When the seismometer is affected by a sinusoidal wave train, the output will also be sinusoidal, and the ratio of trace amplitude to the displacement amplitude of the ground is called the 'magnification' of the seismograph. In general, the magnification is a function of the wave period, and is expressed as a curve plotted on logarithmic scales.
For certain purposes it is convenient to consider the earth's velocity or acceleration, rather than its displacement. Taking a simple harmonic component of the earth displacement chi, we write
chi= chi0 cos(omega t),
whereupon the velocity is
dchi/dt= -chi0omega sin(omega t)
and the acceleration is
d2chi/dt2 = -chi0omega2 cos(omega t).
If we use the symbols V, Vv and Va as the seismograph deflection for unit displacement, velocity and acceleration respectively (so that V = magnification) we see at once that V = ~ Vv = ~2 Va where Vv and Va are called the 'velocity sensitivity' and 'acceleration sensitivity' of the seismograph.
If we have a logarithmic plot of magnification as a function of time, values of velocity sensitivity may be obtained by measuring the ordinates of the curve from lines which have a gradient of +1 in the direction of increasing frequency (or decreasing period). Acceleration sensitivities may be derived by measuring from lines of slope 2. The transparent overlay on Fig. 1.1 may be used for this purpose.
We use the response curves to measure the true amplitude of earth motion, notably for the determination of earthquake magnitude. This would be easy if the earth motion was of simple harmonic character, for the single period present in the trace could be measured, and the appropriate sensitivity factor could be read from the response curve.
If the earth motion has some oscillatory character, without being strictly harmonic, a band of seismic frequencies is present. Each harmonic component in the wave train will contribute to the output in proportion to the magnification of the seismograph at the period concerned. If the magnification curve of the seismograph is relatively flat over the range of periods in the wave train, the harmonic components of the seismogram will represent the corresponding components of earth motion in a fairly uniform way, and an estimate of the dominant frequency of the seismogram will yield a representative value of magnification. If, on the other hand, the seismometer produces a sharply filtered record, important characteristics of the earth motion may be excluded, and reconstruction of the wave train will be impossible.
The most extreme condition is that in which the earth motion is a sharp transient. In such cases, a wide band of frequencies is present, and a single assigned 'period' is far from representative.
Date created: 1/7/97 Last modified: 9/9/97 Copyright © 1997, Global Seismological Services Maintained by: Eric Bergman firstname.lastname@example.org