Geology Reference
In-Depth Information
Fig. 10.15
Focal sphere,
S
, about an earthquake focus.
The first-motion polarity recorded at two seismic stations
R
1
and
R
2
depends from their azimuth, from the take-off
angle (™
1
or ™
2
), and from the orientation of the compres-
sional and dilatational quadrants on the focal sphere
M
11
D
M
0
,
M
22
D
M
0,
M
33
D
0. Consequently,
the new
x
1
and
x
2
axes are termed, respectively,
the
tension axis
,
T
,andthe
pressure axis
,
P
.
These axes indicate the directions of minimum
and maximum compressional stress, respectively,
and their orientation with respect to the far-field
radiation pattern is illustrated in Fig.
10.12
.
The polarity (up or down) of first motion
associated with the arrival of
P
waves, measured
on several vertical-component seismograms dis-
tributed around a seismic source, can be used to
infer the radiation pattern and the focal mecha-
nism of an earthquake. A convenient approach
is to consider a small sphere around the source,
which is called the
focal sphere
. The polarity of
a
P
phase arrival determines whether the corre-
sponding seismic ray left the focal sphere from
a compressional (upward first motion) or dilata-
tional quadrant (downward first motion). The
next step is then to determine the points on the
focal sphere that are crossed by the seismic rays
linking the earthquake focus to each receiver
R
(Fig.
10.15
). The location of one of these points
on the focal sphere can be specified assigning
a take-off angle, ™
0
, and a ray azimuth ¥
0
.The
former quantity can be easily determined using
Snell's law (9.42), while the latter parameter
depends from the location of the seismic station
relative to the epicenter. The results from many
observations are plotted using stereographic or
equal-area projections and specific software is
run to determine the best-fitting orientation of
the compressional and dilatational quadrants on
the focal sphere. The distribution of first-motion
polarities on the focal sphere constrains the ori-
entation of the primary and auxiliary planes as-
sociated with the focal mechanism. Usually, this
is displayed plotting the lower hemisphere of
the focal sphere, but in some cases it could be
convenient to show a lateral view. These plots are
referred to as
beach ball
plots and represent the
standard way to describe the pattern of seismic
deformation in a region. Some basic beach ball
plots associated with common focal mechanisms
are illustrated in Fig.
10.16
.
So far, we have focused on the
kinematic
parameters associated with an earthquake, which
are grouped in the focal mechanism variables
(¥,•,œ), and on a description of the procedure
used for determining them. Therefore, it is time to
consider the parameters that describe the
strength
of earthquakes, which are representative of the
seismic energy released in the coseismic phase.
An earthquake results from sudden slip along
a fault plane, with
finite
average magnitude
u
and direction represented by the rake parameter
œ. Such dislocation, which is unrecoverable and
cannot be considered as an elastic displacement,
determines a
stress drop
along the fault plane, as
illustrated in Fig.
10.17
. Burridge and Knopoff
(
1964
) showed that if
S
is a fault plane and
u
(
x
,
y
)
is the coseismic displacement at a point (
x
,
y
)on
S
,
then a measure of the earthquake size is given by:
M
0
D
Z
S
u
.x;y/dS
D
u
S
(10.52)
where
S
is the fault area and is the rigidity mod-
ulus. The quantity
M
0
is called the
scalar seismic
moment
and has dimension [Nm]. It represents
the most important parameter in seismology for
the specification of the strength of earthquakes
