Summation of the Born series for the normal modes of the Earth



We derive an explicit formula for the response of a laterally heterogeneous, self-gravitating, rotating, dissipative, and physically dispersive earth to an earthquake. The long-period spectrum at a single station is given by u(ω) =RH0 [V(ω) +ωW−ω2T]−1·S0(ω) where R0 and S0(ω) are the unperturbed source and receiver vectors and H denotes the Hermitian transpose. The quantities V(ω), W, and T are the potential energy, Coriolis, and relative kinetic energy matrices, and V(ω) +ωW−ω2T is the Lagrangian matrix whose eigenvalues are the perturbed complex eigenfrequencies. This simple result is obtained by formal summation of the Born series.