Eigenvibrations of laterally inhomogeneous Earth models

Abstract

Summary. A realistic three-dimensional Earth having both radial and lateral variations in seismic properties has been modelled using finite elements. Numerical algorithms are produced that yield efficient estimations of the non-degenerate eigenfunctions and eigenfrequencies of this laterally in-homogeneous spheroid. These serve as quite independent alternatives to previous mathematical methods used to study this problem. The algorithms rely upon Rayleigh's Variational Principle to solve the finite element eigen-problem. The Earth model used is perfectly elastic and non-rotating. The finite element formulation is demonstrated numerically in the case of eigenvibrations of low order torsional oscillations, 0T2 to 0T10, but also applies to the spheroidal oscillations. Theoretical splitting of the eigenfrequencies due to the Earth's ellipticity and to the structural contrasts between continents and oceans is obtained. Eigenfunctions of a realistic laterally inhomogeneous Earth model are computed for this suite of torsional oscillations. It is seen that the relative positions of the continents establish patterns of displacement which may provide explanations for the variations in eigen-spectra observed from one seismographic station to another. The deviations in eigenfrequencies due to the structural contrasts between the continents and the oceans are found to increase from 0.01 per cent of the unperturbed frequency for 0T2 to 0.1 per cent for 0T10. Over the same range of eigenvibrations, the effects on the eigenfrequencies of the Earth's ellipticity are about 0.2 per cent. The amplitudes of the eigenfunctions of these oscillations are seen to have significant global variation. The lower order modes show a latitudinal dependence of the amplitudes that would be expected were the Earth a simple ellipsoid. The higher order modes have more longitudinal amplitude dependence as a result of the increased perturbation effects of the continents. This work demonstrates that the finite element method can be used as a practical scheme for calculating theoretical eigenspectra for global seismographic stations.