The earliest free oscillation recordings were made on strain seismographs - large quartz bars in mine tunnels that provided a fixed reference length to detect motion between two piers imbedded in the ground. Free oscillations are studied by finding the Fourier power spectrum of seismograms, in which the peaks correspond to the resonant frequencies of the earth. The frequencies of these 'tones' thus provide a natural average of the properties of the earth. This refinement is accomplished as a linearized iterative process that is widely known as 'linearized inversion' - adjusting a starting model to fit a new set of data. It is widely used in all areas of geophysics, and involves minimizing some measure of misfit between the observed data and the data that would be predicted from the current model iteration.
Least squares is a commonly used measure of misfit.
To appreciate free oscillations, it is worth noting the analogy with a simpler system - a vibrating guitar string. Here L is analogous to the size of the earth, linear density and volume density play the same role, and tension plays the role of the elastic constants. Of course the earth is much more complicated and has more possibilities for free oscillations than a guitar string. The principal free oscillation types are spheroidal, poloidal, and radial. The simplest type of spheroidal mode is the 'football' mode.