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in calculated magnitudes are not uncommon even under the most favourable
conditions.
Because deep-focus earthquakes are not effective at generating surface waves,
a better magnitude scale for them is based on body waves (P- and S-waves). A first
suggestion by Gutenberg (1945) for
body-wave magnitude m
b
of shallow-focus
earthquakes for P, PP and S waves of period 12 s was
m
b
=
log
10
A
12
+
0
.
01
+
5
.
9
(4.17)
Many
,
h
)in
Eq. (4.13)have been made since then, but the one generally used today is still the
Gutenberg-Richter (Gutenberg and Richter 1956) calibration function:
m
b
=
log
10
A
T
more
recent
determinations
of
the
calibration
function
q
(
max
+
q
(
,
h
)
(4.18)
In this tabulation,
q
(
,
h
) for P-waves is not strongly dependent on focal depth
10
◦
through
80
◦
to
and increases from
∼
6.0 at
=
∼
6.5 at
=
∼
8.0 at
110
◦
. The period
T
of the first arriving P-wave that has travelled through the
crust and mantle will generally be 0.1-5 s.
Values of
m
b
and
M
S
calculated for an earthquake are not usually the same:
the two magnitude scales will not agree on the magnitude of an earthquake. This
should not be a complete surprise; after all the two scales are based on different
signals:
m
b
on body waves with period 1-5 s and
M
S
on surface waves with period
18-22 s. Generally, for a shallow-focus earthquake, the surface-wave magnitude
will be more reliable than the body-wave magnitude. However, that the two scales
are related is clear from Fig. 4.11(a). The data are scattered, but
m
b
and
M
S
appear
to be linearly related. The
m
b
-
M
S
data plots vary appreciably from one earthquake
region to the next, but a worldwide average of
m
b
-
M
S
relations is
=
m
b
=
2
.
94
+
0
.
55
M
S
(4.19)
The two magnitude scales coincide at magnitude 6.5. For small magnitudes,
m
b
is greater than
M
S
, and for large magnitudes,
m
b
is less than
M
S
.
The
seismic moment M
0
of an earthquake is defined by
M
0
=
µ
Au
(4.20)
where
is the shear modulus,
A
the area of the fault and
u
the average displace-
ment on the fault. The seismic moment is a physical quantity with a unique value
for any earthquake. It can be determined by observations and estimates of the
fault-plane area and displacement but can also be expressed in terms of the low-
frequency (
µ
0.005 Hz) amplitude spectra of surface waves. This is important
since it means that
M
0
,ascalculated from seismograms, can be used as the basis
for a consistent magnitude scale.
M
0
is replacing
M
S
as the standard measure of
the long-period source spectra. Plots of log
M
0
against
M
S
and of log
A
against
log
M
0
and the fault length against log
M
0
are shown in Fig. 4.12. The linear trend
<
