422 Inverse Problems
iterations, is close to, but not exactly, the model used to generate the data. This simple example thus has some features of real
situations.
The uncertainties in the final model are shown by the model
variance–covariance matrix
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
m
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
.
xx
xy
xz
xt
yx
yy
yz
yt
zx
zy
zz
zt
tx
ty
tz
tt
(35)
To see that the results seem reasonable, we compare the final
inversion model, taking into account its uncertainty, to the true
model. The standard deviations of each parameter are given by
the square roots of the diagonal terms of the model variance–
covariance matrix, so the final model (x = 0.2 ± 0.25 km,
y = −0.4 ± 0.28 km, z = 12.2 ± 1.08 km, t = −0.2 ± 0.10 s) is an
acceptable representation of the true model.
The model variance–covariance matrix shows some interesting features. The variance of the depth estimate, σ 2
zz is larger
than the corresponding terms σ 2
xx and σ 2
yy , indicating that the
depth is less well constrained than the epicenter. This situation
is common, and arises because all the seismometers are at the
surface. 2 In some cases when the depth is poorly constrained,
it is regarded as fixed, and only the epicenter and the origin are
inverted for. The results of multiple inversions, each with the
depth fixed at a different value, are compared to see which best
fits the data. It is also possible to determine the depth from
other criteria, such as the times of surface reflections (Section
4.3), and then invert with the depth fixed.
The uncertainties in the model parameter estimates are
correlated, because the off-diagonal elements of the model
variance–covariance matrix are nonzero. σ 2
zt , the covariance of
the depth and origin time uncertainties, is negative, indicating
a trade-off between the focal depth and the origin time. At any
station, similar arrival times result if the earthquake occurred
earlier (t smaller) but deeper (z larger). Similarly, σ 2
xy , the
covariance of the x and y location uncertainties, is nonzero,
so the uncertainties in these two parameters are correlated.
A method often used to illustrate this is to extract the 2 × 2
submatrix
σ
σ
σ
σ
xx
xy
yx
yy
2
2
2
2
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
(36)
and diagonalize it by finding the eigenvalues λ (1) and λ (2) , and
the associated eigenvectors (x 1
(1) , x 2
(1)
) and (x 1
(2)
, x 2
(2)
). The uncertainty in the epicenter can then be thought of as an ellipse with
semi-major and semi-minor axes λ (1)1/2 and λ (2)1/2 , oriented in a
direction given by tan
−1 (x 1
(1) /x 2
(1)
). In this case, the semi-major
and semi-minor axes have lengths of 0.29 and 0.24 km, and the
semi-major axis trends N22°E. An interesting feature of the
error ellipse is that its shape and orientation depend on the
(G T G) −1 matrix, whereas the variance of the data, σ d
2 , controls the size of the ellipse. Because the shape of the error ellipse
depends on the geometry of the receivers, it can be examined
without reference to specific data. As written, the ellipse is for a
confidence level of 1σ (68%), but ellipses are sometimes also
given for 2σ (95%), or 3σ (99%).
We have shown that the model variance–covariance matrix
depends on the variance–covariance matrix of the data. In the
example, we knew the standard deviation of the data and that
the errors were uncorrelated. This information would not be
available for a real experiment. However, we could estimate
the standard deviation of the data from the misfit between the
data and the best-fitting model, given by the sample variance s 2 ,
σ
2
≈ s
2 = − =
∑
1
1
n k i
n
(d i
′ − d i )
2
.
(37)
Here, d i ′ are the observations, d i are the values of the data predicted by the best-fitting model, and k is the number of model
parameters determined from the data. Division by n − k, the
number of degrees of freedom, rather than by n, the number
of data, compensates for the improvement in fit resulting from
the use of model parameters determined from the data. Thus,
for our example, the final squared misfit is 0.4 s 2 , and four
parameters were determined from the data, so the sample
standard deviation is s = (0.4/(10 − 4)) 1/2 = 0.08 s, a value close
to the true σ, 0.1 s.
7.2.4 Earthquake location for more complex geometries
This formulation is not restricted to locating earthquakes in
a homogeneous halfspace. Velocity variations can be incorporated in the function relating the arrival time at the i th station
to the origin time t and travel time T(x, x i ),
d i ′ = T(x, x i ) + t.
(1)
For example, a model for locating local earthquakes could have
a series of layers. As a result, even for a source at the surface,
the travel time curve is a more complicated function of distance
(Section 3.2). At close distances, the first arrival is the direct
wave. At greater distances, the first arrival becomes a head
wave from an interface at depth, with the relevant interface
being deeper as the distance increases. The situation is similar,
but more complicated, for a source at depth, because at zero
distance the travel time is nonzero.
The travel time curve can be found either analytically or by
tracing rays. If the receivers are on the surface at (x i , y i ), the
travel time curve T(r, z) depends on the horizontal distance
between source and receiver,
r i = [(x − x i ) 2 + (y − y i ) 2 ] 1/2 ,
(38)
2 Similarly, vertical positions determined using the GPS (Section 4.5.1) by a process
analogous to earthquake location are less precise than the horizontal positions.
iterations, is close to, but not exactly, the model used to generate the data. This simple example thus has some features of real
situations.
The uncertainties in the final model are shown by the model
variance–covariance matrix
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
m
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
=
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
.
xx
xy
xz
xt
yx
yy
yz
yt
zx
zy
zz
zt
tx
ty
tz
tt
(35)
To see that the results seem reasonable, we compare the final
inversion model, taking into account its uncertainty, to the true
model. The standard deviations of each parameter are given by
the square roots of the diagonal terms of the model variance–
covariance matrix, so the final model (x = 0.2 ± 0.25 km,
y = −0.4 ± 0.28 km, z = 12.2 ± 1.08 km, t = −0.2 ± 0.10 s) is an
acceptable representation of the true model.
The model variance–covariance matrix shows some interesting features. The variance of the depth estimate, σ 2
zz is larger
than the corresponding terms σ 2
xx and σ 2
yy , indicating that the
depth is less well constrained than the epicenter. This situation
is common, and arises because all the seismometers are at the
surface. 2 In some cases when the depth is poorly constrained,
it is regarded as fixed, and only the epicenter and the origin are
inverted for. The results of multiple inversions, each with the
depth fixed at a different value, are compared to see which best
fits the data. It is also possible to determine the depth from
other criteria, such as the times of surface reflections (Section
4.3), and then invert with the depth fixed.
The uncertainties in the model parameter estimates are
correlated, because the off-diagonal elements of the model
variance–covariance matrix are nonzero. σ 2
zt , the covariance of
the depth and origin time uncertainties, is negative, indicating
a trade-off between the focal depth and the origin time. At any
station, similar arrival times result if the earthquake occurred
earlier (t smaller) but deeper (z larger). Similarly, σ 2
xy , the
covariance of the x and y location uncertainties, is nonzero,
so the uncertainties in these two parameters are correlated.
A method often used to illustrate this is to extract the 2 × 2
submatrix
σ
σ
σ
σ
xx
xy
yx
yy
2
2
2
2
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
(36)
and diagonalize it by finding the eigenvalues λ (1) and λ (2) , and
the associated eigenvectors (x 1
(1) , x 2
(1)
) and (x 1
(2)
, x 2
(2)
). The uncertainty in the epicenter can then be thought of as an ellipse with
semi-major and semi-minor axes λ (1)1/2 and λ (2)1/2 , oriented in a
direction given by tan
−1 (x 1
(1) /x 2
(1)
). In this case, the semi-major
and semi-minor axes have lengths of 0.29 and 0.24 km, and the
semi-major axis trends N22°E. An interesting feature of the
error ellipse is that its shape and orientation depend on the
(G T G) −1 matrix, whereas the variance of the data, σ d
2 , controls the size of the ellipse. Because the shape of the error ellipse
depends on the geometry of the receivers, it can be examined
without reference to specific data. As written, the ellipse is for a
confidence level of 1σ (68%), but ellipses are sometimes also
given for 2σ (95%), or 3σ (99%).
We have shown that the model variance–covariance matrix
depends on the variance–covariance matrix of the data. In the
example, we knew the standard deviation of the data and that
the errors were uncorrelated. This information would not be
available for a real experiment. However, we could estimate
the standard deviation of the data from the misfit between the
data and the best-fitting model, given by the sample variance s 2 ,
σ
2
≈ s
2 = − =
∑
1
1
n k i
n
(d i
′ − d i )
2
.
(37)
Here, d i ′ are the observations, d i are the values of the data predicted by the best-fitting model, and k is the number of model
parameters determined from the data. Division by n − k, the
number of degrees of freedom, rather than by n, the number
of data, compensates for the improvement in fit resulting from
the use of model parameters determined from the data. Thus,
for our example, the final squared misfit is 0.4 s 2 , and four
parameters were determined from the data, so the sample
standard deviation is s = (0.4/(10 − 4)) 1/2 = 0.08 s, a value close
to the true σ, 0.1 s.
7.2.4 Earthquake location for more complex geometries
This formulation is not restricted to locating earthquakes in
a homogeneous halfspace. Velocity variations can be incorporated in the function relating the arrival time at the i th station
to the origin time t and travel time T(x, x i ),
d i ′ = T(x, x i ) + t.
(1)
For example, a model for locating local earthquakes could have
a series of layers. As a result, even for a source at the surface,
the travel time curve is a more complicated function of distance
(Section 3.2). At close distances, the first arrival is the direct
wave. At greater distances, the first arrival becomes a head
wave from an interface at depth, with the relevant interface
being deeper as the distance increases. The situation is similar,
but more complicated, for a source at depth, because at zero
distance the travel time is nonzero.
The travel time curve can be found either analytically or by
tracing rays. If the receivers are on the surface at (x i , y i ), the
travel time curve T(r, z) depends on the horizontal distance
between source and receiver,
r i = [(x − x i ) 2 + (y − y i ) 2 ] 1/2 ,
(38)
2 Similarly, vertical positions determined using the GPS (Section 4.5.1) by a process
analogous to earthquake location are less precise than the horizontal positions.