Uniqueness in the inversion of inaccurate gross Earth data

A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation (Q) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.