| Some theoretical problems concerning the terrestrial spectroscopy, or the system of the Earth's normal modes of infinitesimal oscillations, can be studied in a general manner by formulating the equations of motion in terms of a single, linear "elastic-gravitational" operator, with no regard to its actual form and any details of the earth model built therein. Thus, by means of group theory, Chapter 2 studies the structure of the terrestrial spectra (particularly the degrees of degeneracy in the spectra) belonging to earth models having different symmetry properties. The analogy to quantum mechanical systems is emphasized. In addition, the influence of perturbations (particularly the pattern of splitting in the spectrum) and the selection rules for the perturbation matrices are obtained. This effort unifies the relevant observations derived otherwise with regard to the terrestrial spectroscopy that have appeared in the literatures over the years. Chapter 3 deals with the excitation of the normal modes on non-rotating and rotating earth models. It is seen that the rotation introduces a fundamental difficulty in the solution of the equations of motion for forced oscillations; and the classical methods, although applicable in the non-rotating case, are no longer adequate. However, using the method of spectral decomposition, exact, closed-form formulas for the amplitudes of the normal modes excited by any arbitrary source (for example, an earthquake) are obtained. The algebraic structure of these formulas is carefully examined based on the group-theoretical results of Chapter 2.; Also in this thesis a numerical/data processing method for the estimation of complex eigenfrequencies of the terrestrial spectrum from seismograms is developed. Thus, Chapter 4 describes the so-called "autoregressive (AR) method", which is based on Prony's method and is closely related to the linear prediction theory. It is established that the complex eigenfrequency of each normal mode corresponds to one complex conjugate pair of poles in the autoregressive model of the seismogram. The method is formulated in the frequency domain, thereby spectral peaks can be analyzed individually, or in small groups. Statistical analysis and some examples from seismic records of some major earthquakes illustrate the method.; Areas of research that the efforts of this thesis will naturally lead to include (i) the study of behavior and the observation of splitting with respect to the gravest elastic modes, (ii) the analysis of the Slichter mode and possibly its observation, and (iii) the analysis of components in the Earth's polar motions that delineate the irregularities in the Earth's rotation--particularly the Chandler wobble. |
