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√
f
−
√
2
√
f
+
√
2
,
S
1
1
o
S
1
h
2
f
f
=
−
i
k
=
=
S
1
,
η
=
1
o
S
1
h
2
.
−
i
f
=
2
), we return to the three-segment model examined
in Sect. 7.2.3. This model contains a near-surface inhomogeneity, which distorts the
transverse impedance in the low-frequency range.
Consider the transverse impedance
Z
⊥
in a model with conductive faults.
Let us begin with the
S
1
- interval relating to the ascending branch of the
apparent-resistivity curve
In the absence of faults (
2
=
Z
⊥
⊥
o
.Here
o
S
1
h
2
1. Then
f
≈
√
−
i
o
S
1
h
2
, whence
η
≈
. Hence the coefficients
A
,
B
,
C
and
D
defined by
(9.5) are close to zero, from which
⎧
⎨
1
S
1
Z
N
=
|
y
| ≥
v
+
q
Z
⊥
(
y
)
≈
(9
.
6)
1
S
1
⎩
Z
N
=
|
y
| ≤
v.
Here we arrive at the same representation (7.77) inferred in Sect. 7.2.3 for the
three-segment model without conductive faults. In the
S
1
- interval the transverse
impedance is not distorted.
Now we pass on to the
h
- interval relating to the descending branch of the
apparent-resistivity curve
2
o
.Here
=
Z
⊥
⊥
o
S
1
h
2
1. Then
f
≈
1,
whence
η
≈
1. Thus,
2
)
e
[
g
1
(
w
+
q
)
−
g
2
q
]
f
−
1
2
1
ke
−
2
g
2
q
f
1
ke
−
2
g
2
q
f
coth
g
g
2
g
f
v
Z
N
(1
A
≈
+
k
)(1
−
+
+
−
2
e
g
2
w
f
−
1
2
1
ke
−
2
g
2
q
f
1
ke
−
2
g
2
q
f
coth
Z
N
1
g
g
2
g
f
v
B
≈
−
+
+
−
2
e
−
g
2
(
w
+
2
q
)
f
−
1
2
1
ke
−
2
g
2
q
f
1
ke
−
2
g
2
q
f
coth
Z
N
1
g
g
2
g
f
v
C
≈
−
+
+
−
1
−
1
ke
−
2
g
2
q
f
2
1
ke
−
2
g
2
q
f
1
ke
−
2
g
2
q
f
coth
2
)
g
2
sin h
g
g
(1
−
g
g
2
g
f
v
Z
N
D
≈
−
+
+
−
.
f
v
(9
.
7)
The normalizing role of the faults is seen from two asymptotic estimates.
Let
. In this model the lithosphere is of infinitely high resis-
tivity and the faults are absent. Then
A
2
→∞
,
f
→∞
2
=
B
=
C
=
0 and
D
=
−
1, whence
Z
N
=−
i
o
h
|
y
| ≥
v
Z
⊥
(
y
)
≈
.
(9
8)
2
Z
N
=−
2
h
i
o
|
y
| ≤
v.
