On Foundations Of Seismology: Bringing Idealizations Down To Earth

Bringing Idealizations Down to Earth

James Robert Brown, Michael A Slawinski;;;

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813207028
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

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This remarkable collaboration between a mathematical physicist and a science philosopher concerns foundational and conceptual issues in seismology. Their aim is to present mathematical, physical and philosophical topics in a clear and concise manner. They provide an extensive philosophical discussion of the methods of science and show how seismology fits in. They explain with care and precision the basic structure of seismology, which is built on classical continuum mechanics. Not only do they explain how various models work in seismology, they also include an extensive discussion of the nature of models and idealizations.

--> Contents:

• Preface
• Aspects of Seismological Theory
• Nature of Science and Its Methods
• On Continua and Models
• Continuum Mechanics: General Principles, Constitutive Relations
• Hookean Solids
• Forward and Inverse Problems
• Intertheory and Intratheory Relations
• Afterword
• Bibliography
• Subject Index
• Name Index
• About the Authors and Illustrator

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Readership: Students and researchers in natural sciences and in philosophy of science.
-->Seismology;Seismic Ray;Idealization;Model Key Features:

• This book is unique in the field; it is the only work devoted to foundational issues in seismology
• It provides a serious introduction to the philosophy of science, which many working scientists will find both interesting and useful in their own research

Chapter 1

Aspects of seismological theory

Given any rule, however “fundamental” or “necessary” for science, there are always circumstances when it is advisable not only to ignore the rule, but to adopt its opposite.

Paul K. Feyerabend (1975)

Preliminary remarks

Seismology is a quantitative physical science that examines mechanical properties of our planet. Descriptions of these properties result from an interplay of mathematical strategies, computational techniques and observational apparatus. Beside its intrinsic interest in describing the Earth’s interior, seismology is of practical importance in earthquake predictions as well as mineral and petroleum exploration.

We begin this chapter with comments on seismology and its history. Then, we describe the conceptual issues involved in its theory, including the relation between theories and models, as well as the choices available to a theorist. We conclude this chapter with a discussion on scientific methodologies.

1.1 Seismology

Seismological theory underlies the study of the propagation of mechanical disturbances within materials composing the Earth and Earth-like bodies, such as planets and moons. There are also seismological studies of stars, referred to as asteroseismology, particularly of the Sun, which is referred to as helioseismology.

From the viewpoint of its inferences, seismology is a branch of geosciences. From the viewpoint of its methodology, seismology is a branch of physics, and—in particular—mechanics.

The term seismology was coined in the nineteenth century by one of its founders, Robert Mallet, in the context of the study of earthquakes; in Greek, σεισµóζ means earthquake and λoγíα means study. Presently, the term is used in a generic sense, not unlike, say, nuclear physics, allowing for a variety of different theories. It does not imply a single and coherent structure as, say, general relativity.

A cluster of theories superseding each other forms the history of seismology. Let us mention a few of its highlights.

Since before the time of Aristotle, it has been known that the Earth is spherical. It was only natural to wonder about its interior and why there were such events as earthquakes. Religious explanations have been offered all along, declaring earthquakes to be expressions of God’s anger, as illustrated in Figure 1.1. Perhaps surprisingly, such explanations are not only a thing of the past.

Fig. 1.1

Apart from supernatural explanations, there is also a long history of naturalistic accounts. One of these is that earthquakes are the product of a decaying Earth. Aristotle and his followers thought of the Earth as a living thing, a biological organism whose various parts had a function in sustaining the whole. Decay of the Earth could be part of the natural process.

Much thinking about the Earth’s interior and about earthquakes is based on analogies with other well-known or newly discovered phenomena. Thus, in the eighteenth century, earthquakes were seen as similar to, and maybe caused by, electrical phenomena akin to thunder and lightning. Edmond Halley, in the late seventeenth century, proposed a hollow Earth. He suggested that it consisted of an 800-kilometer thick shell with two inner concentric shells that rotated in different directions. The reason for this concoction was to explain some anomalous magnetic phenomena. Leonhard Euler also proposed a hollow Earth with its own Sun at the centre and civilized people living on the inside.

Contemporary seismology is largely a product of research of the last two centuries. During that period, mathematical formulations for examining wave phenomena have been advanced and the quality and quantity of measuring instruments have been greatly augmented. Mathematical concepts of P wave, S wave, their surface, guided and interface counterparts, namely, the Rayleigh, Love and Stoneley waves (Love, 1944), as well as their anisotropic extensions, namely, qP, qS₁ and qS₂ waves, became part of the scientific terminology. Also, standards, such as the Richter (1935) scale,¹ developed in collaborations with Gutenberg, and the Preliminary Reference Earth Model (PREM), developed by Dziewoński and Anderson (1981), under the auspices of the Standard Earth Model Committee, were established.

There is, unfortunately, no recent comprehensive history of seismology, only earlier (Davison, 1927/2014) or brief (Dahlen and Tromp, 1998) accounts, as well as monographs and treatises with a particular historical, philosophical, social or cultural focus (Anduaga, 2016; Brush, 1980, 2009; Guidoboni and Poirier, 2004; Placanica, 1997).²

Seismology, in its current form, is a discipline of a quantitative nature, and hence, presents challenges in combining its mathematical and physical content. The rich variety of mathematical models and idealizations makes seismology an interesting theory to study. In light of these considerations, it is surprising that only a few have addressed the complex conceptual structure of seismology. Among them, the philosophers Aitor Anduaga, Mario Bunge and Sheldon Smith have contributed to a better understanding. Smith (2007b, p. 58) remarks that

[ ... ] the micro-structure that continuum mechanics attributes to bodies is not the structure that real-world bodies have. However, one would never model real-world macroscopic bodies via quantum mechanics since that would involve an enormous n-body problem that is less likely to reveal the salient features of, say, actual billiard ball collisions than continuum mechanics. Thus, a modelling of billiard ball collisions that wants to take into account and predict the largest range of the features of actual billiard balls such as their deformation and constitutive responses including thermodynamic responses will tend to be formulated within classical continuum mechanics. Because of its robust contact with the actual world, this branch of physics can serve as a check on what is intuitively plausible when it comes to bodies.

Also, this robust contact makes an intuitive distinction between the continuum-mechanics models and the actual world less obvious, which might render the evaluation of seismological theory with its relation to the Earth more difficult, an issue emphasized in this book.

Bunge (1998, pp. 568-571) points out the particular importance of seismology.

A nice illustration of the intertwining of empirical and theoretical events in the actual practice of science is offered by seismology, the study of elastic disturbances of Terra. [ ... ] in order to read a seismogram so that it may become a set of data regarding an event (e.g., an earthquake) or an evidence relevant to a theory (e.g., about the inner structure of our planet), the seismologist employs elasticity theory and all the theories that may enter the design and interpretation of the seismograph.

Again, the interplay between theory and observations is brought to our attention, and—within observations—the theory of measuring devices is also included.

With notable exceptions, there is little in the way of conceptual analysis of the Earth sciences. Kleinhans et al. (2005) attempt to explain why it is so.

Earth science has received relatively little attention from philosophers of science. [ ... ] most of Earth science is Terra Incognita to philosophers. Apparently, it is generally believed that Earth science cannot offer much excitement [ ... ].

It turns out that intellectual excitement can indeed be found in seismology, though perhaps not immediately. The conceptual problems of quantum theory can be grasped early on, even if solutions seem out of reach. It is easy to be enticed by the uncertainty principle, by spooky action-at-adistance, or by Schrödinger’s cat that is both alive and dead, as illustrated in Figure 1.2.

Fig. 1.2

Yet, seismology’s conceptual foundations are as challenging as those of quantum theory or general relativity, though less obvious. For instance, one has to engage in the study of continuum mechanics, which underlies seismology, before appreciating its conceptual richness.

Another reason for a philosophical lack of interest in seismology is its perceived dedication to mere practical applications. Charles Sanders Peirce (c.1896/1955, p. 53) understood this situation.

Persons who know science chiefly by its results—that is to say, have no acquaintance with it as a living inquiry—are apt to acquire the notion that the universe is now entirely explained in its leading features; and that it is only here and there that the fabric of scientific knowledge betrays any rents.

Peirce is correct; seismology, in particular, seems far from the frontiers of science. All that remains is to use this theory to learn further details about the Earth. Peirce is also correct that this is an illusion.

Seismologists are far from completely understanding the Earth and perhaps even farther from understanding the theoretical foundations of their science. We find scientific practice of paramount interest and—in the spirit of Bunge (1998), Smith (2007b) and Peirce (c.1896/1955)—we see that seismology exhibits conceptual issues involving the intertwining of empirical, theoretical and mathematical concepts.

We formulate seismology by postulating four foundational principles.

1. Earth is a granular body, intrinsically composed of discrete particles, but—for the purposes of seismology—assumed to be a physical continuum.
2. Continuum mechanics is the quantitative framework for seismology.
3. Constitutive equations of continuum mechanics specify properties of a material within a given model of the Earth.
4. Contact between the constitutive equations and the Earth is made through seismic measurements whose interpretations are within the theory of seismology based on the quantitative framework of continuum mechanics.

In this book, as is common in continuum mechanics, the terms ‘constitutive equation’ and ‘constitutive relation’ are synonymous.

The justification for the four principles is provided by considerations such as that the wavelength of seismic disturbances is orders of magnitude larger than the size of grains within a medium through which they propagate. Hence, a continuum is an average of properties of granular materials over a wavelength. Thus, we can adopt the framework of continuum mechanics and bring to bear the formal apparatus of this theory. Observations need to be appropriately interpreted by the theory to provide the required evidence, where the theory of continuum mechanics mediates between observations and interpretations.

To learn about the Earth’s interior by observing effects of earthquakes, seismologists study wave propagation. Measured results depend on properties of the materials through which waves propagate. However, deformations that travel below the surface of the Earth are not observable. One could ask if there exist disturbances that relate an earthquake to a measurement of vibrations at a distant seismograph. Different philosophical approaches to such a question are possible. An empiricist, for instance, might deny such disturbances, and be interested only in the correlations between observable events in one location with observable events in another.

1.2 Theories and models

In geosciences, one commonly learns about the two types of waves that propagate in the Earth, which are referred to as P and S. Strictly speaking, this is false.

P and S waves do not propagate in the Earth but are contained in the equation of motion within an abstract medium, which is a Hookean solid, used by seismologists to model the Earth. The existence of such waves in abstract media was formulated at the beginning of the nineteenth century by such mathematical physicists as Siméon Denis Poisson. Its applicability to interpret seismic measurements is credited to the work of the geologist Richard Dixon Oldham. In his quantitative interpretation, Oldham followed qualitative concepts of the eighteenth century geologist John Michell. As stated by Charles Davison (1927/2014, p. 23–24), in his historical treatise, Michell’s permanent contribution to seismology was

his distinction between the phenomena which are and are not essential to his theory. In separating the vibratory motion from the wave-like motion or visible waves, Michell was in advance of his time. He was one of the first, if not the very first, to assign the vibratory motion in earthquakes to the propagation of elastic waves in the earth’s crust.

At first sight, emphasizing the difference between the Earth and its model may appear as a pedantic distinction, since it goes beyond ordinary scientific usage. However, in foundational studies it is important to understand the status of the entities in question. There are material entities, such as rocks, and there are abstract entities, such as numbers and rays, which are just as real, though they are neither material nor located in space and time. This is the realm of abstract entities, which holds not only mathematical objects but also models of the physical world. The distinction between the physical and abstract realms is referred to as Platonism, in honour of the ideas of Plato. The aim of a foundational study is partly for its intrinsic interest in the subject matter and partly to address conceptual problems that arise in the theory and which, in turn, give rise to misunderstandings and impede progress.

A medium commonly used in seismol...

Table of contents

• Cover page
• Title page
• Copyright
• Dedication
• Contents
• List of Figures
• Acknowledgments
• Preface
• 1. Aspects of seismological theory
• 2. Nature of science and its methods
• 3. On continua and models
• 4. Continuum mechanics: General principles, constitutive relations
• 5. Hookean solids
• 6. Forward and inverse problems
• 7. Intertheory and intratheory relations
• Afterword
• Bibliography
• Subject Index
• Name Index
• About the Authors and Illustrator