The effect of even-order aspherical structure on displacement for each isolated multiplet in the Earth's seismic free oscillation spectrum can be represented by a discrete set of coefficients which characterize the interaction between singlets. Each interaction coefficient is linearly related to aspherical structure of a given harmonic degree and azimuthal order. Although interaction coefficients are nonlinearly related to displacement, they can be estimated iteratively by Newton's method, a technique that we call spectral fitting. We have applied spectral fitting to approximately 350 recordings taken from 18 large or deep events and have estimated the degrees 2 and 4 interaction coefficients for 38 low harmonic degree (l≤9) multiplets. An error analysis, based on misfit, is also presented.

These coefficients and errors provide linear constraints on aspherical structure and are tabulated for use in future inversions. The estimated coefficients partition naturally into two subsets: those for the 10 anomalous multiplets and those for the 28 multiplets dominantly sensitive to mantle heterogeneity. On average, the degree 4 coefficients lie below the error estimates, and we interpret only the degree 2 coefficients. The degree 2 coefficients for the mantle sensitive multiplets behave smoothly, within observational error, along low radial order dispersion branches (0S, 1S, 2S, 5S), indicating that a simple mantle model exists that will fit them. Comparison between the estimated interaction coefficients and those predicted by models expressed as perturbations to &ngr;s or &ngr;p is complicated by the necessity of assuming empirical scaling relationships between &ngr;s, &ngr;p, and &rgr;. The numerical values in these relationships are currently subject to debate, but in any reasonable case the degree 2 coefficients predicted from the models M84A and L02.56 of Dziewonski and Woodhouse agree qualitatively with the estimated coefficients. A wide range of mantle models will quantitatively fit the mantle sensitive coefficients, the fundamental mode coefficients computed from muliplet center frequency observations (0S20--0S40), and the geoid simultaneously. The characteristics of these models are strongly dependent on allowed model characteristics with trade-offs between volumetric and boundary perturbations being particularly important.

In agreement with the results described in an earlier paper using a different technique, the anomalous multiplets are dominated by degree 2, axisymmetric structure somewhere in the core. Inferences concerning the nature and location of this heterogeneity are also dependent upon the scaling relationships and constraints placed on mantle structure. Not surprisingly, the resolution of the cause of anomalous splitting awaits confidence in long-wavelength mantle models and more and different kinds of data. ¿ American Geophysical Union 1988