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The integral relations of this kind can be considered as three-dimensional mag-
netotelluric analogues of the Kertz formulae (Berdichevsky and Jakovlev, 1984;
Berdichevsky and Zhdanov, 1984). The integrals are taken in the sense of their
principal values.
In the scalar form
1
2
∞
∞
dx
o
dy
o
E
x
(
x
E
o
x
(
x
o
,
,
y
,
z
)
=−
y
o
)
(
x
a
π
z
−
x
o
)
2
+
(
y
−
y
o
)
2
+
z
2
−∞
−∞
1
2
∞
∞
dx
o
dy
o
E
y
(
x
E
o
y
(
x
o
,
,
y
,
z
)
=−
y
o
)
(
x
b
π
z
−
x
o
)
2
+
(
y
−
y
o
)
2
+
z
2
−∞
−∞
(5
.
44)
and
1
2
∞
∞
dx
o
dy
o
H
x
H
o
x
(
x
o
,
(
x
,
y
,
z
)
=−
y
o
)
(
x
a
π
z
−
x
o
)
2
+
(
y
−
y
o
)
2
+
z
2
−∞
−∞
1
2
∞
∞
dx
o
dy
o
H
y
H
o
y
(
x
o
,
(
x
,
y
,
z
)
=−
y
o
)
(
x
b
π
z
−
x
o
)
2
+
(
y
−
y
o
)
2
+
z
2
−∞
−∞
∞
∞
1
2
dx
o
dy
o
H
z
(
x
H
o
z
(
x
o
,
,
y
,
z
)
=−
y
o
)
(
x
z
2
.
c
π
z
−
x
o
)
2
+
(
y
−
y
o
)
2
+
−∞
−∞
(5
.
45)
These formulae make it possible to extend the anomalous field from the Earth's
surface to any higher level in the air and weaken in this way the influence of near-
surface inhomogeneities.
From (5.45) we can easily derive the integral relations between vertical and hori-
zontal components of the anomalous magnetic field. Note that in the nonconducting
air
curl
H
A
=
0
.
Hence
H
y
H
x
=
H
z
=
H
z
x
,
y
.
(5
.
46)
z
z
Substituting (5.45
c
) in (5.46) and taking into account that
H
x
,
y
→
0as
z
→−∞
,
we obtain
1
2
∞
∞
dx
o
dy
o
H
x
H
o
z
(
x
o
,
(
x
,
y
,
z
)
=−
y
o
)
(
x
π
x
−
x
o
)
2
+
(
y
−
y
o
)
2
+
z
2
−∞
−∞
(5
.
47)
1
2
∞
∞
dx
o
dy
o
H
y
H
o
z
(
x
o
,
(
x
,
y
,
z
)
=−
y
o
)
(
x
z
2
.
π
y
−
x
o
)
2
+
(
y
−
y
o
)
2
+
−∞
−∞
