Aspherical structure constraints from free oscillation frequency and attenuation measurements
Authors: Smith Mark F. , Masters Guy 
Published: February 2008

We consider some techniques for approximately modeling the effect of aspherical structure on the spectra of unresolvably split multiplets. The spectrum of such a multiplet is usually well modeled by a resonance function whose center frequency and attenuation rate are perturbed from the degenerate values. These variations can be predicted using a generalization of Jordan's location parameter. In particular, we develop an expression for the instantaneous complex frequency shift of the seismogram of a multiplet due to aspherical structure. We term this shift the generalized location parameter and denote it by λ(t); λ(0) is equal to the location parameter λ. We find that weighted averages of λ(t) may be used to predict the frequency and attenuation perturbations due to aspherical structure quite successfully, and since λ(t) is usually a slowly varying function of time, it may be adequately approximated by its low‐order Taylor series expansion. The viability of these approximations enables us to use both center frequency and attenuation measurements to estimate so‐called structure coefficients, which are linear functionals of aspherical structure. We have made resonance function fits to nearly 3000 data spectra for modes 0S20–0S45 and obtained approximately 1200 high‐quality measurements per mode. To estimate the structure coefficients, we linearize the dependence of the frequency and attenuation perturbations on them and iteratively solve the resulting nonlinear problem. When we estimate degree 2–8 elastic structure coefficients for the real Earth, we find that the attenuation measurements are not well modeled when only frequency measurements are used to constrain the solution. The attenuation variance decreases only slightly when the attenuation measurements are used as additional constraints. The use of attenuation values does help to stabilize the structure coefficient estimates, however, as evidenced by an increased correlation between coefficients for modes of adjacent 𝓁. We also find evidence of weak degree 2 anelastic structure which correlates negatively with degree 2 elastic structure, as is geologically reasonable. We believe that higher‐order (above degree 8) elastic structure and along‐branch coupling may be responsible for some of the remaining unmodeled scatter in both the frequency and attenuation data. Our preferred set of estimated structure coefficients is compared with those from previous surface wave, waveform, and free oscillation studies. Quantitative agreement is good only for the strong degree 2 component of aspherical structure, though correlations are also high for degree 6. Our structure coefficients correlate at the 75–80% confidence level with the degree 2–8 geoid coefficients. The correlations are not all of the same sign, which may be related to sign changes in dynamic geoid kernels or to compensating upper and lower mantle heterogeneities. There are strong correlations between the degree 2 structure coefficients and subducting slab models and between the degree 6 structure coefficients and the hotspot distribution, supporting previous speculation of some relationship between them. Correlations with structure coefficients derived from a tectonic regionalization are poor and imply that surface tectonic features are not strongly related to upper mantle structure below 150 km in depth to which modes 0S20–0S45 are most strongly sensitive.