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inversion of the descending branches of the apparent-resistivity and phase curves
may yield a real depth to the conductive basement.
Next consider the anomalies of the vertical magnetic field. In the integral rela-
tion (7.49b) the Green function affects the derivative of the excess current. It is
evident that the intensity of
H
z
−
anomalies depends on how fast
E
x
and
S
1
change.
Physically it means that
H
z
reflects the asymmetry of the excess current. The max-
imum anomalies of
H
z
are observed above discontinuities of
S
1
, say, above the
vertical interfaces. Anomalies of
H
z
like anomalies of
E
x
and
H
y
appear within the
S
1
-interval and disappear in the
h
-interval. This specifies the shape of the frequency
responses of the tipper. The mechanism of attenuation of the near-surface magnetic
anomalies is easy to understand: it follows from (1.49) that with lowering frequency
the current in the upper inhomogeneous layer decays (
J
1
/
J
3
→
0).
In conclusion, we will derive two useful estimates.
1. At what frequency do the induction effects attenuate? To get an answer, it
would be enough to define the boundary of the
h
-interval. From (1.48) we can derive
the following rough estimate:
1
o
h
2
max
S
1
.
T
>>
2
π
o
h
2
max
S
1
,
<<
(7
.
54)
2. How far do the induction effects extend? Turn to (7.48) and (7.50) and repre-
sent the functions
G
(
y
y
) and
G
(
y
y
)as
−
−
∞
h
2
cos
mud m
h
1
h
2
m
2
G
(
u
)
=
h m
π
h
eff
+
+
1
/
f
2
0
(7
.
55)
∞
h
2
m
cos
mud m
h
1
h
2
m
2
G
(
u
)
=
f
2
,
h m
h
eff
π
+
+
1
/
0
where
y
y
/
h
1
=
h
2
=
h
u
=
−
h
eff
,
m
=
mh
eff
,
h
1
/
h
eff
,
h
2
/
h
eff
,
=
h
/
h
eff
,
and
h
eff
Z
N
|
/
o
is the effective penetration depth defined by the normal
impedance. Successively integrating (7.55) by parts, we obtain the asymptotic
decompositions of the functions
G
(
u
) and
G
(
u
)for
u
= |
→∞
.Let
|
y
| −
v
>>
h
eff
.
Keeping the first terms in decompositions of
G
(
u
) and
G
(
u
), we write
O
1
u
4
O
1
u
4
h
2
hf
4
π
h
2
f
2
π
1
u
2
+
1
u
2
+
G
(
u
)
G
(
u
)
=
,
=−
.
(7
.
56)
h
eff
h
eff
Substitute (7.56) in (7.47) and (7.49). Taking into account (1.44) and (7.20), we
obtain the components of the anomalous electromagnetic field far away from the
edge of the inclusion:
