There are two basic types of seismic sensors: inertial seismometers which measure ground motion relative to an inertial reference (a suspended mass), and strainmeters or extensometers which measure the motion of one point of the ground relative to another. Since the motion of the ground relative to an inertial reference is in most cases much larger than the differential motion within a vault of reasonable dimensions, inertial seismometers are generally more sensitive to earthquake signals. However, at very low frequencies it becomes increasingly difficult to maintain an inertial reference, and for the observation of low-order free oscillations of the Earth, tidal motions, and quasi-static deformations, strainmeters may outperform inertial seismometers. Strainmeters are conceptually simpler than inertial seismometers although their technical realization may be more difficult. Chapter 5 is concerned with inertial seismometers only.
An inertial seismometer converts ground motion into an electric signal but its properties cannot be described by a single scale factor, such as output volts per millimetre of ground motion. The response of a seismometer to ground motion depends not only on the amplitude of the ground motion (how large it is) but also on its time scale (how sudden it is). This is because the seismic mass has to be kept in place by a mechanical or electromagnetic restoring force. When the ground motion is slow, the mass will move with the rest of the instrument, and the output signal for a given ground motion will therefore be smaller. The system is thus a high-pass filter for the ground displacement. This must be taken into account when the ground motion is reconstructed from the recorded signal, and is the reason why we have to go to some length in discussing the dynamic transfer properties of seismometers.
The dynamic behaviour of a seismograph system within its linear range can, like that of any linear time-invariant (LTI) system, be described with the same degree of completeness in four different ways: by a linear differential equation, the Laplace transfer function (2.2), the complex frequency response (2.3), or the impulse response of the system (2.4). The first two are usually obtained by a mathematical analysis of the physical system (the hardware). The latter two are directly related to certain calibration procedures (7.4 and 7.5) and can therefore be determined from calibration experiments where the system is considered as a "black box"(this is sometimes called an identification procedure). However, since all four are mathematically equivalent, we can of course derive each of them either from a knowledge of the physical components of the system, or from calibration experiments (sections 7 and 8). The mutual relations among the four basic representations are explained in section 2. Practically, the mathematical description of a seismometer is limited to a certain bandwidth of frequencies that should at least include the bandwidth of seismic signals. Within this limit then any of the four representations describe the system's response to arbitrary input signals completely and unambiguously. The viewpoint from which they differ is how efficiently and accurately they can be implemented in different signal-processing procedures.
In digital signal processing, seismic sensors are often represented with other methods that are efficient and accurate but not mathematically exact, such as recursive (IIR) filters. Digital signal processing is however beyond the scope of this section. Excellent textbooks are available both on analog and digital signal processing, for example [Oppenheim & Willsky 1983] for analog processing, [Oppenheim & Schafer 1975] for digital processing, and [Scherbaum 1996] for seismological applications.
The most commonly used description of a seismograph response in the classical observatory practice has been the "magnification curve", i.e. the frequency-dependent magnification of the ground motion. Mathematically this is the modulus (absolute value) of the complex frequency response (2.3), usually called the amplitude response. It specifies the steady-state harmonic responsivity (amplification, magnification, conversion factor) as a function of frequency. However, for the correct interpretation of seismograms, also the phase response of the recording system must be known. It can in principle be calculated from the amplitude response, but is normally specified separately, or derived together with the amplitude response from the mathematically more elegant description of the system by a complex transfer function or a complex frequency response.
While for a purely electrical filter it is usually clear what its amplitude response means - a dimensionless factor by which the amplitude of a sinusoidal input signal must be multiplied to obtain the associated output signal - the situation is not always as clear for seismometers because different authors may prefer to measure the input signal (the ground motion) in different ways: as a displacement, a velocity, or an acceleration. Both the physical dimension and the mathematical form of the transfer function depend on the definition of the input signal, and one must sometimes guess from the physical dimension to what sort of input signal it applies.
Calibrating a seismometer means measuring (and sometimes adjusting) its transfer properties and expressing them as a complex frequency response or one of its mathematical equivalents. For most applications the result must be available as parameters of a mathematical formula, not as raw data; so determining parameters by fitting a theoretical curve of known shape to the data is usually part of the procedure. Practically, seismometers are calibrated in two steps.
The first step is an electrical calibration (section 7) in which the seismic mass is excited with an electromagnetic force. Most seismometers have a built-in calibration coil that can be connected to an external signal generator for this purpose. Usually the response of the system to different sinusoidal signals at frequencies across the system's pass-band (steady-state method, 7.4), to impulses (transient method, 7.5), or to arbitrary broadband signals (random signal method, 7.6) is observed while the absolute magnification or gain remains unknown. For the exact calibration of sensors with a large dynamic range such as employed in modern seismograph systems, the latter method is most appropriate.
The second step, the determination of the absolute gain, is more difficult because it requires mechanical test equipment in all but the simplest cases (section 8). The most direct method is to calibrate the seismometer on a shake table. The frequency at which the absolute gain is measured must be chosen so as to minimize noise and systematic errors, and is often predetermined by these conditions within narrow limits. A calibration over a large bandwidth cannot normally be done on a shake table. We will at the end of this chapter propose some methods by which a seismometer can be absolutely calibrated without a shake table.