1 Introduction

In front of the number of theoretical and numerical approaches that have been proposed, rotational ground motions have not been observed directly until the last decade due to the lack of appropriate instruments. The direct measurement of point rotations has been attempted through “ad hoc” designed devices, sensitive to torsion as tilt-meter, ring lasers and broadband rotation meters (Kalkan and Grazier 2007; Takeo 1998; Igel et al. 2007; Spudich et al. 1995; Zhang et al. 2004; Liu et al. 2009; Spudich and Fletcher 2009; Jaroszewicz et al. 2012; Lee and Liang 2008; Geological Survey Open-File Report 2007). However such devices are not in common use, and suitable applications in earthquake engineering are still rare and an open-challenging task.

Valid progress in the knowledge of the spatial distribution has been achieved through statistical studies (Kawakami and Sharma 1999; Lee and Trifunac 2009; Lavallée and Archuleta 2003; Zerva and Zhang 1997). A well organized state-of-the-art of the different facets of rotational seismology, concerning theoretical investigations, instrumentation, analysis of ground motion observations and engineering applications is provided by Lee et al., in an introductive paper of Bulletin of Seismological Society of America (Geological Survey Open-File Report 2007) and by Todorovska et al. (2008). It is worth to remember that in the application of Lee and Trifunac (2009), both average and point rotations should to be considered at the same time, for proper formulation of dynamic response of discrete systems.

Measurements in the near-field of earthquakes in Japan, Takeo (1998) indicate that rotational ground motions may be many times larger than expected from classical elasticity theory based on a propagation model. Direct observations have been published, Igel et al. (2005), Igel et al. (2007), Spudich et al. (1995), Zhang et al. (2004), Shabestari and Yamazaki (2003), and in particular Liu et al. (2009), Fig. 1.

Fig. 1
figure 1

Elaboration of data presented by (Liu et al. (2009). Ratio between the peak rotation velocity PRV and the peak horizontal acceleration PGA, in mrad × s/m. Average value 1.43

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Data presented by Liu et al. (2009), concerning 52 earthquake records collected at a single station in Taiwan, are presented in Fig. 1. The PRV is the maximum among the three components, and the PGA is the maximum between the two horizontal components. The ratio is shown in function of the distance from epicentre. The same ratio, computed according to array-derived cross spectra of accelerations (Castellani et al. 2012), is in good agreement with the average value of these data.

Statistical estimate from the set of data of Fig. 1 does not identify a marked dependence of the ratio on the distance from epicentre. Meanwhile, no one record is referring to a distance equal or less to the focal depth, and only 18 to a distance less than 1.5 the focal depth. Consistently, this amount of data, which is the richest so far available, does not provide a clear information for the recurrent conjecture that in the near field rotations might be higher, for the same peak horizontal acceleration (Stupazzini et al. 2009; Takeo 1998).

The paper mentions much higher amplitudes for the two horizontal components of the translation than that for the vertical component, everywhere mention is explicit. This circumstance is generally characteristic of medium to far region. For a same PGA, if for PRV only rotations in a vertical plane is considered, the dependence on distance would be more pronounced.

A direct observation of ground rotations caused by the M = 6.0, 2004 Parkfield earthquake and aftershocks has been published by Spudich and Fletcher (2009). The paper is mainly devoted to the mathematical process to ascertain reliable values for rotations. Results are not in a format suitable for a direct comparison to those of Fig. 1, however the order of magnitude of rotation rate is respected.

Array-derived rotations, and direct measurements at single points, are in principle a parent experimental measure, being the punctual rotation equal to the derivative, with respect to space, of the translations. One weakness of the array-derived rotations, that of being sensitive to “noise”, is likely to be present in point rotation instruments as well. In any case, the practical application dealt with in a previous study (Castellani et al. 2012), has shown that the high frequency content of the rotation motion has little impact on civil engineering structures. On the other hand, excitation provided by rotation in the frequency range 1–5 cps, may be meaningful for tall buildings (Di Cesare et al. 2014; Castellani et al. 2012), chimneys (Castellani and Zembaty 1996), and bridges. Underground tubes are sensitive to excitations at quasi-static frequency. For this application, strains have been computed by Paolucci and Smerzini (2008).

A comparison between rotations from dense-arrays and point measurements, during the same event, and at the same location, has been provided for the first time by Suryanto et al. (2006). The overall fit is estimated to be satisfactory, although factors of 2–5 between such measures have been observed. Authors explain them in terms of local site conditions.

A comparison of rotational waveforms generated by point rotation measurement to those derived from a translational array has been made by Lin et al. (2012). It is of interest both for testing the performance of the rotational sensor and evaluating the rotational motions derived from measurements by arrays of translational seismometers.

2 Representation of the ground motion

In a Cartesian space, let X, Y, and Z be the orientation axes, the latter being vertical. Along these axes the acceleration components during a ground motion are u(t), v(t), and w(t). In the following the attention is focused on the vertical w(t) acceleration, but the elaboration to follow can be applied to each of the three space components.

The set of records will be represented through an interpolation function,

$$w\left( {t,x,y} \right) = d_{1} \left( t \right) + d_{2} \left( t \right)x + d_{3} \left( t \right)x^{2} + d_{4} \left( t \right)y + d_{5} \left( t \right)xy$$
(1)

where x and y are the coordinates, and d 1, d 2, …, d 5 are unknown functions of time, to be computed instant by instant. If the interpolation function is reliable enough, it is possible to evaluate rotations, and shear strains, deriving Eq. 1 with respect to the spatial coordinates.

The quantities d 1, d 2, … are obtained by solving at each instant of time the system of equations,

$$w_{k} \left( t \right) = d_{1} \left( t \right) + d_{2} \left( t \right)x_{k} + d_{3} \left( t \right)x_{k}^{2} + d_{4} \left( t \right)y_{k} + d_{5} \left( t \right)x_{k} y_{k}$$
(2)

where w k (t) is the vertical acceleration recorded at the k-instrument, k = 1, 2, …, N, N being the number of instruments.

The procedure to obtain d 1, d 2, … is explained in the next paragraph. It is repeated instant by instant, independently from an instant to the other. An important worth of the procedure is to find out rotations as continuous functions of time.

Continuum mechanics defines rotations of the elementary volume around the Y-axis, ψ y :

$$\psi_{y} \; = \frac{1}{2}\;\left( {\frac{\partial w}{\partial x}\; - \;\frac{\partial u}{\partial z}\;} \right)$$
(3)

and shear strain γ xz as:

$$\gamma_{xz} = \frac{1}{2}\left( {\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}} \right)$$
(4)

These quantities for a soil deposit under earthquake excitation are considered. The free surface condition at the top of the deposit implies that the shear strain γ xz is equal to zero: therefore the rotation along the free surface is given, equivalently, by:

$$\psi_{y} \; = \;\frac{\partial w}{\partial x},\quad {\text{or}}\quad \psi_{y} \; = \; - \frac{\partial u}{\partial z}$$
(5a, 5b)

This quantity for a soil deposit acted by an earthquake excitation is considered.

The dense array of earthquake motion recorders in Taiwan is composed by N = 29 instruments, i.e., k = 1, 2, …, 29. In our hope all records collected in some seismic events (Fig. 2) will be utilized to build up w(t, x, y). However only seven records are available up to now to the present Author.

The way the function w(t, x, y) is determined in function of the coordinates is dictated by two requirements: (1) fitting, and (2) smoothing.

The fitting is solved in the least square sense over the set of records, as it will be explained in the following. Fitting requires that at least the records collected at positions at hundreds meter distance (C00, I03 and I06) should be reproduced correctly. Fitting is tested for the entire time history. For testifying the good performance only a short time windows is offered, Figs. 4 and 5, where the inaccuracy is of the order of a few percent.

The smoothing is attempted by truncating the series (1) to the first five terms. In theory the series could be of seven terms being available seven stations and thus seven equations for seven unknowns. The analytical functions so determined would be more irregular than expected, because higher terms are likely to introduce spurious effects that might affect the representation between station and station. In the present case the limited amount of accelerograms available, N = 7, has obliged to limit the unknowns to 5. This condition establishes the need to have available a higher number of records.

The first 1500 time accelerations have been considered, for t = 0…15 s, a common time scale for all stations. If it where only for the terms of first order (in x, y), the rotation, instant by instant would be the same along the entire region. The space dependence (Eq. 1), is such that rotation and its first order dependence on coordinates can be inferred.

Observing the records of the seven stations for which records are available, a component of propagation along the X axis may be suggested, as may be seen by observing carefully the peaks in Fig. 3.

3 Interpolating function

For each instant of time, Eq. (2) can be written in matrix form:

$${\mathbf{w}} = {\mathbf{A}} \cdot {\mathbf{d}}$$
(6)

where A is the matrix of coefficients, given by the coordinates of the stations,

$${\mathbf{A}} = \left| {\begin{array}{*{20}l} 1 \hfill & {{\text{x}}_{1} } \hfill & {{\text{x}}_{1}^{2} } \hfill & {{\text{y}}_{1} } \hfill & {{\text{x}}_{1} \times {\text{y}}_{1} } \hfill \\ 1 \hfill & {{\text{x}}_{2} } \hfill & {{\text{x}}_{2}^{2} } \hfill & {{\text{y}}_{2} } \hfill & {{\text{x}}_{2} \times {\text{y}}_{2} } \hfill \\ \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill & \ldots \hfill \\ \ldots \hfill & {{\text{x}}_{7} } \hfill & {{\text{x}}_{7}^{2} } \hfill & {{\text{y}}_{7} } \hfill & {{\text{x}}_{7} \times {\text{y}}_{7} } \hfill \\ \end{array} } \right|$$

x1 y1 are the surface coordinates of the station C00,

x2 y2 are those of I12,

…………

x7 y7 are those of the station O07,

w(t) is the column vector of the measured accelerations,

w1(t) = w(C00, t),

w2(t) = w(I12, t),

w3(t) = w(I06, t),

w4(t) = w(M01, t),

w5(t) = w(M07, t),

w6(t) = w(O1, t),

w7(t) = w(O7, t).

d is the column vector of the five unknown, and the dotted point · denotes the product of the matrix to the column vector d.

The coefficients of “best fit”, d 1 d 2, …, d 5 … are function of the instant of time t. There are more equations, 7, than unknowns, 5. The best fit is sought, the one which comes closest to satisfying all equations simultaneously. If closeness is defined in the least square sense, i.e. that to sum of the squares of the differences between the left and the right-hand side of Eq. (2) be minimized, then the over-determined linear problem reduces to a usually solvable linear problem, called the linear least-square problem.

The reduced set of equations to be solved can be written as the set of 5 × 5 equations:

$$\left( {{\mathbf{A}}^{\text{T}} \;{\mathbf{A}}} \right)\cdot{\mathbf{d}} = \left( {{\mathbf{A}}^{\text{T}} \cdot{\text{w}}} \right)$$
(7)

where A T denotes the transpose of the matrix A, and the dotted point · denotes the product of the matrix to the column vector d or w.

The paper shows that the procedure can be valid, meanwhile a larger number of records is necessary to reach more reliable data about rotations, as already said (Fig. 2).

Fig. 2
figure 2

Stations of the Taiwan dense array of instruments. Named are the seven stations for which records are available to the Author, events of 1981 and 1986. The central station is named C00. Radius of the inner circus is 100 m, that of the intermediate one is 500 m, and that of the outer one is 1000 m

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4 Some results: fitting

In Fig. 3 the vertical acceleration at given points is shown. Some time correlation is noticeable. However there is a systematic larger amplitude at the station M01 with respect to the other stations. A similar condition applies for the other stations, i.e. time correlation is visible, and the amplitude is rather different. Table 1 reports the peaks of the records, that range from 277 to 831 mm/s2.

Fig. 3
figure 3

Time window between 8 and 9 s, stations M01, M07 and I06. In the ordinate, vertical acceleration in mm/s2. Some time correlation is noticeable. However there is a systematic larger amplitude at the station M01 with respect to the other stations

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Table 1 Peak vertical acceleration at the stations, during the entire time history
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The fitting requirement is respected for the stations I12 and I06, Figs. 4 and 5. A short time window is shown, of 2 s duration. However the quality of fitting is similar for the entire time duration. Less satisfying agreement is remarked for the central station C00, where the time evolution is respected, i.e. the time position of all peaks are respected, but the ordinates of the accelerations built according to the interpolation procedure are around 80 % of the original ones.

Fig. 4
figure 4

Time window of the vertical acceleration at station I12. In continuous line the original function. By small squares the function built up by the present procedure, Eq. 7

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Fig. 5
figure 5

Time window of the vertical acceleration at station I06. In continuous line the original function. By small squares the function built up from the present procedure, Eq. 7

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For the central station this is because the interpolation procedure depends also on the variability of the amplitude, which should be recognized as rather random. If we derive rotations by the differences from one another record, the ratio PRV/PGA, reported in Fig. 1, will be random itself. The quality of the interpolation relies thus on the number of equations that it involves. The fitting requirement is even less respected for the more distant stations.

The procedure implied by Eq. 7 ignores the time correlation because the 5 × 5 equations are solved independently instant by instant. However, as it will be seen, rotations appear to be continuous functions of time, and the time correlation present in the original data is kept in the rotations, Fig. 6.

Fig. 6
figure 6

Stations I06 and I12. Window of the time history of rotation accelerations, between 5 and 6 s

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5 Rotations

Rotations around the stations I06 and I12 are investigated, for which the representation of the vertical acceleration w(t) is reliable (Fig. 7). Rotation is measured through the expression in Eq. 5a. Figures 8 and 9 allows a more detailed view of the two rotations.

Fig. 7
figure 7

Entire time history of rotation accelerations, in mrad/s2, at the two stations

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Fig. 8
figure 8

Window of the time history of rotation accelerations, between 7 and 8 s. The average time lag between the peaks of one and the other station is 0.058 s. The time lag between peaks at a same station is 0.11 s

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Fig. 9
figure 9

Power-spectrum of rotation acceleration, at stations I06 and I12. Again the spectrum at I112 is higher that that at I06. However the highest peak is located at a same frequency of 8.3 cps and shows a similar amplitude of the two spectra

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The main frequency of both signals is around 8.2 cps (cycles per second). Figure 6 is the picture of both acceleration rotations. Figures 8 and 9 show a correlation between the two signals, with a time lag of approximately 0.058 s. The time lag between peaks at a same station is 0.11 s, in substantial agreement with the peak of the power spectrum, discussed later on. Again the amplitude of signal at station I12 is higher than that at station I06.

6 Power spectrum of rotation accelerations

There are more functions under the name of power spectrum. In the following, the soil rotation is firstly transformed in Fourier series, in the standard way:

$$w(t) = a_{1} \cos (w_{1} \times t + \varphi_{1} ) + a_{2} \cos (w_{2} \times t + \varphi_{2} ) + \cdots + a_{i} \cos (w_{i} \times t + \varphi_{i} ) + \cdots + a_{N} \cos (w_{n} \times t + \varphi_{n} );$$
(8)

where t is time in seconds, \(\omega {\kern 1pt}_{1} = \frac{2\pi }{T}\) rad/s, where T is the duration of record, \(\omega {\kern 1pt}_{2} = 2 \times \omega {\kern 1pt}_{1} , \ldots ,\omega {\kern 1pt}_{i} \, = i \times \omega {\kern 1pt}_{1} , \ldots ,\omega {\kern 1pt}_{n} {\kern 1pt} = n \times \omega {\kern 1pt}_{1} ,\) \(a_{i} = \sqrt {b_{i}^{2} + c_{i}^{2} }\) with \(b_{i} = \frac{1}{\pi }\int\limits_{0}^{T} {w(t) \times \cos (\omega_{i} t)\;dt}\); \(c_{i} = \frac{1}{\pi }\int\limits_{0}^{T} {w(t) \times \sin (\omega_{i} t)\;dt}\), φ i are phases, with \(\varphi_{i} = {\text{a}}\,{ \tan }\left( {\frac{{c_{i} }}{{b_{i} }}} \right) \, .\)

Given the input signal as a Fourier series, the power-spectrum is:

$$S_{w} \,(\omega_{i} ) = \left( {\frac{{a_{i} }}{1/T}} \right)$$
(9)

Notice that in the literature the same quantity is alternatively expressed as:

$$S_{w} (\omega_{i} ) = \left( {\frac{{a_{i}^{2} }}{1/T}} \right)$$
(10)

In the present formulation the spectral ordinate is in (mrad/s2)/cps units. In all formulations, the power-spectrum is suitable to identify the range of frequencies where the signal is meaningful. Figure 10 shows that the soil rotation so far computed are meaningful in the range 4–11 cps, and the peak is at around 8.5 cps, consistent with the time lag between peaks at a same station, remarked in Fig. 8. It confirms that the soil rotation may constitute an important component of the seismic motion for some construction. In the papers (Di Cesare et al. 2014; Yin et al. 2016) and in previous studies (Castellani et al. 2012; Castellani and Zembaty 1996), the problem has been attempted on how much this signal is effective with respect to the excitation provided by translation motion.

Fig. 10
figure 10

Detail of the power-spectrum of rotation acceleration, at stations I06 and I12

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7 Comparison with other results, and conclusions

Figure 1 presented data by Liu et al. (2009), on the basis of the ratio between the peak rotation velocity PRV, and the peak horizontal acceleration PGA, in mrad × s/m. Average value 1.43.

The picture concerns 52 earthquake records collected at a single station in Taiwan. The PRV is the maximum among the three components, and the PGA is the maximum between the two horizontal components. The ratio is shown in function of the distance from epicentre. The same ratio, computed according to rotations obtained from cross spectra of array-derived accelerations, is in good agreement with the average value of these data (Castellani et al. 2012), Table 2.

Table 2 PRV/PGA reported in Castellani et al. (2012)
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In the present evaluation the same ratio is between 0.11 and 0.216, i.e. from 5 to 10 times less than the same ratio in Fig. 1. There are more reasons that justify why this procedure has led to a lesser ratio PRV/PGA.

  1. 1.

    There is a sensible scatter between the amplitude of acceleration records, as shown in Table 1. This establish the need to have available a larger set of records, for a suitable application of the least square procedure, Eq. 7, as already remarked.

  2. 2.

    The direction of propagation, implied by the line around which station are aligned in Fig. 2, is some how arbitrary. It allows to observe the rotation in the plane XZ and it forces to disregard the rotation in the perpendicular plane YZ.

  3. 3.

    In the Liu et al. paper, the PRV is the maximum among the three components. In the present application PRV in a vertical plane is considered, and besides in one precise plane. The vertical accelerations are in general less ample than the two horizontal components, and thus, for a same PGA, rotations in a vertical plane are less than the corresponding one in the horizontal plane.

  4. 4.

    In the general case, rotations are provided as well by the local and neighboring topographic effects. In this elaboration they are absent.

Under these premises, the interpolation of the array-derived records may be proposed as a promising tool to elaborate rotations, free from assumptions on a propagation model.