ALBERT

Notes: On an aspherical earth, the normal mode amplitude pattern is perturbed by along-branch coupling. In the geometrical optics approximation, the effect of aspherical earth structure on normal mode amplitudes through along-branch coupling is equivalent to a perturbation Φ in the spatial phase of the amplitude pattern. This observable quantity is related to earth structure through a WKBJ phase integral along the source-receiver minor arc. The amplitude pattern is strongly sensitive to odd-degree earth structure and complements the information contained in the frequency shift pattern, which is sensitive to only even-degree structure. We have developed a method to estimate the amplitudes of unresolvably split multiplets from long-period spectra and have applied it to a large number of IDA and GEOSCOPE recordings to obtain a global data set of normal mode amplitudes for spheroidal modes 0S26–0S43. Normal mode amplitude estimation requires that the first surface wave arrival R1 be available in the seismogram, and coherent amplitude patterns have been obtained for 172 data records.The analysis of normal mode amplitudes affords several advantages not shared by surface wave phase velocity analysis or time-domain waveform fitting. Our method allows the estimation of errors in the model and the ability to cull away modes in the frequency domain, and it makes use of the information contained in the entire seismogram. In addition, our method makes no assumptions concerning the spectral shape of isolated multiplets in the frequency domain. The small data set employed here is only a fraction of the size of previous global data sets, but has the advantage that fewer assumptions are built in to the theory on which the structure inversion is based. Several factors, including uncertainty in the moment tensor scaling and focusing and defocusing effects, prohibit recovery of Φ for individual multiplets and limit us to the recovery of the average Φ over the mode band l= 26–43, for each source-receiver pair. Through a preferential weighting of subsets of this mode band, we may obtain estimates of Φ within narrower mode bands centred, respectively, on l= 31, l= 34, and l= 37. Least squares inversion of Φ within the various mode bands, utilizing approximately 100 source-receiver pairs, yields the geographic distribution of local eigenfrequency perturbation δωlocal(θ, φ) in a truncated spherical harmonic expansion. Inversion is first performed for the best fitting degree-6 earth model using all available modes within the band l= 26-43 with equal weight. The degree-6 earth model is obtained by inverting for degrees 1, 3, and 5 with degrees 2, 4 and 6 fixed according to published results from the frequency studies. Narrower band inversions are performed within l= 26-36, l= 29–39, and l= 31–43. Comparison with model M84A shows that the odd degree-1 and 3 components of aspherical earth structure may be reproduced from the amplitude data set. The degree-5 structure differs significantly from that of model M84A.

Notes: Expressions are developed for the frequencies and amplitudes of spheroidal and toroidal modes, taking into account the rotational coupling between modes nSl - nTl+1. and nSl - nTl-1. Starting from quasi-degenerate perturbation theory, we derive coupled wave equations which clearly show the additional restoring force due to Coriolis coupling. We employ the subspace projection method to obtain asymptotic expressions for the perturbed amplitudes and frequencies, accurate to first order in coupling strength ε and zeroth order in 1/l. The resulting expressions may be used to correct free oscillation amplitudes and frequencies in the fundamental Coriolis coupled band for the spheroidal-toroidal coupling. Data examples using preferentially aligned source-receiver paths verify the predicted patterns of amplitude and frequency perturbations.