A numerical calculation of the electromagnetic field from a vertical and a horizontal magnetic dipole above a homogeneous earth

A numerical calculation of the electromagnetic field from a vertical and a horizontal magnetic dipole above a homogeneous earth I.

Thick conductor BY

DAVID

MALMQVIST

Boliden Mining Company,

Boliden, Sweden

Abstract In view of the high accuracy which can be reached in determining the relative field ratio of an electromagnetic dipole field by means of modern measuring equipments, the previous calculations (especially those by Wait 1955, 1956 and by Meyer 1962) of the responses from conducting media of different geometrical form are incomplete, especially if responses of small or very small magnitudes are taken into account. In this first issue of articles concerning magnetic dipole fields a calculation of the responses above a conducting infinite half-space (thick conductor) will be given for three different orientations of transmitter and receiver. All these three cases are formerly treated by Wait (lot. cit.). Due to the many different systems of symbols used by different authors when treating the magnetic dipole field, a brief recapitulation of the theoretical background using more consistent symbols is considered to be necessary. A closer analysis of Wait’s integrals To and Ta follows as well as more complete tables of the integrals mentioned. On the basis of these calculations more detailed vector diagrams have been drawn whereby special attention is paid to responses of small magnitudes. Finally a practical field test above the ice of the Gulf of Bothnia made by means of beam-slingram correspondence is good.

instruments between the

1. Introduction Among geophysicists geoelectrical exploration,

is presented and described. As will be seen the measured and the theoretically calculated values

who are working in research and field surveys concerning growing attention has been paid to methods and systems

DAVID

176 where

a magnetic

relatively

small

dipole

is used

coil, through

MAI.MQVIS’l’

as transmitter.

the windings

Theoretically, the field from a of which a time dependent current

(generally a sine wave current) is flowing, is almost the same as the field from an alternating magnetic dipole. The correspondence to the ideal dipole field is the more exact the smaller the coil dimensions are, compared with the wavelength of the transmitted field, and the greater the distance (relative to the coil diameter) is to the point of interest. In practice the first of these conditions can be considered to be very well fulfilled if an alternating current of audio-frequency is used. From technical considerations the second condition is easily established if only the receiving system is arranged at a distance of say 20-30 times the radius of the transmitting coil. Graf (1934) has, for instance, shown that already at a distance of 13 times the radius of the transmitting coil, the discrepancy between the resulting field from such a coil and the field from a corresponding magnetic dipole, placed at the center point of the coil, is less than 1 %. One of the earliest systems for practical purposes, in which a real magnetic dipole field was introduced, seems to be the one described in the above mentioned paper by Graf (1934). In this paper Graf proposed to modify the Koenigsberger c(Ringsendemethode# (Koenigsberger 1930) in the manner that the field far outside a relatively small loop should be measured. Concerning field surveys, it seems that Graf at that time made tilt angle measurements determining only the direction of the dipole field. Such angle determinations in a horizontal magnetic dipole field were further practised in the so-called vertical loop methods, first described by Broughton to a practical Edge and Laby (1931) and by Mason (1929), but later brought field technique by Slichter (Slichter 195.5 or Eve & Keys 1956) and used on a larger scale during the 1940’s and 1950’s both in Canada and the U.S.A. Early field measurements utilizing relatively small horizontal loops (40 to 80 feet) are further reported to have been practiced in surveys carried out by the ABEM Company of Sweden (Hedstram and Parasnis 1959). During these surveys the electromagnetic field, which at the outer parts of the measuring areas (at distances up to 600 feet from the centre) can be considered to have been a magnetic dipole field, was completely measured by means of the so-called compensator method. More systematic field measurements, where a true magnetic dipole field was used, were started with the introduction of the so-called ccslingramo system, from the beginning characterized by a dipole transmitting coil with a diameter of little more than 1 m and a smaller receiving coil kept at a distance of 40-100 by Holm times the radius of the transmitting coil. This system was introduced in the routine prospecting work of and Werner as early as 1936 (Granar 1960) the Geological Survey of Sweden and was later described in a paper by Werner (1947) as well as in different patent descriptions from the middle of the 1940’s

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD ETC.

177

(Canadian Patents 510,909 and 568,402, Finnish Patent 24,506, Norwegian Patent 79,611, Swedish Patent 149,166 and U.S. Patent 2,532,368). The interest in the slingram technique became more evident when modifications of it proved to be most suitable for airborne EM surveys. Therefore, in the last 10 to 15 years, a couple of airborne dipole systems have been described in different patent issues, the earliest of which seem to be those described in the British Patent 697,933 and the U.S. Patents 2, 623,924 and 2,642,477. Many of these systems are also briefly and incompletely described in advertising articles in mining journals and in small pamphlets from contracting companies. The first tests with an airborne magnetic dipole system for ore prospecting purposes were apparently carried out in Canada by the Lundberg Exploration 1950; U.S. Patent 2,559,586). Another early modification, Ltd. (Lundberg, using two frequencies, is sometimes called the (Canadian method)) (Anonymous 1956; U.S. Patents 2,839,721; 2,929,984; 2,931,973 and 3,042,857; Canadian Patent 651,447). They have been used for many years in airborne surveys in Canada, Finland and elsewhere. If the so called INCO-system (U.S. Patents 2,623,924; 2,953,742 and 2,955,250) should be included in the (Canadian method)) is not defined further in reports but to the author it seems likely. Activity in constructing new equipments for airborne EM-surveys was indeed very strong in the beginning of the 1950’s and almost simultaneous efforts to manufacture different systems of this kind seem to have been made in Canada, Finland, Sweden and certainly also in Russia. For instance in 1952 the Boliden Mining Company started to build an airborne ((slingram)) equipment with a vertical magnetic dipole system which was in practical use in 1954. In the first airborne EM system the magnetic dipole transmitter was generally vertical. The transmitter was mounted in the aircraft while the receiver coil was placed in a special ((bird, which was towed behind the aircraft by means of a tow-line, some 50-200 m in length. An interesting further development in the technique in the late 50’s was further the introduction of the so called rotary field systems where a combined horizontal and vertical dipole was used. These systems have been tried with the crossed coils (mutually perpendicular coils) placed in a towed bird (Tornqvist 1956, Kyzovkin 1964) or in a second aircraft (Tornqvist 1958; U.S. Patent 2,794,949) equipped with a receiving coil and recording instruments, which was flown behind the first aircraft at a distance of some 100-300 m. It could be mentioned in this connection that systems of similar kind, utilizing three mutually perpendicular coils (orthogonal coils) have been proposed (U.S. Patents 2,919,397; 2,953,742 and 2,955,250). The more recent systems for airborne EM surveys are characterized by a remarkable reduction in coil separation. This reduction allows the transmitter and receiver to be rigidly connected to the same aircraft body like, for instance, the latest construction of the Boliden Mining Company and that of the Mullard Radio Valve Ltd. (U.S. Patents 2,995,699 and 3,127,557; Canadian Patents

178

DAVID

653,283

and 653,286).

MALMQVIS’I

In the constructions

(Canadian

Patent

667,836)

mentioned

systems

are used by many

emanating

from the Varian

Associates

and from Nucom Ltd. (Canadian Patent 684,082 and U.S. Patent 3,015,060) the coils are mounted on a rigid boom suspended below the aircraft which in this case should rather be a helicopter. These last companies

in Canada

and U.S.A.

(Pember-

ton

1961). ‘I’o make this list of different airborne magnetic dipole systems even larger but nevertheless incomplete, it can be added that lately the so-called transient, pulse or INPU’l’ methods have been introduced.

2.

Recent magnetic dipole systems for ground EM surveys. Comparison between these systems and airborne systems.

Compared with the many airborne magnetic dipole systems the developments and improvements of ground EM instruments are relatively few. The original slingram system has been manufactured in a couple of designs, by different companies and under different names, and has during the last ten years probably been the most common system in ore prospecting EM surveys. Using experiences from airborne dipole systems, the Boliden Mining Company has recently developed equipments for ground EM surveys called boomor beam-slingram (Westerberg 1966). In the different beam-slingram systems the coil separation is reduced to about 8 m with transmitter and receiver rigidly connected to the same beam in three different positions. The equipments are so light that they can without much difficulty be carried by two men. Under relatively favourable conditions the field ratio - which means the ratio H/Ho of the true magnetic field intensity H and the same field intensity in free space Ho - can be measured with a relative accuracy of 1 x 1O-5 - 2 x 10-s or lo-20 ppm. This is possible both for the in-phase (real) and out-of-phase (imaginary) component of H. If, for instance, in the original slingram with a coil separation of 40 m we can determine the field ratio with a relative accuracy of 0,1-l yh, we should necessarily have an accuracy of l/l25 this value -- that means 8-80 ppm - in the corresponding determination by the beam-slingram systems in order to Technically it is evident that this obtain the same interpretation possibilities. desired accuracy has been achieved. however, we have mostly to expect a When reducing the coil separation, higher ccgeologic noise)) (Slichter 1960) from disturbing bodies of largely minor interest located near the surface in the surveying area. These bodies of different geometries and of larger and smaller size may be situated in the moraine, in other soil material or in the rock formation. Compared to the surrounding medium they are characterized by a different magnetic permeability as, for example, boulders in the moraine which are rich in magnetite, or by a higher conductivity

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD ETC.

179

as for example bodies of slowly moving electrolytes in fissures of the rock formation or electrolytes in more or less inclined layers of the soil. In using the beam-slingram, the geologic noise is often considerable compared to the small field ratios which it is possible to determine by means of the instrument. However, a couple of practical tests have shown that the ratio between geologic noise and the magnitude of responses from orebodies of ordinary size as observed with the beam-slingram is frequently of the same order or only slightly higher than that of the common slingram with coil separations of 40 or 60 m. So far, these notions are based on too few comparisons and these were made in a restricted ore region. Therefore, it is advisable to use instruments of the beam-slingram type with much caution concerning the physico-geological properties of both rocks and overlying soil materials. But, in fact, just the higher geologic noise can on the other hand be used for characterizing rocks and soils,especially that part of them which lies near to the surface, as to their electrical and magnetical properties. The beam-slingram has, for instance, been tested for locating ore boulders in the moraine and determining their depth (Malmqvist 1965). In this special case the local disturbances from neighbouring ore boulders change the appearance of the larger anomaly coming from a more deep-seated orebody only moderately. Contrary to the geologic noise, the magnitude of the magnetic noise (Slichter lot. cit.) is independent of the coil separation. This noise represents only an absolute level of field intensity above which, for a given receiving device and a given frequency interval, the signal field intensity must lie. When the coil separation is reduced which, as previously mentioned, makes it technically possible to measure very small field ratios, some problems enter which must be taken into consideration. Firstly, there is a general need for computing the field ratios of small magnitudes, which has not been possible by means of the tables now available. Secondly, more attention must be paid to disturbances from the upper part of the ground and, therefore, a closer knowledge of the nature of small responses is necessary, in order to correctly interpret the character of these disturbances which can be of magnitudes of several hundred ppm and naturally are especially pronounced in ground EM surveys. Small values of responses are received when the height- and spread-parameters (see below) are simultaneously small, which commonly depends on insignificant conductivity in the ground. The same is the case if the height-parameter is large which it has a tendency to be for small coil separations especially in airborne EM-systems. Finally, another problem of instrumental nature may be mentioned in this connection. Field tests have shown that it is frequently difficult to determine the real zero position of a beam-slingram system, that means the reading of the system in free space. Due to small stray fields, essentially caused by eddy-currents in the metallic parts of the equipment, an adjustment of the real zero position

180

DAVID

4

MALMQVIS’I

I and !iii

--jq--kJ--

--+

C>

J----+--

Pos.

1

Pos.

D

Pos.

a

Qi and X

-q-J.---~-h

Fig. 1. Positions

‘---(

of transmitter

and receiver.

must be done; for example, by adjusting the system on the ice of a lake with known water conductivity. It is then important to have a tool (vector diagrams) to compute the field ratios in a simple way. We shall return to this problem later. The problem does not exist in airborne EM-systems, since, during adjusting, it is easy to fly to a sufficiently high altitude where no response from the ground is to be expected. 3.

The scope of present and intended calculations.

As mentioned, the aim of the present investigation is to provide more complete and accurate tables for computing the field ratios of magnetic dipole systems placed above an infinite half-space with a conductivity > 0 . The relative magnetic permeability is assumed to be = 1 , and the field is proposed to be quasi-stationary, that means the influence of displacement currents is neglected. The calculations are made for the following orientations of transmitter and receiver which are called positions I, II and III: Pos. I (fig. la). A vertical magnetic dipole or a horizontal transmitting coil at height h above the half-space and a co-planar receiving coil at the same height. This position is called I and VII by Wait. The case with the lower number is the special case for h = 0 .

A NUMERICAL

CALCULATION

OF THE

ELECTROMAGNETIC

FIELD

ETC.

181

W

Fig. 2.

Positions

of beam-slingram

systems giving responses nearly corresponding

to those

at

height h above an infinite half-space.

Pos. II (fig. lb). A horizontal magnetic dipole or a vertical transmitting coil at height h above the half-space and a co-axial receiving coil at the same height. The position is called II and IX by Wait. Pos. III (fig. lc). A horizontal magnetic dipole or a vertical transmitting coil at height h above the half-space and a co-planar receiving coil. This position is called III and X by Wait. The model which, geometrically, can be described as a discontinuity along a plane surface of infinite extent, is not very common in nature, The nearest approach to the model is surely the geometrical conditions on the ice of deep lakes or muskegs or on the ice above sea water if the water depth is comparatively large in relation to the coil separation. We can further expect similar conditions above dikes and sheetlike orebodies with significantly higher conductivity than the surrounding rock and with considerable width compared to both height and coil separation (fig. 2a). If subsurface measurements in drill holes or in adits are considered, the geometrical conditions for position II are almost fulfilled ad-

182 jacent higher

DAVID

MALMQVIST

to a boundary of two different rocks, conductivity than the other (fig. 2b).

However, for other

it is intended

models.

of which

one has a significantly

to extend

the calculations with the same accuracy of infinite extent (thin conductor) is a little higher with a magnetic permeability

A calculation

for a sheet

on the way. Calculations for models than 1 are also meant to be executed. 4.

Theoretical background

In order to avoid too many reiterations in the following formulas; and in order to most of the symbols are chosen according to on the subject even if they sometimes do The symbols presume the use of rationalized

we start with a list of symbols used avoid too many symbol confusions Wait (1955, 1956) in his last treatise not seem to be the most adequate. MKS-units.

List of symbols Symbols

Names

and notations

concerning

the symbols

Units

used

e,F,z

Cylindrical

x,y,z

Cartesian

h

Height of transmitter and receiver above discontinuity, i.e. above homogeneous ground surface. Always positive h > 0 .

m

Coil separation or distance between transmitter and receiver (as such mostly used at the last stage of calculations). Always positive e 2 0

m

e

h/e

m

coordinates

m

coordinates

Dimensionless stance between sometimes

parameter transmitter

called

the

denoting ratio height/diand receiver. In the text

height/spread-ratio

PO

Absolute magnetic permeability in homogeneous ground = 47~ . 10-7

(T

Conductivity

mho - m-i

w

Angular

radians

Y

Propagation

free

space

and

velocity factor.

Only

the

quasi-field,

where

y ~-

(ipow) b is treated A

3 = (2,oeoo) ” h . According to Wait (1955) a dimenon the height h , fresionless parameter depending

hegry * m-i ohm - s - m-i

* sag1

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD ETC.

183

-

Symbols

B

B H,H ffo

E,E %

I

Names and notations concerning the symbols used

-I

Units

quency and physical properties of half-space (homogeneous ground (I L 0). Wait has denoted the Avalues for equal height h of transmitter and receiver above discontinuity A’ No such distinction is made in this paper. In the text sometimes called the heightparameter. = (4 poow)’ e , A dimensionless parameter depending on the distance transmitter receiver e , frequency and physical properties of half-space (a > 0) . In the text sometimes called the spread-parameter Magnetic flux density or induction (only used as a vector) Magnetic field intensity above and in homogeneous half-space (u > 0)

weber * m-s orV*s*m-2 A-turns - m-i

Magnetic field intensity in free space (C = 0) at corresponding location and direction of transmitter and receiver Electric field intensity

R D Q V * m-l

According to Wait (1955) the mutual impedance between transmitter and receiver above and in a homogeneous halfspace (o > 0)

ohm

Mutual impedance in free space (u = 0) Magnetic dipole moment Iertzian vector of the magnetic II* but abbreviated to II)

A * ms type (often denoted

Integrating parameter = ile/B . Dimensionless integrating formerly by Wait (1951 and 1955) :A2+ P) :o1 uo + :fwo

-

R

V-S

m-l parameter,

used

t aou1) sow)

Eigenfunctions, depending on I, h and physical propertions of half-space. Used in solving partial differential equations connected to specific boundary value problems

m-l mho * m-s -&--

184

MALMQVIS’I

DAVID

The

most

convenient

manner

to treat

the potential

field

dipole seems to be to express it by means of a Hertzian type. The vector equations can then be written:

(1)

E -:= -

from

vector

a magnetic

of the magnetic

curl II

and B == poH = grad div II -

(2)

By using sine wave current

.

ysn

of time factor exp (iwt) the equation

(4.1) is simpli-

fied to (3) and

because

curl

E = -

E :- -

ice curl II

B 7-y-

hB

a vector

equation

including

only

EI is

obtained grad div II -

(4) which

can be simplified

that means, plied

the Laplacian

by the square

curl curl II -

to the well known

of the Hertzian

of the propagation

+I

r. 0

wave equation

vector is equal to the vector itself multifactor.

5.

The vertical magnetic dipole. The most important contributions to the theory of the vertical magnetic dipole are given by Sommerfeld (1926), Buchheim (1952), Wait (1955), Meyer (1962) and Bhattacharyya (1963) and above a layered earth by Wait (195 l), Slichter and Knopoff (1959) and Kozulin (1963). Only a brief recapitulation of results will be reported here. We assume that the vertical dipole IS situated at the origin (0 , 0 , 0) of a righthanded cylindrical coordinate system above a surface of discontinuity at x = - h (fig. 3). Positive x is directed upwards and the dipole is directed along the z-axis. The task is to deduce an expression for Hz at any point P with the coordinates (Q , y , z) when z 5 0 and especially when z = 0 (Pos. I). Under the given conditions it has been shown that, due to the geometrical symmetry, the only component of the Hertzian vector different from zero is II Z . Therefore, we can drop the vector denotations and directly write the different vectors as differential operators. From eq. (4.3) we obtain for the only component of the electrical field intensity that is not zero

A NUMERICAL

Fig. 3.

Chosen

CALCULATION

coordinate

systems.

OF THE

Cylindrical,

mitter at (0 , 0 , 0) above a discontinuity

ELECTROMAGNETIC

rectangular at (0 , 0 , -

FIELD

or Cartesian.

ETC.

Position

185

of trans-

h) .

(1) and from eq. (4.2) the components

of the magnetic field intensity

and (3) Eq. (4.5) gives the partial differential

equation

(4) In solving the eq. (5.4)

we write llz = I?,(e)

17, - functions, the one only depending two variabels, eq. (5.4) can be written

(5)

- n,(z)

as a product

of two

on e , the other on z . Separating

the

DAVID

186

MALMQVIS’I

If, in the usual manner, the first part of eq. (5.5) is set .; -- 2s and the second = + 12 we get two ordinary differential equations of which the first gives a Bessel function Js(&)/the the second an e - function

second kind Bessel function exp (A 1/A” + ysz) = exp (f

all values from 0 to 00, the sum of the particular

solutions

Ys(+) is rejected/and UZ) . Since L can have must

be:

m -1

(6)

142, h) exp(31~Jo(@W

f

6

where an eigenfunction y is included in order the boundary value problem below. The eq. (5.4) h as also a singular solution:

a nontrivial

solution

o

expt- YW

(7) which

to obtain

according

to Sommerfeld

can be written

as an infinite

integral

(8) and which must be added to the equation for that medium where If we now denote all quantities by an index 0 , and those in the index 1 we can write the Hertzian

integral (5.6) in forming the boundary value the transmitter is located. in the upper medium with the conductivity uo lower medium with the conductivity a , by an potentials

(9)

(UOZ)

uo exp

i

y0(2

,

h) exp (-

uoz)]J~(@)dA for02

zz

-h

and m

(10)

/I~,

= h

\/a

yl(L , h) exptw)Jo(h)~~ forzd

--h

From the boundary conditions that E,, := E,,i and H,, ~- Id?, (the permeability in the two media are equal) at the boundary z -~- --- h , it follows that 17 20 = 17 21 and gzo

bz

of the integrals to z are equal,

317z1 =-

at the

same

boundary.

Hence,

the

integrands

32

in eqs. (5.9) and (5.10) and their which cannot be fulfilled unless

partial

derivatives

with respect

A NUMERICAL

CALCULATION

wo=

OF THE ELECTROMAGNETIC

I

uo-Ul

* ___ exp (uo + Ul

G

187

FIELD ETC.

22&z)

and y1=

If we set us = 0 and consequently half-space except l7,i we obtain

u = il and drop the indexes

for the lower

m

17~ = k

(11)

+A‘--u + u exp (-

exp (M

ilz -

21h)] Jo(2~)dl for02

zl

-h

and m II,1

(12)

= k

- 23, 1+u

s”

exp (uh + uz -

;vt)Jo(L~)dn

for z 5 -

h,

and for z = 0 Zl,,=k

(13)

1+

.I[

b

Differentiating

the eq. (5.13) according

k/p0 [l/e3

I-I, = -

(14)

21.41 Jo(WA

exp (-

s

to the scheme

+ 112

$$

exp (-

of eq. (5.3) gives

2Uz)Jo(d@r3]

0

Setting obtained

a = 0 and

consequently

(15)

HO

which shows that of the dipole. Eqs. (5.14)

which

-k/,uo$

the arbitrary

and (5.15) HZ -=1+es Ho

(16)

=

ratio naturally

u = ), , the field

constant

=

intensity

in free space

is

-m/poe3,

k is equal

to the magnetic

moment

m

give

s

m i12(u - 1) u + 1 exp (-

2fiJt)JdW~

0

is equal to the ratio -?

often

used

by Wait.

20

We introduce and obtain l

now the dimensionless

parameters

A = 2fi/g

and B = k/g

It is of interest to state that Kozulin (1963), in his study of the vertical dipole

*

above a layered

earth, has introduced two complex parameters p and z which obviously are equal to (1 + i)B and (1 + i)A

. Their moduli /p/ and/z/ are 1/rB and 1/F A respectively.

DAVID

188

MALMQVIS’I

4

1 -‘r Ba, I 0

l

g2(d -

6)

Cd +- d

exp (-

gA)Jo(gB)dg

where d = (ga + 24+. 6. Numerical calculation of Wait’s integral To(A , B) and the field ratio HZ/Ho . The infinite integral in the expression (5.17) is denoted by Ta(A , H) by Wait. It is especially convenient for numerical calculations. If we want to compute the field ratios Hz/Ho w,th an accuracy better than 1O-5 (10 ppm), we must, according to eq. (5.17) compute To(A) B) with at least the same accuracy for a B value ~~ 1 . For lower values of B the accuracy may be lower. In the case of higher values of B the accuracy must accordingly be higher or much higher. Very high values of 13 , however, have little interest in practice. For instance we can state that over a half-space of pure graphite (the best conductor in nature common in large quantities) the B value would be only about 8 for a conductivity o = 2.lOsmho.m -1, a frequency =~ 1000 c/s and a coil separation = 8 m. Therefore higher values of B than this have only theoretical interest but are nevertheless of some importance as support for drawing diagrams. 6.1. Special cases 6.1.1. Lim To(A) B) = To(O) B) A+0 In this case, corresponding to field responses very near to the surface of the half-space, we may set exp (-gA) = 1 in eq. (5.17). Further we set CC * = -

B3 ./g’Jo(gB)dg

and obtain

0

( ‘1

2 The

= B3 [i ,,f (g51dJJoCgB)dg last integral

2 ,of (g31d).Jo(gB)dg -

in eq. (6.1) = 9/B5 ( see below).

The

can be deduced in closed form from the above mentioned /eq. (5.8)/ which, with B as parameter, is m

I

(2) wherej=

‘o =(g,d)Jo(gB)dg =

1/z=

1 + i.

exp L-jB),

i ,i: gdJo(gB)dg] other

two integrals

Sommerfeld’s

integral

A NUMERICAL

Repeated

CALCULATION

differentiations

OF THE ELECTROMAGNETIC

of both

sides

of eq. (6.3) with

FIELD ETC.

respect

189

to B give

cc

1

(3)

k3/d)_l&B)&

= -

“‘(gjB)

[js + j/B + l/B21

0

and co

,/’ (g5,d),o(gB)dg

(4)

= exp(;jB)

[j4 +

2j3/B + 5jz/Bs

+ 9j/Bs + 9/B4]

0

When

summing

up the integrals

with their

coefficients

according

to the eqs.

(6.1), (6.3) and (6.4) we get HZ = l/B2 [(iPI Ho

(5) where

Pi and Ps are polynomials

Ps) exp (--jB)

-

9i]

in B

Pr=9+9B-2Ba P2 = B(9 + 8B + 2Bs) . The expression (6.5) corresponds to that published earlier by Wait (1951, 1955) but is expressed by him only in y and Q . We may further separate it in a real and an imaginary part and obtain

(6a)

Re (2)

(6b)

Im

= l/Ba(Pr

HZ = l/Ba[(Pr ( Ho 1

If B is small it is difficult formulas (6a) and (6b) unless and and

(‘4

exponential obtain

functions.

Re

sin B -

Ps cos B) exp (-

cos B + Pa sin B) exp (-

B)

B) -

91

to compute the field ratios according to the exact we use very accurate tables for the trigonometric

It is then

8 HZ =l+BsTS-T+ ( Ho 1 (

better

B

to expand

the functions

in series

16Ba B3 --... 1 105 720

and

0)

I,(-$-)

= Bs(k---g+E

. . . ...)

which formulas are well suited for B 5 0.25 . The formulas of this section concern field ratios. If wanted, however, it is easy to transform them in Wart’s integral by means of the following expression:

190

DAVID MALMQVIS’I

(8)

To(A , B) = l/Bs [$

6.1.2

Lim B+O

case is largely e = 0 , which

real part

of To(A)

Re[To(A, WI =

(9)

of theoretical means

theless some formulas, useful general case, are deduced. 6.1.2.1 The

l]

Ts(A , B) = To(A , 0)

This special be zero unless

-

interest,

because

that we no longer

as computing

checks

if A > 0 , B cannot

have a dipole field. Never-

for the calculations

,/^P exp(-_gA)Jo(gB>dg 0

2

exp(- s-VJ&Bk&

+ 4)& -PI

./ 0

-__@ rgqg4 + 2 i

4)$-

a g”l exp

gA)J&Wk

(-

For B = 0 we get Jo(gB) = 1 . The first integral on the right-hand 2 __ and consequently the following expression can be obtained: As

Re[Ta(A

,0)]= $-T,

l/z

i‘

I,

t--------

-

(11) which

1/2 t and Q =

Ii = after expansion

$

[gs[(g” ‘1

4 _/’ ts[(t4 + l)‘-

__

--__-_.+

+ 4)’ -gs]‘cxp

r/2 A , the integral

of the algebraic

side is

g3[(g4+ 4)'-gg2]‘exp(-gA)dg

0

If we set g =

the

B) can be written

-__vz ag3W

(IO)

under

Ii

(-gA)dg

is

t2] ‘exp (-

part of the integrand

at)&

in series gives

A NUMERICAL

(12)

I1 = -

CALCULATION 4[hFs(a)

OF THE ELECTROMAGNETIC

+

k2F5(a)

+

K3F7(a) +

....

FIELD ETC.

191

knFsn+i(a)]

where 1 F,(a)

=

.1' tY

exp(-

at)&

0

and where the coefficients are calculated most easily by means of the recurrence formula k n+l =

(_

&$

l)n+l

k,

;,

1 4”

integer >’

Likewise the algebraic part of the integrand Is can be expanded in series giving

(13)

12 = -

$

2[2 [Az(a) -

Ela(a)

& E2(a) + k

Es(a) -

g

Em(a) +

- $$ @18(a)-iE22(a) +(i)”Ezs(a). ...>]

where m E,(a) = SW’

exp (-

at)&

1

The integral F,(a) can be integrated in closed form, but the expressions are complex for large values of Yand, therefore tedious to use for computing purposes. m Tables of the integral A,(a) =

s 1

tY exp (-

at)&

with eleven figures, for

Y = 0 - 15 and a = 0.25 - 12, are calculated by Kotani et al. (1938) and from that integral F,(a) is computable according to the formula

(14)

F,,(a) =

(’ ~ + l)! - A,(a) (p+1

The integral E,(a) is tabulated to 7 decimals by Pagurova (1961) for Y = 0 - 20 and for a = 0 - 10 in increments of 0.01. The tables mentioned are not complete enough for computations

with smaller values of A (= q)

and the for-

mulas have been used upto now for A > 1 only. For A = 1 the formulas (6.10), (6.12) and (6.13) g ive Re[To(A , 0)]= 0.112704 instead of 0.112718 in the table II. For higher values of A the discrepancies are much smaller.

DAVID

192 6.1.2.2 The

imaginary

(15)

Im[To(A

MALMQVIST

part of To(A , B) can be written

as

1/Y p , B)] = -2- ;/ g3[(g4 t- 4)‘$- g”]‘exp

I’R”

-

;I

(-

gA)Jo(gB)dg

exp (-

gA)Jo(gB)dg

and further

(16)

Im[To(A

, 0)]

In the same manner is separated

as above the integral

in two parts

(17)

= 4 ; / ’ t3[(t4 + l)+ -$- t2]+ exp (on the right-hand

-$-

side of eq. (6.16)

11 and 12 . Hence

II = 4[kiF3(a)

I- k2Fs(a)

-t-

k3F7(u)

. . . . . k,&‘zn+,(u)]

where k,,.i

ur)& -

=x (-

1))’ v

k,

t,_:

7 an integer I

;:, 0

Es(o) -;&9

Ei2(o)

and (18)

12 = q 243 1 -+ 215

[; A4(o) + A,,(U) -;

E4(o) + g 4

28497

(E30(4 - J E24(o)

219

+

(u) . . . .

>I

As in the case of Re[To(A , 0)] the formulas above have been used upto now for A values > 1 only. For A = 1, Im[To(A, 0)] can be calculated to be 0.199593 instead of 0.199630 according to the table 11. 6.1.3 (19)

Lim

$

= 1 + (2s2 -

1) (s2 + 1)):

A+rx B -+ 03 where s = A/B = 2h/p . This limit value corresponds to field ratios good conductivity. The proof for the formula general case.

above a half-space of infinitely will be given below under the

A NUMERICAL

CALCULATION

6.2 The general case 6.2.1 Small or medium

OF THE ELECTROMAGNETIC

FIELD ETC.

193

A values

both Most of the calculated values of To(A , B) in the table II are computed by means of Simpson’s rule and a modification of it. This modification (Kiepert 1918) implies

that parabolas

according

to the normal

integrand

(instead

of the 4th degree (instead

Simpson’s

of 3 points)

rule),

going

and replacing

of the 2nd degree parabolas

through

5 adjacent

the integrand,

points

of the

are determined

and

integrated. 6.2.1.1 Another

(20)

manner

of writing

Re[To(A , B)] =

According to the scheme eq. (6.20) has been calculated in the intervals shown: 04 1220 -

the real part of To(A , B) in eq. (6.9) is the following:

4 12 20 60

60 - 100 > 100

in increments D H D B 1) D D 0

H D

[(g” -l- 4)

s

-g21

B } exp (-gA)Jo(gB)dg

for Simpson’s rule calculations, with 7 decimals for the following

of 0 H H

the integrand of values of g lying

0.1 0.2 0.5 1.0

0 5.0 % 10.0

The calculations The calculations made by using the

are carried out until the value of the integrand is < 10-V . of the algebraic part of the integrand for g values > 4 is series

(21)

i(l 2gs

-Gg;+

z---E_ 8g6

64gla

The values for the e-function of the integrand Hayashi (1937) and those for the Bessel function the Harvard University Press (1947). 6.2.1.2 The

(22)

integrand

of the imaginary

part

. . . . ...) are taken from from the tables

of the integral

1/2g3 C(&?+ 4+ + gs16 { -2-

g4} exp (-

the tables published

by by

Ts(A , B) gA)Jo(gB)

is calculated in a similar way and according to the same scheme as mentioned above. For g values > 4 the algebraic part of the integrand (6.22) is calculated according to the series

DAVID

194

MALMQVIS’I

;(l_--5_i_2L429_

(23)

4g4

6.2.2. Large values

8g”

)

6W . . . . . ’

of A

For large values of A , the computing technique above is unnecessary. Besides, if the B values are simultaneously moderately large, the technique is impractical or impossible the integrand.

to use owing

to rapid

oscillations

of the Bessel

6.2.2.1 We can write the real part of To(A , R) according

function

part

of

to the eq. (6.20) as follows: 02

(24)

, B)] = j’

Re[Ts(A

g2 exp (-

gA)Jo(gB)&

-

;j0 (N rt -

0

N&g"

exp (-

e‘VJokW&

where (25)

N,

= ; (5$ + 2)&ga(s) = ; [(g” + 4)i + 214 g

1 + $

N,

= ;(5+-

Y$

and (26)

2)+ga(g) = ; [(g4 + 4): _

01(0) and40)=

2 - a(,) are here variable exponents from 0 to cc , ctcgj varies from 0 to 1 and pCg) from and N, are approximately equal to the expressions eqs. (6.25) and (6.26). We consider now the infinite integral

214 z

depending on g . If g varies 2 to 1 . If g is small, N, to the extreme right in the

* (27)

I(Y

, A, B) = bj’g” exp (-gA)Jo(gB)dg = Id{gvJo(gB)j

which can be integrated in closed form if Y is = 0 or an integer > 0 . Since the integral in (6.27) is the Laplace transform of g”Jo(gB) , it is most easily deduced according to a theorem in the theory of the Laplace transform. If (according to Lipschitz’s formula) (28)

I,( Jo(gB)}

= F(A) = (Aa + B’)-’

7: Go

then 3

(29)

L{gJo(gB)}

= -

v

z ;

~2

_+_ I)--2

__ G,

A NUMERICAL

CALCULATION

L{gsJs(A

OF THE ELECTROMAGNETIC

, B)} = $$

= v

(~2 + 1)

--

FIELD ETC.

195

“, = Gs

and so on. If we denote Gi to be the negative derivative of GO with respect to A, Gs the negative derivative

of Gi and so on, we can write the following expressions

for

G up to index 5: (31)

Ga = 3s/B(2.r” -

3) (2.9 -

1))l(sa + I)-iG2

(32)

G4 = 3/B2(&4 -

249 + 3) (2.9 -

(33)

Gs = 15s/B3(&4 -

1))l(ss

40~2 + 15) (2.9 -

+ 1))sG2

1))l(s” + l)-3G2

Because I( Y , A , B) is a continuous function of Y , and 0 $ (x(I) 4 1 , we say that (34)

may

co G3 2 ./’ gs+=(g) exp (-gA)Js(gB)dg 0

2 G4

where the upper inequality signs are applicable if I(V) A , B) is continuously increasing in the interval Y = 3 - 4 , and the lower ones if I(Y , A , B) is decreasing in the same interval. The same is true for cc (35)

G4 2 s

ga+p(g) exp (-

gA) Jo(gB)dg z G 5 .

0

If A is relatively large (> 10) , high values of the integrating parameter g are of no significance for the magnitude of the integral. Hence atgb g 0 and p,,, E 2 and the first integral (6.34) above Therefore (36)

the integral in (6.24)

Gs and the second (6.35) g Gs .

can be written as

Re[Ts(A , B)] E Gs -

6.2.2.2 In a similar way as described

s

k (5’ + 2)‘Ga

+ i (.5* -

above the imaginary part of Ts(A , B)

s

co

(37) WTo(A , B)l =

s

g3(N.

2)*G 5

lx.7

+

Np) exp (-gA)Jo(gB)dg -

g4

0

0

expC--gA)J&B)& can, for larger values of A , be written as approximately

(38)

Im[Ts(A , B)] E i (SB + 2)+Gs + l(5’

-

2)‘Gs

-

G4

DAVID

196

~Ai~~lQVI~l

Table I. Comparisions

of the Simpson’s

rule approximation To(A

and the derivative approximation

in computing

, B) and T$A , B). A =: IO

-

-

Simpson’s

rule

Derivative

Simpson’s

rule

Derivative

appr.

appr.

appr.

wpr.

0

-

0.0014288

0.0014291

0.0003902

~.0003892

0.5

10

0.0014206

0.0014209

0.0003864

0.0003854

1.0

5

0.0013963

0.0013966

0.0003754

0.0003744

2.0

23

0.0013044

0.0013041

0.0003341

0.0003333

3.0

18 1

0.0011668

0.0011662

0.0002753

0.0002747

0.~08296

0.0008281

0.~01453

0.~1458

5.0

-

Re[Tz(A

-

, @I

IO

0.0003561

0.0003~62

0.~~972

0.0~0969

1.0

5

0.0007059

0.0007064

0.0001918

0.0001908

2.0

29

0.0013648

0.0013654

0.0003616

0.0003606

0.0019376

0.0019383

0.0004936

0.0004930

0.0027410

0.0027408

0.0006236

0.0006227

0.5

3.0 5.0

/

1Q 1

-

6.23 The close agreement between the computation results by means of the modified Simpson rule and by means of the above described approximation technique, which can be called the derivative approximation, is evident from table I for A = 10, The calculations of Ta(A , B) for A values > 10 (tabte 11) have been made by means of the derivative approximation technique only. 6.2.4 Concerning the table II it may be mentioned further that only a few calculations have been made for medium values of A . As shown in the table the results of these calculations agree very well with those tabulated by Wait (1955), which are also included in table II. 6.2.5 Suppose that in accordance with eq. (5.17) we multiply the expressions (6.36) and (6.38) by B3 . If now A and B tend towards infinity while their ratio remains constant (= al1 the terms of (4.36) and 6.38) will vanish except the term

s},

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD

ETC.

197

BsGs of (6.36) b ecause G2 contains a third power of B in the denominator while Gs , Gq and Gg all contain higher powers of B [compare the expressions (6.31), (6.32)

in the denominator.

and (6.33)]

Hence

HZ = 1 + B3Gz = 1 + (2~2 Ho

(39)

which corresponds

1) (~2 + 1,:

to the limit value of eq. (6.19).

Table II The integral To(A

, B) -

A

According B

h/e

Re[To(-%B)I

ImTo(A,B)]

’- 1

Re[To(A,

0.00

0,05

OJ

0,oo

00

0,02

0,523394

0,05

0,508714

9,467048

0,lO

0,484856

4,468191

0,20

0,439428

1,971962

0,50

0,320746

0,496866

1300

0,176402

0,061895

24,466728

0,448871

9,485585

0,02

11/P

0,447947

8,770393

0,05

‘la

0,443589

6,556977

0,lO

114

0,431464

3,958207

0,20

‘10

0,401398

1,915836

0,50

'I10

0,301376

0,508176

l,oo

'/xl

0,170143

0,073967

0,oo

0,oo

0,397600

4,503240

0,397003

4,406622

0,05

2’18 1

0,394807

3,975662

0,lO

l/a

0,387707

3,041100

0,20 0,50

l/4 l/l0

0,365483 0,283113

0,499866

l/20

0,164456

0,089331

0,02

1 ,oo w

0,533333

1,743647

0,325538

2,035320

0,02

5

0,325341

2,020296

0,os

0,324315

0,lO

2l/a 1

1,960928 1,772382

0,20

l/P

0,308133

1,306627

0,50

‘16

0,228385

0,484374

l,oo

'I10

0,151866

0,095124

0,oo

0,320784

to Wait

WI ( WTo(A, WI

198

DAVID

~AL~QVIS~

B

A

-iIie[To(A,B)]

Accordi] RelTn(A,B)J

-

0,s

0,oo

0,204321

0,02

0,204265

0,611498

0,05

0,203966

O&O7448

0,lO

0.202910

0.593393

0,202910

0,593397

0,20

0,198812

0,542901

0,50

0,174481

0,332304

l,oo

0,119304

0,105494

1,50

0,070718

0,01713O

2.00 3,00

0,036311

- 0,012953

0,002766

- 0,017954

-- ___-_

OS@ 0,02

0,112718

0,199630

0,112702

0,199539

0,os

0,112611

0,199075

0,lO

OJ12284

OJ97405

0,20

0,111005

0,191023

0,50

0,102660

0,153718

0,079333

0,078759

1,SO

0,053875

0,028509

2,00

0,032612

0,003314

0,007381

- 0,009467

0,079333

1,oo

3,00 5,00

0,078752

0,0073812

- 0,0094642

- 0,0032313

- 0,0037232

--

_Cl,069468

O,OO OJO

0,090803 0,069282

0,090232

0,20

0,068737

0,50

0,065065

0,088545 0,077796

l,oo

0,053888

0,050340

1,50

0,040106

0,025267

2,00

0,027138

0,~8758

0,009132

- 0,003929

0,lO

0,045733

0,047977

0,20 0,50

0,045460

0,047365 0,043327

3,00 5,00 --

- 0,0039273

- 0,0015154

- 0,0030115

0,045824

0,048183

--

0,043607

1 ,oo

0,037714

0,031788

1,50

0‘029891

0,019227

2,OO 3,00 5,00 --

0,0091329 ..-

0,00

2x0

WT&%fol

0,612276

_-

I,0

to Wait

[mlTo(A,Bfl

--

0,0093140

- 0,0007549

- 0,0002488

- 0,0022558 --_II

_-

0,021888

0,009272

0.009314

- 0,000755

A NUMERICAL

CALCULATION

OF THE

ELECTROMAGNETIC

FIELD

Accordh A

B

330

0,oo 0,lO 0,20

530

to Wait

h/e

0,022822

0,017513

15

0,022792

0,017469

0,022705

0,017340

0,022102

0,016461

0,50 l,OO I,50

I’/2 1

0,020101

0,013689

0,017220

0,010089

2900

ai4

0,013950

0,006553

3300 $00

‘12

0.0078427

0,0015375

0,007842

0,001538

sllo

0,0011936

- 0,0010274

0,007896

0,003988

0,007891

0,003983

0,007891

0,003983

0,007875

0,003968

0,007765

0,003870

0,007387

0,003537

0,oo 0,lO

25

0,20

121/a 5

1 ,oo

2112

I,50

1213

0,006803

0,003046

2900 3,00

1114

0,006074

0,002470

61s

0,0044396

0,0013384

0,004440

0,001339

5,00

‘12

0,0017424

0,0000360

0,007387

0,003538

O,OO 0,lO

0,003000

0,001065

371/a

0,002999

0,001064

0,20

IV/4

0,002996

0,001062

0,50

711s

0,002975

0,001048

1 ,oo

3314

0,002892

0,000999

I,50

2112

0,002764

0,000923

2,00

IT/S

0,0025951

0,0008263

3m 5,00

I’/4

0,0021766

0,0006014

"I4

0,0012805

0,0002048

0,001429

0,000389

-_-

-l&O

199

71/a 3

0,50

7,5

ETC.

0,oo 0,lO

50

0,001429

0,000389

0,20

25

0,001427

0,000388

0,50

10

0,001421

0,000385

1900 I,50

5

0,001397

0,000374

3’13

0,001357

0,000358

2900 3,00

2’12

0,0013041

0,0003333

1*/3 1

0,0011662

0,0002747

0,0008281

0,0001458

5,00

DAVID

200

MALMQVIST

hg B

A

ii!‘Q

Rel’Co&Wl

:?W)l

0,000476

0,000090

0,000475

0,000089

7’12 5

0,000471

0,000088

0,00~65

0,000086

0,oo

15.0

0,50

1s

1,oo 1,50 2m 3,00

33/s

0,0004557

0,0~82Y

2112

0,0004310

0,0000754

$00

l’/z

0,0003625 --._-___

0,0000560

0,000213

0,00003 1

--

_0,oo

20,o

-__- --

0,50

20

0,000213

0,000030

l,oo

10

0,000212

0,00~30

63/a 5

0,000210

0,000030

0,0002074

0,0000291

3113 2

0,0002007

0.0000276

0,0001809

0,0000231

1

0,0001144i

0,00000996

O,OOQO15

O,~O~l

1,50 2,00 3,00 5,00 10,oo

--__-

.-_-

-0,oo

50,O

to Wait

MTofA,W

0,50

SO

0,000015

0,000001

1 ,oo

25

0,000015

0,000001

1,50

162/3

0,000015

0,000001

2,00

12’12

0,00001 SO

0,0000009

81/3 J ______--

O,OoOO149

~,00~009

3,043 5,OO

_- -_-

0,~00146 --I_-

~),OO~)OO~

7.

The horizontal mqnetic dipole Important contributions to the study of the horizontal magnetic dipole have been made by Sommerfeid (1929), Belluigi (1949) and Wait (1953, 1956). We assume that the horizontal dipole is situated at the origin (0 , 0 , 0) of a right-handed

Cartesian

coordinate

system

above a discontinuity

surface

at z = -

(fig. 3). Positive z is directed upwards and the dipole is directed along the y-axis. For this case Sommerfeld has shown that II, = 0 ) which condition simplifies the differential operators to:

I according (lc) and

to eq. (4.3)

h

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

(24

an;: poH,=+$$-+&

Pb)

3 MT, x7, ,usH, = --y=lTu 3y ( 3y + 32 1

PC)

POH,= ;

At the boundary, that of eq. (7.lb),

317, (3 + =) z = -

201

\

.

according to eq. to (4.2)

I

y=~,

h we have, owing to the condition

IT,0 =

(3)

FIELD ETC.

E ys = E yr and

1721

and, owing to the condition E,s = E,I and that of eq. (7.la) Ml,0 -=3x

(4) Further,

(5)

owing to the condition

m,o 3y

317$/l 32

Hz0 = Hz1 and that of eq. (7.2a), we have

3nzo

+3x=

-3l7,l JY

+

_XI,1

32:

and, owing to the condition H,s = H,I and that of eq. (7.2b), ]‘02flyO =

y1=fl,1

or (6)

col7yo =

OlD,l

which tells us that at the boundary adjoining free space (us = 0) we have 17, = 0 . According to the eq. (4.5) the partial differential equation in Cartesian coordinates can be written

Setting Q = 1/3cs+ ys , x = Q cos tp and y = Q sin q~, we rewrite (4.7) as a sum of two partial differential equations

(8) and (9)

the eq.

DAVID

202 Because $

MALMQVIST

= 0 , the first of these equations is almost identical with that

for the vertical magnetic dipole (5.4). In solving it, we have only to observe that the boundary condition (7.6) is different from that in the formerly treated case. But, nevertheless, we can almost immediately write the solution for the y-components

(10)

f17,e = k if

& [exp (ZQ.Z)+ f exp f-

2~s~-

Z&z)] Je(+)dl?for

0 1 x 2 --h

and

It was further proved by Sommerfeld that the second differential equation (7.9) could be solved by a kernel function sin p exp (~~)J~(d~) included in the integral. Hence we can set

(12)

Iii%0

=

k

sin 93j’f0(1,/2)

The boundary

condition

fo

(14)

Setting ~0

I=

exp (-

uox -

~oh)Jl(Ae)dA for 0 2 z 2 -

(7.3) makes fe = fl

c

fi and the condition (7.5) gives

212(no - @I)exp (_

u&)

(uo + Ul)~

0 , uo = A and z == 0 in eq. (7.10) gives

17~o =k

(15)

J‘cl + exp(-

Zfi)]Js(A~)dn

and m

(16)

an?/0

---=-ksine, ‘sy

J 0

41

+

h

q

(-

2~)]J0(~e)~a

A NUMERICAL

Likewise

CALCULATION

setting

and (7.14)

L I+u

2iVt)Ji(le)d1

which

after partial

2k sin p

differentiation

3flZO

(18)

-= 32

If we observe to the scheme

exp (-

with respect

2ksinv

s 0

O3 12 -J+u

of eq. 7.2b (with

izz -

to z and then

exp (-

that 2 = sin 9 cP + cos v i $ 3Y

po&,o=-

(19)

ss

203

FIELD ETC.

(~0 = 0 , uo = ), in the eqs. (7.12)

Uzo = -

(17)

OF THE ELECTROMAGNETIC

gives

setting

z = 0 gives

2%)Js(+)d1

we

obtain

finally

according

y = 0) + -3fl1,o 1 = 2k sin2q -~ k cos‘$ 32 e3 e3

3 ( -3n7,o 3y 3y

co

k sinsp, s

-

n2rF1i)

exp (-

2ti)Ja(il~)dil

0

k sinsg,

+e

-- k co&p

* n(u - A) u + il exp (.I”

2fi)JdhW

0

f

Q b

O”A(24- 1) u + il exp (-

2fih)JdW~

7.1 If we set p7 = 90° in eq. (7.19), We change

the notation

H,s

an expression to 1 H 111and write

po[ Hlrr=$+k

(20)

F

Qi

for the position

m n(u - n) u + 1 exp (-

2fi)

II is obtained.

Jl Gk.W

(0

-k

which

expression

after division

and after introduction

f b

P(u - 2) u _ A exp (-

by the free space induction

of the parameters

A and B gives

23Jo(%)dn

2m

,UO1 H ! II = F

=

2k

2

204

(21)

DAVID MALMQVIST

~

H Ho

II

z - = 1 + ; [T&4, B) - BTo(A , B)] Zo

where co

(22)

T&4 , B) =

dd-d I‘0

g4JlW)dg

exp (-

d+g

is Wait’s integral of second order.

7.2 By setting p = 0 in eq. (7.19), the position III:

(23)

obtain an expression

for

k k *n(u- A) exp(- 2~h)Jl(QW tcolHI111= - -3 ./ u+a e - -@O

and after division by ,UOj HO /III

(24)

we correspondingly

=

-

s

=

-

k

p

z

i H -; - L 1 $ BsTz(A , B) i Ho 1111 Z.

Wait (1956) has derived the expressions (7.21) and (7.24) by using the scalar magnetic vector y = - div II . This method obviously leads quicker to the solution for these special components, but is less clear if the other components involved are desired to be deduced. 8. Numerical calculation 1 H and I &!l-lHo II I HO h ’

of Wait’s

integral

‘l?z(A , B)

and

The calculation of the field ratios in the positions II and III analysis of Ts(A , B) . We start with two special cases.

the field

ratios

needs a closer

8.1 Special cases 8.1.1 Lim Ts(A , B) = Ts(0 , B) A-+0 The integral Ts(A , B) in the expression (7.22) in 4 infinite integrals as follows:

can, for A = 0,

be split up

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC m

m

(1)

J

T2(o, w = Jgh(gsjdg - i 0

FIELD ETC.

205

- 2sk214JdgW& m

g3JdgW&

0

o

If we recognize that the last two integrals in (8.1) are found by differentiating the integrals (6.2) and (6.3) with respect to B , the integrals on the right-hand side of (8.1) are successively equal to

Tz(O,33)= l/B2 + 3ilB4 - 2 exp ;jB)(j

(2)

_ a’exp (-jB)

B

+ l/B)

(ja + 2ja/B + 3j/Bs + 3/Bs)

and after simplifying (8.2) we find that

Tz(O, B) = l/B2 + 3ifB4 - exp (;j”

(3)

(iP 5 -

Ps)

where Ps = 3(1 + B) Ps = B(3 + 2B). From the expressions (6.Q (7.21) and (8.3) it is then easy to show that

(41

Re

I

# o IT = 2 -

l/Bs(Ps sin B -

1

P4 cos B) exp (-

B)

and Im

(5)

I

-#= l/Bs[6 0 III

(P s cos B + P4 sin B) exp (-

B)]

where p

3

=p1+-p5

2

=6+6B-Bs

p* = Ps + Ps = B(6 + SB + Ba) 2 These formulas give, after expansion of (8.4) and (8.5) in series (for B 5 0.25), (6) and

Re & i

0 III

= 1-

B3(2/15 --B/6

+ 2Bs/35 I..,..

.)

DAVID

206

= B”(2/15 -

(7) From (7.24) manner : (8)

MALMQVIST

2B”/35 + Bs/36 . . . .)

and (8.3) we can deduce

the following

Re

=2--l/Bs(PssinB-PscosB)exp(-B) H’ I I%” IIII

Im

I- l/Bs[3 -(P -EI Ho I III

expressions

in a similar

and (9)

6 cos

B +

p6

sin B) exp (-

B)]

and as series (for B 5 0.25):

P)

Re

= 1 + B3(4/15 -B/6 -k!_. I Ho I III

Im

= Ba(1/2 If_ 1 Ho I III

+ 4Bsj105

. . . .)

and (11)

; (252 I

4B/15 + 4B3/105

1) ($2 +

1 + ; (2 Ki will be defined

1,.%

. . . .)

:=

$2) (9 + q-4

later.

8.2 The general case 8.2.1 Small and medium A values. Comparing the integrals Ts(A , B) and Ta(A , B) we can see that the algebraic part of the integrands is almost the same except that To contains ,gs instead of the g in Ts . All the formulas of the integrals under 6.2 can thus be used for Tz except that the power of the first g must be a unit lower in Tz than in To and that the Bessel function is Ji(gB) in Ts instead of Ja(gB) in To .

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD ETC.

207

In computing the integrands of Ts for discrete values of g according to Simpson’s rule, and after the scheme described above, the calculations of the integrand of Ts are easy to do and the values fall out almost as by-products of the calculations of To . All that is necessary is to divide the algebraic part of the integrand of To by g and then multiply the product of the algebraic part and the e-function by J&B) instead of Js(gB) . Wh en summing up the integrand values in order to calculate the integral the same scheme and the same technique as that described under 6.2 is then used. In table III the calculations of Ta are made more complete compared with those for Ts because the provisional calculations made by Wait (1956) have been carried out to only 4 decimals besides which there are in some cases too large discrepancies between them and the more correct values. 8.2.2 Large values of A . By writing the integrals Ts(A , B) in similar manner as (6.24) and (6.36) m

(14)

Ke[Ts(A, WI = Jg

exp (--gA)Jl(gB)& - J’ g2W. - N,d exp(- $1 0

0

J M94z

and m

m

(15)

+ NB) exp (--gA)JUWg-~‘g3

Im[TstA , WI = Jg2(N,

exp

0

0

6

s9JWWg

we can apply exactly the same reasoning as under 6.2.2 in order to deduce approximate expressions for the real and imaginary parts of T2 . The difference is that in this case we have to take the Laplace transform L(g’Jr(gB)} which, for Y = 0 , is equal to K. = (A2 + BY*-4 B(As + Bs) B In a corresponding manner we denote Kr as the negative derivative of Ks , Ks as the negative derivative of Kr and so on. The most convenient way to compute these negative derivatives is by means of the following recurrence formulas: AGO) = l/B[l -

(16)

Ko = l/B(l

(17)

Kr = l/B(Gs -

(18)

K2 = l/B(2Gr -

-

(ss + l)-$

AGr) = l/Bs(ss + 1)-i AGs)

DAVID

208

MALMQVIS’I

(1%

KS = l/B(3Gs

-

AGs)

(20)

Kq = l/B(4Gs . . . .

-

AGJ)

For large values

of A (;> 10) we can thus

Re[‘l’s(A , B)] g

(21)

Ki -

k(Sf

write

the approximation

$- 2)‘Ks

-j- i(56

-

formulas

as

2)$K4

and Im[Ts(A , B)] G h (5’ + 2)$Ks

(22)

+ i (5$ -

2)‘Kd

-

Ks

An example of the close agreement between this derivative approximation and the Simpson rule approximation for Ts is shown in table I. If we multiply these last expressions (8.21) and (8.22) by Bs , it 1s easy to show, that if A --f CO and B ---f cx and the ratio between them is held constant, all the terms will vanish except BsKr which gives the limit of (8.12) and (8.13). Table III The

-

-

integral TZ (A, B)

-

According

A

We

B

0.00

-

0,005267

0,494667

0,os

0,012921

0,486671

0,lO

0,025038

0,473371

0,20

0,046971

0,446972

0,50

0,096371

0,370626

1 ,oo

0,136471

0,259604

0

0

--

--

0,oo 0,02

0,004484

0,091117

0,os

0,011149

0,194251

0,lO

0,021973

0,283312

0,20 0,50

0,042072 0,088867

0,339260 0,327451

1 ,oo

0,128605

_0,oo

Re[Tz(A, B)

0,500OOO

0

0,02

_61

-

-

0,oo

-0,05

to Wait

Re[Tz(A,B): 1 Im[Tz(A, B)‘I

0,242555

_-

----

0

0

0,02

0,003974

0,os

0,044542 0,105620

0.10

0,009906 0,019624

0,20

0,038009

0,268621

0,50

0,082140

l,oo

0,121243

0,288917 0,226108

0.182348

A NUMERICAL

CALCULATION

OF THE

ELECTROMAGNETIC

FIELD

T A

ETC.

Accordi

3 RelTa(A,B):

0,2

0,s

190

1,5

0

0,oo

0

0,02

0,003254

0,020291

0,os

0,008123

0,049959

0,lO

0,016156

0,094823

0,20

0,031651

0,160813

0,so

0,070655

0,224990

1,oo

0,107965

0,196239

0,oo

0

0

0,02

0,002043

0,006119

0,os

0,005104

0,015246

OJO

0,010181

0,030137

0,0097

0,0302

0,20

0,020154

0,0.57627

0,019o

0,059s

0,50

0,047141

0,111816

0,0439

0,1051

l,oo

0,077592

0,127322

0,0767

OJ234

1,SO

0,090328

0,106129

0,0887

0,1035

2,00

0,090173

0,079003

0,0879

0,0769

3,00

0,072843

0,037032

0,0705

0,037o

0,009o

0,oo

0

0

0,02

0,001127

0,001995

0.05

0,002817

0,~983

0,lO

0,005599

0,009925

0,0053

0,20

0,011186

0,019527

0,0104

0,0192

0,50

0,026892

0,043805

0,0248

0,044O

1 ,oo

0,047303

0,063525

0,046O

0,0617

1,50

0,058860

0,062975

0,057o

0,0617

2,00

0,062634

0,053207

0,0601

0,052s

0‘0542

0,0304

3,00

0,056225

0,030445

5900

0,032554

0,007574

O,OO 0,lO

0

0

0

0

0,003469

0,004526

0,0032

0,0037

0,20

0,006910

0,009006

0,0063

0,0087

0,so

0,016810

0,021022

0,0152

0,0214

l,oo

0,030623

0,034099

0,029O

0,0332

1,50

0,039825

0,037875

0,0378

0,0372

2,00

0,044372

0,035358

0,0423

0,0353

3900

0,043276

0,023852

0,0417

0,0237

5m

0,028493

0,007528

209

DAVID

210

MALXQVIST

A

3

T ReITdA,B)l

IdTdAJ31

-

I -

Accordi Re[‘J?z(A,B)I

0,OO

0

0

0

0

0,to

0,002289

0,002404

0,0019

0,0017

0,20

0,004564

0,004777

0,004l

0,0043

OJO

0,011176

0,011426

0,0096

0,0121

I,00

0,020803

0,019622

0,0192

0,0189

I,50

0,027878

O&)23519

0,0259

0,0229

2,00

0,032132

0,023643

0,0304

0,024o

3,00

0,033663

0,018229

0.0319

0,0183

5,00

0,024694

0,007058

0,oo

0

0

0

0

0,lO

0,001140

0,0@0874

0,0007

0,0~2

0,20

0,002265

0,001742

0,0017

0,0012

0,so

0,005615

0,0039

0,0050

1 ,oo

0,010715

0,004245 0,007758

0,0093

0,0068

I,50

0,014902

0,010087

0,0129

0,0098

2,00

0,017966

0,011153

0,0167

0,0116

3,00

0,019540

0,010441

0.0189

0,0104

----

--

--3.0

5,00

0,005627

0,018267

-- -_-_

-

-5.0

0,oo

0

0

0

0,lO

0,000395

0,0~199

0,0001

0,20

0,000789

0,000398

0,0002

0,50

0,001959

0,000982

0,0022

0,0002

I,00

0,003822

0,001878

0,0022

0,0009

1,50

0,005505

0,002620

0,003l

0,0022

2,00

0,006944

0,003166

0,0057

0,0029

3,00

0,008974

0,003651

O,OO68

0,0033

5,00

0,009977

0,003007

0,oo

0

0

0,lO

0,000151

0*0~53

0,20

0,000301

0,~106

0,50

0,000749

0,000263

1 ,oo

0,001477

0,000514

I,50

0,002166

2,00 3,00

0,002803 0,003857

0,000742 0,000937

5,00

0,005024 ---

-.-

_.--

h[Tz(A,B)I

-

0,001201 0,001316 --~.-

0 _

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD

Accordir A

ETC.

211

! to Wait

B

ReP”dA, WI 10,o

0

0,oo

50

0,000072

0,000020

0,20

25

0,000143

0,000039

0,so

10

0,~356

0,~97

1,oo

5

0,000706

0,000191

1,50 2,00

3x19

0,001043

0,000282

2112

0,001356

0,000361

12j3 1

0,001938

0,000493

0,~27~8

0,~6227

‘iz

0,0030068

0,0004524

0

0

3,00 5,OO 10,oo 15,o

0

0,lO

0,oo 0,50

15

0,000119

0,000022

0,~237

l,OO 1,50

7118 5

0,000353

0,~ 0,000066

2,00

3514

0,000466

0,000086

3,00

2112

0,000681

0,000123

$00

1112

0,0010408

0,0001782

~20,O

OS00 0,so 1 ,oo 1,50 2900 3,00 5,00 10,oo

50,o

0

0

20

0,000053

0,000008

10

0,000106

0,000015

62/3 5

0,000158

0,000023

0,~210

0,~30

3x/3 2

0,000310

0,~~4

0,0004909

0,0000665

1

0,0007866

0,0000925

OS00 0,50

0

0

50

0,~~38

0,0~2

1 ,oo

25

0,0000075

0,0000004

1,50

162/3

0,0000113

0,0000006

2sJO 3,00

121/z 81/s 5

0,00001 so

0,0000009

0,0000224

0,0000013

0,~0371

0,0~21

5900

9.

The nectar diagram Using the tabulated number of interpolated vector diagrams which made by means of the

values of the tables II and III combined with a large values of TO and Ts , it is possible to draw a number of are shown in Figs. 4 to 8. The interpolations are mostly Lagrange interpolating formula.

DAVID

212

MALMQVIST

____-- ---_=--_--/’

/’

Fig. 4

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD ETC.

213

Fig. 4 a) b) c). Schematic vector diagrams for the positions I (a), II (b) and III (c).

The schematic diagrams of Fig. 4 are drawn in order to give a general idea of the character of the responses for the different coil positions. The quantity LReG$-

1 is utilized 1

as abscissa

in these and following

diagrams.

This

quantity corresponds to the response of the real component reduced by the free space response and also corresponds to the adjusted reading of the real component for most instrument of the slingram type. For the imaging component no Z corresponds directly to such reduction is necessary because the ratio Im ( zo 1 the reading of the imaginary component

on the instrument.

The ratios (reduced

or not) can be expressed in parts of the free space response as in Figs. 4 a/, b/ and cl or in per cent or in ppm as in the other vector diagrams. From Fig. 4 it is evident that in positions I and II we have domains of readings where interpretation

of the responses

due to a thick conductor

is ambiguous;

this means that inside these domains the same pair of readings (real and imagina~) can result from two different conductivities of the conductor. In pos. I this ambiguous domain is connected to relatively large readings (5 y0 to 20 %) while in pos. II the domain lies in that part of the diagram (Fig. 4 b/) of small or medium readings which is of special interest for our purpose (compare Figs. 6 a/ and 6 b/). Inspecting the diagram in Fig. 6 a/ we can establish that, for example, a particular pair of readings (Re = - 12000 ppm and Im = - 4000 ppm) corresponds to a B value equal either to 0.75 for A/Q= l/4 or to 2.0 for h/e = 1.1 . If we know the distance to the upper boundary of the conductor (ale) we can decide which of these two alternatives is the correct one. In pos. III we do not have any ambi-

214

DAVID

0

I:ig. 5 a).

4000

Vector

6000

diagram

MALMQVIS’I

6000

for position

10000

I for medium

12000

14000

ppm

readings.

guous domains of this kind and in this case a given pair of readings corresponds to a given B value (or conductivity if h , p and frequency are known) above a homogeneous thick conductor. In Figs. 5, 6 and 7 the diagrams (a) are especially applicable for medium readings mostly met with in ground surveys using beam-slingram or common slingram. The diagrams (b) of the same number are more applicable in cases where airborne systems are used. The area of readings included in the diagrams (b) is indicated on the diagrams (a).

A NUMERICAL

CALCULATION

OF THE

ELECTROMAGNETIC

FIELD

ETC.

215

1600

Fig. 5 b).

Vector diagram for position I for small readings.

The transformation diagrams of Fig. 8 can be used to transform a given pair of readings in pos. I (the coordinates) to the corresponding pair of readings in pos. III. This transformation is of special interest in co-planar systems since it can be used if we take four readings (two pairs) with the same co-planar equipment. Firstly a pair of readings is taken in pos. I as a vertical dipole system and then the system is rotated 90s around the axis connecting the transmitting and receiving coils giving pos. III in which the next pair of readings is taken,

DAVID

MALMQVIST

PPm +10000

rm00

+6OOO

+‘OOO

+*oclo

0

- 2000

- iOOO

Fig. 6 a).

Vector diagram for position II for medium readings,.

Examination of the transformation diagrams of Fig. 8 shows that for small and medium readings the responses of the real components in pos. I are approximately twice the corresponding responses in pos. III. This is also true for the imaginary components if h/e > 14 . For smaller values of h/p the ratio between the responses in pos. I and pos. III varies between 2 and 1 but can be evaluated more closely by using the diagrams of Fig. 8. Such ratio evaluations enable us to adjust a co-planar beam-slingram system to the correct zero position on the reading scale or to make corrections to that zero position if the determinations are made above a homogeneous ground with a conductivity somewhat higher than 0 . An example will be shown in the next section.

A NUMERICAL

CALCULATION

- 600 ..’

1

t

\,

OF THE

I

, ,: .. !

ELECTROMAGNETIC

,.?

.

FIELD

ETC.

217

,:

.:

- 800

-2400

Fig. 6 b). Vector

diagram for position II for small readings.

Transformation diagrams can of course be drawn for the other two possible combinations, namely pos. I and II as well as pos. II and III.

218

DAVID

MALMQVIST

pm 16OOC

14000

__=$ik /

12000

10000

8000

/’

/

/ 6000

,

/’

4000

2000

0

’

1 2000 htjq,=5

Fig. 7 a).

4000

Vector

6000

diagram

8000

for position

10000

III for medium

12000

readings.

14000

ppm

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD ETC.

1400

1200

1000

800

600

400

200

0

Fig. 7 b). Vector diagram for position III for small readings.

219

DAVID

MALMQVIST

wm 1600C

____ A’

/ / __----_------

___-

-

--

___ _

/-

6000

+-_ PI! and X Coplanar

0

0

2000

4000

6000

6000

10000

12000

WOO0

ppm

[Re ($J-11d&L Fig. 8~3). Transformation

vector

diagram

from pas.

I to III for medium readings.

A NUMERICAL

CALCULATION

OF THE

ELECTROMAGNETIC

FIELD

ETC.

221

_

wm

II

1600

111 l”,”

0

5.

8

s:

Li-

I

I

AZ ID

II

ID

___ _

:

II

:

II

($)d?%

Irn

,I

II

II/l

___ ___

5

=^^^ 0””

s--m--------

___.--0

_____

_______--<_

ppm

-_ /

1400

: 700 ppm

:600

ppm

=500 ppm

=400

600

=

pplll

300 ppm

400

200

0

I

0

h/p:15 t

200 h/p=10 \ 400 h&71/2 /

600

800

1000

1200

1400 [Re

Fig. 8 b). Transformation

10.

vector diagram from pos. I to III

($)

ppm

-11

Z$ie

for small readings.

A practical field test This paper will be concluded by describing a field test, chosen to give an instructive example of the usefulness of the diagrams just reported. The measurements were taken on April 14, 1964 on the ice of the Gulf of Bothnia in the mouth KAgefjirden about 10 km northeast of the town of SkellefteH in Northern Sweden. In this survey two different beam-slingram instruments were used, one of the co-planar type the other of the co-axial type. The instruments were mounted in turn on two wooden stages which permitted the measurements to be taken at different heights (up to more than 9 m) above the level of the ice. The thickness

DAVID

222

MALMQVIS’T p

Reading

Frcq.

pm

:

8.201~3

= 800

c/s

x 1000 r2

20

* q _,

h

?&

* q ‘(3y

V = vertical

dipole

H = horizontal

dipole

=

&I

\ -

15

10

\

L

n

V

n

H

e

V

5

\

4

Height c H ~ 1clrn

--i--

ice

above level

lr Boundary i:e-water

Discontinuity surface

Fig. 9.

Measurements

on the ice in KPlgefjlrden with a co-planar

beam-slingram

instrument.

Curves are drawn through the measured points pIotted on the figure. Interpolated are read for 5 different 3r determined

under the assumption

values

that the discontinuity

lies

1,3 m below the ice level

of the ice sheet was about 0.5 m and the depth of sea water about 30 m or almost 4 times the spread e of the coits. The results of the measurements are shown in the Figs. 9 and 10. Interpretation was complicated by the fact that a layer of fresh water, from melting ice, had been formed above the normal sea water. The surface of discontinuity

A NUMERICAL

CALCULATION

OF THE

ELECTROMAGNETIC

FIELD

ETC.

223

Table IV Beam-slingram Measured

measurements on the ice of the Gulf of Bothnia mouth of Klgefjfrden,

by K. Westerberg,

A. Co-planar

instrument

e = 8.20 m

Corrections:

freq. = 800 c/s

Re + 2280 ppm Im f

Vert. dipole

We

Ii/4

Lat. 64”48’.

April 14 1964. Readings in ppm.

450

Hor. dipole

Re

Iffl

mess.

1480

7090

20

3580

corr.

3760

7540

2300

4030

--

0

I

Re

Im

1

meas.

1990

9580

300

5060

corr.

4270

10030

2580

5510

meas.

2740

13160

700

7480

corr.

5020

13610

2980

7930

---

--~=I4

-

--“Is _-____

meas.

3040

14800

830

8640

corr.

5320

15250

3110

9090

meas.

3750

18350

1180

11580

corr.

6030

18800

3460

12030

--

‘!a

--~

B. Co-axial

instrument

(, = 8.15 m

Corrections

freq. = 800 c/s

h/e

11/4

: Re Im-

I

Re

600 ppm 330

Im

meas.

-

320

-

1430

corr.

-

920

-

1760

meas.

-

500

-

1960

1

corr.

-

1100

-

2290

a/s

meas. corr.

-

810 1410

-

2770 3100

meas.

-

900

-

3070

corr.

-

1500

-

3400

-

‘Is

u

224

DAVID

MALMQWS’I

Discontinuity surtace

p = 6.15

I I

Frep.

= BOO

m c/s

&undary ice-

water

Height ice

above level

Re

-

2000

-

3000

I:

,’

ppm

I=’ Reading

Fig. 10.

Measurements Curves

on the ice in Klgefjlrden

are drawn

through

the measured

with a co-axial points plotted

values are then read for 4 different h/e determined continuity

beam-slingram

instrument.

on the figure.

Interpolated

under the assumption that the dis-

lies 1.3 m below the ice level.

was thus not the bottom of the ice sheet but was diffuse and lay somewhat lower. Three water samples, taken at I, 2 and 3 m depth on the day of the tests and afterwards measured in the laboratory at 23OC had reststivities of 4.7, 2.5 and 1.9 ohm - m respectively. At a later opportunity (August 22, 1964), when the sea water had been homogenized two water samples, taken at the same place at 1 m and 20 m depth, showed resistivities of 2.0 and 2.1 ohm - m at 22OC. For readings with the co-planar instrument the minimum deviation from points on Figs. 5 a/ and 7 a/ (corrected values are denoted as small rings on the diagrams) is obtained if the discontmuity surface is chosen to lie 1.3 m below the ice level and the corrections of the real and imaginary components are chosen to be + 2280 ppm and + 450 ppm respectively. For readings from the co-axial instrument

A NUMERICAL

CALCULATION

OF THE ELECTROMAGNETIC

FIELD ETC.

225

the corresponding corrections are - 600 ppm and - 330 ppm. These corrections, which are used in calculating the corrected values in table IV, are surprisingly large particularly if the real components are considered. Both instruments were adjusted to a zero scale position at an observation point lying on the ground some hundred metres inland from the shore. The B values which can be read from the diagrams (see Figs. 5 a/, 6 b/ and 7 a/) lie between 0.28 and 0.31. If we take the resisitvity of the sea water at 23sC to be 1.9 ohm-m this should correspond to a conductivity (T= 0.53 mhomm-1 . According to the tables of Landolt-Bornstein (Aufl. 6, Bd. II, Teil 7, pag. 85 and 116) a 0.050 n solution of NaCl (0.29 y0 NaCl) has such a conductivity at 23OC. At OOC the conductivity of the same solution should be 0.29 mho. m-1 and at 4OC 0.31 mhomm-1. For e = 8.2 m, frequency 800 c/s and u = 0.29 mhoem-1 a B value equal to 0.25 can be calculated. The corresponding B value for G = 0.31 mhomm-1 is 0.26. Considering the not wholly ideal conditions at the time of the survey, the agreement between the B values calculated from the resistivity determinations of the sea water and those calculated from the beam-slingram measurements must be regarded as rather good.

The author wishes to thank the Boliden Mining Company for permission to publish this paper. He is also indebted to Dr. D. S. Parasnis and Mr. S. Brooks for correcting the English text and to Mr. E. Stenlund who has made most of the often very tedious numerical calculations.

DAVID

MALMQVIS’I

REFERENCES Amemiya,

A. 1938:

Anonymous,

:

See Kotani.

1956:

New

Airborne

Geophysical

Method.

Engineering

and Mining

Journ.,

Vol. 157, No. 3, pag. 84. Belluigi,

A. 1949:

physics,

Inductive

Vol. XIV,

Bhattacharyya,

B.

K.

Above the Earth’s Broughton

Edge,

Prospecting.

Press,

Granar,

Electromagnetic

Fields

with a Vertical

of a Vertical

Vol. XXVIII,

T.

University

Coil. Geo-

H. 1931:

Magnetic

Dipole

Placed

No. 3, pag. 408.

The

Principles

&Practice

of Geophysical

Press, pag. 70-73.

: Beitrage zur Theorie der geoelektrischen D. A. 1956:

Cambridge,

Aufschlussverfahren.

Freiberger

Applied

Grundlagen

Erg.-Hefte,

Geophysics

in the Search

for Minerals.

The

Uni-

pag. 176.

A. 1934: Theoretische Geophysik.

Ground

C 6, pag. 41.

A. S. and Keys,

Graf,

of a Homogeneous

Geophysics,

A. B. and Laby,

Cambridge,

Forschungshefte versity

1963:

Surface.

B uc h h e im, W. 1952 Eve,

Coupling

pag. 501-507.

der Ringsendemethode.

Gerlands Beir. zur angewandten

Bd. 4, pag. l-75.

L. J. 1960: Apparatur

zur elektromagnetischen

Prospektierung.

Freiberger

Forschungs-

hefte C 81, pag. 140. Harvard

University,

1947:

Tables

of the Bessel Functions

and One. By the Staff of the Computation bridge,

K. 1941:

Fiinfstellige

tionen ez und e-5 Hedstrom,

Prospecting,

D. A. 1956:

Kiepert,

1918:

Hannover

1918.

Knopoff,

I,.

Physik.

Zeitschr.

und

by N. Ii. Paterson.

Integral-Rechnung,

der elektr. T. 1938:

Physico-Math.

(Title

for Different

Geophysical

translated

Sot.

1,eitfahigkeit

Tables

of Japan,

Can. Min.

M. 1929:

translated

Series

Geo-

Xufl.

11, pag. 418,

der Erde durch

lnduktion.

of Integral Ser.

Useful for the Calculations

3, Vol. 20, Extra

Number

pag.

and Metall.

Method

Earth,

lsvest.

from Russian) Parameters

for Calculating Akad. SSSR,

Quantitative

at Airborne

Calculations EM

Surveys.

Series

of Anomalies lsvest.

Akad.

Edge.

Electrical

Surveys for Regional Studies in Oil and Ore Prospecting.

Bull. Vol. 43, pag. 190.

En geofysiker

ser pa malmprospekteringens

nuvarande

Annaler Vol. 149, No. 8, pag. 443. Trans.

the Electromag-

Geophysical

No. 10, pag. 1513-1521.

See Broughton

D. 1965:

from Russian) Stratified

Geometrical

H. 1950: Airborne

Jernkontorets Mason,

to Comments

pag. 487498.

A. and Simose,

Field above a Horizontally

T. H. 1931:

Malmqvist,

sowie der Funk-

Tokyo.

Obtained

The

Reply

der Differential-

No. 3, pag. 4322443. Kyzovkin, S. K. 1964: (Title

Lundberg,

und Hyperbelfunktionen

No 4, pag. 451.

Zur Messung

Energies.

J. N. 1963:

SSSR,

D. S. 1959:

31 Jahrgang,

M., Amemiya,

netic

der Kreis-

See Slichter.

J. 1930:

of Molecular

Laby,

Tafeln

Vol. VII,

Grundriss

1959:

Koenigsberger,

Kozulin,

Press, Cam-

See Eve.

L.

l-70.

Kind of Orders Zero

. W de Gryuter & Co. Berlin 1941.

E. H. and Parasnis,

physical

Kotani,

of the First

111, Harvard University

Massachusetts.

Hayashi,

Keys,

Laboratory,

A.I.M.E.

Geophysical

Prospecting

No. 13.

liige och framtid.

A NUMERICAL Meyer,

J.

CALCULATION

1962:*

OF THE

Electromagnetische

einem leitenden homogenen

ELECTROMAGNETIC

Induktion

Halbraum.

eines

vertikalen

ETC.

227

Dipols

iiber

FIELD

magnetischen

Mitt. aus dem Max-Planck-Institut

fiir Aeronomie,

No. 7, pag. 1-118. Pagurova,

V. I. 1961:

Tables

Parasnis,

D. S. 1959:

Paterson,

N. R. 1959:

Electra-Magnetic

of the Exponential

Comments

Prospecting

R. H.

E&v).

Pergamon

Press.

on Paper Entitled

1961:

aSome Model

Experiments

Relating

to

with Special Reference to Airborne Work by E. H. Hedstrijm

and D. S. Parasnisn. Geophysical Pemberton,

Integrals

See HedstrGm.

Prospecting,

Airborne

Vol. VII,

Electromagnetics

No. 4, pag. 435.

in Review.

Geophysics,

Vol.

XXVII,

No. 5, pag. 691. Simose,

T. 1938:

Slichter,

See Kotani.

L. B. 1955:

Geophysics Applied to Prospecting

for Ores. Econ. Geol. 50th Anniversary

Volume, pag. 938. Slichter,

L. B. and Knopoff,

of a Layered Sommerfeld,

Earth.

L. 1959:

Geophysics

Field of an Alternating

Vol. XXIV,

A. 1926: Uber die Ausbreitung

Magnetic Dipole on the Surface

pag. 77-88.

der Wellen in der drahtlosen Telegraphie.

Annalen

der Physik 81, pag. 1135-1153. Tornqvist,

B. H. 1956:

Forschungshefte, Tijrnqvist,

Prospecting

J. R. 1951: of Physics,

Wait,

J. R. 1953

Wait,

vom Flugzug

aus. Freiberger

The

Prospecting

in Sweden.

No. 2, pag. 112-126.

Magnetic

Dipole over the Horizontally

: Induction by a Horizontal Earth.

J. R. 1955: Geophysics,

Trans.

Mutual

Vol. XX,

Amer.

S. 1947

Manganese

Oscillating

Geophysical

Stratified Earth.

Can. Journal

Electromagnetic

Magnetic

Union,

Coupling

Dipole over a Conducting

Vol. 34, No. 2, pag. 185-188.

of Loops

over a Homogeneous

Ground.

No. 3, pag. 630-637.

J. R. 1956: Loops over Homogeneous

Werner,

Prospektion

Vol. 29, pag. 577-592.

Homogeneous Wait,

elektromagnetische

G. 1958: Some Practical Results of Airborne Electromagnetic

Geophysical Wait,

Uber

C 29, pag. 19-25.

Ground,

: Appendix 1 (Geophysical

Geophysics,

Investigations

Ores in the Parish of Jokkmokk) in Odman,

in the Ultevis District,

Jokkmokk,

North

Sweden,

Vol. XXI,

in Connection 0. H. 1947:

No. 2, pag. 479-484. with Prospecting Manganese

for

Mineral&.

Sveriges Geologiska Undersiikning,

ser.

C, No. 487, pag. 68. Westerberg, specting,

K. 1966:

The

Geoexploration

*Beam-Slingram,: Vol.

III,

* A very complete list of references paper.

No.

a New Portable 3, pag.

concerning

EM

Instrument

for ore Pro-

00.

the vertical magnetic dipole is given in this