ON THE
THE RELATION
RELATION
BETWEEN
TELLURIC
CURRENTS
ON
BETWEEN
TELLURIC
CURRENTS
AND THE
THE EARTH'S
EARTH’S
MAGNETIC
FIELD*
AND
MAGNETIC
FIELD*
JAMES R.
R. WAITt
WAITt
JAMES
ABSTRACT
ABSTRACT
The validity
validity of
of Cagniard's
Cagniard’s analysis
analysis of
of the
the behaviour
behaviour of
of telluric
telluric earth
earth currents
currents is
isquestioned
questioned in
in view
view
The
of the
the fact
fact that
that the
the harmonic
harmonic components
components of
of the
the electric
electric field
field and
and the
the magnetic
magnetic field
field tangential
tangential to
to the
the
of
ground are
are only
only proportional
proportional to
to one
one another
another ifif the
the fields
fields are
are sufficiently
sufficiently slowly
slowly varying
varying over
over the
the sursurground
face of
of the
the ground.
ground. His
His result
result is
is extended
extended to
to include
include the
the effects
effects of
of aa layered
layered ground
ground with
with both
both conconface
ductivity and
and susceptibility
susceptibility variations.
variations. Finally
Finally the
the corresponding
corresponding transient
transient problem
problem is
is solved
solved for
for aa
ductivity
two-layer horizontally
horizontally stratified
stratified earth.
earth.
two-layer
In aa recent
recent interesting
interesting and
and important
important paper
paper by
by Cagniard
Cagniard (1953),
(19p,),
the relation
relation
In
the
between telluric
telluric earth
earth currents
currents and
and variation
variation of
of the
the earth's
earth’s magnetic
magnetic field
field is
is
between
studied. In
In the
the present
present paper
paper the
the validity
validity of
of his
his results
results is
is discussed
discussed and
and several
several
studied.
extensions to
to the
the theory
theory are
are pointed
pointed out.
out.
extensions
The impedance
impedance concept
concept in
in electromagnetism,
electromagnetism, as
as first
first disclosed
disclosed so
so lucidly
lucidly by
by
The
Schelkunoff (1938),
(1938), is
is well
well suited
suited to
to this
this type
type of
of problem.
problem. If
If the
the electric
electric and
and magmagSchelkunoff
netic forces
forces tangential
tangential to
to aa surface
surface SS at
at aa point
point PPare
mutually perpendicular,
perpendicular, the
the
netic
are mutually
ratio of
of the
the tangential
tangential electric
electric force
force to
to the
the tangential
tangential magnetic
magnetic force
force is
is termed
termed
ratio
the field
field impedance
impedance normal
normal to
to SS at
at P.
P. In
In general
general the
the tangential
tangential electric
electric and
and magmagthe
netic forces
forces are
are not
not perpendicular
perpendicular so
so that
that the
the field
field impedance
impedance does
does not
not exist;
exist; even
even
netic
doesexist,
exist, it
it may
may depend
depend upon
upon the
the system
system of
of sources
sourcesby
by which
which the
the fields
fields have
have
ifif itit does
been set
set up
up (Monteath,
(Monteath, 1951).
1951).
been
Let the
the x,
X, yy plane
plane be
be the
the boundary
boundary between
between two
two semi-infinite
semi-infinite homogeneous
homogeneous
Let
media,
the
fields
being
set
up
by
sources
in
the
upper
medium.
It
can
be shown
shown
media, the fields being set up by sources in the upper medium. It can be
that
in
this
plane
the
tangential
electric
fields
E,
and
E,
are
related
to
the
tangenthat in this plane the tangential electric fields Ex and Ey are related to the tangential magnetic
magnetic fields
fields by
by
tial
I
a2H,
-+2-
+
terms in y4, etc.
+
terms in y+, etc.
dxay >
and
a2H,
-++----
ayax>
where, 'Yy == irTJ.Lw
iupw-- fJ.LW2
tpw2is
is the
the intrinsic
intrinsic propagation
propagation constant
constant of
of the
the lower
lower medium
medium
where,
and 'r]=iJ.Lw/'Y
7=&w/r
its intrinsic
intrinsic characteristic
characteristic impedance.
impedance. The
The constants
constants rT,
u, f,E, J.Lp
and
isis its
are the
the conductivity,
conductivity, dielectric
dielectric constant,
constant, and
and permeability
permeability of
of the
the lower
lower medium.
medium.
are
Equation (I)
(I) was
was obtained
obtained by
by regarding
regarding the
the distribution
distribution of
of Hx
H, and
and Hy
Hgover
over
Equation
Manuscript received
received by
by the
the Editor
Editor October
October 29,
so, 1953.
1953.
** Manuscript
Radio Physics
Physics Laboratory,
Laboratory, Defence
Defence Research
Research Board,
Board, Ottawa,
Ottawa, Canada.
Canada.
tt Radio
281
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JAMES
JAMES R. WAIT
WAIT
282
the x, y plane
angular spectrum
plane as
as an aperture
aperture distribution
distribution giving
giving rise to an angular
spectrum of
plane waves (Monteath,
sufficiently slowly
slowly over the
(Monteath, 1951). If
If Hx
H, and Hy
ZZ, vary
vary sufficiently
x, y plane, or if 'I'
term in each of the formulae
formulae
X,
y is
is sufficiently
sufficiently large, only
only the first term
sec)
need be retained.
retained. For example, if the frequency
frequency is 0.1 cycle (a period of 10
IO set)
and the conductivity
= 35 kilometers.
kilometers.
/ 1/
I/T'1'11 =
conductivity of the ground is
is 1010-i1mho/meter
mho/meter then 1
Then
greatly between
between points
points 35 km
km
Then for this case
case if Hx
H, and Hy
H, do not change greatly
apart,
equation (I)
(I) becomes:
becomes:
apart, the terms in '1'-2,
ym2,etc. may
may be neglected and equation
Ex
E
~
T/H
=
qH,y
E,
= -
1:
(2)
qHz.
Thus
inThus the field impedance
impedance normal
normal to the boundary
boundary exists and is equal to the intrinsic
than about
about 10
sectrinsic impedance
impedance of the lower medium.
medium. For
For periods longer than
IO seconds
amount in the distance
onds it is
is probable
probable that
that Hx
H, and Hy
H, vary
vary by an appreciable
appreciable amount
corresponding to 1
1'-11 since it is believed
sources of the variations
variations of the
1'y-l)
believed that
that the sources
earth's
kilometres as
as evidenced
earth’s magnetic
magnetic field are at heights of the order of 100
IOO kilometres
from rocket measurements
Martyn, 1952). This
This
measurements (Mitra,
(Mitra, 1952, p. 574; Baker
Baker and Martyn,
would vary
activity.
vary considerably
considerably during
during periods of magnetic
magnetic activity.
Equation
medium (air)
(air) give rise to plane
plane
Equation (2) is
is exact if the sources
sources in the upper medium
waves incident
(ground). For
For most cases
cases the
incident normally
normally on the lower medium
medium (ground).
sources
incident plane
plane waves.
sources in the atmosphere do not give rise to normally
normally incident
Cagniard
equations exactly
exactly
Cagniard recognizes
recognizes this fact and also
also arrives eventually
eventually at equations
equivalent
that the plane
plane wave
equivalent to equation
equation (2). He
He does
does not recognize, however, that
spectrum can contain
incidence (Y.
a. He
He states
contain plane
plane waves with
with a complex angle of incidence
2
that
acyis
course incorrect.
incorrect.
that sin
sin2
is never greater than
than unity
unity in magnitude,
magnitude, which is of course
The
greatly in a distance 1
'1'-\11 is
The limitation
limitation that
that Hx
H, and Hy
H, should not vary
vary greatly
( y-‘
therefore not evident
evident from Cagniard's
Cagniard’s analysis.
When
stratum
When the ground is
is horizontally
horizontally stratified,
stratified, consisting of sayan
say an upper stratum
of thickness h with
'1'1 and intrinsic
intrinsic impedance
impedance
with an intrinsic
intrinsic propagation
propagation constant
constant -yi
T/I
'1'2 and Q,
T/2, the surface
vi and a lower stratum
stratum of infinite
infinite thickness with
with constants y2
impedance
impedance is
is given by
T/
=
+
tl +
+ T/2
q2 tanh
tanh ('Ylh)
(nh)
T/I
h)
T/2
vz + T/I
~1 tanh
tanh ('Yl
(rrh)
T/I
’ = ”
.
(3)
This
transmission
This formula
formula can be arrived
arrived at directly
directly by
by considering the analogous transmission
line problem
line is of length
length h with
with a propapropaproblem (Schelkunoff,
(Schelkunoff, 1938). The
The equivalent
equivalent line
gation
1'1 and characteristic
T/I and is terminated
terminated in an imimgation constant
constant 'y1
characteristic impedance
impedance ql
pedance whose
is then
then 11
T/ as
as given
given by
by
whose value
value is
is T/2.
72. The
The input
input impedance
impedance of the line is
equation
conductivities u1
0"1 and u2,
0"2,
equation (3), which can now be written
written in terms of the conductivities
and magnetic
fJ.2 of the upper and lower strata
as follows:
magnetic permeabilities
permeabilities fJ.I
~1 and ~2
strata as
7 =
mQ
(4)
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TELLURIC
CURRENTS
AND THE
THE EARTH'S
EARTH’S
MAGNETIC
FIELD
TELLURIC
CURRENTS
AND
MAGNETIC
FIRLD
where
283
283
+
/2 + tanh (iO"IJ.1.1W)l/2h
(J.l.20"t!J.l.I0"2)1
(P~u~/P~u~)~~~
(i~io)~/*h
Q =
Q=
+
tanh (iO"IJ.l.IW)1/2h
+ (J.l.20"t!J.l.I0"2)l/2
(WJllW2) 1/Z tanh
I1
(i~~p~w)~/~h
has been assumed
assumed that
that tlW«O"I
c~w<<u~
czw<<uz.This
This equation
equation now reduces
reduces
and it has
and t2W«0"2.
to Cagniard's
Cagniard’s equation
equation (30) for the special
special case
case when the permeability
permeability contrast
contrast
is zero (i.e. J.l.l
~l=~Z-&.
It is
is therefore pointed
pointed out that
that for
between the strata is
=J.l.2-J.l.O). It
cases of formations
formations having
having magnetic
magnetic susceptibility,
susceptibility, ratios O"d
uZ/ul,
0"1, given in his
cases
8, should be replaced by the ratio
ratio J.l.lud
~luz/~2u1,
horizontal
Figures 7 and 8,
J.l.20"1, and the horizontal
scale (inverse frequency)
frequency) should be shifted to the left by a factor J.l.d
~JF~.
J.l.l.
scale
To apply
apply Cagniard's
Cagniard’s results to interpretation
interpretation of sub-surface stratification
stratification it is
is
To
necessary to carry out a harmonic
harmonic analysis of both the electric field function
function
necessary
e%(t)and the magnetic
magnetic field function
function hy(t).
&(t). The
The harmonic
harmonic components
components Ex(w)
E,(w) and
ex(t)
HvCw)
H,(w) at an angular
angular frequency
frequency ware
w are then compared to obtain
obtain the surface imimpedance function
function ?j(w).
q(w). Apparently
Apparently this procedure requires a fair
fair amount
amount of labor
labor
unless the analysis be carried out by electronic
electronic means. A suggested
suggested alternative
alternative is
unless
examine the relation
relation between the time
time functions
functions ex(t)
e=(t) and
to examine
and h@(t)
hy(t) directly.
directly.
If the magnetic
magnetic field hy(t)
h,(t) is
is in the form of a step-function,
step-function, that
If
that is,
is, it
it suddenly
suddenly
changes from one level H
Ho0 to another
another level H
HofAHo
changes
o+!:1H 0 at t=o
t = 0 then the frequency
frequency
H,(w) associated
associated with
with this change is
spectrum Hy(w)
H,(w) =
+S
h,(l)e-i”tdt
m~Hoe-i~t&
= ~AHo
=
S
--m
(5)
iW
0
The corresponding frequency
frequency spectrum
spectrum of the tangential
tangential electric
The
electric field is
IS then
then
by:
given by:
Ez(w) =
= ?j(w)!:1Ho/iw
v(w)AH,,/iw .
Ex(w)
The
v(w) is given
The frequency
frequency function
function '1(w)
given by
by equation
equation (3) for the two-layer
two-layer ground
ground
and it
it can be rewritten
rewritten in the following
following form.
form.
'1(W) = iJ.l.IW[1
1'1
+ 2f(~- (3)ne-2n'Ylh]
n~1 I + (3
(6)
where 8=
~*. The
(3= (~~~J~zur)‘
(J.l.IO"dJ.l.20"1) 1/2.
The time
time function
function e=(t)
ex(t) for the tangential
tangential electric
electric field is
then given
given by:
by:
= A-I f<Xl E,(w)e-iwtdw
Ex(w)ciwtdw
m
e=(t)
ex(t)
=
23r
271"
S
--m
_<Xl
()1/2
~ !:1Ho
271" 0"
=-
I
f
t-"
_<Xl
l
I
-(iw)1/2
- (3)n e- (iW)1/2J eiw1dw.(7)
+ 2 L1 (I--I + f3
(iw)1/2
2na
<Xl
These integrals
integrals are of a known
known type
type (Campbell
(Campbell and
and Foster,
Foster, 1948,
1948, p,
p. 93,
93, formula
formula
no. 807) so
so the integrations
integrations can be readily
readily carried
carried out to yield
yield
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JAMES R. WAIT
WAIT
JAMES
284
e,(t)
= F(T)/&‘ah
\vhe:re
where
F(T)
F(T) =
T-22
n=l [I + (~$Pl.]
(8)
(8)
2
and
T = t/a 2 = t/a&L2.
The
The function
function F (T)
(T) is therefore
therefore characteristic
characteristic of the shape of the electric
electric field
time
It is plotted
time function.
function. It
plotted in
in Figure
Figure I as
as aa function
function of the normalized
normalized time
time factor
factor
T. The
= II corresponds,
The case
case where @
(3=
corresponds, of course,
course, to the homogeneous ground
ground where
l12.It
It is noted
electric field decays as
as TT-l/2.
noted that
that for highly
highly conducting
conducting lower
lower laylaythe electric
ers e,(t)
ex(t) decays more rapidly
rapidly than
than for the homogeneous ground.
ground. Conversely,
Conversely, ifif
lower layer
layer is very
very poorly
poorly conducting
conducting the decay is somewhat
somewhat slower.
the lower
The
The corresponding electric
electric field function
function ez(t)
ex(t) for any
any other
other type
type of magnetic
magnetic
3.5,-------,----------,-------,--------,-------r----------,
-iI
THE
THE
\
I
FIELD
2.5t-----T---1--FIELD
Z
o
DECAY OF THE
DECAY
THE ELECTRIC
ELECTRIC
FOR
FOR
A STEP
STEP MAGNETIC
MAGNETIC FIELD1
FIELD
~
F:
u
Z
Y
\
::l
z 2.0t-------j-~~---_j_-----t_­
2.0
LL
on
~
ii 1.5
1.5f----.---+
LL
ii
u
F
0
~
1.0
1.0
~
\I
U
W
Y
i: 0.5~---~-----~--~~r_-~~
0.5
~
o0
0.1
0.1
0.2
0.5
T==
T
1.0
10
JJ'
I
•3
2
5
10
/qp,h2
t fa;
f-L.h2
I. Electric
Electric field variation
variation with
with time for two-layered
two-layered earth when magnetic field varies
variesstepFIG. I.
stepwise at t=o.
t=o. The parameter
parameter {J=C;t,IT2/J.L20,)'/2
j3= &u.&o~)‘/~ where ITl
~1 and J.Ll
gLIare conductivity
conductivity and magnetic permeperme
wise
ability,
ability, respectively, of the upper layer, IT2
g2 and J.L2
~2 of the lower layer.
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TELLURIC
CURRENTS
AND THE
MAGNETIC FIELD
FIELD
TELLURIC
CURRENTS
AND
THE EARTH'S
EARTH’S
MAGNETIC
285
285
field function
For example
example if the magmagfunction Hy(t)
HU(t) can be obtained
obtained in a similar
similar manner.
manner. For
netic
at time
time tt=O
then
netic field is
is suddenly
suddenly decreased
decreased in value
value by an amount.:lH
amount AHoo at
= 0 and then
is
eXl (t) is given
given by
by
= tl, the corresponding electric field ezl(t)
is re-established at t =
ez,(t) =
e,(t -
h) - ez(t)
es(t) is
is a step-function
step-function response
response given
given by equation
equation (8).
where eit)
(8).
For
For a magnetic
magnetic field function
function h(t) given by
+
h(t)
l(t) = Ho
Ha + .:lHoJ(t)
AHpf(t)
= Ho
for
for
t >
0
> o
< 0o
t <
function cxCt)
Z,(t) is expressed
then the corresponding electric field function
expressed in terms of the
step-function
step-function response
response ex(t)
e=(t) by:
by:
dftf‘ (l -
z,(t) =
= -f
7)ez(7)d7.
cx(t)
J(t - r)exCr)dr.
dt S 00
The limitation
limitation that
that Hx
H, (and Hy)
H,) varies slowly
slowly over the x, y plane
The
plane also
also applies
transient case.
case. An equivalent
equivalent statement
statement is
is that
that H,
to the transient
Hz should not
not vary
vary appreappreciably in a distance (tI27ruj.I.)I/2.
(t/z7ru~()r’~. This
This means that
that the transient
transient curves for cl(t)
ciably
ex(t) are
probably not valid
valid for any
any value
value of tf greater
greater than
than about
about 10
IO seconds.
seconds.
probably
like to thank
thank Mr.
Mr. D.
D. A. Trumpler
Trumpler for his assistance
I would like
assistance with
with the calculacalculations for Figure
Figure I.
I.
REFERENCES
REFERENCES
Baker, W. G. and Martyn,
Martyn, D.
D. F., 1952, The
The conductivity
conductivity of the ionosphere: Nature,
x70, p. 1090·
1090.
Baker,
Nature, v. 170,
Cagniard, Louis, 1953, Basic theory
theory of the magneto-telluric
magneto-telluric method of geophysical
Cagniard,
geophysical prospecting:
prospecting:
18, p. 605-635.
605-635.
Geophysics, v. 18,
1948,
Fourier integrals
integrals for practical
practical application:
Campbell, G. A. and Foster, R. M.,
M., 1948,
Campbell,
Fourier
application: New
New York,
York, D.
D.
Van Nostrand
Nostrand Co., Inc.
Inc.
Mitra, S.
S. K.,
K., 1952, The
The upper atmosphere: The
The Asiatic
Asiatic Society.
Mitra,
Monteath, G. D.,
D., T95I,
~951, The
The application
application of the compensation theorem to certain
Monteath,
certain radiation
radiation and
propagation problems: Proc. Inst.
Inst. Elect. Eng.,
Eng., Pt.
Pt. IV,
IV, v. 98, no. J.
propagation
J.
S. A., 1938,
7938, The
The impedance concept and its application
application to problems of reflection,
Schelkunoff, S.
reflection, refracrefrac‘7, p. 17.
tion, shielding, and power absorption: Bell System Tech. Jour., v. 17,
J7.
CORRESPONDENCE
BETWEEN
PROF. CAGNIARD
CAGNIARD
AND
CORRESPONDENCE
BETWEEN
PROF.
AND DR.
DR. WAIT
WAIT
submitting the manuscript
manuscript of the paper above to GEOPHYSICS,
Before submitting
GEOPHYSICS, Dr.
Dr.
Wait sent a copy of it
it to Prof. Cagniard
Cagniard in France.
France. The
The correspondence
Wait
correspondence between
between
that followed has
has been made available
available by
by its authors for publication
them that
publication along
with the paper and the letters that
that were exchanged are presented herewith.
with
herewith.
Acknowledgment is
is made to Dr.
Dr. Robert
Robert G. Van
Van Nostrand
Nostrand for translating
Acknowledgment
translating Prof.
Prof.
Cagniard's
EDITOR
Cagniard’s letters.-THE
letters.-THE
EDITOR
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286
JAMES R.
R. WAIT
WAIT
JAMES
Paris
24 October,
October, 1953
IQ53
Dr.
Dr. James R. Wait
Wait
Defence Research Board
Board
Ottawa,
Ottawa, Canada
Canada
Dear
Dear Sir:
I have only recently
thank you very
very much. II assure
assure
recently received your
your interesting
interesting work for which
which II thank
you that
that I have not yet had the necessary
necessary time
time to study carefully
carefully its contents nor to refer to the
bibliographic
delay giving
giving you my
my first impresbibliographic references
references which you cite. However,
However, I do not want
want to delay
sions.
sions.
With
61 I)
I) that
that II did not believe that
that they
they
With respect
respect to transients, I stated (GEOPHYSICS,
(GEOPHYSICS,XVIII,
XVIII, p. 61
present great practical
it would please
please me if
if you were to find
practical interest. It
It is
is to be understood that
that it
a convenient and effectiVl'
But, until
until II can be convinced of a
effective interpretive
interpretive method based upon them.
them. But,
better
notion of apparent
apparent resistivity.
resistivity. Do
Do
better technique, I prefer the harmonic
harmonic analysis and the use
use of the notion
not reproach me for not yet having
having explained the means by
having told everything
everything nor for not yet having
which I envisage making
making automatically
automatically the analysis in question. II am not in a position at present
to furnish the details of the apparatus
apparatus which I use
use to this end.
My
telluric sheet
sheet and a horizontally
horizontally
My calculations depend on the dual hypothesis of a uniform
uniform telluric
stratified
The merit
merit of your
your work,
work, merit
merit
stratified geologic
geologic structure.
structure. They
They are, of course,
course, only approximations.
approximations. The
appreciate greatly,
greatly, is to furnish a critique
critique of the value
value of this approximation.
which I appreciate
approximation. However,
However, II
be very happy
happy if you would show
show me how you establish equation
would be
equation (I)
(I) so
so that
that II can
ran judge
judge the
hypothesis upon which it is founded. It
would publish that
that at
at
It would be equall}
equally agreeable to me if you would
some time or another.
some
I would also
also like to make two remarks to you:
I. In the examplt
example which 1 h?ve
heve taken of a plane incident
incident wave,
I.
wave, II had no other
other aim
aim than
than to show
show
causes could give place to the flow of the same type
type of current
current sheet
sheet in the earth.
earth.
how very diverse causes
Since I considered
considered only real plane waves, waves of the physicist
physicist and not those
Since
those of the mathematician,
mathematician,
angles were real and I was not "incorrect"
“incorrect” in stating
stating that
that the
my angles
thp absolute value
value of their
their sines
sines was less
less
I. In your
your case,
case, you chose
chose to consider a source
source situated at a finite
than 1.
finite distance and, fcllowing
fcllowing Weyl,
We)"l,
decompose the dipole emission
emission into plane waves which are only mathematical
to decompose
mathematical abstractions
abstractions and for
approach) but
but do not criticize
criticize me for
which arise complex angles. I do not reproach you (for this approach)
what
what I am doing; we are not treating
treating the same
same problem.
2. Your
Your argument
argument concerning the lack of uniformity
uniformity of telluric
telluric sheets
2.
sheets seems
seems to me to be open to
criticism on
on the ground that
that magneto-telluric
magneto-telluric perturbations
perturbations found at a given place are certainly
criticism
certainly not
emitted by some
some isolated dipole situated in the ionosphere near the zenith.
emitted
zenith. One would
would not explain
remarkable uniformity
uniformity of telluric
telluric sheets,
sheets, a uniformity
uniformity which
which is not simply
simply a point
point of
then all of the remarkable
less adventuresome hypothesis but a proven
proven fact of daily
view nor a more or less
daily experience.
experience. When
When we
simultaneously in the Paris region and in the Midi
Midi of France,
France, at two stations separated by 600
record simultaneously
600
kilometers, over geologic
geologic structures which have absolutely
absolutely nothing
nothing in common,
kilometers,
common, rapid
rapid variation
variation (a few
seconds to a few minutes) of the horizontal
horizontal components of the magnetic
seconds
magnetic field, the two graphs obstriking in their near identity.
identity. Telluric
Telluric recordings made simultaneously
tained are striking
simultaneously in France,
France, Morocco,
Morocco,
Madagascar are certainly
certainly not identical-which
identical-which
and Madagascar
would he contradictory-hut
contradictory-but never the less
less presindisputable "family
“family resemblance."
resemblance.” All
All of this proves, if it
ent an indisputable
it is necessary,
necessary, that
that magneto-telluric
magneto-telluric
perturbations are not generated by isolated dipoles but by vast systems of ionospheric electric curperturbations
whose dimensions
dimensions are on a global scale.
scale. The
The altitude
altitude of these
rents whose
these sheets
sheets of ionospheric current
current can
very well be of the order of only ten kilometers
kilometers without
without introducing
introducing very
very
very much distortion
distortion in the
uniformity
kilometers.
uniformity of the telluric
telluric sheet
sheet over distances
distances of several tens of kilometers.
sending a copy of this letter
letter to Dr.
Dr. Milton
Milton Dobrin.
Dobrin. II thank
I am sending
thank you again for the interest that
that
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TELLURIC
AND THE
FIELD
TELLURIC CURRENTS
CURRENTS AND
THE EARTH'S
EARTH’S MAGNETIC
MAGNETIC FIELD
287
187
you have shown in my work and the useful additions
it. Until
Until I hear
hearfrom
additions which you have made to it.
from
you again, please
please accept, dear sir, my best wishes.
wishes.
L. CAGNIA~D
L.
CAGNIARD
397, Rue
Rue Vaugirard
Vaugirard
Paris, France.
France.
Defence Research Telecommunications
Telecommunications
Establishment,
Establishment,
Radio Physics Laboratory,
Radio
Laboratory,
Defence Research Board,
Board,
Shirley Bay,
Shirley
Bay,
Ottawa, Ontario.
Ottawa,
Ontario.
October 29, 1953
October
Professor
Professor Louis Cagniard,
Cagniard,
397 Rue de Vaugirard,
Vaugirard,
Paris IS,
13,
France.
Dear
Dear Professor
Professor Cagniard:
Cagniard:
Thank you for comments on my note entitled
entitled "On
“On the Relation
Relation Between
Thank
Between Telluric
Telluric Currents
Currents and
the Earth's
Earth’s Magnetic
Magnetic Field."
Field.” I am glad that
that you found this of interest.
that the harmonic
harmonic analysis of the earth current
current and the tangential
I agree that
tangential magnetic
magnetic field records
records
interpretation if the Fourier
Fourier components could be extracted
would lead to a more refined method of interpretation
extracted
In any case
case the transient
transient and the frequency
frequency response
by electronic means. In
response data
data are equivalent
equivalent from
mathematical standpoint.
standpoint. In
In addition,
addition, I believe that
that the "raw
“raw data"
data” from the field records
records could be
a mathematical
directly and compared to the idealized transient
transient curves to ascertain quickly
examined directly
quickly the general
nature
nature of the crustal layers.
don’t believe there is
is any doubt as
as to the existence
existence of complex angles of incidence in the plane
I don't
radiation from a dipolar
dipolar source.
source. This
This question has been
wave spectrum of radiation
been discussed
discussed at length
length by
Clemmow.’ Of course
course at large distances
distances from the dipole the wave front
Booker and Clemmow.1
front is almost plane
plane
and the angle of incidence com>sponds
vertical. At
At
corresponds to the angle hetween
between the line of sight and the vertical.
shorter distances
The E and H fields
fields are no longer
distances the field of the dipole becomes
becomes very
very different.
different. The
mutually
related by the characteristic
characteristic imimmutually perpendicular
perpendicular to the radial
radial direction
direction and they are not related
space. In
In this case
case the spectrum in terms of phne
plane waves contains complex angles of
pedance of free space.
2 a
so that
that sin
sin2
01is not always
always less
less than unity.
unity. Of course,
course, if the source
incidence so
source were a current
current sheet
sheet of
infinite
radiation, obtained
obtained by integrating
integrating
infinite extent and uniform
uniform in amplitude
amplitude and phase, the emanating
emanating radiation,
over all the horizontal
wave.
horizontal electric dipoles, would be a normally
normally incident
incident plane wave.
The current
current system in the upper atmosphere giving
giving rise
rise to short
short period (less
The
(less than
than 60 set)
sec) variavarialimited region as
as observed by
tions of the magn'!tic
magnetic such
such as
as "micro
“micro pulsations"
pulsations” are confined to a limited
2 •3 •4 In
several investigators.
is not permissible. It
would
investigators.2s3T’
In this case
case the infinite
infinite sheet representation
representation is
It would
seem
distribution of electric and magseem reasonable that
that the currents could he
be represented by a finite distribution
netic dipoles
dipoles and therefore one
one must admit
admit complex angles of incidence. On the other hand there have
been
that do occur simultaneously
simultaneously
been short period (5 to 10
IO sec)
set) pulsations of small amplitude
amplitude reported,5
reported,l that
latitudes. At
At northern
over large parts of the world at lower and middle latitudes.
northern latitudes,
latitudes, however, the
variations are generally
generally due to local conditions in the ionosphere. In
short period variations
In any
any case,
case, it
it would
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JAMES
R. WAIT
WAIT
JAMES
R.
288
288
necessary to ensure
ensure that
that the magnetic field components do not change appreciably
appreciably in a
always be necessary
distance equal to the reciprocal of the propagation
propagation constant of the ground.
request I shall outline briefly
briefly the derivationS
derivationa of equation (1)
(I) in my note.
At your request
rectangular coordinate system
system is
is employed with the (x,
(x, y) plane coinciding with
with the ground
A rectangular
axis pointing
pointing downward.
downward. The ground is
is assumed
assumed to be homogeneous
homogeneous with
with electrical conconand the ze axis
O, E,
e, and p..
p. The field
field component H.
H, (and Hu)
Ry) in
in the ground can be set
set up as
as aa superposition of
stants <1',
elementary waves so
so that
H&r, y,
y, z)
2) = JJI~ J_!(K"
J_,/(&,
Kz)e-‘
H.(x,
K
(K,z+K'l1I+ K ")(Ki/K)dK,dK,
2)e-(iK12+~~+K”)(K,/K)dR,dK?
wheref(Ki, K,)
Kp) is
is an
an aperature
aperature distribution
distribution and
and
wheref(K"
K,= +
I_ K22
Kz2
K,2
+ K3
K?2 =
+
2
w2pe
W
P.E --
iopc~
i<1'}JM
P.
== K2.
The
The electric
electric field
field Eu
E, is
is then
then given
given by
by
as ]
aH.
Ey=2.-Eu
= ---.- - I
<1'
Q
in the
the plane
plane z.z = O.
o.
in
++ 2WE
iwe az
a.2 1._0
I_o
It is
is noted
noted that
that under
under the
the integral
integral sign
sign
It
a
a,=-
zKa =
-
idKZ
-
(K12 + K3)
which can
can be
be expanded
expanded in
in aa Taylor
Taylor series
seriesas
as
which
:za
_=-=
a.2
-
iK
[1 - K,22~2K,2 + 0 (~. ) ] .
The
The electric
electric field
field tangential
tangential to
to the
the ground
ground can
can then
then be
be written
written
Ev(x,y,o)
=
-,~_~S,[~-~(Kl’+K2’)
f(K1,
+...I
Kz)e-i(K1”+K~)(K~/K)dKldKp
which is
is equivalent
equivalent to
to
which
-E.=qH+&
2%)
+0(S).
similar equation
equation relates
relates E.
E, and
and Hu
Hu in
in the
the plane
plane .'1:=0.
z=o.
AA similar
S will
more rigorous
rigorous derivation
derivation of
of these
theserelations
relations”
will appear
appear in
in the
the Journal
Journul of
ofApplied
Applied Physics
Physics in
in ua
AA more
copy of
of this
this letter
letter to
to Dr.
Dr. Dobrin.
Dobrin.
forthcoming paper
paper by
by W.
W. J.
J. Surtees
Surteesand
and myself.
myself. II am
am also
alsosending
sendingaa copy
forthcoming
Yours very
very truly,
truly,
Yours
JAMES R.
R. WAIT
WAIT
JAMES
REFERENCES CITED
CITEDINDR.
WAIT’S
LETTER OF
OFOCT~BER
29
REFERENCES
IN DR. WAIT'S
LETTER
OCTOBER 29
H. G.
G. Booker
Booker and
and P.
P. C.
C. Clemmow,
Clemmow, Jour.
Jour. Inst.
Inst. Elect.
Elect. Engs.
Engs. Pt.
Pt. III,
III, 94,
94, 171,
171, 1947
1947 and
and Pt.
Pt. III
11197,
1.I. H.
97,
II, 1950.
1950.
II,
D. La
La Cour,
Cour, Terr.
Terr. Mag.
Mag. and
and Atmos.
Atmos. Elect.,
Elect., 47,
47, 265,
265, 1942.
1942.
2.2. D.
L. Harang,
Harang, Terr.
Terr. Mag.
Mag. and
and Atmos.
Atmos. Elect.,
Elect., 37,
37, 57,
57, 1932.
1932.
3·3. L.
B. Rolf,
Rolf, Terr.
Terr. Mag.
Mag. and
and Atmos.
Atmos. Elect.,
Elect., 36,
36, 7,7, 1931.
1931.
4.4. B.
B. Angenheister,
Angenheister, Terr.
Terr. Mag.
Mag. and
and Atmos.
Atmos. Elect.,
Elect., 25,
25, 26,
26, J920.
1920.
5.5. B.
W. J.J. Surtees,
Surtees,Defence
Defence Research
Research Board
Board Report,
Report, University
University of
of Toronto,
Toronto, October
October 1952.
1952. (D.
(D. R.
R. B.
B.
6.6. W.
Grant No.
No. 67)
67) and
andJ.J.R.
R. Wait,
Wait, Radio
Radio Physics
PhysicsLab.
Lab. Project
Project Report
Report 19-0-4,
19-0-4, Sept.,
Sept., 1953.
7953.
Grant
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TELLURIC
CURRENTS
AND THE
THE EARTH'S
EARTH’S
MAGNETIC
FIELD
TELLURIC
CURRENTS
AND
MAGNETIC
FIELD
289
2%
Paris
16 November,
November, 1953
16
James R. Wait
Wait
Dr. James
Research Telecommunications
Telecommunications Establishment
Establishment
Defence Research
Ottawa, Ontario
Ontario
Ottawa,
Dear Sir:
Thank you for your last letter
letter of October 29,
zg, 1953 and for the interesting
interesting proof which you have
Thank
will make aa brief reference to your work in the chapter
chapter "Electricite
“Electricit
Tellurique” which
TeUurique"
given me. I will
Gottingen has requ€'sted
requested me to write
write for the next edition of Handbuch
Handbuch der
der Physik.
Pkysik.
Prof. Bartels of Gottingen
entirely in accord with
with you except regarding
regarding the question of the micropulsations
micropulsations being
I am entirely
limited to aa very restricted region. Only
Only yesterday,
yesterday, M.
M. Migaux,
Migaux, director
director of the Compagnie
Compagnie Generale
G&&ale
limited
de Geophysique, the man who "sees"
“sees” the most micropulsations,
micropulsations, told me that
that they
they do not exist except
de
industrial disturbances. Along
Along with
with St. Thomas,
Thomas, we have to see
see to believe.
where there are industrial
Please accept, dear Dr.
Dr. Wait,
Wait, this expression
expression of my best wishes.
wishes.
Please
L. CAGNIARD
CAGNIARD
L.
397, Rue de Vaugirard
Vaugirard
397,
15, France
France
Paris IS,
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