48)

so the horizontal electric field can be written as:

.coscosh2

)sinh()(cosh

),(

1

,1,11,

1

101,11,

+

−−=

∑∞

= hzn

hyq

ah

n

hkzhkHk

zyE

n

nnyy

yyy

ππρ

ρ

(3.49)

At the air-half space interface ( ), equation (3.47) reduces to: 0=z

01, )0,( HyH x = , (3.50)

and is constant ∀ . For , equation (3.49) reduces to: y 0=z

60

∑∞

=

+

−=

1

,1,11,

1

101,11, .cosh2

)sinh()cosh(

),(n

nnyy

yyy h

yqa

hn

hkhkHk

zyE πρρ

(3.51)

The surface impedance is defined as:

−= ∑

∞

= hyq

Ha

hnhkkyZ nn

nyyyyyx

,1

0

,1

11,11,1 cosh2)coth()0,( πρρ . (3.52)

In equation (3.52), the term:

)coth(sincos)coth( 11,

12

1,

12

11,1 hkjhkknt

yy

+=

σα

σαωµρ , (3.53)

is identified as the surface impedance of a laterally homogeneous layer above a perfect

magnetic conducting basement (see Chapter 2). Once equation (3.44) is substituted into

(3.52), one obtains:

.

2sinh

2cosh

cosh

)(2

)coth(sincos)0,(

1 ,1

,2

,11,

,12,2

222

12

222

2

,12

1,2,2

12

23

2

11,

12

1,

12

∑∞

=

+

+

+

×

−+

+=

n n

n

nzz

nzz

n

yyzz

ntyx

hlq

qq

hlq

hnk

hnk

hyq

n

kkh

hkjyZ

ρρππ

ρρπ

σα

σαωµ

(3.54)

For 2ly ≥ and

2ly −≤ , the constant term is the same as for medium 1, however

the subscripts are interchanged. One writes the anomalous magnetic field as:

−=∑

∞

= hzn

hyq

azyHn

nn

ax

π1

,2,22, sinexp),( , (3.55)

61

where . The total magnetic field can then be written as: 0Re ,2 >nq

−+

−= ∑

∞

= hzn

hyq

ahk

zhkHzyH

n

nnx

π1

,2,2

2

202, sinexp

)sinh()(sinh

),( . (3.56)


zH

E xyyy ∂

∂= 2,

2,2, ρ , (3.57)

so the horizontal electric field is written as:

.cosexp

)sinh()(cosh

),(

1

,2,22,

2

202,22,

−+

−−=

∑∞

= hzn

hyq

ah

n

hkzhkHk

zyE

n

nnyy

yyy

ππρ

ρ

(3.58)


02, )0,( HyH x = , (3.59)


.exp

)sinh()cosh(

),(

1

,2,22,

2

202,22,

∑∞

=

−+

−=

n

nnyy

yyy

hyq

ah

n

hkhkHk

zyE

πρ

ρ

(3.60)


−−= ∑

∞

= hyq

Ha

hnhkkyZ nn

nyyyyyx

,2

0

,2

11,22,2 exp)coth()0,( πρρ . (3.61)


62

)coth(sincos)coth( 22,

22

2,

22

22,2 hkjhkknt

yy

+=

σα

σαωµρ , (3.62)


magnetic conducting basement (see Chapter 2). Once equation (3.45) is substituted into


.

2cosh

2sinh

2sinh

2expexp

)(2

)coth(sincos)0,(

1 ,1

,1

,22,

,11,2

222

12

222

2

,1,2,22

1,2,2

12

23

2

22,

22

2,

22

∑∞

=

+

+

+

×

−+

+=

n n

n

nzz

nzz

nnn

yyzz

ntyx

hlq

qq

hlq

hnk

hnk

hlq

hlq

hyq

n

kkh

hkjyZ

ρρππ

ρρπ

σα

σαωµ

(3.63)

Equation (3.63) is the complete exact solution for the surface impedance of a

vertical dyke with inclined anisotropic conductivity embedded in an otherwise

homogeneous layer above a perfect magnetic conductor.

3.3.2. Fundamental anisotropic conductivity

The formulations derived in Section 3.3.1 will now be reduced to the problem

of a vertical dyke with fundamental uniaxial anisotropy. In this case, it can be assumed

that 021 ==αα . Firstly, it is noted that the wave number for medium m is now given

by:

mtm jk ,ωµσ= , (3.64)

where to prevent exponentially divergent solutions. Now equation (3.28) is

satisfied and a solution to equation (3.30) is sought. Equation (3.31) is still a valid

solution, and equation (3.32) reduces to:

0Re >mk

63

mt

mnmznm nhkq

,

,2222,, σ

σπ+= , (3.64)

where:

mnmz jk ,, ωµσ= .

We can re-write equation (3.64) as:

2

2222

,m

mnm

nhkqλ

π+= , (3.65)

where mλ is the coefficient of anisotropy of medium m. We note that equation (3.65) is

identical to the solution by Obukhov [3.10]. It follows that:

+

+

+

−−=

hlq

qq

hlq

hnk

hnkh

kknHan

n

n

n

nnn

2sinh

2cosh

)(

,1

,2

,1

1,

2,,12

222

12

222

22

21

220

,1

σσππ

π,

(3.66)

+

+

+

−−

=

hlq

qq

hlq

hnk

hnkh

hlq

hlq

kknHa

n

n

n

n

nn

nn

n

2cosh

2sinh

2sinh

2exp)(2

,1

,1

,2

2,

1,,12

222

12

222

22

,1,221

220

,2

σσππ

π,

(3.67)

and that for 22lyl ≤≤− , the surface impedance can be written as:

64

.

2sinh

2cosh

cosh

)(2

)coth()0,(

1 ,1

,2

,1

1,

2,,12

222

12

222

2

,12

1,2

12

23

2

11,

∑∞

=

+

+

+

×

−+

=

n n

n

n

n

nn

n

n

tyx

hlq

qq

hlq

hnk

hnk

hyq

n

kkh

hkjyZ

σσππ

ρπ

σωµ

(3.68)

For 2ly ≥ and

2ly −≤ , the surface impedance can be written as:

.

2cosh

2sinh

2sinh

2expexp

)(2

)coth()0,(

1 ,1

,1

,2

2,

1,,12

222

12

222

2

,1,2,22

2,2

12

23

2

22,

∑∞

=

+

+

+

×

−+

=

n n

n

n

n

nn

nnn

n

tyx

hlq

qq

hlq

hnk

hnk

hlq

hlq

hyq

n

kkh

hkjyZ

σσππ

ρπ

σωµ

(3.69)

Equations (3.68) and (3.69) were identically obtained by Obukhov [3.10].

Further, it can be observed that when mmnmt σσσ == ,, , then for 22lyl ≤≤− , the

surface impedance is given by:

65

.

2sinh

2cosh

cosh

)(2

)coth()0,(

1 ,1

,2

,1,12

222

12

222

2

,12

12

12

23

2

11

∑∞

=

+

+

+

×

−+

=

n n

n

nn

n

yx

hlq

qq

hlq

hnk

hnk

hyq

n

kkh

hkjyZ

ππ

ρπ

σωµ

(3.70)

For 2ly ≥ and


,

2cosh

2sinh

2sinh

2expexp

)(2

)coth()0,(

1 ,1

,1

,2,12

222

12

222

2

,1,2,22

22

12

23

2

22,

∑∞

=

+

+

+

×

−+

=

n n

n

nn

nnn

tyx

hlq

qq

hlq

hnk

hnk

hlq

hlq

hyq

n

kkh

hkjyZ

ππ

ρπ

σωµ

(3.71)

which are identical equations to those derived by Rankin [3.11]. It is noted that

Rankin�s [3.11] solution was derived using the cgs electromagnetic units (emu) in

which µ is dimensionless and equal to unity in free space.

3.4. Exact formulation: perfectly conducting basement solution

3.4.1. Inclined anisotropic conductivity

If ∞=bσ , then at , and this boundary condition is

equivalent to the top of the basement being a perfect electric conductor. It follows (see

Appendix 3) that the coefficients for equation (3.15) are given by:

0),(, =zyE my hz =

66

)cosh(2)exp(0

hkhkH

Am

mm = , (3.72)

)cosh(2)exp(0

hkhkH

Bm

mm

−= , (3.73)

where is the magnetic field magnitude at , and is constant which may be

complex. Equation (3.15) can then be written as:

0H 0=z

)cosh()(cosh

)( 0, hk

zhkHzH

m

mbmx

−= . (3.74)

It is easily observed that 0)(, =

∂∂

zhH b

mx . Similarly, 0)(, =

∂∂

zhH a

mx . Also,

since . By separation of variables, the anomalous

magnetic field can be written as:

0)0,(, =yH amx 0, )0,( HyH mx =

)()(),(, zgyfzyH mma

mx = , (3.75)

where can be expressed as a Fourier series of sine terms with an argument of )(zgm

hn

22( z)1 π+ :

∑∞

=

+=

0, 2

)12(sin)(n

nmm hznAzg π ,

where n is the mode number (0,1 ) and where are the Fourier

series coefficients, and should not be confused with in equation (3.72), a coefficient

of the magnetic field wave equations. As shown in Figure 3.3, this series satisfies the

boundary conditions that the anomalous magnetic field vanishes at and that

∞ ...., 3, 2, ,

A

nmA ,

m

0=z

0)(, =

∂∂

=hz

amx

zzH

.

67

Equation (3.75) can then be written as:

+=∑

∞

= hznyfzyH

nnm

amx 2

)12(sin)(),(0

,,π . (3.76)

For , each term of equation (3.76) must satisfy a form of

equation (3.13). From equation (3.76), we have the derivatives:

∞= ...., 3, 2, ,1 ,0n

+

∂∂

=∂

∂h

zny

yfy

zyH nma

nmx

2)12(sin

)(),(2

,2

2,,

2 π , (3.77)

++−=

∂∂

hznyf

hn

zzyH

nm

anmx

2)12(sin)(

4)12(),(

,2

22

2,,

2 ππ , (3.78)

+

∂∂+=

∂∂∂

hzn

yyf

hn

zyzyH nm

anmx

2)12(cos

)(2

)12(),( ,,,2 ππ . (3.79)

z = 0 z = h n = 0 n = 1 n = 2 n = 3

Figure 3.3. Fourier series representations, satisfying the boundary

conditions that for and

)(, zg nm

0)(, =zH amx 0=z 0

)(, =∂

∂z

hH amx at . hz =

68

The form of equation (3.13) that the anomalous fields must satisfy is then

written as:

.02

)12(sin)(

2)12(cos

)()12(cossin 11

2)12(sin)(

4)12(cossin

2)12(sin

)(sincos

,

,

,

,2

22

,

2

,

2

2,

2

,

2

,

2

=

+−

+

∂∂+

−+

++

+−

+

∂∂

+

hznyfj

hzn

yyf

hn

σ

hznyf

hn

hzn

yyf

nm

nmmm

mtn,m

nmmt

m

mn

m

nm

mt

m

mn

m

πωµ

ππαασ

ππσ

ασ

α

πσ

ασ

α

(3.80)

At the boundary, 0=z 02

)12( =

+

hzn πsin and the shearing term:

02

)12(cos)()12(cossin 11 ,

,

=

+

∂∂+

−

hzn

yyf

hn

σnm

mmmtn,m

ππαασ

. (3.81)

At , 0=z 12

)12(cos =

+

hzn π , so equation (3.81) can only vanish if either

equations (3.27) or (3.28) are valid, or else if:

ny

yf nm 0)(, ∀=

∂∂

, (3.29)

is valid. The same arguments as used in Section 3.3 to justify the property of equation

(3.29) are valid. This means that equation (3.80) can be reduced to the equation:

.02

)12(sin)(

2)12(sin)(

4)12(cossin

2)12(sin

)(sincos

,

,2

22

,

2

,

2

2,

2

,

2

,

2

=

+−

++

+−

+

∂∂

+

hznyfj

hznyf

hn

hzn

yyf

nm

nmmt

m

mn

m

nm

mt

m

mn

m

πωµ

ππσ

ασ

α

πσ

ασ

α

(3.82)

69

A solution for must satisfy both equations (3.29) and (3.82). To satisfy

equation (3.82), a solution to is of form:

)(, yf nm

f )(, ynm

+

−=

hyq

bh

yqayf nm

nmnm

nmnm 2exp

2exp)( ,

,,

,, , (3.83)

provided that:

++

++=mmnmmt

mmnmmtmznm nhkq

ασασασασ

π 2,

2,

2,

2,2222

,, cossincossin

)12(4 , (3.84)

where is the vertical wave number: mzk ,

1

,

2

,

2

,sincos

−

+=

mt

m

mn

mmz jk

σα

σαωµ .

This solution satisfies the condition that equation (3.76) vanish for ∞=y , if we only

take the a-type terms for positive y and b-type terms for negative y both in m = 2, whilst

both positive and negative exponentials can exist in the finite region of m = 1.

Symmetry conditions at the 2ly ±= boundaries then permit us to write:

nmnm ba ,, = , (3.85)

which will ensure that is an even function about y = 0; i.e.,

. Employing this boundary condition is equivalent to using

one of the boundaries for solving the continuity of the magnetic field components, with

the remaining boundary condition to be available for solving the remainder of the

coefficients. If the

),(, zyH mx

),(),( ,, zyHzyH mxmx −=

2ly = boundary is considered, then following from equation (3.85),

the use of symmetry implies that:

70

∑∞

=

−=

+

−−

012

,2,2

,1,1 2

)12(sin4

exp4

cosh2n

nn

nn HH

hzn

hlq

ahlq

a π , (3.86)

where:

hkzhkH

HHm

bmxm sinh

)(sinh0,

−=≡ ,

for m = 1, 2. The expansion of into a sine series of argument 12 HH −h

zn2

)12( π+ is

written as:

+=− ∑

∞

= hznCHH

nn 2

)12(sin0

12π , (3.87)

where is a complex constant yet to be determined. At the boundary, term-by-term

must be equated, so both equations:

nC

nn

nn

n Ch

lqa

hlq

a =

−−

4

exp4

cosh2 ,2,2

,1,1 , (3.88)

04

exp4

sinh2 ,22,,2,2

,11,,1,1 =

−+

h

lqqa

hlq

qa nzznn

nzznn ρρ , (3.89)

must be satisfied, where equation (3.89) is obtained from the Maxwell equation:

yH

E mxmzzmz ∂

∂−= ,

,, ρ , (3.38)

stating the continuity of the tangential electric field across the boundary 2ly ±= .

Solutions for the a-type coefficients are then:

71

+

=

hlq

qq

hlq

Ca

n

n

nzz

nzz

zznn

4sinh2

4cosh2 ,1

,2

,11,

,12,

2,,1

ρρ

ρ, (3.90)

+

−

=

hlq

hlq

hlq

hlq

Ca

nzz

nzz

nnzzn

n

4sinh

4cosh

4sinh

4exp

,11,

,12,

,1,21,

,2

ρρ

ρ. (3.91)

Following the expansion of into an odd Fourier series with argument 12 HH −

hzn

2)12( π+ , one obtains:

∫

+−=

h

n dzh

znHHh

C0

12 2

)12(sin)(1 π , (3.92)

where:

)cosh()(cosh

)cosh()(cosh

1

10

2

2012 hk

zhkHhk

zhkHHH

−−

−=− . (3.93)

Following through with equation (3.92) using integration by parts (see

Appendix 5), the solution for is: nC

++

++

−+−=

2

222

12

222

22

21

220

4)12(

4)12(

)()12(

hnk

hnkh

kknHCn ππ

π. (3.94)

Substituting equation (3.94) into equations (3.90) and (3.91), solutions for the a-

type coefficients now take the form:

,

4sinh

4cosh

4)12(

4)12(2

)()12(

,1

,2

,1

2,

1,,12

222

12

222

22

21

220

,1

+

++

++

−+−=

hlq

qq

hlq

hnk

hnkh

kknHan

n

n

zz

zznn

ρρππ

π

(3.95)

72

.

4cosh

4sinh

4)12(

4)12(

2sinh

2exp)()12(

,1

,1

,2

1,

2,,12

222

12

222

22

,1,221

220

,2

+

++

++

−+−

=

hlq

qq

hlq

hnk

hnkh

hlq

hlq

kknHa

n

n

n

zz

zzn

nn

n

ρρππ

π

(3.96)

For 22lyl ≤≤− , the anomalous magnetic field is written as:

+

=∑

∞

= hzn

hyq

azyHn

nn

ax 2

)12(sin2

cosh2),(0

,1,11,

π . (3.97)

The total magnetic field can then be written as:

+

+

−= ∑

∞

= hzn

hyq

ahk

zhkHzyH

n

nnx

π)12(sin2

cosh2)cosh(

)(cosh),(

0

,1,1

1

101, . (3.98)


zH

E xyyy ∂

∂= 1,

1,1, ρ , (3.48)

so the horizontal electric field can be written as:

.2

)12(cos2

cosh2

)12(

)cosh()(sinh

),(

0

,1,11,

1

101,11,

+

++

−−=

∑∞

= hzn

hyq

ah

n

hkzhkHk

zyE

n

nnyy

yyy

ππρ

ρ

(3.99)


01, )0,( HyH x = , (3.100)


73

∑∞

=

++−

=0

,1,11,

1

101,11, .

2cosh)12(

)cosh()sinh(

),(n

nnyy

yyy h

yqa

hn

hkhkHk

zyE πρρ

(3.101)


+−= ∑∞

= hyq

Ha

hnhkkyZ nn

nyyyyyx 2

cosh)12()tanh()0,( ,1

0

,1

01,11,1

πρρ . (3.102)


)tanh(sincos)tanh( 11,

12

1,

12

11,1 hkjhkknt

yy

+=

σα

σαωµρ , (3.103)


electric conducting basement (see Chapter 2). Once equation (3.95) is substituted into


.

2sinh

2cosh

4)12(

4)12(

cosh)12(

)(2

)tanh(sincos)0,(

0 ,1

,2

,1

2,

1,,12

222

12

222

2

,12

1,2

12

23

2

11,

12

1,

12

∑∞

=

+

++

++

+

×

−+

+=

n n

n

n

zz

zzn

n

yy

ntyx

hlq

qq

hlq

hnk

hnk

hyq

n

kkh

hkjyZ

ρρππ

ρπ

σα

σαωµ

(3.104)

For 2ly ≥ and

2ly −≤ , the constant term is the same as for medium 1, however

the subscripts are interchanged. One writes the anomalous magnetic field as:

+

−=∑

∞

= hzn

hyq

azyHn

nn

ax 2

)12(sin2

exp),(0

,2,22,

π , (3.105)

74

where . The total magnetic field can then be written as: 0Re ,2 >nq

+

−+

−= ∑

∞

= hzn

hyq

ahk

zhkHzyH

n

nnx 2

)12(sin2

exp)cosh(

)(cosh),(

0

,2,2

2

202,

π . (3.106)


zH

E xyyy ∂

∂= 2,

2,2, ρ , (3.57)

so the horizontal electric field is written as:

.2

)12(cos2

exp2

)12(

)cosh()(sinh

),(

0

,2,22,

2

202,22,

+

−++

−−=

∑∞

= hzn

hyq

ah

n

hkzhkHk

zyE

n

nnyy

yyy

ππρ

ρ

(3.107)


02, )0,( HyH x = , (3.108)


.2

exp2

)12(

)cosh()sinh(

),(

0

,2,22,

2

202,22,

∑∞

=

−++

−=

n

nnyy

yyy

hyq

ah

n

hkhkHk

zyE

πρ

ρ

(3.109)


−+−= ∑

∞

= hyq

Ha

hnhkkyZ nn

nyyyyyx 2

exp2

)12()tanh()0,( ,2

0

,2

01,22,2

πρρ . (3.110)


75

)tanh(sincos)tanh( 22,

22

2,

22

22,2 hkjhkknt

yy

+=

σα

σαωµρ , (3.111)


electric conducting basement (see Chapter 2). Once equation (3.96) is substituted into


.

4cosh

4sinh

4)12(

4)12(

4sinh

4exp

2exp)12(

)(

)tanh(sincos)0,(

0 ,1

,1

,2

1,

2,,12

222

12

222

2

,1,2,22

2,2

12

23

2

22,

22

2,

22

∑∞

=

+

++

++

+

×

−+

+=

n n

n

n

zz

zzn

nnn

zz

ntyx

hlq

qq

hlq

hnk

hnk

hlq

hlq

hyq

n

kkh

hkjyZ

ρρππ

ρπ

σα

σαωµ

(3.112)

Equation (3.112) is the complete exact solution for the surface impedance of a

vertical dyke with inclined anisotropic conductivity embedded in an otherwise

homogeneous layer above a perfect electric conductor.

3.4.2. Fundamental anisotropic conductivity

The formulations derived in Section 3.4.1 will now be reduced to the problem

of a vertical dyke with fundamental uniaxial anisotropy. In this case, it can be assumed

that 021 ==αα . Similar to Section 3.3.2, it is noted that the wave number for medium

m is now given by:

mtm jk ,ωµσ= , (3.64)

where to prevent exponentially divergent solutions. Now equation (3.28) is

satisfied and a solution to equation (3.80) is sought. Equation (3.83) is still a valid

solution, and it is noticed that equation (3.84) reduces to:

0Re >mk

76

mt

mnmznm nhkq

,

,2222,, )12(4

σσ

π++= , (3.113)

where:

mnmz jk ,, ωµσ= .

We can also write equation (3.113) as:

2

2222

,)12(4

m

mnm

nhkqλ

π++= , (3.114)

where mλ is the coefficient of anisotropy of medium m. We note that equation (3.114)

is identical to the solution of Obukhov [3.10]. It follows that:

,

4sinh

4cosh

4)12(

4)12(2

)()12(

,1

,2

,1

1,

2,,12

222

12

222

22

21

220

,1

+

++

++

−+−=

hlq

qq

hlq

hnk

hnkh

kknHan

n

n

n

nnn

σσππ

π

(3.115)

+

++

++

−+−

=

hlq

qq

hlq

hnk

hnkh

hlq

hlq

kknHa

n

n

n

n

nn

nn

n

4cosh

4sinh

4)12(

4)12(

4sinh

4exp)()12(

,1

,1

,2

2,

1,,12

222

12

222

22

,1,221

220

,2

σσππ

π,

(3.116)

and that for 22lyl ≤≤− , the surface impedance can be written as:

77

.

4sinh

4cosh

4)12(

4)12(

2cosh)12(

)(2

)tanh()0,(

0 ,1

,2

,1

1,

2,,12

222

12

222

2

,12

1,2

12

23

2

11,

∑∞

=

+

++

++

+

×

−+

=

n n

n

n

n

nn

n

n

tyx

hlq

qq

hlq

hnk

hnk

hyq

n

kkh

hkjyZ

σσππ

ρπ

σωµ

.

(3.117)

For 2ly ≥ and


.

4cosh

4sinh

4)12(

4)12(

4sinh

4exp

2exp)12(

)(2

)tanh()0,(

0 ,1

,1

,2

2,

1,,12

222

12

222

2

,1,2,22

2,2

12

23

2

22,

∑∞

=

+

++

++

+

×

−+

=

n n

n

n

n

nn

nnn

n

tyx

hlq

qq

hlq

hnk

hnk

hlq

hlq

hyq

n

kkh

hkjyZ

σσππ

ρπ

σωµ

(3.118)

Further, it can be observed that when mmnmt σσσ == ,, , then for 22lyl ≤≤− ,

the surface impedance is given by:

.

4sinh

4cosh

4)12(

4)12(

2cosh)12(

)(2

)tanh()0,(

0 ,1

,2

,1,12

222

12

222

2

,12

12

12

23

2

11

∑∞

=

+

++

++

+

×

−+

=

n n

n

nn

n

yx

hlq

qq

hlq

hnk

hnk

hyq

n

kkh

hkjyZ

ππ

ρπ

σωµ

. (3.119)

78

For 2ly ≥ and


,

4cosh

4sinh

4)12(

4)12(

4sinh

4exp

2exp)12(

)(2

)coth()0,(

0 ,1

,1

,2,12

222

12

222

2

,1,2,22

22

12

23

2

22,

∑∞

=

+

++

++

+

×

−+

=

n n

n

nn

nnn

tyx

hlq

qq

hlq

hnk

hnk

hlq

hlq

hyq

n

kkh

hkjyZ

ππ

ρπ

σωµ

(3.120)

which are identical equations to those derived by Rankin [3.11]. Again, it is noted that

Rankin�s [3.11] solution was derived using the cgs electromagnetic units (emu) in

which µ is dimensionless and equal to unity in free space.

3.5. Discussion

In all discussions here, we have considered only the propagation of

homogeneous TM-type waves. In Chapter 2, it was demonstrated that for a half space,

and for a horizontally stratified half space, the inclined uniaxial anisotropic

conductivity tensor can be written as a corresponding fundamental bianisotropic

conductivity tensor where:

mxxmxxmx ,1

,, σρσ == − , (3.121)

mzz

mzymyzmyymyy

mn

m

mt

mmy

,

,,,

1,

1

,

2

,

2

,sincos

σσσ

σρσ

ασ

ασ −==

+= −

−

, (3.122)

myy

myzmzymzzmzz

mn

m

mt

mmz

,

,,,

1,

1

,

2

,

2

,cossin

σσσ

σρσ

ασ

ασ −==

+= −

−

. (3.123)

79

It has been proven in this chapter that this same tensor equivalence exists for

horizontally inhomogeneous media. It has been demonstrated that the shearing term in

the rotated Helmholtz equation vanishes for horizontally inhomogeneous media, as in

the case for horizontally homogeneous media. Further, if one compares equations

(3.54) and (3.63) with equations (3.68) and (3.69) respectively, it should be noticed that

the equations are identical with the exception that in equations (3.54) and (3.63), the

conductivity terms are given by equations (3.122) and (3.123). Hence, it is concluded

that two-dimensional problems with inclined uniaxial anisotropic conductivity can be

equivalently described as a two-dimensional problem with fundamental biaxial

anisotropic conductivity. This has important applications to approximate methods of

solution. These applications will be discussed in Chapter 4.

3.6. References


[3.2] P. C. Clemmow, Phil. Trans. R. Soc. Lond. A 246, 1 (1953).

[3.3] I. d�Erceville & G. Kunetz, Geophysics 27, 651 (1962); 28, 490 (1963).

[3.4] V. I. Dmitriev & E. V. Zakharov, Phys. Solid Earth 6, 719 (1970).

[3.5] R. G. Geyer, Geophysics 37, 337 (1972).

[3.6] D. Grubert, presented at 12th Workshop Electromagn. Induction Earth, Brest,

France, Aug. 1994.

[3.7] F. W. Jones & A. T. Price, Geophys. J. R. astron. Soc. 20, 317 (1970).

[3.8] G. V. Koschlakov, Geol. Geofiz. 11, 119 (1970).

[3.9] G. G. Obukhov, Izv. Akad. Nauk SSSR Fiz. Zemli 4, 89 (1969).

[3.10] G. G. Obukhov, Izv. Akad. Nauk SSSR Fiz. Zemli 4, 106 (1969).

[3.11] D. Rankin, Geophysics 27, 666 (1962); 28, 490 (1963).

[3.12] I. K. Reddy & D. Rankin, Pure App. Geophys. 105, 847 (1973).

[3.13] R. C. Robertson, IEEE Trans. Geosci. Remote Sensing 27, 369 (1989).

[3.14] E. E. S. Sampaio & D. Dias, Geophys. Prosp. 49, 107 (2001).

[3.15] E. E. S. Sampaio & J. T. Fokkema, J. Geophys. Res. 97, 1953 (1992).

[3.16] U. Schmucker, in Protokoll Kolloqium Elektromagnetische Tiefenforschung

Hochst im Odenwald, edited by K. Bahr & A. Junge (Deutsche Geofys.

Gesellschaft, Germany, 1994), pp. 3-26.

[3.17] R. Truemann, Acta Geod. Mont. Acad. Sci. Hung. 5, 61 (1970).

[3.18] J. R. Wait & K. P. Spies, J. Geomagn. Geoelec. 26, 449 (1974).

80

[3.19] J. T. Weaver, Geophysics 28, 1386 (1963).

[3.20] J. T. Weaver, Can. J. Phys. 41, 484 (1963).

[3.21] J. T. Weaver, B. V. Le Quang & G. Fischer, Geophys. J. R. astron. Soc. 87, 263

(1985).

[3.22] J. T. Weaver, B. V. Le Quang & G. Fischer, Geophys. J. R. astron. Soc. 87, 917

(1986).

81

Chapter 4

Approximate solutions for two-dimensional electrical anisotropy problems

4.1. Introduction

The methods of approximate solutions for modelling a system use a

mathematical model that can be solved numerically to represent the system developed.

The results obtained from the approximate solution are then interpreted in terms of the

original system and serve to develop an understanding of the physical processes

involved [4.20]. As discussed in Chapter 2, one-dimensional exact solutions for

inclined anisotropy, where the directions of anisotropy are at arbitrary angles to the

fundamental co-ordinate system employed, have also been presented previously and

used to explain experimental observations. Further, in Chapter 3, exact solutions were

derived for a two-dimensional dyke embedded in a host layer. In all of these solutions,

the conductivity boundaries correspond to continuous analytical functions in the

electromagnetic fields.

In practical problems however, the inclined anisotropy occurs in structures with

arbitrary shape for which an exact solution may not be obtained easily, if at all. For

such problems, it is essential to employ approximate (or numerical) methods of

solution. Finite-element, finite-difference and integral equation solutions have all been

employed in modeling arbitrary two- and three-dimensional geological structures

[4.19]. However in these instances, the inhomogeneous models were considered as a

spatial juxtaposition of different media with isotropic conductivity. The modelling of

inhomogeneous media with isotropic conductivity is physically intuitive as far as

understanding multi-dimensional electromagnetic induction in complex objects is

concerned. Of specific interest to the geophysical community now is the ability to

model inhomogeneous media with anisotropic conductivity. Frequency-domain

differential equation methods of approximate solutions for such media have been

previously presented using finite-element [4.4, 4.9, 4.11, 4.18] and finite-difference

methods [4.1, 4.2, 4.6-4.10, 4.14, 4.15, 4.19].

It has been demonstrated in Chapters 2 and 3 that two-dimensional problems

involving TM-type incidence on media with inclined uniaxial anisotropic conductivity

were equivalent to problems involving TM-type incidence on media with fundamental

biaxial anisotropic conductivity. The effective horizontal and vertical conductivity

82

values were obtained from the diagonal components of the Euler rotation of the

resistivity tensor into the horizontal and vertical planes respectively, and were shown to

be functions of the diagonal and non-diagonal terms from the Euler rotation of the

conductivity tensor into the horizontal and vertical planes respectively. In this chapter,

we will introduce the self-consistent impedance method of Thiel & Mittra [4.13], based

on a finite-difference approximation of an RC circuit analogy, to model inhomogeneous

media with inclined uniaxial anisotropic conductivity in two-dimensions. RC network

analogies are quite lucid tools for simulating induction problems, particularly when one

is faced with some of the more abstract mathematical concepts behind finite-element,

integral equation and hybrid methods of approximate solution. The work presented in

this chapter has been published as [4.17] and submitted for publication as [4.16].

4.2. Formulation: self-consistent impedance method

A two-dimensional self-consistent form of the impedance method was recently

published and its applications to surface impedance modeling of isotropic media were

discussed [4.13]. The definition of self-consistent is that the magnetic field is assumed

to be unknown everywhere in the solution space (with the exception of the source

terms) and is independent of the model. Previous formulations of the impedance

method [4.3] assumed that the magnetic field everywhere was known and was

dependent of the model.

The self-consistent method requires the solution space to be divided into

rectangular cells bound by four impedance elements as

shown in Figure 4.1. The applied field is a magnetic field impressed on one or more

cells with an implied time variance of exp(

kikikiki ZZZZ ,,4,,3,,2,,1 and , , ,

)tjω . In the formulation presented in this

chapter, it is assumed that 0µµ = everywhere in the solution space. For a quasi-static

solution, displacement currents are ignored everywhere in the solution space. However,

displacement currents can be introduced simply by considering a complex conductivity.

Adopting the finite-difference approach (see Appendix 6), we introduce the notation

(i,k) for the y-z directions, to avoid using ( j,k) since j is the 1− in this thesis. Each

cell has sides and lies in the yz plane. By applying Faraday�s Law to the

(i,k)

k,iki zy , and ∆∆th cell, one can write:

83

, )( )()()(

,,,0,,4,1,

,,31,,,,2,1,,,11,,

kikikikikiki

kikikikikikikikiki

zyHjZIIZIIZIIZII∆∆−=−+

−+−+−

−

++−

ωµ (4.1)

where is the current in the (i,k)kiI ,th element, is the magnetic field component H

normal to the (i,k)

kiH , x

th cell, and are the cell dimensions and , for m = 1, 2,

3 and 4, are the element impedance values given by:

kiy ,∆ kiz ,∆ kimZ ,,

kikikim

kikim zx

yZ

,,,,

,,, ∆∆

∆=

σ, (4.2)

where kim ,,σ is the conductivity of the material in the (i,k)th cell and is the

direction normal to the cell. By applying a finite-difference formulation of Ampere�s

Law in integral form along a rectangular path in the xz plane around (see Figure

4.1), the current in impedance element is given by:

kix ,∆

kiZ ,,1

kiI ,,1

kikikikikiki xHHIII ,1,,1,,,,1 )( ∆−=−= −− , (4.3)

which is recognized as a finite-difference form of boundary condition (1.39). Using

equation (4.3), equation (4.1) can be re-written as:

( ) ( ) ( ) ( )

( ) ( ) ,0

122

2,,,

,12

,,,

1,

2,,,

,12

,,,

1,,2

,,,2

,,,

=∆

−∆

−∆

−∆

−

+

∆+

∆

−+

+−

kikiy

ki

kikiz

ki

kikiy

ki

kikiz

kiki

kikiykikiz

ykH

zkH

ykH

zkH

Hykzk

(4.4)

which can be identified as the finite-difference approximation to the Helmholtz

equation, identical in form to equation (3.30), where and k are the propagation

coefficients in the y and z directions respectively, written as:

kiyk ,, kiz ,,

kiykiy jk ,,0,, σωµ= , (4.5)

kizkiz jk ,,0,, σωµ= . (4.6)

84

∆zi,k

∆zi,k

Z2,i,k Z4,i,k

i-1,k

i,k-1 Z1,i,k

∆x

i+1,k

Figure 4.1. Schematic

convention

integration

Law (after [

The wave numbers a

divergent solutions. For a s

can write the matrix equatio

0JH S = ,

i,k

Z3,i,k

i,k+1

∆yi,k

diagram in the y-z plane showing the numbering

for the impedance mesh. The dotted line shows the

path in the x-z plane for the application of Ampere�s

4.13]).

re chosen such that to prevent exponentially

olution space consisting of N cells, from equation (4.4), one

n:

0Re ,),,( >kizyk

(4.7)

85

where H is a 1×N matrix of unknown magnetic field elements in the solution space and

is the 1×N matrix of applied current densities (i.e., source terms) which must have

non-zero terms to prevent non-trivial solutions in H. For example, in plane wave

incidence, the impressed magnetic field is generated from a series of applied current

densities in the x-direction. S is a sparse square matrix N

0J

2 in size, and is called the

propagation matrix, since it is expressed in terms of the propagation coefficients and

cell dimensions. The unknown magnetic field values are solved with the matrix

equation:

01JSH −= ,

where can be solved by any number of matrix inversion algorithms. The

formulation presented in this thesis was written in Matlab, and solves for the matrix

inversion using a standard LU decomposition method. Dirichlet boundary conditions

are used to define the magnetic field as the incident homogeneous plane wave.

Neumann boundary conditions are employed to terminate the boundaries in the other

directions. From equation (4.4), it is possible to observe that the diagonal elements of S

have the form:

1−S

( ) ( ) 1222

,,,2

,,,

+∆

+∆ kikixkikiz ykzk

, (4.8)

and the non-zero off-diagonal elements have the forms of:

( )2,,

1yk kiy ∆

− , (4.9)

( )2,,

1zk kiz ∆

− . (4.10)

It is clear then, from equations (4.8) to (4.10), that the self-consistent impedance

method can accommodate fundamental uniaxial and biaxial anisotropic media, where

. The current through is calculated from equation (4.2) so the

horizontal electric field can be calculated using the field form of Ohm�s Law:

kizkiykix kkk ,,,,,, ≠≠ kiZ ,,1

kiE ,,1

86

kikiy

kikiki z

HHE

,,,

1,,,,1 ∆

−= −

σ. (4.11)

The surface impedance Z at the earth-air interface, defined at the top of the

(i,k)

yx

th cell, can now be written as:

kikikiy

kiki

x

yyx zH

HHHE

Z,1,,,

1,,

∆−

==−

−

σ. (4.12)

It should be noted that the term in equation (4.12) is measured above the

surface of the conducting half space and the term is measured at the surface of the

conducting half space. It is observed that having at least two rows of air cells above the

surface of the conducting half space provides accurate surface impedance results. Since

the medium above the half space is perfectly insulating, is approximately uniform

over distances less than one-tenth of the wavelength in free-space above the surface. If

one seeks to introduce a semi-implicit approximation for the magnetic field, similar to

the finite-difference time-domain method [4.12]:

1, −kiH

kiE ,,1

xH

21,,

2/1,−

−

+= kiki

ki

HHH , (4.13)

then it is easily observed that equation (4.12) reduces to:

)()(2

1,,,,,

1,,

+

−

+∆−

==kikikikiy

kiki

x

yyx HHz

HHHE

Zσ

.

In the above form, the self-consistent impedance method can solve for

fundamental biaxial anisotropy problems, and those inclined uniaxial anisotropy

problems that reduce to fundamental biaxial anisotropy problems. For the inclined

anisotropy problem, the substitution of equations (2.59) and an analogous vertical

conductivity value obtained from the substitution of instead of from equation

(2.45), into equations (4.5) and (4.6) reduces the problem to a fundamental one.

zE yE

87

Other authors who have considered two-dimensional numerical modelling of

inclined anisotropic media, such as Shize & Shengkai [4.11], have considered

approximate solutions to equation (2.19), the rotated anisotropic Helmholtz equation.

If we were to consider the simulation of horizontally inhomogeneous media with

inclined anisotropy, then according to other authors, equation (4.4) must be replaced by

the finite-difference approximation to equation (2.19):

( ) ( )

( ) [ ]

( ) [ ]

[ ] .0112

cossin

cossin1

sincos1

1cossin2sincos2

1,11,11,11,12,,

2,,,,

,,

1,1,2,,

,2

2,,

,2

2,

,1,12,,

,2

2,,

,2

2,

,2,,

,2

2,,

,2

22,,

,2

2,,

,2

2,

=−−+

−

∆∆−

−

+

∆−

−

+

∆−

+

+

∆+

+

∆

−−+++−−+

−+

−+

kikikikikitkinkiki

kiki

kikikit

ki

kin

ki

ki

kikikit

ki

kin

ki

ki

kikit

ki

kin

ki

kit

ki

kin

ki

ki

HHHHxz

HHz

HHx

Hzx

γγαα

γα

γα

γα

γα

γα

γα

γα

γα

(4.14)

Specifically, it should be noted that in equation (4.14), the so-called shearing

term:

[ ]1,11,11,11,12,,

2,,,,

,, 112

cossin−−+++−−+ −−+

−

∆∆ kikikikikitkinkiki

kiki HHHHyz γγαα

, (4.15)

vanishes unequivocally (see Chapter 3) and equation (4.14) reduces to equation (4.4)

identically. Hence, there is no need to consider the partial term of (4.15) in any two-

dimensional finite-difference formulation for TM-type waves. It should be noted that in

this case, equation (4.14) defines the magnetic fields in both fundamental biaxial

anisotropic media, as well as inclined uniaxial anisotropic media.

The basic field approximation, as given by equation (4.4), is formally identical

to that resulting from a finite-difference approximation with a cell-centered TM-type

magnetic field. This approach is not common in two-dimensional finite-difference

modelling practice, but standard three-dimensional staggered-grid modelling

approaches [4.8] reduce to this case when homogeneity is assumed in one horizontal

direction. Thus, based upon the approximation of equation (4.4), two-dimensional

accuracy tests for staggered grid solutions could be made.

88

4.3. Results: inclined anisotropy in inhomogeneous media

To demonstrate that an inclined uniaxial anisotropic conductor can be simulated

as a fundamental biaxial anisotropic conductor, a number of cases involving inclined

anisotropic conductivity are presented in which the surface impedance of the

conducting half space can be expressed as an analytical function to assess the accuracy

of this new formulation.

4.3.1. Homogeneous layer above a perfect electric conductor

The exact solution for the surface impedance of a TM-type wave normally

incident upon a horizontally layered half space was presented in Chapter 2. For a

homogeneous layer with intrinsic impedance:

122

1sincos

−

+=

nt

jZσ

ασ

αωµ , (4.16)

terminated at z = h by a perfect electric conductor, the surface impedance at z = 0 is

given by the transmission line analogy:

)tanh(sincos1

122

,1 hkjZnt

s

−

+=

σα

σαωµ , (4.17)

where:

+=

nt

jkσ

ασ

αωµ22

1sincos , (4.18)

and to prevent an exponentially divergent solution in . A comparison

between the surface impedance responses obtained from equation (4.17) and with the

self-consistent impedance method is shown in Figure 4.2 for frequencies in the range

, and is presented as a function of the angle of inclination of the

0Re 1 >k

10Hz ≤≤ f

sZ ,1

kHz 10

89

anisotropic conductivity, . The anisotropic layer was assumed to have o900 ≤≤ α

001.01, =tσ S/m and 01.0=1,nσ S/m. The approximate model used 3 air cells, and

earth cells, and the depth between the air-half space interface and the perfect

electric conducting basement was constant at 1 000 m. The cell sizes selected were

m and m.

2×

1003×

20=∆y 10=∆z

.0=

01.01, =nσ

o90≤0 ≤ α


anisotropic layer of 1 000 m thickness with 0011,tσ S/m and

S/m above a perfectly conducting basement. The

anisotropic conductivity tensor is rotated through the range

. The impedance method solutions for 10 Hz (+), 100 Hz

(x), 1 kHz (o) and 10 kHz (*) are presented. The exact solutions for

10 Hz (dotted), 100 Hz (solid), 1 kHz (dash-dotted) and 10 kHz

(double-dashed) are also shown. Note that at 1 kHz and 10 kHz, the

surface impedance is equal to the intrinsic impedance of the layer

and the phase is equal to 45o.

90

The surface impedance magnitude agrees very well with the corresponding

exact solutions for all frequencies considered. The surface impedance phase however,

is more accurate at lower frequencies than at high frequencies. This difference is

suggested to be a resultant from the cell size chosen. For the lower frequencies, the cell

size was chosen to be several orders of magnitude less than the wavelength in the half

space. At the higher frequencies presented, the cell size was approximately in the same

order of magnitude as the wavelength considered. Such discretisation errors are

introduced in all approximate techniques and are not therefore unique to the self-

consistent impedance method.


anisotropic layer of 20 m thickness with 01, 15εε =n , 01.01, =nσ S/m,

01, 5εε =t and 001.01, =nσ S/m above a perfect electrically. The

anisotropic complex conductivity tensor is rotated through the

range . The impedance method solutions for 10o900 ≤ ≤α 4 Hz (o),

105 Hz (*) and 106 Hz (∆) are presented. The exact solutions for 104

Hz (solid), 105 Hz (dashed) and 106 Hz (double dashed) are also

shown. Note that at due to the effects of displacement currents at 106

Hz, the surface impedance is equal to the intrinsic impedance of the

layer and the phase is not equal to 45o.

91

To demonstrate the ability of the self-consistent impedance method to include

complex conductivity, the surface impedance of a homogeneous layer with 01, 15εε =n ,

01.01, =nσ S/m, 01, 5εε =t and 001.01, =nσ S/m above a perfect electrically

conducting basement, for f = 104, 105 and 106 Hz are presented in Figure 4.3, as a

function of the angle of inclination, 0 . The thickness of the anisotropic layer

was assumed to be 20 m and a uniform cell size of ∆x = ∆z = 0.5 m was used. The

solution space was 3 cells wide and 42 cells deep, allowing for 2 air cells above the

anisotropic interface for surface reflections. The corresponding exact solutions are also

presented [4.17].

o90≤≤ α

Figure 4.4. Surface impedance magnitude and phase response at 100 Hz for two

anisotropic layers overlying a perfectly conducting basement using

the self-consistent impedance method (o) and corresponding exact

solutions (solid line). The upper layer thickness was varied and the

total depth was kept constant at 1 000 m. The upper layer has σt,1 =

0.01 S/m, σn,1 = 0.001 S/m and α1 = 30o. The lower layer has σt,2 =

0.05 S/m, σn,2 = 0.5 S/m and α2 = 60o.

92

4.3.2. Horizontal homogeneous layers above on a perfect electric conductor

One may extend the transmission line analogy as discussed in the previous

section to any number of horizontal, homogeneous layers. Figure 4.4 presents a

comparison of the exact and approximate results for a two-layered, horizontally

homogeneous, inclined anisotropic earth above a perfectly conducting basement for a

frequency of 10 kHz for a variable upper layer thickness . The second layer thickness

is varied in each case such that h m. The upper layer has

1h

2h 000 121 =+ h 01.01, =tσ

S/m and 001.01, =nσ S/m. The second layer has 05.02, =tσ S/m and 5.0=2,nσ S/m.

The angle of inclination in the upper layer is 30o and in the lower layer is 60o, and is

kept constant in all models. The cell sizes selected were m and ∆ m,

and the total solution space was 3 × 102 cells where 2 air cells were assigned in the

model. There is good agreement between the impedance method and the corresponding

exact solutions, though at one particular point ( = 100 m), the phase does differ by as

much as 16% from the corresponding exactly derived value and the magnitude varies

by 19%. All other points have variations less than these just described, and reduce to nil

variance for the single layer case when = 1 000 m.

10=∆y 20=z

1h

1h

4.3.3. Vertical dyke embedded in a homogeneous layer above a perfect electric

conductor

The exact solution for the surface impedance of a homogeneous plane wave

incident upon a vertical dyke with inclined electrical anisotropy embedded in an

otherwise homogeneous layer, above a perfect electric conducting basement was

presented in Chapter 3. It was demonstrated in that chapter that the inclined electrical

anisotropy problem is equally posed as a fundamental electrical anisotropy problem. In

Figure 4.5, the surface impedance at 10 kHz is presented for a dyke that is 500 m wide

with 01.0=tσ S/m and 001.0=nσ S/m embedded in an otherwise homogeneous layer

with 00.0=t 1σ S/m and 01.0=nσ S/m. The common depth of the dyke and

homogeneous layer to the perfect electrically conducting basement is 200 m. The cell

sizes selected were m and ∆ m, and the total solution space was 100 × 42

cells where 2 air cells were assigned in the model. There is good agreement in the

surface impedance magnitude between the impedance method and the corresponding

exact solution, but the surface impedance phase has some inaccuracies.

5 z=∆y 5=

93

Figure 4.5. Surface impedance magnitude and phase response at a single

boundary at 10 kHz for a 500 m wide vertical dyke with 01.01, =tσ

S/m and 001.01, =nσ S/m embedded in an otherwise homogeneous

layer with 001.01, =tσ S/m and 01.01, =nσ S/m, terminated with a

perfectly conducting basement. The depth of the dyke and layer is

200 m. The impedance method response (dots) is shown above the

exact response (solid line).

4.5. Discussion

In this chapter, the principle of modelling the surface impedance of an

inhomogeneous half space with inclined uniaxial electrical anisotropy as an equivalent

half space with fundamental electrical biaxial anisotropy has been demonstrated

successfully. The self-consistent impedance method has been introduced, and shown to

accurately model the surface impedance response of these two-dimensional induction

problems over a broad range of frequencies. Whilst the impedance method has been

introduced for this modelling concept in this chapter, it is strongly emphasised that the

modelling principles introduced in this chapter can be applied to any two-dimensional

94

approximate method. The surface impedance magnitude can be accurately modelled

using the impedance method, but the surface impedance phase is shown to be

inaccurate for laterally inhomogeneous and anisotropic models. It is importantly noted

here that this chapter presents the first comparison of any approximate technique with

the corresponding exact solution for a laterally inhomogeneous and anisotropic half

space.

4.6. References

[4.1] M. Eisel, V. Haak, J. Pek & V. Cerv, J. Geophys. Res. 106B, 16061 (2001).

[4.2] W. Heise & J. Prous, Geophys. J. Int. 147, 610 (2001).

[4.3] D. A. James, PhD thesis, Griffith University, 1998.

[4.4] Y. Li, Geophys. J. Int. 148, 389 (2002).

[4.5] R. L. Mackie, J. T. Smith & T. R. Madden, Radio Sci. 29, 923 (1994).

[4.6] G. A. Newman & D. L. Alumbaugh, Geophysics 67, 484 (2002).

[4.7] A. M. Osella & P. Martinelli, Geophys. J. Int. 115, 819 (1993).

[4.8] J. Pek & T. Verner, Geophys. J. Int. 128, 505 (1997).

[4.9] I. K. Reddy & D. Rankin, Geophysics 40, 1035 (1975).

[4.10] P. D. Saraf, J. G. Negi & V. Cerv, Phys. Earth Planet. Int. 43, 196 (1986).

[4.11] X. Shizhe & Z. Shengkai, Acta Seismolog. Sinica 7, 80 (1985).

[4.12] A. Taflove, Computational Electrodynamics: The Finite Difference Time

Domain Method (Artech House, Norwood, 1995).

[4.13] D. V. Thiel & R. Mittra, Radio Sci. 36, 31 (2001).

[4.14] T. Wang & S. Fang, Geophysics 66, 1386 (2001).






[4.18] C. Yang, Xibei Dizhen Xuebao 19, 27 (1997).

[4.19] C. Yin & H. M. Maurer, Geophysics 66, 1405 (2001).


J. App. Geophys. 37, 133 (1997).

95

Chapter 5

Exact solutions for three-dimensional anisotropy problems

5.1. Introduction

As presented in the previous chapters of this thesis, the physical basis for the

surface impedance of a homogeneous plane wave polarised parallel or perpendicular to

the strike of a two-dimensional half space is that the electric and magnetic fields

generated are orthogonal. In any other situation, the electric and magnetic field vectors

are not orthogonal, indicating that either the source field or half space is three-

dimensional and inhomogeneous, or that the source field is not polarised parallel or

perpendicular to the strike of the half space [5.4, 5.5, 5.17]. Below 1 kHz, the source

fields are elliptically polarised [5.16] and this can lead to some difficulties in surface

impedance data reduction and interpretation. Above 1 kHz, the principle source field

for surface impedance measurements is the radiation from lightning discharges or

artificial signals from navigation beacons and radio transmitters [5.22].

These very low frequency (VLF) waves propagate with very little loss in the

earth-ionosphere waveguide as a series of waveguide modes. Cloud-to-ground lightning

discharges and VLF antennae are effectively vertical electric dipoles, and as such only

launch linearly polarised TM-type waves. At imperfect ionospheric and terrestrial

boundaries, the TM-type waves can be reflected as TE-type waves, which propagate

independently and with higher attenuation than the lower order TM-type waves that

dominate VLF propagation over very large distances in the earth-ionosphere waveguide

[5.24]. However, the extent of this mechanism for elliptical polarization has been

calculated to be less than 1% [5.12], and the minor/major axis ratio of the magnetic

field polarization ellipse has been measured to less than 1% [5.14].

The one-dimensional surface impedance of VLF radio waves is commonly

measured using portable surface impedance meters [5.1, 5.18] that measure the

horizontal electric and magnetic field magnitudes, and the phase difference between

them. In the interpretation of the VLF surface impedance, the earth is usually assumed

to be isotropic. However, some authors have presented methods of interpreting one-

dimensional VLF surface impedance data above a laterally anisotropic half space [5.11,

5.18, 5.19]. However, in these formulations, it has been assumed that the horizontal

magnetic field component maintains its linear polarization. This is not exactly true in

96

the presence of an arbitrary anisotropic half space. Further, no analyses have ever been

presented for the surface impedance of an inclined anisotropic half space at arbitrary

skew angles.

In this chapter, the method of auxiliary potentials, developed by Chetaev &

Belen�kaya [5.2, 5.9, 5.10], will be applied to solve for the fields of a linearly polarised,

homogeneous plane wave incident at a skew angle of incidence upon an inclined

anisotropic half space. In the anisotropic half space, all field components will be

coupled and hence a TM-type wave incident upon the half space will generate a

reflected TE-type wave in addition to the reflected TM-type wave. This will influence

the measured electric and magnetic field components in one-dimensional surface

impedance measurements. This work is a logical extension of Chapter 2, where the

source field was assumed to be parallel and perpendicular to the strike of the

anisotropic half space. Expressions for all elements of the impedance are derived, and

their applications to VLF surface impedance measurements are further discussed.

5.2. General solutions for the electromagnetic fields in an anisotropic medium

By developing Chetaev�s method of auxiliary potentials [5.7], we will consider

the surface impedance of homogeneous, monochromatic plane waves characterised by

, incident at a skew angle upon a homogeneous half space with inclined

uniaxial anisotropic conductivity (Figure 5.1). Solutions will be presented for both TE-

and TM-type incident plane waves. This class of problem is a natural extension to the

class of models presented in Chapter 2 of this thesis.

)exp( kz−

Given the Maxwell equations (1.1) to (1.4) for monochromatic fields in the

inclined co-ordinate system of a uniaxial anisotropic medium, characterised by a

uniaxial tensor of complex conductivity given by equation (1.54), where no free

charges or extraneous currents exist, we introduce the potentials:

'' AB ×∇= , (1.31)

Φ−∇−= '' AE ωj . (1.33)

Substituting equations (1.31) and (1.33) into the Maxwell equations (1.2) and

(1.3) results in:

97

x x� θ z = 0 y α α z� y� z

Figure 5.1. Geometry for the homogeneous plane wave incident at a skew angle

θ to the anisotropic half space inclined at angle α about the x-axis.

0�'�)'(' =Φ∇−−⋅∇−∇∆ σµσωµ AAA j , (5.1)

and substituting equations (1.31) and (1.33) into the Maxwell equation (1.1) and (1.4)

results in:

0'�� =⋅∇+Φ∇⋅∇ Aσωσ j , (5.2)

where σ�⋅∇ can not be abbreviated when σ� is anisotropic [5.23]. We introduce the

optimal Lorentz gauge condition [5.6]:

'1 A⋅∇−=Φtµσ

, (5.3)

and substitute equation (5.3) into equation (5.1) to obtain:

0)'( 1�

'�' =⋅∇∇

−+−∆ AAA

t

jσσσωµ . (5.4)

98

for the vector potential A in {x� ,y�, z�} co-ordinates. For and , which are

associated with

'xA 'yA

tσ , equation (5.4) reduces to the homogeneous Helmholtz equations:

0'' =−∆ xtx AjA ωµσ , (5.5)

0'' =−∆ yty AjA ωµσ . (5.6)

From the optimal Lorentz gauge condition, the full condition on the vector

potential is stated as:

Φ=∂∂

−'

1 '

zAz

tµσ, (5.7a)

0'''' =

∂∂

+∂∂

yA

xA yx . (5.7b)

For , which is associated with 'zA nσ , equation (5.4) reduces to:

0'

)1( 2'

22

'' =∂

∂−Λ+−∆

zAAjA z

znz ωµσ , (5.8)

where λσ

σ 1==Λt

n , the reciprocal of the coefficient of anisotropy. When we rotate

the co-ordinates from the fundamental co-ordinate system to the inclined co-ordinate

system (see Appendix 1), equations (5.5) and (5.6) are invariant, whilst we introduce

the functional rotations:

θα sinsin' z

fxf

∂∂−=

∂∂ ,

θα 222

2

2

2

sinsin' z

fxf

∂∂−=

∂∂ ,

99

θα cossin' z

fyf

∂∂−=

∂∂ ,

θα 222

2

2

2

cossin' z

fyf

∂∂−=

∂∂ ,

αcos' z

fzf

∂∂−=

∂∂ ,

α22

2

2

2

cos' z

fzf

∂∂−=

∂∂ .

Equation (5.7) will take the form of:

0sincossinsin '' =∂

∂+

∂∂

− αθαθz

Az

A yx , (5.9)

and equation (5.8) takes the form of:

0cos)1( 22

'2

2'' =

∂∂

−Λ+−∆ αωµσzAAjA z

znz , (5.10)

since all partial derivatives of the homogeneous plane wave field with respect to x and y

are equal to zero. When one considers equations (5.5) and (5.6) with the spatial

variance described by , then and are observed to satisfy the wave

number:

)exp( 1zk− 'xA 'yA

tjk ωµσ=21 , (5.11)

which is the wave number for an ordinary wave, provided Re to prevent

exponentially divergent solutions in and . As a result of the spatial variances,

from equation (5.9), the relation between the vector potential components of the

ordinary wave:

01 >k

'xA 'yA

100

θtan'' xy AA = , (5.12)

is obtained. From co-ordinate rotations (see Appendix 1), one can write for the ordinary

wave:

αθθ cossincos '' yxx AAA += , (5.13)

αθθ coscossin '' yxy AAA +−= , (5.14)

αsin'yz AA = . (5.15)

By substituting equation (5.12) into equations (5.13) to (5.15), the vector

potential components in fundamental co-ordinates can be written as:

+=θ

αθθcos

cossincos 22

'xx AA , (5.16)

)1(cossin' −= αθxy AA , (5.17)

αθ sintan'xz AA = . (5.18)

As a consequence of equation (5.7), it is observed for the ordinary wave that

, which implies: 0=Φ

AE ωj−= . (5.19)

As a result, the components of the electric fields of the ordinary wave in

fundamental co-ordinates can be written as:

+−=θ

αθθωcos

cossincos 22

'xx AjE , (5.20)

101

)1(cossin' −−= αθω xy AjE , (5.21)

αθω sintan'xz AjE −= . (5.22)

From equation (1.31), the components of the magnetic flux density of the

ordinary wave in fundamental co-ordinates can be written as:

)1(cossin'1 −= αθxx AjkB , (5.23)

+−=θ

αθθcos

cossincos 22

'1 xy AjkB , (5.24)

0=zB . (5.25)

After considering equation (5.10) with the spatial variance described by

, then it is observed that satisfies the wave number: )exp( 2 zk− 'zA

αλωµσ

222

2 sin)1(1 −+= tj

k , (5.26)

which is the wave number for an extraordinary wave, provided to prevent

exponentially divergent solutions in . For the extraordinary wave, it is observed that

the scalar potential is not zero but is given by:

0Re 2 >k

'zA

αµσ

cos'2

zt

Ajk=Φ . (5.27)

From equation (5.27), it follows that:

k� cos'

22

=Φ∇ α

µσ zt

Ak

. (5.28)

102

From co-ordinate rotations (see Appendix 1), one can write for the

extraordinary wave:

αθ sinsin'zx AA −= , (5.29)

αθ sincos'zy AA −= , (5.30)

αcos'zz AA = . (5.31)

The components of the electric field for the extraordinary wave are obtained by

substituting equations (5.29) to (5.31) into equation (1.33):

αθω sinsin'zx AjE = , (5.32)

αθω sincos'zy AjE = , (5.33)

αµσ

cos'

22

21

zt

z Akk

E

−= . (5.34)

From equation (1.31), the components of the magnetic flux density of the

ordinary wave in fundamental co-ordinates can be written as:

αθ sincos'2

zx AjkB −= , (5.35)

αθ sinsin'2

zy AjkB = , (5.36)

0=zB . (5.37)

The observed fields will be the sum of the ordinary and extraordinary fields:

aryextraordinordinary EEE += , (5.38)

103

aryextraordinordinary HHH += . (5.39)

From equations (5.20) to (5.25) and (5.32) to (5.37), we can write the

components of the total electric and magnetic fields in fundamental co-ordinates in the

inclined anisotropic half space as:

αθωθ

αθθω sinsincos

cossincos'

22

' zxx AjAjE +

+−= , (5.40)

αθωαθω sincos)1(cossin '' zxy AjAjE +−−= , (5.41)

αµσ

αθω cossintan '

22

21

' zt

xz Akk

AjE

−+−= , (5.42)

αθαθ sincos)1(cossin '2'1 zxx AjkAjkB −−= , (5.43)

αθθ

αθθ sinsincos

cossincos'2

22

'1 zxy AjkAjkB +

+−= , (5.44)

0=zB . (5.45)

Equation (5.45) is expected, following from equation (2.41). Also, equations

(5.40) to (5.32) are observed to satisfy equation (2.42):

0=++ zzzyzyxzx EEE σσσ , (2.46)

provided the elements of the conductivity tensor are given by:

ααθσσσ sincossin)( ntzx −= , (5.47)

ααθσσσ sincoscos)( ntzy −= , (5.48)

104

ασασσ 22 cossin ntzz −= , (5.49)

which states that the vertical current density of a homogeneous plane wave is equal to

zero.

5.3. General solutions for the electromagnetic fields in air

In the upper half space, the general solution for waves characterised by equation

(5.4) are the superposition of the TE- and TM-type waves, which propagate

independently but satisfy the same wave number:

εµω220 =k . (5.50)

where . From the Maxwell equations (1.1) and (1.2), the components of the

homogeneous TM-type waves will be:

ℜ∈ 0k

zE

tB xy

∂∂

−=∂

∂, (5.51)

zB

tE yx

∂∂

−=∂

∂µ

ε 10 . (5.52)

We will choose the general solutions for the magnetic field component of the

homogeneous TM-type waves to take the form:

)exp()exp( 00 zjkNzjkMBy +−= , (5.53)

so the corresponding electric field component can be written as:

)}exp()exp({ 000 zjkNzjkM

kEx −−=

ωµε, (5.54)

105

where M and N are (complex) constants independent of {x, y, z, t}, representing the

coefficients for the down- and up-going homogeneous plane waves. From the Maxwell

equations (1.1) and (1.2), the components of the homogeneous TE-type wave will be:

zB

tE xy

∂∂−=

∂∂

µε 1

0 , (5.55)

zE

tB yx

∂∂

−=∂

∂. (5.56)

We will choose the general solutions for the electric field component of the

homogeneous TE-type waves to take the form:

)exp()exp( 00 zjkQzjkPEy +−= , (5.57)

so the corresponding magnetic field component can be written as:

)}exp()exp({ 000 zjkQzjkP

kBx −−=

ω, (5.58)

where P and Q are (complex) constants independent of {x, y, z, t}, representing the

coefficients for the down- and up-going homogeneous plane waves.

5.4. The problem of a TE-type incident field

Let us consider a TE-type wave incident upon the anisotropic half space at a

skew angle of incidence. Since the ordinary and extraordinary waves in the anisotropic

half space couple all field components, then a TM-type wave must also be reflected, in

addition to the reflected TE-type wave. The problem is to determine the constants N, Q,

and in terms of the amplitude of the incident TE-type wave, P. The continuity

of is ensured by the continuity of across the interface. We now have four

continuity conditions for the horizontal field components. First, we must express in

terms of from continuity of the horizontal components of the TM-type wave that is

generated in the lower half space and reflected into the upper half space. So, we

'xA 'zA

'xA

zB yE

'zA

106

consider the boundary and set in equation (5.53), so only an up-going

TM-type wave in the upper half space is considered. The fields of the reflected TM-

type wave are then equated with the corresponding field components in the anisotropic

half space. From equations (5.53) and (5.54), one can write:

0=z 0=M

2 sinsinθ

αθ

−

++

111

kcosαθ

−

1

21 k

kk

NBzy ==0

, (5.59)

Nk

Ezx µωε0

00

−==

. (5.60)

We now equate equations (5.59) and (5.60) with equations (5.44) and (5.40)

respectively to obtain the relation:

'21

2' sincos

cossin11

zx AkkA

αθθ

+++

= . (5.61)

From equations (5.43) and (5.61), one can write:

'xTMx AFB = , (5.62)

where

+=2

12

22

cossincos

kk

kjFTM θθ . (5.63)

Similarly, one can write:

'xTMx AGB = , (5.64)

where

++

= θθα sin11

cossin 2kjGTM . (5.65)

107

Now, all field components for the electromagnetic field at the surface of the half

space can be expressed in terms of : xB

xTMTM

x BG

jF

jE

+

+−= αθωθ

αθθω sinsin1cos

cossincos1 22

, (5.66)

xTMTM

y BG

jF

jE

+−−= αθωαθω sincos1)1(cossin1 , (5.67)

xTMtTM

z BG

kkF

jE

−+−= α

µσαθω cos1sintan1 2

22

1' , (5.68)

xTMTM

y BG

jkF

jkB

+

+−= αθθ

αθθ sinsin1cos

cossincos12

22

1 , (5.69)

0=zB . (5.70)

It is noticed that when =θ 0o or 180o, then:

txy

jZσωµ−= , (5.71)

}sin)1(1{ 22 αλσωµ −+−=

tyx

jZ . (5.72)

When =θ 90o or 270o, then:

}sin)1(1{ 22 αλσωµ −+=

txy

jZ . (5.73)

0=yxZ . (5.74)

108

Equations (5.71) to (5.74) agree with the expected values of the surface

impedance when the source fields are parallel and perpendicular to the strike of the

anisotropic half space.

5.4. The problem of a TM-type incident field

Let us now consider a TM-type wave incident upon the anisotropic half space at

a skew angle of incidence. Since the ordinary and extra-ordinary waves in the

anisotropic half space couple all field components, then a TE-type wave must also be

reflected, in addition to the reflected TM-type wave. The problem is to determine the

constants P, N, and in terms of the amplitude of the incident TM-type wave, M.

The continuity of is ensured by the continuity of across the interface. We now

have four continuity conditions for the horizontal field components. First, we must

express in terms of from continuity of the horizontal components of the TE-

type wave that is generated and reflected. So, we consider the boundary and set

in equation (5.57) so only an up-going TE-type wave in air is considered. We

equate the fields of the reflected TE-type wave with the corresponding field

components in the anisotropic half space. From equations (5.57) and (5.58), we write:

'xA

B

'zA

'xA

z yE

'zA

0=z

0=P

QEzy ==0

, (5.75)

Qk

Bzx ω

00

−==

. (5.76)

We now equate equations (5.75) and (5.76) with equations (5.41) and (5.43)

respectively to obtain the relation:

'10

20' )1(cossin)(

sincos)(zx A

kkkk

A−−

−=

αθαθ

. (5.77)

From equations (5.44) and (5.77), one can write:

'xTEy AFB = , (5.78)

109

where

ααθ

µσαθ

tan)()1(costan)(

)1(cossin20

102

22

11 kk

kkkkjjkF

tTE −

−−

−+−= . (5.79)

Similarly, one can write:

'zTEy AGB = , (5.80)

where

αµσ

αθ cossincos)(2

22

1201

−+−=

tTE

kkjkkjkG . (5.81)

Now, all field components for the electromagnetic field at the surface of the half

space can be expressed in terms of : xB

xTETE

x BG

jF

jE

+

+−= αθωθ

αθθω sinsin1cos

cossincos1 22

, (5.82)

xTETE

y BG

jF

jE

+−−= αθωαθω sincos1)1(cossin1 , (5.83)

xTEtTE

z BG

kkF

jE

−+−= α

µσαθω cos1sintan1 2

22

1' , (5.84)

xTETE

y BG

jkF

jkB

+

+−= αθθ

αθθ sinsin1cos

cossincos12

22

1 , (5.85)

0=zB . (5.86)

It is noticed that when =θ 0o or 180o, then:

110

}sin)1(1{ 22 αλσωµ −+=

txy

jZ , (5.87)

tyx

jZσωµ= . (5.88)

When =θ 90o or 270o, then:

0=xyZ , (5.89)

}sin)1(1{ 22 αλσωµ −+=

tyx

jZ . (5.90)

Equations (5.87) to (5.90) agree with the expected values of the surface

impedance when the source fields are parallel and perpendicular to the strike of the

anisotropic half space.

5.6. Discussion

From equations (5.66) to (5.70), and (5.82) to (5.86), the elements of the

impedance tensor can be calculated:

x

xxx B

EZ

µ= , (5.91)

y

xxy B

EZ

µ= , (5.92)

x

yyx B

EZ

µ= , (5.93)

y

yyy B

EZ

µ= . (5.94)

111

As we are principally concerned with TM-type propagation problems related to

VLF propagation and surface impedance measurements, polar diagrams of one of the

principal components and additional components of the surface impedance tensor for a

10 kHz homogeneous plane wave at all angles of skew incidence upon a homogeneous

half space with 001.0=tσ S/m, 01.0=nσ S/m, and =α 60o are presented in Figures

5.2 to 5.5. It is observed that the values converge to the expected impedances for

=θ 0o, 90o, 180o and 270o. It should be observed that Figures 5.2 to 5.5 correlate to the

additional impedance polar diagrams expected from a two-dimensional structure [5.3].

The principal impedance polar diagram exhibits the phase of a one-dimensional half

space, but have the magnitude characteristics of a two-dimensional half space [5.3].

This reinforces the well-known principle of the existence of ambiguity in surface

impedance measurements, where it is difficult to differentiate between the surface

impedance response of a one-dimensional homogeneous and anisotropic half space, and

a two-dimensional homogeneous (and anisotropic) half space.

We will now examine the effect of an anisotropic half space on the measured

electric and magnetic fields in an ideal one-dimensional surface impedance

measurement. We introduce ψ as the angle between the orientation of the surface

impedance meter and the direction of propagation, i.e., the angle of the instrument with

respect to θ (see Figure 5.6).

112

Figure 5.2. xyZ of a 10 kHz homogeneous TM-type plane wave as a function of

θ for a half space with 001.0=tσ S/m, 01.0=nσ S/m, and =α 45o.

113

Figure 5.3. of a 10 kHz homogeneous TM-type plane wave as a function

of

xyZ∠

θ for a half space with 001.0=tσ S/m, 01.0=nσ S/m, and

=α 45o.

114

Figure 5.4. yyZ of a 10 kHz homogeneous TM-type plane wave as a function of

θ for a half space with 001.0=tσ S/m, 01.0=nσ S/m, and =α 45o.

115

Figure 5.5. of a 10 kHz homogeneous TM-type plane wave as a function

of

yyZ∠

θ for a half space with 001.0=tσ S/m, 01.0=nσ S/m, and

=α 45o.

The fields induced in the horizontal electric and magnetic dipoles, short

insulated wires and small loops respectively [5.18], will be proportional to the electric

and magnetic field components of the total fields in the direction of the dipole axis.

Ignoring the influences of the radiation patterns of the electric and magnetic dipoles, for

any ψ , components of the observed electric and magnetic fields will be:

ψψ sincos yxobs EEE += , (5.95)

ψψ cossin yxobs HHH +−= . (5.96)

If the homogeneous half space was isotropic, then equations (5.95) and (5.96)

would simply reduce to:

116

x x� ψ y θ y�

Figure 5.6. Angle of orientation of the surface impedance meter ψ with respect

to the skew angle of incidence θ and strike of the anisotropic half

space in the xy plane.

ψsinxobs EE = , (5.97)

ψcosyobs HH = , (5.98)

and the observed surface impedance at any sZ ψ would be given by:

ψtany

xs H

EZ = , (5.99)

which exhibits asymptotes when 90=θ or 180o, as expected for linearly polarised

fields. We will now consider a 10 kHz homogeneous TM-type plane wave incident

upon a homogeneous half space with 001.0=tσ S/m, 01.0=nσ S/m, and =α 60o,

where θ = 45o, 90o and 135o. The fields can be calculated using the expressions derived

in Section 5.4. From equations (5.97) and (5.98), the one-dimensional surface

impedance can be calculated, and the corresponding polar diagrams for the normalized

117

magnitude of the magnetic and electric fields are shown in Figures 5.7 and 5.8

respectively.

Figure 5.7. Normalized observed (measured) magnetic field of a 10 kHz

homogeneous TM-type plane wave as a function of ψ for a half

space with 001.0=tσ S/m, 01.0=nσ S/m, =α 45o and θ =60o.

It is observed in Figure 5.7 that at VLF frequencies, the polarization of the

horizontal magnetic field of a homogeneous TM-type plane wave is independent of the

anisotropy of the half space. This verifies the assumptions of previous authors [5.11,

5.12, 5.15, 5.19] who assumed that the magnetic field maintained linear polarization

above an anisotropic half space.

118

Figure 5.8. Normalized observed (measured) electric field of a 10 kHz

homogeneous TM-type plane wave as a function of ψ for a half

space with 001.0=tσ S/m, 01.0=nσ S/m, =α 45o and θ =60o.

In Figure 5.7, it is observed that at VLF frequencies, the polarization of the

horizontal electric fields of a homogeneous TM-type plane wave are elliptically

polarized and are dependent upon the anisotropy of the half space. For the model

presented here, the major axis of the electric field ellipse is observed to rotated

approximately 12o about the horizontal plane with respect to the magnetic field,

maximizing for values of θ = 45o and 135o. It is suggested that by measuring the polar

radiation fields of the electric field and magnetic fields in one-dimensional surface

impedance measurements, information about the presence of anisotropy in the earth

may be ascertained.

To practically demonstrate this effect of anisotropy in a typical VLF survey, the

surface impedance of a VLF radio wave was measured at 10o instrument orientations

with respect to the direction of propagation of the VLF fields [5.13]. Measuring the

19.8 kHz fields of the North West Cape (Western Australia) VLF transmitter, the

surveys were conducted at Callide Mine, Biloela, Queensland, Australia in July 1997 at

different times of day (5:15 am, 6:15 am, 7:15 am, 8:15 am, 9:15 am, 4:25 pm). The

119

magnitudes of the electric and magnetic fields, and the phase difference between them

were measured using a meter similar to [5.18]. The magnetic field was measured using

a ferrite cored multi-turn loop antenna. The electric field was measured using an

electrically short, insulated dipole antenna, which has been demonstrated to be an

effective antenna for measuring the horizontal electric field [5.20]. Figures 5.9 and 5.10

respectively present the normalized magnetic and electric fields from these surveys.

Figure 5.9. Normalized observed (measured) magnetic field of a 19.8 kHz VLF

wave as a function of ψ measured at different times of day at the

Callide Mine, Trap Gully B Area in July 1997 from [5.13].

120

Figure 5.10. Normalized observed (measured) electric field of a 19.8 kHz VLF

wave as a function of ψ measured at different times of day at the

Callide Mine, Trap Gully B Area in July 1997 from [5.13].

It is noted that during the measurements between 6:15 am and 8:15 am relate to

the sunrise period along the 3.5 Mm propagation path between the transmitter and

receiver sites, explaining degradation in the magnetic and electric field polarizations at

those times. This is a result of the destructive interference that occurs between TM01

and TM02 modes during periods of sunrise and sunset [5.21]. However is observed in

Figure 5.9 that the magnetic fields demonstrate a linear polarization for stable

waveguide propagation conditions (5:15 am, 9:15 am, 4:25 pm). Comparing Figure

5.10 to Figure 5.9, it is observed that the horizontal electric field is elliptically

polarised, with the major axis of the electric field polar pattern rotated approximately

10o from the magnetic field polar pattern. As discussed earlier in the model study, this

suggests the presence of anisotropy in the local earth. However, from the surface

impedance data available, it is not possible to determine whether the anisotropy is due

to inclined or lateral anisotropy in the local earth.

121

5.7. Conclusions

In this chapter, the general expressions for the fields of TE- and TM-type

homogeneous plane waves at a skew angle of incidence upon an inclined anisotropic

half space have been derived. Previous analyses have only considered fields of

homogeneous plane waves in the problems of a laterally anisotropic half space (i.e.,

=α 90o), and have not considered the problem of an inclined anisotropic half space.

Further, previous analyses have not considered the effect of mode conversions at the

air-half space boundary, and in the case of TM-type wave, have assumed linear

polarization of the magnetic field is maintained. The results presented in this chapter

have shown that the assumption that the linear polarization of the magnetic field is

maintained, and further the solutions obtained have been shown to converge on the

expected values for the special cases presented in Chapter 2 of this thesis and can be

considered as general solutions for the homogeneous plane wave incident upon an

inclined anisotropic half space.

5.8. References

[5.1] S. A. Arcone & A. J. Delaney, Radio Sci. 15, 1 (1980).

[5.2] B. N. Belen�kaya, Phys. Solid Earth 8, 252 (1972).

[5.3] M. N. Berdichevsky & V. I. Dmitriev, Magnetotellurics in the Context of the

Theory of Ill-Posed Problems (Soc. Explor. Geophys., Tulsa, 2002).

[5.4] D. E. Boerner, R. D. Kurtz & A. G. Jones, Geophysics 58, 924 (1993).




[5.8] D. N. Chetaev, Sov. Phys. Dokl. 12, 555 (1967).

[5.9] D. N. Chetaev & B. N. Belen�kaya, Phys. Solid Earth 7, 212 (1971).

[5.10] D. N. Chetaev & B. N. Belen�kaya, Phys. Solid Earth 8, 535 (1972).

[5.11] G. Fischer, B. V. Le Quang & I. Muller, Geophys. Prosp. 31, 977 (1983).

[5.12] J. Galejs, Radio Sci. 4, 1047 (1967).

[5.13] W. J. F. Nichols, MAppSc thesis, Central Queensland University, 2001.

[5.14] G. S. Parks, G. H. Price, A. L. Whiston & H. W. Parker, Measurement of VLF

Wavefront Components over Long Paths (Stanford Res. Inst., Rep. No. 3590,

1964).

122

[5.15] I. B. Ramaprasada Rao, R. R. Mathur & N. S. Patangay, in Deep

Electromagnetic Exploration, edited by K. K. Roy, S. K. Verman & K. Mallick

(Narosoa Publ. House, New Delhi, 1998), pp. 607-614.

[5.16] D. Rankin & I. K. Reddy, Pure App. Geophys. 78, 58 (1970).


[5.18] D. V. Thiel, Geoexpl. 17, 285 (1979).

[5.19] D. V. Thiel, Explor. Geophys. 15, 43 (1984).

[5.20] D. V. Thiel, IEEE Trans. Antennas Propagat. 48, 1517 (2000).

[5.21] D. V. Thiel & I. J. Chant, Geophysics 47, 60 (1982); 48, 1697 (1983); 49, 1389

(1984).

[5.22] D. V. Thiel, M. J. Wilson & C. J. Webb, Geoexpl. 25, 163 (1988).

[5.23] A. N. Tikhonov, Sov. Phys. Dokl. 4, 566-570 (1959).


Oxford, 1970).

123

Chapter 6

Conclusions

6.1. Contributions

This thesis has investigated the electrodynamics of plane wave propagation in

electrically anisotropic and inhomogeneous media for applications in the surface

impedance methods of electromagnetic geophysics. The thesis has been written in a

theoretical context, emphasising physical principles over technical details or model

studies. No geological aspects of interpreting electrical anisotropy from experimental

data have been presented.

For a homogeneous TM-type wave propagating in a half space (where no

extraneous currents or charges exist) with both vertical and horizontal inhomogeneities,

where the TM-type wave is aligned with one of the elements of the conductivity tensor,

it has been shown using exact solutions (Chapters 2 and 3) that the shearing term in the

homogeneous anisotropic Helmholtz equation:

.0cossin 11 2

cossinsincos

2

2

222

2

222

=−∂∂

∂

−+

∂∂

++

∂∂

+

xx

tn

x

tn

x

tn

Hjzy

Hσ

zH

yH

ωµαασ

σα

σα

σα

σα

(6.1)

unequivocally vanishes and one need only seek solutions to the homogeneous

Helmholtz equation:

.0 cossinsincos2

222

2

222

=−∂

∂

++

∂∂

+ x

x

tn

x

tn

HjzH

yH ωµ

σα

σα

σα

σα (6.2)

This implies that those problems posed with an inclined uniaxial conductivity

tensor can be identically stated with a fundamental biaxial conductivity tensor [6.13,

6.15], provided that the conductivity values are the reciprocal of the diagonal terms

from the Euler rotated resistivity tensor:

124

xxxxx σρσ == −1 , (6.3)

zz

zyyzyyyy

nty σ

σσσρ

σα

σασ −==

+= −

−1

122 sincos , (6.4)

yy

yzzyzzzz

ntz σ

σσσρ

σα

σασ −==

+= −

−1

122 cossin . (6.5)

Exact solutions have been derived for a family of one- and two-dimensional

problems involving a homogeneous, linearly polarised TM-type field [6.11, 6.13, 6.15].

It has been demonstrated that problems with vertical inhomogeneity (i.e., horizontally

stratified), where the each layer exhibits inclined uniaxial anisotropy, can be solved

with a transmission line analogy provided the conductivity in each expression is given

by equation (6.4). The surface impedance response of a vertical dyke in an otherwise

homogeneous layer has also been derived using Fourier series expansion, where both

the dyke and the layer exhibit inclined uniaxial anisotropy.

These exact solutions have provided the physical intuition to identify the

inclined uniaxial anisotropy equivalence to fundamental biaxial anisotropy present in

two-dimensional TM-type propagation problems. This implies those approximate

methods of solving arbitrary two-dimensional problems for a homogeneous TM-type

wave need only to solve the homogeneous Helmholtz equation and can neglect the

corresponding shearing term. It is identified that this inherent non-uniqueness will pose

a significant challenge for any inversion routine, and the requirement for significantly

more detailed geological a priori information to be included in the different

regularisation methods will be essential.

The self-consistent impedance method, a two-dimensional finite-difference

approximation based on a network analogy, has been demonstrated to accurately solve

for such TM-type propagation problems [6.9, 6.12, 6.14]. Whilst other approximate

techniques more accurate than the self-consistent impedance method exist, the context

of its introduction has been to demonstrate the principle of modelling two-dimensional

inhomogeneous media with inclined uniaxial anisotropy.

The problem of a homogeneous plane wave at skew incidence upon an inclined

anisotropic half space has also been treated in this thesis. When a wave is incident upon

a half space with inclined uniaxial anisotropy, both TM- and TE-type waves are

125

coupled and the linearly polarised incident TM- and TE-type waves reflect both TE-

and TM-type (i.e., elliptically polarised) components. Expressions for all elements of

the impedance tensor have been derived for both TM- and TE-type incidence as

homogeneous waves. This simple model offers potential as a method of predicting the

direction of anisotropic strike and dip from a circular survey of one-dimensional

surface impedance measurements, making it particularly applicable to VLF surveys in

sedimentary environments.

6.2. Future research

This thesis has provided a contribution to the fundamental knowledge of the

electrodynamics of plane wave propagation in electrically anisotropic and

inhomogeneous media. However, there is significant scope for continued research in

this interesting discipline. We conclude this thesis by reviewing some of the possible

topics for future research.

6.2.1. Geophysical applications

In the realistic application of geophysical methods to mineral exploration, the

effects of electrical anisotropy and inhomogeneity must be considered. Studies into

these phenomena will be essential for the future interpretation of resistivity, induced

polarization, surface impedance, borehole induction, and transient and airborne

electromagnetic surveys. To this end, there are three main directions of possible future

research.

To able to fully understand the extent and range of electrical anisotropy in

different geological environments, particularly Australian, the first direction must be

targeted towards acquiring high quality petrophysical data from different rocks and

environments than is currently available. It is well known that all sedimentary rocks

exhibit a form of uniaxial anisotropy (though some biaxial anisotropy has been

observed) [6.5], and most minerals, at the crystalline scale, exhibit uniaxial, if not

biaxial anisotropy. Without this basic appreciation of the existence of electrical

anisotropy, future research and applications of more advanced interpretation methods

may end up to be futile.

The second direction of future research is the continued development of

efficient three-dimensional approximate forward modelling techniques that can

126

simulate the electromagnetic response of complex objects with arbitrary shape and

arbitrary anisotropy for any electromagnetic survey method. This is already an active

research area for magnetotelluric [6.10] and borehole induction logging problems

[6.16], but very little attention has been given to the transient electromagnetic

techniques, particularly airborne methods. To be truly useful to the geophysicist, it is

critical that the approximate forward modelling methods be able to capture the full

geological complexity of the earth, such as topography, induced polarisation effects as

captured in frequency-dependent complex conductivities, high conductivity contrasts

and non-ideal transmitter/receiver characteristics. Integral equations offer some

advantages over differential equation methods, but difficulty exists in being able to

construct (and solve) truly arbitrary and complex models with high conductivity

contrasts. Whilst finite-difference methods are conceptually simple to construct, they

cannot necessarily capture the entire geological complexity of the earth, such as

irregular boundaries. Modern finite-element methods [6.7] are able to capture the entire

geological complexity through curvi-linear cell structures, and probably offer the best

opportunity for realising the goals of developing a general and computationally

efficient approximate forward modelling method.

The third, and most industrially relevant direction of future research is the

development of three-dimensional inversion methods that can solve for inhomogeneous

anisotropic conductivity. Whilst this is currently an intense research area for borehole

induction logging problems, relatively little focus has been given to the inversion of

anisotropic parameters in other electromagnetic techniques. Currently, the majority of

inversion methods developed for surface and airborne techniques seek only those

inverse solutions with isotropic conductivity distributions. Ultimately, as we seek to

solve for realistic inverse models from survey data, it will be essential to have inverse

methods that can capture the necessary geological complexity that exists in reality. The

most challenging problem for inverting data with anisotropic and inhomogeneous

conductivity distributions will be the inherent non-uniqueness that exists, in addition to

the problems of instability and existence. To be able to extract such useful information

though, the ability to input sufficiently more geological a priori data into the inverse

models through different regularisation methods [6.15] will be critical.

127

6.2.2. Complex media

The study of microwave and optical communications has recently been subject

to rapid growth, in terms of both fundamental theory and industrial applications. New

techniques for guiding waves for communication purposes have been of significant

interest. This particular field of study has renewed studies in the broad discipline of

high frequency wave propagation in complex media, inclusive of anisotropic,

bianisotropic and chiral media, photonic crystals, photonic band-gap structures and so-

called metamaterials.

At microwave frequencies, there has been significant interest in the studies of

printed circuits (antennas and transmission lines) on anisotropic substrates [6.1, 6.3,

6.6]. At optical frequencies, there has been significant interest in the studies of glass

[6.2] and silicon-on-insulator [6.8] waveguides. Underlying this whole discipline are

the theoretical investigations into the electrodynamics of complex media [6.4].

Whilst this thesis has not concerned itself with studying these specific problems,

it is observed that there are similarities between these problems, and those geophysical

ones considered in this thesis. Possible future research efforts could include

investigations into integrated optical propagation problems, such as turning mirrors and

waveguides, in anisotropic silicon-on-insulator. Also, metamaterial-filled waveguides

for optical and microwave propagation will provide an interesting research topic.

Electromagnetic wave propagation in periodic and infinite structures (e.g., photonic

crystals) for optical applications has been studied for many years now. However, there

is difficulty in being able to manufacture such devices and correlating the experimental

results to theory. This will continue to be an area of active research. Other possible

research efforts could include further investigations of planar antennas and transmission

lines on anisotropic and bianisotropic substrates.

6.3. References

[6.1] V. Losada, R. R. Boix & F. Medina, IEEE Trans. Antennas Propagat. 49, 1603

(2001).

[6.2] C. J. Markman, Y. Ren, G. Genty & M. Kristensen, IEEE Photonics Tech. Lett.

14, 1294 (2002).

[6.3] F. Mesa, D. R. Jackson & M. J. Friere, IEEE Trans. Microwave Theory Tech.

50, 94 (2002).

128

[6.4] F. Olyslager & I. V. Lindell, IEEE Antennas Propagat. Mag. 44, 48 (2002).

[6.5] E. I. Parkhomenko, Electrical Properties of Rocks (Plenum Press, New York,

1967).

[6.6] R. Pregla, IEEE Trans. Microwave Theory Tech. 50, 1469 (2002).

[6.7] F. Sugeng, Explor. Geophys. 29, 615 (1998).

[6.8] Y. Z. Tang, W. H. Wang, T. Li & Y. L. Wang, IEEE Photonics Tech. Lett. 14,

68 (2002).

[6.9] D. V. Thiel & G. A. Wilson, presented at URSI Nat. Sci. Meet., Boulder, CO,

Jan. 2002.





[6.13] G. A. Wilson & D. V. Thiel, J. Electromagn. Waves Applic., submitted.



[6.15] M. S. Zhdanov, Geophysical Inversion Theory and Regularization Problems

(Elsevier, Amsterdam, 2002).

[6.16] M. S. Zhdanov, D. Kennedy & E. Peksen, Petrophys. 42, 118 (2001).

- The End -

129

Appendix 1

Co-ordinate rotations

�The angles ( ),, ψθϕ are called the Euler angles. Their definition varies widely

� the probability is small that two distinct authors� general rotation matrix will be the

same.� [A1.1]

Let us consider another Cartesian co-ordinate system that is represented by {x�,

y�, z�} where the y�z�-plane is inclined at an angle α about the x-axis, the x�z�-plane is

inclined at an angle β about the y-axis and the x�y�-plane is inclined at an angle θ about

the z-axis. We call this the inclined co-ordinate system (Figures 1.2-1.4). α, β and θ

represent the three elementary Euler angles about the x, y and z axes respectively,

which can be used to rotate one frame of reference to another. When 0==θβ , we

can write:

'xx = , (A1.1)

αα sin'cos' zyy −= , (A1.2)

αα cos'sin' zyz += , (A1.3)

which can be written in matrix notation as:

=

'''

)(zyx

zyx

αR , (A1.4)

where:

−=

ααααα

cossin0sincos0001

)(R . (A1.5)

130

Similarly, when 0== βα , we can write:

θθ sin'cos' yxx += , (A.16)

θθ cos'sin' yxy +−= , (A1.7)

'zz = , (A1.8)


=

'''

)(zyx

zyx

θR , (A1.9)

where:

−=

1000cossin0sincos

)( θθθθ

θR . (A1.10)

Similarly, when 0==θα , we can write:

ββ sin'cos' zxx −= , (A1.11)

'yy = , (A1.12)

ββ cos'sin' zxz += , (A1.13)


=

'''

)(zyx

zyx

βR , (A1.14)

131

where:

−=

ββ

βββ

cos0sin010

sin0cos)(R . (A1.15)

When 0≠α , 0≠β and 0≠θ , one can write:

.coscoscossinsin

sincossincossinsinsinsincoscoscossinsincoscossinsinsinsincoscossincoscos

)()()(),,(

−+−

−−−=

=

βαβαβαθβαθβαθαθβθβαθαθβαθαθβθ

αβθαβθ RRRR

(A1.16)

Note that we can also rotate from the fundamental co-ordinates into the inclined

co-ordinates analogous to equation (A1.16), but would have to rotate the fundamental

co-ordinates by the negative of the Euler angles.

Reference

[A1.1] F. W. Byron & R. W. Fuller, Mathematics of Classical and Quantum Physics

(Dover Publications, New York, 1992).

132

Appendix 2

Tensor rotation

We can write the field form of Ohm's Law in {x�, y�, z�} co-ordinates as:

' '�' EJ σ= , (A2.1)

where '�σ is the second rank conductivity tensor, and the primes denote quantities in the

inclined co-ordinate system. To write equation (A2.1) in the fundamental co-ordinate

system, we can write equation (A2.1) as [A2.1]:

' '� ),,(' ),,( ERJR σαβθαβθ = . (A2.2)

Since , then we can write equation (A2.2) as: IRR =),,( ),,(T αβθαβθ

ERRJ ),,('� ),,( T αβθσαβθ= , (A2.3)

or as:

EJ � σ= , (A2.4)

where , and where J and E are the current density and

electric field intensity, respectively, in the fundamental co-ordinate system. We note

here that

),,('� ),,(� T αβθσαβθσ RR=

),,( αβθR is an orthogonal matrix, i.e., R . ),,( ),,( -1T αβθαβθ R=

Reference

[A2.1] M. S. Zhdanov, D. Kennedy & E. Peksen, Petrophys. 42, 588 (2001).

133

Appendix 3

Solution to one-dimensional magnetic field coefficients

Consider the expression for the background magnetic field:

)exp()exp()(, zkBzkAzH mmmmb

mx +−= . (3.15)

At the air-half space interface for z = 0, we have:

mm BAH +=0 , (A3.1)

where is the (complex) amplitude of the magnetic field in free space. If 0H 0=bσ

then the vanishing magnetic field at z = h is equivalent to the statement that the top of

the basement is a perfect magnetic conductor. For z = h:

( ) ( hkBhkA mmmm expexp0 +−= ) . (A3.2)

From equation (A3.1), we can obtain the relations and

. Respectively substituting these into equation (A3.2), we then obtain the

solutions to the field equation coefficients above a perfectly insulating basement as:

mm AHB −= 0

mm BHA −= 0

( )( )hk

hkHA

m

mm sinh2

exp0= , (A3.3)

( )( )hk

hkHB

m

mm sinh2

exp0 −−= . (A3.4)

If ∞=bσ then at z = h which implies that 0, =myE 0, =∂

∂

=hz

mx

zH

which we can

write as:

( ) ( hkBhkA mmmm expexp0 −−= ) , (A3.5)

134

and equation (A3.1) still applies. Using the same relations as used before we then

obtain solutions to the field equation coefficients above a perfectly conducting

basement as:

( )( )hk

hkHA

m

mm cosh2

exp0= , (A3.6)

( )( )hk

hkHB

m

mm cosh2

exp0 −−= . (A3.7)

135

Appendix 4

Solutions to the Helmholtz equation

To solve Helmholtz equation (3.25) of the anomalous magnetic field in an

inclined uniaxial anisotropic medium rotated into the fundamental co-ordinates, we

must satisfy both equations:

,0sin)(

sin)(cossin

sin)(sincos

,

,2

22

,

2

,

2

2,

2

,

2

,

2

=

−

+−

∂∂

+

hznyfj

hznyf

hn

hzn

yyf

nm

nmmt

m

mn

m

nm

mt

m

mn

m

πωµ

ππσ

ασ

α

πσ

ασ

α

(A4.1)

and

. 0)(, =

∂∂

yyf nm (A4.2)

We will consider a solution of the form:

+

−=

±±

hyq

bh

yqayf nm

nmnm

nmnm,

,,

,, expexp)( , (A4.3)

where the ± superscript above correspond to the positive and negative roots of

to be determined. From equation (A4.3), we can observe that:

nmq ,

2,nmq

)()(

,2

2,

2,

2

yfh

qy

yfnm

nmnm =∂

∂. (A4.4)

Substituting equation (A4.4) into equation (A4.1) results in the equation:

136

,0sin)(

sin)(cossin

sin)(sincos

,

,2

22

,

2

,

2

,2

2,

,

2

,

2

=

−

+−

+

hznyfj

hznyf

hn

hznyf

hq

nm

nmmt

m

mn

m

nmnm

mt

m

mn

m

πωµ

ππσ

ασ

α

πσ

ασ

α

, (A4.5)

where the and )(, yf nm

hznπsin terms cancel, leaving us with the equation:

.0 cossin

sincos

2

22

,

2

,

2

2

2,

,

2

,

2

=−

+−

+ ωµπ

σα

σα

σα

σα

jh

nh

q

mt

m

mn

mnm

mt

m

mn

m (A4.6)

The coefficient q can be isolated from equation (A4.6) by first writing: nm,

,0sincos

cossinsincos

1

,

2

,

2

2

22

,

2

,

21

,

2

,

2

2

2,

=

+−

+

+−

−

−

mt

m

mn

m

mt

m

mn

m

mt

m

mn

mnm

j

hn

hq

σα

σαωµ

πσ

ασ

ασ

ασ

α

(A4.7)

where we identify the wave number of the medium:

1

,

2

,

22 sincos

−

+=

mt

m

mn

mm jk

σα

σαωµ , (A4.8)

and then re-arrange equation (A4.7) to the form of:

0cossinsincos 22

,

2

,

21

,

2

,

2222

, =−

+

+−

−

hknq mmt

m

mn

m

mt

m

mn

mnm σ

ασ

ασ

ασ

απ . (A4.9)

We can identify that:

137

mmnmmt

mmnmmt

mt

m

mn

m

mt

m

mn

m

ασασασασ

σα

σα

σα

σα

2,

2,

2,

2,

,

2

,

21

,

2

,

2

sincoscossincossinsincos

++

=

+

+

−

,

which simply states that:

yy

zzzzyy σ

σρρ =−1 ,

so we can then write equation (A4.9) as:

++

+±=±

mmnmmt

mmnmmtmnm nhkq

ασασασασ

π 2,

2,

2,

2,2222

, cossincossin

. (A4.10)

Now, we seek to demonstrate when equation (A4.3) satisfies equation (A4.2).

We have the derivative:

+

−−=

∂∂ ±±±±

hyq

bh

qh

yqa

hq

yyf nm

nmnmnm

nmnmnm ,

,,,

,,, expexp

)(, (A4.11)

and from equation (A4.2), we can then write:

−=

±±

hyq

ah

yqb nm

nmnm

nm,

,,

, expexp , (A4.12)

which is satisfied when we write:

−=

+−

hyq

ah

yqb nm

nmnm

nm,

,,

, expexp ,

where and are the negative and positive roots of respectively to

prevent exponentially divergent solutions, since it is later observed due to symmetry

conditions that a .

−nmq ,

+nmq ,

nm, =

2,nmq

nmb ,

138

Appendix 5

Solution to Fourier series coefficients

Let us consider the integral:

( )∫

−=

h

n dzh

znHHh

C0

12 sin2 π , (A5.1)

where ( ) ( )

−−

−=−

hkzhk

hkzhk

HHH1

1

2

2012 sinh

sinhsinh

sinh . We can write equation (A5.1)

as:

( ) ( )

−

−

−

= ∫∫ dzh

znhk

zhkdz

hzn

hkzhk

hH

Chh

nππ

0 1

1

0 2

20 sinsinh

sinhsin

sinhsinh2

. (A5.2)

Let us now consider the solution to the general integral:

( ) dzh

znkh

zhk

−

∫πsin

sinhsinh ,

which we will write as ( ) dzh

znzhkkh

−∫

πsinsinhsinh

1 . Let

and

( )zhku −= sinh

dzh

zndv

= πsin so that we can use the integration by parts method,

∫ ∫−= vduuvudv , to then obtain:

( ) ( )

( ) .coscoshsinh

cossinhsinh

sinsinhsinh

1

dzh

znzhkkhn

khh

znzhkkhn

hdzh

znzhkkh

−−

−−=

−

∫

∫π

π

ππ

π

(A5.3)

139

Now let u and ( zhk −= cosh ) dzh

zndv

= πcos so that we can use the

integration by parts method, ∫ ∫−uv=udv vdu , to then obtain:

( ) ( )

( )

( ) ,sinsinhsinh

sincoshsinh

cossinhsinh

sinsinhsinh

1

22

22

22

2

dzh

znzhkhn

hkh

znzhkkhn

khh

znzhkkhn

hdzh

znzhkkh

−−

−−

−−=

−

∫

∫

πγπ

ππ

ππ

π

(A5.4)

which we can then write as:

( )

( ) ( ) ,sincoshsinh

cossinhsinh

sinsinh1sinh

1

22

2

22

22

Ch

znzhkkhn

khh

znzhkkhn

h

dzh

znzhkn

hkkh

+

−−

−−

=

−

+ ∫

ππ

ππ

ππ

(A5.5)

where we introduce C as a constant of integration. Equation (A5.5) can then expressed

as:

( )

( ) ( ) ,sincoshsinh

cossinhsinh

sinsinhsinh

12

222

Ch

znzhkkh

kh

znzhkkhh

n

dzh

znzhkh

nkkh

+

−−

−−

=

−

+ ∫

πππ

ππ

(A5.6)

so then we have solved for the general integral as:

140

( ) ( )

( ) .sincoshsinh

cossinhsinh

sinsinhsinh

1

2

222

2

222

Ch

znzhk

hnkkh

k

hznzhk

hnkkhh

ndzh

znzhkkh

+

−

+

−

−

+

−=

−∫

ππ

ππ

ππ

(A5.7)

Now we apply the bounds between 0 and h to obtain the general definite

integral:

( )

+

=

−∫

2

2220

sinsinhsinh

1

hnkh

ndzh

znzhkkh

h

πππ . (A5.8)

We can now solve equation (A5.2) as:

( )

+

+

−−=

2

222

22

222

12

21

2202

hnk

hnkh

kknHCn ππ

π. (A5.9)

An identical solution is obtained for the integral:

( )∫

+−=

h

n dzh

znHHh

C0

12 2

)12(sin1 π , (A5.10)

which has the solution:

( )

++

++

−+−=

2

222

22

222

12

21

220

4)12(

4)12(4

)12(

hnk

hnkh

kknHCn ππ

π. (A5.11)

141

Appendix 6

Finite-difference operators

Consider an arbitrary function f (x, y) in a curvilinear co-ordinate system.

Consider a Taylor series expansion about the point (x, y) such that:

...!3)(

!2)(

!1)(

3

3

32

2

2

+∆+∆+∆+=∆+ xdx

fdxdx

fdxdxdffxxf , (A6.1)

...!3)(

!2)(

!1)(

3

3

32

2

2

+∆−∆+∆−=∆− xdx

fdxdx

fdxdxdffxxf , (A6.2)

...!3)(

!2)(

!1)(

3

3

32

2

2

+∆+∆+∆+=∆+ ydy

fdydy

fdydydffyyf , (A6.3)

...!3)(

!2)(

!1)(

3

3

32

2

2

+∆−∆+∆−=∆− ydy

fdydy

fdydydffyyf . (A6.4)

Subtracting equation (A6.2) from equation (A6.1), then we obtain the first order

derivative with respect to x:

3)(2

)()( xOx

xxfxxfdxdf ∆+

∆∆−−∆+= . (A6.5)

Subtracting equation (A6.4) from equation (A6.3), then we obtain the first order

derivative with respect to y:

3)(2

)()( yOy

yyfyyfdydf ∆+

∆∆−−∆+= . (A6.6)

Adding equations (A6.2) and (A6.1), then we obtain the second order derivative

with respect to x:

142

422

2

)()(

)()(2)( xOx

xxfxfxxfdx

fd ∆+∆

∆−+−∆+= . (A6.7)

Adding equations (A6.4) and (A6.3), then we obtain the second order derivative

with respect to y:

422

2

)()(

)()(2)( yOy

yyfyfyyfdy

fd ∆+∆

∆−+−∆+= . (A6.8)

If we neglect and higher order terms in equation (A6.5) and re-write it

as:

3)( xO ∆

xxxf

xxxf

dxdf

∆∆−−

∆∆+=

2)(

2)( , (A6.9)

then apply equation (A6.6), in which we have also neglected O , then we obtain

the mixed derivatives with respect to x and y:

3)( x∆

zyyyxxfyyxxfyyxxfyyxxf

zyf

∆∆∆−∆−+∆+∆−−∆−∆+−∆+∆+=

∂∂∂

4),(),(),(),(2

.

(A6.10)

143

Appendix 7

Inhomogeneous plane waves

Definition 1: A homogeneous plane wave is: a wave for which the planes of constant

amplitude and constant phase are parallel. Homogeneous plane waves are sometimes

called uniform plane waves [A7.1].

Definition 2: An inhomogeneous plane wave is: a wave for which the planes of

constant amplitude and planes of constant phase are not parallel. Sometimes called a

heterogeneous plane wave, but this use is depreciated [A7.1].

Consider a plane wave A incident upon a half space in orthogonal {x, y, z} co-ordinates,

at an arbitrary (complex) angle θ with respect to (from) the normal of the x-y plane.

Definition 3: The wave vector is: the complex vector kr

in expressions for wave

propagation using the exponential factor )](exp[ rkrr

⋅− j [A7.1].

For the plane wave, one can write:

}�'exp{ 'zkzjA −= , (A7.1)

where:

xx =' , (A7.2)

θθ sincos' zyy −= , (A7.3)

θθ cossin' zyz += . (A7.4)

Equation (A7.1) is a homogeneous plane wave in the ' -plane. We can re-write

equation (A7.1) in {x, y, z} co-ordinates as:

' yx

}�)cossin(exp{ 'zk θθ zyjA +−= ,

144

which we will re-write as:

}exp{}exp{ })cos(exp{})sin(exp{

zjkyjkzkjykjA

zy −−=−−= θθ

(A7.5)

where θsink=yk and θcosk=zk are the vector components of k. Equation (7.5) is

an inhomogeneous plane wave in the xy-plane. By virtue of equation (7.5), in the xy-

plane, the plane of constant amplitude is not parallel to the plane of constant phase. It is

specifically noted that equation (A7.5) is of the form of the Zenneck surface wave

[A7.2].

Remark: At the surface of a half space, an inhomogeneous plane wave can sometimes

be defined by a homogeneous plane wave at oblique incidence upon the surface of the

half space.

References

[A7.1] IEEE Std 211-1997TM, IEEE Standard Definitions of Terms for Radio Wave

Propagation.

[A7.2] J. R. Wait, IEEE Antennas Propagat. Mag. 40, 7 (1998).

Plane Wave Propagation Problems in Electrically ... · Plane Wave Propagation Problems in...

Documents

Transcript of Plane Wave Propagation Problems in Electrically ... · Plane Wave Propagation Problems in...