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where
Z
N
is the
normal impedance
, that is, the Tikhonov-Cagniard impedance of
the horizontally layered host medium. Within the inhomogeneous domain V, the
normalized field
H
N
excites the excess currents with densities
j
1
(the first polar-
ization) and
j
2
(the second polarization).
Consider a normal field with arbitrary magnetic components
H
x
o
=
E
N
,
H
x
(0) and
H
y
(0) on the Earth's surface. Using the principle of superposition and sum-
ming the effects of excess currents, we determine the associated anomalous field.
According to (1.7):
H
y
o
=
H
x
o
V
H
y
o
V
G
E
(
r
r
v
)
j
2
(
r
v
)
dV
G
E
(
r
r
v
)
j
1
(
r
v
)
dV
E
A
(
r
)
=
|
+
|
,
H
x
o
V
H
y
o
V
G
H
(
r
r
v
)
j
2
(
r
v
)
dV
G
H
(
r
r
v
)
j
1
(
r
v
)
dV
H
A
(
r
)
=
|
+
|
.
In compact form
E
A
(
r
)
H
x
o
J
E2
(
r
)
H
y
o
J
E1
(
r
)
=
+
,
.
(1
12)
H
A
(
r
)
H
x
o
J
H2
(
r
)
H
y
o
J
H1
(
r
)
=
+
,
where
G
F
(
r
r
v
)
j
λ
(
r
v
)
dV
J
F
λ
(
r
)
=
|
V
with F(field) = E, H and
(polarization) = 1, 2.
And finally, taking into account (1.11), we get
H
y
o
Z
N
+
J
E
x
E
x
=
E
x
+
E
x
=
H
x
o
J
E2
+
a
x
H
x
o
J
E
y
E
y
=
E
y
+
E
y
=
−
Z
N
+
+
H
y
o
J
E1
b
y
(1
.
13)
H
x
o
1
J
H
x
+
H
x
H
x
H
y
o
J
H1
H
x
=
+
=
+
c
x
H
y
o
1
J
H
y
H
y
H
y
H
x
o
J
H2
H
y
=
+
=
+
+
d
y
Eliminating
H
x
o
,
H
y
o
from (1.13
c
,
d
) and substituting these values in (1.1
3a, b
),
we establish:
E
x
=
Z
xx
H
x
+
Z
xy
H
y
,
(1
.
14)
E
y
=
Z
yx
H
x
+
Z
yy
H
y
,
where
