Annals 47, 1, 2004, 01/07def


191

ANNALS  OF  GEOPHYSICS, VOL.  47, N.  1, February  2004

Key  words ULF – ionosphere – Alfven – seismicity

1. Introduction

While influence of plasma irregularities is
important for radio-wave propagation through
the ionosphere, the same influence for ULF
waves is usually neglected. In the conventional
approach to many geophysical problems related
to ULF magnetic pulsations (magnetospheric
diagnostics, wave-particle interaction inside ra-
diation belts and so on), the ionosphere is ap-
proximated by a layer with homogeneous con-

ductivity, which is connected with the regular
conductivity profile integrated on height. How-
ever there are some indications that such a sim-
ple model is not valid even for the ULF fre-
quency range taking into account the existence
of ionospheric perturbations in reality. Alper-
ovich et al. (2002) using both theoretical com-
putations and laboratory experiments showed
that rather small (∼ 10-30%) variations in plas-
ma density can produce a several times increase
in «effective» conductivity, included in consid-
eration of ULF wave characteristics. Molcha-
nov et al. (2003) reported the results of ULF
magnetic field observations (0.003-5.0 Hz) at
Kamchatka region during a long period of seis-
mic activation. They found a remarkable and
statistically reliable ULF intensity depression
several days before strong seismic shocks. The
effect was especially clear at nighttime and for
the filter channels 0.01-0.1 Hz and it was absent
at daytime. They interpreted the effect in as-

Depression of the ULF geomagnetic
pulsation related 

to ionospheric irregularities

Valery M. Sorokin (1), Evgeny N. Fedorov (2), Alexander Yu. Schekotov (2), Oleg A. Molchanov (2)
and Masashi Hayakawa (3)

(1) Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation (IZMIRAN),
Russian Academy of Science, Troitsk (Moscow Region), Russia

(2)  Schmidt United Institute of Physics of the Earth, Russian Academy of Sciences, Moscow, Russia
(3)  Department of Electronic Engineering, The University of Electro-Communications, Tokyo, Japan

Abstract
We consider a depression in intensity of ULF magnetic pulsations, which is observed on the ground surface due
to appearance of the irregularities in the ionosphere. It is supposed that oblique Alfven waves in the ULF fre-
quency range are downgoing from the magnetosphere and the horizontal irregularities of ionospheric conduc-
tivity are created by upgoing atmospheric gravity waves from seismic source. Unlike the companion paper by
Molchanov et al. (2003), we used a simple model of the ionospheric layer but took into consideration the later-
al inhomogeneity of the perturbation region in the ionosphere. It is shown that ULF intensity could be essen-
tially decreased for frequencies f = 0.001-0.1 Hz at nighttime but the change is negligible at daytime in coinci-
dence with observational results.

Mailing address: Dr. Valery M. Sorokin, Institute of
Terrestrial Magnetism, Ionosphere and Radiowave Propa-
gation (IZMIRAN), Russian Academy of Science, Troitsk
(Moscow Region), 142092 Russia; e-mail: sova@izmi-
ran.rssi.ru 



192

Valery M. Sorokin, Evgeny N. Fedorov, Alexander Yu. Schekotov, Oleg A. Molchanov and Masashi Hayakawa

sumption of atmospheric gravity wave intensifi-
cation induced by changes in temperature and
pressure near the ground due to gas and water
release in a course of earthquake preparation.
Mareev et al. (2002) considered gravity waves
intensification in the ionosphere. An appear-
ance of gravity waves in the ionosphere leads to
depression of downgoing from magnetosphere
ULF waves due to loss of coherency along the
wave front (like scattering) and due to a change
in effective ionospheric conductivity. Here we
are going to investigate the latter process theo-
retically. Some results on this subject are ob-
tained in a companion paper by Molchanov et
al. (2003) in assumption of vertical stratifica-
tion of the ionosphere. Here we consider the
same effect in assumption of lateral inhomo-
geneity but for thin ionospheric layer.

2. Electromagnetic ULF field in horizontally
inhomogeneous ionosphere

We use a simple model, which includes the
following:

i)  Source of the magnetic field ULF pulsa-
tions is situated in the magnetosphere and it
generates downgoing Alfven waves. They prop-
agate in a homogeneous magnetospheric medi-
um above ionosphere along z-axis, which is co-
incident with the direction of the external mag-
netic field. The frequency of the waves ω << ωci,
where ωci is ion cyclotron frequency, therefore
they propagate with characteristic Alfven wave
velocity CA and their vertical wave number kz =
= kA = ω / CA. In addition we suppose their hori-
zontal wave number k >> kA, another words we
consider oblique Alfven waves.

ii)  Ionosphere is a thin layer at z = 0 with in-
tegrated Pedersen and Hall conductivities ΣP ⋅
⋅ (x, t), ΣH (x, t) respectively, which are time-de-
pendent and inhomogeneous along the horizon-
tal x-axis (see fig. 1). For simplicity, we neglect
input due to ionosphere thickness ∆h, because
of kA ∆h << 1 and present

( , ) ( , )x t x t, , ,P H P H P H0= +Σ Σ ∆Σ . (2.1)

iii)  Atmosphere below ionospheric layer is non-
conductive and current-free, but the ground medi-

um is completely conductive and tangential elec-
tric field disappears at the ground surface z = h.

iv)  For simplicity, we assume independence
of the all field components on y-coordinate, i.e.
∂ / ∂y = 0, and suppose large conductivity along
z-axis for magnetosphere and ionosphere that
leads to disappearance of electric field compo-
nent Ez ∼ 0.

Our model is about the same as was dis-
cussed in many other papers (e.g., Lyatsky and
Maltsev, 1983). The only difference is assump-
tion on lateral ionospheric inhomogeneity. 

Alfven wave can reflect from the ionospher-
ic layer and transform in the isotropic mode
wave (so-called fast magneto-sonic wave) in-
side ionosphere. Then both waves penetrate in-
to the Earth-ionospheric cavity. As usual we
consider Fourier expansion 

)i t~-
( , , ) ( , , )E x z t

dk d
E k z e

2 2
(

r r

ikx
=

r r
~ ~

3

3

3

3

- -

# #
(2.2)

where index r = x, y, and the same for magnet-
ic components br. In our model for Alfven wave
in the magnetosphere only Ex and by exist and

1 2

3

B

x

z

h
L

Fig. 1. The model used to calculate the depression
of the ULF geomagnetic pulsations: 1 – conducting
ionosphere; 2 – ionospheric inhomogeneities; 3 –
Earth surface. h is the height of the lower boundary
of the ionosphere; L is the spatial scale of the seismic
region. 



193

Depression of the ULF geomagnetic pulsation related to occurrence of the ionospheric irregularities

these spectral component are described by fol-
lowing equations:

( , , )
( , , )

, .

z

E x z
k E k z

b
i z

E
k

C

0

1

x
A x

y
x

A
A

2

2
2

2
2

2
2

+ =

= =

~
~

~
~ (2.3)

In opposite only component Ey, bx, bz keep in
the isotropic wave

( , , )
( , , )

,

, / .

z

E x z
k E k z

b
i z

E
b

k
k

E

k k k k c

0

1

y

i y

x

y

z y

i A

2

2

2

0
2 2 2

0

2

2

2
2

+ =

= - =

= - =

~
~

~
~

(2.4)

As concerned the situation below ionosphere (at
the atmosphere) we have well-known relation-
ships

k

b E= - = -

( , , )
( , , )

z
E x z

k E k z

ik z
E

k
k

ic
z

E
x
E

k
k

0

1

r
r

x

y z

y

y
x z

z
x

2

2

0

2 2

0 0

0

2
2

2
2

2
2

2
2

+ - =

-

~
~

~b E= =

0

0

^

c

h

m

(2.5)

where k0
2 << k 2 and kz

2

0 = k0
2 − k 2 ≈ − k 2.

Then after matching of the magnetospheric
and atmospheric fields in the ionosphere layer
we obtain integral equation for the fields inside
ionosphere (see Appendix for details)

( , ) ( , , ) ( )

( ) ( , , )

K k k k k
c

dk k k k

E E

E

0 2
4

0

A 2 $

$

)= -

-

~ ~ r~

~∆Σ
3

3

-

# l l l

"

! (2.6)

where

,

, ,
E

E

E
K

k

k

k

k

c

E E
0

4, ,

x

y H

H

P

H

H

P
P i

P A P H P H

1

2

1

2

2

)

)
= = =

-

-

r ~

Σ
∆Σ
∆Σ

∆Σ
∆Σ

Σ

, ( )cothk k k ik kh= = + +∆

k k k k= + =

= = =

=

G G G

G

"

!

, .k c c C4 4A A A A
2 2

= =r ~ rΣ Σ

Its connection with ULF magnetic field at the
ground is also analyzed in Appendix, where
shown bx (ω, h) >> by (ω, h) and

( , , ) ( , , ).sinhb k h i k kh E k 0x y= -~ ~ ~] g6 @# -
(2.7)

3. Change in the ULF spectrum on the ground
induced by ionospheric perturbations

Using conventional approach of perturba-
tion theory we transform (2.6) as following:

( , , )

( ) ( , ) .

k k K
c
k

K

dk k k K k

E E

E

0 2
4

A
A
2 $

$

)

)

= -

-

~
~

~∆Σ
3

3

-

1 1

1

- -

-

# l l l

" "

! "
(3.1)

where inverse tensor K
1-"

is easy to find

, ( ) ( ) .K
k

k

k

k
k k k k k

1
H

H

H

1

2
1 2

2
=

-
= +l l

1-

= G
"

After substitution in (2.7) we have

( , , )
( , ) ( )

( , )

( , ) ( )
( )

( , ) ( ) ( , )

sinh

sinh

b k h i
k kh
ckk k

E k

i
c k kh

kk
dk k k

M k k k M k

2

4

x
A H

A
P

H1 2

$

$

= +

- -

- -

~
~l ~

~

l ~

~ ~

∆Σ

∆Σ

)

)

3

3

-

( , )E k~

# l l

l l l l

"

,

(3.2)

where

( )
( )

( )

( )
( ) .

M k
k

k k k k

k
k k k k

1

1

H

H H

1 1 2

2 1 2

2

= +

-

l

l
( )M k k=

6

6

@

@

ULF magnetic field at the ground we find as the
inverse Fourier transform of (3.2)

( , , ) ( , , ) ( , , )b x h b x h b x hx x x0 1= -~ ~ ~



194

Valery M. Sorokin, Evgeny N. Fedorov, Alexander Yu. Schekotov, Oleg A. Molchanov and Masashi Hayakawa

( , , )
( , ) ( )

( , )

( , , )
( , ) ( )

( ) ( , ) ( )

( , ) ( , ).

sinh

sinh

b x h
ick k

dke
k kh
kE k

b x h c
ik

dke
k kh

k

dk k k M k k k

M k E k

2

x
A H ikx

x
A ikx

P H

0

1

1

2

#

# $

$

=

=

- - -

~ r~ l ~
~

~ r l ~

~

~ ~

∆Σ ∆Σ

)

)

3

3

3

3

3

3

-

-

-

#

#

# l l l l

l l

"

, (3.3)

In (3.3) the second term bx1 is due to presence of
perturbations in the ionosphere and could be
compared with bx 0. As example for solitary initial
wave E

*
(x) = E

*
exp (ikxx) and E* (k) = 2πE*δ (k +

− kx). We obtain

( , , )
( ) ( )

( , , )
( ) ( )

( ) ( ) ( ) ( )

sinh

sinh

b x h E
k k h
ick k k

e

b x h E c
ik

dke
k kh

k

k k M k k k M k

2

4

x
x x

A H x ik x

x
A ikx

P x x H x x

0

1

1 2

x

#

#

=

=

- - -

~
~l

~
l

∆Σ ∆Σ

)

)

3

3

-

#

" ,

(3.4)

where kx = 2π / λx and λx is horizontal scale of
downgoing Alfven wave. Let us consider influ-
ence a moving density variation in the iono-
sphere (e.g., gravity waves). Alperovich et al.
(2002) showed that such a variation leads to
mainly perturbation of Pedersen conductivity.
So in (3.4) we leave only the first term in the in-
tegrand and present the perturbation as follow-
ing:

( )

( )

cos expx
k x

L
x

k L e

e e

2
2 4

2
2
1

( ) ( )

P P

g

P P

k L

k k L k k L

2

2

2

g g

2 2

2 2 2 2

$

=
-

-

= +

+

{

r

∆Σ ∆Σ

∆Σ ∆Σ -

- - - - + +{ {
$

0

0

b l<

7

F

A

&

0 (3.5)

where ∆ΣP0 is averaged amplitude of the pertur-
bation induced by gravity wave, ϕ is random
phase of the gravity wave, L is spatial scale of
the seismic region. Substituting (3.5) in (3.4)
we obtain the relative change of ULF magnetic
field at the ground

)(A kf-1 ( )dkB k
r

b b b L

x

x

x

x x
x

0 0

0 1
$=

-
=

3

3

-
b b

#

e

,
( )

( )

( ) ( )
.

sinh

sinh

e e

e

A
k

k k k
k h

B
kh k k k k

k

2
1( ) ( ) ( )

( )

i k k x k k L k k k L

k k k L

A P

P

x

x
x

H

0

0 1 2

1

2

2

x x g x

g x

2 2 2 2

2 2

$ + +

+

=
+

=
+

=
+

f Σ Σ
∆Σ

- - - - - + -

- + - +

{

{

7

7

A

A

%

.

(3.6)

Finally after averaging we have for the value
β (ω) = b bx x0

2 the following resultant rela-
tion:

.a = +

B =

exp$

A =

=~

, ( , )

, ,

,

Re

sinh
coth

sinh coth

A d B

i

a i

a i

x t kh k h h

L h

c h C h

1

2

4

x

x x

x

x

x x x

x x x

A

A A A
A

P

A

P

A P A

H

2

1

2 2

2

2 2

2

1
0

2

0

2

$= -

- - -

+ - +

+ - +

= = = =

=

+

b f
r
a w w w

w w p a w w

w
w

w w w

w w w w
w

p w w r m

~ ~

r

Ω Ω

Ω Ω

Ω Ω

Σ Σ
Σ

Σ
Σ

Σ Σ Σ
Σ

3

3

-

2

,a = =Ω

0 0

a 1 2= +

#^

^ ^

^
^_

^ ^_

^

h

h h

h
hi

h hi

h

7 A

*

4

(3.7)

In (3.7) we denote undisturbed integrated Ped-
ersen and Hall conductivities ΣP0, ΣH0 respec-
tively. In a case of the large perturbation zone,
α >> 1 after simple calculations we have 

i= +w w

$

, exp Re

sinh coth

sinh coth

a i

a i

1
4

2

x

x x x x

x

2

2

0 1
2

0
2

0 0

0 1
2 2

0 2

$= - -

+ - +

+ - +

b p f
a
p

w w w w w

w w w w w

a
p

Ω

Ω Ω

Ω Ω

` e

_ _

_ _

j o

i i

i i

9

9

C

C

Z

[

\

]]

]]

_

`

a

bb

bb

(3.8)

In the center of the zone, ξ = 0 relation (3.8) re-
duced to the following:

( )
f f

f f f
1

, , x x

x x x x

2

2

3

2

1 2 3

2

1 2 1 2
2 2

3
2 2

= -
+

+

- -

- -

b f

w w

w w

Ω

Ω Ω

Ω Ω

f a= + hΩ

cothf = +h w w

_

_

i

i (3.9)

where η (x ≥ 0) = 1, η (x < 0) = 0. 



195

Depression of the ULF geomagnetic pulsation related to occurrence of the ionospheric irregularities

Fig. 2. Dependence of the relative magnetic field β at the ground on frequency for night and day hours. Thin
lines show the results of calculations in the thin film ionospheric model, bold lines correspond to full-wave cal-
culations in the IRI-90 model. kx = 0.01 km−1 and other parameters are shown on the picture. 

Fig. 3. Spatial distribution of the relative magnetic field β dependence on distance from projection of the per-
turbation center in the ionosphere, h is height of lower boundary of the ionosphere (here h = 100 km).



196

Valery M. Sorokin, Evgeny N. Fedorov, Alexander Yu. Schekotov, Oleg A. Molchanov and Masashi Hayakawa

Appendix

Let us represent the solution of (2.3) as the sum of downgoing and reflecting waves

( , , ) ( ) ( , )

( , )

( ) ( ),

E k z k e R k e

c C e R k e

k dxe x k C

E

E E

x
ik z a ik z

y A
ik z a ik z

ikx
A A

A A

A A

)

)

) )

= +

-

= =

~ ~

~

~
3

3

-

-

-

-

( , , ) ( )b k z kE=~

#

` j

7

7

A

A

(A.1)

where E
*

(x, ω) is amplitude distribution on the wave front and Ra is reflection coefficient.

It is evident that β (ω = 0) = 1 − ε, β (ω →
→ ∞) = 1 − ε (a1 + 1) / (a2 + 1) and β (ω) de-
creases on frequency. At night-time iono-
sphere when ΣP0 << ΣA we have ε ≈ ∆ΣP0 / ΣA,
a1 ≈ 1, and a2 << 1. At day-time ionosphere when
ΣP 0 >> ΣA we have ε ≈ ∆ΣP0 / ΣA, a1 ≈ a2 >> 1.
Dependence of β on frequency under the cen-
ter of inhomogeneity is shown in fig. 2 for the
night and day hours. The values of integrated
Pedersen and Hall conductivities are given in
the figure. For comparison, these depen-
dences calculated for laterally homogeneous
ionosphere of the finite thickness are also
shown in fig. 2 for the same values of inte-
grated ionospheric conductivities. The values
of β (ω) at relatively low frequencies (f < 0.05
Hz) are about 0.6-0.7 for nighttime and 0.9-
0.95 at daytime for both models. At higher
frequencies (0.2-0.3 Hz) β (ω) calculated with
the use of IRI model grows almost up to uni-
ty. Several extrema seen at the curve are
caused by the ionospheric Alfven resonance
and ionospheric MHD waveguide. This effect
is not obtained in the thin ionosphere model
that gives strong monotonous decrease of β
with frequency at f > 0.3 Hz. At daytime β
weakly depends on frequency and is about
unity in all the frequency range considered.
Spatial distribution of β value at the ground
found with the relation (3.7) is presented in
fig. 3. Note that the size of the depression area
is larger than the perturbation scale in the ion-
osphere.

4. Discussion and conclusions 

It seems that theoretical results here coincide
with observational data reported by Molchanov
et al. (2003). Recently Sorokin et al. (2002, 2003)
investigated the possibility of generating seismo-
induced geomagnetic pulsations due to preseis-
mic changes in the background electric field.

Here we try to estimate another possibility in
connection with intensification of the atmospher-
ic gravity waves before earthquakes and follow-
ing the appearance of the ionospheric inhomo-
geneities. While influence of the horizontal
ionospheric irregularities on propagation of the
VLF waves is known (e.g., Shklyar and Nagano,
1998), the same influence on the Alfven waves is
a rather original problem and we are going to
continue this research for a more complicated
model taking into consideration both vertical and
horizontal stratifications of the ionosphere.

Acknowledgements

This research was partially supported by
ISTC under grant 1121, by Commision of the EU
(grant No. INTAS-01-0456) and by RFBR (grant
No. 03-05-64553). Two authors (O.A.M. and
M.H.) are thankful for support from International
Space Science Institute (ISSI) at Bern, Switzer-
land within the project «Earthquake influence of
the ionosphere as evident from satellite density-
electric field data».



197

Depression of the ULF geomagnetic pulsation related to occurrence of the ionospheric irregularities

The similar relationship for isotropic wave as follows:

( , , ) ( ) ( , )

( , , ) ( ) ( , ) ( , , )

E k z k R k e

b k z
k

k R k e
k

E k z

k k k

E

E

y
i ik z

x
i i ik z i

y

i A

2 2

i

i

)

)

=

= =

= -

~ ~

~ ~ ~ ~ ~

-

-

(A.2)

where Ri is reflection coefficient for isotropic waves. As concerned solution of (2.5) we need to take
into consideration that Ex,y (z = h) = 0, and to match with above-mentioned solutions at the lower
boundary of the ionosphere, hence

( , , ) ( ) ( , )

, , ( ) ( , ) ( , , )

, ,

sinh

cosh coth

E k z k T k k z h

b k z i
k

k T k k z h i
k

k z h E k z

k z h

k z
k

k z h
k

k z h

E

E

E

E

x
a

y
a

x

y
i

x
i

y

)

)

)

)

= -

= - - = - -

-

- -

~ ~

~ ~ ~ ~ ~

~ ~ ~

sinh( , , ) ( ) ( , )

( ) ( , ) ( , , )cosh coth

E k z k T k

b i k T k i E k z

=

= =

~ ~

~ ~

^

_ ^ ^

^

_ ^ ^

h

i h h

h

i h h

8

8 8

8

8 8

B

B B

B

B B
(A.3)

where Ta is transmission coefficient for Alfven wave and T i is coefficient of transformation the Alfven
wave into isotropic wave in the ionosphere. It is evident that by / bx = (ω / c)2T a/ (k 2T i) << 1, if the trans-
formation is essential. It means that geomagnetic pulsations observed at the ground surface are main-
ly related to isotropic wave, which is originated from Alfven wave inside ionosphere. 

The electric fields at the upper boundary of the ionosphere are continuous, hence 

( ), ( ).sinh sinhR T kh R T kh1
a a i i

+ = - = - (A.4)

However discontinuity of the magnetic fields equals to surface currents at the boundary

( , , ) ( , , )

( , ) ( , , ) ( , ) ( , , )

dte dxe b x z t dte dxe b x z t

c dte dxe x t E x t dte dxe x t E x t

0 0

4
0 0

i t ikx

x

i t ikx

x

i t ikx

P y

i t ikx

H x

= + - = - =

+
r Σ Σ

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

-

-

- -

-

-

-

-

- -

-

-

~ ~

~ ~
=

# # # #

# # # #
R

T

S
SS

V

X

W
WW

( , , ) ( , , )

( , ) ( , , ) ( , ) ( , , )

dte dxe b x z t dte dxe b x z t

c dte dxe x t E x t dte dxe x t E x t

0 0

4
0 0

i t ikx
y

i t ikx
y

i t ikx
P x

i t ikx
H y

= + - = - =

= - -
r Σ Σ

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

-

-

- -

-

-

-

-

- -

-

-

~ ~

~ ~

# # # #

# # # #
R

T

S
SS

V

X

W
WW

(A.5)

Suppose now that time variation of the ionospheric conductivity is slow in comparison with geo-
magnetic pulsations. For example characteristic frequencies of the atmospheric gravity waves are
ω 0 ∼ (10−3 − 10−4)c−1 << ω. If so, we can simplify time integration in (A.5). Using now (2.1), (A.1-
A.4) and obvious relation 

( , ) ( , , ) ( , ) ( , , )dte dxe x t E x t dxe x t E x0 0, , , ,
i t ikx

P H x y

ikx

P H x y= =~Σ Σ
3

3

3

3

3

3

-

-

-

-

-

~# # #



198

Valery M. Sorokin, Evgeny N. Fedorov, Alexander Yu. Schekotov, Oleg A. Molchanov and Masashi Hayakawa

REFERENCES

ALPEROVICH, L., I. CHAIKOVSKY, YU. GURVICH and A. MEL-
NIKOV (2002): Laboratory modeling of the disturbed
D- and E-layers: DC and AC fields, in Seismo Electro-
magnetics: Lithosphere-Atmosphere-Ionosphere Cou-
pling, edited by M. HAYAKAWA and O.A. MOLCHANOV
(Terra Scientific Publishing Co., Tokyo), 343-348. 

LYATSKY, W.B. and YU. P. MALTSEV (1983): The Magnetos-
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MAREEV, E.A., D.I. IUDIN and O.A. MOLCHANOV (2002):
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MOLCHANOV, O.A., A.YU. SCHEKOTOV, E. FEDOROV, G.G.

( , , ) ( , ) ( , , )E x
dk

k k t E k0
2

0, , , ,P H x y P H x y0= + -~ r
~Σ ∆Σ

3

3

-

# l l l (A.6)

we finally find for electric field in the ionosphere

( , ) ( , , ) ( ) ( , ) ( , , ) .K k k k k
c

dk
k k t kE E E0 2

4
2

0A 2)= - -~ ~
r~

r
~∆Σ

3

3

-

# l l l
" !

(A.7)

Here time t is parameter and definitions are as follows:

, , ,

( ), ,

, .

coth

E

E

E
K

k

k

k

k

k k k ik kh k k k k c

k c c C

E E
0

4

4 4

, ,

x

y H

H P

H

H

P

P i P A P H P H

A A A A

1

2

1 2
2

2 2

)= = =
-

=
-

= + + = + =

= =

r ~

r ~ r

∆Σ
∆Σ

∆Σ

∆Σ

∆Σ

Σ

Σ Σ

)
R

T

S
SS

R

T

S
SS

R

T

S
SS

R

T

S
SS

V

X

W
WW

V

X

W
WW

V

X

W
WW

V

X

W
WW

" !

BELYAEV, M. SOLOVIEVA and M. HAYAKAWA (2004):
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SHKLYAR, D.R. and I. NAGANO (1998): On VLF wave scat-
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SOROKIN, V.M., V.M. CHMYREV and A.K. YASCHENKO
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Earth surface, in Seismo Electromagnetics: Lithos-
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HAYAKAWA and O.A. MOLCHANOV (Terra Scientific
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SOROKIN, V.M., V.M. CHMYREV and A.K. YASCHENKO
(2003): Ionospheric generation mechanism of geo-
magnetic pulsations observed on the Earth surface be-
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