Geoscience Reference
In-Depth Information
All the magnitude scales are of the form
log
10
A
T
M
=
+
q
(
,
h
)
+
a
(4.13)
where
M
is the magnitude,
A
the maximum amplitude of the wave (in 10
−
6
m),
T
the period of the wave (in seconds),
q
a function correcting for the decrease
of amplitude of the wave with distance from the epicentre and focal depth,
the angular distance from seismometer to epicentre,
h
the focal depth of the
earthquake and
a
an empirical constant related to the seismograph, its site location
and subsurface characteristics.
It is important in all magnitude work to use the period ranges for which the
particular equation has been determined; waves of different frequencies have dif-
ferent amplitude behaviours. Many determinations of the constant
a
and function
q
of Eq. (4.13)have been made, for various frequency bands, focal depths, geo-
graphic locations and so on. Bath (1981) provides a detailed review of earthquake
magnitudes.
For shallow-focus (
50 km) teleseismic earthquakes (those with 20
◦
<<
160
◦
)a
surface-wave magnitude M
S
can be defined as
M
S
=
log
10
A
T
<
max
+
1
.
66 log
10
+
3
.
3
(4.14)
The amplitude
A
used is the maximum of the horizontal component of the
Rayleigh wave in the period range 18-22 s. (If the vertical component is used
instead, the empirical constant in the equation depends on the seismic layering
beneath the recording station; and, if there is no layering, a constant term of 3.1 is
appropriate.) Because earthquakes occurring at greater depths are not so efficient
at generating surface waves, Eq. (4.14)gives too small a value for their magni-
tude. To allow for this, a correction must be applied to Eq. (4.14) for earthquakes
with focal depths greater than 50 km:
(
M
S
)
corrected
=
M
S
+
M
S
(
h
)
(4.15)
where
M
S
is calculated from Eq. (4.14) and
M
S
(
h
)isthe correction for focal
depth
h
. The maximum value of
M
S
(h), 0.4, is used for any earthquake with
a focal depth greater than 90 km. At ranges shorter than 20
◦
(approximately
2200 km), a correction,
), must also be made to Eq. (4.14)toaccount for
differences in absorption, scattering, geometrical spreading and dispersion:
M
S
(
(
M
S
)
corrected
=
M
S
+
M
S
(
)
(4.16)
Estimates of this correction vary between 0.6 and 0.1. These corrections illustrate
the fact that magnitude, although easy to measure, is not an exact description of
an earthquake. Many variables are associated with the recording site and passage
of the seismic signal through the Earth, as well as with the earthquake itself,
all of which can have a large effect on magnitude. Differences of 0.2-0.3 units
