Geoscience Reference
In-Depth Information
over both time and space and must include all frequencies. To do this, the signal
must be recorded with broad-band instruments
5
(instruments that record both
high and low frequencies).
Consider a simple plane wave, a simple harmonic with displacement
u
:
u
=
A
cos (κ
x
−
ω
t
)
(4.25)
where
/wave-velocity),
A
the
maximum amplitude and t time. The kinetic energy per unit volume associated
with this wave is
ω
is the angular frequency, κ the wavenumber (
ω
∂
u
∂
t
2
1
2
ρ
KE
=
1
2
ρ
A
2
2
sin
2
(
κ
x
−
ω
t
)
=
ω
(4.26)
where
ρ
is the density of the material. The kinetic energy varies from a maximum
A
2
2
value of
2toaminimum value of zero during each cycle. For a perfectly
elastic medium, energy is conserved (the sum of potential energy and kinetic
energy is constant), so the total energy per unit volume, is equal to the maximum
of the kinetic energy per unit volume,
ρ
ω
/
A
2
2
2. To calculate the total seismic
energy released by an earthquake, it is necessary to integrate or sum an expression
such as this over time, frequency and position around the epicentre. Because
amplitude and magnitude are related logarithmically in Eq. (4.13), the expression
of energy in terms of magnitude is also logarithmic (constants are determined
from integrations as well as assumptions about density, etc.). Empirical relations
between energy and magnitude are
ρ
ω
/
log
10
E
=
4
.
8
+
1
.
5
M
s
(4.27)
log
10
E
=−
1
.
2
+
2
.
4
m
b
where
E
(measured in joules, J) is the total energy of the seismic waves. These
relationships were determined for, and are therefore reliable for, earthquakes with
magnitude
>
5 only. Note also that, as the
m
b
magnitude scale saturates, energy
estimates will be too low at large magnitudes. An earthquake with
M
S
=
7.3 would
release 5.6
10
15
Jasseismic energy. Were this energy released by smaller earth-
quakes, some 75 magnitude-6 events, or 2000 magnitude-5 events, would need to
occur. In comparison, the Hiroshima atomic bomb was approximately equivalent
in terms of energy to an earthquake of magnitude 5.3. A one-megaton nuclear
explosion would release roughly the same amount of energy as an earthquake of
magnitude
×
6.7. A very small magnitude-2 earthquake would release about as
much energy as a bolt of lightning.
∼
5
Ideally, for local earthquakes frequencies in the range 0.1-500 Hz should be recorded, whereas for
teleseismic recordings of surface waves 0.01-20 Hz would be appropriate. In contrast, for vibroseis
(Section 4.4.1) deep reflection data in the range 5-200 Hz should be recorded and for explosion
seismology (refraction lines, not nuclear explosion detection) 2-200 Hz. These frequency ranges
are those suggested by the National Research Council (U.S.A.) in 1983.
