The spectra of unresolved split normal mode multiplets

Summary A seismic spectrum corresponding to an isolated multiplet n S l or n T l on a spherical earth is characterized fully by three parameters: an amplitude, a peak or central frequency and a half‐width. The amplitude depends on the mechanism of the seismic source; it may be complex for single‐station spectra, but it is real and positive for spectral stacks. The central frequency is the degenerate eigenfrequency of the multiplet, and the half‐width is a measure of the decay rate due to the Earth's anelasticity. On an aspherical earth, the multiplet n S l or n T l is split into 2 l + 1 nearly degenerate singlets. In general this splitting cannot be resolved. If the apparent amplitude, central frequency and half‐width of an unresolved multiplet are measured on an aspherical earth, and subsequently interpreted as if the Earth were spherical, there may be a bias introduced by the splitting. Perturbation theory is used here, correct to zeroth order in the eigenfunctions and first order in the eigenfrequencies, to investigate this bias. Correct to this order, single‐station amplitudes, and therefore source mechanism determinations, are unaffected by asphericity. Measurements of Q made on spectral stacks are always biased toward low Q . The limiting case n ≪ l and s ≪ l , where s is the maximum significant degree in the spherical harmonic expansion of the asphericity, is examined in particular detail. Single‐station spectra appear in this limit to consist of a single line broadened by attenuation alone; Q measurements made on these spectra, prior to stacking, are therefore unbiased. A travelling wave decomposition is considered in order to compare the results of normal mode perturbation theory in this limit with those obtained by applying the theory of geometrical optics to the equivalent surface waves. At the level of truncation considered in this paper, perturbation theory predicts that surface waves will propagate on an aspherical earth at a uniform velocity. This discrepancy with geometrical optics is attributed to the neglect of eigenfunction perturbations.