The seismometers can be thought of as having
a mass M attached to a point of the earth's
surface through a parallel arrangement of a spring and a dashpot.
Denote the ground motion by u(t) and the motion of the mass relative
to the earth as 
. The spring will exert a force proportional to
its elongation 
and the dashpot will exert a force proportional
to the velocity 
between the mass and the earth.
Denoting the constants of proportionality by k and D respectively,
the equation of motion for the mass is (Aki and Richards 1980):
Rewriting the displacement relative to the equilibrium
position of the spring 
as , this can be written:
where 
and 
.
then describes the damping of the geophone and 
the eigen frequency. Laplace transforming
both sides of equation 20 and rearranging gives the frequency
response 
of the seismometer as:
In general the geophone will also have some frequency
independent gain, 
.
The frequency response of the geophone will then be:
For the Lennartz geophones 
V/m/s (Lennartz 1990).
The poles of are given by:
Defining the damping constant h as 
, a critically
damped geophone will have h = 1.0. For an underdamped geophone 
and for an overdamped geophone 
(Aki and Richards 1980).
Equation
23 can then be rewritten as:
For a 1 Hz geophone, damped at 
critical damping (i.e. h=0.707),
, where T is the period,
the poles are (from equation 24):
For a seismometer with a 5 sec eigen period and a damping constant of 0.707 (i.e. the LE-3D/5s geophone), equation 24 gives:
The above description is valid for the classical mechanical devices widely used and also for the Lennartz seismometers which simulate these devices. For other active seismometers, such as instruments with displacement transducers, the equations are slightly different but have the same general form. The poles and zeros of all seismometer types currently in use in the SIL network are given in Appendix A. For all seismometers other than the Lennartz instruments the specifications are taken from manuals shipped with the instruments. The frequency response of the six geophones is shown in Figure 2.
In the first three years of operation of the SIL network (1989-1991), the
poles used for the Lennartz LE-3D 1 Hz geophones
( 
) where
obtained from specifications of S13 seismometers with a damping constant of
0.61. This frequency response differs slightly from the actual Lennartz
response for frequencies
around 1 Hz (see Figure 3).
