Rotational motions in seismology: Theory, observation, simulation

Abstract

The rotational part of earthquake-induced ground motion has basically been ignored in the past decades, compared to the substantial research in observing, processing and inverting translational ground motions, even though there are theoretical considerations that suggest that the observation of such motions may indeed be useful and provide additional information. In the past years, interest in this potentially new observable for seismology has risen, primarily because - with modern acquisition technology such as fiber-optical or ring laser gyros - rotational motions have actually been observed, the resolution is steadily increasing, and the observations are becoming consistent with collocated recordings of translational ground motions. Even though the real benefit to Earth sciences is still under investigation, recent results suggest that collocated measurements of rotations and translations may allow the estimation of wavefield properties (such as phase velocities, direction of propagation) that otherwise can only be determined through array measurements or additional strain observations. In this paper we focus on studies of the vertical rotational component (twist, spin, or rotation around a vertical axis) and review recent results on the fundamental concepts that are necessary to understand the current broadband observations of a wide distance and magnitude range, and show that the classical theory of linear elasticity is sufficient to explain these observations. In addition to direct measurements of rotational motions using ring laser technology, we describe the method to derive rotational motions from seismic arrays and present some initial results. Sophisticated 3D modelling of the rotational ground motions of teleseismic events illustrate the accuracy with which observed horizontal phase velocities match with theoretical predictions, even though the precise waveforms are quite different due to inaccuracies in crustal models or kinematic rupture properties. This may have implications for sparse networks or situations where extremely few or even single-station observations are taken (e.g., in remote areas or planetary seismology). To fully characterize the motion of a deformable body at a given point in the context of infinitesimal deformation, one needs three components of translation, six components of strain, and three components of rotation, a vectorial quantity. Rotational motions induced by seismic waves have been essentially ignored for a long time, first because rotational effects were thought to be small (Bouchon and Aki 1982), and second because sensitive measuring devices were not available. Indeed, Aki and Richards (1980, p. 489) point out that the state-of-the-art sensitivity of the general rotation sensor is not yet enough for a useful geophysical application. However, there have been many reports of rotational effects associated with earthquakes (like twisting of tombstones, or statues). It is certainly possible that some of these effects are due to the asymmetry of the construction. Indeed, as is well known, when the center of mass is not located at the geometric center, a mere translation may induce a local rotation of that structure. However, some field evidences suggest that it is at least not always the case (Galitzin 1914, p. 172). The rotational angles calculated by Bouchon and Aki (1982) for realistic cases of earthquake scenarios (about 10-4 radians) seem indeed too small to be responsible for damages, except, maybe for the case of long structures. However, as, roughly speaking, rotations are proportional to displacement divided by the phase velocity (see Section 30.2.3), when the wave velocity becomes smaller, rotations become comparatively larger. This happens in soft or unconsolidated sedimentary and/or fluid-infiltrated porous media, where wave speeds might be as low as about 50 m/s, hence smaller than usual by about a factor of 50. Thus, it is not implausible that, near seismic sources - where rotations and strains become relatively large even in normal media - rotations and strains become really large and be responsible for the above mentioned damages (there is also growing seismological evidence that rotational amplitudes have been underestimated (Castellani and Zembaty 1996)). Obvi ously, in such a situation, the assumption of infinitesimal deformation would then fail and the theory of finite deformations would be necessary. As an attempt to measure rotational motions with high sensitivity, in the past years ring laser gyroscopes were developed, primarily to observe variations in Earth's absolute rotation rate with high precision (Stedman et al. 1995, Stedman 1997). One of these instruments - located near Christchurch, New Zealand - recorded seismically induced signals of ground rotation rate for several large earthquakes (McLeod et al. 1998, Pancha et al. 2000). These observations gave evidence that the optical sensors indeed provide sufficient accuracy to record seismic rotations. However, they were not fully consistent in phase and amplitude with translational motions recorded with collocated seismometers, the limited consistency being obtained only in a narrow frequency band. Earlier attempts to observe ground rotations with other devices (e.g., solid state rotational velocity sensors, fiber- optical gyros) were limited to large signals close to artificial or earthquake sources (Nigbor 1994, Takeo 1998) and did not lead so far to an instrument of general interest. This explains why Aki and Richards (2002, p. 608) note that as of this writing seismology still awaits a suitable instrument for making such measurements. However, the subsequent development of ring laser technology indicates that a significant part of the gap has recently been filled, as demonstrated by Schreiber et al. (2003b) and Igel et al. (2005b) and exemplified later in this paper. Rotations can also be determined with array measurements, but with important limitations; see Section 30.3.1 and Suryanto et al. (2005). The recording of even small, non potentially damaging, rotational motions is expected to be very useful. First, translation recordings are polluted by rotations. There is a purely geometrical effect (the reference axis of the seismometer is rotated), introducing a cosine factor, hence negligible for very small deformations, but which could become significant in case of very strong ground motion. More importantly, there is also an inertial contribution. It is well known that surface tilt (horizontal rotation - see Section 30.2.2) induces a translational signal (Aki and Richards 2002, p. 604). A similar effect exists for vertical rotation (Trifunac and Todorovska 2001). Measuring the three components of rotation allows in principle to correct for these effects. Second, as the measure of rotation provides additional information, it helps constraining physical models. For example, the measure of vertical rotation and horizontal acceleration allows the estimation of the local Love wave phase velocity and of its propagation direction (see Section 30.2.3 and Igel et al. 2005a). It is also expected (Takeo and Ito 1997) that the measure of rotations will allow to better constrain earthquake rupture histories. The above discussion has been deliberately restricted to classical elasticity, for which the stress and strain tensors are symmetric. In some extreme cases (e.g., large stress gradient) near crack tips or in granular materials, the material can no longer be treated as a continuum. A continuous formalism may still be used for an effective medium in which additional - intrinsic - rotations (i.e., not related to displacement) may exist. In such Cosserat/micropolar media, the stress and strain tensors are no longer symmetric. See, e.g., Dyszlewicz 2004, Nowacki 1986, Maugin 1998, Lakes 1995, Teisseyre and Majewski 2001, Teisseyre et al. 2003, and other papers in this monograph. In the following of this chapter we remain in the framework of classical elasticity. © Springer-Verlag Berlin Heidelberg 2006.