- Email: [email protected]
Electromagnetic response of a permeable inhomogeneous conducting sphere
Geoexplora tion, 11 ( 1973): 1- 20.
© Elsevier'Scientific Publishing Company, Amsterdam -- Printed in Ihe Netherlands
ELECTROMAGNETIC RESPONSE OF A PERMEABLE INHOMOGENEOUS CONDUCTING SPHERE *
JANARDAN G. NEGI, CHANDRA P. GUPTA and UPENDRA RAVAI_ Theoretical Geophysics DivisioH, National Geophysical Research Institute, ttyderabad (India)
(Received July 20, 197l)
ABSTRACT Negi, J.G., Gupta, C.P. and Raval, U., 1973. Electromagnetic response of a permeable inhomogeneous conducting sphere. Geoexploration, 11 : 1-20. Electromagnetic response of a two-layered spherical model to a dipolar field has been calculated considering the simultaneous influences of a permeable conducting cover, magnetic permeability contrast with the surrounding medium, and radial conductivity variation inside the core. Computations have been made and representative curves are given to provide an insight of the contribution of these physical parameters. For certain conductivities of the cover, the response of the composite system has been found to decrease with increasing core-conductivity. The increase in the induction number of the covering shell makes it difficult to interpret the response data. The multifrequency response of the system exhibits two peaks which are diagnostic of the two layers. INTRODUCTION The basic problem of the electromagnetic response of conducting spherical bodies has been of interest to the geophysicist due to its possible application in the prospecting of massive sulphide ore-bodies. The prospecting data are interpreted with the aid of analytical results of the boundary-value problems and/or experimental observations on quantitatively simulated electromagnetic models. In both cases, the electromagnetic responses of idealised models are obtained. It is desirable that tile chosen model should represent the geological situation as closely as possible. Some of the important factors which should be taken into account, to have a closer approximation of the natural occurrences, are: (1) The finite conductivity of the halo-zone surrounding the minerals (Roux, 1959), the host-rocks, and the overburden (Ward, 1967). (2) The magnetic permeability contrast of the target from the surrounding medium which may be greater than unity (Ward, 1959), due to the associated magnetite, etc. (3) The inhomogeneity in the conductivity of the target which may arise due to nonuniform mineralisation (Bateman, 1950) during deposition by hydrothermal processes or due to subsequent alterations (Lowell and Guilbert, 1970; and Rose, 1970). * N.G.R.I. Contribution No. 283.
2
J.G. NEGI, C.P. GUPTA AND U. RAVAL
It may also be worthv~hile to consider the non-uniformity of the energising field. This" may be particularly necessary in the cases when the transmitting source is not far from the target. We should, therefore, consider the influences of the higher-order multipoles induced in a spherical conductor when placed in the field due to a dipole (Wait, 1953; Ward, 1959;Wait, 1960; and Nabighian, 1970). On account of the analytical and computational difficulties in the theoretical investigations, only simplified models of the geological situations have been considered hitherto. The response characteristics of a homogeneous conducting sphere in a uniform field were studied by Wait (1951) and extended for a dipolar field by March (1953). Later, Ward (1959) numerically computed the results of Wait and March for the case of a permeable sphere. The inhomogeneous distribution of conductivity of the mineralized zones was examined by Negi (1962). The screening influence of a thick covering shell has been studied by Negi (1967) and Negi and Raval (1969). Wait (1969) has considered a thin concentric covering shell which is not in galvanic contact with the sphere. Fuller (1971) has also investigated Wait's model. In none of the investigations mentioned above, a shnultaneous influence of the dissipative cover, magnetic contrast, and inhomogeneity in the target conductivity has been studied. We shall examine such a generalised case in the present work considering a permeable, two-layered inhomogeneous spherical conducting model. Though the mathematics of the problem is known, still some key-steps to obtain the solution have been put down. The computation of the analytical results is given to illustrate the behaviour of the response of the chosen model. FORMULATION Fig. 1 describes the model under investigation. An inhomogeneous sphere of radius r 1 is covered by a concentric shell of thickness (r 2 - r 1). The medium surrounding the spherical system is assumed to be homogeneous and infinite in extent. The conductivities o2(r ) and ol(r ) of the shell and the sphere are considered to be varying as:
oj (r) -- o oi (r/rp-"j
( 1)
where j = 1 for the sphere and 2 for the shell; Ooj is constant for given j; and uj is a constant integer. A dipolar source (either electric or magnetic) is assumed to be situated on the z-axis of a spherical coordinate system, at a distance h from the centre O of the sphere. The origin of the coordinate system is also at O. The electromagnetic field at any point P(r, 0, ~) may be considered to be made up of two partial fields (~u, J~u) and (~v, •v) which in turn are derivable from two scalar potentials U and V, termed as transverse magnetic (TM) and transverse electric (TE) modes respectively (Debye, 1909), as:
L Or2
k 2(rU/
eS +
" araO J
rsinO "~rOb le~
(2)
ELECTROMAGNETICRESPONSE OF A CONDUCTING SPHERE
3
Z
I
h
fAU
]
,A'-~
SI
\\\
,.G i
',
k l
'
Fig. 1. A covered inhomogeneoussphere in presence of a dipole.
~u
Fojrsin0 +;~,j " a(,O7_, -~; +;,~,j a(~O]_, ~~ -je° r " ~ - j e~
= L
[
ilajw ~('5)7-+
'
~(rS)l--, jeo
e* ~v = Lf~=(rO~ q(RO]-'+[~-a=('O] " ~,-ao z° + Irsilo ' -a2~ r(rV/)l a~ J
(3) (4) (5)
Here kj is the propagation constant for the j-th region (Fig. 1) and is given by:
(6)
k~ = iajui~o - e/~ico2
for harmonically varying (e iwt) fields. It may be noted that k 1 and k 2 are functions ofr. Both the scalar potentials U and V satisfy the equation: (V 2 - k / ) % = 0 where
~r
Wi
represents either
3r ] + sin~" 80
(7)
Uj or V]. Eq. 7 may be rewritten in the spherical coordinates as: sin0 ~ - ]
sin20 3b 2
The primary and secondary fields in various regions satisfy eq. 8. For the sake of brevity, we shall calculate here the response of the spherical system when energized by a radially oriented magnetic dipole. The results for the transverse magnetic dipole and correspond-
4
J.G. NEGI, C.P. GUPTA AND U. RAVAL
ing expressions for the two orientations of an electric dipole have been derived in the Appendix. The field due to an arbitrary orientation of a dipole may be obtained by suitably combining the fields of a transverse and a radial dipole. We may assume, without loss of generality, that the radial dipole lies along the z-axis (Fig. 1). It follows from the symmetry of the problem that UPm = 0 where the superscript p refers to the primary field and m to the magnetic dipole. (For an electric dipole superscript e will be used.) The primary scalar potential V pro, for a unit source is given as (March, 1953): OO
vpm
= h1 n=0 ~ aP,m +
where:
and:
~n (k3r)
r
Pn(c°sO) f°rr
(9)
t
(2,7 1) ~n(k3h) avPnm. . . . k3 h-
(9a)
@n (k3r) = v'~k3~ln+~ (k3r)
(9b)
'~n (k3r) = v~k3r/rr
(9c)
Kn+{ (k3r)
In+½(k3 r) and Kn+ ½(k3r)
are the modified Bessel functions of the first and second kind, respectively (Schell(unoff, 1948). The solutions of eq. 8 will have only scalar potential V.m l of TE mode and may be expressed in various regions as: Region 3, i.e., r 2 < r < h: V~n = VP m + ~3 n=0
am vn ~n (k3r) Pn (cos0)
(lO)
r
The modified Bessel functions of the first kind have been dropped to satisfy the radiation condition at infinity. The second term on the right hand side is the secondary potential and may be termed as V3m. (2) Region 2, i.e., r 1 < r < r 2 : Oo
V~ = n~O = [bmMn(k2r'v2) vn --r + cmvnNn(~r'v2) 1 "Pn(c°sO) The functions Mn and (2a) for vj 4: 2:
N n are
(11)
defined in general as:
2ko/ r
1-vj2
----) 12-~/l \12-~jl rf_Vj/2
Mn (kjr, vj) = x/Tkojr/2 1 2n+l
....
(lla)
ELECTROMAGNETICRESPONSEOF A CONDUCTINGSPHERE
Nn(k/r, vi)= x/2ko/r/rr K 2n+l ( ~kOj . r l-v//2,
(lib)
(2b) for v/= 2:
M n ( k j r, z)/) = r ~ + x/r( n +l / 5 ~++k ~
(llc)
Nn(kjr, uj)
(lld)
= r
g-
(3) Region 1, i.e., r < r 1 '
v3'=
m
Mn(klr, u 1)
i=
Pn(cosO)for u I <
r
2
(12a)
and
vr~= S'~ dvm
Nn(klr, P1 ) r
n=O
Pn(cosO) for u I > ~
(12b)
The value ofMn(klr, ul) in eq, 12a for u 1 < 2 is given by eq. l l a and that for v 1 = 2 is given by eq. 1 lc. The function N n (k 1r, u 1 ) in eq. 12b is defined by eq. 11 b. The functions N n (x) and M n (x) have been suppressed in eq. 12a and 12b respectively, to satisfy the regularity condition at r = 0. The integration constants a m , b m m and dvn m are to be evaluated using the boundary vn, Cvn conditions. CALCULATIONOF CONSTANTS The boundary conditions at the surfaces r = r 1 and r = r 2 are the continuity of the tangential components of the electric and magnetic fields. These may be expressed in general in terms of the potentials U/and V/as:
0 FrUP +rU 3l 0 frU21 ar t-rVP + rV3_l r=r~ = ~r LrV2_I
(13a) r=r 2
(°3+ic°ea)(UP+U3)] =C(o2+i~e2)Uz] U3 (Vp+ V3)
_ fr
_
Or L r V 2 J r = q
-
L
P2V2
J r=r2
L
PzV2
_] r=r 2
fr<] OrLrV I
J r=rl
(13c)
r=rl
t.
(13b)
PlVI
J
(13d)
r=r 1
6
J.G. NEGI, C.P. GUPTA AND U. RAVAL
Substituting the scalar potentials VP and V/m from eq. 10, 11, 12 and 13 in eq. 14, we obtain a set of four simultaneous equations. On solving these equations we get: rn pm Jgn(k3r2) Rv n avn = avn ~n (k3r2)
(14)
b,vnn _ pm r Nn(k2r2,u2) ] - a,,n •n(k3r2) Fv,, - . w T - LNn (k2r 2, v2)
(15)
pm ~n(k3r2)Fv n [ Mn(k2r2'v2) ] --~ . . . . LM n (k2r 2,1)2)_]
(16)
m
Cvn = avn
FUn(k2r2,v2)] M'n(k2r 1 V2) Fvn ~ - - - - ' [_N'n (k2r 2, v2)J and: --
U'n (k2rl, v2) Fvn [ Mn (k2r2, u2) "] LM;, (kzr 2, v 2 ) . J "~Tt~t -
-
- -
dvm = avn pm t~n(k3r2)
(17) or
N'n(klrl,vl)
forv 1 >
where:
[M~(k2r2,v2)-PvnNn(k2r2,v2)] 122 k3 ~'n (k3r2) [M n (k2r2, v2 )_pvnNn (k2r2, v2)] 123 ~n (k3r2) Rvn =
(18) t
122 k3~n(k3r2) 123 ~n(k3r2) [Mn(k2r2'v2)-PvnNn(k2r2'v2)] -
[M'~(k2r2,v2)-PvnN'n(k2r2,v2) ]
with:
121Mn (klrl, Vl ) M,n (k2rl, v2 ) _/14,n (k 1rl, Pl) Mn (k2rl, v2)
122 PPI,I -~ 121
t
t
122Mn(ktrl,vl)Nn(k2rl,v2)-M'n(klrl,V1)Nn(k2rl,v2) for v! ~< 2
(18a)
ELECTROMAGNETICRESPONSE OF A CONDUCTINGSPHERE
7
or;
U~INn ( k l r l , vl ) M;, (k2rl, u2) NI, (k I r I , ~1) M,, (k2rl, u2) g2
/XlNn(klrl,Ul )N'n(k2rl g2
P2)
Nn(klrl'
"Pl)Nn(k2rl,u~) 18b)
forv I > 2 and: ~n(k3r2)
Nn(k2r2,u2 )
I I~2 k3~'n(k3rg) Nn(k2r2'u21
1
[ N" (-k2r~2'v2!~ = +Rvn ~7 ~n(k3r2) N,,(k2r2,u2) Fvn N;,lk2r2,u2).] t-12 . ~ . . . Nn(k2r2,P2) . . ] DI~ 'n(k2r2'P2)>n-(@;;;~) M"(k2r2'u2)
(19)
A similar expression is obtained for:
Fvn LM~ (k2r~2;v2---)J Here:
[M'n(kjr,vjq d
pM,( jr, vk]
N;, (kit i, vi) J = ~ LNn (kir, Us')J
(20) r= 9
NUMERICALCOMPUTATION Eq.10 and 14 with eq. 18-20 give the total scalar potential in the region 3 (wherein lies our interest for the exploration purpose). On combining the expression for g3m~with eq. 2-5 we may obtain the secondary electromagnetic field due to a perineable inhomogeneous covered sphere. We shall be giving here the numerical results for the case of a radial magnetic dipole only. It may be checked that these curves bring out the essential features of other cases also viz. an arbitrarily oriented magnetic or an electric dipole. The expression for the secondary scalar potential, 17111 -3n is suitably simplified under quasi-static assumption of the electromagnetic field ( [ k31 r <~ 1), which in general is justifiabl, for low frequencies used in induction prospecting, to:
8
J.G. NEGI, C.P. GUPTAAND U. RAVAL
OO
=
n=l hn+l r n+l
x~.P,,(cosO)
(21 )
where:
Xvn(O02, oOl,lal,bl2, Vl,v2, r2, rl, co)= r(w) 2n+l Rvn
(22)
and Rvn is given by eq. 18. The complex quantity Xvn, termed as the response factor, is proportional to the moments of the multipoles excited in the sphere-shell system. The present generalized expression for Xvn can be reduced to the cases studied by Wait (1951, 1953, and 1960), March (1953), Ward (1959) and Negi (1962 and 1967) under appropriate simplifying assumptions. We have computed the quantity Xvn for different values of the dimensionless induction number PI (= x/o01glc°r~) of" the sphere. The other variable parameters are the induction number P2 (= x/°02/-/2cdr J) of the shell, the magnetic permeability contrast (3'13 and Y23 = t11,2/3) between the conducting system and the surrounding medium, and the inhomogeneity factor v 1 defining the conductivity variation inside the sphere. Fig. 2 shows the variation of the in-phase (IP) and quadrature (QR) components of Xvn with P1 for n = 1, 2, 3 and 4. The IP and QR components ofXvl (n = 1) have been plotted for a non-permeable system in Fig. 3A and for the permeable system in Fig. 3B. Fig. 4 and Fig. 5 give the amplitudes and phases of Xvl for P2 = 2 and P2 = 5, respectively. Fig. 6 describes a screening factor j S [ which has been calculated on the basis of the definition given by Walt (1969) for an insulated cover and is given in our case as:
Xvn (k l, k 2) -Xvn (0, k2) [ Is t =
xvn (kl, O)
(23)
The functions Xvn (k 1, k2), Xvn (0, k2) and Xvn (k 1, O) correspond to the sphere-shell system, the shell alone (when the sphere is absent), and the uncovered sphere respectively. We have also obtained the multifrequency response of the system tbr different conductivities and thicknesses of the covering shell; it is given in Fig. 7. DISCUSSION OF RESULTS
Non-uniformity of the primary field The contributions of moments of various multipoles (for different values of n) excited in the spherical system, depend upon the relative dimensions of the sphere and the distance of the observer from it. As shown in Fig. 2, the magnitude ofXvn, decreases as n increases. The effective contribution of the higher multipoles to the secondary scalar potential will be still less as it is to be multiplied by a geometric factor r~"~n+l/(h n+l r n+l ) which will always be less than unity. Thus, if the source and the observer are not very close to the conducting sphere the series for V3m converges fast. The values of the first four terms in the series for V3m have
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
9
b.o I
o.9t
o2 - ,
_
~ 3 " / 2 3 "~
/
_
,
// /I/ !I/'
O" 7
o~ cZ O~
'
//:///////
~,-°
o . 8
/ / / U
~I7,1 //:/: ////
o.,/'/
o.6 o.~
u
o.o
~
I
0-5
I.O
_
I0
I00
R Fig. 2. In-phase lIP) and quadrature (QR) components of the response factor Xvn for n = 1,2, 3, 4.
TABLE I Values of the secondary scalar potential V3mnfor n = 1.2, 3 4. The quantity V~ corresponds to a uniform excitation field: 2n+ 1 #l =
l
hn+lrn+l XvnPn (cos0);0 = 0°;h/r 1 = 2;r/r 1 = 1.95;r2/rl = 1.25;p= = 1.0
P1
V~I
2 6 10 50
0.0/)02145 0.0004778 0.0005406 0.0006316
V1~12
0.0000298 0.0000961 0.0001196 0.0001551
V E X 100 V~nl
13.89 20.11 22.12 24.55
V~13
0.00000554 0.00001970 0.00002736 0.00003939
VT~ X 100 VTll
2.58 4.12 5.06 6.23
V~n4
0.000001207 0.000004011 0.00000584 0.000009857
V~n4 X 100 V~ll
0.56 0.84 1.08 1.56
been c o m p u t e d and listed in Table I. In the subsequent figures only Xvl has been plotted to present the influences of various paraineters. It m a y be noted that this will correspond to a u n i f o r m primary electromagnetic field. It has, however, been seen that the basic nature
10
J.G. NEGI, C.P. GUPTA AND U. RAVAL
of these effects on higher order terms also is the same.
Effect of the covering shell Fig. 3A and Fig. 3B show the effect of increasing the shell conductivity on the total response of the spherical system. The curves for P2 = 0 correspond to the response of the sphere alone. As P2 is increased, both IP and QR components of the response factor Xvl of the sphere-shell system are enhanced. The enhancement is more pronounced for low values o f P l. As seen in Fig. 3A, the IP component for a non-permeable system, with a shell induction number P2 = 5, has quite an anomalous shape besides having a general enhancement, It increases with P l , attains a maximum value and then drops with further increase i n P l to finally settle to an induction limit which is higher than that for an un-
1'2 I'0 0"8 0"6 l
0'4
rr w 0 , 2
zo
1"2
0,0
%g l
I.O
~r-
I
0.8
z o
,~ o..
0.4
-0'8 ZZ //
0'2
A
-0"4 -0.6
0-6
O'O O.5
- 0 "2
x
-I.0
I'0
I0
--pj
-I.2
I00
B
0-5 I'0
IO
I00
~pf
Fig. 3. The influence of variation in conductivity of the shell on the IP (in-phase) and QR (quadrature) components of Xvl for (A) non-permeable and (B) permeable systems. The lowermost curve (192= O) indicates the response of an uncovered sphere. The curves corresponding to uI -=+2, 0, - 2 are given to illustrate the effect of the inhomogeneity in the conductivity of the sphere.
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
11
covered sphere. For a permeable system (')'13 = 723 = 5), the corresponding (P2 = 5) IP (in-phase) curve shows a monotonic decrease with P I' The decrease of IP component with increase in Pl for P2 ~> 5 is an interesting finding of the present investigation. It may imply that under certain conditions the IP response of a body of given conductivity is smaller than that of a less conducting body when both are covered by similar shells. We may possibly attribute the decrease in the IP component with increase in Pl to a phase-shift of more than 7T/2 by the shell. Thus the individual IP responses of the shell and the sphere may become in opposite phases. If the response of the shell, having a high induction number as P2 ~> 5, predominates over that of the sphere •alone, the increase in Pl (and consequent increase in the response of sphere) will cause a reduction in the total response of the composite system. Fig. 5A shows that for P2 = 5 and 713 = 723 > 1, the response amplitude also shows a monotonic decrease with p I • Fig. 4B and 5B give the phases of the response factor forP2 = 2 and 5 respectively. It can be easily seen that a change in P l significantly influences the response phase if P2 ~< 2. But for higher values of p2 (say 5), there is only a little phase shift by increasing the value o f P l . This observation coupled with that from Fig. 5A indicates that if the conductivity of the shell is higher than a certain value, the sphere is masked to such an extent that its electrical properties can hardly be delineated. This inference holds even if the radii of the shell and the sphere have the same order of magnitude and the sphere has a nmch
1.3 0°
I.z I \
I
I
"\\ -20"~-
\
~, = + z = 0
I
[.i / --40"
l
-
- -
~13-- ~'23
1.0 -6o* L-
(uw 0
~\ \\\\
0"9
+
~80 °
NS'-- 0'8 -IOO*
0.7
t
/t
1.11=+2 .
ii/i. ~~1
.
.
.
== - 20 --
0.6 -~ , / / 0.5 r-y. I / /
.
.
.
~
120
.
\\\\\ \ \'\
'x\\,
.
~13=~a3t"J Pz=z --160 =
0.4
0"5 I'0 A
I0 _ _
p, ~
-tsO' 0.5
I00 B
I I-0
i I0 - -
Pf
I00 ~-
Fig. 4 and 5. Tile amplitudes (Fig. 4A and 5A) and phases (Fig. 4B and 5B) of the response factor for different permeability contrasts. The induction number (P2) of the shell is 2 in Fig. 4 and 5 in Fig. 5.
Xvl
12
J.G. NEGI, C.P. GUPTA AND U. RAVAL
1.6
)/13- 5
1'5
Y 12 3 ~ , \ -XN 1'4
l+o
\\\\\ NN~X \ \ "~X
\
++-
1"3
'\\ \
O n
t
r.2
I'1
)Jr-- + 2 =
+
-
-
-
0
=-2
'+o7
.....
I'0
\
I
P2- 5 I
0"5
I0
I'0
A
-
Pl
-
-17
I00 )
B
0".~
I'0
I0 - -
I00
Pt -'-~
Fig. 5. For legend see page 1 1. higher conductivity than the shell. Another way of looking at the influence of the shell has been put forth by Wait (1969). Fig. 6 gives the screening factor IS [, as defined in eq.23. According to this definition, the smaller the factor IS l, the more is the masking influence of the shell. For IS I = I there is no influence of the shell and for ISI = O, the inner core is completely undetectable. In Fig. 6 we find that there is always a particular combination of Pl and P2 when IS [ is minimum. As expected, the masking influence increases with increase in P2" For P2 = 5, the factor IS[ is quite small even when Pl is very large. I,O ~
-
~
O.9 ~
1
0'8 l -'-i U3 -
~jl= + 2 . . . . o.7
=-2
....
"¢,3. "¢2~. I
-
0'6 0.5
.
O. 4
0.3L 0-5
///]
/
.,/~ ~"
I 1.0
I IO - -
I00
pu ---~
Fig. 6. The variation of screening-factor ISI for different conductivities o f the shell.
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
13
Effect of the permeability The in-phase component of a non-permeable system is always positive (Fig.3A). This, however, is not the case when the system has a higher magnetic permeability contrast from the surrounding medium, i.e., when 713 = 3'23 :¢: 1. In this case the IP component is negative for low values of both P2 ( ~ 2) and Pl • The negative value of the IP component is due to the induced magnetisation which is in phase with the prhnary field as supported by Fig. 4B and Fig. 5B also. But as either P l or P2 or both increase, the contribution of the permeability is nullified and the IP component tends to become positive so that beyond a certain combination o f P l a n d P 2 , the IP component attains a positive value. F o r p 2 = 5 the IP component for the permeable system (3`13 = 723 = 5) is positive (Fig. 3B) even for very small values o f P l . The phenomenon of decrease of the response amplitude with p I, described above, for a non-permeable system at P2 = 5, is found to exist in the case of a permeable system (Fig. 4A) even for low values of p2. In this case the phenomenon is due to the opposing effects of 1he magnetic permeability and the electrical conductivity. Thus, in the process of IP becoming positive (Fig. 3B), its absolute value first decreases before crossing zero with the increase of P I' It implies that the amplitude which depends upon the absolute values of 1P and QR components would decrease in this range of P l. This effect is manifested in Fig. 4A in the range of 1.2 < P l < 10 for 3`13 = 3'23 = 5 andP2 = 2. In Fig. 5A, since P2 is quite high (=5) the amplitude shows a monotonic decrease.
Effect of bThomogeneity of the conductivity of the sphere In accordance with eq. 1, O01 is the conductivity of the surface of the sphere which remains invariant irrespective of the value of ~1- The conductivity towards the centre is, however, more (or less) as u I > 0 (or u 1 < 0) in comparison with the corresponding value for u I = O. This change in the conductivity of the inner layers, influences both the IP and QR components. It is seen from Fig. 3A that for low values of p2 ( ~ 2) the IP component is more, the larger is the value of v l . This enhancement of IP component with v I is expected since an increase in v I amounts to an effective increase in the conductivity of the sphere. The effect of the inhomogeneity is significant only in the intermediate range of P l (1 < P l < 4). The quadrature component is "also influenced by u I . For low values o f P l , the QR component is more for larger u I . However, for higher Pl values (i.e., a more conducting system) the quadrature component for larger values of v I decreases faster. Consequently, in this range o f P l , the quadrature component is smaller for larger v 1 . If the shell conductivity is increased so that P2 ~ 5. the influence of u 1 on the IP component undergoes an inversion. Thus, in this case the IP component is less for larger u 1 beyond a certain value o f P l . As mentioned above in connection with the relative influences of the magnetic permeability and the conductivity, there is a certain range of Pl and P2 for which the response amplitude of a permeable system decreases with increase in the conductivity of the core. A natural corollary of this observation is that in this particular range an increase in u l also lowers the response amplitude. This is shown in Fig. 4A for ~'13 = 3'23 = 5 and P2 = 2
14
J.G. NEGI, C.P. GUPTA AND U. RAVAL
and for all the curves of Fig. 5A. M u l t i f r e q u e n c y response
Fig. 7A and 7B describe the variation of IP and QR components of Xvl with the frequency of excitation for different values of thickness of the shell and its conductivity contrast with the core. For slowly varying fields and low conductivity of the shell, the field penetration to the sphere is more and we get the first peak in the response curve at 1,000 C/sec (Fig. 7A). This is mainly due to the induction in the sphere. The shell is more or less transparent at this low frequency. The second peak, occurring at a higher frequency, corresponds to the situation when the skin-effect becomes appreciable and the masking influence of the shell becomes high. The response of the system at higher frequencies is mainly due to the shell. The second peak is smaller than the first peak for low values of o02/o01 and r2/r I . However, as the conductivity or thickness (or both) of the shell increases, the second peak goes on increasing and becomes even higher than the first peak. The second peak disappears when the conductivity of the shell is very low and/or its thick2,4
' '
/ / rf 2.2
/ 2.0
Ooo2 -2 ~3- = io
I.e
/
)Ji" 0
I
I'2
//
/
E8
,.o
/
i
/
//
f-l.2
/I
i/ ii
Z~
//~-,.3
i i
/
//
1'4
~
/
//
"/13"Y23-I 1,6
/
J~.<~-l.i
.;..~ .....
o. 4
°'°,d A
~1.4
/
: t:~
,d
,d
Id
Io~
FREQUENCY(IN CYCLES PER SECOND)
Fig. 7. Multifrequency response of a two-layer spherical system for different (A) thicknesses and (B) conductivities of the shell, expressed in terms of the corresponding properties of the sphere.
ELECTROMAGNETIC RESPONSE OF A CONDUCTINGSPHERE
15
1.8 (TO2
=
F2
/ //
rI 1,4
/
%-'/z3
-~
/
1.2
"
/ /
, /
08
. I() I
,
/
/
/~-~'-i~ 2
~/
y
<1:o-
~_~o
]
0-6
/~/
0 4
/l
,A<~"'--..
~_-~ - -
-" -- .......
-s
2H.\-
0.2
o
- ~0
zf~-~-'~
o#
,d FREQUENCY
,o~
,~
,~
(iN CYCLES PER SECOND)
Fig. 7B. ness is very small• If the response is found over a wide range of frequencies, the different layers of a stratified sphere can be delineated on the frequency scale. This fact may be utilized in the study of geomagnetic variations, considering Dst, Sq, L etc. as the primary field and the model of the earth having shells of different conductivities (Price, 1970). ACKNOWLEDGEMENTS The authors are grateful to Dr. Hari Narain, Director of the National Geophysical Research Institute, Hyderabad for his kind permission to publish this work. We sincerely thank Messrs. K.N.N. Rao, Y.M. Ramachandra, M. Ramakrishna Sarma, V. Subrahmanyam and Miss G. Aruna for their valuable help in computation and preparation of the manuscript. The research was financially supported by the Department of Commerce, Boulder Laboratories, U.S.A. under agreement No. E-105-67(N).
APPENDIX Transverse magnetic dipole The magnetic dipole is assumed to be situated on the z-axis and directed perpendicular to it in the x-z plane. In this case, unlike the radial orientation of the dipole, both poten-
16
J.G. NEGI, C.P. GUPTA AND U. RAVAL
tials U and V exist and the field depends upon the q~-coordinate also. Thus, the primary field due to source of unit strength, here, takes the following form (March, 1953): UPIn
oo A Pn m ~'n (k3r)
sin,) =
VPm
Z
A pln =
"vn
r
(AI)
il~3w( 2n+ l )~n (k3h) n(n+l)k3 h
where: AuPn m=-
and:
Pnl (cosO)
- - -
cos4~ n=l A vpm
( 2n+ l )~'n(k3h) n(n+l)h
Accordingly, the expressions of the scalar potential in various regions may be expressed as: (1) Region 3, i.e., r 2 < r < h: U~
=
g~
Upm
+
g pm
sin )
~
cos) n=l
[ Am
un ~n(k3 r) p1n (cosO)
Avmn
] (A2)
h
(2) Region 2, i.e., r I < r < h:
V~
=
E cos,k ,,=1
Byron
r
+
r
Cvmn
Pn1 (cos0)
(A3)
(3) Region 1, i.e., r < r 1 :
uy'= sin,~ ooZ f Dumn Mn(klr'Vl)V•
cos;~
pnl (cosO)] foru 1 ~<2
r
n=l LDvm
(A4)
and
U~n = sinO
vT'
~
FDuinn Nn(klr, Pl)
cos~ ,,=~ LD,,m
;:
] e~ (cos0) for,,q > 2
(A5)
where the functions ~n (x), ~n (x),Mn, andNn are defined by eq.9b, 9c and l l a - 1 ld. The m , Bun in etc. are determined as before from the boundary conditions (eq. 13). constants Aun Thus, on substituting eq.A1-A5 into eq.13 and solving, we obtain: ~n (k3r2) R
Au~ =APm ~n(k3r2)
un
pm UNn(k2r2'p2) ] Burn -_ Aun ~.(k3r2)Fun L ~ v 2 ) j
(A6) (A7)
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE Apm
Cumn='qm ~n(k3r2)Fun
17
FMn(k2r2'P2)l vvg.,---~7
(A8)
LM n (k2r 2, v 2 }J
['%,(k2r2,v2)] M,; (k2r I , u2)Fw,
L~,~ik~r2~v2-~
an d:
1
I M nlk2r 2, v 2)d
(Ag)
D n}, -~ Apmun ~n (k3r2)
Mn(klrl,ul)
f o r v I -<-2
I
L o r N n ( k l r l , u 1) f o r u l .'>2
1
whe re: t
t
•
[Mn (k2r 2, v2) - PunN'n (k2r 2, v2)] a2+icoe 2 k 3 ~'n(k3r2)
o3+iwe 3 gu/I
~n(k3r2) [Mn(k2r2'v2)-PunNn(k2r2'v2)] (A10)
--
°2+ic°e2
°3+iwe3
k
3~nt (k3r2)
~n (k3r2)
[Mn (k2r 2, v2) - PunNn (k2r2, u2)] r
--[114n (k2r 2 , v 2) -PunNn (k2r 2 , v2)] with:
a 1+iooe1 +. Mn(klrl,Vl)Mn(k2rl,u2)-M'n(klrl,Vl)Mn(k2rl,v2) 0 2 t~e 2 Pun
"~
o 1+icoel , , o2+icoe 2 mn (k I r 1 , v I )Nn (k2r 1 , ta2 ) - N n (k2r 1 , v2)m n ( k l r 1 , v I ) for u I ~< 2
or:
°l+iC°el Nn(klrl,Vl)M,n(k2rl,u
o2 +iooe2
Pu?/
)- Nn(klrl,Vl)Mn(k2rl,P2)
-
°l+iC°el N n (k I rl, v l ) N n ( k 2 r l , , ' - , - N~; { k l r l , v 1)N n (k2rl, v2)
o2+icoe 2
foru! > 2 and:
J.G. NEGI, C.P. GUPTA AND U. RAVAL
18
2+ic°e2 k3~'n(k3r2) Nn(k2r2'v2) o3+i~e; " Cn(k3r2) Nn(k2r2,v2)
1]
t
f 02+icOc2"k3~n(k3r2) Nn(k2r2'v2) Fu [Nn(k2r2,v2) ]
+ Run k~'we3
~n(k3r2)
13
"Nn (k2r2, v2) (All)
-
°2+i°0e2 o3+iwe 3
IM' Nn(k2r2'v2) n(kzr2'v2)N~(k2r2,v2) -Mn(k2r2,v2)]
A similar expression is obtained for:
FunfMn(k2r2'u2 ) ]
LM'. (k2r2, v2)
Mn and Nn are defined by eq. 20. The constants A m ' Bvn,mCvnmand D m are obtained from Amn, Bumn,Cum, and Dumn(eq. • pm A6 - A9) by replacing Aun , Run, and (o/+iwe/)/(O/+l+iwe/+1) by Avn, Rvn and t~j/laj+1 respectively. Thus, for example: m_
pm ~n(k3r2)Rv n
Avn-Avn
~n(k3r2)
where Rvn is defined by eq.18.
Electric dipole When the exciting source is an electric dipole the values of the constants are obtained if we interchange (aj +iwej) by ils/w and vice-versa in the expressions A6 - A9. Thus:
4.
= Ape
~n(k3r2)Re (X3r2)
e _ pe fNn(k2r2'u2) 1 Bun - Aun ~n (k3r2)Fen L(k2r2-V" Nn" ~ , v2) ] _ pe e [-Mn(k2r2'v2)-I Ceun- Aun ~n (k3r2)Fun k ~ J and:
(A12)
..
(A13)
(A14)
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
19
M;,(k2r I u2)Fu, , ~Nn (k2r2' u 2 ) l
,
LN,;(k2r2,,;~5 A
-~,v,, (~2rl, ~2 i~,,, LM,', (kzr 2, ~ 2)--1 D~en =AtPen ~ n ( k 2 r 2 )
.
.
.
.
.
.
(A15)
,
! Mn(ktrl,vl)
f o r u 1 ~<2
I or:
iNn(klrl,Vl)
I i
foru I >2
i
A
(o3+icoe3) where:
A pe:_`
(2n+ 1) ~n (k3h) n(n~l)--k3h~ --
Reun occurring in eq.A15 can be directly derived f r o m e q . A 1 0 through re. e Bun, e Ceun and Den can (oj+icoej)/(Oi+l+icoei+l) by/a//ta/+ 1 . Expressions for Aun,
The factor
placing also be obtained similarly.
REFERENCES Bateman, A.M., 1950. Economic Mineral Deposits. Wiley, New York, N.Y., 2nd ed., 916 pp. Debye, P., 1909. Der Lichtdruck auf Kugeln von beliebigem Material. Ann. Phys., 30:57. Fuller, B.D., 1971. Electromagnetic response of a conductive sphere surrounded by a concentric shell. Geophysics, 36: 9-24. Lowell, J.D. and Guilbert, J.M., 1970. Lateral and vertical alteration-mineralisation zoning in porphyry ore deposition. Econ. Geol., 65: 373-408. March, H.W., 1953. The field of a magnetic dipole in presence of a conducting sphere. Geophysics, 18: 671-684. Nabighian, M.C., 1970. Quasistatic transient response of a conducting permeable sphere in a dipolar field. Geophysics, 35: 303-309. Negi, J.G., 1962. Diffraction of electromagnetic waves by an inhomogeneous sphere. Geophysics, 27: 480-492. Negi, J.G., 1967. Electromagnetic screening due to a disseminated spherical zone over a conducting sphere. Geophysics, 32: 69-87. Negi, J.G. and Raval, U., 1969. Negative electromagnetic screening by a cylindrical conducting cover. Geophysics, 34: 944-957. Price, A.T., 1970. The electrical conductivity of the earth. R. Astron. Soc., Q.J., 11: 23-42. Rose, W.A., 1970. Zonal relations of wall rock alteration and sulfide distribution at porphyry copper deposits. Econ. Geol., 65: 920-936. Roux, A.T., 1959. A review of airborne electromagneuc systems, some factors affecting the choice of systems and interpretation of data. Geol. Soc. S. Afr., Trans., vol 62. Schelkunoff, S.A., 1948. Electromagnetic Waves. Van Nostrand, New York, N.Y., 530 pp. Wait, J.R., 1951. A conducting sphere in a time varyine magnetic field. Geophysics, 16: 668-672. Wait, J.R., 1953. A conducting permeable sphere in the presence of a coil carrying an oscillating current. Can..L Phys., 31: 670-678.
20
J.G. NEGI, C.P. GUPFA AND U. RAVAL
Wait, J.R., 1960. On the electromagnetic response of a conducting sphere to a dipolar field. Geophys2cs, 25: 649-658. Wait, J.R., 1969. Electromagnetic induction in a solid conducting sphere enclosed by a thin conclucting shell. Geophysics, 34: 753-759. Ward, S.H., 1959. Unique determination of conductivity, susceptibility, size and depth in multifrequency electromagnetic exploration. Geophysics, 2 4 : 5 3 1 - 5 4 6 . Ward, S.H., 1967. Mino~g Geophysics. The Society of Exploration Geophysicists, Tulsa, Okla. 2 : 7 0 8 pp.
© Elsevier'Scientific Publishing Company, Amsterdam -- Printed in Ihe Netherlands
ELECTROMAGNETIC RESPONSE OF A PERMEABLE INHOMOGENEOUS CONDUCTING SPHERE *
JANARDAN G. NEGI, CHANDRA P. GUPTA and UPENDRA RAVAI_ Theoretical Geophysics DivisioH, National Geophysical Research Institute, ttyderabad (India)
(Received July 20, 197l)
ABSTRACT Negi, J.G., Gupta, C.P. and Raval, U., 1973. Electromagnetic response of a permeable inhomogeneous conducting sphere. Geoexploration, 11 : 1-20. Electromagnetic response of a two-layered spherical model to a dipolar field has been calculated considering the simultaneous influences of a permeable conducting cover, magnetic permeability contrast with the surrounding medium, and radial conductivity variation inside the core. Computations have been made and representative curves are given to provide an insight of the contribution of these physical parameters. For certain conductivities of the cover, the response of the composite system has been found to decrease with increasing core-conductivity. The increase in the induction number of the covering shell makes it difficult to interpret the response data. The multifrequency response of the system exhibits two peaks which are diagnostic of the two layers. INTRODUCTION The basic problem of the electromagnetic response of conducting spherical bodies has been of interest to the geophysicist due to its possible application in the prospecting of massive sulphide ore-bodies. The prospecting data are interpreted with the aid of analytical results of the boundary-value problems and/or experimental observations on quantitatively simulated electromagnetic models. In both cases, the electromagnetic responses of idealised models are obtained. It is desirable that tile chosen model should represent the geological situation as closely as possible. Some of the important factors which should be taken into account, to have a closer approximation of the natural occurrences, are: (1) The finite conductivity of the halo-zone surrounding the minerals (Roux, 1959), the host-rocks, and the overburden (Ward, 1967). (2) The magnetic permeability contrast of the target from the surrounding medium which may be greater than unity (Ward, 1959), due to the associated magnetite, etc. (3) The inhomogeneity in the conductivity of the target which may arise due to nonuniform mineralisation (Bateman, 1950) during deposition by hydrothermal processes or due to subsequent alterations (Lowell and Guilbert, 1970; and Rose, 1970). * N.G.R.I. Contribution No. 283.
2
J.G. NEGI, C.P. GUPTA AND U. RAVAL
It may also be worthv~hile to consider the non-uniformity of the energising field. This" may be particularly necessary in the cases when the transmitting source is not far from the target. We should, therefore, consider the influences of the higher-order multipoles induced in a spherical conductor when placed in the field due to a dipole (Wait, 1953; Ward, 1959;Wait, 1960; and Nabighian, 1970). On account of the analytical and computational difficulties in the theoretical investigations, only simplified models of the geological situations have been considered hitherto. The response characteristics of a homogeneous conducting sphere in a uniform field were studied by Wait (1951) and extended for a dipolar field by March (1953). Later, Ward (1959) numerically computed the results of Wait and March for the case of a permeable sphere. The inhomogeneous distribution of conductivity of the mineralized zones was examined by Negi (1962). The screening influence of a thick covering shell has been studied by Negi (1967) and Negi and Raval (1969). Wait (1969) has considered a thin concentric covering shell which is not in galvanic contact with the sphere. Fuller (1971) has also investigated Wait's model. In none of the investigations mentioned above, a shnultaneous influence of the dissipative cover, magnetic contrast, and inhomogeneity in the target conductivity has been studied. We shall examine such a generalised case in the present work considering a permeable, two-layered inhomogeneous spherical conducting model. Though the mathematics of the problem is known, still some key-steps to obtain the solution have been put down. The computation of the analytical results is given to illustrate the behaviour of the response of the chosen model. FORMULATION Fig. 1 describes the model under investigation. An inhomogeneous sphere of radius r 1 is covered by a concentric shell of thickness (r 2 - r 1). The medium surrounding the spherical system is assumed to be homogeneous and infinite in extent. The conductivities o2(r ) and ol(r ) of the shell and the sphere are considered to be varying as:
oj (r) -- o oi (r/rp-"j
( 1)
where j = 1 for the sphere and 2 for the shell; Ooj is constant for given j; and uj is a constant integer. A dipolar source (either electric or magnetic) is assumed to be situated on the z-axis of a spherical coordinate system, at a distance h from the centre O of the sphere. The origin of the coordinate system is also at O. The electromagnetic field at any point P(r, 0, ~) may be considered to be made up of two partial fields (~u, J~u) and (~v, •v) which in turn are derivable from two scalar potentials U and V, termed as transverse magnetic (TM) and transverse electric (TE) modes respectively (Debye, 1909), as:
L Or2
k 2(rU/
eS +
" araO J
rsinO "~rOb le~
(2)
ELECTROMAGNETICRESPONSE OF A CONDUCTING SPHERE
3
Z
I
h
fAU
]
,A'-~
SI
\\\
,.G i
',
k l
'
Fig. 1. A covered inhomogeneoussphere in presence of a dipole.
~u
Fojrsin0 +;~,j " a(,O7_, -~; +;,~,j a(~O]_, ~~ -je° r " ~ - j e~
= L
[
ilajw ~('5)7-+
'
~(rS)l--, jeo
e* ~v = Lf~=(rO~ q(RO]-'+[~-a=('O] " ~,-ao z° + Irsilo ' -a2~ r(rV/)l a~ J
(3) (4) (5)
Here kj is the propagation constant for the j-th region (Fig. 1) and is given by:
(6)
k~ = iajui~o - e/~ico2
for harmonically varying (e iwt) fields. It may be noted that k 1 and k 2 are functions ofr. Both the scalar potentials U and V satisfy the equation: (V 2 - k / ) % = 0 where
~r
Wi
represents either
3r ] + sin~" 80
(7)
Uj or V]. Eq. 7 may be rewritten in the spherical coordinates as: sin0 ~ - ]
sin20 3b 2
The primary and secondary fields in various regions satisfy eq. 8. For the sake of brevity, we shall calculate here the response of the spherical system when energized by a radially oriented magnetic dipole. The results for the transverse magnetic dipole and correspond-
4
J.G. NEGI, C.P. GUPTA AND U. RAVAL
ing expressions for the two orientations of an electric dipole have been derived in the Appendix. The field due to an arbitrary orientation of a dipole may be obtained by suitably combining the fields of a transverse and a radial dipole. We may assume, without loss of generality, that the radial dipole lies along the z-axis (Fig. 1). It follows from the symmetry of the problem that UPm = 0 where the superscript p refers to the primary field and m to the magnetic dipole. (For an electric dipole superscript e will be used.) The primary scalar potential V pro, for a unit source is given as (March, 1953): OO
vpm
= h1 n=0 ~ aP,m +
where:
and:
~n (k3r)
r
Pn(c°sO) f°rr
(9)
t
(2,7 1) ~n(k3h) avPnm. . . . k3 h-
(9a)
@n (k3r) = v'~k3~ln+~ (k3r)
(9b)
'~n (k3r) = v~k3r/rr
(9c)
Kn+{ (k3r)
In+½(k3 r) and Kn+ ½(k3r)
are the modified Bessel functions of the first and second kind, respectively (Schell(unoff, 1948). The solutions of eq. 8 will have only scalar potential V.m l of TE mode and may be expressed in various regions as: Region 3, i.e., r 2 < r < h: V~n = VP m + ~3 n=0
am vn ~n (k3r) Pn (cos0)
(lO)
r
The modified Bessel functions of the first kind have been dropped to satisfy the radiation condition at infinity. The second term on the right hand side is the secondary potential and may be termed as V3m. (2) Region 2, i.e., r 1 < r < r 2 : Oo
V~ = n~O = [bmMn(k2r'v2) vn --r + cmvnNn(~r'v2) 1 "Pn(c°sO) The functions Mn and (2a) for vj 4: 2:
N n are
(11)
defined in general as:
2ko/ r
1-vj2
----) 12-~/l \12-~jl rf_Vj/2
Mn (kjr, vj) = x/Tkojr/2 1 2n+l
....
(lla)
ELECTROMAGNETICRESPONSEOF A CONDUCTINGSPHERE
Nn(k/r, vi)= x/2ko/r/rr K 2n+l ( ~kOj . r l-v//2,
(lib)
(2b) for v/= 2:
M n ( k j r, z)/) = r ~ + x/r( n +l / 5 ~++k ~
(llc)
Nn(kjr, uj)
(lld)
= r
g-
(3) Region 1, i.e., r < r 1 '
v3'=
m
Mn(klr, u 1)
i=
Pn(cosO)for u I <
r
2
(12a)
and
vr~= S'~ dvm
Nn(klr, P1 ) r
n=O
Pn(cosO) for u I > ~
(12b)
The value ofMn(klr, ul) in eq, 12a for u 1 < 2 is given by eq. l l a and that for v 1 = 2 is given by eq. 1 lc. The function N n (k 1r, u 1 ) in eq. 12b is defined by eq. 11 b. The functions N n (x) and M n (x) have been suppressed in eq. 12a and 12b respectively, to satisfy the regularity condition at r = 0. The integration constants a m , b m m and dvn m are to be evaluated using the boundary vn, Cvn conditions. CALCULATIONOF CONSTANTS The boundary conditions at the surfaces r = r 1 and r = r 2 are the continuity of the tangential components of the electric and magnetic fields. These may be expressed in general in terms of the potentials U/and V/as:
0 FrUP +rU 3l 0 frU21 ar t-rVP + rV3_l r=r~ = ~r LrV2_I
(13a) r=r 2
(°3+ic°ea)(UP+U3)] =C(o2+i~e2)Uz] U3 (Vp+ V3)
_ fr
_
Or L r V 2 J r = q
-
L
P2V2
J r=r2
L
PzV2
_] r=r 2
fr<] OrLrV I
J r=rl
(13c)
r=rl
t.
(13b)
PlVI
J
(13d)
r=r 1
6
J.G. NEGI, C.P. GUPTA AND U. RAVAL
Substituting the scalar potentials VP and V/m from eq. 10, 11, 12 and 13 in eq. 14, we obtain a set of four simultaneous equations. On solving these equations we get: rn pm Jgn(k3r2) Rv n avn = avn ~n (k3r2)
(14)
b,vnn _ pm r Nn(k2r2,u2) ] - a,,n •n(k3r2) Fv,, - . w T - LNn (k2r 2, v2)
(15)
pm ~n(k3r2)Fv n [ Mn(k2r2'v2) ] --~ . . . . LM n (k2r 2,1)2)_]
(16)
m
Cvn = avn
FUn(k2r2,v2)] M'n(k2r 1 V2) Fvn ~ - - - - ' [_N'n (k2r 2, v2)J and: --
U'n (k2rl, v2) Fvn [ Mn (k2r2, u2) "] LM;, (kzr 2, v 2 ) . J "~Tt~t -
-
- -
dvm = avn pm t~n(k3r2)
(17) or
N'n(klrl,vl)
forv 1 >
where:
[M~(k2r2,v2)-PvnNn(k2r2,v2)] 122 k3 ~'n (k3r2) [M n (k2r2, v2 )_pvnNn (k2r2, v2)] 123 ~n (k3r2) Rvn =
(18) t
122 k3~n(k3r2) 123 ~n(k3r2) [Mn(k2r2'v2)-PvnNn(k2r2'v2)] -
[M'~(k2r2,v2)-PvnN'n(k2r2,v2) ]
with:
121Mn (klrl, Vl ) M,n (k2rl, v2 ) _/14,n (k 1rl, Pl) Mn (k2rl, v2)
122 PPI,I -~ 121
t
t
122Mn(ktrl,vl)Nn(k2rl,v2)-M'n(klrl,V1)Nn(k2rl,v2) for v! ~< 2
(18a)
ELECTROMAGNETICRESPONSE OF A CONDUCTINGSPHERE
7
or;
U~INn ( k l r l , vl ) M;, (k2rl, u2) NI, (k I r I , ~1) M,, (k2rl, u2) g2
/XlNn(klrl,Ul )N'n(k2rl g2
P2)
Nn(klrl'
"Pl)Nn(k2rl,u~) 18b)
forv I > 2 and: ~n(k3r2)
Nn(k2r2,u2 )
I I~2 k3~'n(k3rg) Nn(k2r2'u21
1
[ N" (-k2r~2'v2!~ = +Rvn ~7 ~n(k3r2) N,,(k2r2,u2) Fvn N;,lk2r2,u2).] t-12 . ~ . . . Nn(k2r2,P2) . . ] DI~ 'n(k2r2'P2)>n-(@;;;~) M"(k2r2'u2)
(19)
A similar expression is obtained for:
Fvn LM~ (k2r~2;v2---)J Here:
[M'n(kjr,vjq d
pM,( jr, vk]
N;, (kit i, vi) J = ~ LNn (kir, Us')J
(20) r= 9
NUMERICALCOMPUTATION Eq.10 and 14 with eq. 18-20 give the total scalar potential in the region 3 (wherein lies our interest for the exploration purpose). On combining the expression for g3m~with eq. 2-5 we may obtain the secondary electromagnetic field due to a perineable inhomogeneous covered sphere. We shall be giving here the numerical results for the case of a radial magnetic dipole only. It may be checked that these curves bring out the essential features of other cases also viz. an arbitrarily oriented magnetic or an electric dipole. The expression for the secondary scalar potential, 17111 -3n is suitably simplified under quasi-static assumption of the electromagnetic field ( [ k31 r <~ 1), which in general is justifiabl, for low frequencies used in induction prospecting, to:
8
J.G. NEGI, C.P. GUPTAAND U. RAVAL
OO
=
n=l hn+l r n+l
x~.P,,(cosO)
(21 )
where:
Xvn(O02, oOl,lal,bl2, Vl,v2, r2, rl, co)= r(w) 2n+l Rvn
(22)
and Rvn is given by eq. 18. The complex quantity Xvn, termed as the response factor, is proportional to the moments of the multipoles excited in the sphere-shell system. The present generalized expression for Xvn can be reduced to the cases studied by Wait (1951, 1953, and 1960), March (1953), Ward (1959) and Negi (1962 and 1967) under appropriate simplifying assumptions. We have computed the quantity Xvn for different values of the dimensionless induction number PI (= x/o01glc°r~) of" the sphere. The other variable parameters are the induction number P2 (= x/°02/-/2cdr J) of the shell, the magnetic permeability contrast (3'13 and Y23 = t11,2/3) between the conducting system and the surrounding medium, and the inhomogeneity factor v 1 defining the conductivity variation inside the sphere. Fig. 2 shows the variation of the in-phase (IP) and quadrature (QR) components of Xvn with P1 for n = 1, 2, 3 and 4. The IP and QR components ofXvl (n = 1) have been plotted for a non-permeable system in Fig. 3A and for the permeable system in Fig. 3B. Fig. 4 and Fig. 5 give the amplitudes and phases of Xvl for P2 = 2 and P2 = 5, respectively. Fig. 6 describes a screening factor j S [ which has been calculated on the basis of the definition given by Walt (1969) for an insulated cover and is given in our case as:
Xvn (k l, k 2) -Xvn (0, k2) [ Is t =
xvn (kl, O)
(23)
The functions Xvn (k 1, k2), Xvn (0, k2) and Xvn (k 1, O) correspond to the sphere-shell system, the shell alone (when the sphere is absent), and the uncovered sphere respectively. We have also obtained the multifrequency response of the system tbr different conductivities and thicknesses of the covering shell; it is given in Fig. 7. DISCUSSION OF RESULTS
Non-uniformity of the primary field The contributions of moments of various multipoles (for different values of n) excited in the spherical system, depend upon the relative dimensions of the sphere and the distance of the observer from it. As shown in Fig. 2, the magnitude ofXvn, decreases as n increases. The effective contribution of the higher multipoles to the secondary scalar potential will be still less as it is to be multiplied by a geometric factor r~"~n+l/(h n+l r n+l ) which will always be less than unity. Thus, if the source and the observer are not very close to the conducting sphere the series for V3m converges fast. The values of the first four terms in the series for V3m have
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
9
b.o I
o.9t
o2 - ,
_
~ 3 " / 2 3 "~
/
_
,
// /I/ !I/'
O" 7
o~ cZ O~
'
//:///////
~,-°
o . 8
/ / / U
~I7,1 //:/: ////
o.,/'/
o.6 o.~
u
o.o
~
I
0-5
I.O
_
I0
I00
R Fig. 2. In-phase lIP) and quadrature (QR) components of the response factor Xvn for n = 1,2, 3, 4.
TABLE I Values of the secondary scalar potential V3mnfor n = 1.2, 3 4. The quantity V~ corresponds to a uniform excitation field: 2n+ 1 #l =
l
hn+lrn+l XvnPn (cos0);0 = 0°;h/r 1 = 2;r/r 1 = 1.95;r2/rl = 1.25;p= = 1.0
P1
V~I
2 6 10 50
0.0/)02145 0.0004778 0.0005406 0.0006316
V1~12
0.0000298 0.0000961 0.0001196 0.0001551
V E X 100 V~nl
13.89 20.11 22.12 24.55
V~13
0.00000554 0.00001970 0.00002736 0.00003939
VT~ X 100 VTll
2.58 4.12 5.06 6.23
V~n4
0.000001207 0.000004011 0.00000584 0.000009857
V~n4 X 100 V~ll
0.56 0.84 1.08 1.56
been c o m p u t e d and listed in Table I. In the subsequent figures only Xvl has been plotted to present the influences of various paraineters. It m a y be noted that this will correspond to a u n i f o r m primary electromagnetic field. It has, however, been seen that the basic nature
10
J.G. NEGI, C.P. GUPTA AND U. RAVAL
of these effects on higher order terms also is the same.
Effect of the covering shell Fig. 3A and Fig. 3B show the effect of increasing the shell conductivity on the total response of the spherical system. The curves for P2 = 0 correspond to the response of the sphere alone. As P2 is increased, both IP and QR components of the response factor Xvl of the sphere-shell system are enhanced. The enhancement is more pronounced for low values o f P l. As seen in Fig. 3A, the IP component for a non-permeable system, with a shell induction number P2 = 5, has quite an anomalous shape besides having a general enhancement, It increases with P l , attains a maximum value and then drops with further increase i n P l to finally settle to an induction limit which is higher than that for an un-
1'2 I'0 0"8 0"6 l
0'4
rr w 0 , 2
zo
1"2
0,0
%g l
I.O
~r-
I
0.8
z o
,~ o..
0.4
-0'8 ZZ //
0'2
A
-0"4 -0.6
0-6
O'O O.5
- 0 "2
x
-I.0
I'0
I0
--pj
-I.2
I00
B
0-5 I'0
IO
I00
~pf
Fig. 3. The influence of variation in conductivity of the shell on the IP (in-phase) and QR (quadrature) components of Xvl for (A) non-permeable and (B) permeable systems. The lowermost curve (192= O) indicates the response of an uncovered sphere. The curves corresponding to uI -=+2, 0, - 2 are given to illustrate the effect of the inhomogeneity in the conductivity of the sphere.
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
11
covered sphere. For a permeable system (')'13 = 723 = 5), the corresponding (P2 = 5) IP (in-phase) curve shows a monotonic decrease with P I' The decrease of IP component with increase in Pl for P2 ~> 5 is an interesting finding of the present investigation. It may imply that under certain conditions the IP response of a body of given conductivity is smaller than that of a less conducting body when both are covered by similar shells. We may possibly attribute the decrease in the IP component with increase in Pl to a phase-shift of more than 7T/2 by the shell. Thus the individual IP responses of the shell and the sphere may become in opposite phases. If the response of the shell, having a high induction number as P2 ~> 5, predominates over that of the sphere •alone, the increase in Pl (and consequent increase in the response of sphere) will cause a reduction in the total response of the composite system. Fig. 5A shows that for P2 = 5 and 713 = 723 > 1, the response amplitude also shows a monotonic decrease with p I • Fig. 4B and 5B give the phases of the response factor forP2 = 2 and 5 respectively. It can be easily seen that a change in P l significantly influences the response phase if P2 ~< 2. But for higher values of p2 (say 5), there is only a little phase shift by increasing the value o f P l . This observation coupled with that from Fig. 5A indicates that if the conductivity of the shell is higher than a certain value, the sphere is masked to such an extent that its electrical properties can hardly be delineated. This inference holds even if the radii of the shell and the sphere have the same order of magnitude and the sphere has a nmch
1.3 0°
I.z I \
I
I
"\\ -20"~-
\
~, = + z = 0
I
[.i / --40"
l
-
- -
~13-- ~'23
1.0 -6o* L-
(uw 0
~\ \\\\
0"9
+
~80 °
NS'-- 0'8 -IOO*
0.7
t
/t
1.11=+2 .
ii/i. ~~1
.
.
.
== - 20 --
0.6 -~ , / / 0.5 r-y. I / /
.
.
.
~
120
.
\\\\\ \ \'\
'x\\,
.
~13=~a3t"J Pz=z --160 =
0.4
0"5 I'0 A
I0 _ _
p, ~
-tsO' 0.5
I00 B
I I-0
i I0 - -
Pf
I00 ~-
Fig. 4 and 5. Tile amplitudes (Fig. 4A and 5A) and phases (Fig. 4B and 5B) of the response factor for different permeability contrasts. The induction number (P2) of the shell is 2 in Fig. 4 and 5 in Fig. 5.
Xvl
12
J.G. NEGI, C.P. GUPTA AND U. RAVAL
1.6
)/13- 5
1'5
Y 12 3 ~ , \ -XN 1'4
l+o
\\\\\ NN~X \ \ "~X
\
++-
1"3
'\\ \
O n
t
r.2
I'1
)Jr-- + 2 =
+
-
-
-
0
=-2
'+o7
.....
I'0
\
I
P2- 5 I
0"5
I0
I'0
A
-
Pl
-
-17
I00 )
B
0".~
I'0
I0 - -
I00
Pt -'-~
Fig. 5. For legend see page 1 1. higher conductivity than the shell. Another way of looking at the influence of the shell has been put forth by Wait (1969). Fig. 6 gives the screening factor IS [, as defined in eq.23. According to this definition, the smaller the factor IS l, the more is the masking influence of the shell. For IS I = I there is no influence of the shell and for ISI = O, the inner core is completely undetectable. In Fig. 6 we find that there is always a particular combination of Pl and P2 when IS [ is minimum. As expected, the masking influence increases with increase in P2" For P2 = 5, the factor IS[ is quite small even when Pl is very large. I,O ~
-
~
O.9 ~
1
0'8 l -'-i U3 -
~jl= + 2 . . . . o.7
=-2
....
"¢,3. "¢2~. I
-
0'6 0.5
.
O. 4
0.3L 0-5
///]
/
.,/~ ~"
I 1.0
I IO - -
I00
pu ---~
Fig. 6. The variation of screening-factor ISI for different conductivities o f the shell.
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
13
Effect of the permeability The in-phase component of a non-permeable system is always positive (Fig.3A). This, however, is not the case when the system has a higher magnetic permeability contrast from the surrounding medium, i.e., when 713 = 3'23 :¢: 1. In this case the IP component is negative for low values of both P2 ( ~ 2) and Pl • The negative value of the IP component is due to the induced magnetisation which is in phase with the prhnary field as supported by Fig. 4B and Fig. 5B also. But as either P l or P2 or both increase, the contribution of the permeability is nullified and the IP component tends to become positive so that beyond a certain combination o f P l a n d P 2 , the IP component attains a positive value. F o r p 2 = 5 the IP component for the permeable system (3`13 = 723 = 5) is positive (Fig. 3B) even for very small values o f P l . The phenomenon of decrease of the response amplitude with p I, described above, for a non-permeable system at P2 = 5, is found to exist in the case of a permeable system (Fig. 4A) even for low values of p2. In this case the phenomenon is due to the opposing effects of 1he magnetic permeability and the electrical conductivity. Thus, in the process of IP becoming positive (Fig. 3B), its absolute value first decreases before crossing zero with the increase of P I' It implies that the amplitude which depends upon the absolute values of 1P and QR components would decrease in this range of P l. This effect is manifested in Fig. 4A in the range of 1.2 < P l < 10 for 3`13 = 3'23 = 5 andP2 = 2. In Fig. 5A, since P2 is quite high (=5) the amplitude shows a monotonic decrease.
Effect of bThomogeneity of the conductivity of the sphere In accordance with eq. 1, O01 is the conductivity of the surface of the sphere which remains invariant irrespective of the value of ~1- The conductivity towards the centre is, however, more (or less) as u I > 0 (or u 1 < 0) in comparison with the corresponding value for u I = O. This change in the conductivity of the inner layers, influences both the IP and QR components. It is seen from Fig. 3A that for low values of p2 ( ~ 2) the IP component is more, the larger is the value of v l . This enhancement of IP component with v I is expected since an increase in v I amounts to an effective increase in the conductivity of the sphere. The effect of the inhomogeneity is significant only in the intermediate range of P l (1 < P l < 4). The quadrature component is "also influenced by u I . For low values o f P l , the QR component is more for larger u I . However, for higher Pl values (i.e., a more conducting system) the quadrature component for larger values of v I decreases faster. Consequently, in this range o f P l , the quadrature component is smaller for larger v 1 . If the shell conductivity is increased so that P2 ~ 5. the influence of u 1 on the IP component undergoes an inversion. Thus, in this case the IP component is less for larger u 1 beyond a certain value o f P l . As mentioned above in connection with the relative influences of the magnetic permeability and the conductivity, there is a certain range of Pl and P2 for which the response amplitude of a permeable system decreases with increase in the conductivity of the core. A natural corollary of this observation is that in this particular range an increase in u l also lowers the response amplitude. This is shown in Fig. 4A for ~'13 = 3'23 = 5 and P2 = 2
14
J.G. NEGI, C.P. GUPTA AND U. RAVAL
and for all the curves of Fig. 5A. M u l t i f r e q u e n c y response
Fig. 7A and 7B describe the variation of IP and QR components of Xvl with the frequency of excitation for different values of thickness of the shell and its conductivity contrast with the core. For slowly varying fields and low conductivity of the shell, the field penetration to the sphere is more and we get the first peak in the response curve at 1,000 C/sec (Fig. 7A). This is mainly due to the induction in the sphere. The shell is more or less transparent at this low frequency. The second peak, occurring at a higher frequency, corresponds to the situation when the skin-effect becomes appreciable and the masking influence of the shell becomes high. The response of the system at higher frequencies is mainly due to the shell. The second peak is smaller than the first peak for low values of o02/o01 and r2/r I . However, as the conductivity or thickness (or both) of the shell increases, the second peak goes on increasing and becomes even higher than the first peak. The second peak disappears when the conductivity of the shell is very low and/or its thick2,4
' '
/ / rf 2.2
/ 2.0
Ooo2 -2 ~3- = io
I.e
/
)Ji" 0
I
I'2
//
/
E8
,.o
/
i
/
//
f-l.2
/I
i/ ii
Z~
//~-,.3
i i
/
//
1'4
~
/
//
"/13"Y23-I 1,6
/
J~.<~-l.i
.;..~ .....
o. 4
°'°,d A
~1.4
/
: t:~
,d
,d
Id
Io~
FREQUENCY(IN CYCLES PER SECOND)
Fig. 7. Multifrequency response of a two-layer spherical system for different (A) thicknesses and (B) conductivities of the shell, expressed in terms of the corresponding properties of the sphere.
ELECTROMAGNETIC RESPONSE OF A CONDUCTINGSPHERE
15
1.8 (TO2
=
F2
/ //
rI 1,4
/
%-'/z3
-~
/
1.2
"
/ /
, /
08
. I() I
,
/
/
/~-~'-i~ 2
~/
y
<1:o-
~_~o
]
0-6
/~/
0 4
/l
,A<~"'--..
~_-~ - -
-" -- .......
-s
2H.\-
0.2
o
- ~0
zf~-~-'~
o#
,d FREQUENCY
,o~
,~
,~
(iN CYCLES PER SECOND)
Fig. 7B. ness is very small• If the response is found over a wide range of frequencies, the different layers of a stratified sphere can be delineated on the frequency scale. This fact may be utilized in the study of geomagnetic variations, considering Dst, Sq, L etc. as the primary field and the model of the earth having shells of different conductivities (Price, 1970). ACKNOWLEDGEMENTS The authors are grateful to Dr. Hari Narain, Director of the National Geophysical Research Institute, Hyderabad for his kind permission to publish this work. We sincerely thank Messrs. K.N.N. Rao, Y.M. Ramachandra, M. Ramakrishna Sarma, V. Subrahmanyam and Miss G. Aruna for their valuable help in computation and preparation of the manuscript. The research was financially supported by the Department of Commerce, Boulder Laboratories, U.S.A. under agreement No. E-105-67(N).
APPENDIX Transverse magnetic dipole The magnetic dipole is assumed to be situated on the z-axis and directed perpendicular to it in the x-z plane. In this case, unlike the radial orientation of the dipole, both poten-
16
J.G. NEGI, C.P. GUPTA AND U. RAVAL
tials U and V exist and the field depends upon the q~-coordinate also. Thus, the primary field due to source of unit strength, here, takes the following form (March, 1953): UPIn
oo A Pn m ~'n (k3r)
sin,) =
VPm
Z
A pln =
"vn
r
(AI)
il~3w( 2n+ l )~n (k3h) n(n+l)k3 h
where: AuPn m=-
and:
Pnl (cosO)
- - -
cos4~ n=l A vpm
( 2n+ l )~'n(k3h) n(n+l)h
Accordingly, the expressions of the scalar potential in various regions may be expressed as: (1) Region 3, i.e., r 2 < r < h: U~
=
g~
Upm
+
g pm
sin )
~
cos) n=l
[ Am
un ~n(k3 r) p1n (cosO)
Avmn
] (A2)
h
(2) Region 2, i.e., r I < r < h:
V~
=
E cos,k ,,=1
Byron
r
+
r
Cvmn
Pn1 (cos0)
(A3)
(3) Region 1, i.e., r < r 1 :
uy'= sin,~ ooZ f Dumn Mn(klr'Vl)V•
cos;~
pnl (cosO)] foru 1 ~<2
r
n=l LDvm
(A4)
and
U~n = sinO
vT'
~
FDuinn Nn(klr, Pl)
cos~ ,,=~ LD,,m
;:
] e~ (cos0) for,,q > 2
(A5)
where the functions ~n (x), ~n (x),Mn, andNn are defined by eq.9b, 9c and l l a - 1 ld. The m , Bun in etc. are determined as before from the boundary conditions (eq. 13). constants Aun Thus, on substituting eq.A1-A5 into eq.13 and solving, we obtain: ~n (k3r2) R
Au~ =APm ~n(k3r2)
un
pm UNn(k2r2'p2) ] Burn -_ Aun ~.(k3r2)Fun L ~ v 2 ) j
(A6) (A7)
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE Apm
Cumn='qm ~n(k3r2)Fun
17
FMn(k2r2'P2)l vvg.,---~7
(A8)
LM n (k2r 2, v 2 }J
['%,(k2r2,v2)] M,; (k2r I , u2)Fw,
L~,~ik~r2~v2-~
an d:
1
I M nlk2r 2, v 2)d
(Ag)
D n}, -~ Apmun ~n (k3r2)
Mn(klrl,ul)
f o r v I -<-2
I
L o r N n ( k l r l , u 1) f o r u l .'>2
1
whe re: t
t
•
[Mn (k2r 2, v2) - PunN'n (k2r 2, v2)] a2+icoe 2 k 3 ~'n(k3r2)
o3+iwe 3 gu/I
~n(k3r2) [Mn(k2r2'v2)-PunNn(k2r2'v2)] (A10)
--
°2+ic°e2
°3+iwe3
k
3~nt (k3r2)
~n (k3r2)
[Mn (k2r 2, v2) - PunNn (k2r2, u2)] r
--[114n (k2r 2 , v 2) -PunNn (k2r 2 , v2)] with:
a 1+iooe1 +. Mn(klrl,Vl)Mn(k2rl,u2)-M'n(klrl,Vl)Mn(k2rl,v2) 0 2 t~e 2 Pun
"~
o 1+icoel , , o2+icoe 2 mn (k I r 1 , v I )Nn (k2r 1 , ta2 ) - N n (k2r 1 , v2)m n ( k l r 1 , v I ) for u I ~< 2
or:
°l+iC°el Nn(klrl,Vl)M,n(k2rl,u
o2 +iooe2
Pu?/
)- Nn(klrl,Vl)Mn(k2rl,P2)
-
°l+iC°el N n (k I rl, v l ) N n ( k 2 r l , , ' - , - N~; { k l r l , v 1)N n (k2rl, v2)
o2+icoe 2
foru! > 2 and:
J.G. NEGI, C.P. GUPTA AND U. RAVAL
18
2+ic°e2 k3~'n(k3r2) Nn(k2r2'v2) o3+i~e; " Cn(k3r2) Nn(k2r2,v2)
1]
t
f 02+icOc2"k3~n(k3r2) Nn(k2r2'v2) Fu [Nn(k2r2,v2) ]
+ Run k~'we3
~n(k3r2)
13
"Nn (k2r2, v2) (All)
-
°2+i°0e2 o3+iwe 3
IM' Nn(k2r2'v2) n(kzr2'v2)N~(k2r2,v2) -Mn(k2r2,v2)]
A similar expression is obtained for:
FunfMn(k2r2'u2 ) ]
LM'. (k2r2, v2)
Mn and Nn are defined by eq. 20. The constants A m ' Bvn,mCvnmand D m are obtained from Amn, Bumn,Cum, and Dumn(eq. • pm A6 - A9) by replacing Aun , Run, and (o/+iwe/)/(O/+l+iwe/+1) by Avn, Rvn and t~j/laj+1 respectively. Thus, for example: m_
pm ~n(k3r2)Rv n
Avn-Avn
~n(k3r2)
where Rvn is defined by eq.18.
Electric dipole When the exciting source is an electric dipole the values of the constants are obtained if we interchange (aj +iwej) by ils/w and vice-versa in the expressions A6 - A9. Thus:
4.
= Ape
~n(k3r2)Re (X3r2)
e _ pe fNn(k2r2'u2) 1 Bun - Aun ~n (k3r2)Fen L(k2r2-V" Nn" ~ , v2) ] _ pe e [-Mn(k2r2'v2)-I Ceun- Aun ~n (k3r2)Fun k ~ J and:
(A12)
..
(A13)
(A14)
ELECTROMAGNETIC RESPONSE OF A CONDUCTING SPHERE
19
M;,(k2r I u2)Fu, , ~Nn (k2r2' u 2 ) l
,
LN,;(k2r2,,;~5 A
-~,v,, (~2rl, ~2 i~,,, LM,', (kzr 2, ~ 2)--1 D~en =AtPen ~ n ( k 2 r 2 )
.
.
.
.
.
.
(A15)
,
! Mn(ktrl,vl)
f o r u 1 ~<2
I or:
iNn(klrl,Vl)
I i
foru I >2
i
A
(o3+icoe3) where:
A pe:_`
(2n+ 1) ~n (k3h) n(n~l)--k3h~ --
Reun occurring in eq.A15 can be directly derived f r o m e q . A 1 0 through re. e Bun, e Ceun and Den can (oj+icoej)/(Oi+l+icoei+l) by/a//ta/+ 1 . Expressions for Aun,
The factor
placing also be obtained similarly.
REFERENCES Bateman, A.M., 1950. Economic Mineral Deposits. Wiley, New York, N.Y., 2nd ed., 916 pp. Debye, P., 1909. Der Lichtdruck auf Kugeln von beliebigem Material. Ann. Phys., 30:57. Fuller, B.D., 1971. Electromagnetic response of a conductive sphere surrounded by a concentric shell. Geophysics, 36: 9-24. Lowell, J.D. and Guilbert, J.M., 1970. Lateral and vertical alteration-mineralisation zoning in porphyry ore deposition. Econ. Geol., 65: 373-408. March, H.W., 1953. The field of a magnetic dipole in presence of a conducting sphere. Geophysics, 18: 671-684. Nabighian, M.C., 1970. Quasistatic transient response of a conducting permeable sphere in a dipolar field. Geophysics, 35: 303-309. Negi, J.G., 1962. Diffraction of electromagnetic waves by an inhomogeneous sphere. Geophysics, 27: 480-492. Negi, J.G., 1967. Electromagnetic screening due to a disseminated spherical zone over a conducting sphere. Geophysics, 32: 69-87. Negi, J.G. and Raval, U., 1969. Negative electromagnetic screening by a cylindrical conducting cover. Geophysics, 34: 944-957. Price, A.T., 1970. The electrical conductivity of the earth. R. Astron. Soc., Q.J., 11: 23-42. Rose, W.A., 1970. Zonal relations of wall rock alteration and sulfide distribution at porphyry copper deposits. Econ. Geol., 65: 920-936. Roux, A.T., 1959. A review of airborne electromagneuc systems, some factors affecting the choice of systems and interpretation of data. Geol. Soc. S. Afr., Trans., vol 62. Schelkunoff, S.A., 1948. Electromagnetic Waves. Van Nostrand, New York, N.Y., 530 pp. Wait, J.R., 1951. A conducting sphere in a time varyine magnetic field. Geophysics, 16: 668-672. Wait, J.R., 1953. A conducting permeable sphere in the presence of a coil carrying an oscillating current. Can..L Phys., 31: 670-678.
20
J.G. NEGI, C.P. GUPFA AND U. RAVAL
Wait, J.R., 1960. On the electromagnetic response of a conducting sphere to a dipolar field. Geophys2cs, 25: 649-658. Wait, J.R., 1969. Electromagnetic induction in a solid conducting sphere enclosed by a thin conclucting shell. Geophysics, 34: 753-759. Ward, S.H., 1959. Unique determination of conductivity, susceptibility, size and depth in multifrequency electromagnetic exploration. Geophysics, 2 4 : 5 3 1 - 5 4 6 . Ward, S.H., 1967. Mino~g Geophysics. The Society of Exploration Geophysicists, Tulsa, Okla. 2 : 7 0 8 pp.