Macrospatial Correlation Model of Seismic Ground Motions

Transcription

1 Macrospatial Correlation Model of Seismic Ground Motions Min Wang a and Tsuyoshi Takada b It is very important to estimate a macrospatial correlation of seismic ground motion intensities for earthquake damage predictions, building portfolio analyses etc., whereby damage in different locations has to be taken into account simultaneously. This study focuses on spatial correlation of the residual value between an observed and a predicted ground motion intensity, which is estimated by an empirical mean attenuation relationship. The residual value is modeled in such a way that the joint probability density function PDF of seismic ground-motion intensity can be characterized by the spatial correlation model as well as an empirical mean attenuation relationship, assuming that it constitutes a homogeneous two-dimensional stochastic field. Using the dense observation data of earthquakes that occurred in Japan and Taiwan in recent years, the macrospatial correlation model is proposed and the assumption of homogeneity is verified in this paper. DOI: / INTRODUCTION Prediction of strong ground motion in a wide area has played a very important role in earthquake disaster prevention and damage mitigation, as well as in seismic design and assessment of infrastructures spatially spread. Management of widely located building assets, so-called portfolio analysis, has become popular in the field of earthquake risk management in recent years, whereby simultaneous damage of assets in different locations is of major concern Achiwa et al Simultaneity of the ground motion intensity in different sites should be properly treated, taking into account the damage of building portfolio a group of buildings, infrastructures spatially spread, etc. Mean attenuation characteristics of the ground motion induced from a seismic source can be conveniently predicted by using the mean attenuation relationship. The spatial correlation structure of the uncertainty of the empirical ground motion attenuation relationships also has to be adequately modeled, utilizing the relationships, i.e., their mean characteristic. Therefore, the ground motion intensity can be more accurately predicted with the past empirical attenuation relationships and the macrospatial correlation model proposed in this study. Either perfect correlation or perfect noncorrelation of the uncertainty of attenuation relationships has been intuitively assumed in the past researches Ishikawa et al. 2000, Fukushima and Yashiro a Ph.D. student, Graduate School of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan, b Professor, Graduate School of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan, Earthquake Spectra, Volume 21, No. 4, pages , November 2005; 2005, Earthquake Engineering Research Institute

2 1138 M. WANG AND T.TAKADA However, this correlation is, in fact, dependent upon many factors, such as relative distance between two different sites, ground conditions of the sites, directivity of propagating waves, etc. No past systematic research on quantifying the spatial correlation structures of ground motion intensity has been reported so far. This study focuses on the spatial correlation structure of the residual value uncertainty from its mean value. To model the correlation structure, dense observation of earthquake records is needed. In this paper, high-density earthquake records of five earthquakes based on K-NET and KiK-NET deployed by National Institute for Earth Science and Disaster Prevention NIED in Japan, and the data of 1999 Chi-Chi earthquake, which has been published on CD-ROM Lee et al. 2001, are fully utilized to quantify the spatial correlation characteristics of the ground motions. STOCHASTIC MODELING OF GROUND MOTION INTENSITY Ground motion intensity can generally be expressed in terms of the trend component Ā x and the residual component x from the mean, which represents uncertainty of the relationships Kambara and Takada 1996, as in Equation 1: A x = x Ā x 1 whereby these quantities are the function of a spatial coordinate x. A x can be a measure representing ground motion intensities such as peak ground acceleration PGA, peak ground velocity PGV, and spectral acceleration. PGV, which is closely related to building damage, is adopted in this study. The uncertainty x, comprising the interand intra-earthquake uncertainty, is usually treated as the lognormal random variable. Since we consider only six earthquakes in this paper, the uncertainty associated with inter-earthquake cannot be sufficiently dealt with. Only the modeling on the uncertainty associated with intra-earthquake is comprehensively discussed in this study. Taking logarithm of both sides of Equation 1, the logarithmic deviation L x can be L x =ln x 2 In this study, the logarithmic deviation L x is the difference between the log of the observation and that of the value predicted from the mean attenuation relationship. Assuming that L x is a homogeneous two-dimensional stochastic field, it then follows that its auto-covariance function C LL is described as in the following: µ L =E L x 3 C LL x 2 x 1 =E L x 1 µ L L x 2 µ L 4 where E. is an expectation, x 2 -x 1 is a separation distance between the two locations x 1 and x 2, and µ L is the mean value of L x, and it is not a function of location x due to the homogeneity assumption of the field. The assumption of homogeneity is not usually satisfied because of physical complexity such as inhomogeneous ground conditions, directivity effect of seismic wave propa-

3 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1139 Table 1. Profile of earthquakes Shin et al. 2000, JMA, Kanamori 1997, Fukushima et al No. Earthquake Date M j Depth km Source Fault Type No. of Sites Used 1 Chi-Chi 21 Sept Interplate Tottori-ken Seibu 6 Oct Crustal Geiyo 24 Mar Intraplate Miyagi-ken-oki 26 May Intraplate Miyagi-ken 26 July Crustal 246 Hokubu 6 Tokachi-oki 26 Sept Interplate 287 gation, etc. For simpler treatment of ground motion, however, only the mean attenuation relationship is considered to take into account the inhomogeneity of ground motions, and then L x is assumed to constitute a 2-D homogeneous stochastic field. The above homogeneity assumption is examined later in this paper. DATA ANALYSIS OF PAST ATTENUATION RELATIONSHIPS RECORDED GROUND MOTIONS Selected are the 1999 Chi-Chi earthquake and five recent earthquakes in Japan that occurred from 2000 to 2003, with the moment magnitude M w ranging from 6.2 to 8.0. The moment magnitude of the 1999 Chi-Chi earthquake is 7.6 Lee et al The information on the hypocenter and the corresponding fault model used for the attenuation relationships are listed on Tables 1 and 2 in which M j is JMA Japan Meteorological Agency magnitude. The data of the hypocenter in Japan adopted from the report made by JMA, and the fault model of earthquakes in Japan are taken from the Headquarters for Earthquake Research Promotion HERP or Geographical Survey Institute GSI, while the source parameters of the Chi-Chi earthquake are obtained from past research Shin et al. 2000, Kikuchi and Yamanaka The epicenters of the six earthquakes are shown in Figures 1 and 2. Table 2. Fault parameters of earthquakes Kikuchi 1999, HERP, GSI No. Earthquake Latitude Longitude Depth km Strike Dip Width km Length km M w 1 Chi-Chi Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki

4 1140 M. WANG AND T.TAKADA Figure 1. Locations of earthquake epicenters in Japan. All the records of the Japan earthquakes are taken from K-NET and KiK-NET accelerograms. And only those observations whose minimum distance to the fault plane is within 300 km are adopted in this paper. Acceleration time histories are obtained after the records are processed with a band pass filter of Hz and with correction of the baseline drift Midorikawa and Ohtake 2002, Boore And then PGV can be obtained by integrating the acceleration time histories. For the case of the 1999 Chi-Chi earthquake, the records categorized into D quality among all recorded data, and the data recorded with an old-type instrument A800 are omitted. The study then uses 398 observation points in Taiwan, as plotted in Figure 2. PGV is taken as a ground motion intensity measure to construct the macrospatial correlation model. PAST ATTENUATION RELATIONSHIPS FOR PGV The trend component in the stochastic model of ground motion is usually characterized by the mean attenuation relationships. Two well-known attenuation relationships for PGV used in Japan are adopted herein, as described in the following. Then the logarithmic deviation can be obtained from the data observed in the six earthquakes and the value predicted by these two attenuation relationships, respectively.

5 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1141 Figure 2. PGA distribution of the 1999 Chi-Chi earthquake. 1 the mean Annaka relationship Annaka et al log 10 Vel = 0.725M j H log 10 D e 0.653Mj where Vel is a PGV in cm/s, M j is JMA magnitude, D is a minimum distance to the fault plane in km, and H is a source depth in km. The above law is based on the earthquake records observed in Japan on relatively stiff ground whose shear-wave velocity ranges from 300 to 600 m/s. Although the relationship was proposed for the velocity, its uncertainty is approximated from that of the response spectra at natural period of 0.04 seconds. Annaka also proposed e and o, the uncertainties associated with interand intra-earthquakes, to be 0.37 and 0.51 in natural logarithmic standard deviation, respectively. 2 the mean Midorikawa-Ohtake relationship Midorikawa and Ohtake 2002 log 10 Vel = c log 10 D M w 0.002D H 30 km 6a 5

6 1142 M. WANG AND T.TAKADA Figure 3. Data fitting of past attenuation relationships: a the Chi-Chi earthquake, b the Tottori-ken Seibu earthquake, c the Geiyo earthquake, d the Miyagi-ken-oki earthquake, e the Miyagi-ken Hokubu earthquake, and f the Tokachi-oki earthquake. log 10 Vel = c log H M w 1.6 log 10 D M w 0.002D H 30 km 6b where c = 0.65M w H + d i s i and M w is a moment magnitude. The parameters Vel, D, and H are the same as those of

7 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1143 Table 3. Logarithmic deviation with the Annaka relationship o =0.51, e =0.37 Earthquake Chi-Chi Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki logarithmic µ L deviation L L the Annaka relationship. The Midorikawa-Ohtake relationship is based on earthquake records taken within Japan. The PGV values are defined on stiff ground whose shearwave velocity is around 600 m/ s. This attenuation relationship considers the effects of the fault type crustal, intraplate, and interplate and the source depth including shallow earthquakes and deep earthquakes. Equation 7 shows the effect of the fault type, in which variable d = 0.00, 0.05, 0.15 for crustal, interplate, and intraplate, respectively, dummy variable s is given by category of crustal, interplate, and intraplate. The uncertainties e and o, associated with inter- and intra-earthquakes, are 0.37 and 0.55 in natural logarithmic standard deviation, respectively. SITE EFFECT The above two attenuation relationships are proposed for PGV defined on the stiff ground whose average shear-wave velocity is equivalent to 600 m/s, while the PGV observed in those earthquakes are on the ground surface. This site effect is accounted for this study by determining average amplifications for motions AVR. Previous studies Borcherdt and Gibbs 1976, Joyner et al. 1981, Borcherdt et al show that this amplification factor is strongly correlated to the shear-wave velocity near the ground surface, especially the shear-wave velocity averaged over the upper 30 m V30 in m/s that is used as the key variable to represent site effects. Based on the data in the Kanto area in Japan, the empirical relationship for the site effect in terms of V30 was proposed Midorikawa et al as in Equation 8: log 10 AVR = log 10 V30 The uncertainty associated with the site effect in Equation 8 is 0.37 in natural logarithmic standard deviation. In this study, the shear-wave velocities of the sites in Japan are obtained from the K-NET and KiK-NET. The V30s are calculated according to the Si- Midorikawa rule Si and Midorikawa Since the V30s of the sites in the Chi-Chi earthquake are lacking, the average amplification factor of motion AVR is set to 2.0 for 8 Table 4. Logarithmic deviation with the Midorikawa-Ohtake relationship o =0.55, e =0.37 Earthquake Chi-Chi Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki logarithmic µ L deviation L L

8 1144 M. WANG AND T.TAKADA Figure 4. Homogeneity of L x of the Chi-Chi earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship. all sites in Taiwan Takada and Shimomura 2003, Midorikawa 1991, Midorikawa et al DATA ANALYSIS Figure 3 shows observed data along with the predicted results from the above two PGV attenuation relationships. The PGV given in the figures is the maximum value of EW and NS components of PGV. It is found that the attenuation relationships can capture the mean tendency of the observed data in a wider range. Comparing with the Annaka relationship, the Midorikawa-Ohtake relationship gives better fitness with the observed data. This can be explained from the fact that the latter attenuation relationship takes into account the effect of fault type and source depth. Significant scatter around the mean relationships, however, can also be observed from all the relationships. The uncertainties intra-earthquake L of the attenuation relationships are computed in Tables 3 and 4 in a logarithmic standard deviation from 0.51 to 0.64 for the Annaka relationship, from 0.49 to 0.57 for the Midorikawa-Ohtake relationship, which cannot be ignored and should hopefully be reduced from the engineering viewpoint. The other uncertainty associated with the inter-earthquake can be understood as the scatter of the mean attenuation relationship for different earthquakes. Although the number of earthquakes considered here is very small six, the inter-earthquake uncertainty can be computed from the mean values µ L of the L x, and it becomes 0.37 and 0.34 in a logarithmic standard deviation, respectively, for the Annaka and the Midorikawa-Ohtake mean attenuation relationships. The inter-earthquake uncertainties calculated herein are close to the values, 0.37, originally proposed for these two attenuation relationships. HOMOGENEITY OF LOGARITHMIC DEVIATION In the above section, it has been assumed that the logarithmic deviation L x is a homogeneous two-dimensional stochastic field. It is necessary to examine the homogeneity of L x before evaluating the macrospatial correlation model. Figures 4 9 show

9 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1145 Figure 5. Homogeneity of L x of the Tottori-ken Seibu earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship. distance-dependency of L x, i.e., change of L x to the minimum distance to the fault plane D. Note that L x in the figures is corrected with the mean logarithmic deviation µ L, that is, L x is a random variable with zero mean and standard deviation L. The moving average of L x is also shown in the figures. Its window width is 20 km. The mean value ML and standard deviation of L x within the window are also calculated, and the ranges of the ML with plus-and-minus-one-, i.e. ML±, are shown in the figures. The curves of moving average of L x show the dependency of L x with respect to the fault distance D, which means that the goodness of fit has some dependency on the fault distance. It is observed from the figures that this distance-dependency of the Midorikawa-Ohtake relationship is smaller than that of the Annaka relationship. On the other hand, from the curves of ML±, the moving average of standard deviation does not obviously show the distance-dependency. The value of moving average of standard deviation is about 0.5 with small variation. As a result, the assumption that the logarithmic deviation L x constitutes a homogeneous two-dimensional stochastic Figure 6. Homogeneity of L x of the Geiyo earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship.

10 1146 M. WANG AND T.TAKADA Figure 7. Homogeneity of L x of the Miyagi-ken-oki earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship. field can be approximately satisfied, which can lead to simpler treatment of ground motion with the Midorikawa-Ohtake attenuation relationship. PROPOSAL OF NEW MACROSPATIAL CORRELATION MODEL MACROSPATIAL CORRELATION MODEL The data observed from the 1999 Chi-Chi, Taiwan, earthquake and the five recent earthquakes in Japan were first grouped into several bins with the same separation distance h between two sites so that the separation distance in the same bin is within h± h/2 shown in Figure 10, in which only the Chi-Chi earthquake in Taiwan and the Tottori-ken Seibu earthquake in Japan are given as illustration. Equations 3 and 4 can then be rewritten as Equations 9 and 10, respectively, by discrete expression: Figure 8. Homogeneity of L x of the Miyagi-ken Hokubu earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship.

11 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1147 Figure 9. Homogeneity of L x of the Tokachi-oki earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship. µ L = 1 N all N all L x i i=1 9 C LL h = 1 N h L x N h ai µ L L x bi µ L i=1 10 where N all is the total number of observation sites, N h is the number of pairs of sites x a,x b that meet the condition h h/2 x a x b h+ h/2, the interval h is set to 4 km for all earthquakes in Japan in order to keep accuracy in the statistical analysis. If a shorter interval is taken, the number of available pairs N h becomes smaller, which may lead to larger statistical error. Therefore, in the analysis, a relatively large number of data, say, 100, are needed to obtain reliable results. The interval h is also set to 4km for the Chi-Chi earthquake for the comparison with the same interval, although the N h amounts to 98 at h=1 km when h is set to 2km. Figure 10. Histogram of separation distance h a the Chi-Chi earthquake, and b the Tottoriken Seibu earthquake.

12 1148 M. WANG AND T.TAKADA Table 5. Correlation length b km with the Annaka relationship Chi-Chi Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Due to the availability of dense observation in Japan, N h is more than 100 when h is larger than 14 km, while in the Chi-Chi earthquake, N h amounts to 318 even for h=2 km in Figure 10. By the statistical analysis above, the auto-covariance function C LL can be estimated. Normalized auto-covariance function R LL can be obtained by normalizing C LL with the variation of L x. R LL h = C LL h L 2 Now the macrospatial correlation model can be built up by modeling the discrete values of the normalized auto-covariance function R LL with an exponential decay function as R LL h = exp h/b where h is a separation distance between two observations and b is called a correlation length, that is, it can characterize the degree of correlation of ground motion intensities between two locations. This means that when the separation distance h between two sites equals b, R LL becomes 1/e It can be seen from Equation 12 that this exponential function satisfies two essential conditions: R LL 0 =1, and R LL =0. Furthermore, the model in the exponential function is in such a simple form, only with one parameter b that will greatly please the engineers in their analysis. This correlation model in terms of exponential function can be used for other ground motion intensity such as PGA Takada and Shimomura 2004, although it is being proposed for PGV in this paper. RESULTS OF ANALYSIS The data observed from the above six earthquakes are fully used to build up the macrospatial correlation model. Through statistical analyses, the discrete values of autocovariance function C LL are calculated from Equation 10. An exponential decay function is fitted to the correlation model as in Equation 12. The value of b, the so-called correlation length, can be obtained, and the results are listed in Tables 5 and 6 for the two Table 6. Correlation length b km with the Midorikawa-Ohtake relationship Chi-Chi Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki

13 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1149 Figure 11. Normalized auto-covariance function of the Chi-Chi earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship. attenuation relationships Equations 5 and 6. Figures show curves of normalized auto-covariance functions R LL evaluated. DISCUSSION It can be observed from Figures that the values of the correlation length b for the two attenuation relationships are of the same order of magnitude, ranging from 21 to 48 km. This implies that the spatial correlation characteristic of seismic ground motions has approximately similar tendency regardless of the mean attenuation relationships used as well as earthquake types or regions in Japan. Note that in Figure 12, there is an unreasonable data plot in the closer range of separation distance of 2km, which shows negative correlation. This is mainly because the number of data pairs in this range is only eight, which is quite small to determine the correlation, compared with the number of pairs from the Chi-Chi earthquake, 318 pairs. Further examination of the data pairs shows that the ground conditions of relevant sites giving the data plot are quite different. Some of the sites are made of soft soil, while some are made of granite rock. Figure 12. Normalized auto-covariance function of the Tottori-ken Seibu earthquake for a the Annaka Relationship, and b the Midorikawa-Ohtake relationship.

14 1150 M. WANG AND T.TAKADA Figure 13. Normalized auto-covariance function of the Geiyo earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship. These detail and precise ground conditions cannot be reflected in the past empirical attenuation relationships. These data are ignored in the evaluation of the correlation model. Figures show the exponential decay function adopted in Equation 12 can basically fit the data well except in the range h 10 km. The average separation distance between the observatories of K-NET is about 25 km, which leads to fewer data pairs whose separation distance is shorter than 10 km for statistical analysis. The site effect may be dominant on the seismic ground motion in the range of shorter separation distance that has been explained above. Other factors can also affect the variation of the correlation length. There are many attenuation relationships proposed so far. Although they can easily predict the ground motion, the uncertainties associated with them range from 0.4 to 0.7 in natural logarithmic standard deviation Abrahamason and Shedlock To reduce this uncertainty, more physics-based attenuation relationship is necessary. In other Figure 14. Normalized auto-covariance function of the Miyagi-ken-oki earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship.

15 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1151 Figure 15. Normalized auto-covariance function of the Miyagi-ken Hokubu earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship. words, reliable and predictable source, path, and site parameters that correlate well with ground motions can be more effective than using more complicated functional forms or regression procedures using only magnitude, distance, and crude site characterization Douglas and Smit This study proposes a spatial correlation model of the uncertainty inherent to the earthquake as a method to improve the prediction. The model is directly dependent on the homogeneity of the logarithmic deviation. The factors, such as source characteristic, wave propagation, and site effect, affecting the uncertainty of attenuation relationship will have effects on this model. The fault type associated with the source characteristic is taken into account in terms of parameter d in Equation 7 in Midorikawa-Ohtake relationship. Although the difference of the logarithmic deviation L x corrected with µ L between two relationships is not obvious in Figures 4, 5, and 8, in which the source depths are shallower than 30 km, the variation of the correlation length in the Midorikawa-Ohtake relationship is smaller than that in the Annaka relationship. Although the effect of rupture directivity on ground mo- Figure 16. Normalized auto-covariance function of the Tokachi-oki earthquake for a the Annaka relationship, and b the Midorikawa-Ohtake relationship.

16 1152 M. WANG AND T.TAKADA tions was reported in the literature, the distance range of this effect is about a quarter of the rupture length away from the rupture, about 20 km Somerville et al. 1997, Ohno et al. 1998, Midorikawa and Ohtake Considering the six earthquakes in this study, there are little data observed within this range except from the Chi-Chi earthquake from Figure 4. As a result, the directivity effect can be ignored in the macrospatial correlation analysis. As is commonly known, the Moho is the boundary between the crust and the mantle in the earth where seismic waves change velocity. When an earthquake occurs beneath this discontinuity, the wave propagation is greatly changed. Equation 6b in the Midorikawa-Ohtake relationship takes into account this effect with coefficient of 1.6 when H 30 km assuming the depth of Moho is 30 km rather than 1.0 in Equation 6a, while this propagation effect is not considered in the Annaka relationship. Figures 6, 7, and 9 show this difference in which the source depths are deeper than 30 km. The logarithmic deviation from the Midorikawa-Ohtake relationship shows more homogeneity than that from the Annaka relationship, which shows obviously negative correlation with respect to distance. Therefore, the correlation lengths with the Midorikawa-Ohtake relationship are smaller than those with the Annaka relationship, as it is shown that the correlation length is overestimated by the Annaka attenuation relationship. Even though these effects are considered in the Midorikawa-Ohtake relationship, the correlation length b calculated comprises uncertainties, such as inter-earthquake uncertainty and site effect. Because the PGV of the two attenuation relationships is defined on the engineering bedrock, the PGV obtained from the acceleration time history on the ground surface has to be transferred to that on the engineering bedrock by the average amplification for motion AVR in Equation 8. The uncertainty of site effect associated with Equation 8 is suggested as These uncertainties can propagate to the correlation model. The inter-earthquake uncertainty is magnitude-dependence and usually relates with the different propagation path in the difference area. It is suggested as 0.37 in natural logarithmic standard deviation in two attenuation relationships. Due to the tectonic structures of the Japan islands, the abnormal Q structure in the subduction zone of northeast Japan contributes to the anomalous seismic ground motion, and the additional correction term, the distance between the observatory and the trench axis, is suggested for the correction of the attenuation relationship Morikawa et al The interearthquake uncertainty also contributes to the slight difference of the correlation models in northeast Japan from those in southwest Japan. As discussed above, there are many factors affecting the correlation model. The source characteristic, wave propagation, and site effect are dominant. The contribution of inter-earthquake uncertainty to the variation of the correlation model is also apparent. In the probabilistic seismic hazard analysis, the uncertainty of the prediction with the attenuation relationship is still great. Incorporating this correlation model associated with those physical phenomena inherent to earthquakes not described in the attenuation relationship, the prediction will be greatly improved. Although the attenuation relationship can be improved by introducing new parameters, more complicated attenuation relationships will increase the difficulty in the usage, and additional information is necessary, such as the correction term in northeast Japan that is difficult to obtain. From the

17 MACROSPATIAL CORRELATION MODEL OF SEISMIC GROUND MOTIONS 1153 analysis, the correlation length calculated with the Annaka relationship in Table 5 is overestimated compared with the results in Table 6, because the logarithmic deviation L x is more homogeneous with the Midorikawa-Ohtake relationship. In this study, the macrospatial correlation model associated with the Midorikawa-Ohtake attenuation relationship is proposed. The parameters in this relationship can also be easily found from the web sites of the JMA, K-NET, KiK-NET, HERP, and GSI. The simplicity of the correlation model along with the empirical attenuation relationship will be utilized with great ease in the engineering. CONCLUSIONS AND FURTHER STUDIES Focusing on the logarithmic deviation between observed and previously derived mean attenuation relationship, the macrospatial correlation of residual value is proposed using records observed in the recent earthquakes in Japan as well as in the 1999 Chi-Chi earthquake in Taiwan. From this study, the assumption that the logarithmic deviation L x constitutes a homogeneous two-dimensional stochastic field can be approximately satisfied with the Midoricawa-Ohtake attenuation relationship. The proposed macrospatial correlation model with a one-parameter exponential decay function can basically meet the data well. This simple correlation with the empirical attenuation relationship can be applied to many fields with great ease, especially from the view of engineering. The correlation length b controlling the spatial correlation of ground motion intensity can then be calculated and falls within the range of 20 to 40 km, although the correlation of ground motion intensity is highly dependent on many factors, such as wave propagation, ground conditions, etc. The uncertainty of the correlation length can be attributed to these factors. Due to the complexity of the effect of these factors, it is difficult to quantify the uncertainty for each factor, and Monte Carlo simulation is expected to evaluate the uncertainty of the macrospatial correlation model in the future. The proposed model along with the mean attenuation relationship proposed in previous studies can be effectively utilized with great ease in various earthquake engineering fields. The spatial distribution of ground motion intensity, which is considered to be induced from a particular earthquake source, can be easily evaluated with emphasis on simultaneity of ground motion intensities at two different sites. The macrospatial correlation model can easily be incorporated into the probabilistic seismic hazard assessment Fukushima and Yashiro 2004, Wesson and Perkins 2001 for multiple sites. Another possible application of this model is stochastic prediction of ground motion at an unobserved site using the records observed nearby the site since the joint probability density function of ground motion intensities at two arbitrary different sites has already been obtained by using this macrocorrelation model Shimomura and Takada 2004, Field et al These interesting applications will be studied further. ACKNOWLEDGMENTS Substantial technical support from Mr. Tetsuto Shimomura, a former master s student at the University of Tokyo, is deeply appreciated. The authors would also like to ac-

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