We present a scheme for calculating the partial derivatives with respect to elastic properties of the eigenfrequencies and eigenfunctions of a laterally heterogeneous, rotating, anelastic and elliptical earth model. We use these results to obtain a formula for the partial derivatives of synthetic seismograms (or spectra) with respect to the elastic properties. These partial derivatives, which are based on the variational solution of the forward problem, make it possible to perform iterative linearized waveform inversion for the earth's laterally heterogeneous structure. ¿ American Geophysical Union 1987