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∞
∞
H
y
)
z
=
h
1
=
h
y
)
z
=
h
1
e
−
i
(
k
x
x
+
k
y
y
)
dk
x
dk
y
1
H
τ
(
H
x
,
h
τ
(
h
x
,
4
2
−∞
−∞
(11
.
29)
∞
∞
Y
e
τ
(
e
x
,
e
y
)
z
=
h
1
e
−
i
(
k
x
x
+
k
y
y
)
dk
x
dk
y
,
1
=
4
2
−∞
−∞
which opens the way to evaluating
S
1
(
x
y
) by (11.24). At practical computation
we apply the Fourier-Bessel transformation and reduce the double Fourier integral
to the more convenient Hankel integral, which provides the better accuracy and
robustness. The calculations are performed on the frequencies related to the ascend-
ing branch of the apparent-resisrivity curves.
A model example of mapping
S
1
(
x
,
y
) is shown in Fig. 11.57. The subsur-
face layer with background conductance of 10 S contains a large-scale
,
shaped
anomaly with conductance of 100 S and a small-scale square anomaly with con-
ductance of 15 S. It is underlaid with a 20 Ohm
−
m homogeneous basement. The
calculation has been performed for the period of 0.1 s. We see that the Singer-
Fainberg method gives sharply defined images of the
S
·
anomalies with distortions
below 10%. This is much better than in the maps of the apparent resistivity shown
in Fig. 11.45.
−
11.5.2 The Obukhov Method
The idea of this method is derived from the same model as in the Singer-
Fainberg method (Obukhov et al., 1983). Ignoring the basement conductivity (
2
→
∞
,
layer is located in the free
space. Such simplification is justified on the frequencies related to the
S
1
−
h
2
→∞
), we assume that a thin horizontal
S
1
(
x
,
y
)
−
interval.
as a sum of the external field
H
ext
τ
Let us represent the magnetic field
H
τ
created
layer and the internal magnetic field
H
int
τ
by primary currents above the
S
1
(
x
,
y
)
−
,
−
created by currents induced within the
S
1
(
x
y
)
layer:
H
ext
τ
H
int
H
τ
=
+
τ
.
(11
.
30)
Applying the Price-Sheinmann conditions (7.15), we write for the internal magnetic
field
H
int
:
H
in
x
(
x
H
in
x
(
x
y
)
E
y
(
x
,
y
,
h
1
)
−
,
y
,
0)
=
S
1
(
x
,
,
y
,
0)
(11
.
31)
H
in
y
(
x
H
in
y
(
x
y
)
E
x
(
x
,
y
,
h
1
)
−
,
y
,
0)
=−
S
1
(
x
,
,
y
,
0)
.
It follows from the Bio-Savart law that the horizontal components of
H
int
(
x
,
y
,
h
1
)
and
H
int
(
x
,
y
,
0) are antisymmetric, that is, they have the same moduli and the
H
in
x
(
x
H
in
x
(
x
H
in
y
(
x
opposite
phases,
,
y
,
h
1
)
=−
,
y
,
0)
and
,
y
,
h
1
)
=
−
H
int
y
(
x
,
y
,
0). So, on the Earth's surface
