example, the depth is sometimes estimated (Section 4.3.3) from
half the product of the time difference between the direct P and
pP phases (see Fig. 1.1-3) and the velocity. If the time difference
is measured to 0.25 s, and the velocity is 8 km/s, the method
of propagation of errors (Section 6.5.1) shows that the uncertainty in depth is about 1 km, so it makes little sense to report
the depth to greater precision. In reality the uncertainty will
be greater, because the velocity also has some uncertainty. It is
important to bear in mind that assigning a single value to an
earthquake depth may exceed the relevant accuracy because
faulting extends over a finite area that may be large (on the
order of 10 km for a magnitude 6 earthquake). Moreover,
when we have alternative models with which to estimate
a parameter (for example, the earthquake stress drop estimated from body waves depends on the assumed geometry of
the fault), the uncertainty associated with an estimate using
any particular model underestimates the uncertainty due to
the fact that we do not know which model is best. It is thus
useful to examine how the estimate depends on the precision
of the observation, the model parameters, and the choice of
models.
Seismologists generally assume that the best estimates of
values and uncertainties come from studies by different investigators using multiple datasets and techniques. Ideally, studies
using the same data increase precision by reducing random
errors, and studies using different data and techniques increase
accuracy by reducing the effect of systematic errors. For example, for the well-studied Loma Prieta earthquake, seismic
moment estimates vary by about 25%, and M s values vary by
about 0.1 units.
However, statisticians have long noted the difficulties in assessing probabilities and uncertainties. Two famous examples
are the Titanic, described as “unsinkable” (probability zero)
and the space shuttle, which was lost on its twenty-fifth launch,
surprisingly soon given the estimated probability of accident of
1/100,000. Other examples come from the history of measurements of physical constants, which shows that the reported
uncertainties underestimate the actual errors. For example, the
27 successive measurements of the speed of light between 1875
and 1958 are shown by subsequent analysis to be consistently
in error by much more than the assigned uncertainty. It appears
that assessments of the formal or random uncertainty often
significantly underestimate the systematic error, so the overall
uncertainty is dominated by the unrecognized systematic error
and thus larger than expected. As a result, measurements of
a quantity often remain stable for some time, and then change
by much more than the previously assumed uncertainty. One
possible explanation, termed the “bandwagon effect,” is the
tendency to discount data that are inconsistent with previous
ideas, but later prove more accurate than those included.
Another effect appears to be the discarding of outliers: for
example, although R. Millikan reported using all the observations in his Nobel prize-winning (1910) study of the charge of
the electron, his notebooks show that he discarded 49 of 107
oil drops that appeared discordant, increasing the apparent
precision of the result. Until a method is developed that
excludes obviously erroneous data without discarding real
disconforming evidence, making realistic uncertainty estimates
will remain a challenge. Although such analyses are more
difficult in the earth sciences a for example, an earthquake is a
nonrepeatable experiment a they are useful to bear in mind.
This discussion brings out the fact that although we often
speak of “finding” or “determining” quantities like earthquake source parameters or velocity structure, it might be
better to speak of “estimating” or “inferring” these quantities.
There is no harm in the common and more upbeat phrasing
so long as we remember that these values reflect uncertainties
due to random noise and errors of measurement (sometimes
called aleatory uncertainty, after the Latin word for dice)
and systematic (sometimes called epistemic) uncertainty due
to our choice of model to describe the phenomenon under
consideration.
Although these caveats sound worrisome, seismological
models are far from useless. We can usually develop models
that not only describe the data used to develop them, but to
predict other data. For example, earthquake source models derived only from seismology often predict the observations
made using field geology and geodesy (ground deformation),
both for the specific earthquake studied and for others in the
same region. Moreover, the seismological results often give
useful insight that is consistent with other lines of evidence. For
example, seismology, gravity, and geomagnetism all favor the
earth having a dense liquid iron core chemically different from
the rocky mantle. This idea is also consistent with the fact
that meteorites a thought to be fragments of small planets a
are divided into stony and iron classes. Hence seismologists
use this modeling approach to understand the earth, while
recognizing its limitations.
For several reasons, our models usually improve with time.
First, the data improve in both quantity and quality. Second,
new observational and analytical techniques are introduced.
As a result, long-standing problems such as the velocity structure of the earth are repeatedly reassessed. Successive generations of models seek to explain additional types of data, and
often contain more model parameters in the hope of better representing the earth. Using statistical tests, we find that in some
cases the resulting improvements are significant, whereas in
others the new model improves only slightly on earlier ones. An
important point is that more complicated models can always fit
data better, because they contain more free parameters, just as
a set of points in the x–y plane can be better fit by a quadratic
polynomial than by a straight line. Thus we can statistically test
models to see whether a new model reduces the misfit to the
data more than would be expected purely by chance due to the
additional parameters. Another useful test is whether the new
or old models do a better job of describing data that were not
used in deriving either, a process called pure prediction. When
new models pass these tests, we can accept them a and then
look again to see which data are still not described well and try
to do better.
1.1 Introduction 7
half the product of the time difference between the direct P and
pP phases (see Fig. 1.1-3) and the velocity. If the time difference
is measured to 0.25 s, and the velocity is 8 km/s, the method
of propagation of errors (Section 6.5.1) shows that the uncertainty in depth is about 1 km, so it makes little sense to report
the depth to greater precision. In reality the uncertainty will
be greater, because the velocity also has some uncertainty. It is
important to bear in mind that assigning a single value to an
earthquake depth may exceed the relevant accuracy because
faulting extends over a finite area that may be large (on the
order of 10 km for a magnitude 6 earthquake). Moreover,
when we have alternative models with which to estimate
a parameter (for example, the earthquake stress drop estimated from body waves depends on the assumed geometry of
the fault), the uncertainty associated with an estimate using
any particular model underestimates the uncertainty due to
the fact that we do not know which model is best. It is thus
useful to examine how the estimate depends on the precision
of the observation, the model parameters, and the choice of
models.
Seismologists generally assume that the best estimates of
values and uncertainties come from studies by different investigators using multiple datasets and techniques. Ideally, studies
using the same data increase precision by reducing random
errors, and studies using different data and techniques increase
accuracy by reducing the effect of systematic errors. For example, for the well-studied Loma Prieta earthquake, seismic
moment estimates vary by about 25%, and M s values vary by
about 0.1 units.
However, statisticians have long noted the difficulties in assessing probabilities and uncertainties. Two famous examples
are the Titanic, described as “unsinkable” (probability zero)
and the space shuttle, which was lost on its twenty-fifth launch,
surprisingly soon given the estimated probability of accident of
1/100,000. Other examples come from the history of measurements of physical constants, which shows that the reported
uncertainties underestimate the actual errors. For example, the
27 successive measurements of the speed of light between 1875
and 1958 are shown by subsequent analysis to be consistently
in error by much more than the assigned uncertainty. It appears
that assessments of the formal or random uncertainty often
significantly underestimate the systematic error, so the overall
uncertainty is dominated by the unrecognized systematic error
and thus larger than expected. As a result, measurements of
a quantity often remain stable for some time, and then change
by much more than the previously assumed uncertainty. One
possible explanation, termed the “bandwagon effect,” is the
tendency to discount data that are inconsistent with previous
ideas, but later prove more accurate than those included.
Another effect appears to be the discarding of outliers: for
example, although R. Millikan reported using all the observations in his Nobel prize-winning (1910) study of the charge of
the electron, his notebooks show that he discarded 49 of 107
oil drops that appeared discordant, increasing the apparent
precision of the result. Until a method is developed that
excludes obviously erroneous data without discarding real
disconforming evidence, making realistic uncertainty estimates
will remain a challenge. Although such analyses are more
difficult in the earth sciences a for example, an earthquake is a
nonrepeatable experiment a they are useful to bear in mind.
This discussion brings out the fact that although we often
speak of “finding” or “determining” quantities like earthquake source parameters or velocity structure, it might be
better to speak of “estimating” or “inferring” these quantities.
There is no harm in the common and more upbeat phrasing
so long as we remember that these values reflect uncertainties
due to random noise and errors of measurement (sometimes
called aleatory uncertainty, after the Latin word for dice)
and systematic (sometimes called epistemic) uncertainty due
to our choice of model to describe the phenomenon under
consideration.
Although these caveats sound worrisome, seismological
models are far from useless. We can usually develop models
that not only describe the data used to develop them, but to
predict other data. For example, earthquake source models derived only from seismology often predict the observations
made using field geology and geodesy (ground deformation),
both for the specific earthquake studied and for others in the
same region. Moreover, the seismological results often give
useful insight that is consistent with other lines of evidence. For
example, seismology, gravity, and geomagnetism all favor the
earth having a dense liquid iron core chemically different from
the rocky mantle. This idea is also consistent with the fact
that meteorites a thought to be fragments of small planets a
are divided into stony and iron classes. Hence seismologists
use this modeling approach to understand the earth, while
recognizing its limitations.
For several reasons, our models usually improve with time.
First, the data improve in both quantity and quality. Second,
new observational and analytical techniques are introduced.
As a result, long-standing problems such as the velocity structure of the earth are repeatedly reassessed. Successive generations of models seek to explain additional types of data, and
often contain more model parameters in the hope of better representing the earth. Using statistical tests, we find that in some
cases the resulting improvements are significant, whereas in
others the new model improves only slightly on earlier ones. An
important point is that more complicated models can always fit
data better, because they contain more free parameters, just as
a set of points in the x–y plane can be better fit by a quadratic
polynomial than by a straight line. Thus we can statistically test
models to see whether a new model reduces the misfit to the
data more than would be expected purely by chance due to the
additional parameters. Another useful test is whether the new
or old models do a better job of describing data that were not
used in deriving either, a process called pure prediction. When
new models pass these tests, we can accept them a and then
look again to see which data are still not described well and try
to do better.
1.1 Introduction 7