Amplitudes of the earth's split normal modes

This paper develops the theory necessary to explain the amplitudes of the earth's split normal modes. Expressions are derived for the amplitudes of the free oscillations of a laterally homogeneous rotating and elliptical earth excited by a point double couple. The eigenfunctions for this problem are the complex vector spherical harmonics about the north pole. The amplitudes of the normal modes are obtained by transforming Saito's (1967) results, expressed in vector spherical harmonics about the earthquake source, into geographic coordinates. We explicitly show the dependence of the excitation of each singlet within a multiplet on geometric fault parameters, seismic moment, source depth, earth structure and the geographic coordinates of the source and receiver. We present synthetic torsional and spheroidal displacement and strain spectra for low order fundamental modes (0S2-0S2, 0T2-0T5) excited by four basic fault geometries. Our results are suitable for the synthesis of observed spectra and time domain records in which splitting is an important effect. We have applied these results elsewhere to the Chilean and Alaskan earthquakes and have obtained very good agreement with observations.