Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

Electromagnetic Induction
in the Earth
Lectures Notes Aarhus 1975
Ulrich Schmucker and Peter Weidelt
Laboratoriet for anvendt geofysik
Aarhus Universitet

Electromagnetic Induction -- in the Earth
Contents
1. Intrcduction, bas5.c equati.ons
1.1 . General ideas
1.2. Basic equations
2. Electromagnetic induction i r ~ one-diniensional structures
2. ? . The yenerai solution
2.2. Free modes of d-ecay
2-3. .Ca.lculation of the TE-field for a layered structure
2.4. Particular source fields
2,5. Definition of the transfer function C (w ,K)
2.6. Properkies of C(W, K)
3. I.4odel calculations For two-dimensional struc-tures
3.1 . General equations
3.2, Air-half -space and conductor as basic domain
3,3. A~lomalous slab as basic domain
3.4. Anoinalous region as basic domain
4. Model. calculations for three-diwensional structures
4-7. Introduction
4.2. Integral equation metb-06
. 4.3. The surface integral approach to the mndell-iny
p?:obl.em
5. Conversion formu:lae for a t~~7o-d:i1aensi.ol1al TE-field
5.1 . Separation formulae
5.2, Conversion formulae for the f ield components of
a two.-dimensional YE-fri.el-il at the surface of
a one-dimensional st-ructure
6. Approaches to the inverse problem of electromagnetic
iaduction by linearization
6.1. The Backus-Gilbert method
6. 1 . I. Lntroiiuction
..G , I .2. The linear inverse prol~l.em . .
6.1.3. The non~linear inverse problem

6.2. Generalized matrix inversion
6.3. Derivation of the kernels of the 1-inearized inverse
problem of electromagl~etic induction
6.3.1. The. one-dimensional case
6.3.2. Pa.rtial. derivatives in two and three
diniensions
6.4. Quasi-linearization of the one-dimensional inverse
problem of electromagnetic inducti.0~ (Schlfiucker's
Psi-algoritlun)
7, Basic concept of geomagnetic and magnetotell.u~ic
depth sonding
7.1. Gelleral chaxacteristics of the method
7.2. The data and physical properties of internal
matter which arc involved
7.3, Electromagnetic wave propagation and diffusion
through uniform do~nains
Appendix to 7.3.
7.4. Penetration depth of various .types of geomagnetic
variations and the overall distribution of con-
ductivity within the Earth
8. Cats collection and analysis
8.1. Instruments
8.2. Organisation and objectives of field operatLons
8.3. Spectral arlaLysis of ge0magneti.c' induction data
8.4. Data analysis in the time domain, pilot studies
9. Data interpretation as the basLs of selected models
9.1. Layered half-space
9.2. Layered sphere
9 .'3. Non-uniform thin sheet abo-$e layelred substructure
9.4 Non-uniform Layer above layered subs.bructure
10. Geo;?hysical and geological rel.evance ofgeornagnetic in-
duk.kion studies
11. References for general reading

- 1. Introduction. basic equations
-- I.. 1.. General. ideas
The fo1.lowing rnodel j.s the basis ~fall e1ectromagne'ti.c lne-thods
which try to infer the electrical conductivity stl2ucture in the
Earth's subsurface from an anal.ysis of the natural or art.lfici.a~l
electromagnetic surface field: On or above the surface of -the Earth
is situated a tine-dependent electr'omagnetic source. By Fiiraday's
1.a.w the time-varying inagnetic field - 13 induces an elec.trica1 field
- -
this cu~,rent
F, which drives within the conductor a current 1. By kinp5re's law
has a magnetic fi.eld which again is an inducing ageri-t,
and so on. Hence, there is the closed chain
. induction
__I_ i ->
.-source
l5
//
E
-. -
*mp.res '011.ct:o.
law
-
The mathematical exprfssion of this closed chain is a second orde-r
~art?.al. differen-tia:l equation resulting after the e1iminati.c.n of
two field quantities from the two first order equations 1 + 3 and
At the surface the ~lectrornagnetic field of the source is dl.sturbcd
by the internal fie1.d~. This di.sirurbance depends on the conduc'ti-

A1~Lernativel.y
p = 30. m, where T in h. (lb 1
Eqs, (la,b) can also be used as a rule of thumb if p varies with
depth and an average resistivity is inserted.
The natural magnetic source fields of the ionospheric and magneto-
spheric currents offer periods approxiniately between 0.1 see and 20
permitting crustal and upper mantle surveys down to a depth of
approximately 800 - 1000 km. On the other side periods between 0.91
and 0.002 sec are most commonly used for Sounding of? -?he first 300 c
with artificial fields.
Electromagnetic methods are applicab1.e for two objectives:
1) ~nvestigation of the change of conductivity with depth, in par.ti-
cular detection and delineation of horizontal interfaces marking
a change of stratigraphy or temperature.
2) Investigation of lateral conductivity variations, in particular
search for local regions wiCh abnormal conductivities (e.g.
meta1lj.c ore deposi-ts, salt domes, sedimeni-dry' basins, zones of
eleva-ted temperature).
.The results of the first investigation are often used. to construct
a normal. conductivi-ty model from which local deviations are measure1
In an interpretation of natural fields the c:-.ange of the electro-
magnetic field quantities both with frequency and with position are
used. Broadly speaking .the dependence on the position provides the
lateral reso1.u-tion and the dependence on frequency gives the reso-
lu-tion with depth.
1.2. Dasic equations -
Let - r be .the position vector and let H, g, 2nd + I he the vectoi-s of
the magnetic field, electrical field and curl-2n.t density, ~?especti-
vely. Using SI u11.i.-ts and a vacuurrl permeability throughout:, the
0
pertinent equations are
cupl - H = - I i- L
11.2)
. -e
CUI?].' - E = --kt H
(1.3)
0-
. .
'- 1 = 0- C
(l.4)
div - !i = 0
(1.5)

All field quanti-ties are functions of position _r and time t. Al.so tl:c
(isotropic) conductivi.-t)~ u is a func.tion of position. -e I 3,s the
current density of the external cilrrent sources being different from
0 only at source points. The displacement current c 0- E is generally
neglected in induction studies. This is justified as follows: within
the conductor -- .the conduc-tion current UC - exceeds the displ.acement
current even in the case of ].owest periods (0.001 sec) and highest
5 2
resis1:ivities (10 Rm) still 10 times. In the vacuum where .the
conduction current is absent the inclusion of the, displacsment
current merely introduces a slight phase shift of the external field
at different points. for in this case the solution of (1.2) and (1.3
(including at the RFIS of (1.2) ) are electromagnetic waves, e. g.
the plane wave e i(k*r - - - u ~ ) where , w is the angular frequency and
.- k the wave vector along the direction of propagation. The phase
difference
4
between P and P2 is
1
,/ case
- .
PI Ax P2
MAX
l,j&t = -- coscl
C
.Z in the very pessimistic
of Ax = 1000 km and T = lse
- H ar~d - .I can he eliminated fPonl (1.. 2 - (1.4). It results
L Induction equation
----- - I
6!t using C1.2) and (1.4) the elec-trical charge density p(1') - is
given by
p = E div - E = - c E - grad I.oga, (1.7)
0 0 -
i.e. there is charge accumulated if an electrical field compunclnt :
parallel to the gradj.en.t. of the conductivi-ky. Physically 'this is
cl.ear,, since the normal. componen-t of .the current density is conti-
nuous whereas the charges accoun-i: for tl-~e corresponding discon-ti-
nuity of the normal electri.cal. field.

The e1ectrica:L effect - of the changes is vel?y important. They modify
by a-ttraction and repulsion .the 'curren-t flow; in particul.ar surface
changes deflec-t the current lines in such a way tllat the norinal
current coriiponent vanishes a-t the surface. Iil con-tras-t the magnetj-c -
effect i.s of the ordel- of that of the disp1acemel-i-t current and
can be neglected.
2. E1ectromagneti.c induction in I-di.mensiona1 structures
2.1. The q,eneral solution
The following vector analytical. identities are used in the sequel
(A
- is a vector, $ a scalar).
div curl - A = 0
(2.1)
curl grad + = 0 (2.2)
div(A+) - = @ div - A -1. & a grad 41 (2.3)
curlCA+) -- = + curlA - Ax - grad @ (2.4)
curl"+ - = grad divA - - AA - (2.5)
In this cha-pter it is assumed -1-hat the el.ec.trica1 conductivity o is
a fun~tion of depth z (positive downwa~ds) on1.y. In this case -the
electromagnetic field inside and outside the conductor can be re-
presented in terix of two scalars and $N(~) denoting the elec-
L -
tric and magr,e"ric. type of solution. These fields are defined as
'% .
follows IS the vector in z-direc-tion):
h A
On using t11.e. ident::.t)i- curlC$z) - = '-z - x gl~adC, Cfrom (2. 4 ) 1, i-t is se@!
.that -the ?$-type solu.!:ion has no nagne-tic z.-component and .the E-type
solution has no electrical .- z-component.

.
The electric field of the E-type solution is tangential to the planes
6 = const. IIence, 2-t is called a TE-mode. (tangei~tial -- e:!ectric). Conversely
the M-type solutj.on is name3 a TM-mode.!!~ required from (1.5)
-the 111agnetic field. of both modes is sol.enoidal (i.e. divl-I - = 0). In
addition the E-field of the TE-mode is also solenoidal.. In this
mode there is no accumulation of charge, all current flow is parallel
to the (x,y)-plane.
ldow we have to derive the differen.ti.al equatior~s for the scala~s
GE and $M. The TE-field (2.8a,b) satisfies already (1.. 3) - (1.5).
It remains to satisfy
~u1~l.H~ = 0 g*.
Using (2.5), (2.2), and (2.4) it results
from which Eollows
(fop if afiax = 0 and af/ay : 0 and fi.0 for x,y -> -, then f=O).
From -this equation fol.lows with the same argurnc1-i-ts
lil urliform domains, (2.9) and (2.10) agree. Using (2.9), (2.10),
(2.5), and (2.4) the TE- and TM-fields in the vacuun oiltside the
sources are
11 = 0, 11: = grad -
--M -PI a L
wL 0.
-E H = grad - az' -.E
E = y z x grad $E
O-
Fields on-tside the conductor
-- II
. ..
(2.12

The vari.ables x,y, and t which do not explici.tly occu~ i.n (2.9)
and (2.10) can be separa-ted from Q E and OM by exponential factors.
Let for OE or OM
- x - Y
A
where K = K x + K and K~ = -
,-.
Then (2.3) and (2.10) mad
K = - K . (2.14a,l
The general solution of (1.2) - (1.5) for a. one-dimensional conduc-
tivity structure is the superposition of a TE- and TM-field. The
total field then reads in components:
--- .--a
a20E
H = -Co$ ) + --x
ay M ax;)~
a ae QE
H = - -(uQ ) +
Y ax M ay a z
H = -(- a
z
ax2 ay2
WE
-
1 a "~l$~) -
Ex-o axaz a~
., 7
1
E = -
y o ayaz
3 - "8,
a I - (-- + a
E~ :)OM
ax2 ay
+ "0 ax
General expression of fie1.d components
- -
At a l>.orizontal di.scon-l-inuity the tangential. compone~~fs of - E and - H
and ille ndriaal. con~ponenl: of I1 arc coiitinuous. Let [ ] deno-te
Z
the j~unp of a p;lrticulaT quanti-ty, Theil frorr~ (2.15~) ]?OF )I z
(2.15a)
(2.15b)
(2.15~
(2.16a)
(2.16b)
(2.16~

[$J satisfies the two-dimensional Lapl-ace equa-tion, is bounded
and \rani-shes at infinity because of the finite extent of the
sources. Hence, from Liouvi.lleLs theorem of function theory
[Q,] = 0, i.e. @ C continuous. Differentiating (2.15a) with respect
to x and (2.15b) with respect to y and adding we infer along the
same lines thxt a$ /az j.s coiltinuous. Conversely differentiating
C
C2i15a) with respect to y and (2.15b) with respect to x and sub-
tracting one obtains oQM continuous. Fina1l.y differentiate (2.16a)
with rbspec-t to x and (2.1.6b) with respect to y and add. It re-
sults that ('licr) a(oQ )/az is con.tinuous 01- 34 /2z is
r! M
continous
if cf tends to a constant value at both sides.
Summarizing: - 1
+EJ a~ M
-- 1
a az
continuous
I Conti.nuity conditions I
This result shows that the TE- and TM-field satisfy disioinl ingab;
k.bLrJ
boundary conditions. Hence, they are comp1etel.y independent and
not coupled.
Frc~n (2.17~) follows
@ (z = 4-0) = 0 (2.18;
M
This boundary coilditioll has a serious drawback for the TM-field:
As a result within the conductor no TM-field can be excited by
external sources.
From (2.18) follows via (2.13) fM(o) = 0. Now
K.iultip:lyi.ng (2.9a) by ~ c f and ~ integrating ) ~ over z from 0 Co z,
integra-tion by parts yj-elds in virtue of fM(o) = 0
From (2.20) and (.2.19). fo1.3.oi..ls for yet?]. Er,equencies

- d
IufM12 > 0 .
dz - -
On the other hand f has to tend to a limit f0r.z -t m. Iience,
M
- f (z)
M
0,
-
O

1
2.2 Free modes of decay-
Whereas external fields can exci-te only the TE-mode within the
conductor, there exist free modes of current decay for both TE-
and TM-fields, Consider the typical potential
where @ and f stand for OE, $M and f,, fM, respectively. First it
L
wlll be shown that the decay constants B (ei.genvalues) must be
positive quantities. Inserting (2.23) in (2.9) and (2.10) we ob-
tain
fECz) =. { K ~ - f3~~0(z)l 'fE(z), (2.24)
where K' = K' -1- K'. The ej-genfunctions have -to satisfy the boundary
X Y
conditions
fE = (km) = 0, f (to) = 0, fr(m) = 0
M M
If there is a perfect conductor at finite depth d then
(2.26)
fE(d) = 0, fi(d) = 0 (2.2Ea)
Multiplying (2.24) and (2.25) by f" aid of" and integrating over z'
E M
from -m to +m and 0 to -, respectively, we obtain on integrating
by parts and using (2.26)
Hence, all decay constants f3 are real and positive. The currcnt
flow for TE-decays is in horizontal planes, whereas the TM-decay
currcnts flow predominantly in vertical planes.
Example :
u = O
0 + 0,-
- z - 0

a) TM-mode
From (2.251, (2.26); (2.26a) foll.ous
f (0) = 0 yields fM[z) = sinyz, whereas f' (d) = 0 requires cosyd=O.
M M
Hence,
The unnormalized eigenfunctions are
Theye exists also an electrical (po-ten.tj.al) field outside the con-
ductor. Find it!
b) TE-mode -
From (2.24), (2.251, (2.26~1) follows
"CZ) = K" fd), Z < 0
fE E
2.. .
f;[z)+Y.'f (z)=O, 0 < z < d, y -f3000 - K2 .
E
with f ( -m) = 0, fE, fc continuous across z = 0, f (dl = 0.
E E E
Then f (d) = 0 leads to the eigenvalue condition
E
casyd -I- (a/y)sinyd = 0.
For I: = 0 the ei.genval.u.es zgree with those of 'i'JI, For 1:c1

The TE-decay systems occur as transients to fit the initial con-
ditions, e.g, whcn sn external fie2.d is switched on.
2.3, Calculation of f,Cz) for a layered structure
Let .the half--space collsist of L uniform layers with conductivi-ties
ul., u2, -", u
the upper. edge of layer m being at h (h = 0). Let
L n~ 1
the index 0 reier to the air half-space.
0 =o
hl=O o Then given in z - < 0 the potential of the source
h~-l
we have to solve (2.9a). i.e.
L- l f"(z) E =.{~"tiw~i a(z))fE(z;.
With the abbreviations
and .the understandin g that 11
I* 1
0
- -, we have
In Eq.. (2.33) h > 0 is -the height of the lo!

with
"
the abbreviation
+ - 1 - ?
=
: - 1
- 2' gm 2 exp{i.a m ( 1 1 l m
, m = L-I , . . . , 1 .
Having computed u:, Eqs. (2.32) and (2.33) yield
-
&=f;/Bo.
Thus the field is specified.
E -.
to calculate B: and 6"
m
separately. Instead only the ratio
~f we are interested in f (2) for z < O only, it is not necessary
y m
- +
-- nm / B;
is requi-red. Because of (2.33) ar?d (2.32) f (z) is given by
E
-KZ 4% Z
f (z) = f-{e + y e ), O

2.4. ~articul'ar source fields
Notation: Xn the sequel only fields with a periodic time function
, --
exp(iwt) are considered. For any field quantity .A(s,t)
,+
we write A(r,w) - with the understanding
that the real or imaginary part of A(g,t) exp(iwl;) is
meant. Furthermore, since only the TE-mode is of interest
we shall drop the subscript "E".
In the last section an algorithm for the cal.culation of the $-poten-
tial in a layered half-space has been given. As input occurs
fe(~,K;w) = fi (Krw) exp(-KZ) , tIie representation of - the source - potential.
in t.he wave-nurnber space. Frolu (2.12a) , i .e. - 13=grad (a@/az)
-
follows that - a$/az is the magnetic scalar potential. Between
f- (K, w) and the source potential be (2,~) exist the following reci:-
0 -
- -- 1
2 SS ie(r,w)e dx dy
--. General field
------
- -
If I$ i.s symmetrical. wi.th respect to a vertical axis, i.e. @ is a
(2.35)
function of r = (x" +y2)1'2 only, e4s.. (2.35) and (2.36) simplify t:o
In the derivation of (2.35a) and (2.3Ga) the cartesian coordinates
were replaced by circular. polar coordinates
x = rcos0, y = rsin0, K = KCOS$, K~ = ~siilq
>:
and use was 1nad.e of the identity

2n -i~cr cos (8-$1 -iKrcU
J e ~B=J e c's8d0-2.fcos (~rcos0)ci0=2n~
0 0 0
11
(icr)
0
- 2.
) 1/2
2 1/2
Froin (2.35a), (2.3Ga) follovis that if 4 is function of r= (x2-I-Y
then fo is a function of K = (I:* f K ) . In certain cases (e.cj.
X Y
example h! below), it is on1.y possible to obtain the potential on
the clrlj-ndrical axis (r = o) in closed form. Then (2.35a) reads
Eq. (2.35b) can be considered as a Laplace-Transform for which the
inversion is
where E is. an appropriate real constant (thus that all singill.arj.-
e
ties of 41 (o,z) are at the left of z = E). Simpler wou.ld be the use
of a table of Laplace transforms.
The spectxal representation of the source is now calculated. for
simple sou-rces :
' &) Vertica.:L magnetic dipo1.e ,-
Locate the dipole at r = (o,o,-h), ki > o and let i.ts moment be
-0 -
*
M - = Mz. - When it is produced by means of a small current loop then
!4 = current x area of the loop. ($5 i.s positive if the di.rect:ion of
the current forms with - 2 a right-handed sys::.em, and negative else.)
From the scalar potential
follows
-
Since 4 shows cylindrical symmetry, v7e obtain front (2.3Sa) on using
the resu1.t
I
Ver kical. magnetic dipole

.
b) CLrcular current loop
Let a and I be the radius and the current of the loop. The potential
has cylindrical syr~.nlctry,but can be expressed by means of simp1.e
functi.ons only on the. axis r =: 0. For t.he n~oment we assume that
the current loop is in the plane z = o. For symnetry reasons there
is only a H z component on the axis.
1 Biot-Savart's law yields
whence
- 1a2 A8
' I a h 6 a
c4 . . .
ILZ = -
4n (z2+a 2) 3/2
I a2
z 2 (z2-ka 2)3i2
- .
l\roli xZ = -av/az = + a2+e/az2. Kence integrating twice on using
Vcm) = 0 = ie(m) we obtain
Now we have to solve (2.35b) , i. e.
Looking up in a table of Laplace transforms:
Sf the plane of thc loop is z = -hI 1 - > 0, then a si1npl.e change
of origin yiel-ds
1 Circular current loop I
I
I

In the 1irnl.t a -+ o, I +- m, M = va2 I fixed,this leads to the result,
(2.37) for the vertical ~nagnetic dipole (5 (x) =x/2 -I-
1
0 (x3) ) .
- - %---
C) IIorPzontal maqnetj-c d:ii>ole
A
IIere M = M s and
- --
-e - . . . ,
' ' -- a$ = - Ms
M: ' '
7.- ,$ €2 =
P
a z
41iR3 4n R (R->z-l-11)
With this result, suppressing al.1 further details the eval.uation of
(2.35) vields
d) Line current
--
Current I at y = 0, z = -h. It results the scalar potential
The corresponding source function is
Line current
- J
The delta function. 6 (K ) occurs since there is no dependence on x.
X
6) ' "P1:arie
--- wave"' ' (elementary\indula-Led - 'field)
Assun:e a wavenumher rc = w and let the scalar magnetic source po-
Y
tential be
leading to a magnetic field
- 1 ~ 1 " lIe=;-TJ - 1 ~ 1 1
H~ = H cos (vy) e I si-n (ply) e
Z 0
Y 0
z
sgn(w).

IIence,
- I
---
"e Q -- sin (wy) e- I w ( z
I{o
and -
2 8 ' ~ ~
f--.(~) =
~ ( K ~ )
0 - j.w 1 ~5~ 1
' { ~ C+w) K
Y
- 6 ( ~ -w)} ,
Y
Elementary undulated field
-- -
tghen the source can be considered as elementary harmonic.field, as
in this case, the Fourier integral representation (2.35) cori>plicates
things only.
In the limi-k w + 0 we obtain a uniform source field in y-d.i.rection
For the induction process a strictly uniform source field is useless
since only a vertical magnetic field component induces. Hcnce w
must be non-zero, no matter how small. If we confine our attention
to a finite part of the infinite horizontal. plane, the dimensions
of that part being smaller than l/lw(, then --.- formally we may put
.w = o and can profit from the particularly simple resulting equations.
Such a source field is called a - quasi-uniform - fie1.d. To the
unrealistic uniform field belongs the source potential
which can no longer be represented in terms of (2.35)

-- 2.5. Definition -- of the transfer function C (10, K)
The TE-potentia.1 is
With this potential we define as the basic transfer function for a
half-space with one--dimensional conductivity structure
C
Note that f depends on K and K separately via, the
X Y
function as an amplitude factor. In the ratio f/fl this factor
v,
drops out and it remains only the dependence on K = since
f is a solution of (2.9a), i.e.
How can C be determined from a. knowledge of the surface electro-
magnetic field, at least theoretically?
From (2.8a,h) results
- -, -I-- i~*r - -
- I I = C U ~ (QZ)=IJ{~K~'+K~~Z~~
~ - ~ - - dx dK
- (0 * Y
Let
A +a, - ..- i ,< * r -- -
- H(K,w) - = -- I /I - ~(x,y~~,w)e dxdy
( 2 ~ -m ) ~
- -
A
t-m , -i~c *r
1
- E(_K,w) = /I g(x,y,O,w)e dxdy (2.46b
( 2 ~ -m ) ~
denote the surface fields in the wavenumnber-frequency space. Then
from (2.45a,b), (2.46arb)
* A
- -
' + K = ~ f' K .i- K~ f z
Y

There are several ways to determine C:
a) From the ratio bctween two orthogonal components - of tj>h
- horizonha1 electric and magnetj.~ fisld -
*
Vectorial multiplication of (2.47a) with z and use of (2.43) yields
* A A
E = -iwu C z x H
.- 0 - -
A
or with any horizontal unit vector - e
A A
- -
e - E
C = .. A*- (2.48)
iwyo'(~xg) *' - K .
A * A A
In particular - e = - x and - e L- yields
b) Prom the ratio of vertical and horizontal - l~agne'iic fi~ld
components
(2.47a) yields immediately
c) Froln the ratiq of internal to external -- part of. a lnagnetj-c
horizontal component
i
In z - < 0 f reads,
(2.49a
- -KZ -,- f+
. , f = f e (2.51
0 0
A
A A
Let Xxe and H be the source part and internal part of Hx. Then
xi
from (2.47a) and (2.51)
Defining
A *
s (K, w) = Hxi (5,'" //IS'xe (5, (6)

we arrive at -
v
The same appl-ies to H Y'
. ~
d) Prom the ratio of vertical. gradients 'at' z = 4-0 --- to the
corresponding field compone~lts at z-o
I'/
Let H~(E,~)
be the vertical gradie!.lt of IIx at z = -1-0 in the wave-
number - frequency domain. Then
On applying (2.44) Eq. i2.47a) yields
A
The same applies to I*,,. For applications of (2.55) the surface
1
value of u must be known. C can also be obta.ined from other field
ratios involving vertical. gradients.
The 111et11ods a) and d) are also appl-icable for a quasi-unifozrr!
source fie,ld; b) ancl c) break down in this case.
The apparent resistivity of magnetotellur~cs is defined as
.- A
According to (2.49a,b) its re:.ation to C is
A

2.6. Properties of C(W~K)
a) Signs, limiting values
According to (2.43) the response function C is defined as
where f satisfies
f" (2) =' I K 1- ~ iwu0u (z) 15 (z)
C has the dimension of a lenqtl~.. Let
-i+
C = g - ih or C = [~[e
Then g - > 0, h - > 0 or 0 - c - < s/2
(2. GO)
Proof: Talce the comp1.e~ conjugate of (2.59) , mu1 tip1.y by f and in-
-
tegrate over z. Integration by parts yields
-f(o) frX(0) = I C [fl
m
0
[2-i-(~2--iw~00)
[fl21dz.
Division by 1 f' (0) 1 leads to the result.
The limiting values of C for w -* 0 and w + CQ are
1
f ; tanh (KH) for W+O
I I for LO+ m -
Ji.wuocr (0)
- In (2.62a) H is the depth of a possib1.e perfect conduc-tor. If
absent, then 1-1 i. 'm, C I jk.
Proof: For w=O the solution of (2,59) vanislling at z=X is
f - sinlz~c(H-z), - whence (2.G2a). - For high frequencj.es f tends t(
.---
'the solutio~l for a uniform ha1.f-space, i.e. f -. exp{fidll,~)~ yield:
(2.62b). This limit is attainecl, if the penetration depth for a
uniform halfspace with u - ~'(oJ,
V
is small co~npared with the scale leng-Lh I/K of the external fie12
and the scale l.engt11 lu(o)/ol (0) 1 of corducti-vity variatiol?.

) Computation of ---- C for a laxezed half-space --
When interested in the electric field within a layered structure,
we have to compute a set of coefficients Bf according to the rul.es
m
of Sec. 2.3. These coefficients can be used also to express C:
The latter form is also applicable for K'O, whereas a limiting pro-
cess is involved in the former one in this case. If we are only
interested in C then we may proceed as follows: C can be con-
sidered as a continuous function of depth. Then
-+ 1
where % = - exp{?a d 1,d -
2
Hence, (2.65) and (2.66) yield
- hm+l-l~ ~2 = ~ ~ + i ~
m m m m' 1n o 1%
+ tanh(c, d )
I "mCm+l m in -
m a 1 ~ . u ~ C tanh , ~ - (a ~ d ~ )
m
m m
c = --
Starting with C- = l/a baclcvrard recursion using (2.67) leads to
L L
c) A=rox:i.inate . - interpretation of a - one-dimensional - con+uctivity ---
structure
--
If we can assume K = 0, i.e. scale length of external field large
0 .
compared with penetration depth (as generally done in maqneto-
tellurics) then there exists a simple method tc chtain from
C a first approxi.mation of the underl-ying conductivity structure
(Schmucker-I

X X
Let C = g - 5-11. Then a first approximation u (z ) of u(z) is ob-
tained by setting
(2.G8)
This cannot be proved rigorous1.y but the' following arguments are
in favour of it:
1) zX = g can be considered as the depth of the "centre of gravity"
.of the in-phase induced current system.
2) It w ill be shorvn below that z" continuously increases when the
frequency decreases. According to a) its maximum value is 3.
3) For a uniform half-space and K=O we have h = T. I-rence,
X
ax is correct in this case. For perfect conductions u +m since
h+O.
This approximate method performs prrticularly well when there is a
monotone increase in con6uctivi.ty. The following .two figures (P- 24)
illustrate capabilities and limitations of the method.
dl Properties of C in the comul'ex frequency plane
For the following considerations it is useEull to coilsider the fre-
quency w as a comples quan-tity. Then in the coniplex frequency
plane outside the -- positive - imasinar~ .> axis C is an analytical func- --
tion of frequency. - For s proof multiply (2.59) by fX and inteqrate
over z. Then the integration by parts yields
Hence, for w not on the positive imaginary axis f' (0,w) cannot
vanish. There are neither isolated poles nor a dense spectrun of
poles (branch cut).
Division of (2.63) by 1 f ' ( 0,~) 1 yields
We can easily deduce Eroni (2.70) that

The approximate interpretation of C using (2.60). In
the left figure .(monotone increase of conductivity)
the zero order approximation interpretes .the data
already complete1.y. When there is a resistive layer
(left hand) the zero order interpretation needs re-
finement.. In the dott.ed line at the left hand, an
approxj-mate phase of C was used. This approximate
phase has been obtained by differentiation of the
double-logarithmic plot of p a (T) (cf. Eq. 2.771.

AS a consequence of (2.71a,b)Cis also free of zeros outside the
positive imaginary axis.
e) Dispersion rel.ntions .
Because of the analytical properties of C, its.rea:L and imaginary
part are not indepc!ndent functions of frequency. Let wo be a point
in the upper w-plane and L be a closed contour consisting of the
real. axis and a large semi-circle Ln the lower w half-plane.
Then ~wO=w+ic
1 C(u')dwr =
-J
n i L w'-wo
since the integrand.is analytical
in L. Due to (2.6213) the large
semicircle does not contribute w ' -plane
and the contour can he confined
to th.e real axis, tIere &ut wl=x and let DJ~=w+~.E (W real, > 0)
tend to the real axis. T.len
where
1 ' . E
lim - = SIX - w)
r+4-0 " ~~+(x-w)
has been used. f denotes the Cauchy principal value of the integral
1.e.t for real frequeilcies
Here g is an even and h an odd function of frequency, i.e.
g(-w) = ~ ( w ) P h(--w) = -h(~).
This is a consequence of (2.70). lience a separation of (2.72) in
its real a.nd imaginary part yiel-ds

I
~ispersi-on relations I
Dispersion relations of this kind occur in many branches of physics.
They are a direct consequence of the causalLty requirement.
Relations corresponding to (2,73a.,b) exist also for modulus and
phase of C. Due to (2.71a) C is also free of zeros in the lower
frequency halfplane. Hence the function
....
log{ C (w) 1
is analytical there and vanishes for lwl + due to (2.62b). Let
-i+ (w)
C(w) = [ ~ ( w ) l e
(2.74)
and assume w > 0. Then the relation corresponding to (2.73b) is
or introducing the apparent resistrivi.ty from (2.57)
where p = l(~(o). There exists a simple approximate version of
0
C2.75). Integration by parts yields
dlog pa(x)
= - w-xdx
J - ' l o g l ~ ~ l - ,
27r 0 dlog x
-1
or since x loglW - '1 is an integrable function pea1:ed at w=x,
w -1- X
in an approsimate evaluation one may draw out of fhe integral the
term dlogpajdlogx! . Hence
>:=w
where the result

has been used. With T = 2 dlogw=-dlogT. The final result is
W'
Since in general double-logarithmic plots of pa(T) are used, a first
approximation of the phase can immediately be obtained from the
slope of a sounding curve. The degree of accuracy can he asserted'
from two examples given at the
left. True phase in broken
. . lines, approximate phase in
nwe,.' -7
full lines.
The simple approximate method
- - - - -. - -
60' ,..-
4). --__ of inversion described in c)
.
w 80.
'. /'
(Schmucker-Kuckes relation)
---.-.wc m.3.
II. ..------,' I,. - ".","rn , combined with the approximation
\
-I
' sis&ivity.
Q. I 1 (2.77) provides an extremely
simple tool to derive a zero
'/ ipL'
,m
Ea
70
!D
100 ,000 IOOo3ar< .-- 1 0,
a '? v-
order app?:oximati.on of the true
conductivity from appa.rent re-
f) Inequalities for the freguencyependence of - C
Cauchys formula is

where L is a positively oriented closed contour enclosing only a
domain where C is analytical and the point w. We choose the par-
ticular contour shown at the left. When the radius of the circle
. .
tends to infinity the circle does
not contribute since C (w) = 0(l/&)
for Iwl+-. Hence the contour can be
confined to both sides of the posi-
tive-imaginary axis. On the right
hand side put w'=ih+~, E >' 0. Then
(2.79) yields
m m
= - 1 Im C(iA+E) dh = a q(?,)dA
liin - r x+iw 71
X+i.w '
E++O 0 0
1
where q(h) = - lim - Im C(iA-*E) > 0
1T
E"+O
in virtue of (2.71h) . Summarizing:
The non-negativity of q(X) has the consequence that C must he a
smooth function of frequency. Again let w be a positive frequency
and 1.et
- C=g-'ih.
DeZining
Df: = w -- df - df - d f
dw dlogw dl.0gT
-
- - --
then the fo1lowLn.g constraints apply

(2.82a1b) simply results when (2.80) is split into real and. imagL- ,
nary; it has already been given above (Eq. (2.61 a,b) ) . (2.83) is
proved as follows
The other constraints are proved in a similar way. There are other
constraints involving second and higher derivatives. In terms of
apparent resistivity and phase $ Rqs. (2,83b1a) read:
The slope of a double-logarithmically plotted sounding curve is
- Dpa/pa. As a consequence of (2.85a,b) we have alvrays
The monotone decrease of the real part of C with frequency is a
consequence of
The following figure shows data (full lines) which are inconsistent
a
on the basis of one-dimensional model., since the constraints (2.83a, b) :
are partly-violated. Then the least corrections to the data are
determined that the inequal-ities are satisfied. Since this is only
a necessa-ry coridition, interpretability is not yet granted.

1 2 3 4 CPD 1 2 3 4 CPD
-
g) Dependence of interpretation - on wave-ru~S7er
The fundamental equation is
By the transformations - 1
z = - tanh(~z)
K
it is transformed into
in such a way that
remains unchanged. Hence any C can first be interpreted by a uni-
- -
form external field (K=o) and the result @(z) is then transformed
to the true conductivity by
- -1 1
o(z) = sech4 (ez) .a (- tan11 (KZ) )
K (2.87)
-

For this interpretation the condition
1
C(O) ( K
has to be satisfied on C (w) .
An application of (2.87) is given in the following figure*
When K increases attenuation is interpreted
by geometrical dampping at the expense of
electromagnetic damping due to a perfect
conductor.
.3... M.o.del. .cal.c.ul.at.ions. for. .tw,o-.dimension.al. structures
.-- ~- .- ~ ~~
-- -
3. I.' General. 'equations
We are considering now induction prbblems, where both the conducti-
vity structure and the inducing field are independent of one hori-
zontal coordinate, say x. Compared with Ch,. 2, the class of in-
ducing fields has become more restricted, hut the class of ad-
mitted conductivity structures has been enlarged.
For x-independence, Maxwell's equations
are split into two disjoint sets

The TE-mode has no vertical electric field, ?he TM-mode has no ver-
tical magnetic field. In the treatment of the;+? modes the use of
- -
electromagnetic potentials is not necessary since E x and H x can serve
as pertinent potentials. For conciseness let
- -
H: = Hx I E: = Ex (3.4b,a!
Then E and H satisfy the equations ,
In uniform domains both equations. agree. Eq. (3.5b) resembles the
equation of heat conduction in a non-unifor15 heat conductor.
The continuity of the tangential electric and magnetic field components
at conductivity discontinuities leads to the conditions
-- ..
I -1
E, -- continuous :
an
TE
(3.6a)
H, - 1 - aH
continuous : TM
a -
(3.613)
an
- a is the derivative in direction to the normal of the discontinuity.
an
The E- and H-polarization - shores very different patterns. From (3.lb),
(3.2b) follows that Ex is constant in the air half-space (u=o). Hence
the Tbl-mode admits only a quasi.-uniform inducing magnetic field. In
contrast in the TE-mode any two-dimensional inducing magnetic field
is allowed.
Hence, t17.e source terms to he added on the PJKS of (3. I a, b) are
je(y1z) and Ho 6 (z+h),
-

assuming that the TM magnetic source field j.s due to a uniforrn
sheet current at height z = -h, h > 0. T1lj.s assun~ptj-on, however,
is immaterial for the following.
In the sequel all field quantities are split into a normal and
anoinalous part, denoted by the subscripts "nu and "a" , respectivel-y.
The normal part refers to a one-dimensional conductivity structure.
Let
u(y,zl = un(z) + U,(Y,Z) (3.7)
H(y,z) = Hn(z) + Ha(y,z)
E and H are defined as solutions of the equations
n n
(3.3b)
e
AEn = k2 E + j (3.10a)
n 11
d l d
-(- - 1-1 ) = H z > Or H (0) =
dz ,?2 dz n n ' - n
vanishing for z + m.
In virtue of (3.5a,b), (3.9arb), and (3.103,b) Ea and H satisfy
a
1 d 1. 1 dIln
div (- gradH ) = H a - - - , z > 0
k
a
k * k2 dz -
n
If the anomalous domain is of fin2te extent, E has to vanish uni-
a
forinly at infinity. Under the same condition Ha has to vanish uni-
formly in the lower half-space. At z=o H is zero.
a
If the anomalous domain is of infi.nite extent in horizontal direc-
tion, we can demand only that Ear H +O for z + m.
a
For a numerical solution of (3.11 a) the following three clioices of
a basic dornain are possible (boundaries hatched).
In approach A, (3.11 a) is solved by Finite differences subject to
the boundary condition Ea=O or better subject to an inpedance
boundary collclition (below). In approach B (3.11a) is solved by
finite differences only in the anomal.ous slab. At the ho~izontal
boundaries boundary conditions involicity the normal structure
above and below the slab are applied. I approach C (3.11~~) is re-
duced to an integral equation over the anon~alous domain. These
approaches will now be discussed in details.

i
I Earth
Anomalous
domain
3.2. Air half-space and conductor. as basic domain
(Finite difference method)
For the TE- and TM-mode we have to solve the differential equa-
tions (3.Sa,b), i.e.
with the boundary condltion that the differences E =E-E and
a n
Ha=H-HI> vanish' at infinity. E has to be computed for any given
n
two-dimensional external source field al.ong the lines of Sec.2.3.
The H -field belongs to a uniform external magnetic field.
n
In the finite difference method, the differential operators in
(3.12a,b) are reduced to finite differences. For simplicity a
d
square grid.with grid ~11th h is assumed. Consider the following
configuration of a nodal point 0 and its four neighbours:

Then the evaluation of (3.12a) in uniform subdomains yields
or
E 4-E CE CE - 4Eo = ll2k2E0
1 2 3 4
Within uniform subdomains the same formula qplies to H. Differences
occur if the nodal point 0 is at an interface. Consider as exarrple
a vertical discontinuity:
li
~egio'. I. Region 2
In the absence of region 2 one can write a central difference equa-
tion for E at nodal point O as
where the bracketed superscript indicates the region. In the ab--
sence of region 1 the central difference equation at O is
At the vertical discontinuity we have because of the continuity of
Since also the normal gradient of E is continuous,
The field values E2 (2) and E4 (I) are fictitious and have to be elimi-
nated with the aid of (3.16) and (3.17) fro12 (3.14) and (3.15). The
result is
- where E2 = E2
(1
and E4 Z4
(2)

Hence, the conductivity is to be averaged in the TE-case. Now con-
sider the TM-case:
1
The continui.ty of H and 2 dH(dn yields
I , ,
I~(~)=H(~)=II
H;l 1 =KA2) =
H1 -1 0 0 0' -H3
from where we obtain on eliminating 1-1
.2
(1
and H4
Similar formulae hold for a horizontal discontinuity.
The normal values 05 En and Hn are used as starting as initial
values. At the boundaries we have two choices
a) The boundary values are kept fixed, i.e. E = E
n'
b) Or free boundary values are uked as - ieance - boundary condition,
where n is the direction of the -- outward normal.
--
(3.24) is obtained under the assumption that the anomalous fields
diffuses in form of plane waves outwards, a valid approximation only
if the local penetration depth is small compared with the scale
length of coaductivity changes. It performs poorly at edges and in
isolators. However, better then Ea=O.
Eq. (3.24) yields ascondition at the upper edge of the air layer:
aEa(az = 0 Ci.e, constant horizontal magnetic field). '

The iteration is carried out either alonq rows or colums. Generally
the GauO-Seidel iteration procedure is used with a successive over-
relaxation factor to speed up convergence..
3.3. P.nomalous slab as basic domain
In practice it is not necessary to solve the diffusion equation by
finite differences in t.he total conductor and the air half-space.
Instead it is sufficient to treat the equation only in that slab
which contains the anomalous domain.
Let the anomalous slab be confined to the depth range zl 5 z 5 z2.
Within this domain we have to solve the inho3nogeneous equation
(considering for the moment only the TE-case)
subject to two homogeneous boundary conditiolls at z = z and z2,
1
~Thich involve a for z < z and z > z2 respectively and account for
n 1
the vanishing anomalous field for z -t 2 m. Yihen (3.25) is solved
by finite differences, the discretization invol.ves also the field
values one grid point width above and below the anomalous slab.
The idea is to express these values in terms of a line integral
over Ea at z = z and z2 respectively.
I
Let V and V2 be the half-planes z - < zl and z -- > z2, respectively.
~ e G t r [r). 2, r, be Green's functions which satisfy
" "m
(-0 -
subject to the boundary condition
In V and V2, E is a solution of
1 a
Now Green's formula for two-dimensions states that

The RHS i.s a closed line integral bordering the area over which the
LIIS integral is performed. I11 the RHS differentiation in direction
to the outr,?ard normal is involved.
From (3.28) and (3.26) fol.lows
Identi.fying U with GI V with Ea, Eq. (3.29) yields in virtue of
(3.27)
E -
a Q
around Vm
I
r - a an G ( ~ (&IL)ds.
)
Now G'") and its normal derivative vanish at infinity. Hence, only
the part of the line integral along the axis z = z contrihut-.es.
a a m
Por m=? : - a - - m=2 : - a =--
2n az' an az'
Hence,
Because of (3.26) , Eq. (3.30) depends only on the difference y-yo.
Defining
q . (3.30) reads shorter
For a layered structure in Vm, the kernels R (m)' are easily deter-
mined:

Let us first consider the case m = 2. Assume that there are L uni-
------ (3 ------
I
z =o form layers below z = z2 with conz=zl
ductivities (3 I' (3 2~ * s t a1,r and
upper edges at z = hlr h2, ..., hL
' (hl = z2). In applications, the
z=z,=h, vertical grid width z - z2 will be
'. 4
' so small that zo is in the first
z=z
o uniform layer. Then a solution of
?=h - -2
,r (3.26) having the correct singu-
U
2 z=h3 larity is
.,
IIowever, the boundary condition (3.27) is not yet satisfied. and the
normal conductivity structure has not yet been taken into account.
To achieve this let in
m . o: (z-z ) -a (z-z )
z

4.
Having determined B;, the coefficients 6o and 6L are determined
from the fact that the difference between upward (downward)
travelling waves of (3.33a) ar:d (3.33b) at z = zo must be due to
the primary excitation given by (3.32). Hence,
whence
From (3.30~~)
+ --
Since (3.35) involves only the ratio B1(B1, it can be expressed
in terms of the transfer function C'at z = jz2 (cf. (2.64)):
For a uniform half space 13.35) is simply
(2
x . . .~y-y~.,.z~)
1 - -a1 (2-2 ) (zo-z2)k
= ; 1 e O cos~(y-y~)iix= 1 ~ 1
o
. r: - -0
For kl + 0 (isolator) this yields
[ r-r. [. . . . . . .
(kl l_r_-hll

The case m = 1 can be treated in a quite analogue way. If the
d
anomalous slab extens till the surface, i.e. z = 0, the pertinent
1
kernel is easily derived from (3.36a):
(Because of (3.30a) there has been a change of sign. 1
If there are more normal layers (in addition to the air half-space),
the problem is treated as for m = 2, with the air half-space as
+
last (L-th) layer, we have to calculate 6L and hence B; separately.
We can't use C.
The kernels K (m) are nicely peaked functions. The halfwidth is
approximately 2[zm-zo[, i.e. ttrice the vertical grid width. For an
insulator the tails are comparatively long (-l/y2), for a conductor,
an exponential decrease is inferred from (3.36). In general. two
points to the left and the right of the central point will give a
,
satisfactory approximation:
where
h /2
CO
y: 3h /2
'm)=2 i K(~) (u,zo)du, p1 (m)= JY ~(~)(u,z~)du, pp)= K(~)(U,U~
Po
0 3h /2
J2 Y
-
PWi - Pi# C pi = 1, hy = horizontal griqwidth.
(3.38) expresses in any application of the finite difference formu1
the anoma1.ous part of the electric field outside the anomalous slab
in terms of anomalous .field values at the boundary. At the vertical
boundaries the impedance boundary condition (3.24) is applied.
So far only the TE-case has been considered. The TI$-mode can be
handled similarly, taking only the different boundary condition
into account. The approprate formulas can be worked out as an
exercise.
I

3.4. Anomalous region as basic domain
Inteqral equation method
In the integral equation approach Maxwell's equatiol-Gare transformed
into an integral equation for the electric field over the anomalous
domain. With the usual splitting
o = \ + V a' E = E n + E ~ , k2 = k2 n + k2 a'
we have
or subtracting
AE = k2n + iwv j ,
0
(3.39)
Let Gn be Green's function for the normal conductivity structure, i.e.
AG (r Ir) = ki(g)~~(&l~) - &(L - 5)'
n-o-
G (r [r) can be conceived as the electric, field of a unit line
n -o -
.current placed at % and observed at - r.
Multipl-y (3.41) by Gn, (3.42) by E subtract and intesjrate with
a
respect to - r over the whole space: It results
Greent s theorem (3.29) yields
since Ea and Gn vanish at infinity. Hence, introducing into (3.43)
E instead of Ea,we obtain the integral equation
In (3.44), the inhomogerleous tern En can be computed for a given
normal structure and given external field in a well--known way.
It remains to determine the kernel. G 11 (r -a [r). - It satisfies the reci-
prdcdty relation

i.e. source and receiver are interchangeable. For a proof write
(3.42) for G (r 1 rt ) and a corresponding equation for G (r 1 Y:* ) .
n -a - 1 - -
Multiply these equations crosswise by G (r (r' ) and Gn (rolr' ) , subn-tract
and integrate over the full space. Then the result is ob-
tained on using Green's theorem.
For a sol.ution of (3.44) we have to put a line current at each point
r of the anomalous domain and have to compute the resulting elec-
--O
tric field at each point of this domain replacing its anomaious
structure by the normal conductivity. Assume a rectangular anomalous
don~ain wLth NY cells in. y-direction and iqZ cells in z-direction.
Because of horizontal isotropy the field depends in horizontal d.irec-
tion only on the distance between source axid receiver. Hence, we
need field values only f0r.N.Y horizontal distances. Due to the
layering there is no isotropy in vertical direction. Here we have
to put the line current into the center of each cell. Because of
reciprocity the sorresponding field v.alues have to be calculated
in and below the depth of the source. Hence, for the kernel Gn a
1
total of NY - TNZa(NZ+l) field values has to be calculated.
A second set of kernels is required which transform on using (3.44)
the electrical field in the anomalous domain into the electromagne-
tic surface field for all three components E xr H yr H z . The number of
required kernel data depends on the range where the field is to be
ev~luated.
It remains to calculate G (r ir) for a layered structure. Assume L
n -a -
layers with conductivities oo = 0, al, ...r csL and upper edges
h, ~ 2 ' "'I hL' hL+, - O0.
-
Let the source and observation point be placed in the m-th and p-
layer, respectively, and let in the m-th layer
+ +
6 A-f-(z),
omm
z - < z
0
+ + 6 F3-f (Z),
. Lmm Z > Z - 0

-
6 and 6= can be so adjusted that A: = BL = 1. Since there are no
0
sources in z - < 0 and in z - > zo if z is in the L-th layer,
-
0
= B' = 0. With these starting values the continuity of G and
A. L n
aG /az across interfaces yields the forward and backward recurrence
n
relations
- -
+ + + - 4 -
B-=(l+u /am)qm
m mi- I
Bm,~l+(li-am+l/amlgm Bmtl, m = L-I, ..., 1-1
with
+ +
g-=I /2, g-= (I /2) expf'am (hmtl
0 m
-11 )If m = I, ...f L-I.
In the case )J = L 110 recurrence is required for the B-terms. The
coefficients 60 and 6L are determined from the fact that in (3.46)
the difference in the upward (downward) travelling waves for z -- > zo
and z - < z must be due to the primary excitation given by
0
+ +
where f- = f-(zo). The nominator (including a ) is a Wronskian of
?J u Fr
the differential equation
which is a constant thus ensuring reciprocity. (Proof?)
In applications the anomalous domain is split into rectangul-ar cells,
the electric field is assumed to be constant wlthin each cell. Then
C3.44) reduces to a system of linear equations, which is easily
solved because of its dominant diagonal due to the logarithmic
singularity of the kernels. Either direct elimination or GauB-Seidel
iteration can be applied, the latter being in general quickly
convergent. When E is assumed to be constant withi-n each cell., the
integration over the kernel is easily effected by adding in (3.46)
the factor
4sin (Xhy/2) sinh(a ,h /2) / (XU
* , . 1- z 11 '
_I. r. . . " , _

The integral equation method for the H-polarization case looks
sligh-tly nlore complicated. This is related to the fact that even
for three-dimensional structures Maxiqell's equations l.oo?c simp1.e
when formulated for - E, whereas in the 2-formulation additional gra-
dien'ts of the conductivity arise:
The pertinent equa.tion for H-polarization is (3.5) , i. e.
with the usual splitting
1
div (- gradB) = iwpoH
u
and the additional definitions
the equations for the normal and anomalous part are
and
d l d
-(--- H ) = iwll H , H (o) = H~
dz an dz n o n n ,
1 --
div(pll gradHal = iwpoHa - div(pa gradH)
This equation corresponds to (3.41) for the E-polarization case.
Green's function appropriate to (3.51) is defined as
Physically G can be interpreted as the magnetic field due to an
11
infinite straight line of oscillating magnetic dipoles along the
x-axis.
(3.48)
Multiply (3.51) by Gn and (3.52) by IIa, subtract and integrate over
the whole space. Then

From the generalized Green's theorem
= I ${u - av au
a 11 - v -
an Ids
follows that the LHS of (3.53) vanishes. Applying (3.54) also to
the KHS of (3.53) we obtain, introducing H =
a
N - Hn:
r 7
This is an integral equation for H . It call be cast in a slightly
a
different form, which is particularly suitable for applications
since the high degree of singularity due to a two-fold differentiation
of Gn at - r = r is removed. Because of
-a
div(p a grad~~)=-{iwp~G~-6 (r-r ) 3+pllgradGnegrad (pa/",)
"n - -0
Eq. (3.55) reads alternatively ,
The kernel. of the integral equation consists of two parts. The first
part takes account of the changing concentration of current lines
in anomalous domains. It is a volume effect. The second effect
results from the bending of current lines where conductivity changer
It is essentially a surface effect.
In the case of discontinuous changes in conductivity, which is the
most common assumption, (3.56) needs a slight modification. Assul~c
that the anomalous domain consists of rectangular cells, where the
conductivity is allowed to differ from cell to cell. H is assumed
to be constant in each cell.

Then the discontinuity between cell (i, j) and 1 j contribu-tes
to the integral
It has been used that H and p E are continuous across interfaces.
an
The contributioll frcm the i + i j + I interface is
The integral equation is decomposed into a set of linear equations
for the Ha-values at each cell. The electric fLeld is then obtained
by differentiating (3.56) with respect to the coordinates of E ~ .
-
It remains to calculate Gn. At z = 0 G has to sat.i.sfy the boundary
n
condition of Ha, i.e. Ha = 0, : Gn = 0. The boundary condi.tions at
interfaces are G and(l/cr)a~~/az continuous. Assume aqain L uniform
n
layers with conductivities ol, 02, . . . , and upper edges at
OL
hl = 0, h2, ...I hLI h -
~4.1
- -'.
Let the field in the m-th layer be
where

and y can again be so adjusted that C; = DL = 1. = 0 for
Yo L - Gn +
z = 0 requires then C = - I, and Gn + 0 for z + m demands DL = 0
1
Starting with these initial values, the boundary conditions yield
the forward and backward recurrence relations
where
c 1
@ = a /a and gn, = - exp{?a -h 1)
~n in m 2
m h m
p is the source layer. There is no recurrence required. for the C-
term if u = 1 and for the D-terms if y = L.
The free factors follow again frorn the source.represcntation
which must account for the difference for upward and downward
travel-ling waves at z = z . Hence,
0
With the present determination of the field Gn satisfies again
the reciprocity relation G .(r Ir) = G (r 1 r ) . (Proof?)
1 -0 n - -0
-

4. Model calculations. for tl1.r,ee--d~nei~sioi~a1. structures
- -
4.1. Introduction -
In the three-dimensional case the TE- and TM-mode become mixed and
cannot longer be treated separately. Now the differential equation
for a vector field instead of a scalar field is to be solved. In
numerical solutions questions of storage and computer time become
important. Assume as example that in approach A a basic domain with
20 cells in each direction is chosen. In this case, only the storage
of the electric field vector would require 48000 locations. For an
iterative improvement of one field component at least 0.0005 sec
are needed for each cell. This yields 12 sec for a complete itera-
tion, and 20 min for 100 iterations. This appears to be the ].east
time required for this model. Hence methods for a reduction of com-
puter time and.storage are particularly appreciated in this case.
The equation to be solved is
cur1213 - (r) - -t k2 (r) E (r) = - i ~ l(g) i ~ ~ ~
---
where k2.(r) = i~p~c(g).
-
1 (r) is the source current density.
-e -
After the splitting
where E is that solution of
-11
curlZE (r) + k2(r)F, (r) = -impo&
-11 - n--n-
(4.3)
which vanishes at infinity, we obtain for the anomalous field the
two alternative equations
c ~ r l . + ~ k2E ~ = - k2E
-a n-a a
Eq. (4.4a) is the starting point for the volume integral or integral
equation approach, Eq. (4.4b) is the point or-' starting for the sur-
face integral approach.

4.2. In3ral - --- equation method
Let -1 G.(r --a [r), -- i = 1,2,3 be a solution of
cur12c. .-I (r --0 lr)+k2 - n (r)G. - --I (r --0 Ir)
,.
A
- = zi6(g-%), (4.5)
vanishing at infinity. Here, the x. are unit vectors along the
A h -l h A A A
cartesian coordinate axes: x =x, x =
-1 - -2 1, z3 = - 2. 14u'iul&31y (4.5) by
E (r) and (4.4a) by G.
-a - (r I?:) and integrate the difference with re-
-1 -0 -
spect to - r over the whole space. Green's vector theorem
where d-r i.s a volume element, dA a surface element and - 1 the out-
ward normal vector, yields
since and G vanish at infinity. After combining all three compo--
-i
nents and introducing - E instead of E , the vector integral equation
-a
1 -gtg)d~ (4.7)
--
is obtained. Here is Green's tensor being defined as
wj(rJ
% --O
3 3 A h
A
t(r Ir)= C x.G. (r [r)= Z G.. (r .Ir)x: x.
- -1-1 -0 - ir j=1
13 -0 -- -1 -3
i= I
(using dyadic notation). The tensor elements G.. admit a simple
= 7
physical interpretation: Gij (r _o [r) - is the j-th electric field coin-
ponent of an oscillating electric di-pole of unit moment pointlng
in x.-direction, placed in the normal conductivity structure at &;
1 --
the point of observation is - r. Note that the first subscript and
argument refer to the source, the second subscript and argument to
the receiver. Because of the fundamental reci.p~:ocity in electro-
mag~~etl.sm, source and observer parameters axe interchangeable, i.e.
For a proof replace in (4.5) - r by - r', write an analogous equation
for G.(rlrl), multiply cross-wise by G and Gi, integrate the
-3 - - - j -
difference wit11 respect to - r' over the whole space, and obtain (4.9)
on using (4 .G) . Due to (4.9) , the equation (4.7) is alternatj.vely
written

Eq. (4.7) lnvolves integration over the coordinates of the receiver,
(4.10) requires integration over the coordinates of the source.
The kernel G and the inhomogeneous term Ell of the integral equation
(4.7) or (4.10) depend only on the normal. conductivity structure.
To determine the kernel 9 replace first the conductivity within
the anomalous domain by its normal val-ues. Then place at each point
of.the domain successively two n~utually perpendicular horizontal
and one verti-cal dipole and calculate the resulting vector fields
at each point of this domain. At a first glance the work involved
appears to be prohibitive, but it is sharply reduced by the reci-
prociky (4.9) and the isotropy of the normal conductor in horizon-
tal di.rection. Because of (4.9) , from the elements of Green's
tensor
G G G
xx xy xz
the three elements Gxz , G xy' Gyz need not to be calculated when
G G G is computed. From the remaining six elements G has
zy' yxr zy Y Y
the same structure as G xx' only rotated through 90'. The same re-
lation holds between G and GZx. Hence, there are only the four
z Y
independent elements G G G (say). The particular
XX' yx' GZxr zz
symmetry of the Gxxr Gyx, Gzz-conponent in connection with the re-
ciprocity (4.9) then shows that these components need to be eva1ua.-
ted only for points of observation above source points. Consider
for example a vertical dipole at xo=yo=O, zo. Then (4.9) yields
Because of the isotropy of the conductor in horizontal di.recti.on
(4.11 ) is alternatively written
GZZ (O,O,zOI~,y,~) = GZZ (O,O,Z~-X~-Y~Z~).
Now, GZZ has circular symmetry around the z-axis. Hence,

The element GZx is needed for all z and zo. Assume a rectangular
anomal.ous domain, which is decomposed in-to cells with quadratic
horizontal cross-section. It is sufficient to pose the dipoles in
one corner of the ano&aly in the (x,y)-plane. Then assuming NX, NY,
NZ ce1.l.s in x,y,z-direction the total number of required kernels is
There is still a reduction up to a factor 2 possible, if instead
of the corresponding conponents only some auxiliary functions de-
pending only on the horizontal distance from the sourced are cad-
culated .
The corresponding kernels are inost easily computed using a sepalza-
tion into TE- and Tbl-fields as done ,in Section 1.
Let the normal structure consist of L uniform layers with conduc-
tivities 5 , m = 1, ..., L with upper edges at z = h m = I, ..., L
m in
(h = 0).
1
The Green vector GL satisfies in the m-th layer the equation
tie try a solution in the form
where @ corresponds to the TM-potential and Qi to the TE-potential.
i
According to Sec. 2.1, at horizontal interfaces @ and @ satisfy the
continuity conditions
I
a @
041 =I
I
a$
'4, continuous
There is no coupling between $J and @ across boundari-es. Within
uniform layers, but outside sources 4 and $ satisfy identical
differential equations
-*. 7-'
The behaviour of 4 and $ near source p0i.nt.s can be obtained from
. .
the particular forms, which these functions show in a uniform whole-
space :
-kR
.(r .[r) = (k28ij-a2 jaxiay le
. Gij -a -
'j 4nr\k2
,
~=lr-rl - -a
(4.15)

For a dipole - in z-direction this is equal to
a e-kR A e -kR
G = (k" - - grad-) = - curl2 (z --
4a~k'
since
-kR
A (e-kR/@~r~) ) = lc2e / (4aR) -
-z a~ 4Ta2 - 1 I
Comparison of (4.16) and (4.12) shows that for a whole space
The absence of a TE-potential is clear from physical reasons, since
the magnetic field of a vertical. dipole must be confined to hori-
zontal planes (i.e. no vertical. magnetic field, which can only
be produced by a TE-field).
Using Somrnerfeld's integral, C4.17) is written
The f5.el.d of a horizontal electric dipole (in x-direction, say) has
both an electric and magnetic component in z-direction. Hence, a
TM- and TE-potential are needed. 9 +
Since ,- J%i.
a2 a2 1
a2 e
--kl?. m
-a/ z-z ]
G =-(- + -7) 4Jx=--y - -=-- I I A2e O J~ (Xr)dX cos$
Xz ax2 ay 41~k axaz R 4ak20
and
we have
and from
then follows
xsiqn(z-z )
0
J, (Xr)dXcos$ * sign (z-zo) (4.19)
f

With this knowledge of the behaviour of $,, @,, QX in the uniform
whole-space, these functions for a layered medium can be easily
obtained.
Let in the m-th layer hm < z < hmt1
where
where
+ -1-
~ ~ A i f i z - < z0
+ -I-
~ ~ B I ~ f i > z o
z -
+
and f-. = (z-h )Il a2 = h2-tk2. Then starting with
n~
_
in n~ m m
- t - + - + -
+ 1, A =O, B =Or BL=l1 C =I, C=O, D =01 D =I.
ko 0 L 0 0 JJ 1,
The boundary conditions(4.13) lead to the recurrence relations

+
m )}(XI g-
. .
where gi = e~pI+-n~(h~+~-h o = I f2 and y is the layer of the
sou-rce. In the case y = L there is no recurrence'required for the
B and D terms. The free factors are determined in the usual way
simply by ccn~paring upwards and dowriwards travelling \,raves at
z -- z taking the source terms (4.18) - (4.20) into accoun-t.
0 +
Dropping the subscript y on , k2 8; I Bi, C' , D', fi for
Fr U' V P U V
conciseness, we obtain from t4,21) and (4.18)
whence
+
where f- meacs ft (zo) . From (4.22) and (4.20) follows
v
whence
Because of (4.19) +x is discontinuous across z = z o . Hence

After having determined Qzr VJ,; @,, Green's tensor G is obtained
from (4.12). Al.so required is the magnetic field in z - < 0. Only
the TE-part of a horizontal dipole' contributes.
Let P (r lr), i = lr2 be the magnetic field at r due to a dipole
-i -0 - -
in x -direction at r . Then in z < 0
i --o -
0
- 1?$) = curl2 (z*.) " 0 = grad a*i .
0 -3. - 1 az
Now the kernels for the integral equation (4.7) are determined a.nd
it remains to discuss how this equation is sol-ved in practice. The
simplest way certainly is to decompose the anomalous d0lr.ai.n into
a set of N rectangular cells and to assume that the el-ectric field
is constant within each cell. Then there results from (4.7) a set
of 31V linear equations for 3N unknown$. The system has the form
where - x is the vector of unknown field components, & the matrix of
coefficients and 5 the given vector of the normal electric field
values. A direct inversion of this matrix is only feasible for
N - < 50 (say), because of the large amount of storage required. Hence
iterative methods must be used in general.
The simpl.est way would be to start an iterative procedure with
x0 = - -. q. This sequence of approximations, however, converyes only
if the eigenvalue of =z5 A with the largest modulus has a modulus less
than 1, a condition which is certainly not satisfied if cr is large.
a .
Better convergence properties shows the GauB-Seidel iterative scheme,
using during iteration already the updated approximation and a
successive overrelaxation factor:
Let the components of - & be aik. Then (4.28) reads

Then a' new approximation to xi is obtained by
1 1
i- I
3N -
new- old,- -- R
new + c aikxk old - x old
xi -x i 1-a. { C
li k=l
aikxk
k=i
i +gi}
The successive overrelaxation factor R - > 1 has to be chosen suitably.
In many cases R = I (i.e. no overrelaxation) is already a good
choice. If the GauB-Seidel method does not converge then one can
apply a s-pectruin displacement technique: As already mentioned,
simple iteration without updating is only convergent, if the largest
modulus of the eigenvalues is smaller than 1. Eq. (4.28) is equi-
valent to .
- - x = (A - al)x - - + ax
- + 5
(4.29)
The iterative procedure (4.30) will be convexgent, if a can be
chosen in such way that the eigenvalues of - B are of 1aodu1.u~ iess
than 1. If A is an eigenvalue of - A then A - 1 is an eigenvalue
of B. - Consider for example the following situation that - A
has negative real eigenvalues from 0 to Amax, lXrnax 1 > 1. Then a
choice of a = - IXmax[/2 is appropriate. The condition
is then satisfied for all eigenvalues A. The requirements which
m has satisfied can be put in that form that a circle around a
has to include all eigenvalues of A - but has to exclude the point
1. The following figure illustrates the case stated above.
This case applies approximately
to the three-dimensional modelli
problem, where at least the
largest eigenvalues are for a
higher conducting insertion to
a good approximati.on negative
real. For a poor conducting in-
sertion the largest eigenvalues
are essentially positive, hut
smaller than 1.

4.3. The surface inteyral approach to the modelling problem
In the surface integral approach, within the anolr[alous slab the
equation (4.4b) , i. e.
cur-2~ + k 2 ' ~ = - k 2 ~
-a .-a a-n
is solved by finite differences or an equivalent method. The re-
quired field values one grid point width above the upper horizontal
boundary at z = z and below the lower boundary at z = zL are
1
expressed as surface integrals in terms of the tangential component
of ga at z = z and z2, respectively.
.
1
~
Le-t V and V2 be the half-spaces z < z and z > z respectively,
1 1
and let S m = 1.2 be the planes z = z . Let C!mi (r r , r E Vm.
mr in -1 -0 - -0
- r E vm U S be a solution of
m
(i=1,2,3; m=1,2) satisfying for - r E S the boundary condition
m
In Vl and V2, Ea is a solution of .
cur12i3 + k2 E = 0
-a n -a
Multiply (4.34) by G!~), (4.32) by E , integrate the difference with
-1 -a
respect to - r over Vm, and obtain on using (4.61, (4.33) and ga + 0
-
for r -t -:
' r E Vm, or in tensor notation
-0'
,
where
E (r) = (-I)~J curl
-a --o
'm
3 ,-.
x.
(m)
curlG. .
-1 -1
A
(r Irl;{z x E (r)}d~
-0- - -a

Eqs. (4.35aIb) admit a representation of the field values outside
the anomalous layer in terms of the boundary values of the con-
tinuous tangential component of E .
-a
A physical interpretation of Green's vector G subject to (4.33)
-i
is as follows: Reflect the normal conductivity structure for
z < z and z > z2 at the planes z = z and z = z and place a unit
1 1 2
dipole in x -direction at r and an image dipole at
i ,. --o " 'm
r' = r + 2 (zm - z0)zI the moment being, the same for the vertical
-0 -0
dipole and the opposite for the two horizonta~l dipoles. Then the
tangential. component of G!") vanishes at z = z and G!~) is a
-1 111' -1
solution of (4.32) for - r E Vm.
Hence, if Vm is a uniform half-space, G (m) is constructed from the
-i
whole-space formula (4.15) . Eq. (4.3'5) then reads :
m
Eaz = (-1) / J? ( ~ {x-x j )E (g)+ (y-yo)Eo
(5) ld~,
ax CIY
'm
where R = [r - - r 1, k2 = iww cr
--O 0
Eqs. (4.36a-c) contain as important subcase the condition at the
air-earth interface (m = 1, zl = 0, k = 0).
0
Because of the limited range of the kernelstin applications of the
surface integral on1y.a small portion of Smm:ust be cocsj.dered.
For Eax and E the contribution of the region nearest to r is
aY -0
most important. Assuming E and E to be cionstant within's sma1.l
ax aY
disc of radius p centered perpendicularly over r the weight from
-0
(4.36a.b) is
where h = lz - zo[ . There is a contributiorr to E onl-y if Ea,, and
m az
E have a gradient along x and y direction kespectively.
aY

F
At the vertical boundaries the condition E = 0
--a
is relaxed to the
impedance boundary condition
A A
kga.t = n x curl zar - n b E =Of
-a
whiere gat is the tangential component and - & the outward normal.
5. Conversion - formulae for a two-dbensional TE-field
5.1. Separation formu1.ae
In this subsection 5.1. it is assumed that the conductivity does
not change in x-direction and that the inducing magnetic field is
in the (y,z)-plane, i.e. the conditions for the E-polarization
case of Sec. 3.1. Then from a knowledge of the H and Hz component
Y
along a profile from -a to +a at z = o it is possible to separate
the magnetic field into its parts of internal and external origin
without having an additional knowledge of the underlying conducti-
vity structure.
The pertinent differential equation is C3.5a), i.e.
which reducing in z - < 0 to
which has the general solution
where A: describes the external and A: the internal part of the fie1
The magnetic field components at z = 0 are (cE. (3.2.3) and (3.3a)):
Let . .
- - - - . .
H =W +H HZ = H i.I.Izi,
Y ye yi' z e

where the subscripts "en and "i" denote the parts of external and
internal origin at z = o.
Further, let the Fourier transform of any one of the above six
- ~ K Y dy.
quantities be
A
1 +m _ H(K) = - J H(y,o)e
2~ -m
Then (5.4a,b) yields
A A A A
H /xZe =.-i
Ye Y 1
sgn (K) , H . /HZi = +isynCK) (5.7alb)
A product between Fourier transforms in the x-domain transforms to
a convolution integral in the y-domain:
Hence we obtain from (5.7aIb1
- . - -.
H =+KxHze, H . = -I< x
Ye Yl Hzi
-. - - .
= - K x H H =+ICxH
'ze ye' z i. .yi
- €*K
Convergence was forced by a factor e . The resulting convolution
integral exists only in the sense of a Cauchy principzl val-ue, i..e.
(5.8a) for example reads explicitly
+m
. -
H (y) = 1 K (~-~)IX~~(TI)~~
Ye -m

The four equations
A A A A
H = i sgn (K) HZe , H = + isign(~) H
Ye Y i z i
A A A A
can easily be solved for H yer Hze' $ir 'zi*
When transformed into the y-domain it results 4
I Two-dimensional separation formulae I
For practisal purpuses these separation formulae are not very con-
venient, since the kernel decays rather slowly requiring a long
profile to determine the internal and external part at a given '
surface point.
5.2. Conversion - formulae for the field components - of a two- -
dimensional TE-fielci at the surface of a one-dimensional structure
For the separation of the magnetic field components no knowledge
of the two-dimensional conductivity structure is required. However,
the conductivity enters if it is attempted to deduce for instance
r
the total ver-tical component of the magnetic field at the surface
from *he corresponding tangential componenk. If the conductivity
structure is one-dimensional, the conversicn between two component.:
can be effected using a convolution integral, where the kernel is
derived from the one-dimensional structure via the trans5er funs-
'?here S([K[,W) is the ratio between internal and external part of
A h
H i.e. H . /H in the frequency wavenuder domain.
Y' Yl yer
From (5.9a-c) the,yarious conversion formuias for the durface com!31
nents can be derived. The following table gives the definition of

the pertinent kernels and their form for two particularly simple
structures. Note that in the case of a vanishing conductor (i.e.
in the examples h -> m, or k + 0) the kerne1.s K- and M have to
agree with the kernels of (5.8a) and (5.8~1, respectively. Tile
function L occurring in the P-kernel of the uniform half-space
2'
model is the modified Struve function of the second order (cf.
Abramowitz and Stegun, p. 498).

6.
Approaches to the inverse problem of electromaqnetic induction
by linearization
6.1. The Backus-Gilbert method
6.1.1. Introduction
The method of Backus and Gilbert is in the first line a method to
estimate the information contents of a given data set; only in the
second line it is a method to solve a linear inverse problem. The
procedure takes into account that observational errors and incom-
plete data reduce the reliability of a solution of an inverse
problem. It is strictly applicable only to linear inverse problems.
Assume that we are going to investigate an "earth model." - m (r),
where m is a scalar quantity which for simplicity depends only on
one coordinate. For the following examples it w ill be chosen as
the distance from the centre of the earth (to be as close aa
possible to the original approach of Rackus and Gilbert). Then in
a linear inverse problem there exist N linear, functionals - ("rules"),
whLch ascribe to m(r) via data kernels G. (r) numbers 7. (m) in the
3. 1
way
a
gi(m) = j m r)Girdr i = I, ..., N. (6.1)
0
The measured values of gi(m) are the N data yi, i = 1, ..., N. The
"gross earth functionals" g. (m) are linear in m, since it is .
1
assumed that the data kernels G.(r) are independent of m. The in-
1
verse problem consists in choosing m(r) in such a way that the
calculated functionals gi agree with the data y The Backus-Gilber.;
i'
method shows how m(r) is constraint by the given data set. Bcfore
going into details let us give an example. Assume that we are
interested Ln the density distribu{:ion of spherically symmetrical
earth, i.e. m(r) = p (r) , and that our data consist in the ?\ass 1.1
and moment of inertia 0. Then
-

6.1.2. The llnear inverse problem
The data may have two defects:
a) insufficient
b) inaccurate
Certainly in the above problem the two data M and 0 are insufficient
to determine the continuous function p(r). Generally, the properties
a) and b) are inherent to all real data sets when a continuous func-
tion is sought. The lack of data smoothes out details and only
some average quantities are available, the observational error in-
troduces statistical uncertainties in the model. (If we were looking
for a descrete model with fewer parameters than data, then incon-
sistency can arise as a third defect.) Because of the lack of data,
instead of m at r we can obta.in only an averaged quantity ,
0
which is still. subject to statistical incertainties due to errors
in the data. Let
a
< m(ro) > = I A(ro[r)m(r)dr, (6.2)
0
where a is subject to
The latter condition ensures that agrees with m, if m is a con-
stant. A(r ( r) is the window, througlil which the real but unknown
0
function m(r) can be seen. It is the - averaqinq - or resolution Punctio -
The more A at ro resembles a &-function the better is the resolution
at ro. Resolvable are only the projections of m(r1 into the
space of the data kernels Gi. The part of m, which is orthogonal
to the data kernels cannot be resolved. Hence, it is reasonable
to write A as a linear combination of the data kernels
where the coefficients a. (ro) have to be determined in such a way
I
that A is as peaked as possible at ro. The aost obvious choice
would be to minimize
subject to (6.3). For'c~rn~utational ease Backus and Gilbert prefer

to minimize the quantity
a
s = 12 / A2(ro[r) (r-ro)'dr.
0
(6.6)
Since (r-r )2 is small near rot A can be large there. The factor 12
0
is chosen for the fact that if A is a box-car function of width L
1, 0, else
then s = L, i.e. if A is a peaked.function then s in the definition
of (6.6) gives approximately the width of the peak. s(r0) is called
the spread at ro.
We have also taken into account that our measurements y have ob-
i
servational errors Ay i.e.
it
Insertion of (6.4) into (6.2) using (6.1) yields
From (6.7) and (6.8)
and the mean variance is
N
is the covariance matrix of the data errors, which in general is
assumed to be diagonal.
Of course, we would appreciate if the error of m(ro), i.e. E' would
be very small. But also the spread (6.6)
a . N
s = 12 / A' (rolr) (r-ro) ' = C ai (ro) %(rO)s. (rO)
lk
o i., k=l
with a ~ ~
SikCro) = 12 / Gi(r)G -(r) (r-r )'dr
0 k 0
(6.11)

should be small. Hence it required to minimize simultaneously the
quadratic forms
N N
s= C ai% sik 7 c2= C a a E .
i k lk
i,k=l
i , k=l
subject to the condition (6.3), i-e.
There does not ex'ist a set of a. which minimizes s and E' separately.
L
As a compromise only a combination
can be minimized. In (6.15) , c is an arbitrary positive scaling
factor which accounts for the different dimensions of s and E' and
W is a parameter
which weighs the particular importance of s and E ~ For . W == 1 the
spread is minimized without regarding the error of the spatially
averaged quantity . Conversely .for W = 0 the spread s is
large and the error E' is a minimum. Hence, in general there is a
trade-off between resoluti.on - and accurac~, which for a particular
is shown in the following figure.
ro
EZ max
E'
2 -- - -L ---min
I
W=O
I --
s
S
min max
Near s = s . the trade-off curve i.s rather steep. Hence, a small
m m
sacrifice of resolving power will largely reduce the error of the
.average . This uncertainty relation between resolution and
accuracy is the central point 05 the Backus-Gilbert procedure.
, S

It remains to show a way to minimize Q, subject to (6.15). The way,
however, is well-known. One simply intrsduces as (N+1 )-st unknown
a Lagrangian parameter X and minimizes the quantity
a
Q' = Q + h (Cai~li - 1 ) , ui = / Gi (r)dr (6.16)
0
The differentiation of (6.16) with respect to the (N4-1) unknowns
yields the (N+1) linear equations
where Qik = W Sik + (l-W) c Eik.
This system of equations is easily solved. The meaning of X is
revealed by multiplying (6.17a) by ak, adding and using (6.17b)
and (6.15) . It results
In the case W = 1, we have X = -2s. With a knowledge of a (roll
i
is obtained from (6.8) with gi = y. and E' from (6.9).
3-
When is inserted in (6.1) instead of m(r) , it w ill in
general not exactly reproduce the data.
The minimization (6.15)
N
subject to the constraint C aiui = 1 admits for two data (N:=2)
i=1
a simple geometrical interpretation: For constant s and E' these
positive definite quantities are represented by ellipses, the con-
straint is a line in the (al,a2)-plane. For uncorrelated errors,
the principal axes of E~ are the al and a2 axis.
hT1a W varies from 0 to 1 the combinatiomof (al ,a2) on the fat
line are obtained. s and. E' are determined from the ellipses
throughthese points,Sj-nce all s-(~~)-elli~ses are similar, s and^
2

are proportional to the long axes of these ellipses.
6.1.3. The nonlinear inverse problem
The Backus-Gilbert procedure applies only to linear inverse
problems, where according to
a
the gross earth functionals gi have the property that
This means for instance that the data are bui1.t up in an aclditive
way from different parts of the model, i.e., that there is no
coupling between these parts. This certainly does not hold for
electromagnetic inverse problem, where each part of the conductor
is coupled with all other parts.
In nonlinear problems the data kernel Gi(r) will depend on m. Here
it is in general possible to replace (6.18) by
.
a
gi.(mr)-gi~m)=f(m'(r)-m(r))~i(x,m)dr + ~(m'-m)~, (6.18a)
0
where m and m' are two earth models. The data kernel Gi(r,m) is
called the Fr6chet derivative or functional derivative at
model m.

Agai-n, from a finite erroneous data set we can extract only averaged
estimates with statistical uncertainties, i.e.
As in the linear case A(r Ir) is built up from a linear combination
0
of the data kernels
N
A(rolr) = C a. (rorm)Gi(ro,m).
i= 1 I
Introducing (6.1 9) into (6.20) we obtain
which for nonlinear kernels is different from (6.8) , since i.11 this
nonlinear case gi (m) qi (m) .
In the linear case, two models m and m' which both sa.tisfy the
data lead to the same average model in1 (r ) >. In the nonlinear case,
0
the average models are different; the difference, however, is of
the second order in (m' -m) . (Exercise! )
The Backus-Gilbert procedure in the nonlinear case requires a model
which already nearly fits the data. Then it can give an appraisal
of the inforn~ation contents of a given data set.
6.2. Generalized - matrix inversion
The generalized matrix inversion is an alternative procedure to
the Backus-Gilbert method. It is strictly applicable only to linear
problems, where the model under consideration consists of a set of
discrete unknown parameters. Nonlinear problems are generally
linearized to get in the range of this method. Assume that we :mnt
T
to determine the M component parameter vector p with - p = (plr...,pb5:
and that we have' N functidnalr, (rules) g. i = 1, . . . , N which
1'
assign to any model E a number, which when measured has the average
value y . and variance var (y . ) :
1 1

Suppose that an approximation & to p is known. Then neglecting
terms of order 0 (E - %) ', we have
. Eq. (6.22) constitutes a system of N equations for the M parameter
changes pk - pko. The generalized matrix inversion provides a so-
lution to this system, irrespective of N = M, N < M, or N > M. If
the rank of the system matrix agi/apk is equal to Min (M, N), the
generalized matrix inversion provides in the case M = N (regular
system): the ordinary solrition,M < N (overconstrained system): the
least squares solution,M > N (underdetermined system): the
smallest correction vector p - %.
The generalized inverse exists also in the case when the rank of
agi/apk is smaller than Min(M, N).
After solving the system, the correction is applied to and this
vector in the next step serves as a new approximation to p, thus
starting an iterative scheme.
It is convenient to give all data the same variance ci2 thus
0'
defining as new data and matrix elements
thus weighing in a least squares solution the residuals according
to their accuracy which makes sense. Let
be the parameter correction vector. Then (6.22) reads
corresponds to the data kernels Gi(r) in the Backus-Gilbert
theory). In the generalized matrix inversion first G - is decomposed
into data eigenvectors u and parameter eigenvectors v :
-1 -j
G = U A
T
~
(6.26)
- =--

Here lJ - is a N x P matrix containing the P eigenvectors belonging
to non-zero eigenvalues of the problem
and 1 - is a M x P matrix with the P eigenvectors of the problem
associated with non-zero eigenvalues. P is the rank of 5. - Finally
= A is a P x P diagonal matrix containing just the P nonzero eigenvalues
X . Then the generalized 'inverse of G .- is
j
- H always exists. In the cases mentioned above, - g specializes as
follows
For the proof one has to take into account that because of the
orthogonality and normalization of the eigenvectors one al.ways has
and in addition for
T v v = I (=P-component unit matrix)
=== = (6.30a)
P
For P < Min(M,N) H - cannot be expressed i3 terms of G. -
The generalized inverse provides a solution vector
- -
< x > = H y .
Its relation to the true solution - x is given by
.=_Ax, -- -
where - A is the (M x 21) resolution matrix

- Only for P M, _A - is the M component unit matrix, admitting an
exact determination of - x. (A corresponds approximately to the resolution
function A (ro 1 r) in the Backus--Gilbert theory. But there
are differences: - is symmetrical, A(s[r) not, the norm of A - is
small if the resolution is poor, the norm of A(ro[r) is always 1. )
In generalized matrix inversion exists the same trade-off as in the
Backus-Gilbert method. Cbnsider the covariance matrix of the change:
of.< - x > due to random changes of x:
In particular the variance of is
P . Vk
var (x ) = a2 ZI (3)
Z ,
j=l j
k 0 k = 1,2, .:., M
showing that the variance of xk is largely due to the small eigenvalues
X By discarding small eigenvalues and the corresponding
j'
eigenvectors, the accuracy of x can be increased at the expense
k
of the resolution, since A - will deviate more from a (M x M) unit
matrix if instead of the required M eigenvectors a smaller number
is used. -
The model will in general not.reproduce the data. The repro-
duced data are
= G - - =GWy - - =Ex -
with the information density matrix
T - B=GH=LJQ. - - - -
(6.36)
. . Only in the case N = PI 2 is a unit matrix. In particular, B -
describes the linear dependence of the data in the overconstrained
case. High diagonal values will show that this date contains
specific information on the model which is not contained in other
data. On the other hand a large off-diagonal value shows that this
information is also contained in another data.
Valuable insight in the particular inverse problem can be obtained
by considering the parameter eigenvectors corresponding to high and

small eigenvalues. The parameter vector of a high eigenvalue shows
the parameter combination which can be resolved well, the parameter
eigenvector of a small eigenvalue gives the combination of para-
meters for which only a poor resolution can be obtained.
The generalized inverse is used both to invert a given data set and
to estimate the information contents of this data set when the
final solution is reached.
hring the inversion procedure one has two too1.s to stabilize the
notable unstable process of linearization:
a) application of only a fraction of the con~puted parameter correc--
tion, leading to a trade-off between convergence rate and sta-
bility;
b) decrease of number of eigenvalues taken into account.
In the final estimation of the data contents one might prescribe
for exh parameter a maximum variance. Then one has to determine
from (6.35) the number of eigenvalues leading to a value nearest
to the prescribed one. Finally the row of the resolution matrix
for this particular parameter is calculated.
6.3. 'mri.vation of the kernels for the linearized inverse problem
of electromagnetic induction
6.3.1. The one-dimensional case
Both for the Backus-Gilbert procedure and for the generalized linear
inversion a knowledge of the change of the data due to a small
change in the conductivity structure is required.
In the one-dimensional case the pertinent differential equation and
datum are
, flr(z,w) = I K ~ + iwp0o(z)}f (zrw)
Consider two conductivity profiles o and o with correspondicg
1 2
fields fl and f2. Multiplying the equation for fl with f2 and the
equation for f2 with flr subtracting the resnltinq equations and
-
integrating the difference over z from zero to infinity, we obtain
.I (f;'f2 - f:fl)dz = iwp o 1 (ol-cr2)flf2 dZ
0 0

Now
m
Division by f !, (0) :f; (0) yields
If the difference 60 = a - o is small, f2 in the integral may
2 I
be replaced by fl, since the difference f2 - fl is of the order
of 60 = a2 - o leading to a second order term in 6C = C2
1'
- C1.
Hence to a first order in 6s
m
6C(w) = - iwp l6a(z){ f (z,w) 12 dz
O 0 f' (0,~)
Therefore in the Backus-Gilbert procedure the FrGchet derivative
of C is -iwpb{f(~)/f'(o)}~. In the generalized inversion the deri-
vatives of C with respect to layer conductivities and layer thick-
nesses is required, if a structure with uniform layers is assumed.
Let there be L layers with conductivity am and thickness d in
m
the m-th layer, h < z 2 hm+l (hL+l = -).
m - Then (6.38) yields
since in the last case all layers below the m-th layer are dis-
placed too.'
' '6':3 :2 : Pa,rt?al - derivatives in two- and 'three-dimensions
For two-dimensions only the E-polarization case is considered. The
pertinent equation is (cf . (3.5a) )
AE = iwp a E .
0
Considef again two conductivity structures a and a2.
1

Then
- 77 -
A(E2 - E ) = iwu U (E -E ) + iwc (G -U )E2
1 0 1 2 1 0 2 1
Let GI (r' - [r) - be Greent s function for u i.e.
1'
- and (6.40) by GI (r' I:), integrate
Multiply (6.41) by E2(r1) - El (rr)
the difference with respect to - r.' over the whole (y,z)-plane. Then
Green's theorem (3.29) yields
With the same arguments as in the one-dimensional case we obtain
for small conductivity changes, neglecting second order terms,
i I
The kernel G(rt - [r) - must be determined from the solution of the
integral equation
-
i-
G(rl[r) - - = G ( r r - f ua(rn)Gn(r'[r")~(r"[r)d~"
n- - o - - - - -
Eq. (6.44) results from (6.42) by replacing
This substitution becomes possible, since the 6-function in the
Green's function equations drops out when the difference A(EZ-El)
is formed.
The changes of the magnetic field components are obtained from (6.43
and (6.44) by differentiation with respect to - r. - Methods for the
computation of Gn(rtlr) - - are given in Sec. 3.4.
In the three-dimensional case one obtains in analogy to (6.43) and
(6.44)
where ?is competed by solving the tensor integral equation
(6.44)
Again the magnetic field kernels are obtained by taking the curl
of (6.45) and (6.46) with respect to - r. Methods for calculatincj /.
are given in Sec. 4.2. At the moment, three-dimensional inversion 6
appears to be prohibitive!
-
0

6.4. Quasi-linearization - of the one-dimensional inverse problem
of electromaqnetic induction (Sc1~mucl:er's PSI-alqorithm)
In the one-dimensional problem it is possible to a certain extent
to linearize the inverse problem by introducing new suitable parameters
and data instead of the old ones.
Assume a horizontally layered half-space with L layers having conductivities
u , , . u and thicknesses d
L 1' d2' ..-, dL-l
CdM = m). Let the upper edge of layer m be at z = h and assume
m
that layer conductivLties and thicknesses satisfy the condition
Let k = q. Then the pertinent equation for a quasi-uniform
m
external field is
The transfer function is, as usual,
C(w) = - f,(o)/£;(o) .
Introduce instead of f the new function
m
In the TE-case f and ff are continuous across boundaries, i.e.
As a consequence of (6.48) and
(6.49) + is discontinuous across
boundaries:
'?
z=h
m
The variation of. £ in the m-th layer is
m
. .
-k (z-h )
m
fm(z) = A; e +
+km ( z -hm)
~ m + e
1

whence
2k d -2k d
where a = CA:(A;)~ mm, y=e . m m
I +a
log t-) I -ay
I +a
log (-) I -a
Because of the condition (6.47), y is independent of m.
The quantity la[ is the ratio between the reflected and incident
wave at z = h . Due to the conservation of energy the amplitude
mi 1
of the reflected wave is never larger than the amplitude of the
incident wave. Hence,
i Equality applies to a perfect conductor or insulator. For (a[

Introducing the new transfer function
where u is an arbitrary reference conductivity, we obtain by con-
0
tinued application of (6.53)
.(To "1
= log - + y log - + y a2(h2)
1 O2 .
0 G u
0 1
2
= log - + y log & + y2 log ;;- + y2 Q3(h3)
"1 2 3
L y(w) = (y-I) C ym-I
m
L- 1 0 o
L-
log - - y
I L
. . . . . . . . . . . 0 . . . . . . .
m=l. 0 0
log -,-
In the derivation of (6.55) it has been used that (hL) = 0. Eq.
L
(6.55) is the desired result: it expresses the new data y(w1
linearly in terms of the new model parameter log(o/cro).
, When (6.55) is used for the solution of an inverse problem, one
chooses first an arbitrary oo, and prescribes the constants
c/a m o
d
m
and L. If the occuning conductivity contrasts are not too
sharp, the inversion of (6.55) will already give good estimates
for the layer conductivities. For these estimates the correct
,
response values are calculated, and with the difference between
the data and these values the inversion of (6.55) is repeated,
giving corrections Alog(u /o ) to the previous outcome. At the
m o
end of the iteration procedure, the true thicknesses of the layers
are calculated using (6.47). The errors of u are finally transm
formed into the errors of d . m

.
7. Basic concepts of geomagnetic and magnetotelluric depth sounding
.
. 7.1. . General characteristics of the method
Two types of geophysical surface data can be distinguished to in-
vestigate the distribution of some physical property m(g) of matter
beneath the Earth's surface. The first type is connected with
static or quasi-static phenomena (gravity and magnetic fields), the
second type with time-dependent phenomena (seismic wave propagation)
or with controlled experiments under vartable experimental con-
ditions (DC-geoelectric soundings). Geomagnetic and magnetotellur7ic
soundings utilize the skin-effect of transient electromagnetic
fields. Their penetration into the Earth represents a time-depen-
dent diffusion process, thus the observation of these fields at
the surface produces data of the second type.
The interpretation of static data y = y(R) ...- is non-unique alld an
arbitrary choice can be made among an unlimited number of possible
distributions m(r), - explaining y(R) - equally well. The interpretation
of transient data is with certain constraints unique in the
sense that oniy - one distribution m(r) - can explain the surface data
y = y(R,t), basically because opeadditional variable t (time,
variable parameter of controlled experiment) is involve?,
If the observation of the transient process or the perforrfiance of
the experiment is made at .a single site, the data y '= .~(t) permit
a vertical sounding of the property m = 'm(z), assumed to be a sole
function of depth z beneath that site, If the observations or ex-

'periments are done with profiles or arrays, the data y = y(R,t)
permit a structural sounding of the property m = F(z) + Am(r)
with particular emphasis on lateral variations Am(2).
- .
Here m(z) represents either a global or regional mean distribution.
It may also be the result of vertical soundings at "normal sLtesn
wheqzthe surface data show no indications for lateral variations
of m. Since the dependence of y(R,t) - on m(r) - will be non-linear,
anomalies Ay(R,t)
and
-
-. = y(R,t) - yn(t) will be dependent on Am -
m(z) , i.e. the inierpretation of. second-type -data must proceed on
the basis of a known mean or nornlal distribution F(z) consistent
with data yn("c at a normal site. It should be noted that in the
case of static data of the first type usually no ii~terdependence
between Ay and will exist, i.e, the interpretation of their
anomalies is gependent of global or regional mean distributions
of the relevant property n.
Suppose that for data of the second type the lateral variations
bm(r) - are small in relation to G(z). Then the results of vertical
soundings at many different single sites may be combined to -
approximate a structural distribution m = m + Am. For geomagnetic
induction data the relevant property, namely the electrical. re-
sistivity has usually substantial lateral variations and a one-
dimensional intezpretation as in the case of vertical soundings
will not be adequate. Instead a truly multi-dimensional inter-
pretation of the data is required which is to be based on "normal
data" at selected sites (cf. 8.2). Such normal sites are her@ rather
the exception than the rule.
7.2 The data and physicai properties of internal matter which
are involved
The Earth's magnetic field is subject to small fluctuations g(rtt).
.
They arise from primary sources in the high atmosphere and from
seconda-y sources within the Earth. At the Earth's surface their '
r. amplitude is
-

orders of magnitude smaller' than the clu'asi-stntic main field.
Their depth of ~enetration into the Earth is limited by the "skin-
4Z.L
effect", i.e. by opposing action of electromagnetically induced
eddy currents in conducting matter below the surface . These
currents produce the already mentioned secondary component of H.
They are driven by the eLectric field - E(r,t3 generated by the
changing magnetic flux according to Farad y 's induction law (cf.
7,3), The the-fluctuations in - Fi are referred to "geomagnetic' variations,
those in - E as geo-electric .or "telluric" variations.
The connection between - ti and - E is established by three phy~icai
properties of internal matter: the conductivity a (or its recipro-
cal the resistivity p), the permeability p asid the dielectric con-
stant E. The permeability of ordinary rocks 2s very close to unity
and the approxirnat5,on p = 1 is adequate. The rnagnetic erfect of
displace~~ent durrents which is proportional to E can be neglected
within the earth (cf, Sec. 7.31, leaving the conductivity a as
,the sole prope-i+~':c~nnecting 2 and K. Bcth fzelds tend to zero At
sufficiently great depth, the skin-depth . -
being a characteristic scale length for the depth of penetration
of an oscillatory field exp(iwt) lnto . a . medhm of, resistivity p.
. Geomagnetic induction data are usually presented in terms of trans-
fer functions which connect as -functions of frequency and locations
certain components of - E and - Ii. Examples are the imagneto-telluric
transfer functions:between the tansentidl components 15 C-el I+
and the magnetic transfer functions between the vertical and hori-
zontal conponents of H only. These transfer functions form input
data for - magneto-telluric - and - geomagnetic - depth soundings,
..abbreviated NTS and GDS. . ,
C.
i4otations and units. Rectangular coordinates (x,y,z) are used with
2 positive down. If x is towards local rlagnetic north and y towards
local rnagnetic east, the f ollor~ing
notations of the magnetic and
. .
electric field components are common:
.

The vertical magngtic component HZ is usually simply denoted as Z.
kn
All equations are SI-units, but for H - the tradional geomagnetic
unit ly = GauA may be quoted. A convenient unit for the
telluric field is (mV/km). The magneto-telluric transfer function
between (Ex,E ) and (Hx,H ) will'have in these units the dimension
Y Y
'{mV/km]/y with the conversion
to SI-units. Measuring the period T = 2s/w of time-harmonic field
variations in seconds,
1
p = - km,
2-
if T is measured in hours,
p = 30.2 fi. km,
the unit of p being in either case (am); po = 4s 10'~ volt sec/
. .
Ampin.
7.3 Electromagnetic wave propagation and diffusion through
uniform domains
Consider a volume V for which the physical properties ~,E,u are
constants. Then - E and - H are non-divergent within V and connected
by Maxwell's field equations (. = .a/at)
.... .
rot E = -ppo H, ,
-

Elimination of - E or .,. H yields a second
.
order partial.differentia1
equation .
. V- 2 ~ = pllo(oF - + E E ~
where - F denotes either the electric or the magnetic field vector
(div - F = 0). This equation can be interpreted as a wave equation
or as a diffusion equation, depending on the fastness of the field
variations in relation to the decay time
of free electric charges within V.
. .
Let w be the angular frequency of harmonic field variations., i,e.
- - - -
F = iwF and F = -w~F, Then the basic differential equation is
conveniently written as a wave equation
for WT >> 1 and as a diffusion equation
0
for w~~

the surface. Only near certai-n sulfidic ow-veins the "effective"
decay time T may be in the order of fractions of seconds due to
0
induced polarisation and the propaga-tion terrrt may not be necessarily
small against unity.
The air just above the ground nas'a resistivity in the order of
loi4. Qm, yielding a time constant T of about 10 minutes, Hence,
0
fast fluctuations "propagate"
>
while slow variations "diffuse"
.from their ionospheric sources to the Earth's surface,
For geomagnetic soundings only the field at and below the Earth's
surface matters, regardless how the primary field reaches the
surface. It is necessary, however, to make --- one definite assumption -
about the nature of the primary field in the following sense:
Non-di'.'ergent vector fields such as .... E and - H 5n uniform domains
can be decomposed into two parts
-
F = fI +
ZI1
* 6
- -
= rot(rT) + rot ro't(rsS)
h
where - r denotes a unit vector in some specified direction, here
the z-direction. T and S are scalar Cunctions of position, It is
readily seen that the so-called "toroidal" part fI is orthogonal
A
to 2, i.e, in tne here considered case XI is tangential to planes
z = const. The remaining so-called "poloidaln' part zII of F has
three components:
rF"
\ I
VI /
\ \ 1 1 / r
Let now PI and gII be a "toroidal" and a "poloidal" diffusion
vector, both satisfying
V'P - = iwpoo(l +. T~)I',
from which the toroidal and poloidal parts of .-. H are derived by
definition as follows:
-

Observing that
rot pot rot g = - rot(~'p-~rad - divP) - = -iwp o a rotg, the electric
vectors follow then from Maxwell's field equations as
2i
P
E = - rot rot PI; E
-I -I I
= rot PII. '
p2 i(l-ki)
Since the field which is derived from PI has a - tangential ~agnetic
field, it is termed the "TM-mode" of the total field, The field
derived from PII , on the other hand, has a - tangential electric
field and represents the "TE-mode" of the total field.
Suppose that the conductivity is a sole function of depth, o = a(a),
and therefore the diffusion vector vertical ?ownward: - P = (O,O,PZ).
Let Pz be in planes a = const. a harmonic osci.llaCing function
of x and y with the "wave numbers" kx and k Y in x and y-direction:
Then
p(r)
ik . x : po(z) -e - - with - - = (k,,k Y ), ij. (",s?*
rot g = (ik -ikx,O) Pa
Y'
rot rot - P = (ikx/C, iky/C, ]klz) PZ
1 =. ap0/az
where - C and lk1
0
= k:. + k2, It can be shown that C is
Y
a scale length for the depth of penetration of the field into the
uni.form domain. Insert now rot - P and rot rot P into the equations
, . as given above and obtain the following relations between field
components setting wro=O:

Since CI and CII cannot be zero for a finite wave number lkl, TM-
modes will have in the here considered quasi-stationary approxi.ma-
tion zero magnetic fields in nonconducting matter as in air above
the ground. Hence, the existence of TM-modes within the primary
field cannot be ruled out by magnetic observations,sj.nce it' w i l l
be seen above the ground only in.the vertical electric field EZ,
provided that this vertical. electric field i.s not due to local
anomalies of the secondary induced field (s, below), In reality
J
the detection of a primary EZ will be nearly impossible because
changes of the vertical electric field due to changing atmospheric
conditions appear to be orders of magnitude greater than those
connected with ionospheric sources, Therefore it is an ---
assumption
that the primary fields in geomagnetic induction studies are TEfields
.
In that case the secondary induced field above and within a layered
substructure cs = cs (z) is likewise a TE-field. The sum of both
n
fields w i l l be referred to as the "normalt1 field H and E for the
-n -n
considered substructure. Its depth of penetration is characterized
by the response function CII = C which according to the basic re-
n
lations given 'above can be derived from the magneto-telluric "im-
pedance" of the field
-
E E
nx = - 3 = iwPocn
H
ny Hnx
or from the geomagnetic ratios
(MTS=magneto tclluric sounding)
H H
. --.. nz - . v
1kxCn , = ik C (GDSrgeomagnetic depth
H
Hnx nY Yn sounding)
Suppose the internal resistivity is within a limited range also
dependent on - one horizontal coordinate, say x:
u =.un(z)'+ aa(y,z).
Now a local anomaly H~(~,Z) and Ea(y,z) w i l l appear within the
secondary field.

Two special types of such anomalies can be distinguished. If the
electric vector of .the primary fields is linearly polarized in
x-direction, i.e.
and consequently
E = (E , 0, 0)
-n nx
H = (0, H , Hnz),
-n nY
the anomaly has an electric vector likewise only in x-direction:
because the flow of eddy currents will not be changed in direction.
Hence, the anomalous field is a TE-field -
This polarisation of the primary field vector is termed 3ol.arisati
If the electric vector of the primary field is 1inearl.y polarised
in y-direction, the normal field is
and consequently
E = (0, Eny' 0)
-n
H = (H 0, 0)
-n nx'
provided that its depth of penetration is small in comparison to
its reciprocal wave number, yielding H = 0. Only with this con-
the n z
straint is flow of eddy currents to vertical planes
x = const. and the resulting anomalous field w i l l be a TM-fie1.d -
with zero magnetic field above the ground:
E = (0, E
-a ayy Eaz)
H = (Hax, 0, 0).
-a
This pola.risation .of the primary field is termed "H-polarisation". - -.
For three-dimensional structures
O = Ci(x,y,z) = u Cz) + Cia(x,y,z)
n
the anomaly of the induced field will be composed of TE-and TM-
fields which cannot be separated by a special choice of coord:. 7 nates.
There is, however, the following possibility to suppress in model
calculation the TM-mode of the anomalous field:

Suppose the lateral variations u are confined -to a "thin sheet".
a
This sheet may bz imbedded into a layered conductor from which it
must be separated by thin non-conducting layers. Then no currents
can leave or enter the non-uniform sheet and the TM-mode of the
induced field is suppressed. Such models are used to describe the
induction in oceans, assumed to be separated from zones of high
mantle conductivity by a non-conducting crust.
Schemat:ic summary:
Source field Induced field
,,,\~. x,'~ - . . normal anomalous
TE {
TEtTM(genera1)
TE (thin sheet)
xAppGn?3iX to'?, 3 : Recurrence formula for the calculation of the
depth of penetration C for a layered substratum (cf. chapter 2)
7,
r
Definition: C = - 0
aPolaz
a2p
Differentjal equation to be solved: -L? = (iw~~~o + lkI2)po
which satisfies
az2
V - ~ = P iwpo~ .
Continuity condictions:
1. TE-field: - H and E must be continuous which implies
that C-is continuous
-
2. TM-field: H and (Ex,E ,oEZ) are continuous which imp]-ies
v
that oC is continuous.

Model :
wo+
Solution for uniform half-space: CI = CII = lk(
(a = oO)
Recurrence formula foY layered half-space'{Cn = C(zn)}:
-
C~~ - C~~~
= Jioy o + [kr2
0 L
The TE-field within the n'th layer at the depth z = zn + E,
Z < Z + E < Z
n+l'
can be calculated from its surface value
n n
according to:
with gtzr,zs) = cosh Bps - sinh Brs/(K,Cr)
h(zr,z ) = cosh @, - sinh B K C
s rs r r
for s.< - n
and gCzn, z~+E) = cosh(Kn~)-sinh(K n E)/K n C n

, ,
The formula for E applies also to E and HZ, the formula for
X Y
Hx also to H .
Y
7.4 Penetration depth of various types of geomagnetic variations
and the overall distribution of conductivity within the Earth
Three types of conduction which vary by orders of magnitude have
to be distinguished:
1. Conduction through rock forming minerals
2. Conduction through fluids in pores and cracks between rockfor-
mine minerals
3. Metallic conduction
li: Since the minerals of crustal and mantle material are expected
to be silicates, the conduction through these minerals will
be that of semi-conductors. Their resistivity is in the order
5
of 10 Om at room temperature but decreases with temperature
T is the absolute temperate, k Boltzmann's constant; a and A
0
are pressure-dependent and composition-dependent constants
within a limited temperature range, for which one specific
mechanism of semi-conduction is predominant. Hence, in plots
of in o versus T-' the o-T dependence should be represented
by joining segments of straight lines. Laboratory experiments
with rocks and minerals at high temperature and varying
pressure have confirmed this piece-wise linear dependence of
lno from T-l. Furthermore, they showed that the composition-
dependence of u is mainly contained in the pre-exponential

parameter u and that the pressure dependence of o and A should
0 0
not be the determining factor for the conductivity down the depth
of several hundred kilometers.
7 (, "C The u-T dependence of Olivin (Mg,
1200 3 b0 60 0 11
Fe ) Si04 has been extensively
T--- 2
investigated by various authors
for temperaturs up to 1400~~ and
pressures up to 30 kbar. This
mineral is thought to be the main
constituent of the upper mantle
with a Mg : Fe ratio of 9 : 1 :
I Si04. The diagram
(Mg0.9 Fe~.l 2
shows the range of u(T)--curves
as obtained for Olivin with 90%
Forsterite CMg2Si041. It is beli.eved that the discrepancies among
-the curves over orders of magni:tude are mainly due to different
degrees of oxydati-on of ~e" to Fe 'I within the olivLn samples
used for the measurement. In fact, the samples may have been oxy-
dized in some cases during the heating experiment as evidenced by
the irreproducibility of the u (TI-curves. However, the r>ar.ge of
possible conductivities for olivin with 10% forsterite is greatly
narrowed in at the high temperature end, where we may expec-t a
resistivity of 10 to 100 Qm for 1400°c, corresponding to a depth
of 100 to 200 km within the mantle.
2.:Electrolytic conduction through salty solutions, filling pores and
cracks of unconsolidated rocks, gives clastic sediments resistivi-
ties from 1 t o 50 Qm. The resistivity of sea water with 3.5 gr
NaCl/liter is 0.25 am. There should be an increase of resistivity
by one or two orders of magnitude at the top of the crystalline
basement beneath sedimentary basins.
Near to the surface the conduction in crystalline rocks is also
electrolytic. Their resistivity may be here as low as 500 Qm, but
it increases rapidly with depth, when the cracks and pores close
under increasing pressure. Below the Mohorovicic-discontinuity
the conduction through grains can be expected to be more effective
than the conduction through pore fluids. If, however, partial

h
melting occurs, the conduction through even a minor molten
fraction of crystalline rocks could Pouer the resistivity by,
say, one order of magnitude.
3.: Metallic conduction is expected to give the material in the oute
core a resistivity of Om. This law value is required by
dynamo theories for the explanation of the main field of the
Earth.
8 6 'f 2 -5 0 + 5 + l'f
.
0 -1
4.0
-&?-xu- 197'
~ ;pa*'?
t
3.2 ES 't
s*nri- ~"r\da~tirr, 1, L- Waf-
I 6. -,b~ loo-l~ok
s~!-,--
4:c ru-,o-r
- ...... > ... .- . -. . -. - - d
3. CI~L~JI.~ KC(;~.L.~~
.
3 7," *\
V"C.L.*
Y.. R;*
The frequency spectrum of geomagnetic variations which can be
utilized for induction studies extends from frequencies of
fractions of a cycle per day to frequencies of 10 to 100 cycles
per second. At t'he low frequency end an overlap with the
spectrum of secular variations occurs which diffuse upwards
from primary sources in the Earth's core.
460
/
j , AMfI:4uL&
/) 0
?
Sc-Qeggti.csp.ectrun1 of geoma~netic variations :
>
SccLLl sx
Su-b~t-r
,Wd"t;0*
59 C?,.rcr-KC+ t. )
4 , ,
..L -

(=disturbed)-variations: After magnetic storms the hor,j.zontal
Dst --
-
H-component of the Ear-th's magnetic field shows a worldwide de-
crease. Within a week after stormbeginn H returns asymptotj.cally
to its pre-storm level. This so-called Dst-phase of storms shows
negligible longitudinal dependence, while its latitude dependence
is well described in geomagnetic coordinates by
3 L
is the geomagnetic latitude of the location considered. The
source of the Dst-phase may be visualized as an equatorial - ring
- current (ERC) which encircles the Earth in the equatorial plane
in geomagnetic coordinates:
Tf the interior of the Earth were a perfect insulator and no eddy
currents were induced, the vertical Dst-coniponent would be ZDst -
H, sin @. In reality, only one fourth to one fifth of this value
is observed due to the field of eddy currents which oppose in Z
the primary field of the ERC. Let LDst denote the inductive scale
r -
ijw Q=kso length of DPt, a be the Earth's radius, then for @ : 45'
.,
- -
- 2C~st
'DS t a H~st4
(cf. sec. 8.2)
Hence, with ZDStfHDSt 0.2 the depth of penetration will be 600
km'.

9 (solar quiet) varia-tions: On the da)l-lit side of the Earth thermal
convection in combination with tidal motions generate large-scale
wind-systems in the high atmosphere. 1; the ionosphere these winds
produce electric currents which - for an observer fixed to the
Earth, - move with the sun around the Earth. During equinoxes there
will be two current vortices of equal strength in the northern and
0 0
southern hemisphere, their centers being at roughly 30 N and 30 S.
Due.-to these currents local-time dependent geomagnetic variations
will be observed at a fixed site at the Earth's surface. They are
repeated from day to day in similar form and are called "diurnal"
or Sq-variations:
--
Aiai-n the Z Ampli-tude is much smaller than to be expected from
SQ-
D for an insulating Earth. Tf and D (") deno-te the amplitudes
s Q SQ s ?
of the m1 th subharmonic of the diurnal varl.;i.tions (m=1,2,3 ,I!
(I7 the
corresponding to the periods T = 24, 12, 8, 6 hours) and C s Q
inductive scale length of this subharmonj.~,

with 4 as geographic latitude. From the %Q : FISQ ratio of continental
stations a depth of penetration between 300 and 500 km is inferred.
The penetration depth of SQ in the -ocean basins is still uncertah.
Bays and - Polar Substorms: During the night hours "bay-shaped" de-
flections of the Earth's magnetic fi.eld from the normal level ave
observed from time to time, lasting about one hour. Their amplitude
increases steadily from south to north, reachi-ng its highest value
in the auroral zone. Similar variafions, but much more intense and
rapid, occur during the main phase of magnetic storms, until about
one day after stormbegin.
The source of these so-call-ed "polar substorms" is a shifting and
oscillating current lineament i.n the ionosphere of the aurora zone.
The current wil].be partly cl-osed th~ough field alligned currents
in the magnetosphere, partly by wide-spread ionospheric return
currents in mid latitudes:
S,'L

polar substorlns is j.n mid-latitudes (e.g. Denmarlc, Germany) much
smaller than the ampliiudes of H and D
-
because the vertical field
of induction currents nearly cancels the vertical field of the
polar jet. Under "normal condirtions" the Z:I-l ratio is about 1: '10.
Assu~ning for the mid--latitude substorm field an effective wave
number of 10- 20 000 km, yielding kx 3-6*10-~ as wavenumber, depth
of penetration i.s
'bay
A
I = 150 to 300 km.
bay kx
There are indications tllat the penetration deptll of bays into -the
oceanic substructure is substantialiy smaller. The ocean itself
produces an attenuation of the H-ar~pl.itude of about 75% at the
ocean bottcm, deep basins filled with unconsolidated sediments can
yield a comparable attenua-tion &-6. North German basin).
Pulsations and VLF-emissions: Rapid oscilla-tions of the Earth's fie1
- -
with periods between 5 minu-tes and 1 second are called pu1.sati.011~.
Their ampli-tude increases 1.iIce the ainpli-tude of substorms strongly
when approaching the auroral zone. Their typical midlati.tude
ampli-tude is 1 gamma. The source field structure of bays and pul.-
sstions is similar, the depth of pene-tra-t:ion of pulsations being
largely dependent on the near-surfack conductivity. Lt may rmge
from many kilometers in exposed shield areas to a few hundred
meters and less in sedimentary basins.
The "normal" Z:H ra-tio o:E pulsations is too small to be determS.~-rccl
with any reliabili-ty outside of the auroral zone. However, "an~!naI.oi
Z-pulsations. are frequent and usually of very local. charak-ter. IF
pulsations occur in the form of las-tine harmonic oscilla-:ions , oftel
with a beat-frequency, they are called "pulsation tr>ains1' pt, sin~J.(
pul-satj.on events are ' called "pi" , pul.satj.ons which rt~arlc the be-
ginning of a bay.dis-turbance are called "pc". All three types !la\'c
a clear local time dependence, occuring almost daily:
'F\ 1-1 m- -7,. -

Very rapid oscillations with frequencies between 1 and, say, 100 cps
(=Hz) are called - from the radio enginieers point of view -
"very - - low - frequency emissions: VLF. Their ampl.itudes lie well below
ly. They occur usually intermittendly in "bursts" and are controlled
.in their intensity by the general magnetic activity. In exposed
shield areas they may penetrate downward a few kilometers, but
everywhere else they w i l l be totally attenuated within the very
surface layers.
- Sudden storm -- commencements and sclar flare effects: The begin of a
magnetic storm is usually marked by strong deflections in H and Z
up to 100y within one or two minutes.. This so--called "sudden - - storm
-
commencemen-t" SSC, signals. the inward :notion of the magnetopause
(separating the magnetosphere from the interplanetary space) under
the impact of a suddenly increased solar wind intensity. SSC's
are a world wide, s imultaneous1.y occuring phenornenon. They are,
however, too rarely occuring events (1 per month) for induction
studies.
The momen-tarily increased solar wave radiation, caused by sun spot
e: eruptions, produces a shortlived (5 minutes) in'tensificatrion of the
Sq-system, its magnetic effect is called a "sol.ar flare event". It:
measures a few gammas and like the ssc is a rarely observed varia-
tion type.

Overall - depth - distribution -. Denth of penetation
of resistivity
. .
& ~ n ]
(0) : oceans
(s) : sedimentary basins
(c) : exposed crys-talline
EEJ:
PEJ:
mid-latitude
obs erva-t ions
equatorj.al. e1.ec'tr.c
polar electz~ojet

!
8. Data Collection - and Analysis
8.1 Instruments -
unetic sensors for geomagnetic induction work on land should meet
as many as possi-ble of the following requirements:
(1) Sufficient sensitivity and time resolution
(2) Directional characteristics which allow observation of the
magnetic variation vector in well-defined, preferably ortho-
gonal components.
(3) Stable compensation of the Earth's permanent field, if re-
quired. The sensors should not show "drift" on a time scale
comparable to the slowest variati-ons to be analysed.
(4) Compensation of temperature effects or linear temperature de-
pendence with well defined temperature coefficients for subse-
quent corrections.
C5) Low power consumpti.on to allow field operations on 57et or dry-
cell batteries (less than 100 n;A at 10 to 20 Volts DC con-
tinuous power drain).
C6 1 Minimurn maintenance, all-owing unattended opera-tions over days,
weeks or months, depending on -the period range to be irlvesti-
gated.
(7) Electric outmpu-t signal, adaptab1.e for digital recording
w
No single sensor system can possib1.y meet all requiremen-ts over thc
full frequency range of natural geornagrletic variations and the
- choice of instruments wil.1 depend on the type of variations under
investigation. They record either the total field M Ho + 6H as
?E
sum of -tl;e s-teady field Ho and the variation field 6H, by coml2ensa.-
tion of Ho, the time--derivative fl : 6f'i or a combina-tion of 61.1 nil
x*
& i1. In geomagnetic latitudes the spectral amplitude distr.Lbution of
6H and 611, averagedoverextended perioss of moderate ma~c5tj.c acti-
vity, is the following:
x the variation field 6H only

The next diagram shows the resulting spectral range of adequate
sensitivi.ty and stability for various magnetic sensors, suitable for
geomagnetic studies: Perfornlance on
point (2) to (7)
D
Torsion fibre
B a s s
0 s . 2 3 5 6 7
magnetometer
graph, ~ough
Reitzel variograph
1. )
i
!
I
- I-
i
I
I
-
I
6H + + + t i - -
I I - I
i
I I !
Fluxgate magne- /
tometer (I'oer- i
. ster Sonde) 3.)
6 1-1 i. - - + + J.
Grenet-vario-
meter 1. )
Induction coils
Proton prece-
I
1 !
Kb-vapor Inagne- -------- -. lio+6H - -I- + - - 4-
tometer 3.)
.I. ) Cornpensation of H by mechai-~$.cal torque
0
2.) Bobrov-quartz variometers, Jovilet variometers
3.) Compensation of H with bias fields
0
The horizontal compone~lts of the surface elec-trjc field E which i.s
connected to 6H and 6k can be estimated either from the knowledge
of the depth of penetration C at the period T or from the knowledge
of the resistivity p of the upper layers, assumed to be uniforw.
Measuring H in (y), C in (km), T in (sec) and p i n (Qm) gives
. . .
!
! i !
i 611,6; + - + + - - I
i
i 66 + - + i- i-
: .
ssion magne-to- - -- fi t&H + + + - - .I-
0
meter 3.)
&
r--
Using the spectral amplitudes as given above and C from'previous
sectLon, the following spectral amplitudes of E/H and E are obtained
for mid-la-ti-l-udes during rno:lerate magnetic aeti-vity :

A horizontal electric -- field component in the direction - s is observed
as the fluctuating voltage Es'd between -two electrodes in a distance
d parallel to 2. Due to possible self-polarisation of the electrodes
a quasi-steady voltage of considerable size may be superimposed
upon the fluctuating voltage, truly connected to geomagnetic varia-
-Cions. This self-poten-tial amounts for electrodes in a sa1.t solu-
tion comparable to ocean water to:
> 1 vol-t for unprotected iron rodes
100 mV Cu-CUSO,~ elec-ti-odes
Kaloniel (= biophysical) e1ec:i:vo~
kg-kgC1 (=oceanographic) electro
2
It may vary slowly under changing conditions in the waterbearing soi
Hence, it should be as small as possible in comparison to the voltafi
truly connected to 6H and 6i1. This applies in particular .to periocis
of one hour and more. There are two options to avoid these unwanted
electrode effects: Large electrode spacing (d - > 10 km), hrigh-quali-i:y
electrodes. Preference should be given -1-0 ttte seco~ld option, using
electrode spacings of, say, 100 in. The use G.? a large electlode
spacing does not imply necessarily tha-t a mean electric field,
avera~ed over local. n~ar-surface non-uni.formlties , wi1.l be observed,
since the observed voltage may still be largely de-ter~nined by local
condi-tions near to either one of the electrodes. In any case, the
site of the electrode installati.on should be surveyed with .dipol-
soundings or DC--geoelectrics to ensure layered conditions at -- and
be-tween electrodes.

8.2 Or~anisation and objectives -. of field opcrations
Geomagnetic induction work can be carried out by (i) single--site
soundings, (ii) simultaneous observations along profiles, (iii)
simultaneous observations in arrays covering extended areas. It may
be possible to replace simultaneous by non-simultaneous observations,
provided that the variation field can be "normalized" with respect
to the mean regional variation field. This "normal" field may be
the field at one fixed site with no indications for the presence of
lateral non-unj-formities, or it may be the averaged f'ield obtained
from distant permanent observatories. Usually the normal.isa'tion is
performed in the frequency domain, introducing se-ts of transfer
functions.
Single site vertical sounding:
The resistivity structure within the depth-distance range of pcnetra.
tion is regarded a sole function depth: p = p(z). Its extent is
given by -the modulus of -the induc-tive scale length C(w) at the con-
sidered frequency, whi.ch increases with
- - , --
the increasing period. If a vertical
........... .......... -. .
-. -
J 1 ., . . . ...
.- ..-

For a data reducti.on in the fr3equency domain (cf. Sec. 8.3) the
following set of transfer or response functions w i l l be defined:
For a frequency-space factor Of the source field
the response function C(w) = C n which - characterises the downward
depth of penetration is connected to these transferfunctions accor-
ding to
A Z
n x y
n ikx iwv0
c =- -
The Cagniard apparent resistivity if given hy
the phase of the impedance by
I$(w) = arg(Z ) = arg(C ) + 5 ,
xY n 2
the modified apparent resistivity by
which can he used as an estima-tor of the true resistivi-ty at the
depth
z"(~) = R~IC~I. (cf. Sec. 2.5 and 9.1)
A cornpairison ,of the various equivalent respcrlse func-l-j.ons for a
. , given substructus&ives the fol.lowing displ.ay:
L? *

Tests for the assumption of a layered dis.tri.bution and the source
In addition there are certain constraints with regard to modu.lus,
phase and frequency dependence of the transfer functions. (cf. Sec.
2.6).
The compatability of MTS and GDS vertical sounding results can be
tested by the requirement that
asreadily seen form the second field equation rotE = -imp H .
-n o -n
If the wave-number structure of the source field is not lcnown, a
-
Horizontal Gradient GDS can be formed: Observing that
-
the differentation of E = Cn impo H with respect to y and of
nx nY
= -iou C H wi.th respect to x yields
.Env o n nx
Since the electric surface field may be distorted by lateral non-
uniformi-ties even when their scale length and dep-th is smal.1 in
comparison -to the depth of. penetration, -the E-field observa.tj.oi~s
may be replaced by observing the tangen-tj.al- magnetic field also
at some dep-th d below -the surface, i.e. by conducting a 'vertical -
-----
Gradient GIIS. No knowledge of the wave-number st~uc-ture of the
source field w i l l be required, but"the resi;tivity po between the
' surface and the subsurface points of observations will be reqci-red.
Assume that the lateral gradients of IJnz are small in comparison
to the vertical gradients of H and H in v5.e~ of the condi.tion
n x nY
I -.
I ~ c [

Let q be the tranfer function between Hnx(H at the surfade and
nY
) at the depth z = d:
Hnx(H nY
Then
H (d) = q Hnx(0) where I q 1 < 1.
nx
Single site geo-letic structural sounding: The source field is re-
garded as quasi-uniform (k = 0) and the ver:tical magne-tic component
therefore as anomalous, arising solely from lateral changes of the
resistivity within the depth-distance range of penetration, where
p = Pn(z) + P,(X,Y,Z)\
In this special case the resulting anomalous magnetic vector
H = (Ha,, H , H = H is linearly dependent on the quasi-uniform
-a ay az z
normal magnetic vector H = (Ilnx, H 0) iii the frequency-distance
-n ny'
domain :
denotes a matrix of linear transfer functions as functions of fre-
quency and surface location. This i.mplies that also linear relations
exist between HZ and the total (=observed) ho17izontal variations:
with
These alternative transfer functions A and B can be del-ived now from
observations at a single site. Their graphical display in the form
of Parkinson-Wiese induction arrows indicates the trend of -the sub-
surface resistivity structure which is responsible for the appearance
of anomalous '2-var.i.ations :

The in-phase induction arrow is defined by (in Parkinson's sense of
orientation) by
h
p = - R&(x A + B]
and the ou-t' of-phase arrow by
q + Imagl; A + Bl
h h
where x and y are unit vectors in x- and y-direction. General-ly
speaking, the in-phase arrows point toidards internal concentrations
of induced currents, i.e. to zones of lower than "normal" re-
sistivi.ty at one particular depth.TheYmay point also away from high
resistiv:ity zones around which the induced currents are diverted.
Vertical soundings with station arrays:
The resistivity structure is regarded as layered, p = p(z),but the
inducing source field as non-uniform. Inducing and induced fields
will have ms.tching wave-number spectren with well defined ratios
between spectral components in accordance to the subsurface re-
s i s t ivi-ty structure . h ea-t-k-+h~++t-~o-~~&e-~~idc~~~
e.ve;i:~-~:gh-i-s-k~ol.a-7,-u~.-f m~rnGty--rnaap-~be--~i-~-~P+f+a
Let U and V be field components of the surface field in the frequency
distance domain: U = UCw ,R) , V = VCw ,R). They are deco~nposed into
h h A
the wave-number spectren U(w,k), VCw,k) according to
with
irnh ikR - -
U(w,R) = I I UCw,L,)e dk dk
- '0 Y X

. A
as transfer function between A and B in the wave number domain, which
contains the information about the internal resistivity distribution.
A vertical sounding of this distribution is made by considering
h
R as a function of frequency, k being fixed, or vice versz.
If the array covers the whole globe, the sphericity of the Earth re-
quires the replacement of the krigonometric functions by spherical
harmonics in spherical coordinates Cr,B,A):
a: Garth' s radius ; p:(cos@ : Associated' spherical function. The trans-.
fer functions have then the form
The first and still basic investigations of the Earth's deep conduc-
tivity structure have been carried out by this approach (SCHUSTER,
CHAPMAN and PRICE).
If the array of stations covers only a region of Limited extent, it
may be !impracticable to de'compose the observed field into wave-
spec-tral components or it may be even impossible because only a small.
section of the source field structure has been observed. In that
case vertical array soundings are carried out preferably with response
functi.ons in the frequency-distance domain.
Suppose the source field is quasi-uniform ,i.n one horizon.ta1 direc-- j
tion, say,
U = UCw,y) and V : V(w,y). Ls't
* . %
be the inverse Fourier transform of R. Then, if V is given by the
,. A
roduct of R with U in the (u,k) domain, V will be given by a
corlvolution 05 R with IJ in the (w,y) domain:
+w
with .R U = 1 RCw,y-n) UCq)dq.
-m

If, for instance E = V and H = U, the magnetotelluric relation
A ,. X Y
EX = WM, C II of the k-w domain will transform& into
Y
with N(w,y) as Fourier
-
transform of C(o,k). Observe that reversely
4
-iky
CCw,k) = I N(w,y)e. dy
and therefore
-03
f-
CCw,O) = J N(w,y)dy.
-m
Hence, if H is quasi-uniform within the range of the kernel M,
Y
which is the CAGNIARD-TIKHONOV relaticn con~monly used for single
1
site sounding-s.
In a similar way the magnetic ratio, of vertical to horizontal' va-
riations can be generalized to
with MCw,y) as Fourier transform of ik C(w ,k ) .
Y Y
The response functions N and M have their highes-t valuei. close to
y -
O
and approach zero for distances y which are large i.n compari-
son to the modulus of C(w,k=O): d.;:.rn,-Lc. do-,,o. :.,
----. - ---. .
A.

-- Structural soundi~z with station arrays: - If not only the source
field but also the resistivity structure are laterally non-uniform,
no generally valid linear relations between normal and anomalous
field components exist. In the f~llowing it will be assumed that
the source field is either quasi-uniform or for a given frequency
well represented by a single set of wave numbers. Then the rela-
tions between the various field components can be expressed by
linear transfer functions in the frequency-distance (w,R) domain.
They can be formulated either for the "normal" horizontal field of
the whole array or, if necessary; also for the local horizontal
field only. In either case it is necessary to remove from the ob-
served vertical magnetic component its normal part in accordance
to the normal resistivity dis.tributi.on and the source field struc-
ture. It is assumed that the normal depth of penetration is smal.1
in comparison to the scale length of non-uniformity of the source
~wI,:~I. ;%pi i

. .
These sets of tPansfe$fUnCtions renresek the input ior the model
calculations to explain the anomalous fields H
-a
a resistivity anomaly
or E
-a
in terms of
pn(z) being known.
P,(x,Y,~) = P - Pn(z),
There exist various constraints about acceptable sets of transfer
functions, for examples the functi.ons in either one column of W
must describe a magnetic field of solely internal origin as dis-
cussed below.
The basic complication in the presence of 3-dimensional structures
is due to the fact that the TE and TM mode of the anomalous field
cannot be separated. This separation is possible, however, if the
resistivity structure is 2-dimensional, say
Pa = Pa(yl>z).
If then the normal E-field is linearly
.!
, ,
.. .. .l.
,,' '
:.
;, ;/ ... Y
polarised in xl-direction, the anomaly ,/' ,{ ,> ,,
contains only TE modes with E parallel 1:; %
-a ,,; 71 Y'
A
to x1 and H in planes XI-constant:
-a
E-polarisatio~, If the normal. magnetic field is paral.lsl to x', the
anomalous field is in the TM mode with zero ~nagnetic fields above
the ground. Hence, Ha = 0 for z = 0 and E lies in planes x1-
-a
const. : H-polarisation Ccf. Sect. 7. 3 ). The new sets of transfer
functions in (xl,yl,z) coordinates are given hy
>?here the subscript (\I refers to E-polarisation and -the subscrip-?
(l.) to ~-~olarisation. The resul-ting symmet~y relations for W and
Z in general (x,y,z) coordinates can be derived as follows: Let
be rotation matrices, transforming .- E and -- H fron: (x,y ,z) -to (x' ,yl ,z)
coordinates, c = cosa-and s = sina:
, ,
, !
..,

Since
it follows that
- E 1 = - T E HI= T H -n . ' -a
H ' = T
H
H
n '
E' = Z'HA = Z'TH = TZHn,
- - - n
Z = T-I Z' T.
In a similar way it is readily vkrified that
Hence,
which implies thkt.
W + W =\ih,, and 2. - z =.Z?-Z-~
XX Y Y XY YX
are invariant against rotations and that
"Skew" parameters which characterise the deriation from true 2-
dimensionality are the moduli of the expressions
If a geomagnetic sounding at a single site has been performed near
such a 2-dimensional structure, the relations which connect A and
.B with WZx and W reduce to
ZY
If -the first relation is multi.plied by c and the second relation by
s and then the sum of both is taken, it follows that
This implies that the in-phase and out-of--phase induction a17ro:.?s
will be everywhere perpendicular to the trend of the structure,
which can thus been found.

In magnetotelluric soundings information about the trcnd comes from
the fact that
However, after Z has been rotated into Z' for the thus found angle a,
no distinction is possible which one of the off-diagonal elements of
Z' refers to E- and H-polarisation. If in addition the direction of
the geomagnetic induction arrow is known, such a distinction becomes
possible. Furthermore (z,, [ > [ zLl ,if the structure is better conduc-
ting than its environment and vice versa. Hence, it can be decided
whether the induction arrow points toward a well conducting zone or
away from a poorly conducting zone. .
The dj.sti.nction between the impedances of E- and H-polarisation is
important, if for,a first estimate a 1-dimensional interpretati.on of
.7
Z by a layered. substratum is made. Such an interpretation may give
meaningful results for Zih, but in general no$ for ZL. A test for the
proper choice of the impedance element for a ?-dimensional interpre-
tation comes from the fact that t~ithin a given area Z,, varies less
from place to place than Z1.
A test for self-consistency of the transfer functjons Wh and Wz alonj
a profile y' perpendicular to the strike of a 2-dimensional struc-
ture arises from the purely internal origin of the anoixaly:
WZ = K x Wh and 9Jh = -K x WZ
denotes a convolution of W (or W with the kernel. function l/ny.
h z
-- 8.3 S~ectral Analysis of Geoma~netic 'InductLon Data -
The objective of the data reduction in the frequency domain is the
calculation of transfer functions. They linearly relate functio!?s
.Iv I-
of frequency a field component Z -to one or more other fiel$compo-
nents X, Y , . . . . Let Z(t), X(t) and Y(t) be the observed time Va-
riations of Z, X, -f during a time i.ni-ervall of length T either fpojn

- - .
the same or from different sites. Let Z(w>, X(w), Y(w) be the Fouri-er
transforms of Z(t), XCt) and Y(t). Then a linear relation of the
form
- -
is established in which A and B represent the desired transfer func- -
tions between Z on the one side and X and Y on the other side; 6Z is
the uncorrelated "noise" in 2, assuming X and Y to be noise-free.
As the best fitting transfer - functions will be considered those whicf
produce minLmum noise < 16zI2 > in the statistical average. Here the
average is to be taken either over a number of records -- or within
extended frequency bands of the width which is L times greater
than the ultimate spacing IL'T of individual spectral estimates. The
noise: signal ratio defines the residual e(w) ,%atio of related to
observed signal the . --- coherence R(w):
The coherence in conjunction with the degree of freedom of the
- -
functions A and B.
averaging procedure estab1ishe.s confidence limits for -the transfer
The averaged products of Fourier transforms are denoted as
S
ZZ
., -
= < Z Z * >: power spectrum of Z.
- -
S = < Z Y * >: cross spectrum between Z and Y
ZY
wi-th S = S* .
ZY Y =
In summary, the data reduction involves the following steps
(a) Fourier transformation of tiine records
(b) Calculations of power and cross-spectra
(c) Calculation of transfer functions
(d) )COlculation of confidence limits for the trans-
ferfunctions.
Steps Ca) and (bf can be substi.cuted by the foll.owing alternatives:
(ax) : Calculate auto-correlation functions ) , .. . and cross-
correlation f.urictf.ons R , . . . with T being a time lag,
ZY

with 0 < T < 'rmax.
R (T) = / Z(+-t) Z(t)dt
Z z T
R (T) = / Z(+-t) YCt)dt
ZY T
x
(b ): Take the Fourier transforms of the correlation functions
and obtain as in (b) power- and cross-spectral estimates,
if the averaging is done within frequency bands of the
width m. There is a formal correspondence between the
-- -1.
maximum lag and Af .
The actual performance of the steps (a) to (d) with one or more
setsof records is now described in detail.
-- a. Fourier transformation. - The time series and their spectra are
given, respectively to be found, at discrete values of t and w,
which are equally spaced in time and frequency. Let Z e-tc. be an
n
.instantenuous value of ZCt) for t = tn, n = 0,1,2, . . . N with
At = tntl - t . Outside of the record, extending from t to t
n o N
Z(t) is assumed to be zero.
-
Let Z be the Fourier transform of Z(t) for the frequency f = f .
n1 IT1
Because of the finite length T of the rec6rd the lowest resolvable
frequency and thereby the frequency spacing w i l l be given by
T-I = Af = f and f has to be a multiple of Af. Because Z(t) is
1 m
gj-ven at discrete instances of -time, A t apart, the hi.ghest resol-
-,
vable frequency, called the Ny quist frequenz, w i l l be the reci-
procal of (At-2), i.e.
and
- I
fH - - = MAf
2At

The Fourier integral
- +- -io t T -iw t
n
z(un) = I Z(t)e
dt = J Z(t)e
n
dt
-@2 0
will be evaluated r.umerica1l.y according to th,e trapezoidal formula
of approximation. Setting
and observing tha-t
zo = $[zct 0 ) + 2(tN)-j
"n?n
= 2a fmtn = -- 2 limn
N'
the discrete Fourier transform of Z(t) is
2
L *
A linear trend of the record within the chosen interval may be
wri.tte11 as
with d = Z(tN) - Z(to).
A cor3rection for this trend i.n the frequency domain implies that
the imaginary Fourier transform of Z1(t),
-
is substracted from Z .
m
A second correction arises from the fact that Z(t) does not v?.nish
necessarily outside of the chosen intervall. In that case the
original time function can he multiplied in the rime domain wirh n
weight function W(t) which is zero for t < to and t > tN. The
Fourier transform of the product
W(t) . Z(t)
-
- - - -
is given by7a convolution of Z with the Fourier transform of W:
- 1
W(o) 35 ZCo) - Ff C Wm-& Z; .
A
m
A frequent.1~
used weight function is
' 1 2li T -
WCt) = -11 + cos --(t-t .-
2 T 0
2

which has 'the Fourier transform T for m = 0
"1% 0 71-
t -5
G({l
where <
'm
> is the discs-te Fourie~ transform of W ( t ) * Z(t). This
smoothing procedure of the origi.na1 spectrum is czlled "harming"
after Julius von Hann.
\
\
\
\
\ / -. L- q-4----&
b , I.\/' 3 1
The convolution of W with descrefe values of Z reduces then to
- 1 - -
m T m-m m
-
1-
-(Z. + ZIY m r O
( 2 0
= - C bi A* ZA = m = 1, 2, . . . PI-1
-(Z 4-2 m = M
2 M-1 M
In certain cases it will. be necessary to apply a numerical fi1.ti.r
to the time series to be analysed beforc -. the Fourier transformation.
This filtering process consi.sts i.n a convolution of 2c.t) with a
filter function \d(t) - in the - time do1naj.n and thus corresponds to a
multiplication of Z with W in the frequency domain. But because
the filtering procedure is intended to prepare the time series for
the Fourier transformation it must be carried out in the time domain
If W denotes the filter weight for t = t17, the discrete form of
n
the convolu-ti.on is
Usually even 'filters are employed, W ( t ) = W(-t), to preserve .the
corr-ec-t phase of the Fourier coniponents, The transform of is chose:
in such a way that it acts as a heigh-, lord- or bandpass fj.1-ter for
fr.equency-independent (="whitet') spectra:

The weigh-t . function W is then found by calculating in Four~ieil tl-ansform
of W:
The puTpose of filtering the -time series before making a Fourie?
transformation is
(a! a prewhitening of the spectrum, levelling off peaks and compen-
sating spectral trends
(b) a reduction of unwanted larger spectral values close to zero
frequency,, arising from a background trend in the time series
(c) a suppression of spectral power beyond the Nyquist frequency
to avoid "a.liasingU . Spectral values for the frequencies
*2~-r' f r and . f3M-r>' fM+r' fH-r etc.
are undis-tinguishable (r = 0,l i2, . . . , M-1) :
. Cal'dulation of porter and cross spectra: Let P denote a product
- X - - %
- - m
ZmZIn, ZmYm etc. for a frequency f m = 1,2 . . . .1q. The calcul.a-t'iorl
m'
of power and cross spectral estimates from real. data requi-res that
such produc-ts are averaged with 2a cer-tain degrees of freedom, !C
Suppose -th.a-t the available ~?ecord leng-th just gives the desired re-
- 1
sol.ution Af 2 T for the transfer functions to be determined. In
this case spectral products Pm have .to be derived for a number cf
individual records and then to be avermgcd. Average the sp~:ctral.
products and - not the spectral values! The resulting mean value
is the desired porier or cross spectral es-ti.mate. If L records have
been used, \? = 2L besaucje the sine- and cosine.tr.-,nsfor.~~ls of the
series c011tr~i.b~-te independently to the calculil'tioris of Sm.

- 1
If the record length T allows a frequency resolution Af = T which
is much smalier than a meaningful resolution for the transfer
functions under consideration, then Pm can be averayed over
L = Z/A~ spectral values within (M/L-1) frequency bands of width
-.
Af between fo = 0 and fM = MAf.
Taking the average of Pm withri.n frequency bands implies that EL nu-
merlical filter Q(f) is applied to P(f), yielding the desired power-
cross spec-trum by convolution:
. +M
S = Q x P .- A f L Qm-; P;'
m
-M
(P = 'pi). The filters to be used are even functions of f, bel-
-m
shaped and with zero values outside a range which is smaller then
M Af. Setting Q(f) = 0 for f > f Q'
m
m~
S = Af{ C QA(P m m+m
G=l
m-m
A + P A) + QoPm} .
I I
0.1. 48(
The resultring smoothed spectral estimates Sm have been derivedwith
-
roughly L = &f[Af degrees of freedom. The effective degrees of free-
dom pmay be somewhat smaller, depending on the frequency dependence
of the filter. They are defined as
.y'=
2 varCu)
var (uq)
w11,ere varCu) denotes the variance (=dispersion) of a random va-riable
uCf) and varCuq) the variance of Q x u, hence

Two convenient filters are
3 sinx I,
.the Parzen filter Q(f) = - T = Min!
'626~ m m
Hence., the derivatives of S6z6z with respeci - to ,. the real and imaginary
parts' of the transfer functions A and B have to be zero:
1% nl
aS~z6~ +i---- aS&z6z = = 0 etc.
aRe(An) a1rn(An),

using the notations as - introduced - above. The subscripts m are ommittec!
Elimination of either A or B yields the basic formula& for the de-
termination of transfer functions in geomagnetic and magnetotelluric
The coherence can be derived from
-. *
R' = (AS x z + BSyZ)/SZz
as it is readily seen from
If only a relation between z and X 01: Y. is sought or if X and Y
are linearly independent (S = O), then the above derived relations
XY
reduce to
~2 = a---
S S S
xx yy zz
d. Cal.culation of confi-dence liniits -a,- for the transfer functions
--
- "
In order to establish confidence limits for the transfer functions
A, B of the previous section it is necessary .to fj.nd -the probabili-ty
density func-tion ("pd f") of their devia-tioils fr>om their "true"
" - - -
values Ao, Do. W; assulne that esti.mates of A and B accordi.ng to the
least-square method of the previous sec-tion are wj:thout bias, i. t?.
bl?th~~t sys-tematic errors (E = expected value):
I-lencefor-th, random variables will be wri.tten with capj:l-a1 letters

(e.g. X), their realizations by observation with lower case letters
(e.g. x), and their true values with the subscript "On (e.g. x0).
Let f(X) be the pdf of a variab1.e X. Then the probabi.lity that a
realisation x exceeds a value G is
m
a = J f(X)dX.
G
Hence, there is a "confidence" of
f3 = (1-a) . 100% that x does not
exceed G.
The following pdf's will be needed in this section:
k" C;
(1) The "Gaussian Normal Distribution" of a normalized variable
with the dispersion 1 and zero mean value:
(2) The X2-distribution for v degrees of freedom:
, (3) The Fisher-distribution for N and V2 degrees of freedom:
1
'These distributions are encountered in the fol.lowing problems:
Consider a random variable X which is normally distributed with
Let - 1 n
X = -
n C Xi
i=l
be the, realized .sample mean value of n realizations of X. These
mean val.ues .are also normally distributed with the dispersj.on
accordj-ng to the "central 1.imi-t theorem".

Let
be the realized dispersion within a sample of n observations.
Then the normalized dispersion
w i l l be a random variable with a X2 distribution for U = n-1
degrees of freedom. Let S: and S: be the dispersions within
samples of nl and n2 observations of two different variables
X1 and X Then the ratio of their normalized dispersions
2'
w i l l have a Fisher-probability density distribution E (n-1,n-2).
F
We regard now the Fourier transforms 2, g, i' as random variables
Z, X, Y and denote their real-ization for a single record or a
single frequency component as x, y, z. In a similar way, a and b
shall be the realized transfer functions for a limited number of
records or a limited frequency band G, their true values being
a and bo. We assume now that Z depends linearly only on X and
0
that X is observed error-free:
Then
z = a x +6zo
0 0
6zo is the "realized" not correlated part of z for a single
record. The variable Z shall be normally distributed with the
dispersion
var (Z) = s:,
which w i l l be also the dispersion of 6Z:
By minimizing the uncorrelated power = S for a number
6262
of records or for a number of frequency lines within a frequency
band of the width a realization of the transfer function A is

Observe that the residual, of which -che power has been brought to
a minimum, is
6z=ax - z
0
in contrast to the "true" residual
The random variable
TI I:
11
has a x2-distribution with n degrees of freedom, where n is the
number of records or the number of lines used to obtain the
average.
If a variable U with a ~~ldistribution of v degrees of freedom is
decomposed into two components U and U2,
1
U1 and U will be likewise x2-distributed and the sum of their
2
degrees of freedom w i l l be v:
This decomposition will be carried out now with U as defined
above: Clearly,
The power of the averaged residual has been mi-nimised by setting
implying that .
u1 -
- .7i
-.
s1
0 . .
has a X2-distril>ution of n - 2 degrecs of fr?eedoni because

must be a distribution of 2 degrees of freedom in v-iew of -the
real and imaginary part of (A - do). The ra-tio
has a Fisher-distribution fF(2, n-2). Using the notations fr'om
above,
and observing that
with R' = [S ["IS S ), the Fisher-distributed ratio becomes
zx zz XX
We have found now the pdf of the deviation of A from its true vzlue
to be in terms of its modulus
Let
G
8 = / fF(2, n-2)dF . .
0
be the probability that F does not exceed the value F = G, observing
that fF is a one-sided pdf. Then there w i l l be the probability f3
that the modulus of A lies within the confidence limits'
in the complex plane.
E
1 = I A I R Jg
The threshold value G for a desired value
of f3 can be obtained in ciosed form because
G
..'
xc [A\
^

Example: n = 12 and @ = 95%:
1
n = 12 and @ = 99%: I/-- G = ~'1003 - 1 = ~43 = 1.22
11- 2 1 ' -
2
5
n = 12 and B = 50%: 4 - - G =
11- 2
42 - 1 = a = 0.39
1
For large n and n; >> lny (y = -1, the following approxiinations are
1-F
valid :
- 2 - 2
G - - lny
n-2 n
This approximation. exemplifies the general propaga-ti-on-of-error law,
namely that the errors are reduced proportional. to the square root
of the number of observations.
Ln the more general case that Z depends on X and Y with a non-zero
coherence between X and Y confidence limits can be obtained in a
similar way: Let
z = a x + boyo + 6zo,
0 0
assuming that now X -- and Y are real-ized error free. Then the Fisher-
distributed ratio, involving the deviation of A and B from a and
0
b ; turns out to be
which has a f (4, n-4) dis.tri.buti.on. The confidence limi-ts for A and
F
R cannot be calculated individuall.y, unless of course S is taken
"jr
to be zero. On the other hand, if we assume that I~-a,l and I R-boI
are equal, Chen

whi.ch allows the determination of cornbined confidence limits for
A and B. The threshold value of F for a given probability fi can
be derived from
1-6 = - 1
(1 t -- 26 - n- 4
) - (- ) with m = -
2 '
2 m
Cl + %G) m+2G
8.4. Data analysis in the time domain, pilot studies
Suppose the time func-tions to be related linearly to each o-ther are
not oscillatory but more like a one-sided asymtotic return to the
undisturbed normal level after a step-wise deflection from it:
It may then be preferable to avoid a Fourier transformation and to
derive i-nductive response functions in the the domain.
If Z depends linearly only on X, i.e.
*
Z(w) = A(w) X(m) + 6ZCw)
in the frequency domain, then Z(t) will be derived from X(t) in
the time_domain by- a convolution of X(t) with the Fourier trans-
form of A,
f"
1 - i.wt
A(t) = - J A(w) e dm.
2s -m
Since ZCt) cannot depend on X(t) at some future time T > 't, the
response function Act) must be zcro for t < 0, yielding
-

-
As a consequence, the real and imaginary part of A(w) wil.1. be rel.ated
as functions of frequency by a dispersion relation:
1
with K(w) = .
The time-domain response Act) is now determined frorn a given record
of Z(t) between t = 0 and t = T by minin>.izing'{6~(t))~ within this
intervall:
. . 1.t is assumed that X(-t) is known also for t < 0. Differentation with
h
respect to ACt=t) gives
Let again Z = ZCt = n.A-t) denote the value of Z at equally spaced
n
instances and let A be the value of ACt') for t' = n'At. Then the
n'
replacement of the integrals by sums yields a
-
system of ].inear
equations for the determination of the A,,, n' 0,1, ... N:
N' N N
c An,EZ x X -1 = c zn xn ;
n-n' n-n +.
n' =o n=o n=o
A A
for n = 0,1,2, ... N.
A
Chosing N < N, the response values can be de+enninc?d by a least-
square fit. 0-therwise a generalized inversior~ of the system of
eyuations can be performed.
Consider the limiting cases that
. .
(1) the frequency response is real ;ind independent of
frequency,
(2) -the frequency response is imaginary and proportional
-1.
to w .
The first case is real-ized by the inductive :scale 1-eng-tl, C above
n
X
a perfect conductor at -the c1cp.th z or by .the i.mpedance for, a thin

sheet of the conductance T above a non-conduc~ing substriit~m:
3:
Cn = z , Zll = 11~. The second case can be realized by the same
models, interchanging Cn and Zn (cf. See. 9-11:
- . -
The Fourier transforms of A1 and A2 are
02
sinwt -
A (t) = b .f x- - sgn(t)nbo
- 2 O-o;,
Hence, in the first case Z depends on X at the same instance of
time only, in the second case on X with equal weight during the
entire past:
These simple relations are very useful to conduct a pilot reduction
of sounding data. Case (1) applies to magnetotelluric soundings in
sedimentary basins, underlain by a crystali5~ne basement, to ver-
tical geomagnetQi.c soundings with very long periods, reachirig the
conductive part of the mantle, and to structural geomagnetic soun-
dings where the perturbed Plow of induction currents is in-phase
with the normal magnetic field.
The baste linear relation for structural geoinagnetic soundings can
be written then si-~nply by products in the ti.rne domain:

This relati-on implies .that .the local geomagnetic d

In an "Untiedt diagram!' the endpoi.nts of the horizol:tal dis-turbance
vector (El H ) are drawn as a curve in (x,y)-coordinates during a
x Y
single event. Lines which connect time instances of equal ver-tical
disturbance M w i l l then be para]-lel to the.intersecting line of
i. z
the prefer~ed plane with the ,(x, y) -plane. The "induction arrows'!
are normal to these lines and their length is given by the ratio
of Haz t o the simultaneous horizontal disturbance vector, projected
onto the direction normal to the connecting lines.
.Parkinson and Wiese diagrams which utilize the H az) I-Ix' H relation:
Y
of nuinerous individual events can be employed also in the case of
2-dimensj.ona1 structures wi-th an anomalous field which is not
exactly in-phase with the normal field. Then more 01' less sinilsoicla'
variations are chosen with readings at well defined times, e.g. at
the time of maximum deviation of Haz from the undisturbed level.

9. ---
Data 5.nterpretatj.on on the basis of selected -- iriodels
9.1 - Layered H a l f - s ~
The interpretation of geomagnetic sounding data by a layered re-
sistivity distribution p = p (z) is appropiate at those single
n
sites or for those arrays which do not show anomalous magnetic
Z-Variati.ons and which have a polarisation-independent magneto-
telluric inwedance:
The transfer functions to be used for the interpretation are
the inductive scale length C (w,O)
n
for zero wavenumber
the impedance
the ratio'of internal to
external parts in the
wavenuinber domain
The transfer function of the Y-algorithm to be uscd for the lineari-
sation of the inverse problem is
where p is arbitrary reference resistivity Ccf. chap. 6.4)
0
Considering these transfer functions a.t one particular frequency >
the parameters of the following models can be derived directly
from tl-tem. These parameters may then be used to resen~ble as func-
tions of frequency certain characteristics of the true resistivity
distribution. All formulas are readily derived from the formulas
for TE-fields at zero wavenumber in chap. 7.3, annex.
(a) Single frequency interpretation of the modulus of transfer-
functions (Cagniard-Tilchonov): the model c0nsi.s-ks. of a uniform
half-space. Its resistivity is

if the period T is inensured in seconds, E in mV/Icm and H in y.
X Y
is the "Cagniardapparent resistivi'ty" of the substrat~uil
pa at
the considered frequency.
(b) Single
1
frequency interpretation of the real part of Cn:
The model consists of a perfect conductor at
W X A
f=m :* layer.
Im(Z
.. .
3:
~ , , ~.
n
, .. ;\
z = R~IC I = --
\ \ \ g ,. = 0 ., , n
\. .,, \\... u"c
is the depth of the "perfect substitute conductor" at the con-
\ .'.., \ . X
the depth z = z beneath a non-conducting -top
sidered frequency, indicating the depth of penetration into
the substratum at that frequency.
(c) Single frequency interpretation of the imaginary part of C :
n
-7- %* The model consists of a thin conductive top -
;' f - '-
layer of the conductance I-~ = 9 cr(z)dz,
\
\. , . .,
0
covering a non-conducting half-space. This ,
\
\ . \ \
apparen-t condoctance is given by
(d) Sing1.e frequency interpretation of amplitude - and phase of resp3nse
functions; 4 = arg{Zn}. 1 0 < @ < ~ / 4 : The model consists of
X
a thin top layer of the conductance T above a uniform substra-
tum OF resistivi-ty p"":
-ER
TX =
p"" = PaS
-Im(C n ) - Real(C n ) , -- Re(Zn) ,- Im(Zn)
--
ullo ! cn I *
1znI2 -
1 + cotZ@
2
2 n/]; < @ < ~ / 2 : the model consists of a non-conducting 'top --
layer of thickness h above a uniform substratum of the re-
X
sistivity p ;
Im(Zn) - Re(Zn)
h = Real(C ) + Im(C,) =
n
u"o
--
p" = 2 wl,;{l:rn(~~)~~ = 2 : Pa 5
, . 1i.tan2@
'

The "modified apparent - - resistivity" p can be used as an estimator
I r.
of the resistivity at the depth zf= Re{Cnl = h 4 ~p with
m
,.
pX = 1 f i k as apparent skin-depth of the substratum. The representation
UlJ0
of the true resistivity distribution p(z) by the modified apparent ,
x
. resistivity distribu-tion p CZ~)'~S usually adquate, if p decreases
with increasing depth. It is-less satisfactory, if p increases with
depth.
(e) Multiple frequency interpretation with a three-layer model:
'Is
-
L
X
This interpretation cornbines the single
frequency in-terpretat3.ons (b) and (c) , i. e.
-TL-.A,.~.-'- I-.-
the model consists of a thin top layer with
2 h dl
the conductance T , a non-conducting inter-
mediate layer of the thickness h above a
perfectly conducting substratum. Then
= wp ~h = is the induction parameter of the model wi-th dl
0
pi
as thickness of the top layer and pl = /l& as its ~%in-depth
W0T
with the requirement that pl > 1, implying that h >>> p the induction is predominan-t1.y
1'
j.n the surface sheet, the response functions
al-e independent of h, thus allowing a unique cleterrr~ination of T.
Ln particular,
Pa
-. -
1
W ~ J ,r2
0
Hence, if y = log pa/p and x = l.og T/To (T = period; po and To
0
are reference values), then
e .
the p vs, period curve in log-log prcsen'cati.on is a straigh-t
a
4 b?i.~:h w as curve parameter
-

0
line ascending under 45 , its intersection point on the x-axis
-
given hy
Po .r2)
= log(21ry -
To
with TI = 1 . lox1 in seconds and po in Qm.
0
If, on the other hand, 11~

Exercise
Geomagne-tic varj.ations. in H (north component), D (east component)
and Z (vertical component) have been recorded at Goettingen from
September 5, 17 h to September G, 2h GMT 1957. During the same time
interval1 records have been obtained also of the telluric field in
its northcomponent EN and its eastcomponent EE. Scale values and
directions of positive change are indicated.
(1) Determine the magneto-telluric impedances EN/D and EE/H for
fast (period 1 10 min) and slow (period 1 1 hour) variations,
evalua-ting peak value readings of pronounced deflections from
the smoothed undisturbed level.
Calculate the apparent resistivity pa and estimate the phase @
0 0 0
of the impedance (0 , f 45 , + 90 ).Interpret pa and @ with
one or two layer models for the two period groups separately.
Then search for a model which could explain both period grLcups
by calculating the response function y and solving a system
o'£ two to four linear equations.
(2) The magnetogram shows pronounced variations in Z which in view
of the low latitude of Goettingen can be interpreted as being
due to a resistivity anoma1.y. Check the correlation of Z with
H. and D by visual inspection. Then choose effects with
distinctly different K : D ratios and determine the coefficient
A and B for the anomalous vertical field Z = AH + BD.
Interpret the result.

. , . ' .,
, . - ~a~neto~ran?;l~
Gottin'gen 1953 September 5-6
.i , $ .

9.2 Layered Sphere -
The sphericity of the real Earth has to be taken into account,
if for the considered frequency the depth of penetration is
- 'comparable to the Earth's radius a divided by N. Here N is the
highest value of the degree n of spherical surface harmonics
which describe the surface field as functions of latitude and
longitude. There is a formal correspondence between the wave
number k and n/a or better Jn(n+l)/a (S. belobr).
Only the long-period variations S and Dst p enetrate deeply enough
9
to make a spherical correction of their plane-earth response
functions necessary. The Dst source field is effectively of the
degree n = 1, while the Sq source field contains spherical har-
monics up to N = 5. In fact, the surface field of the m'th Sq
subharmonic is well described by a spherical harmonic of the
degree n = m + 1 when m ranges from one to four. The ra-ti-os a/N
are therefore roughly 6000 km for Dst and 3000 to 1200 km for Sq.
These values have to be set in relation to the depth of penetration
of Dst from 600 to 1000 km and to the depth of penetration of S 9
from 300 to 500 km.
Hence, in either case there will be the need of a spherical correc-
tion, but it should be noted that it will be bigger for S because
9
the greater spatial. non-uniformity of S outweigh-ts the effect
9
of deeper penetration of Dst.
The theory of electromagnetic induction in conductors of sphecical
symmetry surrounded by non-conducting matter can be summarized
as foll.ows. Let
be the magnetic and electric field vectors of time-harmonic TE-
mode fields in spherical geocentric coordinates; r is the geo-
centric distance, X the angle of longitude and 0 the angle of
' 0
co-latitude (= 90 minus latitude) on a spherical surface. Tlte
diffusion victor from Sec. '7.3 points from external sources
r5adi.all.y inwards toward r = 0:

with
and
rot rotP= -
Pr is expressed on a surface r = const. by a series of spherical
surface harmonics. The diffusion equation V'P - = iwuouv '2 within
the v'the Shel.1 is then solvable by .i separation of variables
(K: = iwpouv):
, .
where the characteristic radial function fm satisfies the ordinary
11
second order differential equation
Its general solution are spherical Besselfunctions jn and n of
n
the first and second kind,
. .- with A~ n and B: as complex-valued constants OF in-tegration. Theym
m
decribe the in--going (B ) and out-going (A ) solu-tion, the latter
n n
being zero below the ul.timate shells, surrounding an uniform
inner core, because nn has a singularity at r = o while jn + 0
for r +- 0.
i
The characteristic scale 1.engi-h of pene-tration is defined as i.
with
m m .
cm = r fn/gn
n
d(r f:)
- -
tz: - dr

The field within the conducting sphere, r - < a, is derived from - P
as described in Sec. 7.3. Observing that the radial component of
rot rot - P reduces by the use of spherical harmonics to
the field components of the spherical harmonic of degree n and
order m are:
and
The impedance of the field at spherical sur'aces, expressed in
terms of c:, is then as in the case of plane donductors given by
For the field outside of the conducting sphere, r - > a, the solution
of the radial function is (in quasi-stationary approximation)
Hence, the surface value of the charlac-terisll-ic scale length for
r = a is found to be
The ratio of internal. -to external. parts of the magnetj.~ surfacc
fleld is by definition

yielding
The ratio of internal to external parts is therefore derived from
cm according to
n
which demonstrates the r$le of n/a, respec-tive1.y (n+l)/a, as equi- c
valent wavenumber of the source fielh in spherical coordinates
(cf. Sec. 9.1).
The spherical version of the input function for the inverse problem,
based on a linearisation according to the $-algorithm, is con-
veniently defined as
with K~ = itopO/pO. It w ill be shown that the spherical ccrrection
0
is of the order (n[~[/a)', if the conductivity is more or less
uniform.
If degree and order of the spherical. harmonic representation of
the source field are independent of frequency (this is true for;D st
but -- not for S ), the inverse problem can be solved by deriving
9
from spherical response functions a prel-iminary plane-earth model
h
This model. is subsequently transformed into a spherical. Earth
model p(r) with WEIDELT's transformation formula:
with

and
An algorithm for the direct problem for spherical conductors can
be formulated as follows. It is designed not to give the transfer
function C: for a given model itself but an auxilary transfer
function
as a direct equivalent to the tranfer function C of plane conm
ductors. To see its connection to C differentiate the characteristic
n'
depth function f: with respect to r, observing that the differen-
tiation of spherical Besselfunctions with respect to their argu-
ment u j.s
and the same for n . Then
n
Hence,
d fm
m = fm + = fm(r/?m - n)
gn n n n
with nC /r as "spherical correction ",
. . n
Continuity of the tangential components of the electric and
A
m
magne.tic field requires that Cn and thereby cm as well are conn
tinuous functions of depth. Let r
L
be the radius of the inner
core of conductivity a surrounded by (L-1) uniform shells, the
L'
-yth shell between r
and rvi-,.
(v = 1,2, ... L).
v

Then
with uL = iKL rL because the in-going solutions are zero within
m
the core. But C for r = r can be expressed also in terms of
n L
thd general solution of the (L-1) 'th shell:
*
m
with ui = iKL-l rL. AS a consequence, C for r = rL-l is now ex-
,. n
pressible in terms of c:(P~) and thus an algorithm can be estah~
~
blished for the calculation of the surface value ~z(r = a). In
1
numerical evaluation it is preferable not to derive the spherical
Besselfunctions for the arguments u and u' -themselves, but to
V v
reformulate the algorithm by expressing jn and nn in terms of
hyperbolic functions (cf. Dover Handbook of Mathematical Function,
FoI?mula 10.2.12):
n
jn(u) = i {gnCz) sinh(z) + g-n-l (z) COS~CZ))
with u = iz and the following recurrence formula for the
gnCn = 0, + 1, + 2 ... ):
- -1
Lf I z 1 >> 1, the approximations z for n = f2, + 4, ... and
-
g n
- -z2 . Cn+l).n/2 for n = + 3, + 5, ... are valid; g-l = 0.
gn
By making use of the addition theorems of hyperbo1.i~ functions
the algorithm can be given a form simular to the recurrence for-
mula of plane conductors (cf. Appendix to Sec.7.3)

with
I -
and- a = gn g-n I
1 6-n-1 En-1
The argument of g is u = i K r , the argument of g' is ul=iK r
n v v v n v v vt.1'
If luvl and [ub[ are large compared to unity, i.e. if the mesa
radius is much~larger than the skin-depth of the v'th shell., the
-1
coefficients a are of the order +(u ul) , the spherical correc-
n -4 :-I or u-l u1-2
tion coefficients E of the order uv uv
I1 V v
Consider for example the case of a uniform sphere of the radius a,
the conductivity a, K = iwu o and u = iKa. Then
0

Assuming the skin-depth to be small in comparison to a,
( I u I >> I), tanh(u) z.1 and with z = u/i = Ka
The resulting approximate spherical transfer function
has its plane-halfspace value except for a spherical correction
of the order Cn/z)'.
9.3. Non-uniform thin sheets above a layered substructure
- ---
The thin-sheet approximation has been introduced by
PRICE to geomagnetic induction problems. It applies to those
variat%on anomalies for which there is reason to believe that
they arise from l-ateral changes of resistivity close to the
surface. For instance, highly resistive basement rocks may be
covered by well conducting sediments of fairly uniform re-
sistivity ps but variable thickness d.
Two conditions have to be satisfied, if this surface cover is
to be treated as a "thin sheet" of variable conductance
.r = d/ps:
(i) Its skin-depth J2ps/wvo at the considered frequency must
exceed the maximum thickness dmax by at least a factor of 2.
(ii) The inductive scald-length c-(u,o) for the matter below the
n
surface cover, assumed to be a layered half-space, must be
large in comparison to dmax.
Under these conditions the electric field within the "-thin sheet1'
may. be. regarded as- constant for 0 .: z < d, reducing the first
field ec~uatlon to its thin-sheet approximation

with
~ = - T E
+
as sheet current density; Htg is the tangential magnetic field
above the sheet at z = 0, H- the tangential magnetic field
;;tg
below the sheet at z = d; z is the unit vector downward.
1 ... , . .
G' 3
for example a sedimentary cover with
I
exceeding 4 km i n thickness. Then condition (i) will be satis-
fied for periods up to 2 minutes and condition (ii) for deep
resi.stivities in excess of, say, 50 Slm.
The response functions for the normal field above the sheet at
z = 0 are readily found from
where T is the constant normal part of T, while C; has to be
n
derived from the given substructure resistivi-ty profile. The
field equations.to be considered for the evaluation of the va- .
. . riation anomaly are
with
rot E = impo EIa,
Z -a

The second field equation implies that the vertical magnetic
field is also to be regarded as a constant between z = 0 and
z = d.
In addition there are boundary conditions for the anolnalous field,
arising from the fact that its primary sourced lie within the
sheet.
Pilot interpretation: In many cases the anomalous field has no
inductive coupling with the substructure because its half-width
is small in comparison to (c,[. In that case no currents are in--
duced by Ha in the subs-tructure and the anomalous magnetic field
is solely due to the anomalous sheet current ja within the sheet.
For reasons of symmetry
Observe that the inductive coupling of -the normal field has not
been neglected, i.e. its internal sources lie within the sheet
as well as within the substructure. Assuming that T~ is known
and that it is possib1.e to identify the normal part of the total
electric field - E = En + Ea either by calculation 01, by observations
outside of the anomaly, the anomalous part of the conductance T., a
t
is readily found from observations of Hatg and
-a
E . In this way,
using the example from above,the variable dep-ti1 of the basement
bel-ow the sedimen.ks,
can be estimated.
If an el-ongated structure is in H-polarisation with respect to
the normal field, the magnetic variation anomaly will disappear
(cf. Sec. 7.3). The pilot investigation of the conductance is done 11~
even inore simply because the sheet current densj.ty must be a
constant in th-e direction normal to the trend:

E~~ T r: j = const. or E T + E a r = j = 0.
n~ a a
This approach has been used by HAACK to obtain a fairly reliable
conductance cross-section through the Rhinegraben.
In the case of E-pol-arisation a different kind of simplification
may be in order: Suppose tHe half-width of the anomaly is
sufficiently small in comparison to (W~~T)-' everwhere. Then no
significant local self-induction clue to HaZ which produces
takes places, i.e. the electric field driving the anomalous
current will be the large-scale induced normal field only:
Assuming E again to be known, the conductance anomaly is now
-n ll
derived from the observation of the tangential magnetic var,iation
anomaly normal to the trend,
In the actual performance with real data all field components
may be expressed in terms of their transfer functions W and Z
and thus be norrnalised with regard to h+
r1tg'
Let T be variable only in one horizontal direction, say, i.n y-
dlrec-tion'which. implies that the anomal.ous fj-eld is also variable
only in that direction, obeying the field equa-tions
CE-polarisation) (H-polarisation)
-
H+ - 1-1- = H - H.k -
ay ay ?ax ax ax - jay
with
and

Here a denotes the conducti.vity at the top of the substratum
0
and ~a~ the anomalous vertical field at z = d which is responsible
for driving currents upwards from the substructure to the sheet
and vice versa.
The boundary conditions, reflecting the "thin-sheet origin" of the
variation anomaly, are
(E-polarisation) (H-polarisation)
for z = 0: H' = -K x H H+ = 0,
a Y az ax
-
for z = d: H- = LII x Haz E = x Eaz.
ay a Y
c
The kernel function of the convolution integrals - are -n 5
H " ' ~ - K * I ~ ' ~ ~ *
1 r "7 ip H
KCy) = -
=-> "' = Q ~ Y ) - : ~ Y ~ . ~ -
"Y
hy?
q*
c(i) U-
+m sin(y k ) -7 LB% HCi; "I
1
LII(w,y) = - I -
dk
+
't H"b x
" oik Y
Ha): !
Y Cn~=(W'!C Y
7
)
I L E,
Y-,.-r-,--* . ./-:
sin(y k )
m'
1
Y
LICw,y) = - I - d k . ' LI" Ea; . 2 - . UCLX = -go.
"-0 ik ~-(w,k)
Y n1 Y {hcifl<
s2x.x:,~.c.r := ----
He-e C-
n I1 is the response function of the substructure for the
anomalous TE-field in the case of E-polarisation and C- the
nI
response function of the substructure for the anomalous TM-field
in the case of H-polarisation.
The kernel K is the separation kernel, introduced in Sec.5.1. It
i
connects K and HaZ in such a manner that the source of the
aY
anomaly is "internal" wh.cn seen from above the sheet. The kernels
L are the Fourier transforms of the response functions, intro-
.duced in Sec.7.3, which connect the tangential and horizontal
field components above a layered half-space. Their application
to the anomalous field at z = d implies that H and E diffuse
-a -a
downward into tlie substructure antidisappear for z -p a. ---
They represent
the.inductive coupling of the anomalous field with the
substructure .'

Field equations and boundary condition together constitute a set
of four linear equations. for an equal number of unlcnown field
components. They are uniquely determined in this way. A solution
toward the anomalous tangential electric field in 8-polarisation,
for instance, gives
A similar solution toward the anomalous current density in H-
polarisation gives
For a given model-distribution these equations are solved numeri-
cally, by setting up a system of linear equations for the unknown
field components at a finite number of grid points along y. The
convolution integrals, involving derivatives of the unknown field
componentswith respect to y, are preferably treated by partial
integration which in effect leads to a convolution of the unknown
field components themselves with the derivatives of the kernels.
It should be noted that the kernels L approach for y -t finite
-1
limiting values, given by' { 2 ~ i ( w ,011 .
Inverse problem for two-dimensional strnc_j;Lms
-.
1.t -is. also possib1.e to consider the anomalous conductance -ra a.s
the unknown quantity to be determined from an observed elongated
variation anomaly, in the actual calculations to be represented by a
set of respective transfer functions. The normal. sheet conductance
T and the resistivity of the substructure, for which the kernels
n
L have to be determined, enter into the calculations as free model
parameters. They can be varied to get the best agreement in ~ ~ ( y )
when uslng more than one frequency of the variation anomaly.

Another point of concern is the reality of the resulting numeri-
cal values of ra. Usually empirical data wi1.l give complex values
and the free parameters should be adjusted al.so to minimize the
imaginary part of the calculated conductance.
In E-polari.sation the elimination of H- gives
a Y
tie can then eliminate either HC and Haz, if -ra is to be determined
aY
from anomalous electric field,or we can derive ra from the anoma-
lous magnetic field by observing tha-t
by integration
. of the second field equation. The norrnal electric field is
derived from the normal magnetic field by setting
when the source field is quasi-uniform and
when the source field is non-uniform; N (w,y) is the Fourier
I1
transform of * '%. (w,k). Cf. Sec. 8.2, "Vertical soundings with
Cln11
station arrays".
In H-polarisation only the anolnalous electric Field is observable
at the surface. For the eli.mination of -the anolnalous magneti.~
field at the lower face of the sheet from the field equations
we use the generalized inlpedance boundary condition for the ano-
malous TM-field at the surface of the substructure (cf. Sec.7.3
und 8.2) :

with N (w,y) being the Fourier transform of C- (w,k 1. Insertion
I n1 Y
above gives
9.4 Non-uniform layers .above and within a layered structure
The source of the surface variation anomaly is assumed to be an
anomalous slab between z = zo and z = z + D in which the re-
0
sistivity changes in vertical &horizontal direction: p=p,+pa.
The region above and below the slab are taken to be layered,
p = pn being here a sole function of depth (cf. Sec.3).
The response functions for the normal variatLon field are defined
for the normal structure as given by p . If the anomaly lies at
n
the transj.tion between two extended normal regions, two normal
sol.utions havk to be formulated. It should be no-Led that in that
case the anomalous field w i l l not disappear bu-t converge outside
of the anomaly toward the difference of the two normal soluti.ons.
Henceforth., the normal structure , the normal response functions J
and the normal fields En and E will be assumed to be known
-n . .
throughou-t the lower conducting half-space. The source-field will
be regarded as quasi-uniform except for source fields in E-polari-

sation in the case of longated anomalies when a non-uniform
source can be permitted (cf. Sec. 9.3, calculation of E from
-nx
H 1).
nY
The principal problem in basing -the interpretation of actual
field data on this type of model consists in a proper choice
of the upper and lower bounds of the anomalous slab. The following
argumepts may be useful. for a sensible choice:
The anomalous region must be reached by the normal variation
field) i.e. z should not be made larger than the depth of pene-
0
tration of the normal field as given by I C (w,0)1 at the highest
n
frequency of an anomalous response. A lower bound for the depth
of the slab is not as readly formulated because the ano~nalous
response does not disappear necessarily when the depth of pene-
tration is much larger than z + D, i.e. when no significant
0
normal induction takes places within the anomalous slab. Only
if the anomaly is elongated and the source field in E-polarisa-
tion, can it be said that the depth zo + 1) must be at least com-
parable to the normal depth of penetration at the lowest fre-
quency of an anomalous response.
Pilot studies: In section 8.2 the general properties of the im-
pedance tensor Z above a non-layered structure have been discussed
and the following rule for elongated anomalies was establi.shec1:
The impedance for E-polarisation does not diverge markedly from
the impedance of a hypothetical. one-dimensional response for the
local resistivity-depth profile, while the impedance for H-pola-
risation will do.so unless the depth of penetration is small in
comparison to the depth of the internal resistivi.-ty anomaly and
the inductive response nearly normal anyway.
Suppose then that an impedance tensor has been obtained at a'lo-
cation y on a profile across a quasi-twodimensional anomaly which
by rotation of coordinates has zero or allnos-t zero diagonal ele-
ments and that a distinction of the offdiagonal elements for E- and
H-polarisation can be made (cf.Sec.8.2). Regarding the E-polari-
sztion response for a first approximation as quasi-normal., a local
inductive scale length

is calculated as function of frequency and location. It is
converted into an apparent CAGKIARD resistivity and phase:
or alternatively into the depth of a perfect substitute con.ductor
and a modified apparent resistivity :
which'can be combined'into a local depth versus apparent resj-sti-
x X
vity profile p (z,, , y).
11
If the magnetic variation anomaly rather than the geoelectric
field has been observed, the anomalous part of El! can be derived
by integration over the anomaly of the vertical magnetic varia-tions,
w11il.e the normal part of E,, is calculated from the normal impedance
outside of the anomaly or derived theoretically for a hypothetical
normal resistivity model:
or in terms of transfer functions with respect to H n l '
using for the transfer function of H the nota-tj.ons of page 113.
az
Magnetotelluric and geomagne-tic depth sounding data along a
profi1.e are in this way readily converted either into CAGNIARD
r~esistivity ,and phase-contours in frequency--distance coordina.i-es ,
'if
into lines of , depth-of-penetration z at a gi.ven frequency,
%
or into modified apparent-resis.tivi-ty- contours in a z -distance
cposs-section Either one of these plots will outline the frequency
range, respectively the depth range, i.11 which the source of the
anomaly can be expected to lie, and provide a rough idea about the
resistivities like1.y to occur within the anona1.ous zone,

Single frequency -- interpretation by perfect conductors - at
--
variable depth
Geomagne-tic variations anomalies show frequc.ntly nearly zero
phase with respect to the normal field, the transfer functions
which connect the cbmponents of H and fin being real func-tions
-a
of frequency and locations. This applies in particular .to two.
types of anomalies. Firstly to those which arise from a non-
uniform surf ace layer, thin enough to allow -the 'ltl~in-sl~ee~t"
approxima-tion of the previous sec-t:ion with predominant .i.nduction
within the sheet (ns>>l), Secondly, it applies to anomalies
above a highly conductive subsurface layer at variab1.e dfpth
benea-th an effectively non-conducting cover.
In the first case En will be in-phase with Hn, in the second
case out-of-phase (cf .Sec.9.1). But it is important to note that
the anomalous variation field H wi1.l be in either casc roughly
a
in-phase -. with Hn.

Both types of anomaly can be explained at a given frequency by
the undulating surface S of a perfect conductor below non-con-
ducting matter, I-ts variable depth below the surface point (x,y)
X x
will be denoted as h (x,y) . Outside of the anomaly h shall be
constant and equal to the real part of Cn at the considered fre-
quenc y .
This kind of interpretation is intended to demonstrate the effect
of lateral changes of in-ternal resistivity on the depth of penetra-
tion as a function of frequency and location, it does not provide,
however, quantitative information about -the resistivities involved
nor does it allow a distinction of the two types of anomalies
mentioned above.
Clearly, the magnetic fie1.d below S must be zero and the magnetic
field vector on S tangential with respect to S because the con-
nor a1
tinui-ty condition for the field conponent toif'?? requires that this
component vanishes just above S.
~irect model problem: For a given shape of S the anoinalous surface
field can be found with the methods of potential field theory,
since Ma will be irrotational and of internal origin above S. If
~ ~
in -particular S has a simple shape independen-t of x, the field
%
' lines for E-polarisation in the (y,z)-plane for z - < h (y) can be
found by conformal mapping as follows:
Let w(y,z) = y(yl,zl) + i.z(yl,zl) be an analytic function whi.ch
maps the line zl=O of rectangular (yl ,zl) coordinates into the line
* z=l> fy) of rectangular (y,z) coordinates. Lines z1 = const, are in-
terpreted as magnetic field lines of a uniform field above a perfect
conduc-tor at constan-t depth, their image in the (y,z)-plane as field
lines of adistorted field above a perfect conductor at the variable

x:
depth h (y), the image of the ultiina.'ce field line zf=O being ta.ngen-
tial to the surface of the conductor as required.
If gn= (Hlly '0) denotes the uniform horizontal field vector at a
point in the original (yl,z') coordinates, the components of the
field vector - H= (H ,Hz) at the image point in the (y,z) coordinates
Y
can be shown to be given by
with
The difference - H - H represents
-n
the anomalous field to be
' Inverse problem: The shape of the surface S can be found inversely
from a given surface anomaly by constructing the internal field
lines of -the field gn+ga. Field lines which have at some distance of
the anpmaly the required normal depth R ~ ( C ) define the surface S,
n
provided of course that the surface thus found does no-t inS:ersec3<
the Earth's surface anywhere.
The ac-tual calculation of internal field lines requires 2 downward
ex-tension of the anomaly through the non-conducting matter above
the perfec-t conductor, using the well developed methods of potential
field continuation towards its sources, In order to obtain sufficien:
stability of thenumericaz process, the anomaly has to be low-pass
filtered prior to the downward con'tinuation with a cut-off at a
reciprocal spa-tial wave number comparable to the maximum depth of
intended downward extrapolation.

A field line segment with the horizontal increments dx and dy has
then -the vertical increment
H H
a z a z
az
dz =-dx +-dy =
ax 3~ +H H +H Q2
nx ax nY aY
and the entire field line is the,,constructed by numerical integra:
tion;
An analy-tically solvable inverse problem: Suppose the observed
anomaly' is 2-dimensional and sinusoi.da1,
H (y,z=O) = c sin(k.y) H .
aY nY.
Because of the internal origin of the anomaly
the internal field increasing exponential1.y with depth,
Hence, the slope of field lines at the depth z i s
by integration
z(y) = z - c/k.sin(k-y1.e k.z(y)
0
as the implicit solution with zo as t5e constant of integration.
Here it w i l l be chosen in such a way that the highest point of the
field line is still below the Earth's surface z=0.
We infer from the fi.eLd line equation z(y) that field lines above
sinusoida!. anomalies oscillate non-symmetrically around the depth -1.
z=z with Lj=27i/k as spatial wave length. Their deepest points z=z
0
are at y=L/4,-3L/4,+. ., . ..
y= -L/4,3L/4 ,T.. ...
. .
. . . . , their highest points 2.2- at
u,z
'jar 7, , :f-, , . / ,- - 0 -3
x
- L ....-a' \ 1 f .- ,
/ ,
-r/ ..-
,/' 7 -k> '../%,./'
\.- A--
s u, j., , 0.- , rr .L
1
Z
Y
- i . . '. \
1
Y 1
F: ,.C'C p,
-3

Suppose the spatial wave length of the anomaly is large in'com-
-1.
parison to the deepest point of penetration z . Then the approxi-
mate field line equation
can be used wi-th the requirement tha-t zo is equal to or larger
than c/k. The mean amplitude of the field line oscillations is
readily expressed now in terrns of. the amplitude of the observed
surface anomaly as given by the factor c, namely
If, for example, a moderate anomaly has the ampli.tudd cr0.25 and
a wave length of 628 km, the mean amplitude of the field line'
oscil.lations are given by (zo+'100)/4km, zo? 25 km. Using z =30 km
0
gives a mean ainplitude of 33 km and'the field line oscill.ates
between 4 km and 70 km, This numerical example demonstrates that
even minor ano~nalies require rather large undula-tions in the depth
of a perfect substitute conductor.
'Multi-frequency - - interpretation buocal. ir~duc$ion in isolated
bodies
Most geomagnetic induction anomalies arise from a local rearrange-
ment of large-scale induced currents, In some cases, however, it
may be justified to assume that the source of the anomaly is a
conducting body which is isolated by non-conducting mat-ter from
the large-scale curcent systems. Such a body can be, fop instance,
a crustal lense of high conductivity within the normal1.y highly
resistive crust.
The direc-t - problem: Consider a body of (variable) conductivity
between z=zl and z= 2. ' The sui7roundi.ng matter. at the same depth is
effectively non-conducting at the considered frequency, i,e, large-
scale induced curren-ts which flow in the normal structure above and

- 160 -
below can neither enter nor leave this body.
The inducing field is the normal field HA between zl and z 2 3
regarded as uniform. For simplicity it is assumed that the nor-
mal structure between z=0 and z represents a "thin sheet" of the
1
conductance T ~ . Then the attenuation of H1 with respect to the
n
normal field H at the surface zz0 is given by
n
as thin-sheet induction parameter (Sect.9'3 . . and Appendix to 7.3).
Three aspects of the arising inductiorl probl.em have -to be con-
sidered: (i) the local induction by H' within the body, (ii) the
n
elec-tromagnetic coupling of this body with the normal subst~uc-Lure
below z (iii) the coupling of the body with the surface ccveri;bove
2'
In order to make problem (i) solvable in a straightforrgard manner
uniform bodies of simple shape such as spheres and horizon1:al.
cylinders of infinite length have -to be selected, The anomalous
field due to -the induc-tion in these bodies is of fr,equ.ency.-indepen-
dent geometry. This fact permits si.lnple solutions of The inverse
prohl.em as seen below.
Consider the case of a horizontal cylinder of infi,nite length, the
radius R and -the conductivity u its axis is parallel to x and
z;
intersects the (y, z)-plane at the point (0, zo). The normal horizon-
tal field is. parallel to y and produces hy i.nduction curren-ts which
flow parall.el to x in .the upper halve and anti-parallel to x in "ilie
lower halve of the cylinder. Their magnetic field outside the cy-
linder is -the field of a central 2-dimenstonal di.pole of the monient

m antiparallel to y. It has the components
with r2 = Y2 + (Z~-.Z)~ and cos0 = zs0/r, sin8 = y/r,
The dependence of the dipole moment on the inducing field will
be expressed in terms of a response function, Let mm be the
dip01 moment of a perfectly conducting cylinder which shields
the external fieldwmpletely frorn its interior. Hencc, the in-
duced field at the point (0, z -R) just above the cylinder
0
must be equal to H' i.e.
n'
which implies that it cancels H1 within the cgl.inder as required.
n
The response function of the cylinder with respect to a uniform
inducing field perpendicular to the axis is now defined as the
complex-valued ratio
which can be shown to be a sole function of the dimensionl.ess
induct ion parameter
-.
. ?he modulus of f(il ) will approa.ch zero for nZ >.I, its argument changing from YO for q7 -t- 0 to 0' for
"z
-t -. Hence, at sufficien-tly high frequencies the anomalous
field from the cylinder w i l l be in-phase with H' and not in-
11
crease beyond a certaiii "induc-tive limit", whj.le at sufficiently
low frequencies the anomal.ous field w i l l be out-of-phase and
small in comparison to the field a-i: the i.nductive 1.irnit.
..I

I '. -
L.. . . - I . --.> . ~
4
The transfer function between the anomal.ous and normal field
are now readily expressed in terms of a frequency-dependent
. response function, a frequency-independmt geometric factor,
and a frequency-dependent attenua-tion factor Q' for the normal
field:
wi-th
Q' = (1 4. i nZ)-' ,
An approximate so1.ution of problem (ii) can be obtained by re-
presenting the conductive substructure below the surface sheet
by a perfect conductor at the (frequency-dependent) depth K~(C;)
and by adding to the anomalous dipole field the field of an
image dipo1.e of the moment R ~H;~ at the depth 2sRe(~i)-z~.

Problem (iii) j.nvol.ves mainly the attenuation of the upward
diffusj.ng anomaly ca by ' uniform surface layers. This a.tte.iiuatri.on
is, however, a second order effect and can be neglected, if
at the considered frequency the widespread normal fieid pene-
trates the surface layers to any extent, i.e. if the modulus
of Q' is sufficiently close to unity.
The inverse problem: Assuming the anomalous body to be a uniform
cylinder, the model parameters zo,'R, cr can be uniquely in-
R
ferred from surface observations, providing the transfer func-
tions for H and FIaz are known for at least two locations and
aY
frequencies. For.ming here the ra-tios
the angles 8 and thereby the position- of the cylinder can be
found, The depth of the cylinder axis being kncwn; the dipole
moment RZf(n) can be calculated. Its argument fixes the size
of the induction parameter n and two determinations of nZ at z
different frequencies the radius R and -the conductivi-ty crz. NOW
R and zo being known, the field of the image dipole can be cal--
culated. The inverse procedure is repeated now with the observed
surface anomaly minus the field of the image dipole until con-.
vergence of the model parameters has been reached.
Interpretation with HIOT--SAVART1s law
. . ~
Let - f he the current density vector within a volume dV at the
Q n n h
point - r = (x, y, z). Then according to UZUT-SAVART1s law in SI
units -the magnetic fieled vector at: the point - r = (x, y , z) due
c'
to the line-current element idV is

This law can be used to i-nterpret an observed surface anomaly
of geomagnetic variations in terms of a subsurface distribution
of anomalous induction currents, which in turn are related to
an internal conductivity anomaly a i,e. to local deviations
a'
of a from a normal. layered distribution o (z),
n
It has been pointed out in Sec.8.3 that the anomalous part of
the geomagnetic and geoelectric variation field can be split
into TE and TM modes (= tangential electric and tangential
magnetic modes) and that only -the TE modes produce an observable
magnetic variation anomaly at the Earth's surface. Consequently,
3
only a TE anomalous current density .distribution c~~~ can be
related to the anomalous magnetic surface field Ha, the sub-
script "11" refering to the solutioll I1 of -the diffusion
equation.
Denoting the tangential components of simply with i and
c
-a11 ax
J. the components of the anomalous surface field are given
ay'
A 0
with r2 = (x-x)' + (Y-~ + G2. The inverse problem, namely to
find from a given surface anomaly the internal anomalous current
distribution, has no unique solution. Within certain constrai.nts,
however, it can be made unique, at least in principle. For in-
stance, it is assumed that the anomalous current flow is limited
to ;I certain depth range which implies that the matter between
this depth range and the surface is regarded as non-conducting
at the considered frequency.
The surface anomaly can be extended no:,? dowr~trard t o the top of
.the anomalous depth range with standard metliods which require
A

only adequate smoothing of the observed anomal.ous surface field.
Then a certain well defined depth dep'endence of the anomalous
current density is adopted as discussed below and the anomalous
current distribution can be derived in a straightforward manner.
Suppose an observed anomaly 51, at a given frequcncy has been ex-
plained in this way by a distribu-tion of anomalous internal.
currents. Its connection to the internal conductivity is esta-
blished by the normal and anomalous electric field vector accor-
ding to
Assuming the normal conductivity distribution un(z) to be known,
E as a function of depth 'is readily calculated. There is no
-I1
simple way, however, to derive the anomalous electric field of
the TE mode except by numerical models as discussed below. There
is in particular no justification to regard it as small. i.n com-
parison to the normal electric field and thus to drop the second
term in the above relation. Instead the following argumentati.on
has to be used:
The anomalous electric field in the TE mode can be thought to
contain two distinct components. The first component may be re-
garded as the result of local self-induction due to EaII. I -.
can be neglected a-t suffici.ently low frequencies, when the ;-lzlf-
width of the anomaly is small in comparison to the minimum skin-
depth value within the anomalous zone.
The second component arises from electric charges at boundari-es
and in zones of gradually changing conductivi-by. These charges
produce a quasi-static electric field norroal to bounclarics and
paral7.el -to in-ternal conduc-tivity gradients which ensures the
continuity of the current across boundaries and internal gradient
zones. Hence, this second component of the anomalous electric
field does not disappear, when the frequency becomes small.
It vanishes, however, if anomalous internal currents do not cross
boundaries of gradient zones, i.e. when the normal field is in
E-polarisa-tion with respect to the -tl>end of elongated s'iructures.

Only in this special case will be jus.tificati.on .to neglect the
term oEa at low fr2quenaies and to use the approximation
.to calculate the internal conducti.vj.ty anomaly o from the ano-
a
malous current density and the normal electric field, St w i l l
be advisable to regard here rsa as a sole functi.on of x and y
and thus to postulate the same depth dependence for the anomalous
current and the normal electric field.
Interpretation with numerical models
So far idealized models for a laterally non-uniform subs-tructure
have been considered. They allowed a simplified treatment of
the induction problem and had in common that only few free para-
meters were involved. Once numerical values were attached to
them, the observed surface anomaly of the variation field could
,be converted rather easily into variable model parameters to
characterize the internal change of subsurface conductivity at
some given depth from site to site.
The introduction of more realistic models le'ads to a substantial
increase of the numerical work involved and should be considered
only, if transfer functions for the anomalous surface field are
known with high accuracy. The inverse problem to derive a set
of variable model parameters directly from observations can be
solved by linearisation, i.e. by the introduction of a linear
data kernel, connecting changes of the surface response to
changes in the subsurface conductivity structure (cf. Sec.6).
As before a normal structure o outside of the anomaly is given,
n
the variable model parameters representing only lateral changes
of o with respect to on within the range of the anomaly. Unless

stated otherwise, source fields of lateral homogeneity are
assumed, yielding in conjunction with u the normal fields Hn
n
and Ell as known functions of depth.
Conductivity anomalies aa = o - a are restricted to an anoma-
n
lous slab above or within a laterally uniform structure, exten-
ding in depth from zl to z2, Within This slab two basic types
of conductivity anomalies may be distinguished:
(i) slabs with gradually changing conductivities in horizontal
direction or (ii).slabs which consist of uniform blocs or re-
gions, separated by plane or curved boundaries.
In the first case electric changes are distributed within the
'?
slab, -yielding closed current circuits divl = 0 as required in
quasi-stationary approximation. Consequently, divE - = -(E*grado)/o
will not be zero. In the second case only bloc boundaries carry
free surface changes and the electric field will be non-diqergent
within the blocs. In either case a solution of the diffusion
equation for
-a
H or
-a
E has to be found under the condition that
the anomalous field approaches zero with increasing distance
from the anomalous slab.
Numerical methods for the actual solution of the direct - problem,
when oa is given and Ha or Ea are to be found, have been discussed
in Sec. 2 and 3. Here some additional comments: Models
should be set up in such a way that either the shape of the anomalous
body or the anomalous conductivity is the variable to
be described by a set of model. parameters. After fixi-ng the free
parameters, in particular z and z2, the model parameters for
1
the description of the anomaly are varied un-ti1 agreement is
reached between the observed and calculated transfer functions.
Suppose that models of type (ii) are used, that only vertical
and horizontal boundaries are permitted, and that the anomalous
slab is subdivided by equally spaced horizontal boundaries into
layers. Then the variable model parameters are either the po-
sitions of verti.ca1 boundaries, encl.osing a body of constant

anomalous conductivity oat OP the anomalous conductLvity values
'am
of the blocs m = 1,2,-. .. M, separated by equally spaced
vertical boundaries:
If the search for a best fitting se-t of model parameters is to
be done by an iterative inverse rnethod rather than by trial-and-
error,%inear data kernel matrix G =(g has to be found for
nm
an initial model. In this matrix the element gnm represents the
change yn of the anomalous transfer function at a certain su~face
point and frequency Rn which arises in linear approximation
from a change xm of the model parameter in the m'th bloc of the
starting model. The resulting system of linear equations
- x is solved with the methods of Sec.6. Its solution
Yn
-
m
6nm m
represents the improvement of the initial model, if yn is the
misfit between observed and calculated transfer functions for
the initial model: The process is repeated with the improved
models until the necessary improvements xm are small enough to
justify a linear approximation.
Which complica-tions arise in the general case that not only the
internal conductivity structure but also the inducing source
field are 1aterall.y non-uniform? First of all, in addition to on
as a function of depth the magnetic field H (e) of the external
source must be a known fu11'cti.on of surface location. In the
special case of the equatorial jet field this source field con-
fi.guration w i l l be more or less the same for all day-time varj.a-';iofis
and thus a normalisation.of the observed surface field by the
....

- i
- - i
field at sollie dis-ti~lguishecl surrace point wil.1 be possible, Tor
instance by the horizontal fie1.d at the dip equator,
The conversion formulas of Sec. 5.2 and the methods described
in Sec. 8.2 ("Vertical soundings with station arrays") are now
used to find the inrernal part M (i) of'the normal magnetic field
n
.and the normal electric field En as functions of the surface
loca-ti.on and frequen~y. Finally, the normal surface field is ex-
Ly c oucl.~4wn
tended downwardVwith the spatial Fourier transforms of the down-
ward extension factors as given in the Appendix to Sec. 7.3.
Now the normal field within the anomalous slab is known and the
diffusion equation for the anomalous field can be solved numeri-
cally in the same way as before, when the slab or the anomalous
region themsel.ves are chosen as basic domains for the numerical
calculations. Here a final. schematic summary of the various steps
of the calculations, when the source is the combined field of the
equatorial jet and the low-latitude Sq:
- Interpretation with scale inodel experiments
. . . . . . . . - . . . . - -
Numerical model calculations may be too elaborate, if the induc-
tion by a non-uniform source in a three-din!ensio:lal anomalous
structure has to be considered. An example is the coast effect
of the equatorial jet, when the dip equator crosses the coast-
line under an angle. - No co~nplications arise at those places,
when the dip equator is parallel to xhe coast and the jet field
in E-polarisation (s. above). -
In si-tuation of great complexity the qualitative and even quan-
titative unders-tanding of surface observa-tions may be fur.thcred
by labora-tory scale model experi.ments, sj.niu1.ating the natural
induction process on a reduced scale. Invariance of .the elcc'tro-

the product WUL' constant with L denoting the length scale,
The primary inducing field is produced by an oscilla-ting dipole
source or-by extended current loops, situated as "ionospheric
sources" above an arrangeme& of conductors which represent
the conducting material below the Earth's surface. Alternatively,
the conductors as a whole can be placed into the interior of
coils, say Helmholtz coils, and thus be exposed to a uniform
source field,
.-
\
I
/ I .
1 i:
/'
.

width, however, forces the induced currents to flow in loops
which in strength and phase may be totally controlled by the
edges of the conductor. In order to avoid this unwanted effect
the source field at the level of the conductors should die away
before reaching the edges of - the scale model. For instance, if
a line current source is used, the half-width of the line current
field on the surface of the scale model' should be considerably
smaller then the width of the model..
These complicati.ons do not arise of course, when no attempt is
made to simulate local anomal-ies of a large-scale induced current
system, i.e. when the scale model is placed into the interior
of coils in order to simulate local induction in isolated bodies
(cf. subsection on this topLc in Sec. 9.4).
Here are to men-tion the scale model experiments by GREWET and
LAUNAY who showed how ,a large-scale induction can be simvlated
also by the induction in the interior of coils. Their objective
was to make a scale model of the coast effect at complicated
coastlines. They nptedthat the i.nductive coupling between the
ocean and highly conduc-ting ma-terial at some depth within the
Earth is well represented by a system of image currents at the
level 2 -hx below the ocean. Here h" is again the depth of a
perfect substi-tute conductor for the oceanic substructure at the
considered frequency.
GRENCT and LAUNAY use as model conductors two thin metallic plates
which are connected along two edges by vertical conducting strips.

One of the thin plates represents the ocean and one of its
edges is given the shape .of a certain coastline .to be studied.
The second plate represents the level of the image currents.
The whole arrangement of conductors is placed into a Helmholtz
coil in such a way that the vertical strips are parallel to
. the. magnetic field. Plates and st~-.ips now form a loop normal
to the magnetic flux within the ~elmhoytz coil and thus currents
are induced which flow in the "oceanic" plate parallel to the
"coastline".
In SPITTA's arrangement for th.e study of the coast effect the con-
ductors are placed below a horizontal band-current closed by
a large vertical loop. The oceanic and continental substructure
is represented by a thick metalic plate, the oceans by a thin
metalic sheet which partially covers the plate. The thickness
of the plate is large in comparison to the skin depth and its
width about twice the half-width of the field of the band-
current at the level of the plate. The induced current systems
form closed loops within plate and sheet and can be assumed
to be largely horizontal. By placing one edge of the covering
sheet below the 'center of the band-curren-t the coast effect of
an ionospheric jet can be studied for any angle between coast
and jet.
1,

The ratio of the length-scales model: nature should be in the
6 7
order 1 : 10 or 1 : 10 . In this way current loops of some
1000 km in nature can be reproduced. In SPITTA's' experiment the
ratio of length-scales is 1 : 4.10~. A 4 km thick ocean is re-
presented by an aluminium sheet of 1 rrn thickness, a highly con-
ducting layer in the mantle at 360 km depth'by an aluminum plate
9 cm below the sheet. The width of this plate is 2 m and is
equivalent to 8000 km in nature.
Model conductors niay be chosen from the following materials:
cu - AL - ~b 7 .- 0.5.107 (am)-'
Graphite 3.10~ ' 11
(saturated)
NaCl solution
(concentration of maximum
conductivity )
In SPITTA's experiment the conductivity-ratio model: nature is
7
2.10 : 4. Hence; with a ratio of length scales of 1/4 a
frequency of 1 kHz in the model corresponds to 1/32.10-~ Hz 'lcph
in nature.
DOSSO uses graphite to represent the oceans and highly conduc-
ting material in the deeper mantle, saturated NaC1-solution to
represent the continental surface layers and the poorly conduc-
ting portions of crust and uppermost mantle. Since his model
frequencies are only slightly higher (1 to 60 kHz),.a one orbder
5
of magnitude pester ratio of length-scales (1:lO 1 has to be
used to simulate natural frequencies between 1 cph and 1 cpm.
,
The lists of available model conductors shows that it is diffi-
cult to simulate. conductivity contrasts of 1:10 or 1:100 which
are of particular importance in the natural induction process.
Only salt solutions of variable concentration could provide a
sufficient range in xodel conductivity, but their relatively

low conductivity requires the use of very high frequencies
3
(10 kHz) and extremely large model dimensions (10 m).
In conclusion it should be pointed out that even those scale
model experiments which do not reproduce the natural induction
in a strict quantitative sense may be useful for a descriptive -
interpretation of complicated variation fields. In those cases
only the impedance or the relative changes of the magnetic
field with respect to the field at one distinguished point above
the model will be considered and compared with actual data.
10. Geophysical and geological relevance of geomagnetic
induction studies
In exploration geophysics the magnetotelluric method,
in combination with geomagnetic depth sounding, has been applied
with some success to investigate the conductivity structure of
sedimentary basins. Electromagnetic soundings with artificial
sources as well as DC soundings which truly penetrate through
a sedimentary cover of even moderate thickness are difficult
to conduct on a routine basis. Hence, it seems that electro-
magnetic soundings with natural fields are more efficient than
-any other geoelectric methods in exploring the overall distri-
bution of conductivity in deep basins.
In particular the integrated conductivity T of sediments above
a crystalline basement is well defined by the inductive surface
response to natural EM fields and can be mapped by a survey
with magnetic and geoelectric recording stations. If in addi-
tion some estimate about the mean conductivity of the sediments
can be made from high frequency soundings, the depth of the
crystalline follows directly from T.
If structural details of sedimentary basins are the main interest
of the exploration, a mapping of the electric field only accor-
ding to st'rength and direction for a given polarisation of the
regional horizontal magnetic field will be adequate. The inter-

pretation is handled like a direct current problem in a. thin
conducting plate of variable conductivity. This so-called
"telluric method" represents a very simple lcind of inductive
soundings, but the preferential direction of the superficial
currents thus found usually gives a surprisingly clear im-
pression about the trend of structural elements like grabens,
anticlines etc. The usefulness of this method arises from the
fact that these structural elements can be detected even when
they are buried beneath an undisturbed cover of younger sedi-
ments.
Geomagnetic and magnetotelluric soundings are less useful for
exploration in areas of high surface resistivity, in particular
in crystalline regions. Even pulsations penetrate here too
deeply to yield enough resolution in the shallow depth range of
interest for mining. Audio-frequency soundings with artificial
or even natural sources will be better adapted and are widely
used in mineral exploration.
The probing of deeper parts of crust and mantle with natural
electromagnetic fields will eventually lead to a detailed
knowledge of the internal conductivity distribution down to
about 1000 km depth. Its relation to the downward rise in
temperature is obvious, in fact electromagnetic soundings pro-
vide the only, even though indirect method to derive present-
day temperatures in the upper mantle.

Other derivable properties of mantle material like density and
elastic parameters are.largely determined by the downward in-
crease in pressure with only a second-order dependence on
temperature, which i.s hardly of any use for estimates of mantle
temperatures.
The remarkable rise of conductivity between 600 and 800 krn depth,
however, should not be understood as temperature-related but
as indication for a gradual phase change of the mantle minerals,
possibly with a minor change in chemical composition, this is
with a slight increase of the fe:Mg ratio of the olivinespinell
mineral assembly.
The conductivity beneath the continental upper mantle from 100 km
down to about 600 km is suprisingly uniform and seems to indicate
-that in this -depth range the temperature gradient cannot
be far away from its adiabatic value of roughly 0.5' C/km. The
.mysterious appearance of highly conducting layers in the uppermost
mantle may be connected to magma chambers of partially
molten material and to regional mantle zones of higher than
normal temperatures in general. The expected correlation of high
mantle conductivity, high terrestrial heatflow and magmatic
activity clearly exists in the Rocky Mountains'of North America.
There are also some indication for high conductivities beneath
local thermal areas. The Geysers in California, Owens Valley in
Nevada, possibly Yellowstone and the Hungarian plains are
examples. It should be pointed out, however, that there exist
also regions of.high subcrustal conduc-tivity with absolutely no
correlation to high heatflow or recent magmatic activity. The
most proainent.inland anoma1.y of geomagnetic variations, which
has been found so far, namely the Great Plains or Black Hills
anomaly in North herica,laclcs still any reasonable explanation
,or correlation to other geophysical observations.
0;le . great unsolved problem in geomagnetic inductions
studies is the depth of penetration of slow variations into the
mantle below ocean basins. There are definite reasons to believe

that the upper mantle beneath oceans is hotter than the mantle
beneath continents down to a depth of a few hundred kilometers.
If this is so, a corresphndingly higher conductivity should exist
beneath oceans which could be recognised from a reduced depth
of penetration in comparison to continents. Once a characteristic
conductivity difference between an oceanic and a. continental
substructure has been established, this could be used tb recog-
nize former oceanic mantle material beneath present-day con-
tinents and vice versa.
First houndings with recording instruments at the bottom of the
sea have been carrred out. They confirmed to some extent the
expectation of h'igh conductivities at extremely shallow depth,
but these punctual soundings may not be representative for the
oceans as a whole. Here the development of new experimental
techniques for expedient seafloor operations of magnetic and
geolectric instruments has. to be awaited.
~~sarvatioAs,on. . mid-oceanic islands provide a less expensive
way to study the induction in the oceans which in he case of
substorms and Sq is strongly coupled to the crustal and subthe
ocea s
crustal conductivities beneath . Buf again oceanic islands
are usually volcanic and their substructure may differ from that
of ordinary parts of ocean basins.
The island-effect itself is no obstacle fo$ soundings into the
deep structure. In fact, this effect represents a powerful tool
to investigate the inductive response in the surrounding open
'ocean, since the.theoretica1 distortion of the varia-tion fields
due to the islands can be regarded at sufficiently Low frequency
as a direct current problem for a given pattern of oceanic in-
duction currents at some distance from the island. Setting the
. ..
.density of these currents in relation to the observed magnetic
field on the island gives the impedance of the variation field
, for the surrounding oceari of known integrated conductivity. To a
first approximation induced currents in the ocean do not contribute
to the horizontal magnetic field on the island. Hence, by
knowing their density the horizontal magnetic field on the sea
s
I
i
I:
floor can be calculated from which the inductive response function
for the oceanic substructure follows.
j
!

11. References for general reading
Electroinagnetic induction in the earth and planets. Special
issue of Physics of the Earth and Planet. Interiors, L,
Xo.3, 227-400, 1973.
BERDICHEVSKII, M.N., (Electrical sounding w2th the method of
-magneto-telluric profiling), Nedra Press, Goscow 1968.
FRAZER, M.C.: Geomagnetic depth sounding with arrays of magneto-
meters. Rev. Geophys. Space Phys. -> 12 401-420, 1974.
JANKOGJSKI, J., Techniques and results of magpi~etotelluric and geo-
magnetic deep soundings, Publications OQ? the Institute of
Geophysics, Polish Academy of Sciences, vo1.51i, Warsaw 1972.
KELLER, G.V. and FRISCHKNECHT, F.C.: Electrical methods in Geophy-
sical prospecting. Int. Ser. Monogr. in Electromagnetic Waves,
10, 1-517, 1966.
-
PORATH, H. and DZIEWONSKI, A,: Crustal resistivity anomalies from
geomagnetic depth-sounding studies. Rev. Geophys. Space Phys.
9, 891-915, 1971.
-
RIKITAKE , T. : Electromagnetism and the '~arth 's interior. Develcprnents
in Solid Earth Geophys. - 2, 1-308, 1976.
SCHMUCKER, U.: Anomalies of geomagnetic varizitions in the southwestern
United States. Bull. Scripps Imst .. Oceanogr. -, 13
1-165, 1970. z .
WARD, S.H.: Electromagnetic theory far geophysical application. In:
D.A. Hansen et al. (Eds.), Mining Geophysics, The Sop. of
Expl. Geophysicists, Tulsa, Oklahoma, T.967.
WARD, S.H., PEEPLES, W.J., and RAN, J.: Analysis of geoelectromagne-
tic data. -Methods in Comnii+a+inn;ll --?h\rx