Folded Pendulum Measurements of Earth's Free Oscillations

Randall D. Peters
Department of Physics
Mercer University
1400 Coleman Ave
Macon, Georgia 31207

Copyright January 2004


The nearly-incessant free oscillations of the Earth (not the larger, long-lived normal modes seen following intense quakes) were first observed by accident in the record of a tilt-sensitive instrument designed to study surface physics [1]. Later tiltmeter studies demonstrated the usefulness of autocorrelation for the routine study of these normally short-lived eigenmodes [2]. More recent studies suggest that many tilt-sensitive instruments are capable of observing these oscillations-if the low-frequency response of their electronics is not suppressed, as is customary with conventional designs.

1  Introduction

The electronics of conventional seismic instruments is a type for which the sensitivity falls off at 20 dB per decade for frequencies below approximately 10 mHz. The frequency response of the instrument is determined by the network characteristics of the force-feedback design. As noted by Wielandt, `` output proportional to ground acceleration is unfavourable. .... The system would soon be saturated by the offset voltage resulting from thermal drift or tilt. What we need is a bandpass response ... like that of a normal electromagnetic seismometer ... with a lower corner frequency.''[3].

The sensor of the `normal electromagnetic seismometer' is one which operates on the basis of Faraday's law. The output voltage, determined by the time rate of change of a magnetic flux, is proportional to the frequency of excitation-since a time derivative is involved. We see that the conventional design of modern seismic electronics is cause for severe attenuation of those frequencies corresponding to the most interesting of Earth's eigenmodes. Consider, for example, the lowest frequency case (S2 spherical mode, at 0.31 mHz, corresponding to a period of 54 minutes). For a conventional instrument having a peak response (corner frequency) at 20 mHz, its sensitivity is reduced 60-fold at the frequency of the S2 mode!

For the studies reported in 1 and 2, the electronics was of a different type than that of conventional seismic instruments. These instruments employed a fully differential capacitive sensor [4] in a non-feedback arrangement, resulting in no 20 dB/decade falloff in sensitivity for frequencies below the natural frequency (mechanical) of the tiltmeter. Care must be exercised in such cases, to avoid the saturation tendencies mentioned by Wielandt for this mode of operation.

Although modern seismic instruments use capacitive displacement sensors, their differential arrangement is of a lower symmetry (and thus less sensitive) than those of the `symmetric differential capacitive (SDC) sensor' patented by the author. Additionally, the mechanical range of operation of conventional seismic sensors is severely limited because they function on the basis of gap-spacing variation. By using area variation (a type of `shadow sensor'), much larger mechanical dynamic ranges are possible. Although a large mechanical dynamic range is not important (nor even considered) with the force feedback instrument, it is important for the `open-loop' mode used to see Earth's free oscillations.

Even with adequate sensitivity, it is rare for free oscillations to be plainly visible in `raw trace output' from a typical tilt-sensitive instrument. In the work of ref. 2, autocorrelation was nearly always necessary to see the oscillations. Although autocorrelation analysis is a powerful means for extracting periodic signals from noise, it requires that offset and drift be removed from a record before computing the autocorrelation. This is a laborious operation as compared to the most recent discovery, which is now described.

2  Apodized Power Spectra of Compressed records

The author's recent studies employ a surprisingly user-friendly A/D converter (16-bit, Dataq's DI-700). The versatile software supplied with the converter allows for easy spectral analysis of records. Particularly useful is the means whereby one may conveniently view either (i) an entire record of arbitrary length, or (ii) any length segment within that record. This is accomplished via the windows-based algorithms that employ both the `mouse' and `hotkeys' to alter the degree of data compression.

When doing Fourier transforms (FFT's), the Dataq algorithm permits the user to select a window (such as Hanning, Hamming,..) with which to apodize the record. The Hanning window, as an example, is a shifted cosine of the form H = 0.5+0.5 cos(arg), where `arg' is selected according to record-length so that H is zero at both the start and end of the record. The record is then multiplied by H, as illustrated in Fig. 1.

Figure 1. Example of a Hanning apodized record. The data of the lower trace were generated with an open-loop folded pendulum.

The maximum number of digital values saved to memory from the A/D converter, for the time span shown in Fig. 1, was 27648 sample points. It is generally inconvenient to do transforms larger than 1024 or 2048 when working with a spreadsheet. Thus the record was compressed by a factor of 27 (by the Dataq software) to yield a reduced total of 1024 points.

There is a significant dc offset in the record, as observed in the bottom trace of Fig. 1. Moreover, the offset increases substantially because of drift in the 14000 s from start to finish of the record. These factors may disallow the detection of any low frequency lines that would otherwise be visible in a spectrum of the raw signal. As will be seen in later analyses of the same record using a spreadsheet, the offset and drift may be readily `subtracted out' before doing the FFT. Such is not necessary when working with spectra only; since the apodized spectrum retains decent low-frequency resolution, as seen in Fig. 2. The reader may find this surprising, since the `useful' information in the top trace of Fig. 1 appears to only `lightly modulate' the shifted (and `amplified') `bell-shaped' cosine apodizer.

Figure 2. Spectra generated with and without a Hanning apodizer.

When the FFT is taken on the raw data having offset and drift, there is a `shoulder' near zero frequency tending to `mask' the presence of adjacent spectral lines. Thus, Earth's free oscillation spectral line at 0.94 mHz (period 18 min, T5 torsional) is more clear in the top trace of Fig. 2 than it is in the bottom trace computed from the raw data.

2.1  Removal of offset and drift

It might be assumed that the T5 line that is so clear in the top trace of Fig. 2 could be an artifact of apodization. That this is not the case is now demonstrated by removing offset and drift from the record, as illustrated in the top trace of Fig. 3.

Figure 3. Record and resulting spectrum, after removing offset and drift.

Subtraction of offset and drift was easily done with Excel, and the FFT without apodization of the result (corrected signal) is shown in the bottom graph. The T5 spectral line is still readily resolved in the lower trace of Fig. 3.

3  Autocorrelation

One of the most powerful and under-utilized operations for extracting periodic signals from noise is the autocorrelation. It is readily calculated by (i) taking the FFT of the record, (ii) multiplying the resulting transform by its complex conjugate, and finally (iii) taking the FFT of the product. The last step of the process (possible because of the Wiener-Khintchin theorem) results in real numbers, but when performed with Excel (results being presently presented), one must use the operator IMREAL( ) on the last column of numbers (presently 1024 single-value rows).

Shown in Fig. 4 is the autocorrelation of the 1024 points which constitute the corrected record (top trace) of Fig. 3. As noted before, a useful autocorrelation cannot be performed on the raw data (lower trace) of Fig. 1, which contains offset and drift.

Figure 4. Autocorrelation of the corrected record (top trace) of Fig. 3.

The T5 oscillation of the Earth is readily visible in Fig. 4.

4  Bandpass filtered data

Whereas the raw data is suggestive of a free oscillation, the amount of noise is too large for one to confidently state this to be true. Any estimate of the period of the suggested harmonicity is fraught with significant error if one were to rely strictly on the raw data.

A means for making the periodicity more visible and reducing the error in a direct estimate of its period, is to operate on the record with a bandpass filter based in a Gaussian `kernel'. A later article will provide details on how to construct such filters using a spreadsheet. Shown in Fig. 5 (top graph) is the result of operating on the corrected record (Fig. 3, top trace) with the bandpass filter whose kernel is shown in the lower curve of Fig. 5. The digital signal processing (DSP, post processing) for this case was accomplished by means of FFT-based, spreadsheet convolution.

Figure 5. Enhanced direct observability of the T5 mode by means of a bandpass filter.

5  Conclusion

We thus see from a variety of analysis tools that the signal to which the folded pendulum was responding in this case was that of a harmonic excitation whose period was 18 minutes (T5) and whose duration was approximately a dozen cycles.

5.1  Mechanism

This author postulates that the two mechanisms primarily responsible for the nearly-incessant free oscillations of the Earth are (i) tidal, and (ii) thermal. Consider our planet as being similar to a multiply-cracked boiled egg that one rolls between the hands. The localized `snap, crackle, and pop' associated with shell fragment interactions is cause for normal mode excitations of the egg. Moreover, the `global' forcing function of the hands is significant insofar as driving `efficiency'. In like-manner, two external forcing functions are important to the Earth by reason of their `global' influence; they are the moon and the sun. Both contribute to the tidal strain of the Earth's crust with a roughly 12-h periodicity, but the moon is of greater influence because of its closer proximity. (The tidal force is proportional to the inverse cube of the distance). Although the sun is less important in a tidal sense, its thermal influence is significant. The solar insolation of approximately 1400 W/m2 is applied only to the illuminated part (hemisphere). As the Earth rotates under this solar `heat lamp', thermal expansion/contraction must be met with large-scale (nonlinear) strains (like the cracked egg) having `discontinuities' that excite free oscillations.

Although this paper has concentrated on a single mode, many of the known (catalogued) spherical and torsional modes of the Earth have been seen with this instrument over the past two months for which data has been recorded. One mode per 24-h record is a rough estimate of the frequency with which these oscillations have been observed. It should be noted, however, that their distribution in time is far from uniform. It is hoped that manpower and time resources will in the future permit frequency-of-occurrence graphs to be generated and published.

6  Folded pendulum Instrument

The instrument used for this study is a crude, home-built unit. Sophisticated construction practice was purposely avoided to see if, in fact, free oscillations of the Earth are to be easily observed when `right' electronics is employed. A follow-on article will provide details (including pictures) of the instrument. A key component of the `right' electronics is the array-form (8-elements) of the SDC sensor used. The folded pendulum is ideally suited to this linear array, which will be described in the follow-on article. The mechanical dynamic range of the sensor is approximately 0.5 cm and the calibration constant is roughly 20K V/m. The instrument is housed at the same site used to collect data reported in ref. 2.

With a period set near 5 s for present purposes, the `background' noise of the instrument is too large for serious earthquake monitoring, because of oceanic microseisms. The period of the pendulum was set at this short value to avoid electronics saturation due to drift. The system is not yet temperature controlled, so thermal expansion and electronics gain change with temperature are responsible for significant diurnal variations in sensor output voltage.

In spite of the above factors, large earthquakes are readily observed, as illustrated in the record of Fig. 6. More important for the present work, and as suggested by the earlier figures of this paper, the instrument is well suited to observation of free oscillations.

Figure 6. Earthquake record produced with the folded pendulum. The magnitude of the Rat Island quake was estimated by others to be 7.8 .

Based on the calibration constant noted earlier, the maximum rms amplitude of ground motion in Macon, Georgia due to the quake is estimated around 3 mm. Spectra (not shown) indicate a primary component of harmonicity with a period of about 17 s. The time difference between arrival of the P and S waves of 560 s is consistent with the distance from source to detector, and assumed (nominal) values of P and S propagation speeds.


Kwon, M. H. & R. Peters, ``The study of eigenmode types and source nonlinearity in the free oscillations'', Saemulli Vol. 35, no. 4, 569 (1995).
Peters, R, ``Autocorrelation analysis of data from a novel tiltmeter'', poster presentation, Amer. Geophys. Union conf., San Francisco, 2000.
Erhardt Wielandt, ``Velocity broadband seismometers'', Seismic Sensors and their Calibration (New Manual of Observatory Practice), ed. Peter Bormann and Erik Bergmann, online at
R. Peters, U. S. Patent No. 5,461,319-described online at

File translated from TEX by TTH, version 1.95.
On 23 Jan 2004, 17:02.