248 Earthquakes
Re( ( , ))
Im( ( , ))
,
V
V
B
ω φ
ω φ
⎛
⎝
⎜
⎞
⎠
⎟ = m
(29)
where m is a vector composed of the moment tensor components that we seek to find,
m
,
=
−
+
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
M
M
M
M
M
M
M
M
xy
yy
xx
zz
xx
yy
yz
xz
(30)
and B is the known matrix
B
P
P
S
N
R
R
R
R
sin
cos
(
)
=
−
+
⎛
⎝
⎜
⎜
2
2
2
1
3
0
0
0
φ
φ
N
S
Q
Q
R
R
R
R
(
)
sin
cos
−
⎞
⎠
⎟
⎟
1
6
2
0
0
0
φ
φ
(31)
containing the excitation functions and azimuthal dependence.
To invert seismograms for the moment tensor, we divide the
Fourier transform of the seismogram from the station at r i by
the propagation and seismometer term H(ω, θ i ) (Eqn 27) to
find the complex amplitude V(ω, φ i ). Data from only one seismic station yields two equations in six unknowns, so we cannot
find m. However, with data at three or more stations, all six
components of m can in principle be found. We form a vector
v from the V(ω, φ i ) values observed at each of the n stations.
We similarly use the values of B for each station, and write a
vector–matrix equation equating the observed amplitudes v to
those predicted by the known matrix B and the moment tensor
m that we seek,
v = Bm,
(32)
where
v
Re ( , )
Im ( , )
Re ( , )
Im ( , )
=
⋅
⋅
⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟ ⎟
V
V
V
V
n
n
ω φ
ω φ
ω φ
ω φ
1
1
(33)
and
B
P
P
S
N
P
P
S
N
R
R
R
R
R
n
R
n
R
R
sin
cos
(
)
sin
cos
(
)
=
−
+
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
−
+
2
2
2
1
3
0
0
0
2
2
2
1
3
1
1
φ
φ
φ
φ
0 0
0
0
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
N
S
Q
Q
N
S
R
R
R
R
R
R
(
)
sin
cos
(
)
−
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
−
1
6
2
0
0
0
1
6
2
0
0
1
1
φ
φ
0
Q
Q
R
n
R
n
sin
cos
.
φ
φ
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
(34)
With more than three stations, there are more equations than
unknowns and Eqn 32 is solved using the least squares solution
(Eqn 25), giving
m = (B T B) −1 B T v.
(35)
This solution gives the moment tensor that best predicts the
observed spectral amplitudes. It is estimated at a given frequency for a seismic source that is a delta function in time.
Time variation in the source can be examined by solving for m
at different frequencies.
An important limitation results from the fact that the middle
two columns in matrix B, corresponding to M zz and M xx + M yy ,
do not contain φ, and so have no azimuthal variation. Therefore, no matter how many stations we use at a given frequency,
we are solving only for the sum
1
3
1
6
2
(
)
(
)(
),
S
N M
N
S M
M
R
R
zz
R
R
xx
yy
+
+
−
+
(36)
which is the same at all stations. The inversion thus cannot
find M xx + M yy and M zz separately, but only their sum, which is
the isotropic portion of the source corresponding to possible
volume changes.
One way to deal with this problem is to use data at different
frequencies, where the coefficients of M zz and M xx + M yy are
quite different. This is often difficult because these coefficients
vary slowly with frequency for shallow earthquakes (consider
S R (ω) for the 11 km-deep earthquake in Fig. 4.3-14). Surface
wave moment tensor inversions thus often constrain the source
to have no isotropic portion, so that M xx + M yy = −M zz . In this
case,
Re( ( , ))
Im( ( , ))
,
V
V
B
ω φ
ω φ
⎛
⎝
⎜
⎞
⎠
⎟ = m
(29)
where m is a vector composed of the moment tensor components that we seek to find,
m
,
=
−
+
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
M
M
M
M
M
M
M
M
xy
yy
xx
zz
xx
yy
yz
xz
(30)
and B is the known matrix
B
P
P
S
N
R
R
R
R
sin
cos
(
)
=
−
+
⎛
⎝
⎜
⎜
2
2
2
1
3
0
0
0
φ
φ
N
S
Q
Q
R
R
R
R
(
)
sin
cos
−
⎞
⎠
⎟
⎟
1
6
2
0
0
0
φ
φ
(31)
containing the excitation functions and azimuthal dependence.
To invert seismograms for the moment tensor, we divide the
Fourier transform of the seismogram from the station at r i by
the propagation and seismometer term H(ω, θ i ) (Eqn 27) to
find the complex amplitude V(ω, φ i ). Data from only one seismic station yields two equations in six unknowns, so we cannot
find m. However, with data at three or more stations, all six
components of m can in principle be found. We form a vector
v from the V(ω, φ i ) values observed at each of the n stations.
We similarly use the values of B for each station, and write a
vector–matrix equation equating the observed amplitudes v to
those predicted by the known matrix B and the moment tensor
m that we seek,
v = Bm,
(32)
where
v
Re ( , )
Im ( , )
Re ( , )
Im ( , )
=
⋅
⋅
⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟ ⎟
V
V
V
V
n
n
ω φ
ω φ
ω φ
ω φ
1
1
(33)
and
B
P
P
S
N
P
P
S
N
R
R
R
R
R
n
R
n
R
R
sin
cos
(
)
sin
cos
(
)
=
−
+
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
−
+
2
2
2
1
3
0
0
0
2
2
2
1
3
1
1
φ
φ
φ
φ
0 0
0
0
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
N
S
Q
Q
N
S
R
R
R
R
R
R
(
)
sin
cos
(
)
−
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
−
1
6
2
0
0
0
1
6
2
0
0
1
1
φ
φ
0
Q
Q
R
n
R
n
sin
cos
.
φ
φ
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
(34)
With more than three stations, there are more equations than
unknowns and Eqn 32 is solved using the least squares solution
(Eqn 25), giving
m = (B T B) −1 B T v.
(35)
This solution gives the moment tensor that best predicts the
observed spectral amplitudes. It is estimated at a given frequency for a seismic source that is a delta function in time.
Time variation in the source can be examined by solving for m
at different frequencies.
An important limitation results from the fact that the middle
two columns in matrix B, corresponding to M zz and M xx + M yy ,
do not contain φ, and so have no azimuthal variation. Therefore, no matter how many stations we use at a given frequency,
we are solving only for the sum
1
3
1
6
2
(
)
(
)(
),
S
N M
N
S M
M
R
R
zz
R
R
xx
yy
+
+
−
+
(36)
which is the same at all stations. The inversion thus cannot
find M xx + M yy and M zz separately, but only their sum, which is
the isotropic portion of the source corresponding to possible
volume changes.
One way to deal with this problem is to use data at different
frequencies, where the coefficients of M zz and M xx + M yy are
quite different. This is often difficult because these coefficients
vary slowly with frequency for shallow earthquakes (consider
S R (ω) for the 11 km-deep earthquake in Fig. 4.3-14). Surface
wave moment tensor inversions thus often constrain the source
to have no isotropic portion, so that M xx + M yy = −M zz . In this
case,