We calculated the correlation coefficients between records of central station COO of the array and the stations in the inner and middle rings. The procedure was the following. Let us consider two records of the same event recorded by two different stations. Because of the triggering mode in which all accelerometers operate the time origins of these records are generally different. That is why we used the origin of the time series equal to the time of the latest triggering among two stations. Moreover, because of the delay in the same wave arrivals to the neighboring stations due to their finite propagation velocity we successively shifted two series one with respect to another to better fit the waveforms. The shift was within the limits of 0.5s for “central station—inner ring” pairs and 1.0s for “central station—middle ring” ones. The correlation coefficients were calculated for each individual shift, and the maximum value was taken as a result.
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Note that in all following calculations the full records were used: the separate analysis of individual wavetrains in corresponding time windows has not been performed.
For a single event we obtain at most 12 values of the correlation coefficients on each ring, this number depending on the total number of stations triggered. We averaged all these values thus obtaining the only one coefficient for an individual event, separately for inner and middle rings.
The Fig. 1 shows the average correlation coefficient rav as a function of the local magnitude ML for an EW horizontal component of accelerations (all data shown below are calculated for this component:) . we see that almost linear dependence exists. The correlation coefficients increase rapidly as magnitude increases, reaching the values of 0.8 for magnitudes about 7.
The linear regression is also shown in Fig.1 satisfying the equation rav =−0.205+0. 124 ML, the standard deviation of the coefficient at ML being σcoeff=0.0167.
We remind that the distance between stations for which the Fig. 1 was calculated is 200m. The Fig. 2 shows the correlation coefficients vs magnitude for the “COO—middle ring” pairs with a distance between stations equal to 1000m. It can be seen that in this case there is no systematic distribution of points with respect to magnitude, and the absolute values of the coefficients rav are approximately half as large as in the previous case. The conclusion is that neither high coherency nor its dependence on magnitude are observed as distance between stations increases up to 1km. We mentioned before that a number of values of correlation coefficients have been calculated for a single event on each ring, depending on the number of triggered stations. The Figs. 1 and 2 show the behavior of their average values. For each event we also calculated the ratio rmax/rmin of maximum and minimum values in
Fig. 1. The dependence of the average correlation coefficient on the local magnitude for “central station—inner ring” pairs. The straight line is a linear regression.
Fig. 2. The dependence of the average correlation coefficient on the local magnitude for “central station—middle ring” pairs.
Fig.3. The dependence of the rmax/rmin ratio on magnitude, where rmax and rmin are the maximum and the minimum values, respectively, of the correlation coefficients calculated for all stations on the ring triggered by a single event. The “central station—inner ring” pairs have been used.
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this range and looked how it depends on magnitude. This ratio describes in some way the dispersion of the correlation coefficient values for a single event caused by different local effects. The data are shown in Fig. 3. At low magnitudes (up to 5) the scattering of data is rather high, the ratios rmax/rmin varying from 1.5 to 4, approximately. But at larger magnitudes (above 6) there is far less scattering, scattering, and the ratios rmax/rmin are in the range of 1.1–1.6. We also investagated the absolute differences between accelerations in the “central station—inner ring” pairs. The average difference can be calculated as follows. At each moment the difference between the corresponding samples of the central station record and the station in the ring is taken, the calculation being continued up to physical end of the shortest trace. Then the maximum difference is selected, divided by the maximum acceleration at the central station for normal ization. The last step is averaging all obtained single values of MD/COO (maximum difference/maximum value at the central station) ratios over the whole ensemble of stations triggered in the ring, giving an AMD/COO (average max difference/COO) .
The data obtained are shown in Fig. 4. The pattern similar to the previous one can be observed: the dispersion of the data points tends to decrease with increasing magnitude. At the same time the tendency is evident to decrease of the
normalized AMD.
The previously given results clearly prove that the spatial coherency of the ground motions strongly depends on earthquake magnitude, so that it increases as magnitude increases.
We tried to arrange data versus hypocentral distance ∆H of the events in order to look for correlation. The example is given in Fig. 5 where rav as in Fig. 1 are plotted vs ∆H. A clear dependence like in Fig. 1 is not observed here because of the greater scattering of data points.
To account for the observed statistical dependence of the spatial coherency of strong ground motions on magnitude let us consider the relation
Fig. 4. The normalized maximum differences between records of central station and inner ring, averaged over all triggered pairs, as a function of magnitude.
Fig. 5. The dependence of the average correlation coefficient on the hypocentral distance for the “central station—inner ring” pairs
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between the predominant period Tmax of the seismograms and magnitude. The predominant periods were calculated from the Fourier acceleration spectra at every single station. In Fig. 6 the Tmax’s are plotted vs ML for the central station. The data distribution shows that the systematic relationship exists between these quantities consisting in the obvious increase of Tmax with magnitude. Note that it is typical with strong motion data sets that the correlation exists between magnitude and distance because the large magnitude events trigger the stations at larger distances than smaller events [2]. It can be naturally supposed therefore that the result obtained in Fig. 6 is simply explained by this fact: larger magnitude events are more distant and have therefore the longer periods because of the trivial attenuation of short periods. To avoid such misinterpretation we plotted Tmax together with hypocentral distance ∆H. The result is given in Fig.7. The correlation is much less clear than in Fig.6.
DISCUSSION
It is evident that the mechanism of the strong motions coherency dependence on magnitude lies in the fact that the increase of magnitude is accompanied by the substantial increase of the predominant period (and wavelength) of seismic field. Hence, the influence of the local inhomogeneities ties of the medium within an array becomes weaker, which results in the increasing coherency.
Two factors determine the predominant period as Figs. 6 and 7 show: the local magnitude and the hypocentral distance. The effect of the first one is connected with the focal mechanism of earthquake and is much more significant (Fig. 6). There is also some weaker influence of the second factor acting through the attenuation of high frequencies with distance. Thus, the spatial coherency is more clearly related with magnitude than with hypocentral distance, as we see reflected in Figs. 1 and 5.
ACKNOWLEDGMENTS
The SMART 1 data are made available by the Seismographic Station of the University of California
Fig. 6. The predominant period of the seismograms at central station as a function of magnitude.
Fig. 7. The predominant period of the seismograms at central station as a function of hypocentral distance.
at Berkeley and the Institute of Earth Sciences of the Academia Sinica in Taipei. I would like to thank Professor B.A. Bolt for delivering data and the benevolent correspondence, and Professor A.V.Nikolaev for his numerous discussions and support of this work.
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REFERENCES
1. Abrahamson, N.A., Bolt, B.A., Darragh, R.B., Penzien, J. and Tsai, Y.B. The SMART 1 accelerograph array (1980– 1987): a review, Earthquake Spectra, Vol. 3, pp. 263–287, 1987.
2. Abrahamson, N.A. Statistical properties of peak ground accelerations recorded by the SMART 1 Array, Bull. Seism. Soc. Am., Vol. 78, pp. 26–41, 1988.
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SECTION 3: