6 Introduction
and what we do not, use statistical techniques to assess what
we can say with differing degrees of confidence from the data,
and develop new data and techniques to do better.
In general, the approach taken is to describe complex problems with simplified models that seek to represent key elements
of the process under consideration. For example, an earthquake is a complicated rupture process that occurs in a finite
volume and radiates seismic energy through the real materials
of the earth. As we will see in the next few chapters, we represent all aspects of this process with simple models. We treat
the complex faulting process as elastic slip on an infinitely
narrow surface. We further treat the rock around it as a simple
elastic material, and thus describe the complex seismic wave
disturbance that propagates through it, using a number of
simplifications.
It is important to bear in mind that these models are only
approximations to a more complicated reality. For example,
although the radiated seismic energy is real (it can destroy
buildings), the mathematical descriptions used to understand it
are human constructs. P waves, S waves, seismic phases like
ScS, seismic ray paths, surface waves, or the earth’s normal
modes are all approximations that make the radiated energy
easier to conceptualize. Similarly, we model a fault as a planar
slip surface and use seismological observations to characterize
the slip geometry and history. However, although this process
nicely replicates the seismic observations, it only approximates
the actual physics of earthquake rupture.
We often use a hierarchy of different approximations, as
appropriate. For example, we might first predict the approximate time when a packet of seismic energy arrives by treating
it as a seismic ray, and then use a more sophisticated wave or
normal mode calculation to predict its amplitude and hence
learn more about the properties of the parts of the earth it
traversed. Similarly, we first describe the earth as isotropic
(having the same properties in all directions) and purely elastic
(no seismic energy is lost to heat by friction) and then confront
the deviations from these simplifications.
A similar approach is often followed when discussing the
tectonic context of earthquakes. Although faults, earthquakes,
volcanoes, and topography are real, we associate these with the
boundaries of plates that are human approximations. We will
see that the questions of when to regard a region as a plate and
how to characterize its boundaries are not simple. The simplest
analyses assume that plates are rigid and divided by narrow
boundaries. Later, we treat the boundaries as broad zones, and
eventually we confront the fact that plates are not perfectly
rigid, but in fact deform internally, as shown by earthquakes
that occur within them.
We often choose a type of model to represent the earth
and then use seismological and other data to estimate the
parameters of this model. Thus a characteristic activity of
seismology, and of the earth sciences in general, is solving
inverse problems. We start with the end result, the seismograms, and work backwards using mathematical techniques to
characterize the earthquakes that generated the seismic waves
and the material the waves passed through. Inverse problems
are more complicated than the conceptually simpler forward
problems in which we use the theory of seismic wave generation and propagation to predict the seismogram that would be
observed for a given source and medium. Inverse problems are
harder to solve for several reasons. Seismograms reflect the
combined effect of the source and medium, neither of which is
known exactly. There are often aspects of the inverse problem
that the data are insufficient to resolve. Thus seismology and
other branches of the earth sciences, to a greater extent than
most other scientific disciplines, often infer a “big picture” from
grossly limited and insufficient data. For example, our images
of the earth from seismic waves suffer from the fact that the
severely limited geographical distributions of both earthquakes
and seismometers leave most of earth’s interior unsampled. This
situation is like a doctor examining a possible broken bone with
only a few scattered bursts of x-rays from random directions.
Moreover, although the forward problem typically can be
solved in a straightforward way, giving a unique solution,
the inverse problem often has no unique solution. In fact, the
data are generally somewhat inconsistent due to errors, so no
model can exactly describe the data. Finally, the fact that solving the inverse problem yields a set of model parameters that
describe the observations well does not necessarily mean that
the resulting model actually reflects physical reality. This nonuniqueness reflects the logical tenet that because a implies b,
b does not necessarily imply a. In fact, we often have no way of
determining what the reality is. For example, we will never
truly know the composition and temperature of the earth’s core
because we cannot go there. This limitation remains in spite
of the fact that over time our models of the core have become
increasingly consistent with seismological data, experimental
results about materials at high pressure and temperature, and
other data including inferences from meteorites about the
composition of the solar system. 6
A consequence of this approach is the need to consider issues
of precision, accuracy, and uncertainty. Estimates of quantities
like the magnitude or depth of an earthquake depend both on
the precision, or repeatability, with which data like seismic
wave arrival times and amplitudes are measured, and on the
accuracy, or extent to which the resulting inferences correctly
describe the earth. For example, earthquake magnitudes are
simple measures of earthquake size, estimated in various ways
from seismograms without accounting for effects like the geometry of the earthquake source or lateral variations in seismic
velocities. Hence measurements at different sites yield various
estimates, so it is of little value to argue whether an earthquake
had magnitude 5.2 or 5.4. Similarly, focal depths are derived
from seismic wave arrival times by assuming a velocity structure near the earthquake, which is often not well known. For
6 Similar difficulties afflict most of the earth sciences. Field geologists will never
know whether their inferences about the past history and environment of a region
are correct; paleontologists will never know how realistic their models of ancient
life are, etc.
and what we do not, use statistical techniques to assess what
we can say with differing degrees of confidence from the data,
and develop new data and techniques to do better.
In general, the approach taken is to describe complex problems with simplified models that seek to represent key elements
of the process under consideration. For example, an earthquake is a complicated rupture process that occurs in a finite
volume and radiates seismic energy through the real materials
of the earth. As we will see in the next few chapters, we represent all aspects of this process with simple models. We treat
the complex faulting process as elastic slip on an infinitely
narrow surface. We further treat the rock around it as a simple
elastic material, and thus describe the complex seismic wave
disturbance that propagates through it, using a number of
simplifications.
It is important to bear in mind that these models are only
approximations to a more complicated reality. For example,
although the radiated seismic energy is real (it can destroy
buildings), the mathematical descriptions used to understand it
are human constructs. P waves, S waves, seismic phases like
ScS, seismic ray paths, surface waves, or the earth’s normal
modes are all approximations that make the radiated energy
easier to conceptualize. Similarly, we model a fault as a planar
slip surface and use seismological observations to characterize
the slip geometry and history. However, although this process
nicely replicates the seismic observations, it only approximates
the actual physics of earthquake rupture.
We often use a hierarchy of different approximations, as
appropriate. For example, we might first predict the approximate time when a packet of seismic energy arrives by treating
it as a seismic ray, and then use a more sophisticated wave or
normal mode calculation to predict its amplitude and hence
learn more about the properties of the parts of the earth it
traversed. Similarly, we first describe the earth as isotropic
(having the same properties in all directions) and purely elastic
(no seismic energy is lost to heat by friction) and then confront
the deviations from these simplifications.
A similar approach is often followed when discussing the
tectonic context of earthquakes. Although faults, earthquakes,
volcanoes, and topography are real, we associate these with the
boundaries of plates that are human approximations. We will
see that the questions of when to regard a region as a plate and
how to characterize its boundaries are not simple. The simplest
analyses assume that plates are rigid and divided by narrow
boundaries. Later, we treat the boundaries as broad zones, and
eventually we confront the fact that plates are not perfectly
rigid, but in fact deform internally, as shown by earthquakes
that occur within them.
We often choose a type of model to represent the earth
and then use seismological and other data to estimate the
parameters of this model. Thus a characteristic activity of
seismology, and of the earth sciences in general, is solving
inverse problems. We start with the end result, the seismograms, and work backwards using mathematical techniques to
characterize the earthquakes that generated the seismic waves
and the material the waves passed through. Inverse problems
are more complicated than the conceptually simpler forward
problems in which we use the theory of seismic wave generation and propagation to predict the seismogram that would be
observed for a given source and medium. Inverse problems are
harder to solve for several reasons. Seismograms reflect the
combined effect of the source and medium, neither of which is
known exactly. There are often aspects of the inverse problem
that the data are insufficient to resolve. Thus seismology and
other branches of the earth sciences, to a greater extent than
most other scientific disciplines, often infer a “big picture” from
grossly limited and insufficient data. For example, our images
of the earth from seismic waves suffer from the fact that the
severely limited geographical distributions of both earthquakes
and seismometers leave most of earth’s interior unsampled. This
situation is like a doctor examining a possible broken bone with
only a few scattered bursts of x-rays from random directions.
Moreover, although the forward problem typically can be
solved in a straightforward way, giving a unique solution,
the inverse problem often has no unique solution. In fact, the
data are generally somewhat inconsistent due to errors, so no
model can exactly describe the data. Finally, the fact that solving the inverse problem yields a set of model parameters that
describe the observations well does not necessarily mean that
the resulting model actually reflects physical reality. This nonuniqueness reflects the logical tenet that because a implies b,
b does not necessarily imply a. In fact, we often have no way of
determining what the reality is. For example, we will never
truly know the composition and temperature of the earth’s core
because we cannot go there. This limitation remains in spite
of the fact that over time our models of the core have become
increasingly consistent with seismological data, experimental
results about materials at high pressure and temperature, and
other data including inferences from meteorites about the
composition of the solar system. 6
A consequence of this approach is the need to consider issues
of precision, accuracy, and uncertainty. Estimates of quantities
like the magnitude or depth of an earthquake depend both on
the precision, or repeatability, with which data like seismic
wave arrival times and amplitudes are measured, and on the
accuracy, or extent to which the resulting inferences correctly
describe the earth. For example, earthquake magnitudes are
simple measures of earthquake size, estimated in various ways
from seismograms without accounting for effects like the geometry of the earthquake source or lateral variations in seismic
velocities. Hence measurements at different sites yield various
estimates, so it is of little value to argue whether an earthquake
had magnitude 5.2 or 5.4. Similarly, focal depths are derived
from seismic wave arrival times by assuming a velocity structure near the earthquake, which is often not well known. For
6 Similar difficulties afflict most of the earth sciences. Field geologists will never
know whether their inferences about the past history and environment of a region
are correct; paleontologists will never know how realistic their models of ancient
life are, etc.