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Fig. 10.6
The
L-representation
the Earth's parameters is infinite-dimensional, the inverse problem is indeterminate,
that is, it has an infinite number of solutions. Indeterminacy (to be more precise,
underdeterminancy) of the inverse problem holds even for ideally accurate initial
data. However, if it is not possible to obtain an exact solution to the problem, it is
possible to find a smoothed (locally averaged) characteristic of the medium which
for a given set of observed quantities is determined uniquely and provides the best
approximation to the parameters of the desired model. Thus, emphasis is placed on
the construction of an optimal smoothing operator having the properties of a spatial
filter.
The theory of Backus and Gilbert is sometimes opposed to the Tikhonov theory
of ill-posed problems. It is said that the Backus-Gilbert method is designed for
underdetermined problems with limited number of sufficiently accurate initial data,
whereas the Tikhonov method is presented to problems that may have a unique solu-
tion but they are unstable because of inaccuracy in the initial data. Is such a demar-
cation justified? With interpolation and extrapolation, a set of discrete samplings can
be represented as a continuous function approximating the true field characteristic,
and then an underdetermined problem reduces to a problem that is unstable due
to errors of approximation. It seems that the theory of Backus and Gilbert may by
considered as an integral part of the general theory of ill-posed problems.
10.4.5 Probabilistic Statement of the Inverse Problem
The inverse problem (10.1), (10.2) can be stated in probabilistic terms (Goltsman,
1971; Kovtun, 1980; Tarantola and Valette, 1982; Yanovskaja and Porokhova, 1983;
Glasko, 1984; Tarantola, 1987; Backus, 1988; Spichak, 1999, 2005). It seems that
the probabilistic approach involving the powerful methods of the probability theory
and statistics may give a simple and convenient tools for solving inverse problems
of magnetotellurics and analyzing solutions obtained.
