A simple derivation of the formula to calculate synthetic long-period seismograms in a heterogeneous earth by normal mode summation

A simple modification of Gilbert's formula to account for slight lateral heterogeneity of the Earth by the first-order Born approximation leads to a convenient formula to calculate synthetic long-period seismograms. Partial derivatives are easily calculated, thus the formula is suitable for direct inversion of seismograms for lateral heterogeneity of the Earth. The derivation is much simpler but is equivalent to the one derived by Woodhouse, who obtained it by working in the time domain.