Information Technology Reference
In-Depth Information
Let the admittance
Y
o
=
Y
(0
,
) be known at the Earth's surface, while the conduc-
tivity
(
z
) be known in the interval 0
<
z
<
z
m
. Then, the successive application of
(10.32) provides the admittance
Y
m
=
) at a depth
z
m
.
Now, we prove the theorem of uniqueness, which is formulated as follows. If
Y
(1)
(
z
Y
(
z
m
,
) and
Y
(2)
(
z
(1)
(
z
) and
(2)
(
z
),
,
,
) are the solutions of problem (10.30) for
then
Y
(1)
o
Y
(2)
o
(1)
(
z
)
(2)
(
z
). This theorem is proven ad
(
)
≡
(
) implies that
≡
absurdum. Assume that
Y
(1)
o
Y
(2)
o
(
)
≡
(
)
a
(1)
(
z
)
(2)
(
z
)at0
<
z
<
z
m
−
1
≡
b
(10
.
33)
(1)
(
z
)
(2)
(
z
)at
z
=
>
z
m
−
1
.
c
Then, applying (10.32) to (10.33
a
) and (10.33
b
) and extending
Y
(1)
0
and
Y
(2)
0
to
the depth
z
m
−
1
, we obtain
Y
(1)
Y
(2)
m
1
(
)
≡
1
(
). Let us determine the high-frequency
m
−
−
asymptotics of
Y
m
−
1
(
). According to (10.31),
i
)
m
Y
m
−
1
(
)
∼
m
=
(1
+
o
.
(10
.
34)
2
→∞
Thus, the identity
Y
(1)
m
Y
(2)
m
(2
m
, which contradicts the
assumption (10.33
c
). Successively increasing
m
, we reach the model basement and
obtain
≡
(1)
m
=
1
(
)
1
(
) leads to
−
−
0. The theorem of uniqueness is proven.
II.
The next step was made by Weidelt (1978), who proved the uniqueness the-
orem for a 2D model excited by an
E
-polarized field. In this model, the electrical
conductivity
(1)
(
z
)
≡
(2)
(
z
)
,
z
≥
z
) is supposed to be an analytical function. It has been shown that
simultaneous observations of horizontal components of the electric and magnetic
fields, carried out in the entire frequency range 0
<
(
y
,
<
∞
along an
y
-profile of a
finite length, provide the unique determination of
z
).
The Weidelt theorem was generalized by Gusarov (1981), who considered a 2D
E
-polarized model with the piecewise-analytical conductivity
(
y
,
(
y
,
z
). The Gusarov
theorem states that the piecewise-analytical function
(
y
,
z
) is uniquely determined
by the longitudinal impedance
Z
=
Z
xy
specified in the entire frequency range
0
<
.
All these proofs have their basis in the skin effect. Due to the skin effect, there
always exists a high frequency such that the field or impedance can be approximated
by a high-frequency asymptotics depending on a local conductivity. Comparison
of high-frequency asymptotics for various geoelectric structures suggests that dif-
ferent distributions of conductivity
<
∞
on an infinite y-profile
−∞
<
y
<
∞
correspond to different fields and different
impedances. Unfortunately, the realization of this simple idea encounters signifi-
cant mathematical difficulties due to complexity of the determination of the field
high-frequency asymptotics in heterogeneous media.
