A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of interia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We show how to determine whether a given finite set of gross Earth data can be used to specify an Earth structure uniquely except for fine-scale detail; and how to determine the shortest length scale which the given data can resolve at any particular depth. We apply the general theory to the linear problem of finding the depth-variation of a frequencyindependent local Q from the observed quality factors Q of a finite number of normal modes. We also apply the theory to the non-linear problem of finding density vs depth from the total mass, moment, and normal-mode frequencies, in case the compressional and shear velocities are known.

A cyclic process of refining models of the mechanical structure of the Earth and models of the mechanism of one or more earthquakes is developed. The theory of the elastic-gravitational free oscillations of the Earth is used to derive procedures for resolving nearly degenerate multiplets of normal modes. We show that a global network of seismographs (W.W.S.S.N.) permits resolution for angular orders l ≤ 76 and for frequencies a) w ≤ 0.090 s -1 . The peak or centre frequency of each nearly degenerate multiplet is interpreted to be a gross Earth datum. Together, the data are used to refine models of the mechanical structure of the Earth. The theory of free oscillations is used further to derive procedures for retrieving the mechanism, or moment tensor, of an earthquake point source. We show that a globa network of seismographs permits retrieval for frequencies 0.0125 s-1 ≤ w ≤ 0.0825 s-1 . We show that refined models of structure and mechanism lead to improved resolution and retrieval, and that an array of sources further complements the resolution of multiplets. We present a ‘standardized dataset’ of 1064 distinct, observed eigenfrequencies ol the elastic-gravitational free oscillations of the Earth. These gross-Earth data are compiled from 1461 modes reported in five studies: 2 modes reported by Derr (1969), 159 modes observed by Brune & Gilbert (1974), 240 modes observed by Mendiguren ( 1973), 248 modes observed by Dziewonski & Gilbert (1972,1973) and 812 modes reported here. It is our opinion that the establishment of a standardized dataset should precede the establishment of a standardized model of the Earth. Two new Earth models are presented that are compatible with the modal data. One is derived from model 508 (Gilbert & Dziewonski 1973) and the other from model B1 (Jordan & Anderson 1974). In the outer core and in the lower mantle, below a depth of about 950 km, the differences between the two models are negligibly small. In the inner core there are minor differences and in the upper mantle there are major differences in detail. The two models and the modal data are compatible with traditional ray data, provided that appropriate baseline corrections are made to the latter. The source mechanisms, or moment tensors, of two deep earthquakes, Colombia (1970 July 31) and Peru-Bolivia (1963 August 15), have been retrieved from the seismic spectra. In both cases the moment tensor possesses a compressive (implosive) isotropic part. There is good evidence that isotropic stress release begins gradually, over 80s before the origin time derived from the onset of short-period P and S waves. During the process of stress release the principal axes of the moment rate tensor migrate. The axis of compression is relatively stable, the compressive stress rate is dominant, and the other two axes rotate about the axis of compression. We speculate that earthquakes, occurring deep within descending lithospheric plates, are not sudden shearing movements alone but do exhibit compressive changes in volume such as would be associated with a phase change. We further speculate that compressive changes in volume may occur without sudden shearing movements, that there may be ' silent earthquakes’.

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Some century-old results, due to Rayleigh and Routh, are used to derive a very compact and simple representation for the transient response of the Earth to earthquakes. In particular, it is shown how the residual static displacement field is naturally represented in terms of the normal mode eigenfunctions. ReferenceRayleigh, Lord, 1877. The Theory of Sound, I, Chapter IV and V, MacMillan, London.

The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order, ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients. In this paper we summarize some of the better known properties of propagators plus numerical methods for their computation. When the dispersion relation is some mth order minor of the integral matrix it is possible to deal with mth minor propagators so that the dispersion relation is a single element of the mth minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided. Propagator equations for SH and for P‐SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator equations for P‐SV waves are given. Matrix polynomial approximations to the propagators, obtained from the method of mean coefficients by the Cayley‐Hamilton theorem and the Lagrange‐Sylvester interpolation formula, are derived.

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We use the stationarity of the Fermat ray path to develop theoretical expressions that relate small, aspherical perturbations in velocity to small perturbations in travel times. For the Earth's ellipticity of figure we derive a compact expression for the perturbation in travel time. If the ellipticity is hydrostatic and is the dominant perturbation, then anomalies in travel times are linear constraints on the radial gradient of velocity. Otherwise, the anomalies are constraints on the (unknown) aspherical perturbations. We are led to an inverse problem in either case.Using recently derived models of the Earth we present calculations of the effect of ellipticity. If the effect of focal depth is neglected in the calculations, then mislocations in both epicentre and origin time can result. Differences between our calculations and those predicted by the approximate formula 6t = ( h + H ) f ( A ) are as large as 0-25 s for P at 90". Nowadays, seismologists are prone to attach significance to anomalies as small as 0.10 s. Consequently, we advocate that the effect of focal depth be considered and that traditional approximations be replaced by the more accurate calculations tabulated in this report. Theoretical developmentThe effect of the Earth's ellipticity of figure on travel times was first studied more than 40 years ago by Comrie and by Gutenberg and Richter. Bullen (1965, p. 173-176) has presented a concise theoretical derivation of the effect.Recent improvements in our knowledge of the mechanical structure of the Earth make it desirable not only to re-examine the effect of ellipticity, but also to consider other small, aspherical perturbations as well. We follow the derivation of Bullen A source is located a (r,,, ,!lo, &) in spherical polar co-ordinates, and a receiver at (1965, p. 173-176). (r, 9,d). Surfaces of constant velocity, v, are specified as w, 4 = r+Jr($, 4) (1) where r is the spherically averaged value of R($,c$) s1 so that u(R(9,4)) = u(r+Br(S, 4)) is constant. * Received in original form 1975 May 1. 7 8 A. M. Dziewonski and F. Gilbert k -0.6-J FIG. 2. Comparison of (P phase) ellipticity correction coefficients from equations (22) and (25) for a surface focus.