An instrument for direct observations of seismic and normal-mode rotational oscillations of the Earth

Abstract

The rotations around the vertical axis associated with the normal mode oscillations of the Earth and those induced by the seismic and other disturbances have been very difficult to observe directly. Such observations will provide additional information for 3D modeling of the Earth and for understanding earthquakes and other underground explosions. In this paper, we describe the design of an instrument capable of measuring the rotational motions associated with the seismic oscillations of the Earth, including the lowest frequency normal mode at Ϸ 3.7 ؋ 10 ؊4 Hz. The instrument consists of a torsion balance with a natural frequency of 0 Ϸ 1.6 ؋ 10 ؊4 Hz, which is observed by an autocollimating optical lever of high angular resolution and dynamic range. Thermal noise limits the sensitivity of the apparatus to amplitudes of Ϸ 1.5 ؋ 10 ؊9 rad at the lowest frequency normal mode and the sensitivity improves as ؊3/2 with increasing frequency. Further improvements in sensitivity by about two orders of magnitude may be achieved by operating the balance at cryogenic temperatures. Alternatively, the instrument can be made more robust with a reduced sensitivity by increasing 0 to Ϸ10 ؊2 Hz. This instrument thus complements the ongoing effort by Igel and others to study rotational motions using ring laser gyroscopes and constitutes a positive response to the clarion call for T orsional oscillations of the Earth, sometimes called torroidal oscillations were predicted in the 19th century and have been calculated with increasing precision during the recent decades. The lowest of these modes involve the oscillations of the entire mantle and have periods of Ϸ3,000 s. Careful measurement of these oscillations is of considerable importance in understanding the Earth and the mechanism of earthquakes. The rotations associated with the normal modes of the Earth have not been directly observed with the present-day instruments. Seismological studies have been mostly responsible for our understanding of the nature and dynamics of the interior of the Earth, with additional information coming from other channels such as paleomagnetism, VLBI, and GPS. At present, the seismological studies are essentially limited to monitoring global and local wave fields at various frequencies, measuring only the three components of the translational velocity of ground displacement and in some cases directly the strains. The Fourier transform of the horizontal translations yields the frequency spectrum of these oscillations, and their correspondence with theoretical calculations carried out with the PREM model of Dziewonski and Anderson allows us to identify the frequencies and Q values associated with the rotational oscillations of the Earth's mantle. In a unique effort, Igel et al. (1) have been able to detect the rotational motions induced by the M8.1 Tokachi-Oki earthquake at frequencies of the order of 30-60 mHz, using ring laser gyroscopes.Theoretical studies, for example, by Cochard et al. (2), indicate that the observations of seismic rotational mot...

Performance Characteristics of a Rotational Seismometer for Near-Field and Engineering Applications

“…We then discuss the root mean square (rms) amplitudes induced by the thermal fluctuations and show that these essentially define the detection threshold. The analysis treatment here closely follows that described by Cowsik (2007).…”

“…These rotational motions must result from 3D structural complexity or anisotropy, and further investigation should lead to constraints on these properties. For these applications, especially for the low-frequency weak signals, it may be advantageous to use the design given by Cowsik (2007).…”

“…The signal recorded by the autocollimator may then be shown to be (Cowsik, 2007) α aω ω 2 0 ω 2 2 4ω 2 β 2 1=2 cosωt δ …”

“…These are easily calculated by noting the interconnection that the fluctuating thermal torques bear with the damping term or the quality factor of the balance, as first explained by Einstein (1905) and worked out with progressively increasing detail by Chandrasekhar (1943) and Kubo (1966) who emphasized that the autocorrelation function of the thermal torques is a Dirac-delta function. These were estimated (Cowsik, 2007) to be…”

“…In this article we outline an alternate scheme for the measurement of the rotations, about the local zenith axis, and show that it is possible to extend this design for the measurement of tilts as well. This new scheme is based on the study of responses of torsion balances of low-natural frequencies and shows great promise with high sensitivity and moderate cost (Cowsik, 2007). Such an instrument could provide complements or viable alternatives to ring lasers for studies in rotational seismology.…”