For the whole time series of observations let us introduce for example a shifting interval S equal to one year and divide the time series into one-year intervals D in which the values of maximum earthquakes are determined. We obtain only one Gumbel approximation. If the same (one-year) shifting interval S is applied to the time series which is divided into two-year intervals D, in which the values of maximum earthquakes are determined, then we obtain two Gumbel
approximations. Likewise, for the one-year shifting interval S applied to the time series which is divided into ten-year intervals D gives us ten different Gumbel approximations.
For a better explanation of this approach the following example is demonstrated. Using the catalogue of
Italian earthquakes [2], we selected the subcatalogue for the Friuli region (northern Italy). In this catalogue the earthquake size is given in epicentral intensity of MCS. The Friuli region under study was delineated by geographic coordinates from 45°50’N to 46°36’N and from 12°50’E to 13°50’E . The subcatalog of observed earthquakes contains 1764 events from the period of 1700–1980 with the maximum observed earthquake of 9.5°MCS. In our calculations we applied only one combination of shifting interval S equal to 1 year and interval D equal to 30 years. Figure 2 shows the distribution of all approximations (23 cases) for which the 3rd Gumbel extreme value distribution has a convergent character. The fact that for 7 cases the statistical process had not a convergent solution was rather surprising and has to be explained in the near
Figure 2.
future. This finding is very important from the point of view of practical applications. It gives evidence that our idea concerning the “representativeness” of the 3rd Gumbel approximation need not be necessarily always valid.
The values of the maximum possible earthquakes (MPE) of all 23 convergent Gumbel approximations were analyzed for three different return periods of 120, 1290 and 15500 years and then they were compared with their asymptotes. Figure 3 shows the changes of the MPE values in dependence on the thirty different positions of the beginnings of intervals D with respect to their first positions. We can see that for return period of 120 years minimum values of MPE were obtained for cases in which the beginning of interval
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D is situated approximately in the centre of the original interval D, i.e. for the total shift equal to 10≈18 interval S. This conclusion is not very important because the original beginning of the earthquake time series could be shifted and then we can obtain quite easily an opposite result. But what is extremely important is the fact that for the same cases we obtain quite opposite results for higher return periods, and consequently, for the asymptotes too, e.g. for these cases the maximum values of the MPE are determined. An explanation of this feature does not seem to be very simple and probably special tests have to be made in order to clarify it.
Figure 3.
For each set of the MPE values obtained for selected return periods the “mean value of the maximum possible earthquake MMPE” and its “standard deviation MSD” were determined. These quantities obtained for the Friuli
seismogenic region by the application of the 3rd Gumbel extreme value distribution are drawn in Figure 4. Such a chart informs us immediately about the representativeness of the 3rd Gumbel approximation for the given subcatalogue of earthquakes. We can see that the best approximations and thus the MMPE value with the highest degree of a
representativeness, because of the lowest values of the MSD, belongs to the return periods which are close to the middle of the observation period; in our case about 350 years. For higher return periods the standard deviations increase and the degree of the representativeness of the resulting MMPE values becomes slightly lower. Numerically, the MSD for the return period of 350 years is around 0.5% of the MMPE, but for the return period of 15500 years it is as much as 6.5%, at-
Figure 4.
taining 9% for the asymptote of the MMPE value. We assume that the obtained accuracy of the MMPE can be accepted as a general degree of representativeness for the maximum possible earthquakes determined by the 3rd Gumbel
approximations.
CONCLUSION
The described statistical approach allows us to estimate the accuracy of the approximation obtained by the 3rd Gumbel distribution in a prediction of the maximum earthquake for given return periods of earthquake occurrences. These estimates do not only define the resulting predicted values but also give their possible variance. Such characteristics are quite important from the economic point of view, because for example, in tasks of the earthquake hazard assessments the determination of the maximum possible earthquake directly affects the total cost of seismic resistant structures. Such predicted values of maximum earthquakes also help in calculations of the seismic risk and can make a contribution in some questions of earthquake mitigation.
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REFERENCES
1. Gumbel E.J., 1954: Statistical Theory of Extreme Values and Some Practical Applications. Nat.Bureau of Standards, Appl.Math.Series 33, U.S.Govt.Print. Office.
2. Postpischl D., Ed., 1985: Catalogo dei terremoti italiani dal’anno 1000 al 1980. CNR, PFG, Quad. Ric. Scie. 114–2B, Graficoop, Bologna.
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Study of an Assessment for Site Effect of Seismic Strong Motion
E.Kuribayashi (*), T.Jiang (**), T.Niiro (*), H.Nagasaka (***), S.Kuroiwa (****), S.Nishioka (*)
(*) Dept. of Civil Engineering/Regional Planning, Toyohashi University of Technology, Toyohashi, Japan (**) Dept. of Geotechnical Engineering, Tonji University, Shanghai, The Peoples Republic of China (***) Kumagai-gumi Co., Ltd., Toyokawa, Japan
(****) Nagano Prefectural Government, Japan
ABSTRACT
Effects of sediment-filled valley on seismic ground motions have become of major interest in earthquake engineering throughout this decade. This paper presents interesting phenomena of both analytical and experimental approaches.
INTRODOCTION
Disasters caused by earthquakes are generally complicated. The earthquake damage strongly depends on the subsoil conditions and topography from the past experience of severe earthquake damage.
In order to prove the effects of topographical and geological conditions in behavior of ground motions, a strong motion observation network so called TASSEM, Toyohashi University of Technology Array System for Strong Earthquake Motions, has been developed since 1989. They are located around Toyohashi city, east part of Aichi prefecture, that is regarded as one of the most vulnerable areas to destructive earthquakes designated by many seismologists.
Several records of the strong motion observation have brought a reasonable results among analytical results using one and two dimensional analyses and consequences in microtremor and strong motion observations. From analytical results, amplification would not be influenced very much by the direction and angle of incident wave, but by the topographical conditions.
In addition, it is clear that Boundary Element Method is an effective tool to estimate the behavior of responses in symmetric valleys subjected to incidental waves.