We present a scheme for calculating the partial derivatives of the earth's free oscillation eigenfrequencies with respect to changes in an arbitrary laterally heterogeneous starting model. Such partial derivatives have previously been calculated only with respect to laterally homogeneous models. Numerical tests of the accuracy of the partial derivatives are presented. By means of the scheme presented in this paper it now is possible to apply linearized inverse theory to data such as singlet frequencies or waveforms to obtain a new laterally heterogeneous model, recalculate the forward solution, determine a new set of partial derivatives and residuals, and repeat this cycle until convergence is obtained. It also becomes possible to use a variety of laterally heterogeneous starting models to eliminate possible artifacts due to the choice of the initial model.