On the Equivalence of Two Methods For Computing Partial Derivatives of Seismic Waveforms

The first term of the Born series is widely used (e.g. Woodhouse & Girnius, 1982; Tanimoto 1984) to compute the partial derivatives of seismic waveforms. On the other hand, Tsuboi & Geller (1987) first use the variational method to compute the eigensolutions for a laterally heterogeneous model. They then compute the partial derivatives of the waveforms using those of the eigenfrequencies and eigenfunctions. In the present paper we show that the partial derivatives of the waveforms obtained by these two different methods are equal. Thus the first term of the Born series fully includes the first-order correction to the eigenfrequencies and eigenfunctions. the present paper does not consider models with degenerate eigenstates, such as laterally homogeneous models.