This present study focuses on three topics: (i) formulation of the firstorder perturbation problem of the Earth's normal modes; (ii) formulation of a variational method based on the solutions of first-order quasidegenerate perturbation calculation; and (iii) numerical results for both ordinary degenerate and quasi-degenerate multiplets using formulation (i). The Earth considered is a rotating, laterally-inhomogeneous Earth. To zeroth order, however, it is approximated by a spherically symmetric, non-rotating, elastic and isotropic sphere. Lateral perturbations and Earth's rotation are treated as perturbations. Examples of lateral perturbations are Earth's ellipticity of figure and lateral inhomogeneities. Because of the symmetry in the Earth model, unperturbed modes form multiplets with varying degrees of degeneracy. For almost all of these multiplets, the perturbation calculation need be carried out to first order. When one multiplet and another nearly coincide in their frequencies, first-order ordinary degenerate perturbation method must be replaced by first-order quasi-degenerate perturbation method. Since the latter is a more general form of the former, the first-order perturbation problem is completely solved. The perturbations considered in this study consist of Earth's rotation, ellipticity of figure, a continent-ocean discontinuity near the Earth's surface, and an artificially contrived discontinuity about the core-mantle interface. When the effect due to rotation becomes dominant, the first-order calculation becomes insufficient, and a variational method is posed. Such a method can be formulated from quasi-degenerate solutions. The numerical results show that ellipticity corrections are important and should always be included in the computations of the Earth's normal modes, that lateral inhomogeneities in the Earth's deep interior become an important perturbation for high-overtone multiplets, and that Coriolis coupling between a spheroidal and a toroidal multiplet overshadows all other types of coupling.( O 2 5 -Y ) ( U Y v) = 0. Note that w2 I and Y here differ from the ones in equation (2.3). There po and C were functions of three-dimensional space co-ordinates; here they are functions of
The scheme leading to the computation of the normal mode eigenfrequencies of the laterally inhomogeneous Earth is derived. Rayleigh's variational principle is used to calculate the first-order corrections to the eigenfrequencies of a spherically symmetric, non-rotating and elastic Earth model due to the lateral variations in the real Earth. The WignerEckart theorem is used, whenever applicable, to simplify many complicated angular integrals involving the strain deviator tensor to more basic and manageable integrals involving three spherical harmonics. In particular, near-resonance multiplet coupling for spheroidal-spheroidal, spheroidaltoroidal and toroidal-toroidal multiplets is presented along with the uncoupled case of a single arbitrary spheroidal or toroidal multiplet.
Theoretical normal-mode spectra for four spheroidal and four toroidal multiplets at six different locations of WWSSN stations are computed for a rotating elliptical Earth with a given dissipation and double-couple source models for Chilean and Alaskan earthquakes. The most important difference between these spectra and those computed for a spherical nonrotating Earth is in the former being susceptible to the effect of mutual constructive or destructive interferences among excitations of individual singlets within a multiplet. Although this effect is expected because of fine-structure splittings within a multiplet, a calculation is necessary to determine the extent of interferences. In some cases, spectral shapes of split multiplets are sizeably narrower, broader, higher or smaller than those of respective degenerate multiplets; it is also not uncommon to find at least two peaks within a multiplet, causing an ambiguity as to where within that multiplet is the appropriate position for assigning an observed period associated with that multiplet. Such ambiguity arises only because description of a whole multiplet is constrained to one single spectral peak -a prevalent practice thus far in interpreting observed spectra-while actually (21 + 1) spectral peaks should be simultaneously considered.However, mutual interferences among singlets render such consideration impossible at present due to limited amount of data of imperfect quality and due to our lack of knowledge on dissipation and lateral inhomogeneities within the Earth's interior. Thus, as shown by the results of this study, care must be administered when one interprets real data by using those methods that are appropriate for studying spherically symmetric, nonrotating Earths. On the average, the relative deviation between the location of the maximum amplitude of a split peak and the respective degenerate one and, similarly, between the two values of Q is found to be 0.1 and 20 per cent respectively, while maximum amplitudes differ by 30 per cent.
Using Gilbert & Dziewonski's retrieved structural parameters, Earth model 1066A, their Q,(r) model, and the moment rate tensor for the Colombian earthquake of 1970 July 31, we produce 75 theoretical seismograms in epicentral co-ordinates by superimposing all the normal inodes (1105 modes) within a period range from 100.1 to 963.8 s. The computed seismograms are compared with the respective observed ones. Such a comparison is possible not only because all the modes in the frequency range are taken into account in the computation, but also because we have a realistic kinematic source mechanism at our disposal. A qualitative, though far from exact, reproduction of actual seismograms was successfully effected, indicating the moment rate tensor represents a reasonable source model for the Colombian earthquake. In particular, on the average we are able to reproduce the observed amplitudes to within 30 per cent. Among surface and body-wave phases identified, multiple S-waves can be identified up to 21 S at almost 8 h after the origin time on both the observed and computed co-latitudinal record sections.
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