### Abstract

We derive an explicit formula for the response of a laterally heterogeneous, self‐gravitating, rotating, dissipative, and physically dispersive earth to an earthquake. The long‐period spectrum at a single station is given by u(ω) =R^{H}_{0} [V(ω) +ωW−ω^{2}T]^{−1}·S_{0}(ω) where R_{0} and S_{0}(ω) are the unperturbed source and receiver vectors and H denotes the Hermitian transpose. The quantities V(ω), W, and T are the potential energy, Coriolis, and relative kinetic energy matrices, and V(ω) +ωW−ω^{2}T is the Lagrangian matrix whose eigenvalues are the perturbed complex eigenfrequencies. This simple result is obtained by formal summation of the Born series.

Original language | English (US) |
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Pages (from-to) | 527-533 |

Number of pages | 7 |

Journal | Geophysical Journal International |

Volume | 100 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1990 |

### All Science Journal Classification (ASJC) codes

- Geophysics
- Geochemistry and Petrology

### Keywords

- Born approximation
- free oscillations
- lateral heterogeneity
- normal modes
- synthetic seismograms

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## Cite this

Tromp, J., & Dahlen, F. A. (1990). Summation of the Born series for the normal modes of the Earth.

*Geophysical Journal International*,*100*(3), 527-533. https://doi.org/10.1111/j.1365-246X.1990.tb00704.x