4.4 Moment tensors 247
and anelastic effects of propagation from the source to the receiver. As in Eqn 4.3.20, we write
u(ω, θ, φ) = V(ω, φ)H(ω, θ),
H(ω, θ) = I(ω)
e
iπ
θ
/
sin
4
e −iω aθ/c e −ω aθ/2Qu e imπ /2 .
(27)
V(ω, φ) is the radiation pattern term reflecting the effect of
source geometry, which we want to find, whereas H(ω, θ) represents the effects of the seismometer and of propagation,
which we treat as known. I(ω) is the effect of the seismometer,
and the remaining terms are propagation effects, including
e −ω aθ/2Qu , the effect of attenuation as the wave travels a distance θ (including any 2π terms). In these expressions, a is the
earth’s radius, m is the number of polar or antipolar passages,
and c, u, and Q are the phase velocity, group velocity, and
attenuation at the frequency ω.
To set up the inversion, we write the radiation pattern, which
shows how the amplitude at a given frequency varies with the
azimuth (φ) of the receiver from the source, in terms of linear
combinations of the moment tensor components
V
P M
M
M
R
x y
y y
x x
( , )
sin
(
) cos
ω φ
φ
φ
= −
−
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
2
1
2
2
+
+
+
−
+
(
)
(
)(
)
1
3
1
6
2
S
N M
N
S M
M
R
R
zz
R
R
xx
yy
+
+
(
sin
cos ).
iQ M
M
R
y z
x z
φ
φ
(28)
This expression is analogous to the radiation pattern for
Rayleigh waves due to slip on a fault (Eqn 4.3.20). The difference is that here the seismic source is written in terms of
moment tensor components, rather than as products of trigonometric functions of the fault strike, dip, and slip angles.
Thus Eqn 28 represents more general seismic sources than
double couples due to slip on a simple fault.
As before, the radiation pattern depends on excitation functions derived from the radial eigenfunctions of spheroidal
modes of the appropriate frequency, which describe how a
source at a given depth causes displacements as a function of
frequency. However, in addition to the excitation functions
in (Eqn 4.3.20) (P R , S R , Q R ), we have the excitation function N R that applies to an isotropic source. To see this, recall
that for an explosion the moment tensor (Eqn 17) has equal
diagonal elements (M xx = M yy = M zz = M 0 ) and zeroes off the
diagonal (M xy = M yz = M xz = 0). Substituting these into Eqn 28
yields V(ω, φ) = M 0 N R , which is a radiation pattern that
depends on N R and is azimuthally symmetric, as expected for
an explosion. Conversely, if the source has no isotropic component (M xx + M yy + M zz = 0), N R drops out of Eqn 28.
We can formulate the inverse problem using Eqn 28. At a
given frequency, separating V(ω, φ) into real and imaginary
parts yields the matrix equation
u
u
u
G G
G
G
G
G
G G
G
G
G
G
G G
G
G
G
G
n
n
n
n
n
n
n
1
2
11
12
13
14
15
16
21
22
23
24
25
26
1
2
3
4
5
6
⋅
⋅
⋅
⋅
⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
=
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟ ⎟
.
m
m
m
m
m
m
1
2
3
4
5
6
(24)
This is an overdetermined system of linear equations with
more equations (n) than unknowns (6). We often encounter
such systems when we invert large quantities of data to estimate a smaller number of parameters. As we noted in Section 2.8,
and will explore in depth in Chapter 7, we cannot invert the
matrix G because it is not square. Instead, we find the moment
tensor that best matches the observed seismograms in a least
squares sense, using what is called the generalized inverse of G,
m = (G T G) −1 G T u.
(25)
Thus, because the seismograms are linear functions of the
moment tensor components, they can be inverted to find the
tensor components.
Although we defer discussing most properties of generalized inverse solutions until Chapter 7, a point worth noting
is that how well we can estimate a moment tensor component
depends on the Green’s function. Equation 22 shows that the
seismogram involves products of moment tensor components
with their corresponding Green’s functions. Thus, if G ij is zero,
m j has no effect on the seismogram, no matter how big it is.
Similarly, if G ij is small, m j has little effect on the seismogram,
Conversely, inverting the seismogram to determine m j essentially involves dividing the seismogram by G ij . Hence, if G ij is
small, dividing by it gives a large number, so any small errors
or noise in the data produce spuriously large values of m j . Put
another way, we get good estimates of components to which
the seismogram is fairly sensitive, but poorer ones for components on which the seismogram depends weakly. 4
We now consider one inversion approach, a method for surface waves corresponding to the forward modeling in Section
4.3.4. In a coordinate system with the source at the north pole,
the vertical component of Rayleigh waves on a seismometer at
r = (r, θ, φ) can be written as an inverse Fourier transform:
u(r, t) =
−∞
∞
1
2π Ύ
U(ω, θ, φ)e iωt dω.
(26)
The spectral amplitude U(ω, θ, φ) is a complex number representing the source, the effect of the seismometer, and the elastic
4 We will formalize this idea using eigenvalues in Section 7.3, but before doing so,
we can see intuitively that an estimate of the number of white cats in a dimly lit room
will be better than that of black cats.
and anelastic effects of propagation from the source to the receiver. As in Eqn 4.3.20, we write
u(ω, θ, φ) = V(ω, φ)H(ω, θ),
H(ω, θ) = I(ω)
e
iπ
θ
/
sin
4
e −iω aθ/c e −ω aθ/2Qu e imπ /2 .
(27)
V(ω, φ) is the radiation pattern term reflecting the effect of
source geometry, which we want to find, whereas H(ω, θ) represents the effects of the seismometer and of propagation,
which we treat as known. I(ω) is the effect of the seismometer,
and the remaining terms are propagation effects, including
e −ω aθ/2Qu , the effect of attenuation as the wave travels a distance θ (including any 2π terms). In these expressions, a is the
earth’s radius, m is the number of polar or antipolar passages,
and c, u, and Q are the phase velocity, group velocity, and
attenuation at the frequency ω.
To set up the inversion, we write the radiation pattern, which
shows how the amplitude at a given frequency varies with the
azimuth (φ) of the receiver from the source, in terms of linear
combinations of the moment tensor components
V
P M
M
M
R
x y
y y
x x
( , )
sin
(
) cos
ω φ
φ
φ
= −
−
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
2
1
2
2
+
+
+
−
+
(
)
(
)(
)
1
3
1
6
2
S
N M
N
S M
M
R
R
zz
R
R
xx
yy
+
+
(
sin
cos ).
iQ M
M
R
y z
x z
φ
φ
(28)
This expression is analogous to the radiation pattern for
Rayleigh waves due to slip on a fault (Eqn 4.3.20). The difference is that here the seismic source is written in terms of
moment tensor components, rather than as products of trigonometric functions of the fault strike, dip, and slip angles.
Thus Eqn 28 represents more general seismic sources than
double couples due to slip on a simple fault.
As before, the radiation pattern depends on excitation functions derived from the radial eigenfunctions of spheroidal
modes of the appropriate frequency, which describe how a
source at a given depth causes displacements as a function of
frequency. However, in addition to the excitation functions
in (Eqn 4.3.20) (P R , S R , Q R ), we have the excitation function N R that applies to an isotropic source. To see this, recall
that for an explosion the moment tensor (Eqn 17) has equal
diagonal elements (M xx = M yy = M zz = M 0 ) and zeroes off the
diagonal (M xy = M yz = M xz = 0). Substituting these into Eqn 28
yields V(ω, φ) = M 0 N R , which is a radiation pattern that
depends on N R and is azimuthally symmetric, as expected for
an explosion. Conversely, if the source has no isotropic component (M xx + M yy + M zz = 0), N R drops out of Eqn 28.
We can formulate the inverse problem using Eqn 28. At a
given frequency, separating V(ω, φ) into real and imaginary
parts yields the matrix equation
u
u
u
G G
G
G
G
G
G G
G
G
G
G
G G
G
G
G
G
n
n
n
n
n
n
n
1
2
11
12
13
14
15
16
21
22
23
24
25
26
1
2
3
4
5
6
⋅
⋅
⋅
⋅
⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
=
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜ ⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟ ⎟
.
m
m
m
m
m
m
1
2
3
4
5
6
(24)
This is an overdetermined system of linear equations with
more equations (n) than unknowns (6). We often encounter
such systems when we invert large quantities of data to estimate a smaller number of parameters. As we noted in Section 2.8,
and will explore in depth in Chapter 7, we cannot invert the
matrix G because it is not square. Instead, we find the moment
tensor that best matches the observed seismograms in a least
squares sense, using what is called the generalized inverse of G,
m = (G T G) −1 G T u.
(25)
Thus, because the seismograms are linear functions of the
moment tensor components, they can be inverted to find the
tensor components.
Although we defer discussing most properties of generalized inverse solutions until Chapter 7, a point worth noting
is that how well we can estimate a moment tensor component
depends on the Green’s function. Equation 22 shows that the
seismogram involves products of moment tensor components
with their corresponding Green’s functions. Thus, if G ij is zero,
m j has no effect on the seismogram, no matter how big it is.
Similarly, if G ij is small, m j has little effect on the seismogram,
Conversely, inverting the seismogram to determine m j essentially involves dividing the seismogram by G ij . Hence, if G ij is
small, dividing by it gives a large number, so any small errors
or noise in the data produce spuriously large values of m j . Put
another way, we get good estimates of components to which
the seismogram is fairly sensitive, but poorer ones for components on which the seismogram depends weakly. 4
We now consider one inversion approach, a method for surface waves corresponding to the forward modeling in Section
4.3.4. In a coordinate system with the source at the north pole,
the vertical component of Rayleigh waves on a seismometer at
r = (r, θ, φ) can be written as an inverse Fourier transform:
u(r, t) =
−∞
∞
1
2π Ύ
U(ω, θ, φ)e iωt dω.
(26)
The spectral amplitude U(ω, θ, φ) is a complex number representing the source, the effect of the seismometer, and the elastic
4 We will formalize this idea using eigenvalues in Section 7.3, but before doing so,
we can see intuitively that an estimate of the number of white cats in a dimly lit room
will be better than that of black cats.