Multiple moment expansion of fully coupled aspherical earth seismograms

An asymptotic theory is developed to generate long-period synthetic seismograms on the aspherical earth. For given source and receiver, the method requires only the knowledge of aspherical earth structure along the source-receiver great circle, as well as the transverse gradients and curvature of this structure along the great circle. The theory is developed here for spheroidal modes observed on the vertical component. Each isolated spheroidal mode _pS_l is represented in the time domain by a truncated Taylor series expansion, with the nth expansion coefficient expressed as the nth moment of the multiplet spectrum about a fiducial frequency. The resulting expansions are valid up to order 1/l and account for all of the effects associated with refraction and focusing of the equivalent surface waves.