Download The Influence of Surface Inhomogeneities On Deep Electromagnetic

Geophys. J. R. astr. SOC. (1987) 90,61-73
The influence of surface inhomogeneities on deep
electromagnetic soundings of the Earth
E. B .
B.
Fainberg and
Sh. Singer Institute of Terrestrial Magnetism,
Ionosphere and Radio Wave Propagation (IZMIRAN), USSR Academy of Sciences,
142092 Troitsk, Moscow region, USSR
Accepted 1986 November 17. Received 1986 November 17; in original form 1985 April 3
Summary. Asymptotic expressions for components of the electromagnetic field
of a grounded electric dipole are considered for the model consisting of a thin
surface-layer overlapping a stratified medium with a highly resistive screen on
the roof. It is shown that the method of spatial derivatives makes it possible to
obtain proper estimates of the impedance at distances of r , lhol from the
nearest edge of the surface anomaly (IXo I being the effective depth of the field
penetration in the underlying section). The magnetotelluric methods allow one
to obtain the true values of impedance, provided r s max {lhol, lF/(S-' +
Z, 11'2} where S is the integrated conductivity of the surface layer, F i s the
transverse resistance of the screen, and& is the Tikhonov-Cagniard impedance
for the medium underlying the surface layer.
Key words: magnetotellurics, electrical dipole, subsurface inhomogeneities,
distortions
Introduction
Difficulties of interpretation of findings yielded by deep electromagnetic-exploration,
associated with inhomogeneities of the surface layer, are well known. The difficulties are of
a common nature both for exploration with use of electromagnetic fields of natural originf'
and for exploration with fields excited by controlled sources of current. To overcome such
difficulties, use should be made of electromagnetic field modelling. Still, it should be noted
that despite a high level of development of physical (Moroz, Kobzeva & Timoshin 1975) and
numerical (Vasseur & Weidelt 1977; McKirdy, Weaver & Dawson 1985; Singer et af.
1984) modelling, it is not readily accessible and flexible instrument of research. In connection with the above, the question of how far one should go away from any inhomogeneity in order to obtain information on the Earth's deep structure, by any method based
on a simple laterally uniform model, gains essential significance. In order t o 'answer this
question, the law governing attenuation of anomalous electromagnetic field components
with distance from a surface inhomogeneity should be elucidated. One should bear in mind
that the distance for which the anomaly distorts the electromagnetic field is dependent on
62
E. B. Fainberg and B. Sh. Singer
both the law of change of the anomalous field and the magnitude of the anomaly. In cases
when the anomalous field is small compared to the normal one (weak inhomogeneity),
sounding may be performed even above the anomaly itself without essential errors. On the
other hand, such vast anomalies as seas, oceans, mountain massifs of regional extension, large
active faults, etc., cause a high distortion of electromagnetic fields, but at sufficiently great
distances, the anomalous fields created by them have a structure that allows a proper
determination of parameters of the medium. Thus, the question posed should be formulated
in the following way: under what conditions will a reliable estimate of the impedance be
obtained by the magnetotelluric method, the method of spatial derivatives, and other
methods, if sounding is performed in the total field created by both the external source and
the anomaly of surface conductance. As regards the problem of determination of the
anomalous field values, it can, with a few exceptions, be solved only with the help of
modelling.
Determination of the law of decrease of the anomalous electromagnetic field is also
important for the methods of numerical modelling, as the numerical grid may be restricted
at distances greater than a typical scale of an anomalous field decrease.
An integral equation (Vasseur & Weidelt 1977) for determining tangential components of
electric field is:
E,(r,O)=Er(r,O)+
6(r -r')SaET(r')ds'.
4! .
This equation is derived for the model consisting of a thin surface layer with conductance S,
which differs from the constant value S, only in an anomalous area A ; S, = S - S, is the
value of the anomalous conductance. The inhomogeneous surface layer is underlain by a
laterally uniform section; C(r) is the Green's tensor of the normal section. Once the electric
field is known, components of the magnetic field may be found with the help of the integral
relation
H(r,O -)=H"(r,O-)+
dH(r-r'))S,E,(r')ds'.
The term E; of equation ( 1 ) represents the electric field excited in the normal model, i.e. in
the model that does not contain lateral inhomogeneities; Hf is the associated normal
magnetic field, the components of the Green's tensor functions d(r) and d ~ ( rrepresent
)
components of the field created by a grounded electric dipole. Therefore, in order lo reveal
the law of anomalous field decay beyond the inhomogeneity it is necessary to study the
asymptotic behaviour of a field of such a dipole at a great distance. A review on 'the
adjustment distance' problem has recently been published by Jones (1983). The analytic
solution for the B-polarization case has been found by Dawson, Weaver & Raval(l982).
1 Electromagnetic fields and impedances in the stratified medium containing a resistive
screen
The problem of computation of a field component excited by a horizontal electric dipole
with a moment p e x d - j u t ) , lying on the surface of a laterally uniform section is one of the
classical problems of electrical prospecting. Thus, Vanyan (1965) has shown that in the
absence of intermediate non-conducting screens in the section, tangential components of the
electric and magnetic fields decrease at a great distance from the source according to the
law l/?, and the vertical component of the magnetic field according to the law l/r4. As
revealed elsewhere (Vanyan 1959; Dmitriev, Skugarevskaya & Fedorova 1970): the presence
Surface inhomogeneities and deep electromagnetic soundings
63
of non-conducting screens in the section leads t o another law E,- l/r2. The question of
which of the laws (namely l / r z or l/r3) operates in practice is of essential significance for
geoelectrics. Assuming that the medium is such that at great distances from the dipole, the
electric field will decrease as E,- I/?, whereas H , - l/r3, it is likely that the MTS results
at considerable distances from inhomogeneities will be distorted. However, in practice,
geoelectric sections d o not include layers, being ideal insulators. Therefore, the asymptotic
behaviour of the dipole field in a medium with a resistive screen is of central importance.
The model to be considered below consists of a uniform surface layer underlain by a resistive
layer of thickness h and a conductivity uo(z) that is dependent (generally) upon depth. At
depths where z > h, conductivity varies in accordance with the law u(z), where u% uo,
referring to the typical values of conductivity at depths z > h and 0 < z < h, respectively
(Fig. 1).
Figure 1. The lateraily uniform model excited by a grounded
electrical dipole.
The electromagnetic field created by a grounded electric dipole at the surface of a
stratified medium (i.e. z = 0 -) may be expressed in the form (Appendix A):
I
d
d
r dr dr
r
Equations ( 3 ) and (4) determine the horizontal components of electric and magnetic fields,
the last equations determining the vertical component of a magnetic field and the density
of leakage currents. Here r is the vector from the dipole to the observation point, e , = r/r,
e9 = n x e,, where n is unit vector of normal to the surface of the Earth being directed
upwards, p = prer i- p 9 e q , p r = p * cos cp, p q = -p * sin 9,cp is the angle between electrical
E. B. Fainberg and B. Sh Singer
64
dipole and observation point directions, S is the conductance of the thin surface layer.
Functions Q, are defined by
Q,(r> =
1-
q v ( k ) J 1( k r )d k / ( 2 n ) ,
v = 1,2,3,4,
0
where J1( z ) is the first order Bessel function,
and
q 4 ( k ) = [ l t SZg(O)]-’
Here
ak =
(9)
2kXk(0)
1 + khk(0) ’
the value &(o) determines the spectral impedance of the underlying medium for the
inductive (toroidal) mode:
ZL(0) = -iapOhk(o).
(10)
The absolute value IXk(0)J is an effective penetration depth of inductive mode for the spatial
harmonic exp (- ikr), ZZ(0) is the spectral impedance of underlying section for the galvanic
(poloidal) mode of the field.
A conventional method of calculation of asymptotic expressions for functions (7)
consists of the following. In integrals (7) values of k < l/r are essential. Accordingly, if r is
great it is possible t o substitute factors before Bessel’s function in the integrals with a power
series in k . It is well known that field behaviour within a uniform layer is controlled by the
value (k2 - iwpoa)”’. Hence, the above expansion is legitimate for such great distances as
k l / r < ( ~ p ~ a is
) ”of~interest for. We shall assume that this condition is satisfied’ for all
layers of the section except for the screen, since for the latter the parameter ( W ~ ~ O ~ is)
small. Accordingly, we shall suggest that
-
’ / ~
lh~(h)l
(a)
i.e. the distance from the observation point to the source is much greater than the depth of
field penetration into the section underlying the screen (for k = 0). The presence of the
screen affects the behaviour of the inductive and galvanic modes of the field quite
differently. Thus, with the assumptions that
r>h
and that the screen conductivity is limited by the condition I ~
(- i ~ p ~ u ~ ) it~ is
’ ’shown
),
in Appendix B that for the inductive mode
hk(0) X ho = h t- ho(h)
(b)
~e 1 l (where
h
K
~
( 1 1)
[ho will be used below instead of h O ( 0 ) ]Thereby,
.
both limitations (a) and (b) for the
required distance from the source could be replaced by r I hol, where I hdl is the effective
*
=
Surface inhomogeneities and deep electromagnetic soundings
65
depth of penetration of the inductive mode into that part of the section lying beneath the
thin surface layer. With the same restrictions, the following expression for the spectral
impedance of the galvanic mode is obtained:
Zg,(O)
=z;
(12)
+Yk2,
where Z , is Tikhonov-Cagniard's
.F=
impedance of the underlying section, while
[
o;'(z)dzi-~*
(13)
is the effective transverse resistance of the underlying section. F * designates the
contribution to the transverse resistance due to the section lying beneath the resistive layer
( z > h). As a result, the following expressions for integrand in (7) are obtained
ak
2kXo( 1 - kho),
(14)
-1
XO
AJ
2k
where Xi1 = -iwpoS and A;' =hi' i- A;'. The value l X s l determines a characteristic size
associated with the horizontal skin-effect; !AoI is an effective depth of field penetration
into the section including the surface thin layer. This value is closely connected with
Tikhonov-Cagniard's impedance 2, = -iwpoAo on the Earth's surface. Expansion (14) is
valid, provided that kl Xo I e 1, and expansion ( 1 5 ) required the condition that
IkXoAo/XsI< 1 , which, as it is evident, is less strict than klhol< 1. Indeed, in the highfrequency range (the S-interval) l X o l % lhsl and, accordingly, A. = As. So the restriction
klhoAo/XsIe 1 leads t o klXol< 1. In the low-frequency range (the h-interval) Ihol< IXsl,
A. = A o andsoklhoAo/XSI=klhoI- Iho/Xsle1 iscertainly fulfilled ifklXoIe 1 .
Integrals appearing after substitution of (14), (15) in (7), (8) can be calculated if use is
made of the known expressions
[Jl(x)dx = 1,
X 2 m + iJ , ( x ) dx
= (-1)"(2tn
+
1)
Integrals included therein should be considered as the limit at E + Oi- of integrals containing
an additional factor exp(-ek) in the integrand. As a result, asymptotic expressions for Q , ,
Q2 and Q3 are as follows
66
E. B. Fainberg and B. Sh. Singer
which are valid at r , Iho I. Substitution of ( 1 2 ) into (9) leads to an integral which can be
calculated with the help of the known expression
where r(z) is Euler's gamma function, K,(z) is McDonald's function of order v. Thus
function Q4 has the asymptotic form
where
hL = (S.EAo/Xo)l/z = [ .E/(S-'
+ 2;)
designates a parameter that controls the effects of current leakage via the resistive layer.
Within the h-interval A. = ho,and hL =
(Ranganayaki & Madden 1980). Within the
S-interval hL = ( F / Z ; ) ' / 2 ~;'(h/h,,)'/* and XL doesn't depend upon S. Substitution of
(18)-(20) and (22) into (3)-(6) makes it possible to obtain asymptotic expressions for the
field components of a horizontal electric dipole, which are valid at distances r > I hoI from
the source:
(,§a1/'
-
Thus, at distances r , Ihol from the source, tangential and vertical components of the
magnetic field decrease in accordance with the laws l / r 3 and l/r4, respectively. At the same
time the electric field behaviour at the distance r s lhol is additionally dependent on the
ratio between r and hL. In the case of r lhLl contributions of terms containing McDonald's
function are exponentially small and expression (23) leads to a l / r 3 dependence of the
tangential component of the electrical field upon distance. This is evident if the 'function
K 1( 2 ) is replaced by its asymptotic representation at large values of the argument:
At small values of z, K l ( z ) = z-'; hence, within the distance range I hol Q r Q I XL I, the
behaviour of electric field components is substantially changed to the form
2nE, = S-'
A0
- ( p r e r - p,e,)
P.
A0
Expressions ( 2 3 ) and ( 2 4 ) make it possible to evaluate relationships between mutually
orthogonal components of the electric and magnetic fields on the Earth's surface
EJH, = -Zo [ l - ( S Z o X o h ~ ) - 'r 3 K ; ( r / X L ) ] ,
E,/Hr = Z o [l -t. (2SZohohL)-' r z K l ( r / l q , ) ] .
(28)
Surjace inhomogeneities and deep electromagnetic soundings
67
It is helpful t o obtain similar ratios for field components on the t o p of the section underlying the surface layer. The latter may be useful, for instance, for ocean bottom MTsoundings or soundings with the help of vertical gradient methods. The magnetic field H;
beneath the surface layer is calculated from (23) and (24) using ratios (A7) and (A8), so
An estimate o f the impedance can also be made with the help o f the spatial derivatives
method (Berdichevsky, Vanyan & Fainberg 1969). Making use of (24), (25) we obtain
Zo = -iwpoHz/ V. H,.
(30)
Thus, the impedance may be determined by the spatial derivative method at distances
r s lhol from the source in case the field is excited by the dipole source, whereas the
magnetotelluric method requires the additional restriction r B I XL I t o be satisfied.
2 Impact of surface inhomogeneities
The magnetotelluric method is based upon determination of the impedance Zo from the
ratio E, = Z o* n x H,, which is valid for the laterally uniform medium excited by a plane
wave. Similar t o the magnetotelluric method are vertical gradient methods with vertically
separated measurements o f tangential components of the magnetic field. The method of
spatial derivatives is based on the use of the vertical component of the magnetic field and
its vertical derivative. The impedance is in practice calculated from the ratio
-iwpoHz = 2,V. H,. This method is valid as long as the spectral impedance remains
practically unchanged in the spatial frequencies domain covering the major part of the
magnetic field energy. Since the magnetic field observed on the Earth's surface belongs to
the inductive mode, whereas both modes make contributions t o the telluric field, it is likely
that the method of spatial derivatives will be less sensitive t o distorting effects of geoelectric inhomogeneities than the magnetotelluric one.
The electromagnetic field in the presence o f surface inhomogeneities may, be determined
from equations ( I ) , (2). The Green's functions included in these equations 6(r - r') and
GH(r - r?, as was already mentioned, have a simple meaning: if the source of a field is a
surface electric-dipole jse(r) = ep(r - r'), i = 1 , 2 located a t point r ' , then the fields created
by it at the point r will be equal t o
,
E, = e l GIi(r - r ') + e2C2'(r - r ')
and
H=elG:(r
- r ' ) + e 2 G $ ( r -r')+e3G$(r - r ' ) ,
where e l , e2 (horizontal) and e 3 = -n (vertical) are the unit vectors o f the Cartesian
coordinate system.
In the case the distance o f the observation point r from the anomalous area substantially
exceeds its geometric size, the Green's function can be taken out of the integrals in ( I ) , (2).
In this case the contribution of the anomaly t o the field observed at point r coincides with
E. B, Fainberg and B. Sh. Singer
68
the field of an equivalent electric dipole having a moment
P=
s,
SaE, ds
and located within the anomalous area. If the size of the anomalous area is comparable with
the distance to the observation point, or exceeds it, then the areaA can be subdivided into
parts A i ( A = UiAi),
which are small compared with the distance to the observation point.
Consequently, the anomalous field at the point r may be considered as the field of the set of
electric dipoles
Pi
=S,:
E, ds
located in points ri within Ai,i.e.
1i Ef(r),
H(r) = H"(r) t. 1 Hi(r),
E,W = E X r ) +
(31)
(32)
i
where subscript i denotes the contribution of the dipole (pi, ri}. We assume that the fields
E:(r) and H"(r) allow the determination of the impedance Z , in accordance with the
formulas to be used in magnetotelluric method and the method of spatial derivatives. Now,
we shall try to answer the question of how far one should get away from the edge of the
anomaly to calculate properly Zo basing on total fields (31), (32). It is not difficult to
answer this question bearing in mind that the ith member of the sum is determined by
formulas differing from (23)-(25) by the substitution of pi for p and r - ri for r. Therefore,
if the distance from the nearest edge of the anomaly is rmin> Ihol, the formula
-iwpoHz = Z , V. H, is valid for each member of the sum, hence, the whole sum satisfies
this formula. Analogous considerations show that at a distance of r,in > max { I ho I, I hLI }
from the nearest edge of the anomaly, the impedance is correctly evaluated with the help of
the magnetotelluric method.
Thus, the method of derivatives poses, generally speaking, less strict constraints upon the
geoelectric structure of the region in which it can be applied, since it yields the true results
at distances exceeding lXol from the nearest edge of anomaly, whereas the use of the
magnetotelluric method requires, not only rmin> Ihol,but also rmins I h ~ l Expressions
.
(29) show that the same statement is valid for determinations of the impedance bkneath the
surface layer (bottom of the ocean) as well.
The ratio between scales IXo I and lhLl is determined by the geoelectric section. Thus, for
crystalline foundations of shields it is feasible to adopt 9-10"Ohm m2 (Jamaletdinov
1982). At S 100--1000 S and in h-frequenty interal hL 1000-3000 km, IhLJ greatly
exceeds the value Iho 1 5 400 km. This result makes clear the wide spreading of impedance
distortions due to the surface inhomogeneities' influence. For oceanic areas hL 450 km,
assuming that 9-1O7 Ohm m2 (Cox 1980) and S 2 1O4 S.
If the values of transverse resistance of the Earth's crust assumed herein do not differ
much from the actual ones, then magnetotelluric observations at the ocean bottom seem to
be more promising compared with those made on continents.
Another advantage of the method of derivatives is that impedance determinations
are independent of both intensity of the source and its geometry and orientation. This
advantage seems to be quite essential for deep electromagnetic soundings with controlled
sources, such as the experiment 'Khibini' in the Kola peninsula, where the geometry of the
effective source is unknown.
-
-
-
-
-
-
Surface inhomogeneitiesand deep electromagnetic soundings
69
Acknowledgments
We are grateful to Drs T. W. Dawson and P. Weidelt for help in preparing this paper.
References
Berdichevsky, M. N., Vanyan, L. L. & Fainberg, E. B., 1969. Magnetovariational sounding according to
spatial derivatives of a field, Geomagn. Aeron., 9, 369-371 (in Russian).
Cox, C. S . , 1980. Electromagnetic induction in the oceans and induction inference on the constitution of
the Earth, Geophys. Surveys, 4,137-156.
Dawson, T. W., Weaver, J. T. & Raval, V., 1982. B-polarization induction in two generalized thin sheets at
the surface of a conducting half-space, Geophys. J. R. astr. Soc., 69, 209-234.
Dmitriev, V. I., Skugarevskaya, 0. A. & Fedorova, E. I., 1970. On high frequency asymptotics of electromagnetic field in the stratified medium. Izv. Akad. Nauk. SSSR, Physics o f the Earth, 2,44-51.
Jamaletdinov, A. A., 1982. Normal electrical section of crystalline foundation and its geothermal interpretation according to the data of MHD-soundings in the Kola peninsula, in Deep electromagnetic
soundings with the use of pulse MHD-generators, pp. 35-48, Apatites, (in Russian).
Jones, A. G., 1983. The problem of current channelling: a critical review, Geophys. Surveys, 6,79-122.
McKirdy, D. McA., Weaver, J. T. & Dawson, T. W., 1985. Induction in a thin sheet of variable conductance at the surface of a stratified earth - 11. Three-dimensional theory, Geophys. J. R. astr. Soc.,
80,177-194.
Moroz, I. P., Kobzeva, V. M. & Timoshin, B. V., 1975. Modelling of electrodynamic processes in the
inhomogeneous conducting medium. Kiev, (in Russian).
Price, A. T., 1949. The induction of electric currents in nonuniform thin sheets and shells, Q. J. Mech.
appl. Math., 2, 283-318.
Ranganayaki, R. P. & Madden, T. R., 1980. Generalized thin sheet analysis in magnetotellurics: an
extension of Price's analysis, Geophys. J. R . astr. Soc., 60,445-457.
Singer, B.Sh. & Fainberg, E. B., 1985. Electromagnetic Induction in Non-uniform Thin Sheet and Shells,
(Chapter 2, 3) Moscow, (in Russian).
Singer, B.Sh., Dubrovsky, V. G., Fainberg, E. B., Berdichevsky, M. N. & Ilamanov, K., 1984. Quasi threedimensional modelling of magnetotelluric fields in the South-Turanian platform and in the SouthCaspian magadepression, Izv. Akad. Nauk. SSSR,Physics o f the Earth, 1,69-81.
Vanyan, L. L., 1959. Some problems of the theory on the frequency soundings of horizontal stratifications, Prikladnaja Geophysika, 23, p. 3-45, (in Russian).
Vanyan, L. L., 1965. Basics of Electromagnetic Soundings, Moscow, (in Russian).
Vasseur, G . & Weidelt, P., 1977. Bimodal electromagnetic induction in non-uniform thin sheets with an
application to the northern Pyrenean induction anomaly, Geophys. J. R. astr. Soc., 51,669-690.
Appendix A
THE FIELD O F A S U R F A C E ELECTRICAL DIPOLE
The electromagnetic field in a laterally homogeneous medium is expressed via two scalar
functions V(r, z ) and W(r, z) using the following relationships (Vasseur & Weidelt 1977)
E, = i w p , , n x ~ , ~ V,
-
H, = V,
av
-az
Ez = a-'V
Hz= - V t V ,
W,
n xV,W,
(
3
u-l - ,
E. B. FainbergandB. Sh. Singer
70
where E,, H, are the tangential components of the electric and magnetic fields; E,, H, are
the corresponding vertical components; a(z) is the conductivity of the medium which
depends only upon the depth z ; n is the unit vector of external normal to the surface of the
Earth, and V, is the operator of spatial differentiation wrt horizontal radius-vector r; time
factor is exp (-iot). The potential V(r, z ) of the inductive (toroidal) mode and the
potential W(r, z) of the galvanic (poloidal) mode satisfies the equations
V' V + iwl.coo(z)V = 0,
V*
(A51
[a'(Z)VW ] + iwpo w = 0
(A6)
and zero boundary conditions at t ++m. In a non-conducting atmosphere (z<O)
W(r, z) = 0. As a consequence, the magnetic field is expressed only via the potential V(r, z).
The tangential component E, of the electrical field and the vertical component of the
magnetic field H, are continuous at the surface thin sheet (Fig. 1) while the tangential
component of the magnetic field suffers a discontinuity
n X[HT(r, 0-) - H,(r, O+) ] = js,
(A71
where the surface current density is
jS=SET+jse.
Here S(r) is the conductance of the surface layer, jse(r) is the density of the external
surface current. From (A7) and (A2) follows the expression for the surface current density
js = - n xV,$ - V, W(r, O+),
('49)
where
aw
aw
+(r) = -(r, 0 + ) - -(r, 0-1
aZ
az
is a current function. (Al), (A8) and (A9) lead to the vector equation for functions $(r)
and W(r, O+):
S-'(-nxV,J/ - V,W)= iwponxVTV- V,
(
z)
a-' -
+ S-ljse.
Simple operations reduce this to a system of two scalar equations (Singer & Fainberg 1985)
V . [ ~ - ' ( v , + - n x V , W ) ] =-iol.coV:c/+ V;
(S-'nxjse),
('41 1)
The first equation can be reduced to the well-known Price's equation (Price 1949) if one
omits the second term in the rhs and the term with W(r, 0+) in the lhs. The latter
corresponds to the absence of leakage currents. For a uniform surface layer, (A1 1) leads t o
two independent equations
V+($ + iopoSv) = V, (nx j*),
(A12)
Surface inhomogeneities and deep electromagnetic soundings
71
It is not difficult to solve these equations, if one makes use of the relationships
d
- V(k, z ) = k V(k, z )
dz
if
z < 0,
d
if z > 0
- V(k, z ) = - hi1( z ) V(k, z )
dz
and
d
u-l - W(k, z ) =- ZB,(z)W(k,z) if z > 0
dz
between Fourier components V(k, z ) , W(k, z) of the potentials V(r, z ) , W(r, z ) and their
derivatives with respect t o z. The value hk(z) determines the spectral impedance of the
inductive mode
ZL(Z) = -iwpoh&)
(A 16)
for the medium lying at depths greater than z, Z:(z) is the spectral impedance of the
galvanic mode. The solutions of(A12), (A13) are
where integration is carried out along the Earth surface, R = r - TI, eR = R/R. Equations
(AIO), (A14) leads also to expressions
-
V(r , 0 ) = LQ, ( R) (n x eR) jse(r ’) ds ’,
a
Q3(R)(n x e R ) * jse(r’) ds
Functions Q,(r), Y = 1 , 2 , 3 , 4 are defined via(7)-(9). Expressions (A17)-(A20) together
with (Al)-(A4) leads to the expressions for the electromagnetic field components at the
Earth surface. Replacement of the external current density jse with p6(r’) results in
expressions (3)-(6) for the field of grounded electrical dipole with a moment equals to p.
Appendix B
IMPEDANCE F U N C T I O N S O F A L A T E R A L L Y UNIFORM SECTION CONTAINING A
RESISTIVE LAYER
We shall obtain an approximate expression for hk(0) under the assumption that the upper
part of the underlying section contains a resistive screen of thickness h and conductivity
u o ( z ) ,beneath which lies a more conducting part of the section u(z), { u % uo}. Changes of
the value Xk(z) within the range of depths 0 < z < h are described by the equation
-d hk -
dz
[k2-k K$(Z)] h2 = - 1 ,
E. B. Fainberg and B. Sh. Singer
72
the solution of which should coincide with the value hk(h) at z = h. The modulus of the
latter determines the effective penetration depth of the inductive mode into that part of the
section lying beneath a resistive layer. At rather low conductivities of the layer, such as
I K o I h < 1,
(B2)
Uo*U
and for such small values of k that
klh,(h))* 1,
k h < 1,
.
it is possible to omit the second term in the lhs of (BI).
~ ’ ~ consequently I ~ ~ h ~ ( h( U) ~l <
/ U ) ” ~ .It
Indeed, Ihk(h)l < Iho(h)l ( w l . ~ ~ u ) -and,
should be noted that we d o not suggest homogeneity of a resistive layer. The value uo or K~
included in the estimates determines only the order of the relevant values within the
boundaries of the resistive layer. Thus, equation (BI) after having been simplified is reduced
to dhk/dZ = - 1 , hence
-
hk(0) = hk(h) h + 0 [K:(k, 2) A:],
Ki(k, Z ) = k Z + K i ( Z ) .
Bearing in mind that the dependence of hk(h) upon k is determined by the terms
[kh0(h>l2one concludes that
-
hk(0) = ho(0)+ 0 [K:(k,
Z)
hi],
(B4)
where
h,(O) = h f A@).
The impedance of the galvanic mode within a resistive layer satisfies the equation
d
- 2;
dz
-
Uo(Z)
(Z;),>’ = -K:(k,
Z)/Uo(Z).
(B5)
In order to determine the dependence of 2; upon k the following expansion should be
substituted in (B5):
Z;(Z> = Z 0 ( z )+ f l ( z ) k 2-I-f 2 ( z ) k 4+ . . .
(B6)
Here the equality Zk(z) = 2 { ( z ) = Z o ( z ) is used (Vasseur & Weidelt 1977). Comparison of
the terms with the same power of k leads to a chain of equations
df1
- 2U020(Z)fi = - U i ’ ( Z ) ,
dz
and so on. The solution of equation (B7) has the form
fl(Z)
=
[fm+ ~(S)u;l(S)~~]/F(z),
Surface inhomogeneities and deep electromagnetic soundings
73
The exponent may be estimated as
in accordance with (A16). This value is small compared to unity since
constrains (B2). Thus, F(z) = 1 and formula (B9) can be simplified to
fi ( z ) =fl ( h ) +
K~
complies with
00' (5) d5'.
The value fi(h) represents the coefficient of k2 in the power series expansion of Z i ( h ) .
Below it is designated a s F * . Thus,
Z8,(h) = Zo(h) + F * k Zt- . . .
(B11)
-
It is not difficult to evaluate that .T* Ao(h)/a.Thus, the coefficient of k2 in the series for
Zi(0) is
yEfl(0)=.T* +
I
a i ' ( z ) d z + O(K;hi).
The solution of (B8) has the form
f2
(h) - f 2 (0) =
I,"
atf: ( z ) dz
-
h/ao
-
[h + Ao(h) a. /uI2.
Taking into account that f 2 ( h ) Al(h)/a we obtain the series
Z i ( 0 ) =Z,(O) + Y k 2 + . . .
The first omitted term is small compared withFk'.