On the equivalence of two methods for computing partial derivatives of seismic waveforms-II. Laterally homogeneous initial model

The perturbation to the synthetic seismogram (i.e., the partial derivative) due to an infinitesimal laterally heterogeneous perturbation to an initially laterally homogeneous model may be computed by either of the following methods. (1) Solving a (2l + 1) × (2l + 1) matrix eigenvalue problem to find the eigenfrequencies and eigenfunctions of the split singlets; summing the split singlets; and then subtracting the original synthetic for the unperturbed model. (2) Using the first-order term of the Born series. In the present paper we show that the partial derivatives computed by the above two methods are equal.