DATA REDUCTION AND STATISTICAL INCONSISTENCY IN LINEAR INVERSE PROBLEMS

An estimator or confidence set is statistically consistent if, in a we ll-defined sense, it converges in probability to the truth as the numb er of data grows. We give sufficient conditions for it to be impossibl e to find consistent estimators or confidence sets in some linear inve rse problems. Several common approaches to statistical inference in ge ophysical inverse problems use the set of models that satisfy the data within a chi(2) measure of misfit to construct confidence sets and es timates. For example, the minimum-norm estimate of the unknown model i s the model of smallest norm among those that map into a chi(2) ball a round the data. We give weaker conditions under which the chi(2) misfi t approach yields inconsistent estimators and confidence sets, Both se ts of conditions depend on a measure of the redundancy of the observat ions, with respect to an a priori constraint on the model, When the ob servations are sufficiently redundant, using a chi(2) measure of misfi t to selected averages of the data yields consistent confidence sets a nd minimum-norm estimates. Under still weaker conditions, one can find consistent estimates and confidence intervals for finite collections of linear functionals of the model. In an idealization of the problem of estimating the Gauss coefficients of the magnetic field at the core from satellite data, using a constraint on the energy stored in the f ield, suitable data averaging leads to consistent confidence intervals for finite collections of the Gauss coefficients.