Experimental and numerical results on earthquake-induced rotational ground. motions

Experimental and numerical results on earthquake-induced rotational ground motions Chiara SMERZINI 1, Roberto PAOLUCCI 2, Marco STUPAZZINI 2 Corresponding Author: Chiara SMERZINI, Ph.D. Student Doctoral School of Earthquake Engineering and Engineering Seismology, ROSE School Via Ferrata 1, 27100, Pavia, Italy Mail:csmerzini@roseschool.it Tel: +39.0382.516911 Fax: +39.0382.529131 AUTHORs AFFILIATIONS 2 Department of Structural Engineering, Politecnico di Milano Piazza Leonardo da Vinci 32, 20133, Milano, Italy 1

ABSTRACT Owing to the lack of direct measurements of earthquake-induced ground rotations, a set of experimental and numerical results on rotational ground motion is illustrated. The results cover a rather broad range of magnitude, from 4 to 6.5, and regard both far filed and near source conditions. Experimentally results are derived through a suitable spatial interpolation procedure of displacement records collected from two dense arrays, Parkway Valley (New Zealand) and UPSAR (California). Validation checks both with other array-derived methods and with numerical simulations are carried out, to verify the reliability of our estimations and the capability of numerical wave propagation analysis codes to provide realistic estimates of rotational ground motions over a reasonable frequency band. Peak Ground Velocity (PGV) and Peak Ground Rotation (PGR) values obtained via the spatial interpolation procedure over dense seismic arrays are then put in comparison with the numerical results computed along two representative cross-sections of the Grenoble Valley (France), in the near field of a M W 6 strike-slip fault. In this case a combination of topographic, rupture directivity and site effects produces rotations of remarkable amplitudes, of the order of 10-3 rad. Finally, a PGV-PGR correlation, extended to both the experimental dataset and the synthetics in the Grenoble basin, is discussed. A common linear trend between PGV and PGR is found and turns out to be in reasonable agreement with other published results. INTRODUCTION Perfectly general motion would also involve rotations about three perpendicular axes, and three more instruments for these. Theory indicates, and observation confirms, that such rotations are negligible (Richter, 1958). Earthquake-induced rotational ground motion has been practically ignored for a long time in earthquake engineering, where seismic actions are generally formulated in terms of three translational ground motion components and neglecting the three rotational counterparts. On one side, ground rotations induced by seismic wave propagation were considered to be too small to have relevant effects on man-made structures, while, on the other side, instruments capable to detect rotational motion larger than 1 µrad/s with sufficient resolution are few (Liu et al., 2008). Only in the last 10-15 years, more attention has been paid to a more thorough study of rotational motions, made possible by the modern acquisition technology such as fiberoptical gyros (see e.g. Lefevre, 1993), solid-state devices (Nigbor, 1994) or ring-laser sensors (Stedman, 1997). Nevertheless such devices are not generally available in common seismic networks and it is common practice to indirectly estimate rotational ground motions by finite- 2

differencing measurements obtained from dense arrays (see e.g. Niazi, 1986; Oliveira & Bolt, 1989; Singh et al., 1997; Huang; 2003; Ghayamghamian & Nouri; 2007; Spudich & Fletcher, 2008). In this paper records from two dense seismic arrays, specifically Parkway Valley, in New Zealand, and UPSAR, in California, have been used to derive earthquake-induced rotations through an experimental procedure based on a suitable spatial interpolation rather than on a classical finite difference scheme. Past studies, either based on direct observations (see e.g. Nigbor, 1994; Takeo, 1998) or on numerical simulations (Bouchon & Aki, 1982; Castellani & Boffi, 1986) underlined that rotational motions may be relevant in the near fault region of an earthquake from the engineering standpoint. In seismological and geophysical applications rotations have been proved to be useful for the recovery of permanent displacements or long-period far-field wavefield (see e.g. Trifunac & Todororovska, 2001; Pillet & Virieux, 2007). From the engineering point of view, rotations may play a role in the dynamic response of certain type of buildings and when soil-structure interaction effects are significant (see e.g. Stratta & Griswold, 1976; Kalkan & Graizer; 2007). Few attempts to estimate rotational response spectra based either on array observations or numerical simulations have been done so far, starting from Castellani & Boffi (1986) who were the first to give a proper definition of the rotational response spectrum. Similarly to the translational response spectrum, it represents the maximum rotation acceleration of a single degree of freedom (dof) rotational oscillator of natural period T subjected to a prescribed rotational ground acceleration. Castellani and co-workers (Castellani & Boffi; 1989; Castellani & Zembaty; 1996) developed empirical expressions for the ratio of rotational vs. translational spectra, based on cross-correlation derived from strong motion records obtained at closely spaced arrays (namely SMART 1, Taiwan, and Shizuouka, Japan) and commented that it tends to be of about one order of magnitude higher in near-fault regions than in far-field conditions. Owing to the relatively poor knowledge on the quantification of rotational earthquake ground motion, the first purpose of this paper is to derive several experimentally-based estimates of rotational ground motions, based on a suitable interpolation technique of displacement records from dense seismic arrays. This approach has been already illustrated by Paolucci & Smerzini (2008) and applied to obtain earthquake-induced transient ground strains during four earthquakes: two weak motion events recorded at Parkway Valley (New Zealand) and two strong motion earthquakes registered at UPSAR (California), namely the December 2003 M W 6.5 San Simeon earthquake and September 2004 M W 6.0 Parfield event. In this work the interpolation method is applied to the same 3

case studies to derive a rather broad picture of the expected values of rotational amplitudes for magnitude ranging from about 4.2 to 6.5 and epicentral distances varying from about 10 to 80 km. In particular, attention is focused on a systematic comparison of our results with those obtained by Spudich & Fletcher (2008) for the 2004 Parkfield earthquake, to check the accuracy of the available array-derived rotational estimations. Furthermore, 2D numerical simulations of seismic wave propagation along a representative geological cross-sections of the Parkway Valley basin, where detailed information is available, are carried out with the aim of verifying the capability of the available numerical codes to predict earthquake-induced ground rotations over a reasonable frequency ranges (up to about 2 Hz). Finally we will address the issue regarding the dependence of surface ground rotations on near fault conditions and complex site effects by analyzing two representative cross-sections of the Grenoble basin, object of large scale 3D numerical simulations as illustrated in Stupazzini et al. (2008b). SUMMARY OF OBSERVATIONS IN ROTATIONAL SEISMOLOGY While a large selection of ground motion prediction equations is presently available in the literature in terms of different shaking parameters of engineering and seismological interest, such as Peak Ground Acceleration (PGA), Velocity (PGV) and Displacement (PGD) or response spectral ordinate, similar relationships are missing in terms of strains and rotations. This is due to the substantial lack of direct measurements of rotations and to the limited studies devoted to the latter parameters. To provide an overview of the available work on rotational ground motion, either from experiments or from numerical simulations, we have summarized in Table 1 a selection of results in terms of both Peak Ground Rotation about the vertical axis (alternatively referred to as torsion), denoted by PGR z, and Peak Ground Rotation Rate, PGRR z. Each row of the table contains information about: i) data type, subdivided into measurements, array-derived and numerical; ii) source parameters: magnitude M, epicentral distance R and source mechanism; iii) type of soil. Referring to the latter point, a simplified classification was assumed between stiff (denoted by 1) and soft (2) soil, roughly delimited by the S- wave velocity in the upper 30 m of the crust V s30 = 300 m/s as a threshold value. Most studies reported in Table 1 are based on the analysis of records obtained by dense seismic arrays (see e.g. Oliveira & Bolt, 1989; Bodin et al., 1997; Huang, 2003). Particular attention is drawn on the seismo-geodetic estimates recently performed by Spudich & Fletcher (2008) based on the array recordings at UPSAR, California, for the 28 September 2004, Parkfield M W 6.0 and 4

three aftershocks (M W 5.1, M W 4.9 and M W 4.7). Referring to this work, we listed only the so-called broadband estimates, calculated from their band-passed rotation data after applying a multiplicative empirical correction factor to embody higher frequency components. A detailed comparison between our experimental method and the results published by Spudich & Fletcher (2008) is illustrated in the sequel and aims at cross-validating these array-derived estimates, at least in the frequency range below approximately 1.5 Hz. As shown in Table 1, few examples of direct measurements of ground rotations are available, mainly in terms of rotational velocities PGRR z, typically providing values larger than expected based either on numerical simulation or array-derived methods. Such discrepancies, likely due to the frequency band limitations of the latter approaches, have already been pointed out and discussed by Huang (2003) and Spudich & Fletcher (2008). Besides array-derived analyses, a significant effort has been devoted recently to perform threedimensional large-scale numerical simulations of realistic seismic wave propagation. Stupazzini et al. (2008) simulated the rotational wave field generated by close-by strike-slip earthquakes with M W 4.5 and M W 6 inside the valley of Grenoble (France) by means of the Spectral Element Method. Similarly, Wang et al. (2008) showed parametric analyses based on finite difference simulations of Los Angeles basin for a target earthquake with M W 7. In spite of their strong differences, such as the maximum propagated frequencies, the size of the model and the chosen scenario, these two studies agree on: i) the strong influence of source parameters, especially the hypocenter location, on rotational motions in near source conditions and ii) the dependence on the basin structure and its location with respect to the strike of the causative fault. 5

Table 1 Selection of available literature data on earthquake-induced rotational ground motion, either from experiments or from numerical simulations. Peak values of horizontal ground velocity (PGV h ), ground rotation (PGR z ) and rotational velocity (PGRR z ) about the vertical axis. Additional information concerning the data type, the source parameters (magnitude M W, epicentral distance R and source mechanism) and type of soil (a simplified classification was assumed between soft and stiff soil). Reference Code Data Type M W EQ. parameters R Source [km] Mech. Type of Soil PGV h [m/s] PGR z [rad] PGRR z [rad/s] B&A82 2 6.6 1 SS 1 1 2 10-4 1.2 10-3 6.6 1 SS 1 1.6 3 10-4 1.5 10-3 L&T85 2 6.6 10 N.A. 1 0.45 1 10-4 1.2 10-3 N86 1 6.6 5 SS 2 0.203 2.75 10-4 7 10-4 5.6 6 N.A. 1 0.15 7.4 10-6 N.A 5.7 30 N.A 1 0.12 8.5 10-6 N.A O&B89 1 5.8 22 N.A 1 0.30 1.46 10-5 N.A 6.7 84 N.A 1 0.06 6.8 10-6 N.A 7.8 79 N.A 1 0.391 3.93 10-5 N.A C&B89 2 6.6 18 SS 1 6.92 10-2 3.06 10-5 1.04 10-4 N94 3 (1 kton) 1 Expl. 1 0.2780 6.6 10-4 2.4 10-2 Bea97; Sea97 1 6.7 311 R 2 0.03 5.6 10-5 N.A 7.5 305 R 2 0.11 2.07 10-4 N.A T98 3 5.7 3.3 SS 1 0.29 N.A. 3.3 10-3 5.3 3.3 SS 1 0.20 N.A. 8.1 10-3 H03 1 7.7 6 T 1 0.33 1.71 10-4 N.A K&G07 2 6.7 17 T 3 1 0.06 0.3 Lea08 3 6.2 133 N.A. N.A. N.A. N.A. 6.3 10-4 5.1 51 N.A. N.A. N.A. N.A. 5.2 10-4 Wea08 2 7.0 1 SS 1 N:A. N.A. 2.75 10-3 Sea08 2 4.5 7 10 SS 2 1.2 10-2 1.86 10-5 1.8 10-4 6.0 14-19 SS 2 1.24 1.69 10-3 8.24 10-3 6.0 8.8 SS 1 0.25 8.81 10-5 1.09 10-3 S&F08 1 5.1 14.4 SS 1 6.02 10-2 2.0 10-5 4.46 10-4 4.9 18.3 SS 1 2.74 10-2 1.36 10-5 2.47 10-4 4.7 14.0 SS 1 1.19 10-2 4.69 10-6 9.44 10-5 B&A82:Bouchon and Aki (1982); L&T85: Lee and Trifunac (1985); N86: Niazi (1986);O&B89:Oliveira and Bolt (1989); C&B89: Castellani and Boffi (1989); N94: Nigbor (1994) ; Bea97: Bodin et al. (1997); Sea97: Singh et al. (1997); T98: Takeo (1998); H03: Huang (2003); K&G07: Kalkan & Graizer(2007); Lea08: Liu et al. (2008); Wea08: Wang et al. (2008); Sea08: Stupazzini et al. (2008a); S&F08: Spudich & Fletcher (2008) Data type: 1 = array-derived 2 = numerical/semi-analytical 3 = measured Source Mechanism SS = strike-slip T = thrust R = reverse Soil Type 1 = stiff (V s30 > 300 m/s) 2 = soft (V s30 < 300 m/s) 3 = includes soil-structure interaction 6

DENSE ARRAY MEASUREMENTS OF GROUND ROTATIONS Referring to Paolucci & Smerzini (2008) for a detailed description of the interpolation procedure, we limit herein to recall some of its basic aspects and to briefly describe the selected arrays: namely, the Parkway Valley dense temporary seismograph network (New Zealand) and UPSAR (California) array. The Parkway Valley, Wainuiomata, New Zealand, is a small and shallow alluvium basin that has already been the object of several detailed studies on the quantification of complex site effects (see e.g. Chávez-García et al., 1999; Paolucci & Faccioli, 2003; Stephenson, 2007). Due to its peculiar local site conditions, the valley was instrumented with a temporary digital weak-motion network (Aug 1-Oct 12, 1995), consisting of 24 stations equipped with 1 Hz velocity meter with average spacing of about 40 m. In this work we analyze the recordings obtained during two events, of M W 4.9 and M W 4.2, denoted as Parkway #1 and #2 from here on, at epicentral distances of about 81 km in both cases. The second case study is based on records from the the UPSAR array (Usgs Parkfield, California, dense Seismograph ARray), in California (35 49 44 N; 120 30 13 W), consisting of 14 irregularly spaced seismograph stations, equipped with a three-component L-22 velocity transducer with a natural frequency of 2 Hz (see e.g. Fletcher et al., 1992). The separation distances are significantly larger than at Parkway valley, being approximately equal to 130m on average. With reference to this array, we have taken into consideration strong motion records of the December 22, 2003, San Simeon M W 6.5, and the September 28, 2004, Parkfield M W 6.0 earthquakes, with epicentral distances R = 65 km and R = 11.6 km, respectively. The interpolation procedure is implemented in a Matlab code relying on the Biharmonic Spline algorithm (Sandwell, 1987) over a regular grid with constant spatial step of x = y = 5m including all the recording stations for each of the 4 events under consideration. It gets as input the displacement records, obtained from the raw signals after an appropriate signal post-processing (basically a constant baseline correction followed by a 4 th order Butterworth high-pass filter, with cut-off frequency equal to 0.05 Hz for the UPSAR records and 0.1 Hz for the Parkway records, were found to be sufficient, as explained in Paolucci & Smerzini, 2008). The interpolation allows us to reconstruct the continuous three-component displacement wave field at the ground surface s( x, y,z = 0;t ) = [u, v, w], from which the rotation tensor Ω can be determined applying a 2 nd order centred finite difference scheme: 7

Ω( x, y,z 0 = 0;t ) = ωz ω y ω 0 ω x z ω y ωx 0 (1) with 1 v w 1 w u 1 u v ω x = ; ω y = and ω z =. (2) 2 z y 2 x z 2 y x In absence of further conditions, only the rotation about the vertical axis ω z can be obtained, since displacement variations with depth are not known. Nevertheless, taking into account the freesurface boundary condition (i.e. vanishing of shear stresses at z=0), allows one to determine the two rotational components about the horizontal axes in the following way: v w w γ yz = 0 = ωx = ; z y y u w w γ xz = 0 = ω y = (3) z x x As discussed by Spudich & Fletcher (2008), the assumption of a flat ground surface, that is behind Eq. (3), can be considered reasonable for the UPSAR array as well, located on a hilly area, owing to the large dominant wavelengths of the input motion. As a concluding remark, behind the proposed computation method there is the assumption of small ground rotations. The interpolation method relies on the recorded translational components obtained from an array of conventional strong motion transducers. The most accurate representation of the dynamic behaviour of the pendulum includes, in fact, not only the translational components but also terms proportional to rotation of ground surface, which are generally neglected. This means that the recorded translational motion is contaminated by rotational components of motion, which affect especially the low frequency part of the spectral response of the instrument (Trifunac & Todorovska, 2001). However, this limitation can reasonably be neglected when ground rotations are small and attention is not focused on the recovery of permanent ground displacements. COMPARISON WITH PUBLISHED RESULTS In this section we focus on a more detailed comparison between our estimates and the results of the seismo-geodetic approach as implemented by Spudich & Fletcher (2008) for the 2004 Parkfield earthquake. 8

Although our interpolation method and the one used by Spudich & Fletcher, commonly referred to as seismo-geodetic approach, belong to the same family of array-based or multi-station technique, there are some basic differences to be underlined. The seismo-geodetic approach has been introduced first by Spudich et al. (1995) and Bodin et al. (1997): the former applied the procedure to strong motion data obtained at the UPSAR array at Parkfield, California, while the latter considered three relatively closely spaced stations in Mexico City. Referring to Bodin et al. (1997) for a detailed explanation of the rational behind this procedure, we limit here to illustrate the fundamental features. The method consists in expressing the components ui( t ) / x j (i, j = 1, 2 and 3) of the displacement gradient tensor D in terms of differences u( t ) of ground displacement recorded by n three-component stations at separation distances x with respect to a selected reference station 0 u as follows: u n i = u n i u 0 i u = x i j x n j u x i j = u x n i n j for i, j = 1,2 and 3 (4) The method obviously requires that at least 3 station recordings are available; a generalized least squares minimization scheme may be efficiently used if more than 3 are available. An important assumption behind this method is the following: the deformation field is considered to be uniform within the networks at any time, i.e. the variations of displacement between two adjacent stations are linearly predictable (Bodin et al., 1997). That means that for a given network, a single strain estimate alone can be inferred. This assumption arises from plane wave propagation hypothesis and it is fulfilled if the array is closely spaced enough. Specifically, the average separation distance of the array should not exceed one quarter of the dominant wavelength λ to have gradient estimates accurate to approximately 90%. Contrarily, in our method the hypothesis of uniform rotation does not apply, as it can use a relatively small grid size (5 m) and interpolation function of higher degree to reconstruct the continuous displacement wave field starting from the original recorded time histories. Specific validation checks, illustrated by Paolucci & Smerzini (2008), indicate that their interpolation technique provides accurate results up to at least 2 Hz, even for average inter-station separation distance up to few hundreds of m, such as in the UPSAR array. This is in agreement with the practical rule suggested by Bodin et al. (1997), according to which the characteristic array dimension should not exceed one quarter of the dominant wavelength. Figure 1 compares the rotational time histories (left), obtained for the Parkfield earthquake, in terms of the three independent components ω x, ω y and ω z, and the corresponding Fourier amplitude 9

spectra (right) as estimated through our interpolation based procedure (thick line) and the seismogeodetic approach (dashed line). Referring to this latter, rotation (recall that a single estimate can be obtained) is computed from sub-array 5-12 and filtered in the band 0.1-1.4 Hz. To have a consistent comparison, we have computed the rotational motions at a central point of the sub-array, specifically at coordinates x=450 m y=550 m and filtered them in the same frequency band (see Figure 1 on the right-hand side). It is apparent that the comparison is very satisfactory at least in the frequency band under consideration. The principal phases of rotational ground motions are reproduced with good accuracy in both methods. The differences in terms of peak values are about 20% for ω x, 4% for ω y and 9% for ω z. This comparison represents a further validation of the reliability of the results of the interpolation procedure used in this work. Figure 1 Comparison of our method and the seismo-geodetic approach as implemented in Spudich & Fletcher (2008) in terms of time histories (left) and Fourier amplitude spectra (right) at UPSAR, during the Parkfield earthquake. The scale of rotation is mrad. The three components of rotations are shown: rotation about the x-axis ω x (up), about the y-axis ω y (centre) and about the vertical z axis ω z (bottom). On the right-hand side the UPSAR array is shown: the dashed circle denotes the sub-array from which the rotation data on the left-hand side are calculated according to the Spudich & Fletcher s procedure (2008), while the cross symbol at (450 m, 550 m) indicates where the rotation time histories are computed by our interpolation method. 10

VALIDATION WITH NUMERICAL SIMULATION RESULTS BY SPECTRAL ELEMENT METHOD As a further validation of our array-based estimates, we show in this section a comparison with the results of a numerical simulation of in-plane wave propagation along a representative longitudinal (NS) cross-section of Parkway Valley (see localization of the selected alignment in Figure 2 on the left-hand side). For the numerical simulations, we refer to the Spectral Element (SE) method developed by Faccioli et al. (1997) and implemented in the software package GeoELSE (Geo-ELastodynamics by Spectral Elements, http://geoelse.stru.polimi.it). Further details can be found in Stupazzini (2004). The Parkway Valley NS cross-section, discretized by SE, is depicted in Figure 3 (top) and the linear visco-elastic mechanical properties are summarized in Table 2. Internal soil dissipation has been introduced by a frequency dependent quality factor Q=Q o f/f o, where Q o is the quality factor at frequency f o ~ 1.6 Hz. The numerical model was subjected to the vertical incidence of a plane SV wave, with time-dependence given by the velocity history recorded at station 25 (see Figure 3, bottom) selected as the reference rock site based on previous studies by Yu et al. (2003). The numerical grid size has been designed to propagate accurately frequencies up to about 5 Hz. Figure 2 Left: Parkway Valley temporary array. Superimposed is the trace of the NS cross-section modelled and simulated in this study. Right: aerial view of the Parkway Valley location. 11

Figure 3 Top: Detail of the numerical model of a representative geological North-South cross-section of Parkway Valley. The model was kindly provided by W. R. Stephenson (Pers. Comm., 2006). The localization alignment A-A is specified in Figure 2 (right). Bottom: time dependence of the vertically incident input P-SV wave impinging on the numerical model as given by the velocity time histories, in terms of NS (left) and UD (right) components, recorded at station 25 (see Figure 2). Note that the scale of the velocity time histories is mm*100/s. Table 2 Mechanical and dynamical properties of the numerical model of Figure 3. Layers are listed from top (ground surface) to bottom (bedrock). Layer # Material V P [m/s] V S Q S Density Thickness [m/s] [-] [kg/m 3 ] [m] 1 Swamp deposits 1650 155 20 1800 11.5 2 alluvium 1650 345 20 1800 13.5 3 Old lacustrine deposits 1650 133 20 1800 7 4 old alluvium 2400 1200 80 2000 13 5 bedrock 3500 1750 200 2600 - As a validation of the numerical simulation, Figure 4 compares the horizontal rotations ω y (note that this is the single non-vanishing component for 2D simulations), calculated by the spatial interpolation procedure, with the numerical results, at selected receivers on the ground surface. Namely we consider receivers 09 and 17, located along the A-A alignment (see left-hand side of Figure 2). The comparison is carried out in terms of both time histories (left) and Fourier amplitude spectra (right), after applying a band-pass filter between 0.5 and 2 Hz, in order to make consistent the results of both procedures. Recall, as underlined previously, that the spatial interpolation method is estimated to provide accurate results up to frequencies of about 2-2.5 Hz. 12

The numerical simulations are able to reproduce ground rotational motions at Parkway Valley with a satisfactory accuracy, both in the time and frequency domains. In particular, we note that the dominant peak response around 1.6 Hz is well predicted by a 2D longitudinal model of Parkway Valley and it is associated to the fundamental natural frequency of vibration of the basin (see e.g. Stephenson, 2007). Although the peak values of ground rotation are over-predicted by the numerical simulations, the results of which are affected by the uncertainties related to the selection of the reference station for the input motion, and by the simplified assumptions regarding the soil layering and the corresponding dynamic properties, there is a remarkable overall agreement of results. Together with the previous comparison with Spudich & Fletcher (2008) solution, this is a further support both to the reliability of our predictions via the spatial interpolation procedure and to the capability of numerical techniques to provide realistic ground motion estimates. Figure 4 Validation of the numerical simulation performed by means of the Spectral Element Code GeoELSE. Comparison of horizontal rotation time histories ω y (left) and the corresponding Fourier amplitude spectra (right) obtained through our interpolation procedure (thick black lines) and the synthetic ones (dashed grey lines) at receivers 09 (top panel) and 17 (bottoml). Data are band-passed filtered between 0.5 and 2 Hz (this cut-off frequency is denoted with a small arrow on the left-hand-side plots). PGV-PGR CORRELATION After having illustrated some validation checks, we focus now on the issue regarding the correlation between translational and rotational ground motions. From here on we will refer only to PGR z, as representative of the amplitude of rotational ground motion. In fact, we have noted that, for the 13

Parkway and the Parkfield earthquakes, PGR z are greater than either PGR x or PGR y of a factor of ~2.6, on average. Differently, for the San Simeon earthquake vertical rotations are about 60% of horizontal components, most likely due to its reverse focal mechanism. Based on elastic wave theory (see e.g. Lee & Trifunac, 1987; Igel et al., 2005), for a harmonically propagating plane wave with displacement u(x;t), polarized in the horizontal direction, the vertical rotation is related to the transverse translational velocity through the following relationship: 1 u( x;t ) ω z ( x;t ) = (5) 2 Va t where V a is the apparent propagation velocity. In terms of peak values Eq. (5) simplifies as follows: 1 PGRz ( x ) = PGV( x ) (6) 2Va Eq. (6) provides a simplified tool to correlate the maximum amplitude of rotations from the corresponding translational components. The main drawback is that its applicability is conditioned to the fulfilment of the assumption of plane wave propagation in a homogenous medium and to an appropriate estimation of the prevailing wave type and the phase velocity. These parameters are not, in fact, generally available to the designer. In this section, we thus aim at discussing the applicability of Eq. (6), making use of the rather broad set of data obtained thanks to the experimental procedure described above. Figure 5 displays the PGV h -PGR z pairs for the 4 earthquakes under consideration, where 2 2 PGV h = max Vx + Vy. It is interesting to note that all datasets tend to be aligned along a straight t line in log-log space with equation: Log10 PGRz = ηlog10pgvh κ (7) where η ~ 0.98 and κ = 3.37. The coefficient of correlation between the maximum particle velocity and vertical rotation is close to 1, so that, when the parameter η is forced to be unity, the Least Squares best fit line turns out to be: PGRz = PGV / χ (PGV in m/s) (8) h where χ = 2120 m/s is the median value, while 1434 m/s and 3130 m/s correspond to the 16 and 84 percentile, respectively, σ = 0.17 being the standard deviation. We incidentally note that Spudich & Fletcher (2008) found a ratio of horizontal PGV (or, similarly, PGA); over PGR z (or 14

peak ground torsion rate, PGRR z ) equal to 2c, with c = 1000 m/s. The scaling factor 2c is very close to the parameter χ derived from Eq. (8). It is worth underlining that the fact that the relation between PGR z and PGV h is close to linear does not imply that rotations scale with the actual apparent propagation velocity, as also commented by Spudich & Fletcher (2008). In fact, apparent propagation velocity V a ~ 2.5 km/s estimated at the UPSAR array by spatial cross-correlation analyses of the Parkfield earthquake records (Fletcher et al., 2006) is much larger than the value V a ~ 1000 m/s that would be deduced by application of Eq. (8) and Eq. (6) with χ = 2120 m/s. This means that care should be taken if Eq. (6) is used to estimate the apparent velocity of a wave field. In Figure 5 the broadband estimates given by Spudich & Fletcher (Pers. Comm., 2008) for the same M W 6.0 Parkfield event and three aftershocks are also superimposed (see Table 1 for a detailed list of the data). For a consistent comparison with our method, the broadband estimates have been considered, as they reasonably cover the same frequency band. The comparison of our estimates and Spudich & Fletcher s data is satisfactory. Firstly, the value of PGR z determined by Spudich & Fletcher for the Parkfield event (8.81 10-5 ) practically coincides with the mean value of the set of pairs calculated by our interpolation procedure (8.98 10-5 ). Furthermore, the broadband estimates associated with the aftershocks tend to align along the same line of Eq. (8), supporting the indication that the relationship between PGR z and PGV is poorly dependent on earthquake magnitude and source-to-site distance. The very different site conditions at Parkway Valley and UPSAR arrays suggest a poor influence of site conditions as well, although this issue will be discussed in more detail in the following section. 15

Figure 5 PGV vs. PGR z pairs, obtained through the spatial interpolation procedure for the 4 earthquakes under consideration. Superimposed are the best-fit LS line corresponding to the median value χ = 2120 m/s (solid line) and the two best-fit lines (dashed line) associated with the 16 and 84 percentile, χ- = 1434 m/s and χ+ = 3130 m/s, respectively. Superimposed are also the broadband estimates of Spudich & Fletcher (2008) for the M W 6 Parkfield earthquake and three aftershocks (denoted by filled triangles) and the synthetics PGV-PGR z pairs obtained inside the Grenoble valley though the SE simulations (see following section for further details). NUMERICAL RESULTS FOR THE GRENOBLE VALLEY, M W 6 EARTHQUAKE The numerical simulations of the seismic response of the valley of Parkway have shown that the assumption of vertical plane wave incidence provides satisfactory results in terms of rotational ground motion in far-field conditions. However several studies (see e.g. Stupazzini et al., 2008a) have pointed out that the combination of two main factors, i.e. source directivity in the near field and complex 2D/3D site effects, may induce remarkably larger values of PGR. Starting from this consideration, we aim now at exploring how PGR, PGV and the ratio PGV/PGR vary along two representative cross-sections of the valley of Grenoble, located in the near field of a M W 6 earthquake, simulated by the 3D Spectral Element Code GeoELSE (Stupazzini et al., 2008a), in the framework of an international benchmark of earthquake ground motion numerical simulations (Chaljub, 2006). The combination of source directivity coupled with topography and deep basin effects, may produce a realistic picture of the rotational wave field. Figure 6a shows the 3D mesh used for the computations and Figure 6b depicts a top view of the Grenoble basin along with the localization of the causative strike-slip fault and of the two alignments under consideration: S1, located along the eastern tip of the Y-shaped basin, at hypocentral distances R hypo from about 5.6 to 13.6 km and S2, located along the southern tip, where the maximum amplification effects occur, at 11.5 <R hypo < 22.5 km. Our results refer to a homogenous slip distribution along the fault with hypocenter located at the centre of the fault itself. A detailed description of the various assumptions made for the simulations in the Grenoble area can be found in Stupazzini et al. (2008a). The PGV-PGR pairs obtained from the 3D numerical simulation of M W 6 Grenoble valley, that will be illustrated in more detail in the following section, are superimposed to those obtained by the spatial interpolation of dense array measurements in Figure 5. It is noted that: i) owing to the complex site conditions, the PGV-PGR z values in the Grenoble valley are remarkable larger than those estimated during the 2004 Parkfield event, although the magnitude and the epicentral distances are similar. However, ii), the PGV-PGR z ratio tends to align along the same correlation line as for the previous data. 16

Figure 7 shows the simulated PGV h, PGR z and the ratio PGV/PGR z at equally spaced receivers along the alignment S1 (left-hand side) and S2 (right-hand side): we have chosen to analyze the results in terms of vertical rotation for the 3D simulation in the Grenoble valley, since ω z has been proved to be the largest component for strike-slip events (see e.g. Stupazzini et al., 2008a; Wang et al., 2008). All data have been filtered between 0.1 and 2 Hz. As a preliminary observation, it is noted that PGR z vs. distance X does not show a clear trend: the pattern is, in fact, much more irregular than PGV since rotations are affected by smaller wavelengths. PGV h 0.6 m/s and PGR z 4 10-4 rad are calculated along the cross-section S1, while they reach values up to 1 m/s and 10-3 rad, respectively, in the southern tip of the basin due to the unfavourable combination of source directivity and complex site/topographical effects. These effects are apparent from the velocity time histories at equally spaced receivers along the two alignments: the incoming wave field remains trapped inside the basin with a remarkable amplification of ground motion at surface and increase of duration. In particular, the response of cross-section S2 turns out to be strongly three-dimensional, where trapping and amplification of wave is particularly relevant inside the easternmost side of the basin. Nevertheless, in both cross-sections the PGV h /PGR z ratios stabilizes around values of approximately 2000 m/s inside the alluvial deposits, while it increases of at least 4-5 times at outcropping bedrock. The same value of 2 km/s has been inferred by a best-fit linear correlation on a broad set of synthetic PGV-PGR pairs inside the whole alluvial basin in the recent work of Stupazzini et al. (2008a) and it agrees also fairly well with the value χ = 2120 m/s obtained from the results of the experimental interpolation procedure (see Figure 5 for further details). This suggests that the complexity of wave field achieved inside the basin by 3D numerical simulations is realistic enough to obtain reliable predictions of rotational ground motion, while in the surrounding bedrock the assumption of homogenous material, together with a oversimplified model of the fault, may have increased artificially the coherence of ground motion. As a consequence, the rotational components are reduced, since they come from a spatial differential operation and they are more sensible to the high-frequency part of ground motion. 17

Figure 6 a) 3D mesh used for the computation of the Grenoble scenario with the GeoELSE software package. The causative strike-slip fault (M W 6, neutral directivity), the mechanical properties of the whole model and the cross-sections S1 and S2 analyzed in this work are pointed out. b) top view of the valley of Grenoble. 18

Figure 7 Up: surface ground response along the cross-sections S1 (left-hand side) and S2 (right-hand side) inside the valley of Grenoble in terms of PGV h (in m/s), PGR z (in mrad) and the ratio PGV h (in m/s). PGR z. A sketch of both cross-sections is also depicted. Bottom: velocity time histories at equally spaced receivers (the thick lines indicate the edge of the basin). CONCLUSIONS Owing to the relatively poor knowledge on the quantification of rotational earthquake ground motion, in this paper we show a broad picture of experimental and numerical results concerning earthquake-induced rotational ground motion. Empirical estimates of rotations during 4 earthquakes with magnitude ranging from about 4.2 to 6.5 and epicentral distances varying from around 10 to 80 km are obtained through the procedure already illustrated by Paolucci & Smerzini (2008), based on 19

a suitable interpolation technique of displacement records from two dense seismic arrays: Parkway Valley (New Zealand) and UPSAR (California). The comparison between rotational ground motions obtained at UPSAR during the 2004 Parkfield M W 6 through our method and the seismogeodetic approach of Spudich & Fletcher (2008) shows a very satisfactory agreement, both in time and frequency domain, at least for frequencies below about 1.5 Hz. As a further validation check, two-dimensional numerical modeling of a representative longitudinal cross-section of the Parkway basin by means of the Spectral Element Method (SEM) is proved to provide realistic results in reasonable agreement with our array-derived estimations, despite the relatively rough assumption of vertical plane wave incidence. The numerical results of the seismic response of two representative cross-sections inside the Grenoble basin at few wavelengths from the hypocenter of a strike-slip event with M W 6 show that the combination of source directivity and complex site effects in near source regions may induce high levels of PGR, of about 10-3 rad, about one of order of magnitude larger than at UPSAR Parkfield, for similar magnitude and epicentral distance. This suggests that the unfavourable interaction of 3D topographical/site effects and source directivity patterns may play a predominant role in affecting rotational ground motion. The assessment of a threshold value beyond which ground rotations should be explicitly included to study the seismic response of complex structures such as high-rise buildings or extended infrastructures is still controversial. While Stratta & Griswold (1976) pointed out that a value of about 5 mrad may influence significantly the response of elongated structures, according to other studies, such as Kalkan & Graizer (2007b), rotations up to few mrad should not be sufficiently large to affect the dynamic response of structures. As a concluding remark, we note that a linear trend with a median value of the ratio PGV/PGR equal to approximately 2000 m/s describes with relatively low dispersion both our experimental datasets and a representative suite of empirical estimates performed recently by Spudich & Fletcher (2008). Furthermore, it is reasonablly close to the general trend of the synthetics PGV-PGR pairs inside the basin of Grenoble. Such a relationship allows a first-order approximation assessment of the amplitude of rotational motions, although further results should be explored, especially to account for variability of surface ground motions in the near field of severe earthquakes. Furthermore, the coefficient of proportionality between translational and rotational motions turns out to be poorly correlated with the actual apparent propagation velocity, as simple theoretical developments based on 1D plane wave propagation in homogenous media, would suggest. 20

ACKNOWLEDGMENTS We wish to thank W.R. Stephenson and G.Q. Wang for kindly providing data of the Parkway Valley and UPSAR arrays, respectively. We also gratefully acknowledge the cooperation of P. Spudich and J. B. Fletcher who kindly made available to us their rotational data at the UPSAR array. Comments by E. Kalkan and by another anonymous reviewer helped improving the manuscript. 21

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