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E
x
H
y
where
E
x
(
y
)
. This relation can be used as the
boundary condition for the problem on the electric field in the air. On applying the
Green theorem to the region
z
→
and
H
y
(
y
)
→
as
|
y
| →∞
≤
0, the problem is reduced to the integral equation
for
E
x
(
y
)
=
E
x
(
y
,
0):
v
E
x
y
)
E
x
(
y
)[
S
1
(
y
)
S
1
]
dy
,
=
+
o
−
−
.
E
x
(
y
)
i
G
(
y
(7
47)
−
v
where the Green function
∞
−
y
)
dm
h
2
π
cos
m
(
y
y
)
−
=
.
G
(
y
(7
48)
h
1
h
2
m
2
f
2
+
hm
+
1
/
0
involves the induction parameter
f
defined by (7.21).
Now we turn to the magnetic field. From (7.46), (7.47) and (7.48) we can easily
derive formulas for the magnetic field on the Earth surface,
H
y
(
y
)
=
H
y
(
y
,
0) and
H
z
(
y
)
=
H
z
(
y
,
0):
v
G
(
y
H
y
y
)
E
x
(
y
)[
S
1
(
y
)
S
1
]
dy
H
y
(
y
)
=
+
−
−
a
−
v
(7
.
49)
v
d
dy
{
y
)
E
x
(
y
)[
S
1
(
y
)
dy
,
H
z
(
y
)
=
G
(
y
−
−
S
1
]
}
b
−
v
where
G
(
y
−
y
) is a function obtained by differentiating the Green function
G
:
∞
y
)
dm
h
2
π
m
cos
m
(
y
−
G
(
y
y
)
−
=
f
2
.
(7
.
50)
h
1
h
2
m
2
+
hm
+
1
/
0
The normalized amplitude of the Green function
G
is plotted in Fig. 7.8. Calcula-
tions were done for
h
2
/
h
1
=100
(
h
-interval). The Green function plays the role of a spatial filter. The filter pass-band
at a level of 0
h
1
=
49 with
λ
1
/
h
1
=
25, 50 (
S
1
-interval) and
λ
1
/
|
|
/
o
. It increases monotonously with
lowering frequency. So, if at high frequencies we observe the induction effect of
currents flowing in the vicinity of the observation site, then at low frequencies the
remote zones begin to affect. The same regularities are characteristic of the function
G
that governs the horizontal magnetic field. Once again we see that the magnetotel-
luric sounding is developing not only in the vertical direction, but in the horizontal
direction as well. In passing to the
h
-interval, the induction effects begin to atten-
uate. Really, the contribution of the integral terms in (7.47) and (7.49a) becomes
negligibly small as
.
1max
G
is about 2
h
ef f
=
2
Z
N
→
0 and the magnetotelluric field comes to the normalcy.
