222 Earthquakes
where D
— is the average slip (or dislocation) on the fault with
area S. We often use the seismic moment as a scale factor and
write F(t) = M 0 x(t), where x(t) is the source time function.
The final term, sin 2θ cos φ, describes the P-wave radiation
pattern. It is four-lobed, with two positive, compressional,
lobes and two negative, dilatational, ones. The displacement is
zero on the fault (θ = 90°) and auxiliary (φ = 90°) planes. Thus
the fault plane and auxiliary plane are nodal planes separating compressional and dilatational quadrants. The maximum
amplitudes are between the two nodal planes.
Similarly, the shear wave displacement has two components,
u θ ê θ + u φ ê φ , where
u
r
θ
πρβ
=
1
4
3
F(t − r/β) cos 2θ cos φ,
u
r
φ
πρβ
=
1
4
3
F(t − r/β)(−cos θ sin φ).
(6)
Note that the term involving F(t) corresponds to waves propagating at the S-wave speed β. As shown in Fig. 4.2-6, the
S-wave motion does not have nodal planes, but it is perpendicular to the P-wave nodal planes and is zero on the null axis.
It converges toward the center of the compressional quadrants,
which, as we will see shortly, is the location of the T, or least
compressive stress, axis. It also diverges from the centers of the
dilatation quadrants, known as the P, or most compressive
stress, axis. Thus, although the S-wave pattern does not reflect
the fault plane as clearly as the P-wave pattern, it can also be
used to study the fault geometry. An interesting feature of
Eqns 3 and 6 is that they show why S waves on seismograms are
usually bigger than P waves — the equations predict an average
ratio of α 3 /β 3 , or about 5.
Because the radiated seismic waves vary as a function of
θ and φ, seismograms recorded at different directions from
the earthquake can be used to find the fault geometry. The
P wave is the first wave to arrive from an earthquake, so on a
seismogram it is an isolated arrival whose polarity is often easy
to identify. A set of P-wave first motions thus often makes
it possible to locate the nodal planes that divide the regions of
different polarity. The first S waves are harder to use, because
they arrive later in the seismogram and can be buried in a complicated wave train. It is still possible, however, to use the Swave information. One way to do this is to consider the relative
amplitudes of the two S-wave components.
One additional concept is needed to determine fault plane
solutions using the first motions from various seismic stations.
The radiation patterns show the displacements that would
occur on a sphere with infinitesimal radius about the source.
The observations, of course, are at stations some finite distance
from the source. We thus need to convert the observations
at the stations to hypothetical ones surrounding the source.
To do this, recall that seismic waves do not travel in straight
lines from the earthquake to a station. Instead, because seismic
velocities vary with depth, rays follow curved paths.
Fig. 4.2-8 The angle of incidence at the earthquake source is the angle
from the vertical at which the ray leaves the source, and thus the angle at
which the ray intersects the lower focal hemisphere.
Earthquake
Upper
focal
hemisphere
Lower
focal
hemisphere
Seismic
stations
i 1
i 2
R = 0
As discussed in Section 3.4, the ray paths are given by Snell’s
law, which says that the ray parameter is constant along a ray.
Thus the ray parameter of the ray arriving at a given distance
can be found from the slope of the travel time curve T(∆),
p
r
i
v
dT
d
sin
.
=
= ∆
(7)
Hence taking r as the radius at the earthquake source and v
as the velocity at the source depth, the value of dT/d∆ for this
distance gives the ray’s angle of incidence at the source, often
called the take-off angle. How far a ray travels depends on its
take-off angle (Fig. 4.2-8); rays with large take-off angles leave
the source closer to the horizontal and travel shorter distances
than those with smaller take-off angles.
The distance that a ray has traveled thus gives its take-off
angle. Table 4.2-1 is a sample table relating teleseismic travel
distances and take-off angles for P waves from a surface-focus
earthquake. These distances and angles depend on the velocity
model assumed. In teleseismic first motion studies, stations
at distances greater than 100° are generally not used because
the rays hit the earth’s core, and stations for distances closer
than 30° are often avoided because the take-off angles depend
strongly on the upper mantle velocity structure used. In local
earthquake studies, care is taken to ensure that the velocity
model is appropriate.
Using such tables, the distances to seismic stations can be
converted to take-off angles. Thus the locations of compressions and dilatations can be converted to their positions on
the surface of the lower focal hemisphere, a hemisphere with
infinitesimal radius about the source. A similar approach can
be used for data directly above a deep earthquake, where the
upper focal hemisphere is a natural representation.
4.2.4 Stereographic fault plane representation
We have seen that fault geometry can be found from the distribution of data on a sphere around the focus. Because plotting
where D
— is the average slip (or dislocation) on the fault with
area S. We often use the seismic moment as a scale factor and
write F(t) = M 0 x(t), where x(t) is the source time function.
The final term, sin 2θ cos φ, describes the P-wave radiation
pattern. It is four-lobed, with two positive, compressional,
lobes and two negative, dilatational, ones. The displacement is
zero on the fault (θ = 90°) and auxiliary (φ = 90°) planes. Thus
the fault plane and auxiliary plane are nodal planes separating compressional and dilatational quadrants. The maximum
amplitudes are between the two nodal planes.
Similarly, the shear wave displacement has two components,
u θ ê θ + u φ ê φ , where
u
r
θ
πρβ
=
1
4
3
F(t − r/β) cos 2θ cos φ,
u
r
φ
πρβ
=
1
4
3
F(t − r/β)(−cos θ sin φ).
(6)
Note that the term involving F(t) corresponds to waves propagating at the S-wave speed β. As shown in Fig. 4.2-6, the
S-wave motion does not have nodal planes, but it is perpendicular to the P-wave nodal planes and is zero on the null axis.
It converges toward the center of the compressional quadrants,
which, as we will see shortly, is the location of the T, or least
compressive stress, axis. It also diverges from the centers of the
dilatation quadrants, known as the P, or most compressive
stress, axis. Thus, although the S-wave pattern does not reflect
the fault plane as clearly as the P-wave pattern, it can also be
used to study the fault geometry. An interesting feature of
Eqns 3 and 6 is that they show why S waves on seismograms are
usually bigger than P waves — the equations predict an average
ratio of α 3 /β 3 , or about 5.
Because the radiated seismic waves vary as a function of
θ and φ, seismograms recorded at different directions from
the earthquake can be used to find the fault geometry. The
P wave is the first wave to arrive from an earthquake, so on a
seismogram it is an isolated arrival whose polarity is often easy
to identify. A set of P-wave first motions thus often makes
it possible to locate the nodal planes that divide the regions of
different polarity. The first S waves are harder to use, because
they arrive later in the seismogram and can be buried in a complicated wave train. It is still possible, however, to use the Swave information. One way to do this is to consider the relative
amplitudes of the two S-wave components.
One additional concept is needed to determine fault plane
solutions using the first motions from various seismic stations.
The radiation patterns show the displacements that would
occur on a sphere with infinitesimal radius about the source.
The observations, of course, are at stations some finite distance
from the source. We thus need to convert the observations
at the stations to hypothetical ones surrounding the source.
To do this, recall that seismic waves do not travel in straight
lines from the earthquake to a station. Instead, because seismic
velocities vary with depth, rays follow curved paths.
Fig. 4.2-8 The angle of incidence at the earthquake source is the angle
from the vertical at which the ray leaves the source, and thus the angle at
which the ray intersects the lower focal hemisphere.
Earthquake
Upper
focal
hemisphere
Lower
focal
hemisphere
Seismic
stations
i 1
i 2
R = 0
As discussed in Section 3.4, the ray paths are given by Snell’s
law, which says that the ray parameter is constant along a ray.
Thus the ray parameter of the ray arriving at a given distance
can be found from the slope of the travel time curve T(∆),
p
r
i
v
dT
d
sin
.
=
= ∆
(7)
Hence taking r as the radius at the earthquake source and v
as the velocity at the source depth, the value of dT/d∆ for this
distance gives the ray’s angle of incidence at the source, often
called the take-off angle. How far a ray travels depends on its
take-off angle (Fig. 4.2-8); rays with large take-off angles leave
the source closer to the horizontal and travel shorter distances
than those with smaller take-off angles.
The distance that a ray has traveled thus gives its take-off
angle. Table 4.2-1 is a sample table relating teleseismic travel
distances and take-off angles for P waves from a surface-focus
earthquake. These distances and angles depend on the velocity
model assumed. In teleseismic first motion studies, stations
at distances greater than 100° are generally not used because
the rays hit the earth’s core, and stations for distances closer
than 30° are often avoided because the take-off angles depend
strongly on the upper mantle velocity structure used. In local
earthquake studies, care is taken to ensure that the velocity
model is appropriate.
Using such tables, the distances to seismic stations can be
converted to take-off angles. Thus the locations of compressions and dilatations can be converted to their positions on
the surface of the lower focal hemisphere, a hemisphere with
infinitesimal radius about the source. A similar approach can
be used for data directly above a deep earthquake, where the
upper focal hemisphere is a natural representation.
4.2.4 Stereographic fault plane representation
We have seen that fault geometry can be found from the distribution of data on a sphere around the focus. Because plotting