Synthetic free-oscillation spectra: an appraisal of various mode-coupling methods

Normal-mode spectra may be used to investigate large-scale elastic and anelastic heterogeneity throughout the entire Earth. The relevant theory was developed a few decades ago, however-mainly due to computational limitations-several approximations are commonly employed, and thus far the full merits of the complete theory have not been taken advantage of. In this study, we present an exact algebraic form of the theory for an aspherical, anelastic and rotating Earth model in which either complex or real spherical harmonic bases are used. Physical dispersion is incorporated into the quadratic eigenvalue problem by expanding the logarithmic frequency term to second-order. Proper (re) normalization of modes in a 3-D Earth model is fully considered. Using a database of 41 earthquakes and more than 10 000 spectra containing 116 modes with frequencies less than 3 mHz, we carry out numerical experiments to quantitatively evaluate the accuracy of commonly used approximate mode synthetics. We confirm the importance of wideband coupling, that is, fully coupling all modes below a certain frequency. Neither narrowband coupling, in which nearby modes are grouped into isolated clusters, nor self-coupling, that is, incorporating coupling between singlets within the same multiplet, are sufficiently accurate approximations. Furthermore, we find that (1) effects of physical dispersion can be safely approximated based on either a fiducial frequency approximation or a quadratic approximation of the logarithmic dispersion associated with the absorption-band model; (2) neglecting the proper renormalization of the modes of a rotating, anelastic Earth model introduces only minor errors; (3) ignoring the frequency dependence of the Coriolis and kinematic matrices in a wideband coupling scheme can lead to similar to 6 per cent errors in mode spectra at the lowest frequencies; notable differences also occur between narrowband coupling and quasi-degenerate perturbation theory, which linearizes the eigenvalue problem as well.