Theoretical free-oscillation spectra: the importance of wide band coupling

Illustrative calculations are presented of the effect of the coupling of large groups of Earth's normal modes in a realistic 3-D earth model. In previous work the effect of modal splitting and coupling has been treated either in the self-coupling approximation (SC) or in the group-coupling approximation (GC), also known as quasi-degenerate perturbation theory. In SC a mode is treated as isolated in the spectrum and the theory of degenerate splitting is used to calculate the effect of lateral heterogeneity, rotation and ellipticity on each individual modal multiplet. In GC a small number of modes close together in frequency are treated as a group and the effect of coupling within the group is included. Of course, SC can be considered as a special case of GC in which there is only a single modal multiplet in the group. In principle, modal spectra are affected by coupling among all modes. Given that explicit calculations can be performed only for a finite collection of modes, we are led to consider the question of how large the group of modes considered must be in order to obtain theoretical spectra of sufficient accuracy for a given purpose. The particular purpose that we have in mind is the use of modal spectra to refine 3-D models of earth structure. To do this we carry out calculations in which the results of SC and GC are compared with those of full coupling (FC), by which we mean coupling calculations including large collections of multiplets, for example all modes having frequencies less than a certain upper bound or cut-off frequency. The results indicate that SC and GC often represent a poor approximation to FC. The differences in modal spectra calculated using SC/GC and those obtained using FC are, in many cases, comparable to the differences between observed and theoretical spectra. We compare FC for all 140 spheroidal and toroidal modes up to 3 mHz with GC using the 25 pairs of modes defined by Resovsky & Ritzwoller (1998) and with GC in 33 subgroups of spheroidal and toroidal modes. We find that full coupling is needed to obtain the best synthetic spectra. The matrices for full coupling of 140 modes become large (approximately 2500 x 2500), but are still tractable.