The correlation coefficients
The existence of rotation effects at the Earth surface associated with earthquakes has been observed probably at least from the times when scientific approach to the ground motions during the quake had started. They are described in several classical monographs, such as Hobbs  and Davison , in which cited examples concern, among other things, twisting of some obelisks, tombs and segments of columns. However, early publications explain such phenomena as incidental effects of interference between linear vibrations [3, 4]. For instance, Imamura  explained the rotation effects of some objects at the ground surface by the impact of body/surface waves: due to such impact, an object can be inclined, partly losing contact with the ground surface, and when returning to the vertical some twist occurs with respect to its former position. Hence, from the beginning, the rotational effects have been treated as derivative effects, and it was stated that although such effects are observed, they cannot be explained as effect of any rotational waves - or rotational components of seismic waves - because existence of such waves or components would contradict the ideal elastic theory .
In the second half of last century, it was observed a spectacular development of continuum mechanics including defects, granular structure and other deviations from the ideal linear elasticity. Special interests were concentrated on the micropolar and micromorphic continua. In such elastic continua, the real rotation can be accompanied by other kind of axial motion – the twist-bend motion. On above base, it was theoretically proved that so-called the seismic rotation waves could propagate through grained rocks, initially by Teisseyre  who initially attributed the appearance of rotation components in seismic wave by coupling the seismic waves with the micromorphic response of the medium characterized by the an internal/granular structure [7, 8]. From this time, this possibility was extended to rocks with microstructure or defects [9, 10] or even without any internal structure [11 -13], due to the asymmetric stresses in the medium. On this base, various types of rotational waves have been discussed theoretically [14, 15].
It should be stressed that seismologists share different opinions about the nature of rotation waves – see the preface of a monograph on rotational seismology . Perhaps, as it is underlined in preface of a book , still the majority believes that such rotation motions are not related to inner rotations but are directly related to rotation in the displacement field which may reach much higher magnitudes in materials with an internal structure than in homogenous layers; considering damages in the high buildings, there are many examples indicating enormous increase of rotation effect caused by consecutive impacts of seismic body and surface waves.
Nevertheless, all above aspects can be treated as elements of rotational seismology. It is an emerging field for studying all aspects of rotational ground motions induced by earthquakes, explosions, an ambient vibrations. It should be noticed that nowadays there is observed rapid growth of the rotational seismology interest in many geophysical fields of knowledge  which includes wide seismology disciplines, seismological apparatus, seismic-origin phenomena, physical and engineering aspects of earthquakes as well as geodesy.
However practical aspect of rotational events and phenomena investigation is connected with method of their recording, and different rotational seismology branches need different devices. For example, earthquake physics need devices operating with sensitivity below 10-9 rad/s/Hz1/2, whereas the engineering of a strong-motion seismology devices operating with a frequency range 0.05-100 Hz with sensitivity 10-6-10-1 rad/s/Hz1/2 . In this subject, it should be noticed that the seismic rotation waves were for the first time effectively recorded in Poland in 1976 . Even though, from this time, waves or phenomena of this type have been studied in a few centers over the world, a further experimental verification of the existing rotational phenomena needs a new approach to the construction of the measuring devices, because the conventional seismometers are inertial sensors detecting only linear velocities . Thus, during measurement of the rotation present in the seismic field, with the use of a special array or set of conventional seismometers (for example based on a set of two classical mechanical seismometers ), data are disturbed by linear movements . Therefore, an innovative device is necessary to detect the rotational seismic phenomena/events. According to our knowledge we can confirm that the technical implementation of the Saganc effect  is the most proper way to measure rotation directly. One can find instances of such solution: a ring laser  as well as a fibre optic seismometer [25-27]. It gives the opportunity to carry out the measurements without any reference system.
It should be underlined that all experimental data recorded during earthquakes shows that rotational components are small in comparison to linear motions - less than 10% [14, 19] or have half of above value and exist with some delay regarding last one .
Based on above review in this chapter we present an analysis of a few examples of the rotational seismograms. Authors have concentrated on the local seismic events obtained at the Książ Observatory in Poland. These signals were obtained from two kinds of sensors described in section 2: the micro-array of TAPSs – Twin Antiparallel Pendulum Seismometers (also named rotation seismometers or double pendulum horizontal electromagnetic seismometers) and with the Sagnac interferometer of AFORS - Autonomous Fibre-Optic Rotational Seismometer, constructed at the Institute of Geophysics and the Military University of Technology, respectively. It should be also underlined that signals derived from micro-array include two components which, according to Asymmetric Continuum Theory, have character of rotational wave: rotation
2. Instrumentation for recording rotational components of the seismic events
Figure 1 presents the general view of the measurement devices installed, at the beginning of July 2010, in the Książ Observatory, Poland (located at 50.84380333N, 16.291755 E). There are AFORS-1, the micro-array of seismometers consisting of two TAPSs (TAPS-1 and TAPS-2) oriented perpendicularly in the N-S and E-W directions, and other instruments such as accelerometers (parallel positioned with TAPSs), etc.
The data detected by TAPSs (two channels for each of them) are stored by standard seismological system KST while data detected by AFORS are stored both by FORS-Telemetric Server and KST. The KST system uses sampling of the signals with frequency equals 12,8 kHz. The process of data storing by KST uses frequency of 100 Hz. Figure 2 presents an example of a diagram with data collected on March 11th, 2011 at 6 h 58 min. (after the Honshu earthquake M=9.0 on March 11th, 2011 at 5 h 46 min. 23 s UT, recorded in Książ, Poland on March 11th, 2011 at 5 h 58 min. 35 s UT), used in previously presented analysis .
2.1. Design of the Twin Antiparallel Pendulum Seismometers
The micro-array of seismometers (system of two TAPSs perpendicularly oriented) is an experimental apparatus, devised in the Institute of Geophysics, and manufactured according to description presented below, on the base of short period SM-3 seismometers. This is one of the simplest micro-arrays for measuring the rotation and twist (shear) . It was deployed at two Polish observatories, in Książ and Ojców (see ). The third identical set of sensors was used in Central Italy .
The idea of using the classical short period SM-3 seismometer as a new kind of mechanical rotational seismometer named TAPS is presented in Figure 3 . It is a set of two SM-3 seismometers (named in Figure 3b as left – L and right – R) situated on a common axis and connected in parallel, but with opposite orientation. In the case of the ground motion containing displacements
where sign “+” and “-“ are for R and L seismometer, respectively.
As one can see, in the case of identical two seismometers the rotational motions and displacement can be obtained from the sum and difference of the two recorded signals as:
If the ground could be treated as a perfect rigid body, then the rotational motion recorded by sole one TAPS was identical to rotation. But rocks and the ground surface are not perfectly rigid; they transfer the mechanical waves due to slight, transient deformations which, seen along different axes, may different. Consequently, rotation
Relations (2a) and (2b) remain valid, however, only when both seismometers forming the system have exactly the same response characteristics. Because, as a matter of fact, the pendulum seismometers are, inevitably, slightly different, the special TAPS channels calibration algorithm is used. In this system, for the aim of comparing both sensors, it is possible to rotate the position of one seismometer in such a way that both the pendulum seismometers, suspended on the common axis become oriented in the same directions, one just above the other – this is the test position. The working and test position for the case of the horizontal seismometers are schematically shown in Figure 4. The records obtained in the test positions can differ mainly due to differences in their response characteristics, and to minimize these errors, the following left channel signal calibration procedure is usually applied:
Nevertheless, there is some discrepancies due to the recording procedure in the case of the present of the difference of TAPS’ pendulums attenuation characteristics. Figure 5 presents the simulation which indicates that there can be some errors in the data caused by the recording proceeding. The considered simulation was made for the attenuation difference equals |βL-βR|=0.05 between left and right seismometer attenuation. It is easy to see that the major error of the signal is present when the simulated rotation is characterized by smaller value of amplitude compared to the simulated translation component. However, this is the region where the seismic-origin rotation is expected. For above reason the process of TASP calibration seems to be an essential complication of the system work’s correctness. Moreover, the extremely high sensitivity to the translational motions of the seismometers (preferred for the component of displacement detection) taken into account in their construction can limit the accuracy of such devices, too.
From above mentioned reasons, the additional numerical procedures for improving the TAPS performance may be applied, based on filtering in frequency  or in time  domains. The respective filters can then be applied to records in the normal working position to reduce the influences of non-equal operation of pendulum seismometers, presented above. However, these methods use test position of the TAPS, that generally changes the condition of the TAPS operation. For this reason, another procedure of the recorded data processing, based on smoothing by the spline functions has been also proposed . It should be noticed, that the main disadvantage of all listed methods is that they operate on recorded data, which can limit TAPS usefulness for some applications. In the research presented in this paper, the mentioned methods of signal correction were not used.
2.2. Design of the Autonomous Fibre-Optic Rotational Seismograph
The AFORS-1, used in our research, is one of three such devices existing in Poland and manufactured on the base of fibre-optic gyroscope, all dedicated for direct measurement of rotational components existing in seismic events and having accuracy below 5‧10-9 rad/s for 1 Hz detection band.
The physical principle for these devices is the Sagnac effect  which is a result of difference between two beams propagating around closed optical path, in opposite direction. Figure 6a presents the basic principle of the Sagnac’s experiment. The input light beam is splitted by a beam splitter into a beam circulating in the loop in a clockwise - cw direction (Figure 6a - beam T) and a beam circulating in the same loop in a counterclockwise - ccw direction (Figure 6a – beam R). One can observe the interference pattern, in the output light, caused by the interference phenomenon of the two waves. In the case of the present of rotation with an angular rate represented by vector rad/s then a fringe shift ΔZ is observed in the output of the interferometer. The fringe shift is given by the formula :
in which is a vector of sensor loop area, λ0 is the wavelength of the used light in vacuum, c is velocity of light in vacuum. According to the above formula a fringe shift is proportional to the cosine of angle between the axis of rotation and the normal to the sensor plane. There have been used the indigo mercury wavelength as well as the sensor area S = 866cm2 in the Sagnac’s experiment  which gave the fringe shift equals 0.07 fringes for the rotational speed equal to 2 rps. Nevertheless other papers  indicate that the possible detectable fringe shift was of the order of 0.01 fringe in those time. Thus, perhaps Sagnac has been carried out the research with a maximal accuracy. Sagnac also established that the effect does not depend on the shape of surface area S or on the location of the centre of rotation, whereas future investigation shown that this effect it does not depend on the presence of a commoving refracting medium in the path of the beam .
Figure 6b presents the Sagnac interferometer in the optical fibre solution which uses optical waveguide of the long length L wound on sensor loop of the diameter D which was shown firstly in 1976 . In this approach, a phase shift
where Ω is the rotational component perpendicular to the sensor loop. It is clearly to indicate that the sensitivity can be change by physical dimension of the sensor loop as well as by the length of the applied waveguide. It should be noticed that application of three such systems, which loops are jointly perpendicular, provides data about space vector of the rotation. One can obtain the position change in space by integrating the data in time domain. The above procedure is used in the configuration of the fibre optical gyroscope - FOG which now, nearly 40-years from 1976, is the best recognized interferometric sensor performed in the fibre-optic technology.
However, for a desired rotation rate in the range of 10-6 – 10-9 rad/s, the Sagnac effect generates a very small phase shift, so it is needful to separate and protect this effect from other disturbances so that the Sagnac effect is the unique nonreciprocal effect in the device. For this reason all FOGs use, shown in Figure 7, the reciprocal configuration  which is also called minimum gyro-configuration . This configuration guarantees an ideal equilibrium of two counter-propagating beams in the interferometer by obtaining true single mode operation at the common input-output port of the system. It is not disturbed even by non-single mode operation in the another part of the interferometer.
It is well known that each interferometric devices yield a cosine response. For above reason the detected signal practically do not change during the small changes of the rotation due to slow changes of the cosine function at the zero. In order to obtain higher sensitivity the operation point of the interferometer is shifted by applied additional phase shift modulation. The FOG utilizes the reciprocal phase modulator which is placed in the end of the sensor loop. It caused the modulation of the phase shift by propagation delay without any residual zero offset . In this way one obtain the odd response instead of even one. An ultimate performance is, however, obtained only if the unbiased response is perfectly even and the biasing modulation has only odd frequencies. Therefore, the applied phase modulator is also a delay line filter operating at the eigen-frequency  – the delay in the loop is equal to a half of modulation period which suppresses the residual even harmonic signals. Nowadays the FOG utilizes broad-band light source for eliminating the Kerr effect which produces the phase shift in the optical fiber Sagnac interferometer . Such a broadband source is also needed to remove coherence related with noise and drift due to backscattering and backreflection as well as lack of rejection of the polarizer [44–46]. Finally, for achieving the high scale factor linearization, FOG uses a digital phase step feedback  by the same reciprocal phase modulator as the biasing modulator and all-digital processing procedures where demodulation is carried by a digital subtracting and sampling of the modulated signal is obtained by using analogue-digital converter [48, 49].
Currently, a digital processing of FOG systems is designed to record angular changes instead of rotation rates, thus, the optimization of such system to register the interesting phenomena from the rotational seismology point of view is problematic. Therefore it should be emphasized that the AFORS construction based on experiences according to the FOG development described above, but with system optimization for a direct measurement of the rotation rate only . Such an approach gives a system which through a direct use of the Sagnac effect can limit drift influence on a device operation.
A detailed description of the AFORS system was published previously [29, 30, 50], hence here we summarized the above data regarding AFORS-1 construction, calibration and management. The second device - AFORS-2 is located in Warsaw (Poland) for initial works connected to the investigation of the irregular engineering construction torsional response and the interstory drift . We anticipate that the new device, based on AFORS-1 and -2, AFORS-3 will be construed in the 2014 which gives us the opportunity to mount new innovative system instead of FORS-II assembled in seismological observatory Ojców, Poland .
The AFORS uses the minimum configuration of the FOG, however opposite of it, AFORS operates in open-loop architecture with digital data processing . This technical solution is motivated by the fact that rotation events (Ω) are registered as sudden changes of rotational rate which amplitude is determined in a direct way from the Sagnac phase shift (Δφ) by following equation :
where S0 is the optical constant of interferometer which depends on the fundamental parameters of fibre coil.
Upper part of Figure 8 presents the block diagram of the AFORS-1 optical part configuration. The AFORS-1 construction, designed according to the minimum-gyro configuration, contains of the: SLED diode (ΔB=31.2 nm, λ0 = 1305.7 nm, Pout= 9.43 mW;
It should be emphasised that in the AFORS construction we have applied the special processing unit ASPU (Figure 9), which enables to obtain the detected rate of rotation in a direct way from the measured Sagnac phase shift (7). The ASPU detects the rotation rate (Ω) by selection and conversion of the first (A1ω) and second (A2ω) amplitude of the harmonic output signal [u(t)] using the following formula :
The calibration process was realized basing on the measurement of the defined slow rotation connected to the vector of Earth rotation in Warsaw, Poland (i.e. ΩE= 9.18 deg/h ≡ 4.45∙10-5rad/s for = 52°20’’). AFORS was mounted vertically on a special rotation table (Figure 10a) and then rotated so that the sensor loop was directed to the North, South, East and West. The detected signal was equalled to zero for East-West (sensor loop collinear with the Earth rotation axis) while the signal was maximal, equal to ± 4.45∙10-5rad/s, for North-South (the sensor loop perpendicular to the vector component of Earth rotation). During the calibration procedure in the first step were determined the position for maximal signals, North and South, then, the position of the sensor for the West and East was defined as the midpoint between those two signals because of problems with determination this position basing on searching of signals equal zero. The maximal signals were obtained with accuracy equals 0.5 deg. In order to eliminate the drift phenomenon the 10 night hours averaging signal was applied. The above procedure allows for scaling the system and determining the constant operation parameters – optical
3. Analysis of rotational components of the seismic fields caused by local events
In this section we present analysis of data obtained at Książ Observatory, reveal the rotational components presence in entire seismograms (from P-wave arrival onwards), for the cases of local seismic events, of the mining (Lubin on January 20th, 2011, two events starting at 04 h 59 min. 1 s UT - shown below as Figure 11) and tectonic (Jarocin on January 6th, 2012, the event starting at 15 h 38 min. 10 s UT - shown below as Figure 12) provenience.
The results were obtained directly from KST recording system and they include five plots: channels 1 and 2 for TAPS-1; channel 3 for AFORS-1; channels 4 and 5 for TAPS-2. The TAPSs’ records show linear motions that appeared during the earthquakes, and rotational components are calculated as described in the section 2.1. The channel 3 for AFORS-1 shows rotational oscillations measured in a direct way.
As one can see, the both kinds of devices (AFORS-1 as well as set of micro-array of TAPSs) recorded the events in the same time, which can confirm some correlation between devices. However the following investigation needs an additional data proceedings. To limit noise influence on recorded signals the average procedure in the beginning has been applied. From above mentioned reason the recorded seismic events have been averaged in moving windows of 100 samples (which is equal to period of 1 second).
The results of above operation are presented in Figures 13a-13c respectively for above three events. Consistently, we present five plots for all of them. The first plot, named TAPS-1 channel 2, presents the velocity of linear ground motion in m/s registered by second seismometer in this device. Second diagram shows channel 1 of TAPS-2. It should be noticed that second channel for any TAPS, after change of sign, has very similar plot to the first channel, so they are not presented both in our Figures. Third plot, named AFORS, presents the rotation velocity in rad/s, registered directly by AFORS-1. The result of measurement of the same component, but using the four simultaneous signals from micro-array of TAPSs (the procedure is described in section 2.1), is presented as plot four named Rotation from TAPS. Finally, last plot named Twist from TAPS, presents twist component obtained from the same micro-array in accord with, mentioned in the introduction, the Asymmetric Continuum Theory  – also in rad/s.
Even though the results presented in Figures 13a-13c show similar time of occurrence of the rotational components recorded by the AFORS-1 and calculated from TAPSs, their shape differs, and inevitably the correlation coefficients between them are low (though greater than zero). These coefficients are presented in Table 1, for both events recorded on January 20th, 2011, and for their selected parts. This selection consists of: time-period when P waves arrive – 2 seconds in each case; time period of firsts S waves arrivals – again 2 seconds, and a time-period when great S-type oscillations dominate; here we choose twenty seconds in each case (such unusually high amplitudes of low frequency oscillations, dominating in the late stage of the tremor, characterize the seismic field generated by mining seismic events in the Lubin area and received at Książ observatory).
|Twist TAPS –|
17.00 – 19.00 s
26.00 – 28.00 s
30.00 – 50.00 s
146.00 – 166.00 s
|Twist TAPS –|
Additionally, correlation between both rotational components calculated from TAPSs recordings, that is – rotation and twist – was checked too, for the same chosen time-periods. This appeared generally high (especially in the second event), which confirms previous observations by KP. Teisseyre that in the recordings of seismic events made with a set of TAPSs, there usually occurs conformance between rotation and twist, either direct or reverse.
Dissimilarity between simultaneous rotation recordings obtained from AFORS and the micro-array of TAPSs may be explained by different characteristics of the used instruments or by certain errors or/and noise present in one or the other side of compared results, or in both. Here, there is dramatic difference in spectra (Figure 14), that of AFORS is always much longer. For the analyzed case of mining events, the spectrum of AFORS signal bears high amplitudes in the range of 2 – 8 Hz. Spectra of all the signals and rotational components obtained from TAPSs are short – they practically decrease to zero level at about 20 Hz – and bear sharp maximum at about 0.5 Hz, while the linear signals have also wide area of relatively high amplitudes in the range of 1 – 8 Hz. Moreover, the signals from AFORS bear high percentage of noise, probably of electronic provenience.
To find similarities in the obtained results in other way than just by sight, the following analysis has been applied. First – any of compared data chains was transformed into chain of absolute values (moduli). Then, the chains of moving averages of these moduli are created, with the window length of 100 samples, which is equivalent to 1 s. Here, absolute values of rotation (velocities) were compared – these from AFORS with those calculated from set of TAPSs. Further, analogical moving averages were investigated, but calculated from moduli of all four signals recorded by the TAPSs. If the rotation obtained from AFORS is symbolized with ωo, rotation obtained from the TAPSs with
Results of this analysis are shown in Figures 15-17, which all have the same scheme. Diagrams a) show the plot of moving average of absolute values of the rotation rate obtained from AFORS; diagrams c) – analogical averages of rotation rate calculated from TAPSs’ data; diagrams e) – analogical averages of the mean signals moduli from TAPSs – as in formula (9c). The blue plots presented in diagrams b) represent the moving correlation coefficient between moving averages of absolute rotation from AFORS and its analogue from TAPSs (compare the plots in diagrams a) and c) described mathematically by equations (9a) and (9b)). This coefficient is calculated for a window of 100 samples (1s). Additionally, in the same diagram we present reference thick lines – orange or mauve, and turquoise – which should facilitate the comparison of correlation coefficient with compared chains. The procedure for obtaining the upper reference line is following: create normalized chain (9b) to (9a) by comparing both maximum values, as (9b’). Next, make new chain as sum of (9a) with (9b’), and normalize it to 1. Thus we have obtained a doubly–normalized mean chain which joins shapes of the original two chains. The turquoise reference line is analogical to orange one, but multiplied by -1. Purpose of this line presence is to facilitate comparison of stages of high negative correlations between compared chains, again with the described doubly–normalized averages. Analogically, curves presented in diagrams d) and f) are the plots of the correlation coefficient in a moving window of 100 samples; in d) – between averages plotted in diagrams c) and e), and in f) – between averages plotted in e) and a). Reference lines are produced in analogical way as for diagrams b); the upper one is mauve in diagrams f). Figure 15 shows results of such analysis applied to recordings of first event from Lubin area; Figures 16 and 17 are made for second event from Lubin area and for Jarocin earthquake, in the same methodology.
Comparison of the moving correlation coefficients with the moving averages allows to find certain rule. In time-periods when main seismic phase arrive – as P and S (for local distances, it might be jointly for example Pg and Pb phases and analogically Sg and Sb in the S-type phases family), all investigated correlations between moving averages of the absolute signals are generally high. These are some of the time-periods in which blue line in the diagrams b), d) and f) is near 1, and the upper reference line rises. More interestingly, these time-periods starts slightly before any noticeable rise in initial moving averages, despite facts that all the moving windows had the same length. This phenomenon may have two causes: certain common order in concerned signals starts just before noticeable arrival of seismic waves, or/and this is a mere effect of filtration (which used in every contemporary seismic recording system).
The chosen time-periods of high correlations and rise of the moving averages are as follows.
For the first seismic event from Lubin - two time periods: from 16.2 to 18.5 s, this include arrival of P waves, and from 25.8 to 27 s, which include S waves arrival (see Table 1). For the second seismic event from Lubin – only one time period, from 127.9 to 129.5 s, which include arrival of P waves. For the Jarocin earthquake – one non-continuous time period, for diagrams b) and f): 21.7 – 24.1 s and for diagram d) : 20.9 – 24.1 s. This time-period, in all three diagrams, starts before arrival of P waves and comprises arrival of this seismic phase.
We did not find yet any other rules, especially – we did not find any relation between episodes of negative correlations (near -1) and the initial moving averages. We suppose that such comparison method may be useful in analysis of various chains, and especially their moving averages, as in this work. Crucial point is that not the original signals, but chains of their absolute values are compared.
Simultaneous measurements of the rotations in seismic field with the use of completely different instruments – here AFORS-1 which is the Sagnac interferometer and the micro-array of TAPSs allow for comparison of the used equipment. In this work, such comparison revealed that signals differ significantly, to the degree which complicates analysis. From both kind of instruments, rotations are obtained in the same time-periods, but their plots differ. These differences are attributed partly to difference in instruments spectra and partly to disturbances in the signals, of technical provenience. Nevertheless, an analysis using the moving averages of absolute signals values, and consequently also coefficients of correlation between these averages confirmed common roots of these recorded signals, despite all their imperfections.
Research on seismic rotational effects, especially in buildings and other large constructions is widely recognized as very important and therefore these studies flourish. On the other hand, rotational components in the seismic field are also studied in various ways, but even existence of these components still evoke controversy. Presence of these components in seismograms, especially in their initial part which, according to classical theory of elasticity, contain only compressional waves, is explained in various ways. Authors believe that no one contemporary explanation is complete and proven, but this not preclude usefulness of further studies.
This work was done in 2013-2014 under the financial support of the Polish Ministry of Science and Higher Education under Key Project POIG.01.03.01-14-016/08 “New photonic materials and their advanced application”, the MUT statutory activity PBS-850 and partially the the Polish National Centre for Research and Development under contract No PBS1/B3/7/2012.