The least-squares approach to the problem of inversion of a set of observed free oscillation periods leads to a system of normal equations. A solution of this system becomes nonunique if the determinant of the coefficient matrix is equal to or approaches zero. The occurrence of such a condition is investigated by analysing matrices of auto-correlation and cross-correlation between the partial derivatives of free oscillation periods. This approach is closely related to the Dirichlet kernel method described by Backus and Gilbert (1968).
A simultaneous solution for density and shear velocity is nonunique if presently available data are used because of the high correlations between the shear velocity partial derivatives and the density partial derivatives in the upper mantle (negative correlations) as well as in the lower mantle (positive correlations). The nonuniqueness in the lower mantle could be avoided if the data for higher torsional modes were included.
If a solution for one parameter is attempted assuming that the distribution of the other one is known, then the density solutions are nonunique for the top 100 km of the mantle. The density solutions are also nonunique in the depth range 3700-6371 km if only fundamental mode data are used. Nonuniqueness for the inner core occurs even when the data for the first two overtones are included. Interpretations of shear velocity distribution are highly nonunique in the first 300 km of the mantle.
There are high correlations between the density partials or shear velocity partials for the last 1000 km of the mantle and partials with respect to changes in the core radius. This permits compensation of a change in the core radius by redistribution of densities or shear velocities in the lower mantle such that the free oscillation periods remain unchanged within the limits of the measurement error.