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).