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including the exact model solution. Let values of the impedance tensor [
Z
] and the
tipper
˜
W
be known from observations. Then we can determine the approximate
solutions ˜
Z
(
x
,
,
W
(
x
,
,
y
z
) and ˜
y
z
) of problem (10.1) by minimizing the
misfit
functionals
:
}=
[
Z
]
]
I
Z
Z
Z
(
x
{
˜
−
[
Z
{
x
,
y
,
z
=
0
,
,
˜
,
y
,
z
)
}
M
[
Z
]
]
=
inf
∈
−
[
Z
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
}=
}
(10
.
61)
˜
W
I
W
W
W
(
x
{
˜
−
W
{
x
,
y
,
z
=
0
,
,
˜
,
y
,
z
)
˜
W
.
=
inf
∈
−
W
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
}
M
The misfit minimization procedure is usually iterative. A starting model is con-
structed through the parametrization of the interpretation model. Solving the for-
ward problem for the starting model, we calculate the misfits between model and
experimental values of the impedance tensor and the tipper.Then a new model,
decreasing the misfits, is chosen. The iterations are performed until the misfits
approach the level of errors in the initial values of [
Z
] and
˜
W
. If the misfits cannot
be decreased to the level of errors in the initial data, this implies that the compact
set
M
is overly narrow. In this case, we test successively widening compacta (e.g.,
we increase the density of subdivision of the model). A compactum on which the
equation misfit is equal to the error in initial data is regarded as an optimal com-
pact set. However, an overly wide compactum makes the problem unstable and can
yield a solution that differs strongly from the exact model solution. This limits the
practicality of the optimization method.
It is obvious that separate inversions of the impedance and the tipper make
sense if solutions ˜
Z
(
x
W
(
x
z
) are close to each other. Otherwise
magnetotelluric and magnetovariational inversions call for correlation. We can, for
instance, carry out the magnetotelluric and magnetovariational inversions in parallel,
minimizing the functional of total misfit
,
y
,
z
) and ˜
,
y
,
g
Z
[
Z
]
]
2
{
,
,
}=
−
{
,
,
=
,
,
,
,
}
I
(
x
y
z
)
[
Z
x
y
z
0
(
x
y
z
)
(10
.
62)
g
W
}
2
˜
W
+
−
W
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
and controlling the contributions of magnetotelluric and magnetovariational inver-
sions by means of weights,
g
Z
and
g
W
. Alternatively, we can accomplish successive
partial inversions, minimizing the functionals of magnetotelluric and magnetovaria-
tional misfits
[
Z
]
]
2
I
Z
{
,
,
}=
−
{
,
,
=
,
,
,
,
}
(
x
y
z
)
[
Z
x
y
z
0
(
x
y
z
)
.
(10
63)
}=
}
2
˜
W
I
W
{
(
x
,
y
,
z
)
−
W
{
x
,
y
,
z
=
0
,
,
(
x
,
y
,
z
)
.
