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problems. Let us construct the operators of the forward problem, [
Z
{
x
,
y
,
z
=
0
, that at each iteration step cal-
culate the impedance tensor and tipper from a given conductivity
,
,
(
x
,
y
,
z
)
}
] and
W
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
(
x
,
y
,
z
). Obvi-
ously, these operators depend on the dimensionality of the model.
10.2.1 Forward Problem in the Class of 1D-Models
Let us consider a 1D-model in which the electrical conductivity
(
z
) is a piecewise-
constant function of depth
z
:
(
z
)
=
m
for
z
m
−
1
<
z
<
z
m
,
m
∈
[1
,
M
]
,
z
o
=
0
,
z
M
=∞
,
h
m
=
z
m
−
z
m
−
1
,
(10
.
7)
where
m
and
h
m
are the conductivity and thickness of the
m
th layer, respectively.
At a depth
z
=
z
M
−
1
, the model rests on an infinite homogeneous basement having
a conductivity
const
. The scalar impedance
Z
of this model can be found
directly from the Riccati equation:
dZ
(
z
=
M
,
)
(
z
)
Z
2
(
z
−
,
)
=
i
o
,
z
∈
[0
,
z
M
−
1
]
,
(10
.
8)
dz
where
Z
(
z
,
) satisfies the boundary condition
i
)
o
2
Z
(
z
M
−
1
,
)
=
(1
−
M
and is continuous at boundaries between layers.
10.2.2 Forward Problem in the Class of 2D-Models
Let a 2D-model striking along the
x
-axis contain an anomalous doman
l
where the conductivity is a piecewise-constant function of the horizontal coordinate
y
and depth
z
and let this domain be bordered by infinite normal background
y
<
|
y
| ≤
−
l
and
y
>
l
where the conductivity depends solely on the depth
z
:
⎧
⎨
N
(
z
)
y
<
−
l
=
(
y
,
z
)
−
l
≤
y
≤
l
.
(10
9)
⎩
N
(
z
)
y
>
l
.
The electromagnetic field in a 2D-model can be divided into two independent
modes: the induction TE-mode with the components
E
x
,
H
y
,
H
z
and the galvanic
TM-mode with the components
E
y
,
H
x
. The TE-mode gives the longitudinal
impedance
Z
and the tipper
W
zy
, which reflect the induction effects of geoelec-
tric structures (induction anomalies), whereas the TM-mode gives the transverse
impedance
Z
⊥
, reflecting the galvanic effects of geoelectric structures (galvanic
E
z
,
