Analysis of electromagnetic prospecting data by means of apparent wave numbers

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ANALYSIS OF ELECTROMAGNETIC

PROSPECTING DATA BY MEANS OF

APPARENT WAVE NUMBERS

1.

Het verloop van de ondergrens van een permafrost struktuur kan, onder bepaalde voorwaarden, het verloop van diepere strukturen weerspiegelen. Deze ondergrens zal beter kunnen worden bepaald met een elektromag-netische methode dan met de weerstandsmethode zoals dat door Baulin en Ostry is voorgesteld.

2.

De voortplantingsnelheden die gevonden worden voor diepere lagen bij de verwerking van seismische signalen kunnen, bij de aanwezigheid van ondiepe stnikturen, grote afwijkingen vertonen met de werkelijk a a n -wezige snelheden.

3.

De komponenten van de electromagnetische veldsterkte voor een t w e e -lagen struktuur met een dxpool als potentiaalbron worden door Keiler en Frischknecht met verschillende niet juiste tussenstappen berekend.

Keiler, G.V. , Frischknecht, F . C .

Electrical methods in Geophysical Prospecting. Pergamon Press, 1966,

4.

Een akoestisch signaal in water, van korte tijdsduur, kan slechts relatief weinig lage frekwenties bevatten.

5.

In een isotroop kontinuum, w a a r d e deformatie uitsluitend bepaald wordt door de spanningstoestand, hoeven de richtingen van maximale kompressie en van maximale verkorting niet samen te vallen, (Deformatie en span-ning worden ieder door een tensor van de tweede orde voorgesteld.)

6.

Het kan bewezen worden dat het in het algemeen niet mogelijk is om met een elektromagnetische of elektrische geofysische methode opeen-hopingen van olie of gas in de ondergrond rechtstreeks aan te tonen.

7,

Zowel in het gewone als in het wetenschappelijke spraakgebruik heeft het woord 'toeval' meerdere betekenissen.

Het verdient derhalve aanbeveling om in voorkomende gevallen waar misverstand kan optreden omtrent de bedoelde betekenis, dit woord 6t niet te gebruiken df te voorzien van een toelichting.

8.

Bij gebruik van de elektromagnetische dipoolmethode voor strtiktuur-onderzoek ligt de grootste gevoeligheid voor de dikte van de bovenlaag van een tweelagen struktuur niet altijd bij een zender-ontvanger afstand die binnen de grenzen van twee tot vijf maal deze laagdikte ligt, zo-als doorgaans wordt aangenomen.

9.

Het op een ludieke wijze nastreven van een alternatieve maatschappij, waarin de vrouw gelijke rechten zou moeten hebben als die, welke de man reeds heeft verworven, wordt eerder geschaad dan gediend door de naamkeiue van de aktie Dolle Mina.

ANALYSIS OF ELECTROMAGNETIC

PROSPECTING DATA BY MEANS OF

APPARENT WAVE NUMBERS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR.IR. C.J.D.M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OP DONDERDAG 9 JULI 1970 TE 14.00 UUR

6^ /oié>

19

DOOR

PETRUS JOHANNES MARIA THOMEER

doctorandus in de wis-en natuurkunde geboren te 's-Gravenhage

BI/LIOIHEEK

TECHNlSCHb HÜGtSCHOC

DELFT

1970 DRUKKERIJ BRONDER-OFFSET N.V. ROTTERDAM

Dit proefschrift is goedgekeurd door de promotor PROF. O. KOEFOED.

Aan mijn ouders

Aan mijn vrouw

The results presented in this thesis are based on an investigation carried out at the Konlnklijke/Shell Exploration and Production Laboratory, Rijswijk, and the author is grateful to Shell Research N. V. for the permission to publish them, He wishes to express his sincere gratitude to Dr. W. L. Scheen, head of the geophysical department, for his helpful comments.

He also appreciates the work done by Mrs. S.M. Germans in the prepara-tion of the manuscript.

-CONTENTS

Page

List of Figures 7 List of Symbols 9 Samenvatting (Dutch Summary) 11

Summary 12 I. Introduction 13

II. Analytical computation of the electromagnetic field 16 generated by a dipole source

A. The Maxwellian laws 16 B. The vector potential 18 C. The boundary conditions 20 D. The solution of the Poisson equation 24

1. Homogeneous half space 26 2. Two-layer case 30 3. General multilayer case 33

HI. The apparent wave number 35 A. Definition - 35 B. Determination of the apparent wave number 36

C. Some examples 40 IV. Behaviour of the apparent wave number for a < a„ 41

A. Area of reliability 41 B. Behaviour of the magnetic field strength for large 46

contrasts

V. Behaviour of the apparent wave number for a > a. 47

A. Area of reliability 48 B. Determination of a relation for the apparent 48

wave number

VI. Analysis of electromagnetic prospecting data 50 A. Optimisation of survey parameters 50

1. Frequency sounding 53 2. Distance sounding 55 3. Combined frequency and distance sounding 58

B. Conductivity profiling 58 C. Effect of e r r o r s in the measurement of the magnetic 59

field strength on the interpretation

D. Use of the apparent wave number in computerised 62 interpretation programs

Page

v n . Conclusions 64 References 67 Figures 1-37 73 Appendix I : The electrical properties of subsurface rocks H I

Appendix H: Some considerations on the source field 113

-LIST OF FIGURES

Page

12 Apparent wave number as function of pk. and S for D = 0.70 13 Apparent wave number as function of pk, and S for D = 0.80

Fig. 1 Components of the electromagnetic field strength over a 73 homogeneous halfspace

2 Magnetic field strength (z-component) over a two-layer 74 subsurface

3 Electric field strength (qj-component) over a two-layer 75 subsurface

4 Relation between the apparent wave number and the magnetic 76 field strength (z-component)

5 Apparent wave number as function of pk. and S for D = 0.10 77 6 Apparent wave number as function of pk. and S for D = 0.20 78 7 -Apparent wave number as function of pk. and S for D = 0.25 79 8 Apparent wave number as function of pk. and S for D = 0.30 80 9 Apparent wave number as function of pk. and S for D = 0.40 81 10 Apparent wave number as function of pk. and S for D = 0.50 82 11 Apparent wave number as function of pk. and S for D = 0.60 83

84 85 14 Apparent wave nimiber as function of pk and S for D = 0.90 86 15 True values of pk as function of pk, and S for D = 0.50 87

a X

16 True values of pk as function of pk. and D for S = 100 88 17 pk as function of S and pk. according to relation (49) for 89

D^= 0.50

18 True values of pk f or S < 1 as function of fik for D = 0.125 90 19 True values of pk for S < 1 as function of pk, for D = 0.25 91

a ' ^ 1 20 True values of pk for S < 1 as function of pk, for D = 0.50 92

a, J.

21 I k A-, I as function of D and pk, for S < 1 93 a i. X

22 Sensitivity to D of magnetic field strength as function of S and 94 2tjkj^ for D = 0.0625 and D = 0.125

23 Sensitivity to D of magnetic field strength as function of S and 95 2tjkj^ for D = 0.25 and D = 0.375

24 Sensitivity to D of magnetic field strength as function of S and 96 2tjk^ for D = 0.50 and D = 0.75

25 Sensitivity to D of magnetic field strength as function of S and 97 D for 2t^kj^ = 0.4(1+1) and 2tj^kj^ = 0.8(1+1)

26 Sensitivity to D of magnetic field strength as fimction of S and 98 D for 2tj^k = 1.2(1+1) and 2tj^k = 1.8(l+i)

27 Sensitivity to D of magnetic field strength as function of S and 99 D for 2t^\ = 2.4(l+i) and 2tj^k = 3.6(1+1)

-28 Sensitivity t o D of m a g n e t i c field s t r e n g t h a s function of S and 100 D f o r pkj^ = 2 . 4 ( l + i )

29 Sensitivity t o D of m a g n e t i c field s t r e n g t h a s function of S and 101 D for pkj^ = 3.6(l+i)

30 Sensitivity t o D of m a g n e t i c field s t r e n g t h a s function of S and 102 D f o r pk^ = 5.0(1+1)

31 Sensitivity t o D of m a g n e t i c field s t r e n g t h a s function of S and 103 D f o r pkj = 7.2(1+1)

32 Sensitivity t o D of m a g n e t i c field s t r e n g t h a s function of S and 104 D f o r pk^ = 10.0(l+i)

33 Contour m a p showing l i n e s of equal s e n s i t i v i t y t o D a s function 105 of D and 2t^k for S = 2

34 Contour m a p showing l i n e s of equal s e n s i t i v i t y to D a s function 106 of D and 2tj^k^ f o r S = 10

35 Contour m a p showing l i n e s of equal s e n s i t i v i t y t o D a s fimction 107 of D and 2tj^k for S = 100

36 Relation between r e l a t i v e e r r o r s in t h e field p a r a m e t e r s due to 108 an e r r o r in t h e m e a s u r e m e n t of t h e magnetic field s t r e n g t h f o r

o p t i m a l f r e q u e n c y - s o u n d i n g frequency

37 Relation between r e l a t i v e e r r o r s in t h e field p a r a m e t e r s due t o 109 an e r r o r in t h e m e a s u r e m e n t of the m a g n e t i c field s t r e n g t h for

LIST OF SYMBOLS 1. Latin symbols

A arbitrary vector (only used on page 6) A matrix

a variable, used in Laplace transform a. arbitrary function of x

B magnetic induction (vector) b. arbitrary function of x

c velocity of light in free space c. arbitrary function of x

D electric displacement (only used on page 5) D depth ratio, equal to 2t./p

d radius of source loop d. arbitrary function of x

E electric field intensity (vector)

E , relative electric field intensity (vector); E , = 1 in free space f frequency

g function of x and k

H magnetic field intensity (vector)

H , relative magnetic field intensity (vector); H , = 1 in free space h height of source above structure

I current intensity (vector)

I modified Bessel function of first kind and n order _n i symbol denoting the complex part of a number J electric current density (vector)

J „ Bessel function of first klad and zero order t i l

K modified Bessel function of third kind and n order n

k. wave number of medium j k apparent wave number 1 element of length

M magnetic dipole moment of source loop (vector) m magnitude of M

N number of layers

O. order of magnitude of residual part in expansion j S conductivity contrast, equal to CJo^'^i

s variable, used in Laplace transform

-t variable, used in Laplace -transform (only on page 16) t time

t. thickness of layer j , . T. depth from surface to the interface j—>j+l, equal to T, t

J i=l V integration variable (volume)

y vector

Y_ Bessel function of second kind and zero order z vertical component of cylindrical co-ordinate system 2. Greek symbols

a arbitrary function of S and pk^ 8 arbitrary function of S

V arbitrary function of S 6 arbitrary function of S e permittivity e(x) error- in the variable x X wavelength

H. magnetic permeability of medium j n vector potential (vector)

p electric charge density (only used on page 5) p radial component of cylindrical co-ordinate system

a. electric conductivity of medium j

cp angular component of cylindrical co-ordinate system cp phase angle of apparent wave number

a V. function of x and k. J ] v. • function of x, k. and k. 1 J ' 1 ] UJ circular frequency 3. Vector operators V gradient Vx rotation V* divergence

V Laplace operator, equal to 7" ^

SAMENVATTING (DUTCH SUMMARY)

Een van de vele methoden om gegevens t e verzamelen over structuren onder het aardoppervlak is die, waarbij de verstoring van aangelegde electro-magnetische velden door die ondergrondse structuren, gemeten wordt. Deze verstoringen kan men zich denken tot stand te komen door de invloed van de electrische eigenschappen van ondergrondse gesteenten.

Voor een homogene ondergrond kunnen deze electrische eigenschappen worden uitgedrukt in één physische grootheid: het golfgetal. Berekening van de verstoring van electromagnetische velden is in dit geval betrekkelijk eenvoudig uit te voeren.

In het geval van een inhomogene ondergrondse structuur, zoals bij een meerlagen structuur, kan niet meer worden volstaan met één golfgetal om de electrische eigenschappen in uit te drukken, maar zullen meerdere golfgetallen noodzakelijk zijn (voor elke homogene laag één). De berekening van de versto-ring die een aangelegd electromagnetisch veld van zo'n structuur ondervindt is veel gecompliceerder dan in het geval van een homogene ondergrond. Bovendien blijkt het in de praktijk een zeer tijdrovende en daardoor vaak ook niet meer mogelijke zaak om gegevens, verkregen uit metingen van verstoringen, te ana-lyseren.

Hieraan wordt in dit proefschrift tegemoet gekomen door aan een dergelijk meerlagen structuur toch één golfgetal toe te kennen: het schijnbare golfgetal. Het blijkt mogelijk om hiervoor een benaderende uitdrukking af te leiden, welke benadering voor de meeste, in de praktijk voorkomende gevallen voldoende nauw-keurig is om het gedrag van een meerlagen structuur mee te beschrijven.

In het geval van een tweelagen structuur wordt dit schijnbare golfgetal gebruikt om de componenten van de electromagnetische veldsterkte te berekenen. Bovendien wordt een analyse gemaakt van de invloed van de structuur, te weten de electrische eigenschappen van beide lagen en de dikte van de bovenste laag, op de grootte van de gemeten veldsterkte. Het blijkt dan mogelijk te zijn om, door een juiste keuze van de frequentie van het aangelegde veld en/of de af-stand tussen zender en ontvanger, te komen tot een optimale gevoeli^eid voor de dikte van de bovenste laag.

Tot slot wordt ook de doorwerking van kleine meetfouten op de interpreta-tie van metingen onderzocht.

-SUMMARY

One of the many methods that can be used to gather information about subsurface structures is that in which the disturbing influence of subsurface structures on applied electromagnetic fields is measured. These disturbing influences a r e caused by the electrical properties of subsurface rocks.

In the case of a homogeneous subsurface, these electrical properties can be confined to one physical characteristic - the wave number. The disturbance of the electromagnetic fields can be computed in this case in a relatively simple manner.

In the case of an inhomogeneous subsurface, such as a multilayer structure, the electrical properties can no longer be expressed in terms of one wave

number, but more are needed (one for each homogeneous layer). The computation of the disturbing influence of such a structure on an applied electromagnetic field is much more complicated than in the case of a homogeneous subsurface. Furthermore, a practicable analysis of data obtained from measurements of these disturbances appears to be a most time-consuming, and therefore in many cases impossible, task.

To overcome these difficulties, in this thesis we have nevertheless assigned one wave number to such a multilayer structure: the apparent wave number. It appears possible to deduce an approximated expression for this apparent wave number which In most of the cases that occur in practice is sufficiently accurate to describe the behaviour of a multilayer structure.

This apparent wave number is used to compute the components of the electromagnetic field strength for a two-layer structure. Furthermore, the influence of the structure - in terms of the electrical properties of both layers and the thickness of the upper layer - on the magnitude of the measured field strength has been analysed.

It appears to be possible to optimise the sensitivity for a particular thickness of the upper layer by choosing the most appropriate frequency and/or source-receiver separation.

Finally, the influence of small e r r o r s in the measurements on the inter-pretation of the latter is investigated.

-ANALYSIS OF ELECTROMAGNETIC PROSPECTING DATA BY MEANS OF APPARENT WAVE NUMBERS

I. INTRODUCTION

The investigation of the subsurface by means of a method that is used on or above the surface and measures changes in physical characteristics of that subsurface may be defined as the typical aim of geophysical work. There are various physical characteristics that one can measure, such as the velocity of acoustic waves in rock, gravity, magnetic permeability, heat and electrical conductivity, etc. Each geophysical method is related to one or more of these characteristics. One of the geophysical techniques that is of special interest for mining exploration is the group of methods combined under the general name

[21 43]

'electrical and electromagnetic prospecting methods' ' . They are used to investigate the electrical properties of subsurface rocks: their conductivity o, their magnetic permeability M and their permittivity e.

Whereas the electrical methods employ direct or very low frequency currents as a source of energy and use electrodes both to force these currents to flow in the ground and to measure the subsequent potential differences, the electromagnetic methods normally use higher frequency currents (sub-audio frequency range) and do not necessarily require the planting of electrodes in the ground. The latter is an elaborate and time-consimiing procedure.

The electromagnetic dipole induction method considered in this work employs two loops of wire: one, used as source coil, c a r r i e s a current of a distinct circular frequency iju, thus giving rise to a magnetic Induction field B; the other, the receiver coil, is located at some distance p from the source coll and measures the radial or (depending on its orientation) the vertical magnetic induction at that location. For a sequence of N layers with horizontal interfaces in which each layer has distinct electrical properties, the components of the electromagnetic field strengths depend on the following parameters:

a. the survey parameters; namely the source-receiver separation and the sounding frequency.

b . the field parameters; namely the electrical and geometrical properties of each of the N layers (the N layer is supposed to have semi-infinite thickness).

-Maxwell's laws can be applied to this system to determine the electromagnetic rog]

field components. As early as 1909 Sonmierfeld gave the solution for the special case in which N = 1, the homogeneous halfspace. He computed the components of the electromagnetic field strengths, which, when both the source and receiver were located on the surface, appeared to consist of a finite series of elementary functions.

Much later, In 1951, Wait ' solved the Poisson differential equation (which follows from the laws of Maxwell), from which the field components can be derived by simple differentiation rules, for the general N-layer case. The solution he obtained for the potential distribution of a given subsurface configu-ration is expressed in the form of an infinite integral. This integral cannot be expressed as a finite series of elementary functions (except, of course, for N = 1). Therefore, to either interpret or predict field data, it is necessary to compute this solution integral numerically for all possible cases, so obtaining whole sets of so-called 'master curves'. This is the procedure followed by

fis] Frischknecht in 1967 for two-layer cases .

In principle, these curves might be used in the field to obtain a direct interpretation of measurements, but this process will in general be too time-consuming to be of effective use. Furthermore, it is Important, and in many cases even necessary, to equip the field operator with a criterion that he can use to obtain the set of survey parameters that results in the highest sensitivity of his device for the (or those) parameter(s) one is the most interested in. Finally, it might also be of interest to know the effect of small e r r o r s in the measurements on the interpretation of the data. These possible e r r o r s can either be systematic, such as incorrect orientation of source and/or receiver coil, or may be due to natural and industrial noise, misreadings of the systems output, etc.

To satisfy these needs (which to a greater or lesser extent are present in all geophysical prospecting methods), one can try to approximate the solution integral by expansion of its integrand. In general, this approximation will not give physical insight, as it is a purely mathematical approximation. The approximation can, however, be made both imaginative and meaningful by introducing apparent or effective electrical properties to the multilayer system.

With a more proper definition, this means that those apparent electrical properties are assigned to an N-layer system (and given survey parameters) which, when inserted in the solution of the homogeneous halfspace, result in the same magnitudes for the electromagnetic field strengths as obtained over the N-layer system.

-In most practical cases the permeability appears to be the same as that in free space and since the frequency range we will employ is so low that displacement currents can be neglected in comparison with the conductivity currents that we will encounter in the subsurface rocks, the only electrical properties that we

FS 21 43] have to deal with a r e the conductivities ' '

These (together with the sounding frequency) determine a wave number for each separate layer. The apparent wave number will then be a function of all these separate wave numbers, the thicknesses of the layers and the source-receiver separation.

For a multilayer system with arbitrary interfaces, the thus 4efined apparent wave number will generally be so complicated that its behaviour can only be found by means of model experiments for subsurface structures. For horizontally layered media, however, we will give an approximated relation for the apparent wave number with which the behaviour of the electromagnetic field components can be satisfactorily described. The relation is derived analytically from the solution integral by comparing the N-layer system with the homogeneous halfspace.

Simple expressions a r e then found for the field strengths in the two-layer case, from which the following analyses can be made:

a. Optimisation of survey parameters: the system can be made most sensitive for one specific field parameter - which in most cases is the thickness of the upper layer - by choosing the most appropriate sounding frequency and/or source-receiver separation.

b . First estimate of the influence of changes in the field parameters: in conductivity profiling, a change in the field strength(s) can be translated directly into a numerical change in one (or more) of the field para-m e t e r s .

c. First estimate of the magnitudes of the field strengths: the field strengths can be computed simply using the apparent wave nimiber. d. Influence of e r r o r s in field data on interpretation: an idea can be

gained of the effect of e r r o r s in field data on the interpretation of these data into field parameters.

By considering an N-layered structure as being built up of subsequent two-layered structures - which can, in principle, be effected by choosing the survey parameters in the appropriate way - this same analysis can be used ' However, it will be clear that one has to be very careful in doing this, because the relation for the apparent wave number is only approximate, and there is thus the danger of adding up e r r o r s in interpretation for each new pair of two-layered structures.

-In chapter n , the components of the electromagnetic field strengths a r e computed from the vector potential fields. These potential fields are computed by solving the Poisson equation for each of the N-layers and then inserting the boundary conditions for each interface.

In the following chapter the apparent wave number is introduced, and a relation is derived for the two-layer case that expresses this wave number in terms of the parameters of the upper and lower layers.

In chapter IV, this expression is tested against exact computed values for the apparent wave number for the case in which the lower* layer is more con-ductive than the upper layer. Curves are given that can be used in field surveys to obtain a direct (approximated) solution for the field parameters sought.

In chapter V, the expression for the apparent wave number is tested for the situation in which the lower layer is less conductive than the upper layer. As it appears that this expression has only a small area of reliability in this case, another relation is derived experimentally which covers a much larger area of reliability. Curves can again be made for a direct, approximative interpretation in field surveys.

Chapter VI gives an extensive analysis of the behaviour of electromagnetic prospecting data obtained with the dipole Induction method. A means of

optimising the sensitivity for one of the field parameters is derived and com-puted for various cases, which include almost all those that could possibly occur in the field.

Finally, the influence of e r r o r s in field data on the interpretation is discussed.

n. ANALYTICAL COMPUTATION OF THE E.M. FIELD GENERATED BY A

DIPOLE SOURCE

Using the Maxwellian laws, the Poisson equation is solved with the help of the vector potential. After mserttng the boundary conditions for multilayer structures, the potential and the e . m . field components are computed. A. The Maxwellian laws

In their basic form, Maxwell's equations can be written as:

-VxH = ^ + J ot

V x E ^ - f

(1) ' 7 . B = O V . D = p w h e r e : H = m a g n e t i c field i n t e n s i t y E = e l e c t r i c field i n t e n s i t y B = m a g n e t i c induction D = e l e c t r i c d i s p l a c e m e n t J = e l e c t r i c c u r r e n t d e n s i t y p = e l e c t r i c c h a r g e d e n s i t y

In homogeneous and i s o t r o p i c m e d i a , we a l s o have t h e following r e l a t i o n s :

J = a . E (2.1) D = e . E • . (2.2) B = M.H (2.3) w h e r e : a = e l e c t r i c conductivity e = p e r m i t t i v i t y H = m a g n e t i c p e r m e a b i l i t y

A s we will only c o n s i d e r h a r m o n i c a l l y t i m e v a r i a n t f i e l d s with c i r c u l a r f r e q u e n c y 0), e q u a t i o n s (1) b e c o m e (no s o u r c e s ) : VxH = (iuue + D ) . E (3.1) VxE = -iuuu.H (3.2) V . B = 0 (3.3) V . D = 0 (3.4) 17

-B. The vector potential

From equation (3.4), with the aid of Gauss's differential rule, it follows that the electric field can be written as the curl of an arbitrary vector A:

E = V x A (4) It is convenient to define A as follows:

A = -i(D|i.n (5) where TT is called the vector potential. Using equations (3.1) and (4), it now

follows that the electromagnetic field components can be expressed in terms of this vector, as:

E = -iuJia.VxTT (6.1) and

-icu^.H = Vx(-iu)li.VxTT)

H = 7 7 . TT - V TT (6.2) Taking the curl of both sides of the latter equation, we have:

7xH = VXV7. TT - Vx^TT and according to Stokes' rule:

VX7(V. TT) = 0 Thus with (3.1) and (6.1):

(iuue + a)(-iu)|a.yxTT) = -VXV^TT

or

V n = (-U) ne + iii)taa)TT (7)

-By definition

2 2

k. = iuuu.a. - u) la.e. (8) and hence we obtain the Poisson equation for TT:

V^n. = k^TT. (9) J J J

where the suffix j denotes the medium for which we will solve the potential problem.

Before solving equation (9), something can be said about the k . ' s . They have two components: iiui-ia, which is related to conduction currents, and

2

0) (ie, which is related to displacement currents. In this dissertation we will restrict ourselves to the situation in which the displacement currents can be neglected with respect to the conduction currents. This means that:

I iu)i_ia| » ID (ie

a s> IDS

(10) H = arbitrary

We have introduced this restriction on the basis of the following considerations: 1. The values for a encountered in geophysical surveys lie in the range

—5 8

10 mho/m < a < 10 mho/m (see Appendix I) 2. The values for e encountered in geophysical survey lie in the range 7.2 10"^°F/m > e > 8.8510"^^F/m (see Appendix I) Taking the minimum value for a and the maximum value for e, the frequency f has to satisfy the condition

f « 2.200 Hz (11) which means that we will restrict ourselves to the sub-audio frequencies.

-C. The boundary conditions

Let us consider a multilayer system of media with the following features: 1. The interfaces are horizontal.

2. Each layer j consists of a homogeneous isotropic medium, with thick-ness t..

i

3. Each layer j has a different conductivity with respect to the following and preceding layer: c!._^ * a. *

<^-j.i-4. Let there be N layers; then tvr ~ " •

A magnetic dipole source is introduced at a h e i ^ t h above the first layer, as shown in the figure below. The source itself consists of a loop of wire, the plane of which is parallel to the surface of the first layer, and it carries a

. T iujt current Ie

.. . _^

| z = -h —) U^ a = 0 ^ ^ p-axis ^^ a^ ^2 °2 h z-axis r 1^3 '^3 *3

^^N-1

" N - 1 > ^'N °N *N-1 ^ N -The magnetic dipole moment M of this loop we will define as the product of the area of the current-carrying loop and the magnitude of the current:

1M| = |l|.TT.d^ (12.1) where d is the radius of the circular loop. M is directed parallel to the

direction of plane of the loop. Under certain conditions, the potential field of a current carrying loop is given by (see also Appendix H):

--M

TT = % 1 (12.2) [ p + ( Z + h ) ^ ] 2

Consider the cylindrical co-ordinate system (z, p, co) as sketched in the previous figure. It can easily be seen that the system is symmetric about the z-axis. We are therefore, justified in stating that the vector potential only has a com-ponent in the z direction. In the following we will therefore omit the subscript

TT = TT^ ( 1 3 )

The boundary conditions for the system described are as follows:

1. As the field energy is limited, the components must reach zero at Infinity: Lim E = 0 2 . L i m E = 0 Z >+oo L i m H = 0 p —»<» L i m H = 0 Z — > -H» From (3.3), we have V.B = 0 (14.1)

Taking the volume integral over a box,

as shown,

f|Jv.Bdv = 0

volume

or, with Gauss's theorem: )(B.da) = 0

area

-and hence (as t approaches zero):

(M.H^). = (M.H^).^^ (14.2) at the interface j —» j+1, for j = 0, 1, 2 , N - 1 .

3. The same condition can be found for the electric displacement, with the help of (3.4),

(c.E^). = (e.E^).^^ (14.3) at the interface j —> j+1, for j = 0, 1, 2 N - 1 .

4. Taking the Integral of (3.1) over an area as shown, i — ^

^rEl3ij4

jJ(VxH).da = aJJ(E.da) we obtain: ^ 1 L-»-—1^1 j+1 - d t area and thus:

^(H.dt) = aJJ(E.da)

As I approaches zero, the area on the r i ^ t - h a n d side vanishes, and we obtain the condition:

(H,). = (H^).^, (14.4) at the interface j —» j+1, for j = 0, 1, 2 N - 1 .

5. The same condition is valid for the electric field. Using (3.2),

(Et)j = ( E p j , , (14.5) at the interface j —> j+1, for j = 0, 1, 2 , N - 1 .

Boundary conditions 1 to 5 can be translated into conditions applicable to the Hertzian vector:

1. Lim TT. = 0 (15.1)

p—»=> ^

2 . L i m TT. = O (15.2) Z > +00 ^

'• ^j"j = ^^j+l>l <^^-2)

M S ) j = (g)j+x <--^)

which must be fulfilled for all j = O, 1, 2, N-1 at the interface j — » j + l . Proof

Conditions (15.1) and (15.2) follow directly from (14.1) and (6). Conditions (15.3) and (15.4) follow from conditions (14.2) to (14.5), which state that E , E., B and H. a r e continuous at each interface.

t n t

From (6.1) we see that E = -iuun.vxrr

/ ' I STT 3TT . \

and because the system is cylindrically symmetrical about the z-axis, the potential vector TT must be independent of cp:

3cp thus ^ = 0 (16)

E = iiuia IT- (17.1)

op

From (6.2) ^ (^ TT 1 9^TT _ i arr M i - ^-l] \3p3z' p scpsz' p 3p P 3p " 2 2 /

p" w

and because of (16) .2 H = (^, 0, - i | I I p M ^ (17.2) \ 3 p 3 z ' ' p 9p ^ apy

Conditions (14.2) to (14.5) become, when expressed in t e r m s of TT:

-("•S)l={<)

j+1 (18.1)

(in.) =(A.)

vapaz/j V^paz^j+i

r fi% _^ 1 aTT\i r/^^TT ^ 1 ^y\

(18.2)

(18.3)

for all j = 1, 2 , . . . . , N-1 at the interface j—>j+l. Integration with respect to p gives:

R

I©j'^''=V^^-"3<''o)

and letting R tend to infinity, we obtain, with (15.1): R

Lim I — ) . dp = -TT.(p )

and thus:

R

R ^ l I [(^ir)j+i" (^ 0)j]''p = ^•^<°o)]j - ^•"(Po)] j+1

According to (18.1) the integrand vanishes, and we thus obtain: (^i.n) = (M.n)

I

which is the same as (15.3). The same procedure can be followed with (18.2) and (18.3), resulting in conditions (15.4) and (15.3).

D. The solution of the Poisson equation

The potential equation (9), in cylindrical co-ordinates for the cylindrically symmetric system, reads:

1 a a - ^ P — n . + p ap ao ]

a^n.

o z k ^ . T T . J J (19) 24

To solve this equation, we will try a solution that can be written as:

TT.(p,z) = R.(p).Z.(z) (20)

Inserting this in (19), we obtain:

/ , dR. d^R.x d^Z. „ ^P ^P dp2 ^ J dz^ J J J J In this equation the variables p and z can be separated:

2 d^ï^i '^^i 2 2

P —2^ + P V ^ ^ '^ ^j " ° ^^^^

and ^ ^ ^ i 2 2 5I - (k; + x^) . Z = 0 (22) dz'^ ^ ^

wherc'x is an arbitrary number, independent of p and z. Equation (21) is a Bessel equation, and its solution is given by:

R.(p) = c.(x).J^(px) + d.(x).Y^(px) (23)

where J (px) is a Bessel function of zero order and first kind and Y (px) is a _{o o} Bessel function of zero order and second kind. The functions c. and d. a r e

arbitrary functions of x, determined by the boundary conditions (15). The solution of (22) can be written as:

zVx +k. -zvx +k.

Z.(z) = a.(x).e J + b.(x).e ^ (24) where a. and b. are arbitrary functions of x, determined by the boundary

con-ditions (15) and the character of the source.

It can easily be seen that the general solution of (19) is now given by:

/ 2 ^ , 2 " / 2 ^ , 2'

ZV X • K. ^ZV X "•" iC:-i

TTj(p,z) = j[c^(x)J^(px) + d.(x)Y^j(px)].[aj(x)e ^ + b . ( x ) e ^ J d x

For convenience, and in accordance with (15.1), we will set all d.(x) equal to zero [since Y^(px) becomes infinitely large for p—»»] and insert c.(x) in a.(x) and b.(x), thus giving:

r z^/x^ + k^' n.(p,z) = J J^(px).[a.(x)e ^ + b.(x)e

z^x^+kf^

J ] d x

(26) 1. Case iL_Honi°êeneous_halfspace, _N_= _1_ z = -h U^ a = 0

The potential vector for the halfspace z < 0 is constructed from the potential of the source field (12), the so-called primary vector field and the secondary field as given by equation (26). Thus:

z < 0 n =

° (p2 + (z+h)2)2 r + J J^(oX) [a^ e^^ + b^ e ' ^ ' ' ] dx (27.1)

0 < z

n ^ = J j ^ ( o x ) [ a ^

/ 2 ^ , 2 zvx + k , / 2 , 2 ' zVx + k . - z ^ x + k , . ,

' - ^Jdx

+ b e

(27.2)

As for large P the function J^(px) (:) ~7=^, the boundary condition (15.1) is

/px

already fulfilled.

Condition (15.2) results in: b^ = 0

a^ = 0

(28.1) (28.2)

-Condition (15.3) results in; -M„m

, 2 TTi ^ ^ I V ^ ) [ ^ + boJ*' = ^1 I Jo<P'')[^ -^ \]^

<P ^ ^ ) o

M

2 2

p + h ) is the Laplace transform of J (px) with respect to h, this can be written as:

^o- { \ + \ - m e " ^ = ^^ . (a^ + b^) (28.3) Condition (15.4) can be treated in the same way, resulting in:

( m e " ' ^ + a^ - b^). X = (a^ - b^) A ^ + kj" (28.4) The values for a and h^ can be solved from equation (28):

% = \ =

h =

\ =

-me-^.

0 0

- m e " ' "

/~2 2* ^ H ^ x - n ^ V x +k .. •-li^x + H^yx^ + k^"-'

~Li + / 2 'i']

^1 ki,x + ki vx'^ + k, 1 o 1 dx (29)

Substituting in formulae (27), the potential field becomes:

/ 2 ,2' r> / ii\ r ^ i ^ - n v x +k, -1 ^ ^ ° " o = 2 " " 2 ^ -°» J (px)e<'^"'''''. -^ ° , 2 i J d x (30.1) ° (p^ + (z+h)^)5 J ° '-^^x + M^J^/x^ + k^-' M " - h x - z ^ x ^ + l ^ r Vi^x-n Vx^+kJ" z > 0 TT = - m . - 2 . J (px)e \^-— ^ " T ^ # « ^ (30-2) o l o l 27

-In their general form, these integrals cannot be expressed in a finite series of elementary functions. We will therefore restrict ourselves to the case where the source is located at the origin (h = 0) and where there is no contrast in the magnetic permeability (yi = M,). The potential vector at point (p, z = 0) then becomes:

n(p) = -Si -m f J (px) r ^===ijdx (31)

Putting X = k . . sinh t, the integral becomes:

/ 2 ^ , 2 ' OD . pX - Vx +k -, _ J (px) , „ t dx = -k, J (k, p sinh t). e cosht dt ^ ° ^x + V T T Z - " i J o 1 o 1 o 1[ ~ T I V ' l P sinht)(e"* + e'^S dt

where the Laplace transform is defined as

00 L | f ( t ) ; s } = ff(t)e"^*dt - ^ L { j ^ ( k ^ p s i n h t ) ; l | - y - L {j^(k^p sinht) ;3 [11]. o Since (Bateman):

L {Jjj(a. sinh t) ; s | = I^^^^ (a/2). K^_^ (a/2) (32)

~2~ ~2~

under the conditions: Re(n) > - 1 ; Re(s) > - 5 ; Re(a) > 0, where I and K are modified Bessel functions of the first and third kinds, respectively, the potential vector (31) can be written as:

TT(P) = - f + mk^/2.[li(k^p/2).K_i(kj^p/2) + l3(k^p/2).K_3(k^p/2)j

-Since: I^(z) = y ^ - sinh(z) K_.(z) = ^ . e -l 3 ( z ) ^ ^ . [ c o s h ( z ) - ^ i S k -l ^ ] 2

^3(^)=€-[^-H-"^

~2

this can be transformed into the relatively simple solution:

2 r "•'ih

"rel^P) = — ^ L ^ - (l + k , P ) . e J (33)

"^^^ (k^p)''

where jr (p) has been defined as the ratio of the vector field over the homo-geneous halfspace and the field in free space [ s e e also (12)].

In figure 1 the behaviour of rr ,(p) as a function of k.p has been plotted in an Argand diagram.

With the help of equations (6), the E and H field components can now be computed: ^ = 0 aTT E^ = 0 W a^TT *^p Öpöz " z - p • ap f' ap

It follows that the relative field components are given by: ^"rel<P)

=cprel="rel<P> " P " S ^

(kjp)

^ [ 3 - (3 + 3k^p+k5p2).e^'']

_(34.1)

2 a a "rel<P) z r e l P ap ^ p

* i P l (.\P)'

[9 - (9 + 9k^p + 4k^p^ + k^p^)e ^ ] (34.2)

The component H cannot be derived from equation (33). Therefore we first differentiate equation (30.1) with respect to z and then evaluate the result. It then proceeds in the same way as for IL and Hz:

^ ^ = - ^ { L ( y k j P S i n h t ) ; 0 ) - L (J^(k^p sinh t) ;4)} and for H we can then write:

mk'

m K , 3 r K^P KlP K P K^p ,

«p(P) = V a^Vo (-2^^(-l-> - ¥-Y-)K2(—)}

(34.3)

which values can be found from tables for I and K ' ' . The relative value H ,(p) is indefinite, since in free space the magnetic field has no com-ponent in the p direction.

In figure 1, E , and H , have been plotted as functions of the parameter k^^p.

2. Case__II:_The two-lajrer case, N_= 2 z = -h

a = 0

• 4

-^1 «1

^2 2

The potential vector field in the three regions can be written as:

z $ 0 n

° (p2 + (z+h)2)^

ï4v^)K^"+^^''1dx

(35.1)

f

r

^^1 ^^n

^ z ^ t. TT = J J ( p x ) | a e + bj^e | dx (35.2)

00 2X -ZX

t^ ^ z TTg = J J(px){a2e ^ + bg e ^ | d x (35.3)

where we have defined X. as: 1

X. =v'x^ + k^' 1 J

The boundary conditions, as given in (15), give the following equations for a. and b.: J (15.2) Lim n = 0 : b = 0 ^ ' ^ ^ o o _{z—>- "} L i m TTg = 0 : a . = 0 z—»"

(15.3) H n = P, n , at z = 0 : (a - me~^l-i = (a, +bi)|i,

o o l l o o l l l

t^X, "''i^i ~*1^2 ' ^ l " l " '^2^2 at z = tj^ : (a^^e + b^^e )\i^=ii^h^e

(15.4) an an^ . • a ^ = ^ r at z = 0 : (a^ + m e " ^ x = (a^-b^)X^

^ " l ^"2 , , , "*1^1 . - h \ . V . -^1^2 - r — = - r — at Z = t , : ' - - "" ~ >v _ v i- _

az az 1

at z = t, : (a^e - b^e )X = -X .bg.e

They can be written as a matrix equation:

' ^ - ^ 1 -t-l-,

X X, -X.

O H^e Hj^e -l-igC - t X t X - t X ^0 -X^e ^ ^ ^ 1 ^ >^2^ >

-hx

(36)

or in the abbreviated form: A.y = c

The determinant of A can be computed:

det A = {{\^X^ + l^iX)(U^X2 + [i^X^) + (U^X^ - UjXW{<2'^2^i^ ^ ^1^ (37.1)

2\tirti<\-V

It is nonzero for all x. It therefore follows that the solution of (36) can be [17]

written, according to Cramer's rule , as:

^i

det A. det A

; i = 1, 2, 3, 4

if A = 4 x 4 matrix with det A ^ 0

where y = CH,, "H ), and A.- is defined as the matrix that is obtained on substituting the vector c in the i column. It can be computed:

2t,X,i t,(X,-X2)

detX^ = Vo^i-^^i'')(^^i^+^^^i) + (^o\+^l'''>^^\-^2^i^ ^ 1^^ ^

(37.2) and thus [with (37.1) and (37.2)]:

-- 2 t X

(M^X^-M^x)(^^X^ + M^X^) + (^^X^ +P^x)(^^X^-^^X^)e ^ ^ _^ - 2 t X • ™^

(M„Xi+M^x)(M^X2+M2Xi) + (M„X^-M^x)(H^X2-li2Xi)e ^ ^

With the foi-mal definition:

|i.X. - [xX.

^ i j = ? X T T I x : i ' ^ = ^' 2, 3, . . . . , N (38)

the potential vector (35.1) becomes:

rX_,+X,„e

-2t^X,

n(P,z$0) = „ ••• 1 + m f J (px){ °^ ^^ ö n r f e * ^ " ' ' ^ ' ' dx (39) (p'+(z+h)5- ; ° \ ^ ^ , X , , e ^ ^ ' l

ol 12

From this expression and equation (6) the relative E and H field components can be derived: ^ r e l - 1 - ^ ^ ^ I . i ( P x ) ^ ^ ^ ^ ^ ^ . 2 ! I l x e ( - ^ ) - dx (40.1) 5 » -2tjX^ « z r e l - ^ ^ ^ ^ ^ J j ^ ( P x ) C ° ^ ' ' ^ ^ \ 2 , J x ^ e ( - ^ ) - d x (40.2) « ' - 2 ( - + h ) 2 ) ^ IH-X X e ^ 1 01 12

3. Case III: The ^ n e r a l multilayej case, N >_2

The vector potential in the various layers can be written as:

^ - 0 TT = „ '"^ g i + J J (px)a e^'' d s (41.1) ° (p2+(z+h)2)^ ^ ° ° CD - 1 j f 2X- -Z^.^ ^ t ^ g z ^ S t j ^ TT. = J y p x ) [ a . e J + b.e ^Jdx (41.2) - 33

N-1 , -zXjj

^ ^ , £ , %

" N

= J

VP'^)^N®

^ <^^-^)

o k=l "

The boundary conditions (15) can again be applied for each interface j — j + 1 , resulting In the set of N linear equations:

(15.2) ajj = b^ = 0

T.X. -T.X. T.X. - T . ^ - , 1 (15.3) li.a.e ^ ^ + n.b.e •> ^ = n a.^,e ^ ^ ^ + lJ.^,b.^,e ^ ^

' ] 1 1 1 j+1 1+1 1+1 1+1

T X —T X T X —T X

(15.4) X.a.e ^ ^ - X.b.e J J = X.^,a.^,e ^ ^""^ - ^ . ^ , b . ^ , e ^ ^""^ ' ] 1 3 J 1+1 J+1 J+1 J+1 where j = 0, 1, 2, . . . . , N-1, and T. is defined by:

J

T. = E t, ; T = 0

This set of equations can be written in matrix form [see also eqn.(36)]: A . y = c

where A is a 2N x 2N matrix formed by the coefficients of the a.'s and b . ' s . Furthermore,

y = (a^, b^, a^, h^ a._^, b^, a^_^, h^)

—Jjv ~ l l X

c = (|a me , -xme , 0, 0) The solution is obtained with help of Cramer's rule:

det. A.

y, = é (42)

' det A

where A. is obtained from A by substituting the vector c in the i column of A. - 34

Example

For N = 3 (three-layer case), we obtain for the potential vector above the top layer:

n(p, z ^ 0) = 3 """ g i + m j J (px)e(^ ^^"^ . (p2+(z+h)2)2 i ° -2t X -2t X -2(t X +t X )

r^ol"V^2^23^ ^ 2 ^ ^ 3 ^ Idx

L -2t,X, -2(t,X +t„X„) -2t„X -' l - \ l \ 2 - ^ ^ o l \ 2 - - ^ 2 ^ 2 3 ^ (43) The components of the electromagnetic field can be derived from this relation In the usual way.

m . THE APPARENT WAVE NUMBER

The apparent wave number is first defined, and then analytical computa-tions are carried out to determine its value for the two-layer case.

A. Definition

The potential field above a homogeneous halfspace can be expressed in elementary functions [see (33)]. If more layers are present, however, the expression for the potential becomes an integral, which has no simple solution. As the electric and magnetic field components are derived from this potential function, they do not have a simple solution either. In field surveys, either the electric or the magnetic (or both) components are measured. To interpret them, the measured curves can be compared with numerically computed curves for E and H. To meet all possible situations, a whole set of these so-called master curves is needed. This is a most time-consuming, but still effective, method. However, it does not give a rapid indication of the best choice of survey parameters (frequency and source-receiver separation) for a given (or estimated) geo-electric subsurface configuration. We therefore propose the introduction of an apparent or effective conductivity.

-Definition 1

The apparent conductivity of a geo-electric structure is equal to that conductivity of a homogeneous halfspace that would give the same magnitude for both (complex) components of the electric or magnetic field for a given source-receiver separation and frequency.

This definition enables us to make the following three remarks: 1. In general it is not necessary that the apparent conductivity derived

from the magnitudes of the electric field components is the same as the apparent conductivity derived from magnitudes of the magnetic field components.

2. The apparent conductivity will, in general, be a complex number; thus [with (1)]:

' V E

Electric: a = a „ . e ^ _{app a, E} ii^a M Magnetic: a = a , , . e ' ° app a, M

3. The apparent conductivity will, in general, be a function of the survey parameters (the frequency and sourcereceiver separation) and the d i s -tribution of conductivities with depth in the geologic structure. Following this definition of the apparent conductivity, an apparent wave number can be defined [ s e e also equs. (8) and (10)]:

Definition 2

k = (luuu.o J a ^ ^ app'

where M has been taken the same as in free space.

From the foregoing it follows that there will be two (different) apparent wave numbers; one emanating from the electric field component, and the other from the magnetic field component.

B. Determination of the apparent wave number

Let us consider the two-layer case (N = .2). From Definitions 1 and 2, the apparent wave number is defined by the relation

-H^(N = 2, tj,, k j , k2) = -H^(N = 1, k^) (44.1) for the magnetic apparent wave nimiber, and by

E^(N = 2, t^, k^, kg) = E^(N = 1, k^) (44.2) for the electric apparent wave number.

For simplicity, we thus restrict ourselves to the cases where the mag-netic permeability in the media is the same as that in free space:

^1 = ^^2 = ^^o

Note: This Is not a serious restriction because In most practical situations the permeability Is the same as In free space (see also Appendix I). Using relations (39) and (6.2), the left-hand term of (44.1) can be written as:

H^(N = 2) = H^(N = 0) - f ^ p ^ J j^(px)g2.e-(-'^)dx

where

p ap ap J o^ o

H (N = 0) = z-component of magnetic field In free space 2 , 3m p r + —Ö o T (p^+(z+h)^)2 (p^+(z+h)")^ -2t X, \ l + \ 2 ^ ^2 = -2t,X •' with U2 = M, = M„ (45) ^ ^ \ l \ 2 «

and the r l ^ t - h a n d side of (44.1) can be written [using eqns. (30.1) and (6.2)] as:

CO

where:

-X - Vx^ + k^"

g1 =

_{'1 ^ / 2 ^ , 2'}

X + vx + k

_a

Thus (44.1) can be written as:

ixi

^èpiIVP*2-Je^^^"'^'^ = 0 (46)

o

This relation Is satisfied if, for all x:

^2 h

As, for large arguments, J-(z) will decrease according to

° (2TTZ)2

we will try to find a solution for (46) by expanding g. and g„ as a Maclaurln

series around x = 0 and equating the corresponding coefficients of the two

series.

For g. we then obtain:

g^ ^ -1 + -jf . X - 4 . x^ + ^ . x^ + o (x^) (47)

•^a K

and further

> < o i = ^ - ^ ^ E - - - A - ' - ; i - ' - 0 2 ( x ^ )

^12 - ^^12 - k ^ • ^^ ^ ^y^ ^'^' h2 = (ki-k2)/(ki + k2)

1 ^

exp(-2t^Xj^) ~ exp(-2t^k^) - j ^ exp (-2t^k^). x^ + 0^(x^

1

Substituting these expansions in equation (45), we obtain:

-2 - 1 + -2q q ( - -2 . _ J . _ t ) x ^ ^ ^ ( J . - J . - -2 t ) x ^ + 0 (x^ k^(l-Q)-^ k^(l-Q)-<k^ k2 V ' ^ kfd-Q) •'l k2 'V'^^Oe^'^^ g , c;

with: Q = ]s.^^.exp(-2t^k^) Evaluation of the denominator gives:

4 - l - k ^ ö 7 ^ + - T ^ - 2 [ 2 + 2Q + k,(Q-l)(^+t )]x2

l + . . . . O g ( x ^ k^(l Q) k^(l-Q)^^ ^ k^ I J

- S - ^ r i + 6Q + Q^ - 2 ( l - Q ) ^ ( ^ + k t . ) V + O (x^ ^(1_Q)3L kg I I J 7 and thus g„ becomes:

kJ(l-Q)'"^

( l + 3 Q ) ^ - 4 k Q ( l - Q ) 2 ( ^ + t , )

h- k^-(l-Q)-'^ kf(l-Q)2 k j ' (1-Q)3

0(x^ (48)

Equating the coefficients of x from this relation with those from equation (47) gives the apparent wave number:

1 - k .exp(-2t k )

k = k == (49) •^a 4 l + k ^ 2 - e ^ P ( - 2 t i k i )

2

and it now appears that the coefficients of x are also equal. The coefficients 3

of X , however, are not the same. -2t k

k^ge ^ N 3 - 4 k ^ ( j ^ + t ^ ) ( l - Q ) 2 - 2 Q - Q ^ )

go =i g, + 0^ 5 x^ + 0(x*) (50) ^ ^ k J ( l - Q ) 3

which for sufficiently small values of x will converge to the relation between

-gg and g. that we hoped for.

The approximated apparent wave number (49) that we have obtained is the same as that which we would have obtained by treating the problem with plane, vertically Incident, electromagnetic waves.

The apparent wave number obtained Is also the same as that which we would have arrived at by setting up this derivation for the electric field com-ponent, because essentially we have not equalised the magnetic field comcom-ponent, but rather the potential vector [ s e e eqn. (46)] from which the electric and magnetic components are derived by differentiating with respect to p and z according to (6). Therefore the apparent wave number thus found is Independent of the source-receiver separation p.

C. Some examples

To show the character of the electric and magnetic field components in the two-layer case, some examples of E . and H , have been computed and are shown In figures 2 and 3; both for the exact solution obtained by numerical integration of (40) and for the approximated solution (dotted lines In figs. 2 and 3) that Is obtained by Inserting the apparent wave number - as computed for the different configurations - in the formula for the homogeneous halfspace [eqn. (34)]. For convenience, we have introduced the new dimension-less parameters pk^ and 2t../p.

It can be seen that, for pk. = 1.2(1+1), the two ts^pes of computed values deviate from each other, especially for small values of the parameter 2t.. / p . For larger values of the parameter p k^, we obtain a good approximation for the field components using the apparent wave number. Only for the smaller pk. values and for the lower 2t./p values do we find larger deviations. In the limiting case for tj^ » p, however, the results of the two methods of computa-tion tend to converge, since we have:

k « k^ for tj^ » p and thus:

X ,(k ) « X ,(k,) for t, » p ol^ a' ol^ 1' 1 ^ On the other hand, we have:

-X , , + -X ^ 2 ^ ^ ( - ^ ^ l ^ ) ^ ,^, , , ^ , l+X^^X^2«xp(-2t^^j)"^\l<ki) ior t ^ » p

which confirms the predicted behaviour of both ways of computing in the limiting case.

IV. BEHAVIOUR OF THE APPARENT WAVE NUMBER FOR

a^ <

o„

The magnetic field Is computed with our approximated apparent wave number and tested against exact computed values. Finally, the behaviour of the field for large contrasts Is derived.

A. Area of reliability

Let us consider the cases where the lower layer has a higher conductivity than the upper layer. With the help of the apparent conductivity [ o r wave num-ber k„, see (49)], the magnetic field can be computed. This calculation has

a

been carried out for various values of pk , and the results have been plotted In figure 4. The values obtained In thlg way can be tested against the true values obtained by means of numerical .Integration of relation (40.2)'- ' i These latter computations are also compared with those of Keller and Frischknecht, 1967, and Sllchter and Knopoff, 1959'--^^'^^-'.

A two-layer case is specified by three parameters, which we will call field parameters (as opposed to the survey parameters, which are the source-receiver separation and the frequency f); they are the conductivities of the two layers and the thickness of the first layer. Instead of these three parameters, we will however use the dlmenslonless parameters:

D = 2t./p : depth ratio

S = OoA'i : conductivity contrast

pk. = p.(—5—)(l+i) : product of the wave number of the first layer and the source-receiver separation

The exact computations of the relative magnetic field strength are tabulated in the following pages for a depth ratio D = 0.5, the contrasts:

-S = 3, 10, 30, 100, 300, " and the pk..-values:

pk^ - 1.6, 2.4, 3.6, 5.0, 7.2, 10.0

Rather than compare the field strengths resulting from the two means of com-putation, we compare the values of the apparent wave numbers, obtained with (49) and plotted in figures 5 to 14, and the (true) wave numbers derived from the values of H , (exact) and the curves of figure 4. These latter curves relate the magnetic field strength of a multilayered structure to the appropriate apparent wave number for given parameters, since they are computed by

icp

inserting various values k e ^, instead of k.., in the formula for the magnetic field strength for the homogeneous half-space (34.2), according to the definition of the apparent wave number (see chapter HI).

These exact apparent wave numbers are given (amplitude and phase) in the last two columns of the tables.

S = 3 D = 0.5 pk^/(l+l) 1.6 2 . 4 3 . 6 5.0 7 . 2 10.0 ï^^(«zrel) 1.235 0.929 0.454 0.101 0.024 < 0.001 ^'^(«zrel) -0.320 -0.583 -0.642 -0.477 -0.187 -0.088

| p k j

2.74 3.80 5.25 6.92 9.75 14.30 cp„ (degrees) _a 42 40 39.5 41 44.5 45 S = 10 D = 0.5 pk^/(l+l) 1.6 2 . 4 3 . 6 5.0 7 . 2 10.0 « « ( « z r e P 1.005 0.779 0.488 0.161 -0.026 0.000 ^'«(«zrel) -0.367 -0.443 -0.521 -0.470 -0.201 -0.087

IpkJ

3.25 4.09 5.24 6.67 9.50 14.35 cp„ (degrees) _a 33 31 35 39 45 45 42

-S = 30 D = 0.5 pkj/(l+l) 1.6 2 . 4 3 . 6 5.0 7 . 2 10.0 «^<«zrel) 0.862 0.731 0.512 0.196 -0.028 -0.001 ^^(«zrel) -0.280 -0.341 -0.457 -0.465 -0.210 -0.087

IpkJ

3.55 4.17 5.15 6.61 9.35 14.40 cp^ (degrees) 23 25 33 37 45 4 5 S = 100 D = 0.5 pk^/(l+l) 1.6 2 . 4 3 . 6 5.0 7 . 2 10.0 Re(H ,) _{^ z rel'} 0.789 0.707 0.528 0.220 -0.028 0.000 ^ ( « z r e l ) -0.200 -0.272 -0.415 -0.461 -0.217 -0.086

IPkJ

3.79 4.23 5.14 6.50 9.28 14.41 cp^ (degrees) 18 20 30 37 45 4 5 S = 300 D = 0.5 pk^/(l+i) 1.6 2 . 4 3 . 6 5 . 0 7 . 2 10.0 ««<«zrel) 0.754 0.695 0.536 0.233 -0.029 -0.002 ^ < " z r e l ) -0.153 -0.235 -0.393 -0.458 -0.220 -0.086

|pkj

3.92 4.27 5.11 6.51 9.25 14.40 cp (degrees) a. 14 18 28 36 45 45 S = " D = 0.5 Pkj/(l+l) 1.6 2 . 4 3 . 6 5 . 0 7 . 2 10.0 R«<«zrel) 0.707 0.673 0.548 0.252 -0.029 -0.002 ^ - ( H , , , i ) -0.083 -0.181 -0.360 -0.454 -0.225 -0.086

IpkJ

4.13 4.34 5.04 6.40 9.17 14.40 cp (degrees) _a 7 14 27 35 4 5 45 43

-These values of pk are plotted In figure 15. The best fit of these plots with those from figures 5 to 14 results In the following table:

D (approximate) 0.35 0.40 0.44 0.44 0.44 0.45 D (exact) 0.50 0.50 0.50 0.50 0.50 0.50 S (approximate) 2 . 2 7 20 50 100 500 S (exact) 3 10 30 100 300 CO

As for pk.., the correct value is found in all cases. We can see from the table, by comparing the exact and the approximated depth ratios, that the accuracy of the approximation Increases with the contrast. For very high contrast the e r r o r Is only 10%, and for an intermediate contrast It reaches 12%. As for the con-ductivity contrast, we see that the e r r o r increases with the contrast.

On the other hand, we can also examine the behaviour for a given contrast as function of the depth ratio. In the following tables, the exact apparent wave number is given for S = 100 and D = 0.125, 0.25, 0.50, 0.75 and 1.00. S = 100 D = 0.125 pk^/(l+i) 0 . 8 1.2 1.6 2 . 4 5.0 ««(Hzrel) 0.205 0.156 0.131 0.107 0.080 ^ ( « z r e l > -0.237 -0.137 -0.095 -0.060 -0.036

|pkj

7 . 8 9 . 4 10.6 12.1 14.3 cp (degrees) a 26 21 18 14.5 12 S = 100 D = 0.25 Pk^/(l+l) 0.80 1.2 1.6 2 . 4 3 . 6 5 . 0 R^<Hzrel) 0.474 0.386 0.344 0.299 0.258 0.209 I - ( " z r e l ) -0.267 -0.178 -0.140 -0.114 -0.118 -0.141

IpkJ

5 . 5 6 . 3 6.8 7 . 4 7 . 9 8 . 5 cp^ (degrees) 20.5 15.5 13 12 13.5 18 44

-S = 100 D = 0.50 pk^/(l+i) 1.6 2 . 4 3 . 6 5 . 0 7 . 2 ï^«(«zrel) 0.789 0.707 0.528 0.220 -0.028 ^™<H,,el) -0.200 -0.272 -0.415 -0.461 -0.217

IpkJ

3 . 8 4 . 3 5 . 1 6 . 5 9 . 3 cp„ (degrees) _a 17 20.5 28.5 37 45 S = 100 D = 0.75 pk^/(l+i) 0 . 8 1.2 1.6 2 . 4 3 . 6 5.0 ««<"zrel) 1.144 1.100 1.065 0.956 0.577 0.070 I°^<«zrel> -0.113 -0.159 -0.226 -0.416 -0.673 -0.548

IpkJ

2 . 1 2 . 4 2 . 7 3 . 5 4 . 9 6 . 8 cp^ (degrees) 30.5 28 29 34 40 44 S = lOÖ D = 1.00 pk^/(l+i) 1.2 1.6 2 . 4 3 . 6 5 . 0 7 . 2 ««(«zrel) 1.200 1.184 1.067 0.537 0.013 -0.019 I°^<Hzrel) -0.128 -0.223 -0.498 -0.785 -0.523 -0.163

\^K\

2 . 2 2 . 5 3 . 4 5.0 7 . 0 10.2 cp^ (degrees) 37 37 40.5 44 45.5 45

The values of pk are plotted in figure 16. Again, by trying to find the best-fitting plots from figures 5-14, we obtain the following table of contrast and depth ratios: D (approximate) 0.121 0.23 0.44 0.65 0.75 D (exact) 0.125 0.25 0.50 0.75 1.00 S (approximate) 100 120 50 10 4 S (exact) 100 100 100 100 100 45

-The correct values are again found for p k^.

It follows from this table that the accuracy of the approximate depth ratio decreases from about 3% at D = 0.125 to 25% for D = 1.0. The accuracy of the contrast also decreases with increasing depth ratio.

As a conclusion, it now can be stated that, provided we allow for an e r r o r In the determination of the depth ratio, the apparent wave number - as defined by relation (49) - can be used to determine the magnetic field strength. On the other hand, measurements in the field can be Interpreted to a first-order approximation by using figure 4 to find the apparent wave numbers and the plots of figures 5 to 14 to determine the upper-layer thickness and its conductivity. The conductivity contrast, however, will then be subject to a relatively high e r r o r - especially for large depth ratios. The best approxima-tion for D is given for the cases with higher values for S (S » 1), lower values for D (D < 1.0) and pk > 2 (1+1).

B. Behaviour of the magnetic field strength for large contrasts

From the plots of the apparent wave number as function of the field parameters (figs. 5-15), it can be seen that for high contrasts |pk | reaches

a asymptotic values and only cp changes. This behaviour can also be seen

a

analjftlcally by taking the limit of pk , as given by (49), for an infinitely large contrast: 1 - k e'^^^^^ "^^I'^l Lim pk^ ^^ -2t k^ = P^^l ^ 2 t k^ <51) 1 + k. „e 1 - e Since we have: 1 - v s Lim k, „ = Lim T=S. = - 1 S_»oo 12 s_>co 1 + V s

relation (51) now can be approximated for small values of 2t^k. by evaluation of the exponent, resulting in:

pk„ ^ pk, 2 2 ^,,3,,3,

-i"i -h'^i

2 - 2t,k, + 2t;k7 - 0(t]|^kp

^ ^ 2t^k^ - 2tJkJ + O(tJkJ)

« ^ + P t , k ; (52) 46

-which we, for convenience, will write as: P l ^ a ^ 2 . ^ + i . p k ^ . 2 t ^ k ^

Inserting this approximation in the formula for H , for the homogeneous half-space (34.2) to obtain the field strength for this (special) two-layer case, we find:

H , = — ^ f o - ( 9 + 9pk + 4 p V + p V ) e ^1

_{z r e l , , . 2 L ^ a ^ a ' ^ a ' J}

<PV

which for 2t.k. -*^ pk^ becomes:

Hz rel " - ^ <53)

^^^l 2 2

« _ i (1 . atfkf)

P

for S » 1 and ] 2t^kj^|<K mln.(l, |pkj^|)

This is the same result as we would have obtained by considering the case of a dipole source situated at a height t.. above a layer with an Infinitely high con-ductivity. This source would induce a m i r r o r dipole at a depth t. under the

[lO 2 l l conductor surface, with the same dipole strength but the reverse direction ' •'

Relation (53) implies that the magnetic field strength on top of a two-layer surface with a very large conductivity contrast, a small depth ratio and a small first wave number ( i . e . for low frequencies), to a first-order approxima-tion depends only on the depth ratio.

V. BEHAVIOUR OF THE APPARENT WAVE NUMBER FOR g^^ >

a„

In this chapter, we first show that the relation for the apparent wave number as given in chapter i n cannot be used to describe the behaviour of the true apparent wave number when the upper layer Is more conductive than the lower one. Some master curves for k are computed, from which a relation is derived to describe the behaviour of k for this case.

a

-A. Area of reliability

The true apparent wave nimiber Is derived from the magnetic field strength (40.2) with help of the curves of figure 4 In the same way as for the case a^ < a^. These (true) apparent wave numbers are given In figures 18 to 20 as functions of pk. for the depth ratios and contrasts:

D = 0.125, 0.25, 0.50

S = 0.00, 0.01, 0.03, 0.10, 0.30

Comparing these curves with those computed with relation (49) shows that the

approximated k -values behave entirely differently from the true values for k . _{a a} As an example, the approximated values for D= 0.50 and various contrasts are

given in figure 17. Both the phase and the amplitude of k behave differently. a

It can therefore be concluded that for the case of a more highly conductive upper layer, the relation (44.1) is not fulfilled (approximatlvely) by equalising the integrands of the expressions for H ,(N = 1, k ) and H ,(N = 2, t , ,

z rel a z rei ± k . , k„) for small Integration variables only [see also (50)].

B. Determination of a relation for the apparent wave number

The true values of k can, of course, be used for the interpretation of field data. Althou^ the curves in figures 18 to 20 cover the most commonly used range of depth ratios and conductivity contrasts, for a more precise interpretation they need to be extended to other D- and S-values.

On the other hand, it would be convenient to have some simple relation, describing the behaviour of the apparent wave number, which could be used both in the field and in computerised interpretation programs to give a first approximation of the field parameters. Such an approximate formula will be derived in the following.

As a first step, we seek a relation between the depth ratio and the relative apparent wave number. In figure 21, the ratio Ik /k,I has been plotted

a 1

as a function of D for various contrasts and values of p k^ on semllogarithmic paper. From it, we see that for this range of parameters we can write the approximation:

| k j = a . | k J . h i ( D / B ) (54) where P = P(s) and Is determined by the abscissae In figure 21 where all

I k A . I curves converge. Since we have:

-S = 0.0 S = 0.1 S = 0.3 e = 2 . 3 0 10 8 = 1.25 lO" 8 = 2.65 10

8 will be approximately related to S according to:

8(8) = 0.023(1-8)^ (55)

The variable a is a function of both the contrast and P k • it is determined by the slopes of the various curves In figure 21, from which we can derive the following table of a-values:

p k / ( l + l ) 0.3 S = 0.1 S = 0.0 7 . 2 5 . 0 3 . 6 2 . 4 3.84 3.9.6 4.09 4.22 2.60 2.75 2.89 3.06 2.27 2.39 2.53 2.71

When these values are plotted on logarithmic paper, s t r a l ^ t lines result, which can then be written as:

a(pk^, S = 0.3) « 3 . 0 9 l p k ^ r ° - ^ ^ ° a(pk^, S = 0.1) -«3.50|pk^|"°--^^'^ a(pk^, S = 0.0) «4.58|pk^|"''-^^^

Taking the constants In these relations as functions of the contrasts only, a(pk^, S) « Y(s).|pk^l -5(s)

It can be shown that y and 6 are related to the contrast according to: 1.30S

Y(s) « 3.09 e 5(s) « 0.16 e -0.823S and thus for 1 we obtain:

a(pk^, S ) « 0 . 3 2 4 e ^ - 3 ° ^ . l p k / - ^ ö ^ -0.823S (56)

-Inserting this relation and relation (55) in (54), the relative magnitude of the apparent wave number becomes, finally,

„ , „ -0.823S

^ T s o s — ^ M fiJ

' ' l 3.09 e^-^"^ ^0.023(1-S)^^

(57) 023(1-S)'

which relation has proved to be a fair approximation only for the conditions: 0.125 $ D ^ 0.50

2.4(1+1) $pkj^ $ 7.2(l+i) S < 0.3

These restrictions could, of course, be eliminated by determining a more accurate relation for the apparent wave number. However, the new relation would be more complicated and, moreover, the restrictions do not pose serious limitations on field measurements (for a h i ^ l y conductive upper layer, the separation and frequency are chosen in such a way that D is small and |pk. | is not larger than 10, because of the screening effect of the first layer).

VI. ANALYSIS OF ELECTROMAGNETIC PROSPECTING DATA

With the help of the apparent wave number It Is possible to obtain formulae from which the values for the source-receiver separation and frequency for which the magnetic field Is the most sensitive for the subsurface parameters can be derived. The Influence of e r r o r s in the measurement of the magnetic field on the interpretation is discussed.

A. Optimisation of survey parameters

One of the large group of geo-electrical prospecting techniques, the method in which a current Is Induced In the subsurface by a time-variant magnetic field, is called the electromagnetic induction method > > > > J _ This method is used both at ground level and from the air. Usually a loop of wire, t h r o u ^ which a time-varying current Is forced to flow, is used to generate the magnetic source field. Sometimes this loop Is replaced by a long