Information Technology Reference
In-Depth Information
10.3.1 On the Existence of Solution to the Inverse Problem
At first glance, this question appears to be simple, because the impedance tensor
[
Z
] and the tipper
˜
W
measured on the Earth's surface should correspond to the
really existing distribution of conductivity in the inhomogeneous Earth. However,
the experimental values of the impedance tensor and the tipper are inaccurate, and
they may conflict with mathematical models.
Let [
Z
] and
˜
W
contain measurement and model errors
W
. It is evident that
the real distribution of conductivity in the Earth and the real MT and MV response
functions do not belong to the chosen model class on which the inverse problem is
defined. Such an inverse problem does not have a rigorous solution. To remove this
contradiction, the notion of quasi-solution is introduced: a conductivity distribution
Z
and
z
)issaidtobea
quasi-solution
of the inverse problem (10.1) if the condi-
tions (10.2) are satisfied, i.e., if the misfits of the impedance tensor and the tipper do
not exceed errors in the initial data,
(
x
,
y
,
W
. The inverse problem (10.1) has a set
of quasi-solutions. From this set we have to select a quasi-solution that provides (at
a given level of abstraction) the best approximation to the real geoelectric structure.
This conductivity distribution ˆ
Z
and
z
) is called the
exact model solution
. When
solving the inverse problem, we endeavour to find the exact model solution.
Using the notion of the exact model solution, we can formalize the definition
of measurement and model errors. Let [
Z
] and
ˆ
W
be the impedance tensor and the
tipper obtained from a model that belongs to the chosen model class and has the
conductivity ¯
(
x
,
y
,
(
x
,
y
,
z
). Then, measurement errors are determined as
[
Z
]
[
Z
]
ˆ
W
ˆ
W
ms
Z
ms
W
=
−
,
=
−
(10
.
28)
and model errors are determined as
md
Z
[
Z
]
[
Z
=
−
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
]
(10
.
29)
ˆ
W
ˆ
W
md
W
=
−
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
.
m
W
and applying the triangle rule, we
reduce (10.28), (10.29) to the initial condition (10.2).
ms
Z
md
Z
ms
W
Setting
=
+
and
=
+
Z
W
10.3.2 On the Uniqueness of Solution to the Inverse Problem
We proceed from the following heuristic statement. The inverse problem has a
unique solution if it is defined on a given model class and the impedance tensor
and the tipper belonging to this class are exactly determined on the entire Earth's
surface in the entire frequency range. This statement was proven in four partial
cases.
