442 Inverse Problems
Problems
1. Show the following matrix identities:
(a) For an arbitrary (not square) matrix A, the matrices A
T A and
AA T are symmetric.
(b) For an arbitrary (not square) matrix B and a symmetric
matrix A, (B
T AB)
T = B
T
AB.
(c) For square matrices A and B such that (AB)
−1 exists, (AB)
−1 =
B
−1
A
−1 .
2. Show that if a square matrix G has an inverse, the inverse and generalized inverse are identical.
3. Show that if the variance–covariance matrix of the data is diagonal,
σ
2
d = σ
2
ij δ ij (with no summation implied), its inverse is another diagonal matrix W d = δ ij /σ 2
ij . (Also with no summation implied.)
4. Show that the model variance–covariance matrix (Eqn 7.2.32) σ
2
m =
G
−g σ
2
d (G
−g )
T reduces to σ
2
m = σ
2
(G
T
G)
−1 when the data errors are
uncorrelated and equal, so the data variance–covariance matrix is a
constant times the identity matrix, σ
2
d = σ
2
δ ij .
5. Show that if the data errors are uncorrelated but not equal, such that
the data variance–covariance matrix of the data is the diagonal
matrix σ
2
d = σ
2
ij δ ij with inverse W d (problem 3):
(a) The least squares criterion (Eqn 7.2.14) for the inverse problem gives rise to the weighted least squares solution ∆m =
(G
T W d G)
−1
G
T W d ∆d.
(b) The model variance–covariance matrix is σ
2
m = (G
T W d G)
−1 .
6. For a halfspace with uniform (and known) velocities α and β:
(a) Show how the location problem can be formulated to use
both P-wave and S-wave first arrival times as data. Write the
data vector, model vector, and partial derivatives. How do
these differ from the case for P waves alone?
(b) Show how the location problem can be formulated to use only
the difference between P-wave and S-wave first arrival times
as data. Write the data vector, model vector, and partial
derivatives. How do these differ from the case for P waves
alone? How might you apply this method if only the P velocity
were known? Under what conditions might this method be
useful?
7. For the idealized tomographic experiment in Figure 7.3-2:
(a) Show how one row of the G matrix in Eqn 7.3.10 can be
derived from the others, such that the four teleseismic ray
paths are not linearly independent. Give a physical interpretation of this result.
(b) Find four rows of the G matrix in Eqn 7.3.6 that are linearly
independent, and give a physical interpretation of this result.
Computer problems
C-1. Write a subroutine to find the generalized inverse G
−g =
(G T G) −1 G T of an (n × r) matrix G, using a matrix inversion
subroutine. As a test, check that the solution satisfies the criterion
that for a square matrix G that has an inverse, the inverse and
generalized inverse are identical.
C-2. For a homogeneous halfspace with P-wave velocity α:
(a) Write a subroutine to compute the distance and travel time
between two points (x, y, z) and (x i , y i , z i ). Test this for
some simple cases.
(b) Use the result of (a) to write a program that reads an earthquake location, origin time, and medium velocity and the
locations of n seismic stations, and finds the first arrival
time at each station.
(c) Write a subroutine using the result of (a) to compute the
partial derivatives of the first arrival time at a station with
respect to changes in the model parameters (location,
origin time, and medium velocity).
(d) Modify the result of (b) to compute arrival times for a starting model (assumed location, origin time, and medium
velocity), and then locate the earthquake by inverting these
synthetic data to find the best-fitting model. The result of
C-1 should be useful. Have the program iterate until the
model change between iterations is less than a parameter
you set. The program should have the option to invert for
velocity or hold velocity fixed at an assumed starting value.
C-3. Test the location program with a set of station locations, a “real”
origin time and location, and an incorrect starting model. The
program should retrieve the “real” model. Once this works for
error-free data, add some errors to the travel times, either by using
your computer’s random number function or by simply choosing
some numbers. Invert for the best-fitting model, and see how
the result of the inversion changes as the errors become a larger
fraction of the travel times. How do the results depend on
whether the velocity is held fixed or inverted for?
C-4. Compute and compare χ
2 and χ
2
ν for C-3 for cases in which
you inverted for velocity and in which the velocity is fixed at an
incorrect value. Using the F-ratio test, does the improved fit due to
inverting for velocity seem significant?
Problems
1. Show the following matrix identities:
(a) For an arbitrary (not square) matrix A, the matrices A
T A and
AA T are symmetric.
(b) For an arbitrary (not square) matrix B and a symmetric
matrix A, (B
T AB)
T = B
T
AB.
(c) For square matrices A and B such that (AB)
−1 exists, (AB)
−1 =
B
−1
A
−1 .
2. Show that if a square matrix G has an inverse, the inverse and generalized inverse are identical.
3. Show that if the variance–covariance matrix of the data is diagonal,
σ
2
d = σ
2
ij δ ij (with no summation implied), its inverse is another diagonal matrix W d = δ ij /σ 2
ij . (Also with no summation implied.)
4. Show that the model variance–covariance matrix (Eqn 7.2.32) σ
2
m =
G
−g σ
2
d (G
−g )
T reduces to σ
2
m = σ
2
(G
T
G)
−1 when the data errors are
uncorrelated and equal, so the data variance–covariance matrix is a
constant times the identity matrix, σ
2
d = σ
2
δ ij .
5. Show that if the data errors are uncorrelated but not equal, such that
the data variance–covariance matrix of the data is the diagonal
matrix σ
2
d = σ
2
ij δ ij with inverse W d (problem 3):
(a) The least squares criterion (Eqn 7.2.14) for the inverse problem gives rise to the weighted least squares solution ∆m =
(G
T W d G)
−1
G
T W d ∆d.
(b) The model variance–covariance matrix is σ
2
m = (G
T W d G)
−1 .
6. For a halfspace with uniform (and known) velocities α and β:
(a) Show how the location problem can be formulated to use
both P-wave and S-wave first arrival times as data. Write the
data vector, model vector, and partial derivatives. How do
these differ from the case for P waves alone?
(b) Show how the location problem can be formulated to use only
the difference between P-wave and S-wave first arrival times
as data. Write the data vector, model vector, and partial
derivatives. How do these differ from the case for P waves
alone? How might you apply this method if only the P velocity
were known? Under what conditions might this method be
useful?
7. For the idealized tomographic experiment in Figure 7.3-2:
(a) Show how one row of the G matrix in Eqn 7.3.10 can be
derived from the others, such that the four teleseismic ray
paths are not linearly independent. Give a physical interpretation of this result.
(b) Find four rows of the G matrix in Eqn 7.3.6 that are linearly
independent, and give a physical interpretation of this result.
Computer problems
C-1. Write a subroutine to find the generalized inverse G
−g =
(G T G) −1 G T of an (n × r) matrix G, using a matrix inversion
subroutine. As a test, check that the solution satisfies the criterion
that for a square matrix G that has an inverse, the inverse and
generalized inverse are identical.
C-2. For a homogeneous halfspace with P-wave velocity α:
(a) Write a subroutine to compute the distance and travel time
between two points (x, y, z) and (x i , y i , z i ). Test this for
some simple cases.
(b) Use the result of (a) to write a program that reads an earthquake location, origin time, and medium velocity and the
locations of n seismic stations, and finds the first arrival
time at each station.
(c) Write a subroutine using the result of (a) to compute the
partial derivatives of the first arrival time at a station with
respect to changes in the model parameters (location,
origin time, and medium velocity).
(d) Modify the result of (b) to compute arrival times for a starting model (assumed location, origin time, and medium
velocity), and then locate the earthquake by inverting these
synthetic data to find the best-fitting model. The result of
C-1 should be useful. Have the program iterate until the
model change between iterations is less than a parameter
you set. The program should have the option to invert for
velocity or hold velocity fixed at an assumed starting value.
C-3. Test the location program with a set of station locations, a “real”
origin time and location, and an incorrect starting model. The
program should retrieve the “real” model. Once this works for
error-free data, add some errors to the travel times, either by using
your computer’s random number function or by simply choosing
some numbers. Invert for the best-fitting model, and see how
the result of the inversion changes as the errors become a larger
fraction of the travel times. How do the results depend on
whether the velocity is held fixed or inverted for?
C-4. Compute and compare χ
2 and χ
2
ν for C-3 for cases in which
you inverted for velocity and in which the velocity is fixed at an
incorrect value. Using the F-ratio test, does the improved fit due to
inverting for velocity seem significant?