by at least 1σ, and about 5% should be outside the 2σ range.
Hence a low χ
2
ν value, which we might view as showing an
excellent fit, is more likely to imply that the uncertainties in the
data have been overestimated, and have thus made χ 2
ν appear
too small. For example, χ
2
ν for the NUVEL-1 model is 0.24,
whereas it is expected to lie with 95% probability between 0.93
and 1.07. This effect is also seen for other plate motion models,
suggesting that the assigned data uncertainties are more like
95% (2σ) confidence limits than one standard deviation. If so,
the uncertainties in the model are correspondingly less than
implied by the model variance–covariance matrix. Thus the χ 2
test formalizes the adage that if something seems too good to be
true, it probably is. 1
A second issue is whether the number of model parameters is
appropriate. As discussed in Chapter 5, there are often several
possible plate boundary geometries for an area. Naturally,
more plates can describe plate motions in an area better
because the model has more parameters. Thus we ask whether
the improved fit shown by a lower value of χ 2
ν is more than
expected purely by chance due to the additional parameters.
For example, a set of data in the x–y plane are always better
fit by a higher-order polynomial, such as a quadratic versus a
straight line.
This issue can be addressed using the F-ratio test, which gives
insight into whether a set of data are significantly better fit by a
model with more parameters. The idea is that if a set of n data
are fit by two models, one with r parameters (n − r degrees
of freedom) and a second with p parameters (n − p degrees of
freedom) with p greater than r, the second model should fit the
data better, and χ 2 (p) should be less than χ 2 (r). To test if the reduction in χ
2 is greater than would be expected simply because
additional model parameters are added, we form the statistic
F
r
p p r
p n p
[ ( )
( )]/(
)
( )/(
)
.
=
−
−
−
χ
χ
χ
2
2
2
(11)
Statistical tables or mathematical software give the probability
P F (F, ν 1 , ν 2 ) of observing an F value greater than that observed
for a random sample with ν 1 = (p − r) and ν 2 = (n − p). Thus,
for example, if P F is 0.01, there is only a 1% chance that
the improved fit of the model with more parameters is due
purely to chance. Because this test depends on the ratio of χ 2 ,
it is not affected if the uncertainties are consistently over- or
under-estimated.
We can use F to test whether the fit to n relative motion data
of a model with p + 1 plates is significantly better than that of
one with p plates. The p plate model has 3(p − 1) parameters
(n − 3p + 3 degrees of freedom), whereas the p + 1 plate model
has 3p parameters (n − 3p degrees of freedom). Thus
1 This approach has been used to argue that Mendel’s famous results in 1865 that
established the science of genetics are so good a the probability of observing them is
0.004% a that they are suspect. Similarly, instructors have used χ
2 tests to show that
students’ results reported in laboratory classes are so good that they are unlikely to
have actually been obtained.
F
p
p
p
n
p
[ (
)
(
)]/
(
)/(
)
=
−
+
+
−
χ
χ
χ
2
2
2
1
3
1
3
plates
plates
plates
(12)
is tested using P F (F, ν 1 , ν 2 ) with ν 1 = 3 and ν 2 = (n − 3p). If the
risk that the improved fit would occur by chance is small, perhaps less than 1%, then we treat the additional plate as distinct.
Conversely, if the improved fit is likely to result simply from
the additional parameters, the data do not strongly indicate
the presence of an additional plate. For example, such tests
show that although the boundary between them is indistinct,
North and South America should be treated as separate plates.
This approach is used to investigate complicated regions where
the plate geometry is unclear, such as near Japan and in the
Indian Ocean. Similarly, we can investigate regions of intraplate
deformation to see whether there is resolvable motion.
In many applications these or other statistical tests can be
used to examine how well a model fits the data and to gain insight into whether the model is too simple (underparametrized)
to explain the data or more complicated (overparametrized)
than is required by the data. For example, we can examine
cases when adding more layers to a velocity model significantly
improves the fit to travel time data, when a more complex
earthquake source model fits seismograms significantly better,
or when a more complex model of earthquake recurrence describes an earthquake history better. In these applications the
statistical tests address only the data used, so a more complex
model may be justified based on other data, even if it is not
required by the data tested. Moreover, we often suspect that
the earth is more complicated than we would like when using
simple statistical models. In particular, we often have little a
priori knowledge of how to estimate the random and systematic errors. Even so, it is worth subjecting models to tests and
seeing how well the data support our beliefs. This testing is a
key part of the cycle (Fig. 1.1-8) by which models are refined
using new data and model parameterizations.
Further reading
Many discussions of inverse theory, including ours, are based on Lanczos
(1961). Applications in the earth sciences, especially seismology, are discussed in texts and reviews including Parker (1977), Aki and Richards
(1980), and Menke (1984). Treatments of tomographic methods in seismology are given by Nolet (1987), Thurber and Aki (1987), Spakman and
Nolet (1988), Humphreys and Clayton (1988), and Romanowicz (1991).
Inversion for the properties of stratified media is reviewed by Wiggins
(1972).
Tests for goodness of fit are discussed in statistical texts such as
Bevington and Robinson (1992) and Freedman et al. (1991); the latter
treats the issue of Mendel’s results. Chase (1972) and Minster et al. (1974)
present the inverse problem for plate motions; the latter gives the partial
derivatives. Stein and Gordon (1984) and DeMets et al. (1990) discuss applications of the F-ratio test to plate motions and intraplate deformation.
7.5 Further reading 441
Hence a low χ
2
ν value, which we might view as showing an
excellent fit, is more likely to imply that the uncertainties in the
data have been overestimated, and have thus made χ 2
ν appear
too small. For example, χ
2
ν for the NUVEL-1 model is 0.24,
whereas it is expected to lie with 95% probability between 0.93
and 1.07. This effect is also seen for other plate motion models,
suggesting that the assigned data uncertainties are more like
95% (2σ) confidence limits than one standard deviation. If so,
the uncertainties in the model are correspondingly less than
implied by the model variance–covariance matrix. Thus the χ 2
test formalizes the adage that if something seems too good to be
true, it probably is. 1
A second issue is whether the number of model parameters is
appropriate. As discussed in Chapter 5, there are often several
possible plate boundary geometries for an area. Naturally,
more plates can describe plate motions in an area better
because the model has more parameters. Thus we ask whether
the improved fit shown by a lower value of χ 2
ν is more than
expected purely by chance due to the additional parameters.
For example, a set of data in the x–y plane are always better
fit by a higher-order polynomial, such as a quadratic versus a
straight line.
This issue can be addressed using the F-ratio test, which gives
insight into whether a set of data are significantly better fit by a
model with more parameters. The idea is that if a set of n data
are fit by two models, one with r parameters (n − r degrees
of freedom) and a second with p parameters (n − p degrees of
freedom) with p greater than r, the second model should fit the
data better, and χ 2 (p) should be less than χ 2 (r). To test if the reduction in χ
2 is greater than would be expected simply because
additional model parameters are added, we form the statistic
F
r
p p r
p n p
[ ( )
( )]/(
)
( )/(
)
.
=
−
−
−
χ
χ
χ
2
2
2
(11)
Statistical tables or mathematical software give the probability
P F (F, ν 1 , ν 2 ) of observing an F value greater than that observed
for a random sample with ν 1 = (p − r) and ν 2 = (n − p). Thus,
for example, if P F is 0.01, there is only a 1% chance that
the improved fit of the model with more parameters is due
purely to chance. Because this test depends on the ratio of χ 2 ,
it is not affected if the uncertainties are consistently over- or
under-estimated.
We can use F to test whether the fit to n relative motion data
of a model with p + 1 plates is significantly better than that of
one with p plates. The p plate model has 3(p − 1) parameters
(n − 3p + 3 degrees of freedom), whereas the p + 1 plate model
has 3p parameters (n − 3p degrees of freedom). Thus
1 This approach has been used to argue that Mendel’s famous results in 1865 that
established the science of genetics are so good a the probability of observing them is
0.004% a that they are suspect. Similarly, instructors have used χ
2 tests to show that
students’ results reported in laboratory classes are so good that they are unlikely to
have actually been obtained.
F
p
p
p
n
p
[ (
)
(
)]/
(
)/(
)
=
−
+
+
−
χ
χ
χ
2
2
2
1
3
1
3
plates
plates
plates
(12)
is tested using P F (F, ν 1 , ν 2 ) with ν 1 = 3 and ν 2 = (n − 3p). If the
risk that the improved fit would occur by chance is small, perhaps less than 1%, then we treat the additional plate as distinct.
Conversely, if the improved fit is likely to result simply from
the additional parameters, the data do not strongly indicate
the presence of an additional plate. For example, such tests
show that although the boundary between them is indistinct,
North and South America should be treated as separate plates.
This approach is used to investigate complicated regions where
the plate geometry is unclear, such as near Japan and in the
Indian Ocean. Similarly, we can investigate regions of intraplate
deformation to see whether there is resolvable motion.
In many applications these or other statistical tests can be
used to examine how well a model fits the data and to gain insight into whether the model is too simple (underparametrized)
to explain the data or more complicated (overparametrized)
than is required by the data. For example, we can examine
cases when adding more layers to a velocity model significantly
improves the fit to travel time data, when a more complex
earthquake source model fits seismograms significantly better,
or when a more complex model of earthquake recurrence describes an earthquake history better. In these applications the
statistical tests address only the data used, so a more complex
model may be justified based on other data, even if it is not
required by the data tested. Moreover, we often suspect that
the earth is more complicated than we would like when using
simple statistical models. In particular, we often have little a
priori knowledge of how to estimate the random and systematic errors. Even so, it is worth subjecting models to tests and
seeing how well the data support our beliefs. This testing is a
key part of the cycle (Fig. 1.1-8) by which models are refined
using new data and model parameterizations.
Further reading
Many discussions of inverse theory, including ours, are based on Lanczos
(1961). Applications in the earth sciences, especially seismology, are discussed in texts and reviews including Parker (1977), Aki and Richards
(1980), and Menke (1984). Treatments of tomographic methods in seismology are given by Nolet (1987), Thurber and Aki (1987), Spakman and
Nolet (1988), Humphreys and Clayton (1988), and Romanowicz (1991).
Inversion for the properties of stratified media is reviewed by Wiggins
(1972).
Tests for goodness of fit are discussed in statistical texts such as
Bevington and Robinson (1992) and Freedman et al. (1991); the latter
treats the issue of Mendel’s results. Chase (1972) and Minster et al. (1974)
present the inverse problem for plate motions; the latter gives the partial
derivatives. Stein and Gordon (1984) and DeMets et al. (1990) discuss applications of the F-ratio test to plate motions and intraplate deformation.
7.5 Further reading 441