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anomalies). Thus, we have three independent formulations of the inverse problem,
separating induction and galvanic anomalies of different physical origin.
1. MT inversion (TE-mode): the conductivity
(
y
,
z
) is found from the longitu-
dinal impedance
Z
. To determine the operator
Z
{
y
,
z
=
0
,
,
(
y
,
z
)
}
, the longi-
tudinal impedance is written in the form
z
=
0
E
x
(
y
,
z
=
0
,
)
E
x
(
y
,
z
,
)
Z
(
y
,
z
=
0
,
)
=
)
=
i
o
,
(10
.
10)
E
x
(
y
,
z
,
)
H
y
(
y
,
z
=
0
,
z
where
E
x
(
y
,
z
,
) is obtained from the Helmholtz equations
2
E
x
(
y
,
,
+
2
E
x
(
y
,
,
z
)
z
)
+
i
o
N
(
z
)
E
x
(
y
,
z
,
)
=
0
,
|
y
|
>
l
,
y
2
z
2
2
E
x
(
y
2
E
x
(
y
,
z
,
)
+
,
z
,
)
+
i
o
(
y
,
z
)
E
x
(
y
,
z
,
)
=
0
|
y
| ≤
l
(10
y
2
z
2
.
11)
with the conditions at infinity
E
x
(
z
E
x
(
y
,
z
,
)
→
,
)
,
E
x
(
y
,
z
,
)
→
0
(10
.
12)
z
→∞
|
y
|→∞
and the boundary conditions
E
x
(
y
,
z
,
)
[
E
x
(
y
,
z
,
)]
S
=
0
,
S
=
0
.
(10
.
13)
n
Here,
E
x
(
z
) is the normal electric field, and
n
is the normal to the bound-
ary
S
between blocks or layers of different conductivities. The square brackets in
(10.13) indicate a jump of a function at the boundary
S
. The anomalous electric field
E
x
(
y
,
E
x
(
z
) satisfies in the air the radiation condition.
For the longitudinal impedance we have
,
z
)
=
E
x
(
y
,
z
)
−
E
x
(
z
,
)
Z
(
y
,
z
,
)
→
Z
N
(
z
,
)
=
)
,
(10
.
14)
H
y
(
z
,
|
y
|→∞
) and
H
y
(
z
where
Z
N
(
z
,
,
), are the normal (one-dimensional) impedance and the
normal magnetic field.
2. MV inversion (TE-mode): the conductivity
(
y
,
z
) is found from the tipper
W
zy
. To determine the operator
W
zy
{
y
,
z
=
0
,
,
(
y
,
z
)
}
, the tipper is written in the
form
z
=
0
E
x
(
y
,
z
,
)
H
z
(
y
,
z
=
0
,
)
y
W
zy
(
y
,
z
=
0
,
)
=
)
=−
,
(10
.
15)
E
x
(
y
,
z
,
)
H
y
(
y
,
z
=
0
,
z
