Inverse Theory with Grossly Inadequate Data

When only a few observations are available as data for an inverse problem, it is proposed that the best way to use them is to obtain bounds on various functionals of the structure. To do this, the model is found that has the smallest (or largest) value of the functional. In this way, for example, equations are derived for finding the model value that is exceeded somewhere by all structures satisfying the data, and thus this value must be exceeded in the Earth itself. The same techniques can be used to derive conditions for the existence of a solution, when a certain data set is given; this is an important problem in non-linear inverse theory. Three examples are given, including the non-linear problem of electrical conductivity in the mantle. There, one- and two-data problems are solved and, by means of the existence theory, self-consistency criteria are defined for amplitude and phase measurements and for amplitude measurements at two different frequencies.