TY - JOUR

T1 - Synthetic free-oscillation spectra

T2 - An appraisal of various mode-coupling methods

AU - Yang, Hsin Ying

AU - Tromp, Jeroen

N1 - Funding Information:
We welcome the comments and suggestions of the editor Michael Ritzwoller, David Al-Attar and an anonymous reviewer, which helped to improve the manuscript. HY also thanks Shu-Huei Hung and Li Zhao for discussions and Jose E. Roman for inspiring the idea of auxiliary right eigenproblems. Numerical simulations for this article were performed on a Dell cluster built and maintained by the Princeton Institute for Computational Science & Engineering (PICSciE). Figures were produced based on the Generic Mapping Tool (Wessel et al. 2013). The open source seismic measurement software package FLEXWIN used for this research is freely available for download via theComputational Infrastructure for Geodynamics (CIG; geodynamics.org). HY was supported by National Science Council of Taiwan (NSC) grant 102-2917-I-564-063 during her stay at Princeton University. This researchwas further supported byNSF grants 1063057 and 1112906 andNSC grants 102-2116-M-002-025 and 103-2811-M-002-223.
Publisher Copyright:
© 2015 The Authors. Published by Oxford University Press on behalf of The Royal Astronomical Society.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - 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 ∼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.

AB - 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 ∼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.

KW - Computational seismology

KW - Surface waves and free oscillations

KW - Theoretical seismology

KW - Tomography

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U2 - 10.1093/gji/ggv349

DO - 10.1093/gji/ggv349

M3 - Article

AN - SCOPUS:85024114848

VL - 203

SP - 1179

EP - 1192

JO - Geophysical Journal International

JF - Geophysical Journal International

SN - 0956-540X

IS - 2

ER -