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in the initial data well.
However, we can restrict ourselves to considering the measurement errors and
accept the normal (Gaussian) probability distribution for the function F:
One cannot say that we know the statistics of errors
exp
{}
2
F
−
F
1
s
F
√
2
p
( F)
=
−
,
(10
.
79)
2
s
F
where
s
F
is the root-mean-square (standard) deviation of F.
The situation with statistical description of the Earth's conductivitiy,
,iseven
worse. Here our information is very limited, and depending on a priori data, we can
rely on more or less reasonable hypotheses. Let us show some examples.
1. If a priori information about the medium is rather scanty, then all we can do
is to assume that
. But then, the
inverse problem is unstable and its statement (no matter, deterministic or probabilis-
tic) makes no sense.
2. Let a priori information available allow us to assume that
is distributed uniformly over an infinite space
belongs to a com-
pact set
M
∈
, and that it is distributed in this set with uniform density
const
=
0
∈
M
p
(
)
=
(10
.
80)
∈
.
0
M
Then
{}
2
F
−
F
con
st
s
F
√
2
(
)
=
ln
−
.
(10
.
81)
2
s
F
In this case, the condition for maximum likelihood (10.78) reduces to equation
F
)
=
∈
M
F
)
.
−
F
(˜
inf
−
F
(
(10
.
82)
implying the minimization of the misfit functional
F
2
I
=
−
F
{}
(10
.
83)
as in the optimization method. The problem is conditionally correct and can be
solved directly by the optimization method, using (10.62) or (10.63).
3. Now, let us suppose that the existing a priori information is sufficient for con-
structing a hypothetical conductivity distribution
0
, which belongs to a compact
0
(in probabilistic sense !). We will express this requirement in terms of the normal
distribution
⊂
. The requirement for the desired solution
set
M
is that it must be close to
exp
2
1
s
√
2
−
−
0
p
(
)
=
,
(10
.
84)
2
s
2
