Waves and Rays in Seismology

Answers to Unasked Questions

Michael A Slawinski

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813239890
• Edition: 2
• Topic: Scienze fisiche
• Subtopic: Geofisica

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The author dedicates this book to readers who are concerned with finding out the status of concepts, statements and hypotheses, and with clarifying and rearranging them in a logical order. It is thus not intended to teach tools and techniques of the trade, but to discuss the foundations on which seismology — and in a larger sense, the theory of wave propagation in solids — is built. A key question is: why and to what degree can a theory developed for an elastic continuum be used to investigate the propagation of waves in the Earth, which is neither a continuum nor fully elastic. But the scrutiny of the foundations goes much deeper: material symmetry, effective tensors, equivalent media; the influence (or, rather, the lack thereof) of gravitational and thermal effects and the rotation of the Earth, are discussed ab initio. The variational principles of Fermat and Hamilton and their consequences for the propagation of elastic waves, causality, Noether's theorem and its consequences on conservation of energy and conservation of linear momentum are but a few topics that are investigated in the process to establish seismology as a science and to investigate its relation to subjects like realism and empiricism in natural sciences, to the nature of explanations and predictions, and to experimental verification and refutation.

In the second edition, new sections, figures, examples, exercises and remarks are added. Most importantly, however, four new appendices of about one-hundred pages are included, which can serve as a self-contained continuum-mechanics course on finite elasticity. Also, they broaden the scope of elasticity theory commonly considered in seismology.

--> Contents:

• Science of Seismology
• Seismology and Continuum Mechanics
• Hookean Solid: Material Symmetry
• Hookean Solid: Effective Symmetry and Equivalent Medium
• Body Waves
• Surface, Guided and Interface Waves
• Variational Principles in Seismology
• Gravitational and Thermal Effects in Seismology
• Seismology as Science
• Appendices:
  • On Strains
  • On Stresses
  • On Thermoelasticity
  • On Hyperelasticity
  • On Covariant and Contravariant Transformations
  • On Covariant Derivatives
  • List of Symbols

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Readership: Students, professionals, researchers, and laypersons interested in seismology.
-->Keywords:Elasticity Theory;Inverse Problems;Seismology;Continuum Mechanics;Mathematical PhysicsReview:

"This one-of-a-kind book is refreshing in its presentation of an amazing blend of fundamental scientific and philosophical questions with their practical implications to concrete examples in Seismology. It is refined in its style, in the sophistication of its quotes, in the breadth of its sources and in the many details that reveal a labour of love. As an additional bonus, the book is also extremely useful. It presents the underlying theory of the relevant aspects of Continuum Mechanics in a clear and sufficiently rigorous way, while challenging the reader's intellect at every step of the way... This inspiring book is highly recommended."

Professor Marcelo Epstein
University of Calgary, Canada

"This book provides an extensive and self-contained treatment of the mathematical theory of wave propagation in elastic continua, with special attention to topics, some of them well advanced, which are most important for their applications in geophysics... The author's wide culture, clear style and rigorous approach make this book a first foundation stone of a field which should be called Rational Seismology."

Professor Maurizio Vianello
Politecnico di Milano, Italy
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Chapter 1

Science of seismology

[T]he rich available stock of structures that pure mathematics studies with great generality not only furnishes physicists with a practically inexhaustible source of revolutionary concepts, but it also facilitates the comparison between new theories and their predecessors.¹

Roberto Toretti (1999)

Preliminary remarks

In this chapter, we discuss disciplines that, in a variety of ways, contribute to the science of seismology, such as mathematical physics, continuum mechanics, computational science and electromagnetism. Also, we discuss branches of seismology, such as earthquake seismology, exploration seismology, global seismology and theoretical seismology. Even such a provisional, and admittedly arbitrary, list, with different methods and research strategies for each discipline or branch, indicates the multidisciplinarity of seismology. The purpose of this chapter is to examine such a list. This purpose is distinct from the purpose of Chapter 9, where we discuss features that make seismology a science. Therein, the question is not which disciplines contribute to seismology, but whether or not seismology is a science. Herein, instead of discussing criteria required by seismology to be a science, we examine common threads for seismological subjects.

We begin this chapter with a historical sketch. We conclude with a classification of seismological subjects, and select those pertaining to this book.

1.1Purpose and methodology: Historical sketch

In this section, we examine scientific pursuits that constitute seismology and the place of seismology within a hierarchy of science. We pursue this discussion within the context of a history of seismology.

Seismology could be classified as a discipline belonging to geophysics and, in general, to Earth Sciences, due to commonality of purpose, sensu lato, among the geosciences. Such a classification, however, ignores distinct methodologies, even within geophysics itself. A geophysicist studying earthquakes, to whom we refer as an earthquake seismologist, invokes largely different physical principles from those used by a geophysicist examining the Earth’s magnetic field. Even fewer commonalities of method appear if we compare an earthquake seismologist with a palæobotanist.

In view of commonality of method, seismology—or at least its theoretical underpinnings—could be classified as a branch of continuum mechanics. In an analogous manner, a study of the Earth’s magnetic field could be classified as a branch of electromagnetism. As discussed on page iii, below, both continuum mechanics and electromagnetism are basic sciences, and both introduce an abstract concept of a field as their underlying entity.

The word seismology is composed of Greek terms. Its first part refers to earthquakes and the second to an enquiry. It appeared in the middle of the nineteenth century. The word geophysics is also composed of Greek terms. Its first part refers to the Earth and the second to the science of natural things. It appeared towards the end of the nineteenth century.

Thus, for instance, Edmond Halley, whom we could now describe as an astronomer, mathematician, physicist, meteorologist and geophysicist, would not have been described as a geophysicist in his lifetime. Philosophically, Halley would have objected to the distinction between a physicist and a geophysicist, since there are no fundamental principles of physics that distinguish physics, in general, from physics of the Earth. Removal of such a distinction—which stemmed from Aristotelian physics that distinguished between the changeable terrestrial realm and the permanent celestial realm—was one of the main achievements of the Scientific Revolution, which provided a research platform for Halley. The present distinction between geophysics and physics is based on the focus of enquiry, not on different physical principles.

A difficulty for the taxonomy of seismology and geophysics can be, at least in part, explained by considering the foundational work of Mario Bunge (1967), a philosopher of physics, according to whom neither geophysics nor seismology are basic sciences. According to the nomenclature of Bunge (1967), they are based sciences, since they rely on other disciplines, which themselves are basic, like continuum and quantum mechanics. Neither continuum nor quantum mechanics relies on other disciplines, even though only the latter enquires into fundamental properties of materials; the former is, to a certain degree, a selfsufficient black-box theory.

If we view quantitative seismology as a subject of continuum mechanics, we accept—using the terminology of Bunge (1967)—its hypotheti-codeductive formulation. For instance, the P and S waves, which are commonly—and rather hastily—viewed as physical phenomena within the Earth, are neither exclusive properties of seismology nor—perhaps more importantly—does their concept originate within empirical studies of Earth Sciences: they belong to the mathematical field of mechanics, where they are abstract entities whose properties serve as quantitative analogies for experimental measurements.

Under Robert Hooke’s seventeenth-century assumption of linear dependence between force and deformation, and the principle of balance of linear momentum—together with the stress tensor, formulated in the nineteenth century by the French mathematician Augustin-Louis Cauchy—we derive wave equations in an isotropic continuum. The form and properties of these equations were already known to Jean-Baptiste le Rond d’Alembert, a French mathematician, mechanician, physicist, philosopher and music theorist. Hooke, himself, was a natural philosopher, architect and polymath. Thus, as a subject of continuum mechanics, quantitative seismology is a hypotheticodeductive discipline, where hypotheses are Hooke’s law and balance of momentum, and deduction is the mathematical derivation of the wave equation.

As described by Love (1892, Historical introduction, p. 25), based on these assumptions and using a deductive argument,

[t]he theory of the propagation of waves in an unlimited isotropic elastic medium was first considered by Poisson, who, in his memoir of 1828, shewed that there are two kinds of waves, one waves of compression, and the other waves of distortion, and that these are propagated independently with different velocities.

Thus, in 1828, the mathematical existence of the P and S waves was recognized in the realm of Hookean solids. Strictly speaking, their existence is limited to that realm. Yet, their empirical counterparts can be considered in the physical world.

Richard Dixon Oldham, a British geologist, is credited with having first interpreted a seismogram in the context of separate arrivals of the P and S waves. Such an interpretation, however, does not imply that these waves are intrinsic physical phenomena within the Earth. As illustrated in Exercises 1.1 and 1.2, such an interpretation is tantamount to extracting, from many events that appear on a seismic record, the two events that are in a reasonable agreement with the quantitative predictions resulting from deduction. In interpreting recorded data, Oldham pursued an empirical science, but he achieved this by invoking analogies between physical observations and mathematical entities; the latter resulted from the work of Hooke, d’Alembert, Cauchy and others. In doing so, Oldham demonstrated that Hooke’s assumption and subsequent derivations result in pertinent analogies for physical observations. He justified a deductive continuum-mechanics formulation in the context of experimental seismology.

To understand the meaning of analogy and interpretation in Oldham’s work, let us emphasize that the P and S waves do not have an independent physical existence in a manner akin to Mars or Venus, which exist as planets, regardless of any model of the solar system. Again, as described by Love (1892, Historical introduction, pp. 25-26), following a deductive argument, in 1842,

Green considered the propagation of plane waves in an æolotropic medium and concluded that there are three kinds of waves which are propagated with different velocities.

As in the case of P and S waves, an interpretation of seismological data in terms of three kinds of waves, which we denote by qP, qS₁ and qS₂, where q stands for “quasi”,² was not feasible, from a pragmatic viewpoint, until sufficiently accurate seismograms were developed. However, fundamentally, such an interpretation was impossible prior to the recognition of the mathematical existence of these waves.

The requirement of a mathematical model to precede a specific physical interpretation can be illustrated as follows. In general, each of the three waves in an anisotropic solid exhibits a different velocity than the speed of either of the two waves in an isotropic one. Thus—depending on the choice of the model—a given event on the seismogram is associated with a wave from the model under consideration. In spite of this apparent inconsistency—where the same event can be associated with two distinct waves, say, P and qP —both models lend themselves to interpretations of a seismogram.

With a certain abuse of terminology, we could say that the ontology of a particular seismic wave is derived from its epistemology: the physical existence of a given wave is a function of the method for its enquiry. In other words, among a plethora of mechanical disturbances recorded by a seismograph, one model allows us to identify disturbances whose physical properties exhibit analogies that are pertinent to the behaviour of waves contained within that model, and another model allows us to identify the same disturbances with different waves, which are contained within the second model. Even though one of the models might be more accurate, it cannot be endowed with more physical reality, since this accuracy is a quality of a model, which remains in the abstract realm. The increase of accuracy means only that a given model might provide a better analogy. However, as discussed in Chapter 4, it is not necessarily so; in particular, if data are contaminated with measurement errors.

In the context of various pursuits pertaining to the Earth, it is of interest to note that, based on his seismic interpretation, Oldham is also credited with the first convincing inference that the Earth has a central core, which had been already suggested by ...