GG304 Lecture 21
1
LECTURE 21: SEISMOMETERS
A seismoscope is an instrument that records the occurrence of an earthquake.
The seismograph , which record the ground vibrations that result from the
earthquake, dates from the late-1800s. Most seismographs utilize the inertia of
a heavy object that is only loosely coupled to the ground. As the ground
vibrations, its motion relative to the slowly-moving object is amplified and
recorded by a seismometer . Seismometers are designed to detect either
horizontal or vertical ground motions. The ground motion is recorded either on
paper or digital media.
A strain seismometer
measures small changes
in the distance between
two locations (strains of
10-8-10-10) that occur
during a seismic wave’s
passage.
Inertial seismometers
operate by recording the motion, u, of inertial mass M due to ground motion
q = A cos ωt , where ω is the frequency of a seismic wave and A is its (amplified)
amplitude. Typically, seismic waves are composed of a spectrum of frequencies.
For a restoring force of –ku, the equation of motion is:
∂2
∂2u
∂2q
2
M 2 (u + q ) = −ku
+ ω 0u = − 2
or
∂t
∂t 2
∂t
where ω 0 = k M is the natural frequency of the seismometer. This resonant
frequency will dominate the seismometer’s response unless it is damped. To
avoid this, velocity-dependent damping is added with damping factor λ:
Clint Conrad
21-1
University of Hawaii
GG304 Lecture 21
2
∂2u
∂u
∂2q
2
+
2
λω
+
ω
u
=
−
0
0
∂t 2
∂t
∂t 2
Then the response of the seismometer is:
# 2λωω &
0
is the phase lag.
Δ = tan−1 % 2
2(
$ ω0 − ω '
Aω 2
cos (ωt − Δ )
The seismometer’s motion u is then: u =
12
2
2
2
2 2 2
ω0 − ω + 4λ ω ω0
u = U cos (ωt − Δ )
where
((
)
)
For λ=1, the seismometer is
critically damped . For smaller
λ, underdamping results in
continued periodic oscillations,
while for larger λ overdamping
suppresses the seismometer
response. Critical damping is
ideal and produces a
seismometer response of:
Aω 2
u= 2
cos (ωt − Δ )
ω0 + ω 2
Seismometers are designed to measure seismic energy within a particular
frequency range – denoted by the seismometer’s natural period (2π/ω0).
Long-period seismometers feature small ω0, and thus the phase lag Δ0
and the seismometer displacement equals the ground motion: u = A cos (ωt ) = q
The long-period seismometer is thus a displacement meter , and records
seismic energy with frequencies of 0.01-0.1 Hz (10-100 s periods).
Short-period seismometers feature large ω0>>ω. Here again the phase lag
ω2
1 d 2q
Δ0 and the seismometer motion is: u = 2 A cos (ωt ) = − 2 2
ω0
ω 0 dt
The short-period seismometer is thus an accelerometer , as it measures the
acceleration of ground motion, responding to frequencies of 1-10 Hz (periods of
0.1-1 s). It is useful for measuring strong motion of earthquakes.
Clint Conrad
21-2
University of Hawaii
GG304 Lecture 21
3
Broadband seismometers
use sophisticated electronics
to impose restoring forces
proportional to displacement,
which prevent movement of an
inertial mass. Their bandwidth
spans both long and short
periods (and the 1-10 s range
with significant seismic noise
from micro-seisms and ocean
waves), and features a dynamic
range that spans 105 variation in
amplitude.
A seismogram is a time-record
of ground motion recorded by a
seismometer. The arrival of
seismic energy of different
types (e.g., P- or S-waves) is
often marked on a seismogram.
Each event is referred to as a
seismic phase and can be used
to constrain information about
the Earth’s interior properties.
Clint Conrad
21-3
University of Hawaii