 # Seismic Wave Propagation in Stratified Media

BRIAN L. N. KENNETT
https://www.jstor.org/stable/j.ctt24h2zr

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-x)
B.L.N. Kennett
4. Chapter 1 Introduction
(pp. 1-19)

The surface of the Earth is in constant slight movement and the motion at any point arises from both local effects, e.g., man-made disturbances or wind-induced rocking of trees, and from vibrations arising from afar, such as microseisms generated by the effect of distant ocean storms. If we look at the records from an observatory seismometer in its carefully constructed vault, or from a geophone whose spike is simply driven into the ground, we find that these largely consist of such seismic ‘noise’. From time to time the irregular pattern of the records is interrupted by a disturbance which rises...

5. Chapter 2 Coupled Equations for Seismic Waves
(pp. 20-36)

The incremental displacement u induced by the passage of a seismic wave is governed by the equation of motion

${\rm{\rho }}({\rm{x}}){\partial _{tt}}{u_i}({\rm{x}}) = {\partial _j}{{\rm{\tau }}_{ij}}({\rm{x}})$ , (2.1)

in the absence of sources. The behaviour of the material enters via the constitutive relation connecting the incremental stress and strain. When the approximation of local isotropic behaviour is appropriate, the displacement satisfies

${\rm{\rho }}({\rm{x}}){\partial _{tt}}{u_i}({\rm{x}}) = {\partial _j}[{\rm{\lambda }}({\rm{x}}){\partial _k}{u_k}({\rm{x}}){{\rm{\delta }}_{ij}} + {\rm{\mu }}({\rm{x}})({\partial _i}{u_j}({\rm{x}}) + {\partial _j}{u_i}({\rm{x}}))]$ , (2.2)

which leads to rather complex behaviour for arbitrary spatial variation of the elastic moduli λ,µ. As discussed in Section 1.3.3 we may accommodate slight dissipation within the medium by allowing the moduli λ,µ to be complex functions of frequency within the seismic band.

Even...

6. Chapter 3 Stress-Displacement Fields
(pp. 37-58)

In Chapter 2 we have shown how the governing equations for seismic wave propagation can be represented as coupled sets of first order equations in terms of the stress-displacement vector b. We now turn our attention to the construction of stress-displacement fields in stratified media.

We start by considering a uniform medium for which we can make an unambiguous decomposition of the wavefield into up and downgoing parts. We then treat the case where the seismic properties vary smoothly with depth. Extensions of the approach used for the uniform medium run into problems at the turning points of P or...

7. Chapter 4 Seismic Sources
(pp. 59-80)

A significant seismic event arises from the sudden release of some form of potential energy within the Earth or at its surface. For earthquakes the stored energy is usually associated with the strain built up across a fault zone by continuing deformation. In some deep events it is possible that volume collapse occurs with the release of configurational energy in upper mantle minerals in a phase transition. For explosions either chemical or nuclear energy is released. Surface impact sources dissipate mechanical energy.

In all these cases only a fraction of the original energy is removed from the ‘hypocentre’ in the...

8. Chapter 5 Reflection and Transmission I
(pp. 81-102)

Nearly all recording of seismic waves is performed at the Earth’s surface and most seismic sources are fairly shallow. We are therefore, of necessity, interested in waves reflected back by the Earth’s internal structure. In seismic prospecting we are particularly interested in P waves reflected at near-vertical incidence. For longer range explosion seismology wide-angle reflections from the crust-mantle boundary are often some of the most significant features on the records. At teleseismic distances the main P and S arrivals have been reflected by the continuously varying wavespeed profile in the Earth’s mantle. For deep sources we are also interested in...

9. Chapter 6 Reflection and Transmission II
(pp. 103-128)

In the last chapter we showed how to define reflection and transmission matrices for portions of a stratified medium bordered by uniform half spaces or a free surface.

We now demonstrate how the reflection and transmission properties of two or more such regions can be combined to give the overall reflection and transmission matrices for a composite region. These addition rules form the basis of efficient recursive construction schemes for the reflection matrices. These recursive schemes may be developed for stacks of uniform layers or for piecewise smooth structures with gradient zones separated by discontinuities in the elastic parameters or...

10. Chapter 7 The Response of a Stratified Half Space
(pp. 129-157)

We now bring together the results we have established in previous chapters to generate the displacement field in a stratified half space $z > 0$ , due to excitation by a source at a level ${z_S}$ . We suppose that we have a free surface at $z = 0$ at which the traction vanishes; and ultimately the stratification is underlain by a uniform half space in $z > {z_L}$ with continuity of properties at ${z_L}$ . In this uniform region we impose a radiation condition that the wavefield should consist of either downward propagating waves or evanescent waves which decay with depth, the character depending on the slowness. As...

11. Chapter 8 The Seismic Wavefield
(pp. 158-176)

So far in this book we have shown how we may calculate complete theoretical seismograms for a horizontally layered medium. Such calculations are most useful when the total time span of the seismic wavetrain is fairly short and there is no clear separation between different types of wave propagation processes.

As the distance between source and receiver increases, the wavetrain becomes longer and waves which have travelled mostly as P waves arrive much earlier than those which propagate mostly as S. Also the surface waves which are principally sensitive to shallow S wavespeed structure separate out from the S body...

12. Chapter 9 Approximations to the Response of the Stratification
(pp. 177-208)

We have already seen how we may generate the complete response of a stratified medium, and we now turn our attention to the systematic construction of approximations to this response, with the object of understanding, and modelling, the features we have seen on the seismograms in Chapter 8. We will develop these approximations by exploiting the physical character of the solution and a very valuable tool will be the partial expansion of reverberation operators. The identity

${[{\rm{I - R}}_u^{AB}{\rm{R}}_D^{BC}]^{ - 1}} = {\rm{I + R}}_u^{AB}{\rm{R}}_D^{BC} + {\rm{R}}_u^{AB}{\rm{R}}_D^{BC}{\rm{R}}_u^{AB}{\rm{R}}_D^{BC}{[{\rm{I - R}}_u^{AB}{\rm{R}}_D^{BC}]^{ - 1}}$ , (9.1)

enables us to recognise the first internal multiple in ‘AC’, cf. (6.18), whilst retaining an exact expression for the effect of the...

13. Chapter 10 Generalized ray theory
(pp. 209-233)

In the previous chapter we have developed approximations to the seismic response in which, in essence, we retain frequency dependence in the amplitude of any reflection effects and so a numerical integration over slowness is needed. We now turn our attention to a further class of approximation in which the factorisation of the seismic response is carried even further.

The seismic displacement is represented as a sum of ‘generalized ray’ contributions for which the amplitude depends only on slowness and the phase has a slowness dependent term multiplied by frequency. For each of the generalized rays we are able to...

14. Chapter 11 Modal Summation
(pp. 234-270)

The various expressions which we have derived for the receiver response for general point source excitation have singularities associated with the properties of the reflection and transmission matrices for portions of the stratification and the corresponding reverberation operators. In particular we have a set of poles associated with the vanishing of the secular function for the half space $\det \{ {T_{DL}}(0)\}$ (7.12). This secular function is independent of the depth of the source and depends on the elastic properties in the half space.

For the combinations of frequency and slowness for which $\det \{ {T_{DL}}(0)\}$ vanishes we have non-trivial solutions of the equations of...

15. Appendix: Table of Notation
(pp. 271-274)
16. References
(pp. 275-284)
17. Index
(pp. 285-288)