### Abstract

Smith and Dahlen demonstrated that in a weakly anisotropic earth model the relative surface wave phase-speed perturbation δc/c may be written in the form, δc/c = ∑_{n=0,2,4} (A_{n} cos nζ + B_{n} sin nζ) where A_{n} and B_{n} are frequency-dependent depth integrals and ζ denotes the ray azimuth. In this approximation, surface wave anisotropy is governed by 13 elastic parameters and the azimuthal dependence of the phase speed is represented by an even Fourier series in ζ involving degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters). Jech and Pšenčík demonstrated that in such a weakly anisotropic earth model the relative compressional-wave phase-speed perturbation may be expressed as δc/c = (2c^{2})^{-1}B_{33}, whereas the relative shear wave phase-speed perturbations are given by δc/c = (4c^{2})^{-1}{B_{11} + B_{22} ± [(B_{11} - B_{22})^{2} + 4B_{12}^{2}]^{1/2}. We demonstrate that the coefficients B_{33} B_{11} B_{22}, and B_{12} may be written in the generic form, B_{lm} = ∑_{n=0}^{4}[a_{n}(i) cos nζ+b_{n}(i) sin nζ], where ζ denotes the local azimuth and i the local angle of incidence. For B_{11}, B_{22} and B_{33} the coefficients a_{n} (i) and b_{n}(i) are an even Fourier series of degrees zero, two and four in i, but for B_{12} they are an odd Fourier series of degrees one and three. Like surface waves, the azimuthal (ζ) dependence of body waves involves even degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters), but, unlike surface waves, it also involves the odd degrees one (six elastic parameters) and three (two elastic parameters). Thus, weakly anisotropic body-wave propagation involves all 21 independent elastic parameters. We use spectral-element simulations of global and regional seismic wave propagation to assess the validity of these asymptotic body and surface wave results. The numerical simulations and asymptotic predictions are in good agreement for anisotropy at the 5 per cent level.

Original language | English (US) |
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Pages (from-to) | 1130-1152 |

Number of pages | 23 |

Journal | Geophysical Journal International |

Volume | 168 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2007 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Geophysics
- Geochemistry and Petrology

### Keywords

- Body waves
- Phase-speed perturbations
- Spectral-element simulations
- Surface waves
- Weakly anisotropic earth model