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hypothetical model, ignoring results of observations. At small
, the minimization
of
(
) leads to the dominating minimization of I(
), i.e., the stabilizing effect
of
) is suppressed and an unstable incorrect solution is obtained. An optimum
value of
(
providing a sufficiently small misfit and ensuring sufficiently strong sta-
bilization of the solution is to be found.
The regularization parameter
should be consistent with the error
in the initial
information. The optimum value of
can be chosen by testing a monotonically
decreasing sequence
, variational problem (10.70)
is solved and the iterative sequence of solutions characterized by their misfit is deter-
mined. The parameter
1
>
2
>...>
n
. For each
in the initial
data, is regarded as an optimum parameter. The optimum parameter of regulariza-
tion provides a conductivity distribution fitting best the exact model solution. This
simple technique is applicable if the error
=
opt
, at which the misfit attains the error
is well known. However, we commonly
have a more or less gross estimate:
min
≤ ≤
max
.
(10
.
74)
from interval (10.74)
are tested. Close solutions selected from the resulting set are averaged, providing an
approximation to the exact model solution.
If we know next to nothing of measurement and model errors, the parameter
opt
cannot be chosen by the knowledge of solution misfits. In this case, a quasi-
optimal value of the regularization parameter is determined. For example,
In this case, solutions consistent with various values of
opt
can
be defined as a value
at which the solution of the problem significantly deviates
from requirements of the stabilizer (smoothness or closeness to the hypothetical
model) but yet remains sufficiently stable. This heuristic method for the determi-
nation of
opt
was proposed by Hansen (1998). It is based on the so-called L-
representation. A monotonically decreasing sequence of regularization parameters
1
>
2
> ... >
n
is tested and the misfit I
and the stabilizer
are deter-
. Figure 10.6
mined for various
and a fixed minimum of Tikhonov's functional
presents the curve of
versus I
on a log-log scale. This curve has typically the
L-shaped form, with a fairly distinct bend separating a nearly horizontal branch with
large I
and small
.The
exact model solution is best approximated by assuming that the central point of the
bend, characterized by the largest curvature, defines the quasi-optimal parameter of
regularization,
from a nearly vertical branch with small I
and large
opt
.
10.4.4 A Few Words About the Backus-Gilbert Method
A description of the Backus-Gilbert method usually begins with the following state-
ments (Backus and Gilbert, 1968). The number of observations is always finite, but
the characteristics of the medium cannot be represented a priori by a finite number of
parameters. If the space of the observation data is finite-dimensional but the space of
