Geophysical Research Letters

Observations of Earth's toroidal free oscillations with a rotation sensor: The 2011 magnitude 9.0 Tohoku-Oki earthquake



[1] We report for the first time observations of Earth's toroidal free oscillations recorded on a ring laser system that is sensitive to rotational ground motions around a vertical axis. Because of the high noise level on the horizontal translational components in classical seismometers, long-period toroidal modes are amongst the most challenging observations in seismology. In addition, pure uncontaminated observations of long-period motions are difficult as translational sensors are sensitive to rotational motions also. We show that the toroidal modes associated with rotational motions are complementary to those associated with translations and confirm the relatively spatially compact character of the Mw 9.0 Tohoku-Oki earthquake. The observations presented here complement the translational, strain, and gravitational records. We expect these observations to provide additional constraints on long-wavelength deep Earth structure and earthquake sources.

1. Introduction

[2] Giant earthquakes radiate mechanical energy into and around the Earth leading to world-wide oscillatory ground displacements sometimes in the centimeter range observable for days, sometimes weeks after the event [e.g., Park et al., 2005]. The resulting global standing wave patterns that form after the constructive interference of the seismic wave field – the Earth's free oscillations (or normal modes) – are characterized by discrete frequencies that depend primarily on the motion type (toroidal or spheroidal modes), and the structure of our planet. This implies that observations of free oscillations provide some of the most important large-scale constraints on a variety of elastic parameters, attenuation, and density of the Earth's deep interior [e.g., Gilbert and Dziewonski, 1975; Giardini et al., 1987; Ishii and Tromp, 1999; Laske and Masters, 1999; Beghein and Trampert, 2003; Deuss et al., 2010]. In addition, low frequency normal modes allow putting strong additional constraints on the energy, geometry, and duration of large earthquake sources [e.g., Stein and Okal, 2005; Park et al., 2005].

[3] At most seismological observatories ground motions are measured using broadband seismometers. Classically, these seismometers record three components of translational motions, usually in the vertical, North-South and East-West direction. However, for a comprehensive understanding of seismic waves, it would be of advantage to measure in addition six components of strain and three components of rotation [e.g. Aki and Richards, 2002]. Strain meters have been operating for decades [e.g., Alsop et al., 1961] and have also been used to observe free oscillations [e.g., Kanamori and Anderson, 1975; Park et al., 2008]. The techniques for measuring rotations are still under development [e.g., Schreiber et al., 2006, 2009a, 2009b].

[4] Widmer-Schnidrig and Zürn [2009] convincingly stated that the study of the Earth's free oscillations is one of the areas in which measurements of rotations could be extremely helpful. Because of the observational challenges in connection with free oscillations, any improvement or complement to existing techniques is desirable [e.g., Park et al., 2005; Ferreira et al., 2006]. This is particularly crucial for toroidal mode observations, which until now have been hindered by the high noise levels in horizontal component seismometers. However, the measurement of free oscillations requires sensors with a high sensitivity, and until recently it was not possible to make direct observations of free oscillations with the rotation sensors available [Widmer-Schnidrig and Zürn, 2009].

[5] Furthermore, classical long-period seismology is heading into an instrumentation crisis [Park et al., 2005; Ferreira et al., 2006] as high-quality observatory type instruments like the STS-1 [Pillet et al., 1994] are no longer manufactured. At ground motion periods relevant for free oscillations the cross-coupling effects (e.g., ground rotations leading to apparent translational motions) may exceed the amplitude of the modes to be observed [Widmer-Schnidrig and Zürn, 2009]. In addition to the multiple benefits of having collocated recordings of rotations and translations [McLeod et al., 1998; Pancha et al., 2000; Igel et al., 2005, 2007; Pham et al., 2009; Ferreira and Igel, 2009; Kurrle et al., 2010; Fichtner and Igel, 2009; Bernauer et al., 2009] measuring the complete ground motion including rotations would allow those cross-coupling effects to be removed and the signal quality considerably improved.

2. Ring Laser Measurements

[6] At present, the most sensitive instrument for the measurement of seismic rotations in a broad frequency range is the G-ring laser system at the geodetic observatory at Wettzell in Southeast Germany [Schreiber et al., 2006, 2009a, 2009b]. The measurement with this instrument is based on two laser beams that propagate in opposite directions in a square closed cavity. When the measurement system – rigidly attached to bed rock – rotates, the effective cavity lengths for the two laser beams differ, and a beating occurs. The observed interferogram, caused by the Sagnac effect, is directly proportional to the rotation rate of the sensor around the normal vector of the area enclosed by the laser beams. The G ring laser at Wettzell is a single component rotation sensor and measures the rotation rate around a vertical axis. Due to this special orientation, it is in principle only sensitive to horizontally polarized shear waves, i.e., SH-type waves that correspond to the superposition of toroidal modes. In terms of seismic surface waves, this means that the ring laser can record Love waves, but no Rayleigh waves (assuming a spherically symmetric Earth). Spheroidal modes do not cause any rotations around a vertical axis. In normal mode spectra recorded by horizontal seismometers, it is sometimes difficult to unambiguously identify all spectral peaks due to the cross-axis sensitivity mentioned above [Widmer-Schnidrig and Zürn, 2009].

[7] The G ring laser started operation in 2001 and has been continuously improved and refined ever since. Its ability to detect and record ground rotations from teleseismic earthquakes has already been demonstrated [Igel et al., 2005, 2007]. However, the signal-to-noise ratio was not sufficient to detect multiple-orbit surface wave trains or, equivalently, free oscillations of the Earth. This was even the case for the strongest earthquake during the last three decades, the Mw = 9.1 Sumatra earthquake in 2004. Since then, however, the sensitivity of the ring laser system and its data acquisition system have been improved by about 20 dB compared to the analysis by Widmer-Schnidrig and Zürn [2009]. As a consequence, it was possible to detect the free oscillations of the Earth excited by the Mw = 9.0 earthquake near Honshu on March 11, 2011. This will be demonstrated in the following section.

3. Observations and Modelling

[8] On March 11, 2011, a strong devastating Mw = 9.0 earthquake with subsequent destructive tsunami occurred in the North Pacific Ocean, near the east coast of Honshu. After this earthquake, it was for the first time possible to detect Love wave trains circling the Earth four times (both in the direction to and away from the receiver location) with the G ring laser at Wettzell (WET). Figure 1 shows seismograms of rotational motions around a vertical axis recorded with the ring laser (red). In addition, seismograms of transverse horizontal acceleration, recorded with a collocated STS-2 broadband seismometer, are displayed (black). The time windows shown contain the first eight Love wave trains, which are indicated by the labels G1–G8. The horizontal transverse trace was obtained by rotating the original North-South and East-West component records according to the theoretical back azimuth.

Figure 1.

Observed and synthetic Love waves (G1-8) for the 2011 Mw 9.0 Tohoku-Oki earthquake propagating around the Earth. (top) The direct Love wave propagating from source to receiver in Wettzell, Germany, (G1) and the one propagating around the other side of the Earth (G2) plus subsequent multiples (G3–G4). The left axes and the red traces correspond to rotational motions around a vertical axis. The right axes and the black traces correspond to transverse acceleration. Polarity of rotational motion is chosen such that rotational and transverse motions are correlated for G1. (middle) Graphical illustration of direct and multiple Love wave propagation paths along the great circle paths to an arbitrary receiver. (bottom) Observed and synthetic Love waves for further multiples (G5–G8); see text for more details. The data and synthetics were filtered using a zero-phase band-pass filter between 180 and 280 seconds.

[9] For G1, G3, G5, and G7, the waveforms of rotation rate and transverse acceleration are very similar. The polarity of the rotation rate is chosen such that the G2n+1 (n ≥ 0) rotation-rate waveforms correlate positively with transverse acceleration. As has been shown in previous studies, this is expected for plane SH waves, and the amplitude ratio between acceleration and rotation rate (except for a minus sign) equals twice the horizontal phase velocity [e.g., Pancha et al., 2000; Igel et al., 2005]. This also applies here, the scaling factor between transverse acceleration and rotation rate is 9 km/s, corresponding to an average phase velocity of 4.5 km/s. Following initial studies by Igel et al. [2005, 2007], we investigated if combined amplitude measurements of vertical rotation rate and transverse acceleration can be used to estimate Love wave dispersion curves. This seems possible in principle if the full waveforms are taken into account [Kurrle et al., 2010], indicating the information content in single-station records of both rotational and translational motions further investigated by Fichtner and Igel [2009] and Bernauer et al. [2009]. For wave trains G2, G4, G6, and G8, rotation rate and acceleration are not in phase but show reverse polarities. This is due to the reversed propagation direction (changing sign of the wavenumber vector) of these waves with respect to G2n+1 (rotation of the transverse components by 180° allows toggling between correlation and anti-correlation of both observations).

[10] For comparison, we computed synthetic seismograms by normal mode summation using the MINEOS code [Masters et al., 2007] (code available at on the spherically symmetric Earth model PREM [Dziewonski and Anderson, 1981]. A point source model is assumed using source parameters from the Global CMT catalogue ( These synthetic seismograms are in good agreement with the observed seismograms and show the same polarity flip as the observed data. To analyze the free oscillations, we computed amplitude spectra of the time series displayed in Figure 1. The observed seismograms were corrected for the instrument responses, multiplied by a Hanning taper and then Fourier transformed for a 36 hr time window.

[11] The observed and synthetic spectra are shown in Figure 2 (transverse seismometer component, WET T, and vertical component ring laser, RLAS). The frequencies of fundamental toroidal modes as predicted for the PREM model are indicated by thin vertical lines. As expected, in the transverse components primarily toroidal modes can be identified. As for the transverse component at WET T, the ring laser spectrum (RLAS) is also dominated by toroidal modes. The modes between 0T10 and 0T30 can be clearly identified. However, despite the waveform similarity of transverse acceleration and rotation rate in the time domain, there is a marked difference in the relative spectral amplitudes between these motion components.

Figure 2.

Observed and synthetic amplitude spectra of transverse and rotational motions. (top) Spectra of 36 hour observed transverse accelerations (WET T) in the range 0.5–4 mHz and of rotational motions around a vertical axis (RLAS). Theoretical fundamental-mode toroidal frequencies (blue vertical lines) for the non-rotating, spherically symmetric Earth model PREM are also shown. (middle) Examples of transverse motion patterns for specific toroidal modes. From left to right: 0T52, 0T102, 0T152, 0T202, 0T252. (bottom) Synthetic spectra for transverse acceleration (WET T) and rotational motions around a vertical axis (RLAS) for the PREM model; theoretical toroidal eigenfrequencies are also shown as vertical bars.

[12] To understand these differences, we carry out comparisons with synthetic amplitude spectra shown in Figure 2 (bottom). Although the amplitudes are slightly underestimated in the synthetics, the spectra from real data and synthetic data have very similar differential characteristics. The synthetic rotation spectrum differs considerably from the spectra of transverse acceleration. To illustrate this difference we calculate the complete surface displacement field and the curl of the displacement field (i.e., the rotation field) for individual toroidal modes for a spherically symmetric Earth [Ferreira and Igel, 2009]. In Figure 3 displacement vectors as a function of latitude and longitude superimposing the magnitude of rotational motion around a local vertical axis are shown for four exemplary toroidal modes. The modes are calculated for a uniform excitation at the North pole. The differential behaviour of rotational versus translational spectra can be clearly identified by picking arbitrary locations on the spherical surface. For example, in areas with no translational motions, rotational motions have a maximum and vice versa.

Figure 3.

Illustration of translational and rotational motions for toroidal free oscillations. (a) Displacement vectors (black) and magnitude of left-(red) and right-rotational (blue) motions for toroidal mode 0T21 as a function of latitude and longitude. For individual modes, rotational motions have maxima where the translational motion vanishes. (b, c, d) Same as Figure 3a but for modes 0T32, 0T43, 0T54, respectively.

[13] It is important to note that- with respect to observations- this explanation is somewhat simplified as 1) for each n-l couple there are m = 2l + 1 oscillations with the same frequency and different spatial wave-numbers, that split in case of a rotating, three-dimensional Earth; 2) A realistic double-couple point source will excite the numerous modes with different amplitudes depending on its orientation; 3) Earth's rotation will lead to a weak time-dependence of the free oscillations' nodal lines. Nevertheless, based on the dependence of the rotation rate around the vertical axis on the spatial gradient of the horizontal ground motions the principal interpretation holds.

[14] To illustrate the potential use of our novel observable for Earth's free oscillations we calculate synthetic seismograms and the corresponding spectra for two different point source solutions derived from observations recorded world wide after the Mw 9.0 Tohoku-Oki earthquake. The results indicate that the point source models for the M9 Tohoku-Oki earthquake are compatible with the low frequency part of the free oscillation spectra (see auxiliary material). This clearly demonstrates that the measurement of ground rotations can be used to put additional constraints on earthquake source properties with a sensitivity equivalent to translational measurements.

4. Discussion and Conclusions

[15] While the quantity of seismic stations permanently recording Earth's tremors after earthquakes and in seismically quiet time windows continuously increases there are still some unresolved fundamental issues in connection with the unambiguous observation of translational motions that have been pointed out many years ago [e.g., Aki and Richards, 1980]. This holds in particular for Earth's free oscillations as illustrated in the recent work by Widmer-Schnidrig and Zürn [2009]. In the light of the recent giant earthquakes that shook our planet since the Great Sumatra earthquake in December 2004, seismologists have highlighted the need to maintain and extend our capability for highly accurate measurements of long-period ground motions. The benefits for the imaging of the deep Earth structure- in particular density- but also attenuation and anisotropy, as well as the additional constraints on large earthquake sources [Park et al., 2005] are undisputed.

[16] In conclusion, we present here for the first time observations of rotational motions induced by free oscillations of the Earth after the Mw 9.0 Tohoku-Oki earthquake of 2011. We show that ring laser technology has advanced to a sensitivity level that provides an interesting complement to classical seismological instrumentation with benefits for observations of Earth's long period ground motions and, thus, for enhanced source and Earth structure imaging.


[17] We gratefully acknowledge support from the European Commission (Marie Curie Actions, ITN QUEST, We also acknowledge support from the German Research Foundation (Project Ig16/8) and comments by Ruedi Widmer-Schnidrig. We appreciate the comments and suggestions by Duncan Agnew. We would like to thank Kostas Lentas for helping to verify the modelling results.

[18] The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.