often viewed as traveling waves. However, the longest-period
modes, like 0 S 2 , 0 S 3 , 0 T 2 , 0 T 3 , etc. have such long periods that
we think of them as modes. Higher-order modes are often
thought of in terms of a body wave phase to which they contribute. Of course, the descriptions are equivalent.
2.9.8 Normal mode synthetic seismograms
As we will see in many places in this text, various techniques
are used to create theoretical, often called synthetic, seismograms for the earth. One of these is normal mode summation,
analogous to the way the propagating waves on the string were
generated in Section 2.2. This summation is also the way that
a music synthesizer creates a particular sound by summing the
right combination of harmonic overtones (i.e., modes). 9
For example, torsional mode displacements (Eqn 11) are
synthesized by
u T
r
r
r
r
( , , )
θ φ
= ∑ ∑ ∑
−
=−
( , )
( ) ( , )
.
n l
n l
m
s r n l r
l
m
r
r
i
t
t
Q
m l
l
A r r W r
e
e
n l
m
n l
m
n l
T θ φ
ω
ω
2
(23)
To do this, we need to know the modes’ radial eigenfunctions,
n W l and eigenfrequencies n ω l
m , which are determined by the
earth’s velocity and density structure. These modes are then
weighted by excitation amplitudes n A l
m , determined by the
depth, geometry, and time history of the seismic source and
the depth of the receiver. We also need to know the attenuation, or quality, factor n Q l , discussed in Sections 3.7 and 7.4,
which measures the rate at which the mode’s seismic energy is
lost by friction (without this effect, the earth would ring like a
bell forever). This formulation assumes that all singlets in a
multiplet have the same quality factor.
The modes of the earth are found by computing the
radial eigenfunctions and the corresponding eigenfrequencies.
Although this process is beyond our scope here, several
techniques have been developed to do this. Some involve
propagating the values of stresses and displacements from the
center of the earth to the surface, layer by layer, while satisfying
the boundary conditions at each layer. The frequency of the
mode is iterated until the final surface values satisfy the free
surface boundary conditions. This process is analogous to that
used to determine the periods of the Love waves in the layer
over the halfspace example (Section 2.7.3).
The amplitudes, or excitation coefficients, depend on the
earthquake’s fault geometry. One of the many advantages of
evaluating the normal modes in a coordinate system whose
pole is at the seismic source is that the radiated energy has
strong symmetry. As noted in Section 1.1 and discussed further
in Chapter 4, earthquakes radiate energy in a pattern with fourlobed symmetry about the fault plane. Thus any given fault
geometry is reflected by various combinations of the m = 0, ±1,
and ±2 singlets. The excitation also depends on the depth of the
source, much as that for the string depended on the source position. 10 An earthquake at 600 km depth strongly excites modes
whose eigenfunctions are large at that depth, whereas other
modes are barely excited. However, the relative excitations
will be very different for an earthquake at 10 km depth. For
example, as previously discussed, fundamental mode surface
waves correspond to the fundamental (n = 0) branch of torsional and spheroidal modes, for angular orders greater than
about 20. Because these modes’ radial eigenfunctions are small
at great depths, a 600 km-deep earthquake does not excite
surface waves efficiently.
The modes are summed at a specific receiver location. Thus
the displacements in Eqn 23 are expressed in terms of the
radius of the source r s , the radius of the receiver r r , and the
colatitude and azimuth of the receiver, θ r and φ r . A slightly
disturbing feature of the mode sum (Eqn 23) is that both the
time functions and the vector spherical harmonics are complex
numbers. However, the sum gives the displacement as a real
number. Similarly, although individual modes oscillate everywhere on earth at all times, even before a traveling wave from
an earthquake could arrive, the mode sum yields waves that
arrive after a finite time. Thus, although modes are mathematical objects that are hard to visualize, their sum gives rise to a
meaningful physical displacement (Fig. 2.9-12).
Figure 2.9-13 shows a comparison of observed seismograms with synthetic seismograms created using normal mode
summation. The fits are good enough that many studies use
observed normal mode amplitudes to find the fault geometry
and focal depth of earthquakes, especially when they are large
and remote from seismometers. This process is an inverse
problem, corresponding to the forward problem of generating
a synthetic seismogram.
It is worth noting that while the synthetic receivers are usually placed at the surface (where seismometers are), they can
also be computed for any depth within the earth. Figure 2.9-14
shows a record section that would be recorded at a distance of
70° from an earthquake if seismometers could be placed a
depths ranging from the surface to the core–mantle boundary.
We will use this idea shortly to visualize shear wave propagation (Section 3.5.5) by evaluating normal mode synthetic
seismograms at 100,000 locations in the mantle.
2.9.9 Mode attenuation, splitting, and coupling
So far, we have discussed the modes of a spherically symmetric, nonrotating, purely elastic, and isotropic earth. This
9 Although the fundamental notes for a clarinet, trumpet, and flute might be the
same, playing each instrument excites a different suite of overtones, giving a different
sound. For instance, because the open end of the clarinet allows only odd-numbered
overtones, the absence of even-numbered overtones contributes to its warm, dark
sound. Early synthesizers added only a few overtones, producing a false, tinny sound.
Modern synthesizers sum overtones up to and beyond frequencies of 20,000 Hz, the
limit of human hearing, so the synthesized sounds can be indistinguishable from those
of the actual instruments.
10 This effect is analogous to the way in which bowing at different locations on a
violin string makes different sounds.
2.9 Normal modes of the earth 111
modes, like 0 S 2 , 0 S 3 , 0 T 2 , 0 T 3 , etc. have such long periods that
we think of them as modes. Higher-order modes are often
thought of in terms of a body wave phase to which they contribute. Of course, the descriptions are equivalent.
2.9.8 Normal mode synthetic seismograms
As we will see in many places in this text, various techniques
are used to create theoretical, often called synthetic, seismograms for the earth. One of these is normal mode summation,
analogous to the way the propagating waves on the string were
generated in Section 2.2. This summation is also the way that
a music synthesizer creates a particular sound by summing the
right combination of harmonic overtones (i.e., modes). 9
For example, torsional mode displacements (Eqn 11) are
synthesized by
u T
r
r
r
r
( , , )
θ φ
= ∑ ∑ ∑
−
=−
( , )
( ) ( , )
.
n l
n l
m
s r n l r
l
m
r
r
i
t
t
Q
m l
l
A r r W r
e
e
n l
m
n l
m
n l
T θ φ
ω
ω
2
(23)
To do this, we need to know the modes’ radial eigenfunctions,
n W l and eigenfrequencies n ω l
m , which are determined by the
earth’s velocity and density structure. These modes are then
weighted by excitation amplitudes n A l
m , determined by the
depth, geometry, and time history of the seismic source and
the depth of the receiver. We also need to know the attenuation, or quality, factor n Q l , discussed in Sections 3.7 and 7.4,
which measures the rate at which the mode’s seismic energy is
lost by friction (without this effect, the earth would ring like a
bell forever). This formulation assumes that all singlets in a
multiplet have the same quality factor.
The modes of the earth are found by computing the
radial eigenfunctions and the corresponding eigenfrequencies.
Although this process is beyond our scope here, several
techniques have been developed to do this. Some involve
propagating the values of stresses and displacements from the
center of the earth to the surface, layer by layer, while satisfying
the boundary conditions at each layer. The frequency of the
mode is iterated until the final surface values satisfy the free
surface boundary conditions. This process is analogous to that
used to determine the periods of the Love waves in the layer
over the halfspace example (Section 2.7.3).
The amplitudes, or excitation coefficients, depend on the
earthquake’s fault geometry. One of the many advantages of
evaluating the normal modes in a coordinate system whose
pole is at the seismic source is that the radiated energy has
strong symmetry. As noted in Section 1.1 and discussed further
in Chapter 4, earthquakes radiate energy in a pattern with fourlobed symmetry about the fault plane. Thus any given fault
geometry is reflected by various combinations of the m = 0, ±1,
and ±2 singlets. The excitation also depends on the depth of the
source, much as that for the string depended on the source position. 10 An earthquake at 600 km depth strongly excites modes
whose eigenfunctions are large at that depth, whereas other
modes are barely excited. However, the relative excitations
will be very different for an earthquake at 10 km depth. For
example, as previously discussed, fundamental mode surface
waves correspond to the fundamental (n = 0) branch of torsional and spheroidal modes, for angular orders greater than
about 20. Because these modes’ radial eigenfunctions are small
at great depths, a 600 km-deep earthquake does not excite
surface waves efficiently.
The modes are summed at a specific receiver location. Thus
the displacements in Eqn 23 are expressed in terms of the
radius of the source r s , the radius of the receiver r r , and the
colatitude and azimuth of the receiver, θ r and φ r . A slightly
disturbing feature of the mode sum (Eqn 23) is that both the
time functions and the vector spherical harmonics are complex
numbers. However, the sum gives the displacement as a real
number. Similarly, although individual modes oscillate everywhere on earth at all times, even before a traveling wave from
an earthquake could arrive, the mode sum yields waves that
arrive after a finite time. Thus, although modes are mathematical objects that are hard to visualize, their sum gives rise to a
meaningful physical displacement (Fig. 2.9-12).
Figure 2.9-13 shows a comparison of observed seismograms with synthetic seismograms created using normal mode
summation. The fits are good enough that many studies use
observed normal mode amplitudes to find the fault geometry
and focal depth of earthquakes, especially when they are large
and remote from seismometers. This process is an inverse
problem, corresponding to the forward problem of generating
a synthetic seismogram.
It is worth noting that while the synthetic receivers are usually placed at the surface (where seismometers are), they can
also be computed for any depth within the earth. Figure 2.9-14
shows a record section that would be recorded at a distance of
70° from an earthquake if seismometers could be placed a
depths ranging from the surface to the core–mantle boundary.
We will use this idea shortly to visualize shear wave propagation (Section 3.5.5) by evaluating normal mode synthetic
seismograms at 100,000 locations in the mantle.
2.9.9 Mode attenuation, splitting, and coupling
So far, we have discussed the modes of a spherically symmetric, nonrotating, purely elastic, and isotropic earth. This
9 Although the fundamental notes for a clarinet, trumpet, and flute might be the
same, playing each instrument excites a different suite of overtones, giving a different
sound. For instance, because the open end of the clarinet allows only odd-numbered
overtones, the absence of even-numbered overtones contributes to its warm, dark
sound. Early synthesizers added only a few overtones, producing a false, tinny sound.
Modern synthesizers sum overtones up to and beyond frequencies of 20,000 Hz, the
limit of human hearing, so the synthesized sounds can be indistinguishable from those
of the actual instruments.
10 This effect is analogous to the way in which bowing at different locations on a
violin string makes different sounds.
2.9 Normal modes of the earth 111