Modelling of coupled normal modes of the Earth: the spectral method

Abstract

SUMMARY Modelling of coupled free oscillations or seismograms in an earth with small-scale lateral heterogeneities (a few hundred kilometres) is presently impossible without strong approximations, such as taking into account the coupling effect of the neighbouring modes only. Even within this assumption, first-order perturbation theory is generally insufficient, and variational theory must be used, leading to numerically heavy diagonalizations. An alternative method is presented in this paper. The first characteristic of this method is the use of higher order perturbation theory, which expresses the aspherical normal modes as a power series of perturbations. This perturbation theory generalizes the classical perturbation theory, in order to take into account density heterogeneities and secular terms by a renormalization technique. We show that from the third order on, the aspherical normal modes are computed with an accuracy a hundred times better than normal mode observations usually permit. The second characteristic is the use of a generalization of the spectral method in the tensor (elastic) case. Classically, interaction terms are treated as matrix products and require computations increasing as lmax4, where lmax is the maximum angular order of the modelled modes, when coupling is fully taken into account for an earth model with small-scale heterogeneities. We show that such terms can be computed with a backward and forward Legendre transformation, for which computations increase only as lmax3. This method is thus faster by an order of lmax than the variational approach. It is promising for the study of fully coupled modes and seismic waves in a realistic earth including small-scale lateral heterogeneities associated with narrow tectonic provinces such as in mid-oceanic ridges, subduction zones and continental margins.