Appendix: Mathematical and
Computational Background
If you wish to learn about nature, to appreciate nature, it is necessary to understand the language she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we pay attention.
Richard Feynman, The Character of Physical Law (1982)
A.1 Introduction
The study of seismology follows a pattern characteristic of
many scientific disciplines. We first identify phenomena that
we seek to understand, such as the propagation of seismic
waves through the solid earth. We then consider the physics of
the simplest relevant case, such as the propagation of a wave
of a single frequency through a uniform material, formulate
the problem mathematically, and derive a solution. From this
solution, we build up mathematical solutions to more complex
problems, each of which is ideally a better approximation to
the complexities of the real earth. Although the simpler problems can be solved analytically, eventually the complexities
require numerical techniques.
We thus rely on a set of mathematical techniques often used
in physical problems. Experience suggests that although many
readers are familiar with most of the mathematics required
in this book, a review is often helpful. This appendix briefly
summarizes a broad range of material. The first sections treat
a variety of mathematical topics. The final section reviews
some concepts relevant to the use of computers for scientific
calculations.
In using these mathematical techniques, it is worth bearing in mind that we are invoking the special power of mathematics to deal with physical problems. This power is that if a
physical problem is posed correctly in mathematical terms, then
applying mathematical techniques to this formulation yields
quite different, and often apparently unrelated, statements
that also correctly describe the physical world. For example,
in Section 2.4 we used the equations of elasticity and applied
vector calculus to derive the properties of seismic waves that
we observe. Similarly, in Section 2.5 we derived an observed
physical relation, Snell’s law, starting from three different physical formulations. Conversely, we have seen that different physical phenomena can be described using similar mathematical
approaches and so have some deep similarities. Although in
hindsight such successes may not seem surprising, because
many of the mathematical methods we use were developed to
solve such physical problems, they illustrate the intimate connection between sciences like seismology and mathematics. 1
A.2 Complex numbers
In several of our applications, notably in describing propagating waves and their frequency content, complex numbers are
helpful. We thus briefly review some of their properties.
The complex number z = a + ib, where i = −1, has a real
part, a, and an imaginary part, b. These relations are sometimes
written a = Re (z) and b = Im (z). Complex numbers are typically plotted in the complex plane with their real parts on the
x 1 axis and their imaginary parts on the x 2 axis (Fig. A.2-1).
Alternatively, a complex number can be written in polar coordinate form as
z = a + ib = re iθ = r(cos θ + i sin θ).
(1)
1 Most seismologists are more conservative than Paul Dirac, a leader in the
development of quantum physics, who invented the delta function. Dirac regarded
mathematical beauty as a guiding principle, stating that “it is more important to have
beauty in one’s equations than to have them fit experiment.”
Computational Background
If you wish to learn about nature, to appreciate nature, it is necessary to understand the language she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we pay attention.
Richard Feynman, The Character of Physical Law (1982)
A.1 Introduction
The study of seismology follows a pattern characteristic of
many scientific disciplines. We first identify phenomena that
we seek to understand, such as the propagation of seismic
waves through the solid earth. We then consider the physics of
the simplest relevant case, such as the propagation of a wave
of a single frequency through a uniform material, formulate
the problem mathematically, and derive a solution. From this
solution, we build up mathematical solutions to more complex
problems, each of which is ideally a better approximation to
the complexities of the real earth. Although the simpler problems can be solved analytically, eventually the complexities
require numerical techniques.
We thus rely on a set of mathematical techniques often used
in physical problems. Experience suggests that although many
readers are familiar with most of the mathematics required
in this book, a review is often helpful. This appendix briefly
summarizes a broad range of material. The first sections treat
a variety of mathematical topics. The final section reviews
some concepts relevant to the use of computers for scientific
calculations.
In using these mathematical techniques, it is worth bearing in mind that we are invoking the special power of mathematics to deal with physical problems. This power is that if a
physical problem is posed correctly in mathematical terms, then
applying mathematical techniques to this formulation yields
quite different, and often apparently unrelated, statements
that also correctly describe the physical world. For example,
in Section 2.4 we used the equations of elasticity and applied
vector calculus to derive the properties of seismic waves that
we observe. Similarly, in Section 2.5 we derived an observed
physical relation, Snell’s law, starting from three different physical formulations. Conversely, we have seen that different physical phenomena can be described using similar mathematical
approaches and so have some deep similarities. Although in
hindsight such successes may not seem surprising, because
many of the mathematical methods we use were developed to
solve such physical problems, they illustrate the intimate connection between sciences like seismology and mathematics. 1
A.2 Complex numbers
In several of our applications, notably in describing propagating waves and their frequency content, complex numbers are
helpful. We thus briefly review some of their properties.
The complex number z = a + ib, where i = −1, has a real
part, a, and an imaginary part, b. These relations are sometimes
written a = Re (z) and b = Im (z). Complex numbers are typically plotted in the complex plane with their real parts on the
x 1 axis and their imaginary parts on the x 2 axis (Fig. A.2-1).
Alternatively, a complex number can be written in polar coordinate form as
z = a + ib = re iθ = r(cos θ + i sin θ).
(1)
1 Most seismologists are more conservative than Paul Dirac, a leader in the
development of quantum physics, who invented the delta function. Dirac regarded
mathematical beauty as a guiding principle, stating that “it is more important to have
beauty in one’s equations than to have them fit experiment.”