Calibration with arbitrary signals

The purpose of calibration is in most cases to obtain the parameters of an analytic representation of the transfer function. Assuming that its mathematical form is known, the task is to determine its parameters from an experiment in which the input and the output signals are compared. Since only a signal that has been digitally recorded is known with some accuracy, both the input and the output signals should be recorded with a digital recorder. In comparison with other methods where a predetermined input signal is used and only the output signal is recorded, recording both signals has the additional advantage of eliminating the transfer function of the recorder from the analysis.

Calibration is a classical inverse problem that can be solved with standard least-squares methods. The general solution is schematically depicted in Fig. 27: A computer algorithm (filter 1) is implemented that represents the seismometer as a filter, and permits the computation of its response to an arbitrary input. An inversion scheme (3) is programmed around the filter algorithm in order to find best-fitting filter parameters for a given pair of input and output signals. The purpose of filter 2 is explained below. The sensor is then calibrated with a test signal (4) for which the response of the system is sensitive to the unknown parameters but which is otherwise arbitrary. When the system is linear, parameters determined from one test signal will also predict the response to any other signal.

The approximation of a rational transfer function with a discrete filtering algorithm is not trivial. For the program CALEX (section 9) we have chosen an impulse-invariant recursive filter [Schuessler 1981]. The method formally requires that the seismometer has a negligible response at frequencies outside the Nyquist bandwidth of the recorder, a condition that is severely violated by most digital seismographs; but this problem can be circumvented with an additional digital low-pass filtration (filter 2 in Fig. 27) that limits the bandwidth of the simulated system. Signals from a typical calibration experiment are shown in Fig. 28. A sweep as a test signal permits the residual error to be visualized as a function of time or frequency; since essentially only one frequency is present at a time, the time axis may as well be interpreted as a frequency axis.

When the transfer function has been correctly parametrized and the inversion has converged, then the residual error consists mainly of noise, drift, and nonlinear distortions. At a signal level of about one-third of the operating range, typical residuals are 0.03% to 0.05% rms for force-balance seismometers and 1% for passive electrodynamic sensors.

With an appropriate choice of the test signal, other methods like the calibration with sinewaves, step functions, random noise or random telegraph signals can be duplicated and compared to each other. An advantage of the CALEX algorithm is that it makes no use of special properties of the test signal, such as being sinusoidal, periodic, step-like, or random. Therefore, test signals can be short (a few times the free period of the seismometer), and they can be generated with the most primitive means, even by hand (you may turn the dial of a sinewave generator by hand, or produce the test signal with a battery and a potentiometer, or simply switch a current on and off). A breakout box or a special cable may be required for feeding the calibration signal into the digital recorder.

Some other routines for seismograph calibration and system identification are contained in the PREPROC software package [Plesinger *et al.* 1996].

2002-11-08