The static deformation of an elastic half-space by surface pressure is reviewed. A brief mention is made of methods for solving the problem when the medium is plane stratified, but the major emphasis is on the solution for spherical, radially stratified, gravitating earth models. Love-number calculations are outlined, and from the Love numbers, Green's functions are formed for the surface mass-load boundary-value problem. Tables of mass-load Green's functions, computed for realistic earth models, are given, so that the displacements, tilts, accelerations, and strains at the earth's surface caused by any static load can be found by evaluating a convolution integral over the loaded region.

Deformation of the Earth by surface loads

Publisher Summary This chapter focuses on elastic wave propagation in stratified media. The development of the theory of elastic wave propagation in stratified media has been strongly influenced by the problems of seismic wave propagation and the nature of the seismograms recorded from earthquakes. For purely analytic developments of elastic wave propagation, the level of manageable algebraic complexity is reached in a model with one or two uniform layers overlying a uniform half space. This chapter shows how the excitation of elastic waves, within a horizontally stratified structure, can be conveniently developed in terms of reflection and transmission matrices. This procedure has allowed the construction of the full response of the medium or approximations with desired properties so that theoretical seismograms may be calculated for realistic distributions of elastic parameters. Although this development has been for isotropic media, nearly all the results apply directly to the case of full anisotropy if 3 × 3 reflection and transmission matrices allowing coupling between all wave types are employed. This development of the wavefield for both source and receiver within the stratification may be used for other classes of wave propagation.

Seismic Wave Propagation in Stratified Media

Preface 1. Origin and history of the Solar System 2. Composition of the Earth 3. Radioactivity, isotopes and dating 4. Isotopic clues to the age and origin of the Solar System 5. Evidence of the Earth's evolutionary history 6. Rotation, figure of the Earth and gravity 7. Precession, wobble and rotational irregularities 8. Tides and the evolution of the lunar orbit 9. The satellite geoid, isostasy and post-glacial rebound 10. Elastic and inelastic properties 11. Deformation of the crust: rock mechanics 12. Tectonics 13. Convective and tectonic stresses 14. Kinematics of the earthquake process 15. Earthquake dynamics 16. Seismic wave propagation 17. Seismological determination of Earth structure 18. Finite strain and high pressure equations of state 19. Thermal properties 20. The surface heat flux 21. The global energy budget 22. Thermodynamics of convection 23. Thermal history 24. The geomagnetic field 25. Rock magnetism and paleomagnetism 26. Alternative energy sources and natural climate variations: some geophysical background Appendix A. General reference data Appendix B. Orbital dynamics (Kepler's laws) Appendix C. Spherical harmonic functions Appendix D. Relationships between elastic moduli of an isotropic solid Appendix E. Thermodynamic parameters and derivative properties Appendix F. An Earth model: mechanical properties Appendix G. A thermal model of the Earth Appendix H. Radioactive isotopes Appendix I. A geological time scale 2004 Appendix J. Problems References Index.

Physics of the Earth

Summary
A gross datum of the Earth is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We prove that the collection of Earth models which yield the physically observed values of any independent set of gross Earth data is either empty or infinite dimensional. We exploit this very high degree of non-uniqueness in real geophysical inverse problems to generate computer programs which iteratively produce Earth models to fit given gross Earth data and satisfy other criteria. We describe techniques for exploring the collection of all Earth models which fit given gross Earth data. Finally, we apply the theory to the normal modes of elastic-gravitational oscillation of the Earth.

Numerical Applications of a Formalism for Geophysical Inverse Problems

An extended form of the correspondence principle is employed to determine directly the quasi-static deformation of viscoelastic earth models by mass loads applied to the surface. The stress-strain relation employed is that appropriate to a Maxwell medium. Most emphasis is placed on the discussion of spherically stratified self-gravitating earth models, although some consideration is given to the uniform elastic half space and to the uniform viscous sphere, since they determine certain limiting behaviors that are useful for interpretation and proper normalization of the general problem. Laplace transform domain solutions are obtained in the form of ‘s spectra’ of a set of viscoelastic Love numbers. These Love numbers are defined in analogy with the equivalent elastic problem. An efficient technique is described for the inversion of these s spectra, and this technique is employed to produce sets of time dependent Love numbers for a series of illustrative earth models. These sets of time dependent Love numbers are combined to produce Green functions for the surface mass load boundary value problem. Through these impulse response functions, which are obtained for radial displacement, gravity anomaly, and tilt, a brief discussion is given of the approach to isostatic equilibrium. The response of the earth to an arbitrary quasi-static surface loading may be determined by evaluating a space-time convolution integral over the loaded region using these response functions.

The impulse response of a Maxwell Earth

A study is made of the free and forced oscillations of the earth. The natural periods are determined for radial, torsional and spheroidal types of oscillation. Several models of the earth are used: a homogeneous model, such as was assumed originally by Love, a model consisting of a homogeneous solid mantle enclosing a homogeneous liquid core, Bullen's model B, and Bullard's models I and II. It is found that the spheroidal oscillation of order 2 has a period of about 53$\cdot $5 min in all models, except the homogeneous one, for which this period is only 44$\cdot $3 min. The common period of 53$\cdot $5 min agrees to within the observational error with the period of 57 min observed by Benioff on the seismograms of the Kamchatka earthquake of 1952. In contrast to all the other models, Bullen's model B possesses an additional spheroidal oscillation of order 2 of a period of 101 min. The latter oscillation is confined mainly to the core, its amplitude in the mantle being relatively very small. Benioff's observation of a second oscillation of a 100 min period in the Kamchatka earthquake record might be considered as evidence favouring Bullen's model B. The latter differs from Bullard's models mainly by having a central density of around 18 instead of about 12 g/cm$^{3}$. However, a theoretical investigation of the relative excitation of the various free modes by an impulsive compressional point-source situated at a shallow focal depth, shows that the amplitude of the 100 min oscillation should be more than 1000 times weaker than that of the 53$\cdot $5 min oscillation. It is thus not clear how a near-surface earthquake could have excited the core-oscillation. The spectrum of the free modes of oscillation has also been determined for n = 3 and 4, including the fundamental and the first two overtones for each case. The computed free periods of spheroidal oscillation range from 53$\cdot $5 min down to a period of 8 min for the fourth overtone in the case n = 2. We have also treated the bodily tides for Bullen's and Bullard's models. Love's numbers were determined in the case n = 2 for tidal periods of 6, 6 $\surd $2, 12 h and $\infty $. The dependence of the Love numbers on the period is small, a maximum range of variation of 13% occurring in the k-values between the periods of 6 h and $\infty $.

Oscillations of the Earth

Summary

The propagation of Rayleigh waves in the Earth is investigated in the whole range of periods T from about 10s up to one hour. Three methods are necessary in order to cover this range of periods effectively. The standard flat Earth method, with neglect of gravity, gives values for the phase velocity C correct to within I per cent up to T= 50s only, and for the group velocity U up to T= 250s. The method of the flattening of the Earth, with neglect of gravity, has the I per cent accuracy limits for C and U at 300 and 400, respectively. Inclusion of gravity effects in the flattening of the Earth approximation does not alter the above limits. For T > 300 (n < 25) one must determine the period T(n) of free oscillation of the Earth as a function of the order of the spherical harmonic n. This involves the solution of a system of differential equations of the sixth order, in which the gravitational effects are included. The wave penetrates appreciably into the core already at T= 600. Using the above three methods in their respective ranges of validity, we have evaluated C(T) and U(T) for (I) Bullen's Model B, (2) the Jeffreys-Bullen Model, as modified by Dorman, Ewing and Oliver, and (3) the Gutenberg Model. The observed Rayleigh group velocity data of Ewing and Press for T < 380s and the phase velocity data of Nafe and Brune for T < 300s agree with the values computed for the Gutenberg model, but not for the other models. This substantiates a previous conclusion reached by Takeuchi, Press and Kobayashi and by Dorman, Ewing and Oliver that the observed Rayleigh wave data provide evidence in support of Gutenberg's lowvelocity layer. The few observed Rayleigh group velocities between T= 400 and 600 are substantially lower than the theoretical values for all the three models.

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Propagation of Rayleigh Waves in the Earth

: A theory is developed for the effect of the earth's rotation on its spectrum. Each line is resolved by rotation into a multiplet of (2n+1) lines, as in the Zeeman effect. A theory is also given of the intensity distribution in the lines of the multiplet for the case of a pointsource. Good agreement is obtained between this theory and the doublets observed seismically and gravimetrically.

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Rotational Multiplets in the Spectrum of the Earth

The linearized Boltzmann integral equation for the problem of propagation of sound in a rigidsphere gas was solved by first determining the eigenfunctions of the collision operator and then solving the propagation matrix. Determinants up to order n = 105 were solved, and the results are illustrated. With the maximum order 105 of the matrix solved, the values of the propagation parameters obtained show convergence to within 2% for R > 1.8, where R = 2fc/3πf. The theoretical results are in fairly good agreement with Greenspan's measurements of the speed and attenuation of sound in rarefied helium.

Propagation of Sound in a Gas of Rigid Spheres

In this investigation we solve the problem of propagation of a compressional pulse in a solid half-space which is overlain by a solid layer of different properties. The point source is situated at the depth ½H, H denoting the thickness of the layer. Theoretical seismograms of the vertical displacement w at the surface are evaluated out to ranges r = 20H. The solution is obtained by the exact ray theory. The displacement Wo due to the source was assumed to have a shape which at large distances reduces to a sawtooth. The once-reflected waves from the interface, PP and PS, are strongly marked. The Rayleigh wave is already recognizable at r = 5H and is fully developed at r = 20H. The method of ‘ray theory’ was applied here far into the region where the normal mode theory converges well. The theoretical seismograms are illustrated. The properties assumed for the layer (1) and for the half-space (2) are λ1 = μ1 μ2 = 2μ1cs2 ≡ c2 = 1.1cs1 ≡ 1.1c1cp2 > cp1 > c2 > c1

Z. Alterman

Papers

Oscillations of the Earth

Propagation of Rayleigh Waves in the Earth

Rotational Multiplets in the Spectrum of the Earth

Propagation of Sound in a Gas of Rigid Spheres

Propagation of a compressional pulse in a layered solid.