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The synthetic field is determined in the air as a field excited by a plane wave
vertically incident on the Earth's surface at which the impedance relation (1.14) and
the Wiese-Parkinson relation (4.2) are satisfied.
We begin with analysis of relations between components of the anomalous mag-
netotelluric field in the air.
5.2.1 Anomalous Magnetotelluric Field in the Air
Examine a three-dimensional model where the horizontally homogeneous layered
Earth includes a bounded inhomogeneous domain of arbitrary geometry. The air is
supposed to be nonconductive. Following the traditional geoelectric representation,
we ignore the displacement currents and assume that inhomogeneities of the Earth
cause the anomalous field
E
A
(
E
x
,
H
z
) in the air, which
meets the Laplace equation and consists solely of the TE-mode (Schmucker, 1971a;
Berdichevsky and Jakovlev, 1984).
The anomalous electromagnetic field in the air can be expressed in terms of its
components at the Earth's surface. Let
E
A
E
y
,
,
H
A
(
H
x
,
H
y
,
0)
H
A
,
be known at the Earth's surface
0. Then we have the boundary-value problems for
E
A
H
A
in the air:
z
=
,
E
A
(
x
,
y
,
z
)
=
0 t
−∞
<
x
,
y
<
∞
z
<
0
0as
x
2
E
A
(
x
,
y
,
z
=
0)
=
E
o
(
x
,
y
)
E
o
(
x
,
y
)
→
+
y
2
→∞
(5
.
41)
0as
x
2
E
A
(
x
,
y
,
z
)
→
+
y
2
+
z
2
→∞
and
H
A
(
x
,
y
,
z
)
=
0 t
−∞
<
x
,
y
<
∞
z
<
0
0as
x
2
H
A
(
x
H
o
(
x
H
o
(
x
,
y
,
z
=
0)
=
,
y
)
,
y
)
→
+
y
2
→∞
(5
.
42)
0as
x
2
H
A
(
x
,
y
,
z
)
→
+
y
2
+
z
2
→∞
,
where
E
A
(
x
y
) are the anomalous elec-
tric and magnetic fields in the air and at the Earth's surface. Solutions of these
problems are given by the Poisson integrals for the half-space
z
,
y
,
z
)
,
H
A
(
x
,
y
,
z
) and
E
o
(
x
,
y
)
,
H
o
(
x
,
≤
0 (Tikhonov and
Samarsky, 1999):
∞
∞
1
2
dx
o
dy
o
E
A
(
x
,
y
,
z
)
=−
E
o
(
x
o
,
y
o
)
(
x
,
π
z
−
x
o
)
2
+
(
y
−
y
o
)
2
+
z
2
−∞
−∞
1
2
∞
∞
dx
o
dy
o
H
A
(
x
H
o
(
x
o
,
,
y
,
z
)
=−
y
o
)
(
x
z
2
.
π
z
−
x
o
)
2
+
(
y
−
y
o
)
2
+
−∞
−∞
.
(5
43)
