I n t e r p r e t a t i o n o f t h e s e c o n d a r y e l e c t r o m a g n e t i c field 

by a n a l y t i c c o n t i n u a t i o n 

M. Y. DizioGLU (*) 

Ricevuto il 24 Giugno 1967 

SUMMARY. — Given the values of a component of secondary magnetic 
field on the plane surface of the earth, an attempt has been made to ap-
proach the sources by finding the values of the component on a plane at 
a depth h below the surface, the plane being above the sources. 

To this end a Green's Function for the wave equation, appropriate 
for a semi — infinite space was determined. The integral obtained was 
expanded into series suitable for numerical calculations. 

The average values of the component on the earth plane were assumed 
to satisfy a double series and the coefficients were determined by the least 
square method. 

Formulae were ultimately developed to calculate the values on a plane 
at a depth h, in terms of average values on six concentric circles around 
each point. 

RIASSUNTO. — Dati i valori di una componente di un campo magnetico 
secondario sulla superficie piana della Terra, si è tentato di individuare le 
sorgenti, trovando i valori della componente su un piano posto al di sopra 
delle sorgenti ad una profondità li sotto la superficie. 

A questo scopo è stata determinata una funzione di Green per l'equa-
zione dell'onda, adattata ad uno spazio semi-infinito. L'integrale ottenuto è 
stato esteso in serie, idonee ai calcoli numerici. 

I valori medi della componente sul piano terrestre, sono stati assunti 
per soddisfare una serie doppia, ed i coefficienti sono stati determinati col 
metodo dei minimi quadrati. 

Le formule sono state infine sviluppate per calcolare i valori sul piano 
alla profondità h, nei limiti dei valori medi, su sei cerchi concentrici attorno 
ad ogni punto. 

(*) Chief Geophysicist — Minerai Research and Exploration Institute 
of Turkey. 



4 1 G M . Y . D i Z i o 6 L U 

I t is known that, with the inductive electromagnetic method of 
prospecting, the secondary magnetic field is investigated and thence 
some inferences as to the localities and position of the sources produc-
ing such fields are made (2). These interpretations are usually 
carried out by qualitative methods. Sometimes curve fitting methods 
are also used for this purpose, but such calculations are tedious and 
time consuming and different assumptions made for such calculations 
are usually far from the true state. 

Here, in this paper an attempt is made to solve the revers pro-
blem. That is to say, given the components of the secondary magnetic 
field on the plane surface of the earth, we will try to approach the 
sources by finding the values of the components on a plane at a depth h 
below the surface; the plane being above the sources. 

I t should be stated here that such a procedure would not at all 
improve the limitations of the method or lift the ambiguities of in-
terpretation, which would remain the same (3) (4) (5). But such ap-
proaches towards the sources would enlarge the anomaly and subside 
the background, so that the resolving power of the interpretation 
would be increased. 

Let us now assume that we know the components of the secondary 
field produced by a primary field uniform at every point. In practice 
such a secondary field can be obtained by keeping the distance bet-
ween the antenna and the receiver and moving the arrangement at 
equal intervals along parallel lines. 

These components would satisfy the equation (in Gaussian units): 

4 7i n d Ht u 1c d" Hi 
' = c- o ' It + c- ' d¥ ' 

where H t = any component of the secondary magnetic field 

¡i = permeability of the medium 
Q = resistivity of the medium 
Tc = dielectric constant of the medium 
c = speed of light. 

As, usually the resistivity of the country rock is quite high, the 
coefficient of the first term becomes negligibly small. Therefore it 
can be discarded. 

If we assume that the primary field is sinusoidal, then: 

Hi = H (x, y, z) e , 



I N T E R P R E T A T I O N O F T I I E S E C O N D A R Y E L E C T R O M A G N E T I C F I E L D , E T C 4 1 7 

where, 

q = 2 n f/c. 

The function E must satisfy: 

(V2 + k*)E = 0 , 
where, 

k2 = [i k q2 . 

Approach toicards the source: 

Let any component of the secondary magnetic field be known 
on any plane z — 0. This plane may be the surface of the earth or 
a plane above the surface of the earth as in the case of airborne elec-
tromagnetic prospecting. Let us suppose that z is positive downwards 
in a direction towards the source of the field. 

Let us define a function: 

2n 

H (r, z) = J E (r, z,d)dQ , [ 1 ] 
o 

where E (r, z) is the average value of any particular component of the 
secondary magnetic field around a circle of radius r centered at (o, o, z). 

If the z axis does not pass through any source, it is possible to 
expand E (0, z) into series around z = 0 in terms of the intensities on 
the surface of the earth in the form: 

— — a E (o, o) h2 y- E (o, o) 
E (0, h) = E (0, 0) + h > +— > + ... etc. [2] 

The derivatives in equation [2] can be calculated in different 
ways, but as the calculation of even derivatives are more direct than 
the calculation of odd ones, we can get rid of them by using the up-
ward extrapolation: 

— — s i r (0,0) 7i2 a 2 H ( 0 , 0 ) 
E ( 0 , - f c ) = E (0, 0) _ A — - f - L + 2 7 + • • • e t C " [ 3 ] 

Upon adding equations [2] and [3], we obtain: 

E (0, h)= 2 
— h2 a2 E (o, o) -E(0,-h). [4] 



4 1 8 M . T . D i Z i O f i L U 

We have now to develop a method of calculation for the deriva-
tives in brackets as well as for the last term E (o, - h) which is the 
continuation upward to a height li above the surface. We will first 
deal with the calculation of the derivatives. 

From the wave equation in polar coordinates: 

52 H I V 1 3 , \ ^ 
= — - + + k2 \E 

a z2 l i r ! f i r I 

a4 H _ / a2 

a z* ~ U r" 
1 <> \2 Ti , / 32 i a \ \ E -h 2 7c2 1- -
r a r \a r2 r a r 

E + k'E 

a° E 
a«1 

1 / a 2 1 a y — / a 2 1 a >2 
= — - + E — 3 7c2 - + 6 \ a r 2 r a w \a r2 r a r) 

— 3 + - • r—) S — k° H , 
\a r2 r a rl ' 

E — 

/ a - 1 a \< 
where „ H— • - , means that the operation is to be applied i 

\a r2 r a r] 
times in succession. 

We can show by an application of Green's Theorem, that E (r, 0) 
is an even function, therefore it can be written in the form: 

E (r, 0) = a„ + a2 r2 -f- at r4 + «„ r6 + . . . etc. [5] 

Now if we perform the operations for the derivatives, we find: 

WE ( 0 , 0 ) 
~' 1 = — K + 4 a2) 

a4 E to, 0) 
— - ' — - = a0 7c4 + 8 7c2 a2 + 64 a4 c) 

a6 H (0, 0) 
- V'- = — K 7c6 + 12 7c4 «2 + 192 7c2 »4 + 2304 a,) . 
a z6 

If we replace these in equation [4], we obtain: 

E (0, h) = 2 

h 

ft4 
H (0, 0) — — (Oo 7c2 + 4 a2) + — («„7c4 + 8 fc2a2 + 64 a,) — 

[6] 

1 (a0 7c
0 + 12 7c4 a2 + 1 9 2 7c2 a4 + 2304 a„) + ] — E (0, - 7i) 



I N T E R P R E T A T I O N O F T I I E S E C O N D A R Y E L E C T R O M A G N E T I C F I E L D , E T C 4 1 9 

Since the convergence of the series in brackets is rapid enough, 
only four terms have been used. 

Determination of the coefficients in the series [5]. 

We can take as many terms as we please, and obtain the values 
of their coefficients by means of the method of least squares. We 
have made two trials. In one of them we supposed that: 

5 (r, o) = a0 + a2 r2 + a4 r 1 , [7] 

and solved the normal equations with average values around circles 
of radius 0, 1, 2, 3, 4, 5 and 6. The results are as follows: 

a0 = + 0,386100 5 (0) + 0,386100 5 (1) + 0,250962 5 (2) + 

+ 0,070772 H (3) — 0,086910 5 (4) — 0,127520 5 ( 5 ) + 

+ 0,070610 5 (6) [8] 

a2 = — 0,039186 5 (0) — 0,039186 5 (1) — 0,008508 5 ( 2 ) + 

+ 0,029982 5 (3) + 0,057324 5 (4) + 0,046974 5 ( 5 ) — 

— 0,035196 5 ( 6 ) [9] 

a4 = + 0,000819 5 (0) + 0,000819 H (1) + 0,000006 5 ( 2 ) — 

— 0,000989 5 (3) — 0,001626 5 (4) — 0,001149 5 (5) + 

+ 0,001414 5 (6) . [10] 

If we assume that: 

5 (r, 0) = a0 + a2 r2 + ai r1 + a„ r°, [11] 

then the formulae for the coefficients found by the same method 
become: 

a0 = + 0,02544 5 ( 0) + 0,0220 5 ( 1 ) + 0,1360 5 ( 2) — 

— 0,00217 5 (3) + 0,0970 5 (4) — 0,0442 5 (5) — 

— 0,030112 5 (6) + 0,02616 5 (7) [12] 

a2 = — 0,000495 5 (0) — 0,000361 5 (1) — 0,000211 5 (2) + 

+ 0,0000780 5 (3) — 0,000256 5 (4) + 0,000468 5 (5) + 

4- 0,000343 5 (6) — 0,000374 5 (7) [13] 



4 2 0 M . T . D I Z i O Ô L U 

a, = + 0,0000143 H (0) + 0,0000186 3 (1) — 0,0000195 H (2) + 

+ 0,0000721 3 (3) + 0,0001825 H (4) + 0,000324 H (5) + 

+ 0,0001444 3 (6) — 0,0002727 3 (7) [14] 

a, = + 0,00000124 3 (0) + 0,000000908 H (1) — 0,00000015 3 (2) — 

— 0,00000191 3 (3) — 0,00000378 3 (4) — 0,00000679 3 (5) — 

— 0,0000065 ÏT (6) -(- 0,0000057 3 (7) . [15] 

Evaluation of 3 (0, - h). 

Let us suppose that the singularities of 3 all lie outside a closed 
surface 8 bounding the volume V. Then putting: 

B' = «"¡r — r' I 

(where r is the position vector of the point in consideration and r' is 
the position vector of any other point) and this value of 3 in the 
Green's Theorem and applying it to the region bounded externally 
by 8 and internally by G, a small sphere with center r and radius e, 
we find that if the point (P) lies inside 8: 

f l s 
( 5 e « l r - r ' | etklr-r'l dE(r') 

H ( r ) i n ' Ir—r' | _ \r—r'\ ' H \ds' = 
[16] 

l i m r ( / 1 \ — ô 3 (r') ) e « * i r - r ' i 
— o i l « - E [ r ' ] 

c 

We can show that the value of the limit on the right-hand side of this 
equation is — 4 n 3 (r). 

Determination of an appropriate Green's function for the half-space z < 0. 

Let us suppose that G (r, r') satisfies the same wave equation and 
that it is finite and continuous with respect to either the variables 
x, y, z or to the variables x', y', z' for points r, r' belonging to a region V 
which is bounded by a closed surface 8 except in the neighbourhood 
of the point r, where it has a singularity of the same type as: 

glfclr-r't 

• | r - r ' | ' 



I N T E R P R E T A T I O N O F T I I E S E C O N D A R Y E L E C T R O M A G N E T I C F I E L D , E T C 4 2 1 

as r' r. Then proceeding as in the derivation of equation [16], 
we can prove that, if r is the position vector of a point within F , t h e n : 

f [ 1 7 ] 

where n is the outward-drawn normal to the surface S. 
I t follows immediately from equation [17] that if Gl (r, r') is such 

that it satisfies the boundary condition: 

G, (r, r') = 0 , 

and if the point with position vector r' lies on S, then: 

H ( D = - f [18] 
4 71 J 0 11 

s 

Now for the half-space: 

g U - l r - r ' l g t t l 2 - r ' l 

\l-r'\ 

where I = (x, y, z) is the position vector of the image in the plane 
z — 0 of the point with position vector r = (x, y, - z). For this function 
it is easily shown that if the point with position vector r' lies on N, 
the x, y plane, then: 

a G1 _ 0 a ieikR, 
a TO ~ ~ — a z \ B) 

where: 

B* = (x — x'f + (y— y')2 + a2 . 

I t follows from equation [18] that if E = / (x, y) on z = 0, then 
when z < 0: 

— 1 a f e'*« 
3 (x, y, -z) = — - - / (x', y') — dx'-dy', [19] 

N 

where H (x, y, - z) is the value of a component of the secondary field 
at the point (x, y, - z) above the surface. 

Since the measured field is not known as a mathematical function, 
we wish to devise a practical scheme for the numerical evaluation of 
the integral. 



4 2 2 M . y . D i Z i O Ö L U 

Evaluation of the integral. 

S eLkn 
Putting the real component of „ in the integral; 

ö z 11 ° ' 

- oo -r oo 
c1 r* 

CT, v 3 r f , , , cos fcE + 7c2î sin fcE , H (x, y,-z) = — — I J f(x',y') dx 
-oo — - o o 

let (x' — x) = r cos d ; (y' — y) = r sin 0 , 

.since tte' • dy' = j 7 |'A ^ I 1 dr . dd = r dr • dd . 
r i 

r = oo 6 = 2;r 

Ô 3 F (' -N . M COS A/i + A/i sin lilt , ^ = - ^ * (r, 0) r dr dO . 

If we define a function: 

271 

E, (r) = I Hz (r, d) dd , 
2 n 

u 

the equation [20] becomes: 

„ , = cos le B + le B sin Je B 
H = — I H (r) — r-z-dr , 

this can be written in the form: 

_ _ h 
-= H (0) + H (bt) r cos JcB + kB sin kB 
H = 2 « j _ 3 r.dr + 

o 
_ _ 62 

, H (bt) + H (b2) r cos 7cÄ + 7cE sin kB 
^ 2 * r - d r + e t c - . 

bi 

where bz are the radii of circles around the origin. 



I N T E R P R E T A T I O N OF T I I E S E C O N D A R Y E L E C T R O M A G N E T I C F I E L D , E T C 4 2 3 

I f we expand cos kit and sin kit and make the necessary integra-
tions, we obtain the following series: 

k°- 1 

h (bl + h*)1'* 
k4 / 1 

+ h3 — (b\ + Wf'ï] k° i 1 o \ 5 ! 

3 \3 ! 
1 

6 ! 

h — (bl + A2)1'" 

y) + 
+ 

1 
4 ! 

+ h 
H (bj 

[replace b1 by b2 in the same series] 

+ h 
3 (b2) 

(^ + h,2)'12 (b\ + h2)1'2 

_ {bl + ft.)!/, _ + fc»)!/«!^ ( i _ 1 - j . . . J + % : 

[replace and ¿>2 by b2 and ba respectively in the last series] + ...etc. [22] 

H (bj) 
2 

B y replacing this formula in equation [6], the complete formula 
for the evaluation of the intensity at a depth h can be obtained. B y 
this formula, it would be possible to find the required intensities on 
a plane surface nearer the sources thus the localities of the sources 
producing the secondary field could be inferred. 

R E F E R E N C E S 

( ! ) W E S L E Y J . P . , Response of a dyke to an oscillating dipole. «Geophysics», 
23, pp. 128-133 (1958). 

( 2 ) W A I T J . R . , A conducting sphere in a time varying magnetic field. « Geo-
physics », 16, pp. 666-672 (1951). 

( 3 ) S L I C H T E R L . B . , An inverse boundary value problem in electrodynamics. 
«Physics», 4, p. 418 (1933). 

(•*) S K E E L S D. C . . Ambiguity in gravity interpretation. «Geophysics», 12, 
pp. 43-56 (1947). 

( 6 ) T S U B O I C. and F U C H I D A T . , Relations between gravity values and corre-
sponding subterranean mass distributions. « Bulletin of the Earthquake 
Research Institute », Tokyo, Imperial University, 15, pp. 636-649 (1937).