Calibration by stepwise motion

The movable tables of machine tools like lathes and milling machines, and of mechanical balances, can replace a shake table for the absolute calibration of seismometers. The idea is to place the seismometer on the table, let it come to equilibrium, then move the table manually by a known amount and let it rest again. The apparent ``ground'' motion can then be calculated from the seismic signal and compared to the known mechanical displacement. Since the calculation involves triple integrations, offset and drift must be carefully removed from the seismic trace. The main contribution to drift in the apparent horizontal "ground" velocity comes from tilt associated with the motion of the table. With the method subsequently described, it is possible to separate the contributions of displacement and tilt from each other so that the displacement can be reconstructed with good accuracy. This method of calibration is most convenient because it uses only normal workshop equipment; the inherent precision of machine tools and the use of relatively large displacements eliminate the problem of measuring small mechanical displacements. A FORTRAN program named DISPCAL is available for the evaluation.

  

Figure 29:Absolute mechanical calibration of an STS1-BB (20 sec) seismometer on the table of a milling machine, evaluated with DISPCAL.The table was manually moved in 14 steps of 2 mm each (one full turn of the dial at a time). Traces from top to bottom: recorded BB output signal; restored and detrended velocity; restored displacement.

 The precision of the method depends on avoiding two main sources of error:

1.

Restoring the ground displacement from the seismic signal (a process of inverse filtration) is uncritical for broadband seismometers but requires a precise knowledge of the transfer function of short-period seismometers. Instruments with unstable parameters (such as electromagnetic seismometers) must be electrically calibrated while installed on the test table. However, once the response is known, restoring the absolute ground motion is no problem even for a geophone with a free period of only 0.1 sec.

2.

The effect of tilt can only be removed from the displacement signal when the motion is sudden and short. The tilt is unknown during the motion, and is twice integrated over time in the calculation of the displacement. A longer interval of motion will therefore cause a larger effect of the unknown tilt. Practically, the motion may last about one second on a manually operated machine tool, and about a quarter-second on a mechanical balance. It may be repeated at intervals of a few seconds.

Static tilt before and after the motion produces linear trends in the velocity which are easily removed before the integration. The effect of tilt during the motion can however only approximately be removed by interpolating the trends before and after the motion. The computational evaluation consists in the following major steps (Fig. 29):

1.

The trace is deconvolved with the velocity transfer function of the seismometer.

2.

The trace is piecewise detrended so that it is close to zero in the motion-free intervals; interpolated trends are removed from the interval of motion.

3.

The trace is integrated.

4.

The displacement steps are measured and compared to the actual motion.

  

Figure 30: Calibrating a vertical seismometer on a mechanical balance. When a mass of w₁ grams at some point X near the end of the beam is in balance with w₂ grams on the table or compensated with a corresponding shift of the sliding weight, then the motion of the table is by a factor w₁/w₂ smaller than the motion at X.

In principle a single steplike displacement is all that is needed. However, the experiment takes so little time that it is convenient to produce a dozen or more equal steps, average the results, and do some error statistics. On a milling machine or lathe, it is recommended to install a mechanical device that stops the motion after each full turn of the spindle. On a balance, the table is repeatedly moved from stop to stop. The displacement may be measured with a micrometer dial or determined from the motion of the beam (Fig. 30).

From the mutual agreement between a number of different experiments, and from the comparison with shake-table calibrations, we estimate the absolute accuracy of the method to be better than 1%.