Civil Engineering Reference
In-Depth Information
Seismic moment
M
0
measures the energy
E
released by fault rupture during earthquakes (Scholz,
1990). The following relationship is applicable to all source mechanisms:
Δτ
2
E
=
G
M
(1.14)
0
where Δ
τ
is the stress drop Δ
τ
=
τ
1
-
τ
2
, and
τ
1
and
τ
2
are the shear stresses on the fault before and
after brittle fracture occurs, respectively;
G
is the shear modulus of the material surrounding the fault
as also shown in equation (1.9.1). For moderate-to-large earthquakes, the mean values of Δ
τ
are equal
to about 6.0 MPa. In the defi nition of
M
w
, the stress drop is assumed constant.
Magnitude - moment relationships have been defi ned empirically for periods less than 20 seconds
(Purcaru and Berckhemer, 1978), as below:
log
MM
(
)
=
1.5
+
16.1
(1.15)
0
S
and body wave magnitude
m
b
can be related over a wide range to
M
S
by the following semi- empirical
formula proposed by Gutenberg and Richter (Richter, 1958 ):
m
=
0.63
M
+
2.5
(1.16)
b
S
therefore, combining equations (1.15) and (1.16), seismic moment
M
0
can be related to body waves
m
b
and vice versa. Moreover, Figure 1.15 may be used when relationships between
M
0
and magnitude
scales other than
m
b
and
M
S
are sought.
Expressions correlating magnitude scales and fault rupture parameters can be found in the literature
(e.g. Tocher, 1958 ; Housner, 1965 ; Seed
et al
., 1969 ; Krinitzsky, 1974; Mark and Bonilla, 1977 ). For
example, Bonilla
et al
. ( 1984 ) computed
M
S
as a function of the fault rupture length
L
:
ML
()
=+
6.04
0.71
log
()
L
(1.17.1)
S
where the length is in kilometres. Equation (1.17.1), which is applicable for
M
S
> 6.7, is based on mean
values, while the 95th percentile is given as follows:
MML
0.95
()
+
(1.17.2)
=
0.52
S
Surface wave magnitude
M
S
has also been related to the maximum observed displacement of fault
D
. Empirical relationships are provided as a function of the fault rupture mechanism (Slemmons, 1977 ),
as shown below:
Mab D
S
=+
log
()
(1.18)
where the displacement
D
is in metres, while coeffi cients
a
and
b
are given in Table 1.5 .
Similarly, Wyss (1979) proposed a relationship between the fault surface rupture
S
and surface
magnitude
M
S
given by:
M
4.1=+
()
log
S
(1.19)
S
in which the area
S
should be expressed in km
2
. Equation (1.19) is applicable for
M
S
> 5.6.
In some regions, correlations as given above are of little value since many of the important geologic
features can be deeply buried by weathered materials. Results of studies by Wells and Coppersmith
(1994) are outlined in Table 1.6 for different types of fault mechanisms, i.e. strike- slip, reverse and
