Study on Macro-Spatial Correlation Characteristics of Seismic Ground Motion in Japan

Study on Macro-Spatial Correlation Characteristics of Seismic Ground Motion in Japan by Min WANG 26794 Submitted Aug. 13, 2004 Department of Architectural Engineering Graduate School of Engineering the University of Tokyo

Contents 1. INTRODUCTION 7 1.1. Background of Study 8 1.2. Objectives of Study 11 1.3. Previous Studies 12 1.3.1. Previous studies on macro-spatial correlation of ground motion 12 1.3.2. Previous studies on attenuation relation 14 1.4. Outline of Study 16 2. MACRO-SPATIAL CORRELATION MODEL 17 2.1. Attenuation Characteristics of Ground Motion 18 2.1.1. Past attenuation relations of PGA 18 2.1.2. Past attenuation relation of PGV 19 2.1.3. Past attenuation relations of SA 20 2.1.4. Site amplification factor 21 2.2. Separation of Mean and Residual Component of Ground Motion 22 2.3. Macro-Spatial Correlation Model 24 3. ANALYSIS OF SPATIAL CORRE-LATION OF GROUND MOTION 25 3.1. Database of the Seismic Ground Motion 26 3.2. Macro-Spatial Correlation Analysis for PGA 28 3.2.1. Data fitness for PGA 31 3.2.2.Homogeneity of logarithmic deviation 39 3.2.3.Proposal of new macro-spatial correlation model for PGA 45 3.2.4.Discussions 52 3.3. Macro-Spatial Correlation Analysis for PGV 53 3.3.1. Data fitness for PGV 53 3.3.2.Homogeneity of logarithmic deviation 61 3.3.3.Proposal of new macro-spatial correlation model for PGV 67 3.3.4.Discussions 73 3.4. Macro-Spatial Correlation Analysis for SA 74 3.4.1. Data fitness for SA 74 3.4.2.Homogeneity of logarithmic deviation 82 3.4.3.Proposal of new macro-spatial correlation model for SA 88 3.4.4.Discussions 94 3.5. Results from the Spatial Correlation Analysis 96 4. APPLICATIONS OF MACRO-SPATIAL CORRELATION MODEL 99 4.1. Real-Time Prediction on Seismic Ground Motion Intensity 100 4.2. Evaluation of Joint Exceedance Probability 104

4.3. Evaluation of Seismic Risk for Portfolio on Sites 108 5. CONCLUSIONS AND FUTURE STUDY 112 5.1. Conclusions 113 5.2. Future Research Needs 115 REFERENCE 116 ACKNOWLEDGEMENTS 121

Contents of Figures Figure 2-1. Illustration of Separation of Mean and Deviatoic Component of GM... 22 Figure 3-1. Locations of Earthquakes in Japan... 27 Figure 3-2. Spatial Distribution of PGA of the Tottori-ken Seibu Earthquake... 28 Figure 3-3. Spatial Distribution of PGA of the Geiyo Earthquake... 28 Figure 3-4. Spatial Distribution of PGA of the Miyagi-ken-oki Earthquake... 29 Figure 3-5. Spatial Distribution of PGA of the Miyagi-ken Hokubu Earthquake... 29 Figure 3-6. Spatial Distribution of PGA of the Tokachi-oki Earthquake... 30 Figure 3-7. Fitness of PGA of the Tottori-ken Seibu Earthquake... 32 Figure 3-8. Fitness of PGA of the Geiyo Earthquake... 33 Figure 3-9. Fitness of PGA of the Miyagi-ken-oki Earthquake... 34 Figure 3-10. Fitness of PGA of the Miyagi-ken Hokubu Earthquake... 35 Figure 3-11. Fitness of PGA of the Tokachi-oki Earthquake... 36 Figure 3-12. Homogeneity of L(x) of PGA of the Tottori-ken Seibu Earthquake... 40 Figure 3-13. Homogeneity of L(x) of PGA of the Geiyo Earthquake... 41 Figure 3-14. Homogeneity of L(x) of PGA of the Miyagi-ken-oki Earthquake... 42 Figure 3-15. Homogeneity of L(x) of PGA of the Miyagi-ken Hokubu Earthquake... 43 Figure 3-16. Homogeneity of L(x) of PGA of the Tokachi-oki Earthquake... 44 Figure 3-17. Histogram of Separation Distance h... 46 Figure 3-18. Normalized Auto-Covariance Function of PGA of the Tottori-ken Seibu Earthquake... 47 Figure 3-19. Normalized Auto-Covariance Function of PGA of the Geiyo Earthquake... 48 Figure 3-20. Normalized Auto-Covariance Function of PGA of the Miyagi-ken-oki Earthquake... 49 Figure 3-21. Normalized Auto-Covariance Function of PGA of the Miyagi-ken Hokubu Earthquake... 50 Figure 3-22. Normalized Auto-Covariance Function of PGA of the Tokachi-oki Earthquake... 51 Figure 3-23. Correlation Coefficient LD vs. Correlation Length b for PGA... 52 Figure 3-24. Fitness of PGV of the Tottori-ken Seibu Earthquake... 54 Figure 3-25. Fitness of PGV of the Geiyo Earthquake... 55 Figure 3-26. Fitness of PGV of the Miyagi-ken-oki Earthquake... 56 Figure 3-27. Fitness of PGV of the Miyagi-ken Hokubu Earthquake... 57 Figure 3-28. Fitness of PGV of the Tokachi-oki Earthquake... 58 Figure 3-29. Homogeneity of L(x) of PGV of the Tottori-ken Seibu Earthquake... 62 Figure 3-30. Homogeneity of L(x) of PGV of the Geiyo Earthquake... 63 Figure 3-31. Homogeneity of L(x) of PGV of the Miyagi-ken-oki Earthquake... 64 Figure 3-32. Homogeneity of L(x) of PGV of the Miyagi-ken Hokubu Earthquake... 65 Figure 3-33. Homogeneity of L(x) of PGV of the Tokachi-oki Earthquake... 66 Figure 3-34. Normalized Auto-Covariance Function of PGV of the Tottori-ken Seibu Earthquake... 68 Figure 3-35. Normalized Auto-Covariance Function of PGV of the Geiyo Earthquake... 69 Figure 3-36. Normalized Auto-Covariance Function of PGV of the Miyagi-ken-oki Earthquake... 70 Figure 3-37. Normalized Auto-Covariance Function of PGV of the Miyagi-ken Hokubu Earthquake... 71 Figure 3-38. Normalized Auto-Covariance Function of PGV of the Tokachi-oki Earthquake... 72 Figure 3-39. Correlation Coefficient LD vs. Correlation Length b for PGV... 73 Figure 3-40. Division of Period and Average SA (Damping = 5%)... 74 Figure 3-41. Fitness of SA of the Tottori-ken Seibu Earthquake... 75 Figure 3-42. Fitness of SA of the Geiyo Earthquake... 76

Figure 3-43. Fitness of SA of the Miyagi-ken-oki Earthquake... 77 Figure 3-44. Fitness of SA of the Miyagi-ken Hokubu Earthquake... 78 Figure 3-45. Fitness of SA of the Tokachi-oki Earthquake... 79 Figure 3-46. Homogeneity of L(x) of SA of the Tottori-ken Seibu Earthquake... 83 Figure 3-47. Homogeneity of L(x) of SA of the Geiyo Earthquake... 84 Figure 3-48. Homogeneity of L(x) of SA of the Miyagi-ken-oki Earthquake... 85 Figure 3-49. Homogeneity of L(x) of SA of the Miyagi-ken Hokubu Earthquake... 86 Figure 3-50. Homogeneity of L(x) of SA of the Tokachi-oki Earthquake... 87 Figure 3-51. Normalized Auto-Covariance Function of SA of the Tottori-ken Seibu Earthquake... 89 Figure 3-52. Normalized Auto-Covariance Function of SA of the Geiyo Earthquake... 90 Figure 3-53. Normalized Auto-Covariance Function of SA of the Miyagi-ken-oki Earthquake... 91 Figure 3-54. Normalized Auto-Covariance Function of SA of the Miyagi-ken Hokubu Earthquake... 92 Figure 3-55. Normalized Auto-Covariance Function of SA of the Tokachi-oki Earthquake... 93 Figure 3-56. Difference of b-values for PGA, PGV,and SA... 95 Figure 4-1. Comparison of Existing and Proposal Prediction of Strong Motion... 100 Figure 4-2. Prediction of Strong Motion for Unobserved Site... 101 Figure 4-3. An Example of Prection of PGA... 103 Figure 4-4. Network of Infrustructure... 104 Figure 4-5. Series and Parallel System Model... 105 Figure 4-6. Contour of Joint PDF and Joint Exceedance Probability with h = 10 km... 107 Figure 4-7. Contour of Joint PDF and Joint Exceedance Probability with h = 25 km... 107 Figure 4-8. Contour of Joint PDF and Joint Exceedance Probability with h = 50 km... 107 Figure 4-9. Illustration of Expected Loss and PML... 109 Figure 4-10. Illustration of Regions of Joint Probability... 110

Contents of Tables Table 2-1. Uncertainties of the Attenuation Relations in Natural Logarithm... 20 Table 3-1. Profile of Earthquakes... 25 Table 3-2. Fault Parameters of Earthquakes... 25 Table 3-3. Statistical Characteristics of L between Mean Component of PGA and the Attenuation Relation... 37 Table 3-4. Statistical Characteristics of L between Maximum Component of PGA and the Attenuation Relation... 37 Table 3-5. Correlation Length b (km) for Mean Component of PGA... 45 Table 3-6. Correlation Length b (km) for Maximum Component of PGA... 45 Table 3-7. Statistical Characteristics of L between Mean Component of PGV and the Attenuation Relation... 59 Table 3-8. Statistical Characteristics of L between Maximum Component of PGV and the Attenuation Relation... 59 Table 3-9. Correlation Length b (km) for Mean Component of PGV... 67 Table 3-10. Correlation Length b (km) for Maximum Component of PGV... 67 Table 3-11. Statistical Characteristics of L between Mean Component of SA and the Attenuation Relation... 80 Table 3-12. Statistical Characteristics of L between Vector Component of SA and the Attenuation Relation... 80 Table 3-13. Correlation Length b (km) for Mean Component of SA... 88 Table 3-14. Correlation Length b (km) for Vector Component of SA... 88 Table 3-15. Average of AVS30 in the Different Reigon... 96

1. INTRODUCTION

1.1. Background of Study Japan ranks itself in the few quake-prone countries and with high earthquake risk. There are up to 500 thousand buildings damaged, over 6400 people killed in the 1995 Hyoko-ken Nanbu Earthquake, and the heavy damage to function and structure of urban safety system remains fresh in our memories 1). Just in the last year, great loss was made from the Miyagi-oki Earthquake on May 26th, the Miyagi-ken Hokubu Earthquake on July 26th and the Tokachi-oki Earthquake on September 26th. The future earthquake with the magnitude 8, occurred in the East Sea, South Sea and the Southeast Sea has drawn the great threat. In the world wide, the destructive earthquakes are still active, such as the 1989 Loma Parieta Earthquake and the 1994 Northridge Earthquake in California, the 1999 Kocaeli Earthquake in Turkey, the 1999 Chi-Chi Earthquake in Taiwan, the 2001 El Salvador Earthquake in America, the 2001 Western India Earthquake, and lately the 2003 Southeastern Iran Earthquake. Every year countless damage and loss are suffered from the earthquakes. A big earthquake usually causes directly damage in the structures, people injured or killed, economic losses aftermath the disaster. And the indirect losses and impacts will emerge with the time, especially come from the damage of the urban system, such as traffic infrastructure (the highway system or railway network), lifeline (electric power / gas / water network), communication net etc. The objects suffered from earthquake damage can be characterized with network, i.e., a spatially-spread system. It requires that the seismic design is conducted not only to each member within the system, but also to the network taken as a whole. From the view of disaster prevention, it is essential to keep the seismic performance in the wide area so that the important functions of the disaster prevention can sustain from the earthquake. In general, to improve the seismic reliability of the network during the disaster, the redundancy design is performed though in the wide damage like earthquake. However, it can not be confided that the network reliability with redundancy design is maintained because of the simultaneous damage within tens of kilometers or even wider area. More and more interest is drawn by the government and researchers in the evaluation of earthquake risk for in the future due to the great loss in the past earthquakes. Great efforts have been made on the new principle for disaster preventions and risk management. The prediction of strong motion is the basis of the evaluation of earthquake risk. Portfolio analysis, i.e. management of widely-located building assets, has become popular in the field of earthquake risk management in recent years, whereby simultaneous damages of assets in different locations are major concern 2). It should be incorporated in the urban planning covering the disaster prevention of local municipality, the running of the factory group of the company, and the management of power generations of the power company, etc.. Redundant

portfolio is usually adopted in the management of the main building and its alternative buildings in order to mitigate the risk. Therefore, the simultaneity of the ground motion intensity in different sites should be properly treated for these spatial-spread systems. The simultaneous probability of ground motion between sites which can be characterized by the spatial-correlation of the uncertainty turns out to be important in the evaluation of risk for the widely-spread system. The treatments of the correlation of the uncertainty are various in the past researches due to no systematic study on quantifying the spatial correlation structure of ground 3, 4) motion intensity. Ishikawa et al assumed the correlation is perfect in the seismic risk evaluation of buildings. Fukushima et al. 5, 6,) assumed perfect non-correlation in the portfolio analysis and redundant portfolio analysis. Yamazaki et al. 7) thought that the correlation length characterizing the degree of the correlation of ground motion was 5.0 km depending on the density of the observatory. Based on the study 8) on the macro-spatial correlation of the 1999 Chi-Chi Earthquake, Nishijima 9) adopted 28 km as the correlation length for probabilistic seismic hazard analysis at two sites, and Fukushima et al 10) adopted 20 km as the correlation length for arrangement of alternative buildings. However, the generality and applicability in Japan of the study of Takada et al. 8), only based on the 1999 Chi-Chi Earthquake in Taiwan, need further study with more data. According to the above reviews, the study on the macro-spatial correlation structure turns out to be important in the seismic design, risk assessment, etc. On the other hand, the development of the observatory network in Japan provides an opportunity to evaluate the spatial-correlation model. More information on strong-motion in hypocenter area was greatly helpful on investigating the generation of strong-motions and accessing the disaster for quick mitigate response, while few accelerograms were observed in the Kobe Earthquake 1995. It incentived the Japanese government to launch an action plan to increase the density of strong-motion observation stations, to upgrade the observation network and release future strong-motion records as soon as possible. The Nation Research Institute for Earth Science and Disaster Prevention (NIED), Science and Technology Agency, was given the responsibility to implement the program. Kyoshin Net (K-NET) standing for strong motion is a outcome yielded from this one-year program. K-NET 11) is a system which send strong-motion data on the Internet, which are obtained from more than 1000 observatories deployed all over Japan. The average distance between two stations is about 25km. The seismographs are installed in the free field in order to have a systematic uniform recording condition. Additionally, the soil conditions at each site under 10 to 20 m from the surface ground, including P and S-waves velocity structures, are released in the Internet. As part of the activities of the Headquarters for Earthquake Research Promotion, Kiban-Kyoshi Network 12) (KiK-NET, the digital strong-motion seismograph on the seismic bedrock) has been set up as an annex to Hi-net (the high sensitivity seismograph observation network). It has released the strong-motion data obtained from over 660 observatories in the Internet since Aug.

2000, as well as the underground soil conditions including P and S-waves velocity structures obtained by downhole measurement. Unlike K-NET, the seismographs of the KiK-NET were installed not only in the bedrock as hard as possible so that most of them were installed under the well deeper than 100 m. Also the free field seismographs were also installed in each site. The strong-motion data from the K-NET and KiK-NET can be used for various research fields of earthquake engineering and seismology, and help local governments assess the possible earthquake hazard and manage their disaster response efficiently. The information sent on the Internet form the K-NET and KiK-NET contributes to scientific research such the prediction of strong-motion characteristics at a site, and it contributes to engineering applications such as the development of building codes etc. In this study, contributed to these dense observation networks, the K-NET and KiK-NET, the macro-spatial correlation model was able to be built up.

1.2. Objectives of Study Prediction on the strong motion intensity in the wide area is important for the earthquake prevention of the local municipalities, seismic design of the architecture, civil engineering facilities, also for the earthquake insurance and portfolio assessment. The accumulative studies have contributed actively to the earthquake prevention planning, high-rise buildings and public lifelines. The source model is characterized to simulate the earthquake occurrence, wave propagation so that the spatial distribution is predicted. However, this method is time-consuming and money-consuming. The other technical difficulty is to estimate the time process of slipping of an individual fault. The mean empirical attenuation relations can predict the distribution of the ground motion intensity easily and quickly. The previous attenuation relations are usually expressed as a simple function of magnitude and distance, which are difficult to adequately characterize the complexity of the earthquake. The uncertainty associated with these relations is about 0.5-0.7 in natural logarithmic standard deviation. Also the spatial correlation between the sites can not be taken into account in the attenuation relations, as has great effect on the results predicted. Therefore, the evaluation of seismic ground motion intensity is irrational without taking into account the uncertainties. Regardless of deterministic or probabilistic approach, the seismic hazard map is represented in contours which we can evaluate strong motion intensity at a site, namely, the point which only represent the information at the site rather than the surrounding region which the site locates in. That is, the evaluation of strong motion intensity at a site is dependent from those at the neighborhood since the spatial correlation between two sites is not involved in the hazard map. This study focuses on the spatial correlation of the residual value between an observed value and a predicted ground motion intensity by an empirical mean attenuation relation. The residual value is then modeled in such a way that the joint density function (PDF) of seismic ground motion intensity can be characterized by the spatial correlation model as well as an empirical mean attenuation relation, assuming that it constitutes a homogeneous two-dimensional stochastic field. The seismic design of widely-spread system, the evaluation of the damage on widely-spread traffic infrastructure, lifeline, etc. and the portfolio analysis can be implemented by considering the macro-spatial correlation between different sites. Using the dense observed data of earthquakes occurred in resent years in Japan, the macro-spatial correlation model of the ground motion intensity is proposed. The generality and difference in different seismic intensity measures, attenuation relations, and regions are also discussed. It is expected to play a more effective role in risk management in the future.

1.3. Previous Studies 1.3.1. Previous studies on macro-spatial correlation of ground motion Although there are many previous studies on the spatial correlation or spatial variation of earthquake ground motion, these studies focused on the accelerograms observed from very close locations, the separation distance usually within hundreds meters. Four distinct phenomena give rise to the spatial variability of earthquake-induced ground motions 13) : wave passage effect (difference in arrival times of wave at separate stations), incoherence effect (loss of coherence of seismic waves due to scattering in the heterogeneous medium of the ground, as well as due to the differential superposition of wave arriving from an extended source), local site effect (spatial varying local soil profiles and the manner in which they influence the amplitude and frequency content of the bedrock condition underneath each station as it propagates upward), and attenuation effect (gradual decay of wave amplitudes with distance due to geometric spreading and energy dissipation in the ground medium). Since these effects are well characterized by so called coherency function, which is the normalized cross-power spectral density of the motion at two stations. As the key of the study on spatial variation of ground motion, many coherency functions were proposed to better evaluate the effects in the recent studies. Harichandran and Vanmarcke s model 14) was suggested as the sum of two exponentials with five parameters assuming the homogeneous random field with cross SDF (auto spectral density function). Hindy and Novak 15), Loh 16), Luco and Wong 17) and others have used single exponential function. Hao 18) proposed the model as product of two exponential functions taking into account the anisotropy, one is the function of separation distance, the other is the function of root of separation distance. This model includes eight parameters, which leads to difficultly evaluate these parameters. Nakamura and Yamazaki 19) suggested a Gaussian coherence function model for space-time earthquake ground motion assuming its limited portion to be homogeneous and stationary. More complex models have been proposed by Abrahamson et al 20). Studies show that typically, coherency becomes smaller as the distance between stations increase and decreases with increasing frequency. Also, attenuation is found to have little influence on the coherency function. Since the region studied for spatial correlation or spatial variation of earthquake ground motion is usually within hundreds, it can be called micro-spatial correlation herein. Therefore it can significantly influence internal forces generated in structures with multiple supports such as bridges and viaducts, long-span structures, and large structures such as dams. In contrast to the micro-spatial correlation, the study on the macro-spatial correlation of ground motion draws a graet attention for the wider area. There are distinctive difference in analysis

between macro- and micro-spatial correlation of ground motions. In the macro-spatial correlation analysis, the measures representing ground motion intensity such as peak ground acceleration (PGA), peak ground velocity (PGV) or response spectra are interesting, rather than the accelerogram between the locations. The stochastic model adopted in the macro-spatial correlation analysis is auto-covariance function rather than coherency function for the micro-spatial correlation analysis. The mean attenuation characteristic is significant and should be appropriately considered in the evaluation of the macro-spatial correlation while the attenuation effect is not significance in the micro-spatial correlation analysis. Using the records observed from the Hyogo-ken Nanbu Earthquake, Kambara and Takada 21) proposed the measures of ground motion intensity could be expressed by the product of the trend component and residual component which was assumed to be homogeneous two-dimensional stochastic field. The mean attenuation relation was adopted to characterize the trend component. Then macro-spatial correlation model could be obtained from the normalized auto-covariance function. In the study, the correlation decays close to zero when separation distance is over 10 km i.e. independence on two locations which could be attributed to lack of dense set of observed data for such an analysis. Kawano et al 22) proposed a similar study on the macro-spatial correlation model which did not taken into account the mean characteristic of ground motion in Nagoya area. Although the correlation length was given about 9 km in the study, this result may fail to represent the correlation of ground motion because the mean attenuation characteristic was not taken into account which led the homogeneity assumption was not satisfied. Takada and Shimomura 8) proposed the macro-spatial correlation of seismic ground motion intensity based on the 1999 Chi-Chi earthquake. The correlation length was given about 14-30 km in this event. Since the soil condition in each site is not available, the study can not appropriately consider the site effect so that the result of correlation length may be overestimated. The result only derived from the Chi-Chi earthquake in Taiwan could not represent the correlation in Japan. Its generality and applicability to Japan need to be further examined. The details of the macro-spatial correlation model of are described in the Section 2.

1.3.2. Previous studies on attenuation relation Estimation of an attenuation relation of strong earthquake ground motion intensity has been an interesting research subject in the field of engineering seismology and has played an important role in seismic risk assessments. It is efficacious, for example, in determining design ground motions, and they are particularly essential in probabilistic seismic hazard analysis, in which a wide range of earthquake sizes and locations are incorporated. The design of any engineering structure is based on an estimate of ground motion, either implicitly through the use of building codes or explicitly in the site-specific design of large or particularly critical structures 23). Among various quantitative measures that represent ground motion characteristics, such as peak ground acceleration (PGA), peak ground velocity (PGV), response spectra and spectrum intensity, peak ground acceleration has been most widely chosen for analysis, because it is the most simple representative of accelerogram or acceleration time series, and has long been used for the basis of calculating inertial force in the seismic design of structure. Peak ground velocity has strong correlation with the response of building and damage, and it is also an important measure though it is not frequently used as peak ground acceleration. The most useful measure for the current engineering practice, however, may be response spectrum, because it represents the maximum seismic response taking account of the natural period and damping of structure. The spectrum acceleration (SA) is widely used as a basis of seismic load in designing structures. For instance, it underlies the relationship between lateral design force and period of structure in the static design procedure of various engineered structures. It also prescribes design response spectrum for the dynamic analysis of structures. Systematic reviews of the recently developed attenuation relations for estimating horizontal response spectra, PGA and PGV in the Unite States were given by Joyner and Boore 24, 25). They adopted a two-step stratified regression procedure to produce an attenuation relation which was further applied by a number of investigators. After a long-term, these equations were revised incorporating new data and extending the period range covered by these equations. In the revision 26) of 1994, it significantly improves the treatment of site effects, in which the site-effect term was changed from a constant for each site class to a continuous function of shear-wave velocity at the site averaged to a depth of 30 m. Campbell developed a set of near-source attenuation relations (Distance 60 km) from 1981 27). The 1989 study 28) provided the only coherent set of attenuation relations for both the horizontal and vertical components of PGA, PGV, and SA the 1989 study for soil sites. In 1993, these studies were extended to include hard rock recordings, but only for the horizontal components of PGA and SA 29). In the 1994 study 30), the attenuation relation for the horizontal component of PGA underwent a major revision with the addition of recordings on soil, soft rock, and hard rock from significant worldwide earthquakes the occurred from 1987 to 1992, and from selected worldwide earthquakes that occurred prior to 1987.

In Japan, an acceleration attenuation relation was first proposed by Kanai et al. 31), who used data observed in the Hitachi mine and the source region of the Matsushiro earthquake swarm. Subsequent regression analyses of strong-motion data have been conducted to develop reliable attenuation relations as the number of observed data has increase. Tanaka and Fukushima 32) reviewed acceleration attenuation relations developed for Japan within the last 10 years and examined fundamental characteristics of attenuation for peak horizontal acceleration by transforming those formulas to a unified common-logarithmic functional form. Fukushima and Tanaka 33) examine and clarify the causes of the unexpected low rate of predicted attenuation and to develop a physically meaning new attenuation relation applicable for both short- and middle-distance range (0.1R300km) in Japan. Fukushima 34) revised the acceleration attenuation relations based on the study of 1990, supplementing the new data. Annaka proposed a peak ground acceleration attenuation relation on the engineering bedrock based on the strong-motion records on Kanto area and around. In the 1997 study 35), the peak ground acceleration and spectrum acceleration attenuation relations were developed using the data around all Japan. In addition to these, the attenuation relations have been developed for the specific seismo-tectonic regions, i.e., active tectonic regions, subduction zones and stable continental regions. Midorikawa et al. 36) also derived the attenuation relations for crustal, inter-plate and intra-plate earthquakes based on Japanese strong motion data. The attenuation relation was revised in the 2002 study 37) which took into account the depth of earthquake with addition of the new data. These recent studies of the attenuation relations have mainly laid emphasis on constructing more physically meaningful predictive equations. Persistent efforts have been continuously expended on studying attenuation relations of ground motion. Major interests of the recent studies include the functional form of the attenuation relation, regional variability of earthquakes and effect of site condition. Regarding the functional form of the attenuation relation, various predictive equations have been developed by examining the source and seismic wave propagation characteristics.

1.4. Outline of Study This study consists of 5 chapters. Following the introduction is Chapter 2 which describe the macro-spatial correlation model as well as the stochastic model and the mean attenuation characteristic of the seismic ground motion. Chapter 3 constructs and proposes the macro-spatial correlation structures for PGA, PGV, and SA, respectively, based on the dense observations. The previous attenuation relations are firstly used to characterize the mean characteristic of ground motion, then the homogeneity of the residual value is examined. Finally, the macro-spatial correlation model is built up through statistical analysis. Herein, the macro-spatial model of SA is analyzed in short periods (0.05-0.5 sec.), intermediate periods (0.5-1.0 sec.) and long periods (1.0-3.0 sec.) respectively. The discussions and findings are presented before closing the chapter. Chapter 4 presents some possible applications of the proposed model in the prediction of ground motion intensity, the joint exceedance probability of civil engineering, and the application on earthquake insurance or portfolio analysis etc. so that the model can be explicitly understood. Chapter 5 summaries and concludes the study as well as the future studies.

2. MACRO-SPATIAL CORRELATION MODEL

2.1. Attenuation Characteristics of Ground Motion The trend component in the stochastic model of ground motion is usually characterized by the mean attenuation relations. The attenuation relations of PGA, PGV, and SA used in this study are described in the followings respectively. 2.1.1. Past attenuation relations of PGA The Annaka, Fukushima and Midorikawa-Ohtake attenuation relations are adopted to characterize the mean value of PGA. 1. the Annaka relation 35) 10 j 10 0.653M j log Acc = 0.606M + 0.00459H 2.136 log ( D + 0.334 e ) + 1.730 ( 2-1) 2. the Fukushima relation 34) 0.42 log10 Acc 0.42M log 10 ( 0.025 10 M w w D ) 0.0033D 1.22 = + + ( 2-2) 3. the Midorikawa-Ohtake relation 37) 0.5M w log10 Acc = c log 10( D + 0.0060 10 ) 0.003 D ( H 30 km) 0.5M w log10 Acc = c + 0.6 log 10 (1.7 H + 0.0060 10 ) ( H > 30 km) 0.5M w 1.6log 10( D+ 0.0060 10 ) 0.003D ( 2-3) c = 0.59Mw + 0.0023H + 0.02 for crustal Earthquake c = 0.59Mw + 0.0023H + 0.10 for Inter-plate Earthquake c = 0.59M + w 0.0023H + 0.32 for Intra-plate Earthquake ( 2-4) where Acc: is a PGA in gal; M j is JMA magnitude; M w is moment magnitude; D is a minimum distance to the fault plane in km; H is a source depth in km.

2.1.2. Past attenuation relation of PGV The Annaka and Midorikawa-Ohtake attenuation relations are adopted to characterize the mean value of PGV. 1. the Annaka relation 35) 10 j 10 0.653M j log Vel = 0.725M + 0.00318H 1.918log ( D + 0.334 e ) 0.519 ( 2-5) 2. the Midorikawa-Ohtake relation 37) 0.5M w log10 Vel = c log 10( D + 0.0028 10 ) 0.002 D ( H 30 km) 0.5M w log10vel = c + 0.6 log 10(1.7H + 0.0028 10 ) ( H > 30 km) 0.5M w 1.6 log 10( D+ 0.0028 10 ) 0.002D ( 2-6) c= 0.65Mw + 0.0024H 1.77 for crustal Earthquake c = 0.65Mw + 0.0024H 1.72 for Inter-plate Earthquake c = 0.65M + w 0.0024H 1.62 for Intra-plate Earthquake ( 2-7) where Vel is a PGV in kine defined on the engineering bedrock; M j is JMA magnitude; M w is moment magnitude; D is a minimum distance to the fault plane in km; H is a source depth in km. The uncertainties of these attenuation relations is included in the researches. The uncertainty of the attenuation relation is usually divided into inter- and intra-earthquake uncertainty which are listed in Table 2-1. The uncertainties of the Annaka relations are approximated from those of the response spectra at natural period of 0.04 seconds, while the total uncertainties are only given in the Fukushima relations. 34, 35, 37) Table 2-1. Uncertainties of the Attenuation Relations in Natural Logarithm Uncertainty Annaka Fukushima Midorikawa-Ohtake PGA PGV PGA PGV PGA PGV Inter-earthquake 0.37 0.37 --- --- 0.37 0.37 Intra-earthquake 0.51 0.51 --- --- 0.62 0.55 Total-uncertainty --- --- 0.67 0.67 0.69 0.64

2.1.3. Past attenuation relations of SA The Annaka and Fukishima attenuation relations are adopted to characterize the mean value of SA. 1. the Annaka relation 35) 0.653 M j 10 m j h d 10 o log Sa = C ( T ) M + C ( T ) H C ( T )log ( D + 0.334 e ) + C ( T ) ( 2-8) where Sa is 5% damped absolute acceleration response spectra (SA) defined on the engineering bedrock. C m (T), C h (T), C d (T) and C o (T) are the regression coefficients depending on the period T. The uncertainties of the relation are also the functions of period T. 2. the Fukushima relation 38) 2 0.42M log 10 Sv a1 ( T ) M 2( ) ( ) log 10 ( 0.025 10 w w a T M w b T D D ) = + + + ( 2-9) where Sv is 5% damped velocity response spectra (SV) which is computed by dividing SA by factor 2π/T, where T is the undamped natural period of the oscillator; a 1 (T), a 2 (T) and b(t) are the regression coefficients depending on the period T. The uncertainties of the relation are also the function of period T. Since the Equation (2-9) deals with the SV, it can easily converted to give SA. Then the attenuation relation for SA can be Equation (2-10). 2 0.42M 2 log 10 1( ) 2( ) ( ) log 10 ( 0.025 10 w π Sa = a T M w + a T M w + b T D D + ) + log10 ( 2-10) T where Sa is 5% damped absolute acceleration response spectra (SA) defined on the seismic bedrock.

2.1.4. Site amplification factor The attenuation relations for PGV in Section 2.1.2. are defined on the stiff ground, that is, engineering bedrock whose average shear wave velocity is equivalent to 400~600 m/s, while the PGV observed in the earthquakes considered are on the ground surface. The Fukushima attenuation relation for SA in Section 2.1.3. is defined on the seismic bedrock whose average shear wave velocity is equivalent to 3000 m/s. Therefore, it is necessary to transform the PGV and SA obtained on the ground into those on the engineering bedrock and seismic bedrock, respectively, with a simple calculation such as a site amplification factor. Previous studies 39, 40, 41) show that the site amplification characteristic is strongly correlated to the shear wave velocity near the ground surface and use shear-wave velocity averaged over the upper 30 m (AVS30) as the variable to represent site effects. Based on the data in the Kanto area in Japan, the empirical relationship between the amplification faction of PGV (RPGV) and AVS30 was proposed 42) as, log RPGV = 1.83 0.66 log ( AVS30) ( 2-11) 10 10 Fukushima 38) proposed the ground amplification characteristics from the seismic bedrock to ground surface by the one-dimensional multiple wave reflection theory considering the frequency-dependent Q -1 model. The relation between the amplification factions of SA (RSA) for engineering bedrock and seismic bedrock and AVS30 was proposed with the same function as, log RSA( T) = β ( T ) α( T)log ( AVS30) ( 2-12) 10 10 in which β(t), α(t) are the regression coefficient depending on the natural period. The value of AVS30 is a time-weighted average, computed by dividing 30 m by the S-wave travel time from the surface to 30 m as in Equation (2-17). AVS30 = n ( Hi / Vsi) i= 1 30 ( 2-13) where n is the stratum number over the upper 30 m; H i is the depth of the i-th stratum in m; Vs i is the shear-wave velocity of the i-th stratum. Studies on the average shear-wave velocity over the upper 30 m (AVS30) are available in researches. Fujimoto and Midorikawa 43) proposed the relations between the AVS30 and the geomorphological land classification, and generated a nationwide map of AVS30. Matsuoka and Wakamatsu 44) estimated the AVS30 using Japan Engineering Geomorphological Classification Map.

2.2. Separation of Mean and Residual Component of Ground Motion Since Housner 45) first considered that earthquake ground motions might be treated as random processes, this idea has been widely accepted. For a random phenomenon, although the outcome of each sample of a repeated observation is not predictable, some averages of the samples are usually found to follow some rules, and these averages reflect the effect of the influencing factors. In case of earthquake ground motion, the specific condition of the area, the geometry of the causative fault, the energy distribution, the distance, and the soil condition of the site determine these average. It may be said that there are two groups of factors that control a random phenomenon: one group controls the mean characteristic of the quality; the other group adds the random property to it. Similar to it, the ground intensity is also treated as random process herein. As mentioned in Section 1.3.1., the stochastic model of ground motion proposed by Kambara and Takada 21), and applied in the study of Takada and Shimomura 8), is detailed as following. The seismic ground motion intensity observed at a site can be described as the product of the trend component and the residual component as given in Equation (2-14), illustrated in Figure 2-1. A( x) = A( x) ε ( x ) ( 2-14) in which x is the spatial coordinate of a site; A(x) is an observed seismic ground motion intensity; A( x ) is a predicted seismic ground motion intensity; ε( x ) is a residual value between an observed and a predicted ground motion intensity. ε Figure 2-1. Illustration of Separation of the Mean and Residual Component of Ground Motion

A(x) can be a measure representing ground motion intensities such as PGA, PGA, and SA etc. A( x ) usually can be predicted by the mean attenuation relations. The uncertainty is usually treated in term of the natural logarithm of ε( x ). Taking logarithm of the both sides of Equation (2-14), the logarithmic deviation L(x) is defined as given in Equation (2-15). In this study, natural logarithms are used in the absence of a special description. L( x) = ln ε ( x ) ( 2-15) Then L(x) is modeled following the assumption that L(x) constitutes a homogeneous two-dimension stochastic field. The auto-covariance function of L(x) between different two sites depends only on the relative distance as defined in Equation (2-16). ( ) = E( [ ( ) µ )( ( ) µ )] C h L x L x ( 2-16) LL 1 L 2 L where E[. ] is a spatial expectation; h is a separation distance between the two locations x 1 and x 2, h= x 2 -x 1 ; L is the mean value of the logarithmic deviation L(x), L =E[L(x)].

2.3. Macro-Spatial Correlation Model The data observed from the five recent earthquakes in Japan are used to build up the macro-spatial correlation model whereby the auto-covariance function C LL is estimated through statistical analysis. The observed data were firstly 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. Equation (2-16) can be rewrite as Equation (2-17) by discrete expression: N( h) 1 C ( h) = ( ( ) ) ( ( ) LL L x µ L x µ a ) i L bi L Nh ( ) ( 2-17) i= 1 where N 1 all µ = ( ) L L x ; N(h) is the number of pairs of sites (x i a, x b ) that meet the condition h N i= 1 all h/2 < x a x b < h + h/2; N all is the total number of observation sites; h is the discrete distance whose interval is h. In this study, the interval h is set to 4 km in order to keep accuracy in the statistical analysis. Normalized auto-covariance function R LL (h), auto-covariance function normalized by the variation of L(x), σ, is obtained from C LL (h) as following. 2 L 2 ( ) ( )/ R h C h σ = ( 2-18) LL LL L Now the macro-spatial correlation model can be built up by modeling the discrete values of the normalized auto-covariance function R LL with an exponential decaying function as in Equation (2-19): R ( ) exp LL h ( h ) = b ( 2-19) where h is a separation distance between two observations and b is so-called a correlation length, which can characterize the degree of correlation of ground motions between two locations. It means that when the separation distance h between two sites equals to b, R LL becomes 1/e ( 0.37). It can be seen from Equation (2-19) 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.

3. ANALYSIS OF SPATIAL CORRE- LATION OF GROUND MOTION

3.1. Database of the Seismic Ground Motion The data set used in this study is based on the K-NET and KiK-NET which obtained the strong-motion data from observatories deployed all over Japan and released strong-motion data on the Internet. Events selected are the five earthquakes in Japan which occurred from 2000 to 2003, with the moment magnitude M w ranging from 6.2 to 8.0. The information on hypocenters and fault models used for the attenuation relations are listed on Tables 3-1 and 3-2 in which M j is JMA (Japan Meteorological Agency) magnitude. The data of the hypocenters in Japan adopted from the reports made by JMA, and the fault models of earthquakes in Japan were taken from the Headquarters for Earthquake Research Promotion or Geographical Survey Institute (GSI). The epicenters of the five earthquakes were shown in Figure 2-1. Table 3-1. Profile of Earthquakes 46, 47, 48, 49) No. Earthquakes Date M j Source Depth (km) Fault Type Number of Sites Used 1 Tottori-ken Seibu 2000/10/06 7.3 9 Crustal 315 2 Geiyo 2001/03/24 6.7 46 Intra-plate 370 3 Miyagi-ken-oki 2003/05/26 7.0 71 Intra-plate 230 4 Miyagi-ken Hokubu 2003/07/26 6.2 12 Crustal 246 5 Tokachi-oki 2003/09/26 8.0 42 Inter-plate 287 No. Earthquake Latitude ( ) Table 3-2. Fault Parameters of Earthquakes Longitude ( ) Depth (km) Strike ( ) 50, 51, 52, 53, 54) Dip ( ) Width (km) Length (km) 1 Tottori-ken Seibu 35.35 133.3 1 152 86 10 20 6.8 2 Geiyo 34.1 132.7 38.1 156 52 10 20 6.7 3 Miyagi-ken-oki 38.94 141.81 52 192 68 19 17 7.0 4 Miyagi-ken Hokubu 38.41 141.17 3.7 11 40 5.1 13.6 6.2 5 Tokachi-oki 42.12 144.55 19.7 231 22 83 85.7 8.0 All the records and soil condition of sites were taken from K-NET and KiK-NET. For the study, only the records on the sites whose minimum distance to the fault plane is within 300 km and AVS30 can be estimated are adopted in this paper according to the norm of Fujimoto and Midorikawa 42). The numbers of record used to analysis for PGA, PGV, SA were listed in the Table 3-1. Following the study of Midorikawa and Ohtake 37), and reference to the study of Si and Midorikawa 36), acceleration time histories were obtained after the records processed with a band pass filter of 0.2~10 Hz and with the correction of the baseline drift. Then PGV could be obtained M w

by integrating the acceleration time histories. SA was computed as elastic absolute-acceleration response spectra with 5% damping. PGA, PGV and SA were taken as the ground motion intensity measures to construct the macro-spatial correlation model respectively. Figure 3-1. Locations of Earthquakes in Japan

3.2. Macro-Spatial Correlation Analysis for PGA The PGA components adopted to analysis were mean value and maximum value of EW and NS components of PGA. The results of two components were also examined and compared. The spatial distributions of two components of each earthquake were shown in Figures 3-2~3-6. (a) Mean Component (b) Maximum Component Figure 3-2. Spatial Distribution of PGA of the Tottori-ken Seibu Earthquake (a) Mean Component (b) Maximum Component Figure 3-3. Spatial Distribution of PGA of the Geiyo Earthquake

(a) Mean Component (b) Maximum Component Figure 3-4. Spatial Distribution of PGA of the Miyagi-ken-oki Earthquake (a) Mean Component (b) Maximum Component Figure 3-5. Spatial Distribution of PGA of the Miyagi Hokubu Earthquake

(a) Mean Component (b) Maximum Component Figure 3-6. Spatial Distribution of PGA of the Tokachi-oki Earthquake

3.2.1. Data fitness for PGA Figures 3-7~3-11 showed observed PGA data along with the predicted results with three PGA attenuation relations. The mean component and maximum component of PGA were both fitted by three relations. It was found that the attenuation relations can capture the mean tendency of the observed data in a wider range. Comparing with the Annaka relation and the Fukushima relation, the Midorikawa-Ohtake relation gives better fitness with the observed data. This can be explained from the fact that this attenuation relation takes into account the effects of fault type and source depth. Significant scatter around the mean relation, however, can be observed for all the relations. The uncertainties of the attenuation relation were computed in Table 3-3, Table 3-4. The tables list the mean value and standard deviation of logarithmic deviation L(x) for each attenuation relation. For simplicity, as a linear measure, correlation coefficient ρ LD, which represents the dependency between the logarithmic deviation and fault distance, that is, the goodness of fit of the attenuation relation, are also computed for comparison. With the correction of mean value of logarithmic deviation L in the tables, the corrected mean attenuation relations are also show in the figures.

(a) (b) (c) (d) (e) (f) Figure 3-7. Fitness of PGA of the Tottori-ken Seibu Earthquake

(a) (b) (c) (d) (e) (f) Figure 3-8. Fitness of PGA of the Geiyo Earthquake

(a) (b) (c) (d) (e) (f) Figure 3-9. Fitness of PGA of the Miyagi-ken-oki Earthquake

(a) (b) (c) (d) (e) (f) Figure 3-10. Fitness of PGA of the Miyagi Hokubu Earthquake

(a) (b) (c) (d) (e) (f) Figure 3-11. Fitness of PGA of the Tokachi-oki Earthquake

Table 3-3. Statistical Characteristics of L(x) between Mean Component of PGA and the Atenuation Relation Earthquake L from Annaka Relation L from Fukushima Relation L from Midorikawa- Ohtake Relation Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki µ L -0.024 0.497 0.746 0.598-0.238 σ L 0.628 0.740 0.893 0.835 0.842 ρ LD -0.411-0.624-0.418 0.057-0.508 µ L 0.052 0.347 0.957-0.082 0.000 σ L 0.568 0.665 0.850 0.831 0.762 ρ LD -0.146-0.498-0.275 0.101-0.320 µ L -0.037-0.206-0.050 0.062-0.533 σ L 0.577 0.590 0.814 0.828 0.732 ρ LD -0.217-0.205-0.065 0.058-0.153 Table 3-4. Statistical Characteristics of L(x) between Maximum Component of PGA and the Atenuation Relation Earthquake L from Annaka Relation L from Fukushima Relation L from Midorikawa- Ohtake Relation Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki µ L 0.080 0.591 0.825 0.683-0.156 σ L 0.639 0.751 0.897 0.844 0.847 ρ LD -0.413-0.623-0.416 0.049-0.506 µ L 0.156 0.441 1.037 0.004 0.082 σ L 0.580 0.676 0.855 0.838 0.767 ρ LD -0.154-0.499-0.274 0.093-0.318 µ L 0.066-0.133 0.030 0.148-0.451 σ L 0.588 0.600 0.819 0.834 0.737 ρ LD -0.223-0.218-0.065 0.050-0.153

From the Table 3-3, the intra-earthquake uncertainties of Annaka relation for mean component of PGA range from 0.628 to 0.893 in natural logarithm, from 0.568 to 0.850 for Fukushima relation, and from 0.577 to 0.828 for Midorikawa-Ohtake relation. Although the number of earthquakes considered here is very small, five, the inter-earthquake uncertainties can be computed, and it becomes 0.424, 0.425 and 0.233 in a natural logarithmic standard deviation, respectively for Annaka relation, Fukushima relation and Midorikawa-Ohtake relation. The correlation coefficients of L(x) with respect to D vary from -0.624 to 0.057 for Annaka relation, from -0.498 to 0.101 for Fukushima relation, and from -0.217 to 0.058 for Midorikawa-Ohtake relation. Similarly from the Table 3-4, the intra-earthquake uncertainties of Annaka relation for maximum component of PGA range from 0.639 to 0.897 in natural logarithm, from 0.580 to 0.855 for Fukushima relation, and from 0.588 to 0.834 for Midorikawa-Ohtake relation. the inter-earthquake uncertaintyies can be computed with 0.421, 0.421 and 0.236 in a natural logarithmic standard deviation, respectively for Annaka relation, Fukushima relation and Midorikawa-Ohtake relation. The correlation coefficients of L(x) with respect to D vary from -0.623 to 0.049 for Annaka relation, from -0.499 to 0.093 for Fukushima relation, and from -0.212 to 0.051 for Midorikawa-Ohtake relation. The intra-earthquake uncertainties of attenuation relations for the five earthquake falls in 0.5-0.9, which are not smaller than the values in the Table 2-1. This implied that if the ground motion intensity is predicted only by the mean empirical attenuation relations, the uncertainty is too notable to ignored. The inter-earthquake uncertainties are close to the value, 0.37, originally proposed for these attenuation relations in Table 2-1. It is shown that the Midorikawa-Ohtake attenuation relation fits for PGA better than the other two attenuation relations with the smaller intra-uncertainty L, inter-uncertainties and correlation coefficients LD. Although the Midorikawa-Ohtake relation gives the prediction of maximum component of PGA, the results of statistical characteristics above for mean component of PGA do not largely change for maximum component. And so do the other two relations.

3.2.2.Homogeneity of logarithmic deviation In the Section 2.1, it has been assumed that the logarithmic deviation L(x) constitutes a homogeneous two-dimensional stochastic field. It is necessary to examine the homogeneity of L(x) before evaluating the macro-spatial correlation model. Figures 3-12 to 3-16 show the distributions of L(x) to the minimum distance to the fault plane D. Note that the logarithmic deviation L(x) in the figures is corrected by its mean value L, that is, L(x) is a random variable with zero mean and standard deviation L. The moving averages of L(x) are also shown in the above figures. Its windows-width is 20 km. The mean value ML and standard deviation σ of L(x) within the widow are also calculated, and the ranges of the ML with plus-and minus-one-σ, i.e. (ML±σ), 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 goodness of fit has some dependency on the fault distance. It is observed from the figures that the dependency of the Midorikawa-Ohtake relation is smaller than those of the Annaka relation and Fukushima relation. It is correspondent with the correlation coefficients ρ LD listed in Tables 3-3 and 3-4. On the other hand, from the curves of (ML±σ), the moving averages of standard deviation σ do not show this distance-dependency. The value of moving average of standard deviation is about 0.3 with small variation. As a result, the assumption which the logarithmic deviation L(x) of PGA constitutes a homogeneous two-dimensional stochastic field can be approximately satisfied. It will lead to simple treatment of PGA with the Midorikawa-Ohtake mean attenuation relation.

(a) (c) (b) (e) (f) Figure 3-12. Homogeneity of L(x) of PGA of the Tottori-ken Seibu Earthquake (d)

(a) (c) (b) (e) (f) Figure 3-13. Homogeneity of L(x) of PGA of the Geiyo Earthquake (d)

(a) (c) (b) (e) (f) Figure 3-14. Homogeneity of L(x) of PGA of the Miyagi-ken-oki Earthquake (d)

(a) (c) (b) (e) (f) Figure 3-15. Homogeneity of L(x) of PGA of the Miyagi-ken Hokubu Earthquake (d)

(a) (c) (b) (e) (f) Figure 3-16. Homogeneity of L(x) of PGA of the Tokachi-oki Earthquake (d)

3.2.3.Proposal of new macro-spatial correlation model for PGA The data observed from the five earthquakes are fully used to build up the macro-spatial correlation model. Through statistical analysis, from Equations (2-17) to (2-18), the discrete values of the auto-covariance functions C LL and the normalized auto-covariance functions R LL with respect to the separation distance h are estimated. The number of pairs of sites (x a, x b ) within h h/2 < x a x b < h + h/2 with interval h = 4 km, are shown in Figure 3-17. The number of pairs in the group of h = 0~4 km is very small, ranging from 8~27 since the average separation distance of K-NET observatories is about 25 km. It is natural to consider that small samples in the group have effect on the evaluation of auto-covariance in this group. An exponential decaying function is adopted to evaluate the correlation model as Equation (2-19). The regression curves are show in the Figures 3-18~3-22. The values of b, so-called a correlation length, can be obtained, and the results are listed in Tables 3-5 and 3-6 for mean component and maximum component by three attenuation relations respectively. Earthquake Table 3-5. Correlation length b (km) for mean component of PGA Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Annaka 23.3 42.4 45.7 31.3 60.3 Fukushima 11.6 27.4 41.8 32.5 52.0 Midorikawa-Ohtake 12.9 11.5 38.7 31.4 46.2 Earthquake Table 3-6. Correlation length b (km) for maximum component of PGA Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Annaka 23.1 41.7 45.2 31.0 59.1 Fukushima 11.6 27.1 41.5 32.1 50.9 Midorikawa-Ohtake 12.9 11.7 38.4 31.0 45.2 Note that in the Figure 3-18, there is an unreasonable data plot in the closer range of separation distance of 4 km, which shows negative correlation. This is mainly because the number of data pairs in this range is only eight, which is too small to get reliable results. More careful 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 of the sites are made of granite rock. These detail and precise ground conditions cannot be properly reflected in the past empirical relations. In the calculation of b-values, these data are ignored.

(a) (c) (e) Figure 3-17. Histogram of Separation Distance h (b) (d)

(a) (b) (c) (d) (e) (f) Figure 3-18. Normalized Auto-Covariance Function of PGA of the Tottori-ken Seibu Earthquake

(a) (c) (b) (e) (f) Figure 3-19. Normalized Auto-Covariance Function of PGA of the Geiyo Earthquake (d)

(a) (c) (b) (e) (f) Figure 3-20. Normalized Auto-Covariance Function of PGA of the Miyagi-ken-oki Earthquake (d)

(a) (c) (b) (e) (f) Figure 3-21. Normalized Auto-Covariance Function of PGA of the Miyagi-ken Hokubu Earthquake (d)

(a) (c) (b) (e) (f) Figure 3-22. Normalized Auto-Covariance Function of PGA of the Tokachi-oki Earthquake (d)

3.2.4.Discussions It can be observed, from the Figures 3-18~3-22 and Tables 3-5~3-6, that the values of the correlation length b for the three attenuation relations are of the same order of magnitude, raging from 12 to 60 km. This implies that the spatial correlation characteristic of seismic ground motion has approximately similar tendency regardless of the mean attenuation relations used as well as earthquake types or regions in Japan. The difference of b-values analyzed from mean component and maximum component of PGA seems very small and fall in a quite small range. It implies that although the PGA is defined for the mean component or maximum component in the attenuation relation, the attenuation can be used to analysis of the spatial correlation model for both components. The correlation length b calculated from the Midorikawa-Ohtake relation seems to be slightly smaller than those from the Annaka relation and Fukushima relation. This can be explained from the previous discussions on the fault distance-dependency of L(x), as seen in Figures 3-12~3-16 and Tables 3-3~3-4. The smaller the distance-dependency, the better the attenuation relations fits, and the more homogeneous the logarithmic deviation is. The Midorikawa-Ohtake attenuation relation can better characterize the ground motion, which lead to the reasonable correlation length, while the correlation length could be overestimated with the Annaka attenuation relation or Fukushima attenuation relation. Figure 3-23 shows the relation between b-values and the absolute values of the correlation coefficient ρ LD. ρ ρ (a) (b) Figure 3-23. Correlation Coefficient ρ LD vs. Correlation Length b for PGA Examining the difference in the regions of interest in Japan, the correlation length b in the Northeastern Japan tends to be larger than that in the Southwestern Japan for the three attenuation relations. It can be observed that the b-value ranges from 31 to 47 km in the northeastern region and, from 11 to 13 km in the southwestern region. This will be discussed in the later section of this paper.

3.3. Macro-Spatial Correlation Analysis for PGV The PGV is transformed into the engineering bedrock by the site amplification factor (RPGV). The components used to analyze are mean value and maximum value of EW and NS components of PGV. The results of two components are also examined and compared. 3.3.1. Data fitness for PGV Figures 3-24~3-28 show observed PGV data along with the predicted results with two PGV attenuation relations. The mean component and maximum component of PGV are both fitted by two relations. It is found that the attenuation relations can capture the mean tendency of the observed data of PGV in a wider range. Comparing with the Annaka relation, the Midorikawa-Ohtake relation gives better fitness with the observed data. This can be explained from the fact that this attenuation relation takes into account the effects of fault type and source depth. Significant scatter around the mean relation, however, can be observed for two relations. The uncertainties of the attenuation relations are computed in Tables 3-7, Table 3-8. The tables list the mean value and standard deviation of logarithmic deviation L(x) for each attenuation relation. For simplicity, as a linear measure, coefficient of correlation ρ LD, which represents the dependency between the logarithmic deviation and fault distance, that is, the goodness of fit of the attenuation relation, are also computed for comparison. With the correction of mean value of logarithmic deviation L in the tables, the corrected mean attenuation relations are also show in the figures.

(a) (c) (d) Figure 3-24. Fitness of PGV of the Tottori-ken Seibu Earthquake (b)

(a) (b) (c) (d) Figure 3-25. Fitness of PGV of the Geiyo Earthquake

(a) (c) (d) Figure 3-26. Fitness of PGV of the Miyagi-ken-oki Earthquake (b)

(a) (b) (c) (d) Figure 3-27. Fitness of PGV of the Miyagi Hokubu Earthquake

(a) (c) (d) Figure 3-28. Fitness of PGV of the Tokachi-oki Earthquake (b)

Table 3-7. Statistical Characteristics of L(x) between Mean Component of PGV and the Atenuation Relation Earthquake Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki L from Annaka Relation L from Midorikawa- Ohtake Relation µ L -0.611-0.210 0.213 0.042-0.581 σ L 0.498 0.547 0.563 0.580 0.636 ρ LD 0.230-0.473-0.458 0.360-0.489 µ L 0.033-0.161 0.099-0.041-0.243 σ L 0.521 0.482 0.504 0.565 0.560 ρ LD 0.400 0.073 0.046 0.309 0.005 Table 3-8. Statistical Characteristics of L(x) between Maximum Component of PGV and the Atenuation Relation Earthquake L from Annaka Relation L from Midorikawa- Ohtake Relation Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki µ L 0.462-0.114 0.325 0.140-0.481 σ L 0.511 0.557 0.581 0.583 0.6445 ρ LD 0.215-0.468-0.474 0.352-0.495 µ L 0.182-0.065 0.210 0.057-0.143 σ L 0.532 0.493 0.514 0.569 0.565 ρ LD 0.383 0.067 0.011 0.301-0.010

From the Table 3-7, the intra-earthquake uncertainties of Annaka relation for mean component of PGA range from 0.498 to 0.634 in natural logarithm and from 0.482 to 0.565 for Midorikawa-Ohtake relation. The inter-earthquake uncertainties can be computed with five earthquakes, and it becomes 0.367 and 0.140 in a natural logarithmic standard deviation, respectively for Annaka relation and Midorikawa-Ohtake relation. The correlation coefficients of L(x) with respect to D vary from -0.489 to 0.360 for Annaka relation, from 0.005 to 0.400 for Midorikawa-Ohtake relation. From the Table 3-8, the intra-earthquake uncertainties of Annaka relation for maximum component of PGA range from 0.514 to 0.645 in natural logarithm and from 0.493 to 0.569 for Midorikawa-Ohtake relation. The inter-earthquake uncertainties computed become 0.356 and 0.153 in a natural logarithmic standard deviation, respectively for Annaka relation and Midorikawa-Ohtake relation. The correlation coefficients L(x) with respect to D vary from -0.495 to 0.352 for Annaka relation and from -0.010 to 0.383 for Midorikawa-Ohtake relation. The intra-earthquake uncertainties of attenuation relations for the five earthquake falls in 0.5-0.7, which are not smaller than the values in the Table 2-1. Although PGV is defined on the engineering bedrock corrected with the site amplification factor, making such corrections does not significantly reduce the prediction uncertainty associated with empirical attenuation relations. This implied that if the ground motion intensity is predicted only by the mean empirical attenuation relations, the uncertainty is too notable to ignored. The inter-earthquake uncertainties are close to the value, 0.37, originally proposed for these attenuation relations in Table 2-1. It is shown that the Midorikawa-Ohtake attenuation relation fits for PGV better than the Annaka attenuation relation with the smaller intra-uncertainty L, inter-uncertainties and correlation coefficients LD. Although the Midorikawa-Ohtake relation gives the prediction of maximum component of PGV, the results of statistical characteristics above for mean component of PGV do not largely change for maximum component. And so does the Annaka relation.

3.3.2.Homogeneity of logarithmic deviation In the Section 2.1, it has been assumed that the logarithmic deviation L(x) constitutes a homogeneous two-dimensional stochastic field. It is necessary to examine the homogeneity of L(x) before the macro-spatial correlation model is evaluated. Figures 3-29 to 3-33 show the distributions of L(x) to the minimum distance to the fault rupture D. Note that the logarithmic deviation L(x) in the figures is corrected by its mean value L, that is, L(x) is a random variable with zero mean and standard deviation L. The moving averages of L(x) are also shown in the above figures. Its windows-width is 20 km. The mean value ML and standard deviation σ of L(x) within the widow are also calculated, and the ranges of the ML with plus-and minus-one-σ, i.e. (ML±σ), 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 goodness of fit has some dependency on the fault distance. It is observed from the figures that the dependency of the Midorikawa-Ohtake relation is smaller than those of the Annaka relation. It is correspondent with the correlation coefficients ρ LD listed in Tables 3-7 and 3-8. On the other hand, from the curves of (ML±σ), the moving averages of standard deviation σ do not show this distance-dependency. The value of moving average of standard deviation is about 0.5 with small variation. As a result, the assumption which the logarithmic deviation L(x) of PGV constitutes a homogeneous two-dimensional stochastic field can be approximately satisfied. It will lead to simple treatment of PGV with the Midorikawa-Ohtake mean attenuation relation.

(a) (c) (d) Figure 3-29. Homogeneity of L(x) of PGV of the Tottori-ken Seibu Earthquake (b)

(a) (c) (d) Figure 3-30. Homogeneity of L(x) of PGV of the Geiyo Earthquake (b)

(a) (c) (d) Figure 3-31. Homogeneity of L(x) of PGV of the Miyagi-ken-oki Earthquake (b)

(a) (c) (d) Figure 3-32. Homogeneity of L(x) of PGV of the Miyagi-ken Hokubu Earthquake (b)

(a) (c) (d) Figure 3-33. Homogeneity of L(x) of PGV of the Tokachi-oki Earthquake (b)

3.3.3.Proposal of new macro-spatial correlation model for PGV The data observed from the five earthquakes are fully used to build up the macro-spatial correlation model. Through statistical analysis, from Equations (2-17) to (2-18), the discrete values of the auto-covariance functions C LL and the normalized auto-covariance functions R LL with respect to the separation distance h are calculated. The number of pairs of sites (x a, x b ) within h h/2 < x a x b < h + h/2 with interval h = 4 km, are shown in Figure 3-17. Similarly to PGA, the small number of data pairs in the bin has effect on the evaluation of auto-covariance in this bin. An exponential decaying function is adopted to evaluate the correlation model as Equation (2-19). The regression curves is show in the Figures 3-34~3-38. The values of b, a correlation length, can be obtained, and the results are listed in Tables 3-9 and 3-10 for mean component and maximum component by two attenuation relations respectively. Earthquake Table 3-9. Correlation length b (km) for mean component of PGV Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Annaka 21.3 49.9 37.5 28.9 43.7 Midorikawa-Ohtake 29.5 37.8 21.1 25.1 22.5 Earthquake Table 3-10. Correlation length b (km) for maximum component of PGV Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Annaka 21.0 49.5 39.7 27.7 44.5 Midorikawa-Ohtake 28.6 37.5 21.6 24.0 22.4 Note that in the Figure 3-34, there is an unreasonable data plot in the closer range of separation distance of 4 km, which shows negative correlation. This is mainly because the number of data pairs in this range is only eight, which is too small to get reliable results. 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 of the sites are made of granite rock. These detail and precise ground conditions cannot be properly reflected in the past empirical relations. In the calculation of b-values, these data are ignored as mentioned in Section 3.2.3

(a) (b) (c) (d) Figure 3-34. Normalized Auto-Covariance Function of PGV of the Tottori-ken Seibu Earthquake

(a) (c) (d) Figure 3-35. Normalized Auto-Covariance Function of PGV of the Geiyo Earthquake (b)

(a) (c) (d) Figure 3-36. Normalized Auto-Covariance Function of PGV of the Miyagi-ken-oki Earthquake (b)

(a) (c) (d) Figure 3-37. Normalized Auto-Covariance Function of PGV of the Miyagi-ken Hokubu Earthquake (b)

(a) (c) (d) Figure 3-38. Normalized Auto-Covariance Function of PGV of the Tokachi-oki Earthquake (b)

3.3.4.Discussions It can be observed, from the Figures 3-34~3-38 and Tables 3-9~3-10, that the values of the correlation length b for the two attenuation relations are of the same order of magnitude, raging from 21 to 50 km. This implies that the spatial correlation characteristic of seismic ground motion has approximately similar tendency regardless of the mean attenuation relations used as well as earthquake type or regions in Japan. The difference of b-values analyzed from mean component and maximum component of PGV seems very slight and fall in quite a small range. It implies that although the PGV is defined for the mean component or maximum component in the attenuation relation, the attenuation can be used to analysis of the spatial correlation model for both components.. The correlation length b calculated from the Midorikawa-Ohtake relation seems to be slightly smaller than those from the Annaka relation. This can be explained from the previous discussion on the fault distance-dependency of L(x), as seen in Figures 3-34~3-38 and Tables 3-9~3-10. The smaller the dependency, the better the attenuation relations fits, and the more homogeneous the logarithmic deviation is. The Midorikawa-Ohtake attenuation relation can better characterize the ground motion, which lead to the reasonable correlation length, while the correlation length could be overestimated with the Annaka attenuation relation. Figure 3-39 shows the relation between b-values and the absolute values of the correlation coefficient ρ LD. ρ (a) (b) Figure 3-39. Correlation Coefficient ρ LD vs. Correlation Length b for PGV ρ It shows that the difference in regions of interest in Japan is not as obvious as that of PGA, although the correlation length b in the Northeastern Japan tends to be slightly smaller than that in the Southwestern Japan for the two attenuation relations. It can be observed that the b-value ranges from 21 to 45 km in the northeastern region and, from 21 to 50 km in the southwestern region. This will be discussed in the later section of this paper.

3.4. Macro-Spatial Correlation Analysis for SA The SA components used to analyze are mean component for the Annaka relation and vector components for the Fukushima relation. The period was divided into three ranges, that is, short period range from 0.05 to 0.5 sec., intermediate period range from 0.5 to 1.0 sec. and long period range from 1.0 to 3.0 sec. Periods at 0.3 sec., 0.8 sec. and 2.0 sec. represent short period range, intermediate period range and long period range respectively. The characteristic SAs at these three periods are calculated with the average SA at each of the three period ranges as shown in Figure 3-40. Figure 3-40. Division of Period and Average SA (damping = 5%) 3.4.1. Data fitness for SA Figures 3-41~3-45 show observed SA data along with the predicted results with two SA attenuation relations. The mean component and vector component of SA are both fitted by two relations. It is found that the attenuation relations can capture the mean tendency of the observed data of SA in a wider range. It is difficulty to say which attenuation relation is better since two relations were defined on the different ground condition. Significant scatter around the mean relations, however, can be observed for two relations. The uncertainties of the attenuation relations are computed in Tables 3-11 and 3-12. The tables list the mean value and standard deviation of logarithmic deviation L(x) for each attenuation relation and different period ranges. For simplicity, as a linear measure, coefficient of correlation ρ LD, which represents the dependency between the logarithmic deviation and fault distance, that is, the goodness of fit of the attenuation relation, are also listed in the tables. With the correction of mean value of logarithmic deviation L in the tables, the corrected mean attenuation relations are also show in the figures.

(a) (d) (b) (e) (c) (f) Figure 3-41. Fitness of SA of the Tottori-ken Seibu Earthquake

(a) (d) (b) (e) (c) (f) Figure 3-42. Fitness of SA of the Geiyo Earthquake

(a) (b) (d) (e) (c) (f) Figure 3-43. Fitness of SA of the Miyagi-ken-oki Earthquake

(a) (d) (b) (e) (c) (f) Figure 3-44. Fitness of SA of the Miyagi Hokubu Earthquake

(a) (b) (d) (c) (f) Figure 3-45. Fitness of SA of the Tokachi-oki Earthquake (e)

Table 3-11. Statistical Characteristics of L(x) between Mean Component of SA and the Annaka Atenuation Relation Earthquake Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Short Period Intermediate Period Long Period µ L -0.234 0.369 0.660 0.461-0.362 σ L 0.646 0.717 0.898 0.851 0.921 ρ LD -0.356-0.577-0.380 0.111-0.527 µ L -0.755 0.008 0.361 0.208-0.661 σ L 0.697 0.687 0.677 0.789 0.837 ρ LD 0.104-0.334-0.251 0.302-0.580 µ L -0.632 0.109 0.623 0.419-0.505 σ L 0.638 0.602 0.653 0.737 0.819 ρ LD 0.308-0.215-0.239 0.300-0.523 Table 3-12. Statistical Characteristics of L(x) between Vector Component of SA and the Fukushima Atenuation Relation Earthquake Short Period Intermediate Period Long Period Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki µ L -0.128 0.197 0.908-0.269 0.420 σ L 0.608 0.654 0.861 0.863 0.839 ρ LD -0.040-0.390-0.201 0.186-0.332 µ L -0.343-0.175 0.462-0.456 0.160 σ L 0.666 0.583 0.574 0.730 0.631 ρ LD 0.452-0.135-0.081 0.433-0.370 µ L -0.024-0.154 0.541-0.122 0.051 σ L 0.781 0.581 0.581 0.775 0.657 ρ LD 0.682 0.203 0.116 0.542-0.200

From the Table 3-11, the intra-earthquake uncertainties of short period for mean component of SA by the Annaka relation, ranged from 0.646 to 0.921 for short period range, from 0.677 to 0.837 for intermediate period range, from 0.602 to 0.812 for long period range in natural logarithm. The inter-earthquake uncertainties can be computed and became 0.450, 0.510 and 0.555 in a natural logarithmic standard deviation, respectively for short period, intermediate period and long period. The coefficient of correlations of L(x) with respect to D varied from -0.577 to 0.111 for short period, from -0.580 to 0.302 for intermediate period and from -0.523 to 0.308 for long period. From the Table 3-12, the intra-earthquake uncertainties of short period for vector component of SA by the Fukushima relation ranged from 0.608 to 0.863 for short period range, from 0.574 to 0.730 for intermediate period range, from 0.581 to 0.781 for long period range in natural logarithm. The inter-earthquake uncertainties computed became 0.468, 0.378 and 0.282 in a natural logarithmic standard deviation, respectively for short period, intermediate period and long period. The coefficient of correlations of L(x) with respect to D varied from -0.390 to 0.186 for short period, from -0.370 to 0.452 for intermediate period and from -0.200 to 0.682 for long period. The above results involving intra-, inter-uncertainties and correlation coefficients from the Fukushima attenuation relation shows slightly smaller than those from the Annaka attenuation relation. Especially, the smaller uncertainty from Fukushima relation defined on the seismic bedrock show the site effect is larger in long period than in short period.

3.4.2.Homogeneity of logarithmic deviation In the Section 2.1., it has been assumed that the logarithmic deviation L(x) constitutes a homogeneous two-dimensional stochastic field. It is necessary to examine the homogeneity of L(x) before evaluating the macro-spatial correlation model. Figures 3-46 to 3-50 show the distributions of L(x) to the minimum distance to the fault plane D. The moving averages of L(x) are also shown in the above figures. Its windows-width is 20 km. The mean value ML and standard deviation σ of L(x) within the widow are also calculated, and the ranges of the ML with plus-and minus-one-σ, i.e. (ML±σ), 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 goodness of fit has some dependency on the fault distance. It is observed from the figures that the dependency of the Fukushima relation is slightly smaller than those of the Annaka relation. It is correspondent with the correlation coefficients ρ LD listed in Tables 3-11 and 3-12. On the other hand, from the curves of (ML±σ), the moving averages of standard deviation σ do not obviously show this distance-dependency. The value of moving average of standard deviation is about 0.7 with small variation. As a result, the assumption which the logarithmic deviation L(x) of SA constitutes a homogeneous two-dimensional stochastic field can be approximately satisfied. It will lead to simple treatment of SA with the two mean attenuation relations.

(a) (d) (b) (e) (c) (f) Figure 3-46. Homogeneity of L(x) of SA of the Tottori-ken Seibu Earthquake

(a) (b) (d) (c) (f) Figure 3-47. Homogeneity of L(x) of SA of the Geiyo Earthquake (e)

(a) (b) (d) (c) (f) Figure 3-48. Homogeneity of L(x) of SA of the Miyagi-ken-oki Earthquake (e)

(a) (b) (d) (c) (f) Figure 3-49. Homogeneity of L(x) of SA of the Miyagi-ken Hokubu Earthquake (e)

(a) (b) (d) (c) (f) Figure 3-50. Homogeneity of L(x) of SA of the Tokachi-oki Earthquake (e)

3.4.3.Proposal of new macro-spatial correlation model for SA The data observed from the five earthquakes are fully used to build up the macro-spatial correlation model. Through statistical analysis, from Equations (2-17) to (2-18), the discrete values of the auto-covariance functions C LL and the normalized auto-covariance functions R LL with respect to the separation distance are calculated. The number of pairs of sites (x a, x b ) within h h/2 < x a x b < h + h/2, with interval h = 4 km, are shown in Figure 3-17. Similarly to PGA, small number of data pairs in the group have effect on the evaluation of auto-covariance in this group. An exponential decaying function is adopted to evaluate the correlation model as Equation (2-19). The regression curves is show in the Figures 3-51~3-55. The values of b, so-called a correlation length, can be obtained, and the results are listed in Tables 3-13 and 3-14, for mean component by Annaka relation and vector component by Fukushima relation in different period respectively. Table 3-13. Correlation length b (km) for mean component of SA by Annaka relation Earthquake Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Short Period 16.7 31.3 42.0 32.0 64.0 Intermediate Period 16.9 15.6 17.2 22.4 53.5 Long Period 33.9 34.2 22.4 17.6 45.5 Table 3-14. Correlation length b (km) for vector component of SA by Fukushima relation Earthquake Tottori-ken Seibu Geiyo Miyagi-ken-oki Miyagi-ken Hokubu Tokachi-oki Short Period 12.5 10.2 37.0 33.1 54.9 Intermediate Period 34.6 23.8 17.0 35.0 36.7 Long Period 77.0 48.1 17.0 31.7 24.2 Note that in the Figure 3-51, there is an unreasonable data plot in the closer range of separation distance of 4 km, which shows negative correlation. This is mainly because the number of data pairs in this range is only eight, which is too small to determine the correlation. 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 of sites are made of granite rock. These detail and precise ground conditions cannot be properly reflected in the past empirical relations. In the calculation of b-values, these data are ignored as mentioned in Section 3.2.3

(a) (b) (d) (c) (f) Figure 3-51 Normalized Auto-Covariance Function of SA of the Tottori-ken Seibu Earthquake (e)

(a) (b) (d) (c) (f) Figure 3-52. Normalized Auto-Covariance Function of SA of the Geiyo Earthquake (e)

(a) (b) (d) (e) (c) (f) Figure 3-53. Normalized Auto-Covariance Function of SA of the Miyagi-ken-oki Earthquake

(a) (b) (d) (c) (f) Figure 3-54. Normalized Auto-Covariance Function of SA of the Miyagi-ken Hokubu Earthquake (e)

(a) (b) (d) (c) (f) Figure 3-55. Normalized Auto-Covariance Function of SA of the Tokachi-oki Earthquake (e)

3.4.4.Discussions It can be observed, from the Figures 3-51~3-55 and Tables 3-13~3-14, that the values of the correlation length b for the two attenuation relations are of the same order of magnitude, raging from 10 to 80 km in all period range. This implies that the spatial correlation characteristic of seismic ground motion has approximately similar tendency regardless of the mean attenuation relations used as well as earthquake type or regions in Japan. The variation of the correlation length b calculated in different period from the two relations seems similar. Examining the difference in the regions of interest in Japan, the b-values in the short period are smaller in Southwestern Japan than those in Northeastern Japan, while it is contrast in the long period. The b-value of the Tottori-ken Seibu Earthquake in long period is calculated to 77 km with Fukushima relation. It is possibly overstated because the fault distance-dependency of L(x) is so large, 0.682, that the correlation length b is overestimated. The same problem can be seen from the Tokachi-oki Earthquake with Annaka relation, that the fault distance-dependency of L(x) is over 0.5 to overestimate the correlation lengths which are 64.0 km, 53.5 km and 45.5 km for short period, intermediate period and long period respectively. The above results are consistent with the results of PGA and PGV, that is, the difference in regions of b-values of SA in the long period is similar with the case of PGV, and the difference in regions of b-values of SA in the short period is similar with the case of PGA, which are shown in Figure 3-56. It can be explained that PGA is strongly correlated with short period about while PGV is strongly correlated with long period 55). Also regardless of the ground motion measures defined in the different ground, the tendency is similar, that is, b-value of PGA defined on the surface ground tends similar with that of SA defined on the seismic bedrock, and the b-value of PGV defined on the engineering bedrock tends similar with that of SA defined on the seismic bedrock.

(a) (b) (c) (d) (e) Figure 3-56. Difference of b-values for PGA, PGV and SA

3.5. Results from the Spatial Correlation Analysis In this chapter, using the macro-spatial correlation model described in Chapter 2, the spatial correlations of seismic intensities PGA, PGV and SA were stochastically studied respectively from five earthquakes occurred in resent years in Japan. For PGA, two components, mean component and maximum component, were analyzed with three attenuation relations respectively. The results were listed in Table 3-5 for the mean component and Table 3-6 for the maximum component. The correlation length for PGA falls in 11-60 km, and the correlations of two components are similar with the same attenuation relation. The Midorikawa-Ohtake relation gave better results than other two relations. Difference in the region can be observed that the correlation lengths, 31-47 km, in the Northeastern Japan tends larger than those, 11-13 km, in Southwestern Japan. For PGV, two components, mean component and maximum component, were analyzed with two attenuation relations respectively. The results were listed in Table 3-9 for the mean component and Table 3-10 for the maximum component. The correlation length for PGV falls in 21-50 km, and the correlations of two components are similar with the same attenuation relation. The Midorikawa-Ohtake relation gave better results than other two relations. Difference in the region is not obvious as those of PGA since the correlation analysis is evaluated on the engineering bedrock. The correlation lengths fall in 21-45 km in the Northeastern Japan and 21-50 km in Southwestern Japan. For SA, the period was divided into three regions, short period range, intermediate period range and long period range. Correlation on the mean component was analyzed with the Annaka relation in each period region as well as correlation on the vector component was analyzed with the Fukushima relation in each period region. The results were listed in Table 3-13 for the mean component and Table 3-14 for the vector component. The correlation length for SA falls in 10-64 km for short period range, 15-54 km for intermediate period range, and 17-77 km for long period range. The results of SA are consistent with the results of PGA and PGV. Difference of SA in the region tends to be similar to those of PGA in the short period range and those of PGV in the long period range. Figures of the normalized auto-covariance function of PGA, PGV and SA show the exponential decaying function adopted in the Equation (2-19) 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 less 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 which has been explained. Other factors can also have effect on the variation of the correlation length.

There are many attenuation relations 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 56). To reduce this uncertainty, more physics-based attenuation relationship is necessary. In other word, 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 57). 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 term of parameter c in Midorikawa-Ohtake relation. Though the difference of the logarithmic deviation L(x) corrected with µ L of the relations is not obvious, in the Tottori-ken Seibu Earthquake and the Miyagi-ken Hokubu Earthquake of which the source depths H are shallower than 30 km (Figures 3-12, 3-15, 3-29 and 3-32), the variation of the correlation length in the Midorikawa-Ohtake relation is smaller that that in the Annaka relation or Fukushima relation. Although the effect of rupture directivity on ground motions was reported in the literatures, the distance range of this effect is about a quarter of the rupture length away from the rupture, about 20 km 58, 59, 60). Considering the five earthquakes in this study, there are few data observed within this range from Figures 3-7~3-11, Figures 24~28 and Figures 3-41~45. As a result, the directivity effect can be ignored in the macro-spatial correlation analysis. As is known, the Moho is the boundary between the crust and the mantle in the earth where seismic waves change velocity. When the earthquake occurs beneath this discontinuity, the wave propagation will be greatly changed. Equations (2-3) and (2-6) in the Midorikawa-Ohtake relation take 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, while this propagation effect is not considered in the Annaka relation and the Fukushima relation. Figures 3-8, 3-9, 3-11, 3-25 3-26 and 3-28 show this difference in which the source depths H are deeper than 30 km. The logarithmic deviation from the Midorikawa-Ohtake relation shows more homogeneous than that from the Annaka relationship which shows obviously negative correlation with respect to the fault distance. Therefore, the correlation lengths with the Midorikawa-Ohtake relation are smaller than those with the Annaka relation or the Fukushima relation, as it is shown that the correlation length is over-estimated by the Annaka attenuation relation and the Fukushima attenuation relation. Even though these effects are considered in the Midorikawa-Ohtake relation, the correlation length b calculated comprises uncertainties, such as inter-earthquake uncertainty and site effect. Because the PGV of the two attenuation relation is defined on the engineering bedrock and the SA is defined on the engineering bedrock or seismic bedrock, the PGV or SA obtained from the

acceleration time history on the ground surface has to be transferred to that on the engineering bedrock or seismic bedrock by the average amplifications for motion (RPGV, RSA) in Equations (2-11) and (2-12) respectively. The uncertainties of site effect associated with Equations 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 the attenuation relations of PGV and PGV. For SA, the inter-earthquake uncertainty is suggested as the function of period. Due to the tectonic structures of the Japan islands, the abnormal Q structure in the subduction zone of the 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 61). The inter-earthquake uncertainty also contributes to the slight difference of the correlation models in the Northeast Japan from those in the 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 relation is still large. Incorporating this correlation model associated with those physical phenomena inherent to earthquakes not described in the attenuation relation, the prediction will be greatly improved. Although the attenuation relation can be improved by introducing new parameters, more complicate attenuation relation will increase the difficulty in the usage, and additional information is necessary, such as the correction term in the Northeast Japan which is difficult to obtain. From the analysis, the correlation lengths calculated with the Annaka relation and Fukushima relation are overestimated compared with the results of the Midorikawa-Ohtake relation in the Tables 3-5s, 3-6, 3-9 and 3-10, because the logarithmic deviation L(x) is more homogeneous with the Midorikawa-Ohtake relation. In this study, the macro-spatial correlation models of PGA and PGV associated with the Midorikawa-Ohtake attenuation relation are proposed. The parameters in this relation can also be easily found from the websites 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.

4. APPLICATIONS OF MACRO- SPATIAL CORRELATION MODEL

4.1. Real-Time Prediction on Seismic Ground Motion Intensity Real-time seismic monitoring, especially for strong earthquakes, is an important tool for seismic hazard mitigation. It provides valuable near-real-time information for rapid earthquake emergency response, thereby mitigation the loss. The concept can be illustrated as Figure 4-1. Existing Prediction σ σ Occurrence of Earthquake Quick Report Attenuation Relation Existing Prediction Macro-Spatial Correlation Model σ σ Epicenter and Fault Information Observatories of Strong Motion New Prediction Figure 4-1. Comparison of Existing and Proposal Prediction of Strong Motion Just after the occurrence of Earthquake in some area, the location of epicenter and ground motion intensities in some major cities where observation stations are located can be reported immediately by JMA. Though the ground motion intensity can not be measured directly in the wider area without the observation stations, it can be effectively predicted with the macro-spatial correlation model together with the mean attenuation relation, which reduces the uncertainty more than only with the past attenuation relation. The uncertainty of an attenuation relation can be decomposed into two uncertainties 9) as in Equation (4-1) in general. ε = ε ε (4-1) P S where ε denotes the total uncertainty of an attenuation relation. ε P, the intra-earthquake uncertainty, which is assumed to be log-normally distributed with zero mean and standard deviation σ P, denotes the modelling uncertainty from site-specific factors; and ε S, the

inter-earthquake uncertainty, which is assumed to be log-normally distributed with zero mean and standard deviation σ S, denotes the modelling uncertainty from common factors for all sites. Assuming independence of the two uncertainties, and ε S has a perfect correlation between different sites in a single event. On the other hand, the spatial correlation of ε P can be evaluated by the macro-spatial correlation model proposed in Equation (2-19). Then the total standard deviation, σ, and the co-variances meet the relations Equations (4-2) and (4-3). σ = σ + σ (4-2) 2 2 2 P S Cov( L, L ) = R ( h) σ + σ (4-3) 1 2 2 2 LL P S where L 1 and L 2 are logarithm random components of two sites and σ is the common standard deviation for L 1 and L 2. x 1 x 2 x 3 y =? Observed Site Unobserved Site Figure 4-2. Prediction of Strong Motion for Unobserved Site A seismic motion intensity which is not observed in Figure 4-2. can be estimated stochastically using seismic motion intensities observed nearby. When logarithm residual values at n sites are t observed as = (,,, ) X, the conditional probability density function of the x1 x2 x n logarithm random component at a sites is represented as: f ( y) Y X x = f Y, X f X ( y, x,, x ) ( x,, x ) 1 1 n n = (4-4) where stochastic variable Y is the logarithm random component at a focused site. Y follows a normal distribution. The conditional mean ( Y X=x ) and the conditional standard deviation ( Y X=x ) of Y are obtained:

2 = σy X= x + t Y X= x 2 XY c c XY t 1 1 ( C C ) µ σ x x σ (4-5) 2σ t 1 2 Y = = 1+ Y Cc Y σ σ σ σ σ X x X X (4-6) where C X is the covariance matrix of (x 1,, x n ) which consists of covariance obtained from Equation (4-3); XY and C c satisfy the following equations. C Y, X 2 t σ σ XY = σ XY CX (4-7) t c = Y Y C CX σ X σ (4-8) X where C Y,X is the covariance matrix of (y, x 1,, x n ). Here, the conditional probability density function of the seismic ground motion intensity at a site of interest is formulated as following under the condition that logarithm residual values at n sites are pre-recorded. f 2 1 (ln a ln A µ ) ( a) = exp A X= x A X= x 2 2πσA X= x 2σ A X= x (4-9) where a stochastic variable A is the seismic ground motion intensity at the site and A is the mean value of the attenuation formula at the site. Using spatial correlation model proposed in this study, the PGA of a site can be estimated stochastically from PGA of other n sites nearby. An example was illustrated in the Figure 4-3. The PGA of station TKCH01 was predicted by nearby stations in the Tokachi-oki Earthquake. The parameters of earthquake was taken from Tables 3-1 and 3-2, and the correlation length adopted was from the analysis results in Table 3-8 in Section 3.2. In the case of n=0, the prediction was made only by attenuation relation without nearby stations. From the figure, when the station number n increases from 0 to 3, the predicted PGA becomes closer to actually observed value 136.9 gal. More importantly is on the 41% decrease in the variation from 0.825 to 0.491. This implies that the accuracy of the prediction becomes better increasingly with proposed macro-spatial correlation model. Furthermore, the site without observatory can also be predicted by this model, and the seismic map can be estimated for the all area. Contrary to the traditional prediction, the prediction result of the site includes the correlation information of nearby strong-motion and with a smaller uncertainty.

σ σ σ σ Figure 4-3. An Example of Prediction of PGA

4.2. Evaluation of Joint Exceedance Probability Considering networks of infrastructure shown in Figure 4-3, a control-center consisting of multiple buildings, traffic line with bridges, and radio network with towers, the network will still work if one of members is damaged. In order to keep network on working, it is important to evaluate the joint PDF exceeding certain level when the seismic design are adopted on system reliability. In the previous studies on the probabilistic seismic hazard analysis, the spatial correlation of seismic intensities at multiple sites has not been clearly considered. Either perfect correlation or perfect independency is usually assumed between the sites. The macro-spatial correlation was applied to evaluation of joint exceedance PDF herein. Building 1 Building 3 Bridge 1 Bridge 3 Building 2 Bridge 2 (a) Control Center Tower 1 (b) Traffic Line Tower 4 Tower 2 Tower 3 (c) Radio Network Figure 4-4. Network of Infrastructure For simplicity, the system like Figure 4-4 can be categorized as two cases, a series system and a parallel system as in Figure 4-5 in which either system is made of two elements. The probabilities of system failure can be expressed as CDF in Equations (4-10a, 4-10c) and (4-10b) for series system and parallel system, respectively. The system reliability can be analyzed subsequently.

A1 A1 A2 (a) A Series System A2 (a) A Parallel System Figure 4-5. Series and Parallel System Model For Series System: Prob P{ A a A a } = > > (4-10a) 1 1 2 2 For Parallel syetem: Prob P{ A a A a } = > > (4-10b) 1 1 2 2 For the series system, the Equation (4-10a) can be rewritten into Equation (4-10c). { A a} { A a } { A a A a } Prob = P > + P > P > > (4-10c) 1 1 2 2 1 1 2 2 in which A 1, A 2 are seismic ground motion, a 1, a 2 is the seismic level. As cab be seen from Equation (4-10), the joint CDF or PDF is necessary both for series system and parallel system. Therefore the joint PDF or CDF Prob P{ A a A a } = > > is only focused on here. Notice 1 1 2 2 the Equation (2-14) and (2-15), the joint CDF can be rewritten into Equation (4-11) or (4-12). a 1 a 2 Prob = P ln ε1 > ln lnε2 > ln A1 A2 (4-11) or Prob P{ L l L l } = > > (4-12) 1 10 2 20 in which L is the logarithmic deviation and a 1 a 2. L follows normal distribution l10 = ln, l20 = ln A1 A2 with zero mean value and standard deviation σ L. And the correlation of L 1 and L 2 can be obtained from Equation (2-19) given the correlation length. Then the joint PDF of two sites or the joint exceedance probability can easily be calculated by Equation (4-13). 2 2 1 1 l 1 l 1 l 2 l 2 fll ( l1, l2) = exp 2R 2 2 2 LL + 2πσ ( 1 ) 21 ( RLL ) L R σ LL L σl σl σ L (4-13) Substitute Equation (4-13) into Equation (4-12), the joint probability can be Equation (4-14):

l10 l20 ( ) Pr = f l, l dldl LL 1 2 1 2 (4-14) Through coordinate transformation of l 1 and l 2, Equation (4-14) can be simpler from as Equation (4-15): Pr = 1 Φ + Φ +Φ f ( u )du 2σ 1 σ 1 σ 1 l10 + l20 u1 2l10 u1 2l20 U1 L RLL l10 + l20 L + RLL L + RLL 1 1 (4-15) 2σ L 1 RLL in which () is CDF of the standard normal distribution, f ( u ) is PDF of the standard U1 1 normal distribution, u 1 and u 2 meet the next Equation. u1 l 1 1 =Λ u l 2 2 (4-16) where is transformation matrix, 1 1 1 Λ=. 2 1 1 Figures 4-6~4-8 give examples of contours of joint PDF and joint exceedance probability of two sites, P1 and P2, with the correlation length 25 km. The contours of joint PDF of three cases with separation distance 12 km, 25 km and 50 km show that the smaller the separation distance, the more elongated the contour of joint PDF. Through the same method, the seismic hazard surface can be evaluated for the system mentioned at the beginning of this section, or other similar system that can be expressed as series or parallel system. In the seismic insurance field, earthquake insurers have always ideally needed to construct an exceedance probability surface for loss to a portfolio covering multiple sites distributed over a wide geographical region. The next section will give an example of portfolio of assets using joint exceedance probability.

(a) (b) Figure 4-6. Contour of Joint PDF and Joint Exceedance Probability with h = 10 km (a) (b) Figure 4-7. Contour of Joint PDF and Joint Exceedance Probability with h = 25 km (a) (b) Figure 4-8. Contour of Joint PDF and Joint Exceedance Probability with h = 50 km

4.3. Evaluation of Seismic Risk for Portfolio on Sites Management of widely-located building, so called portfolio analysis, has become prevalent in the field of earthquake risk management in recent years, whereby the joint exceedance probability of sites is of major concern. The exceedance probability of portfolio loss began to be developed in the late 1980 s, however, commercial impetus to improve insurance loss modeling was given by two destructive earthquakes in California 62) : Loma Prieta (1989) and Northridge (1994). There are some definitions of risk. In narrow sense, it is defined as an expected loss, that is, the product of the probability of occurrence and the relevant loss as in Equation (4-17) or (4-18). Risk ( R) = Probability of Loss( P) Loss ( C) (-17) or n R = R = P C i i i i= 1 i= 1 n (4-18) In the broad sense, the risk is defined as the loss under certain probability. So as the measures of risk, PML(Probable Maximum Loss) and expected loss are often adopted in the quantitative evaluation of risk. PML is a financial measure of the seismic vulnerability of a building. It represents a loss estimate that assumes a 90% confidence of not being exceeded during a specified period of time (e.g. 475 years) or earthquake event. In other words, there is only a 10% chance that the actual damage would be higher than the PML for the given time period or earthquake event 63). The PML is an estimate of the cost to restore a structure to pre-earthquake condition, and is expressed as a percentage of the building replacement value. Figure 4-9 shows the relation between the expected loss and PML. The first term in the Equation (4-17) is usually designated as hazard curve or hazard surface. Since the multiple-site risk is studied, the hazard surface is adopted herein. In order to extend the hazard analysis for a single site to a portfolio of multiple sites, the Cornell 64) approach can be followed in a similar way. In this approach, an earthquake hazard is commonly defined as the frequency of exceedance, or probability under a suitable model, of a particular level of intensity or ground motion. In the section 4.2., the joint exceedance PDF or CDF has been derived for multiple sites considering the spatial correlation under an earthquake. Based on the Cornell theory and the macro-spatial correlation model, the hazard surfaced can be obtained as described in the following.

Figure 4-9. Illustration of Expected Loss and PML Note that joint exceedance probability in Equation (4-10) can be rewritten into a conditional probability that an Intensity Measure Type (IMT) will exceed an Intensity Measure Level (IML) given the particular Earthquake (Location X) and presuming the occurrence of a given Earthquake (Magnitude m) as Equation (4-19). It should be understood as a joint probability of multiple sites rather than a single site. Prob( IMT IML m,x) (4-19) Then the annual exceedance probability is determined from a sum over distance and magnitude: λ( IMT IML) = I ν Prob( IMT IML m, X )Prob( M = m m, X )Prob( X =x ) i i i i i i i i i i= 1 M X (4-20) where ν i is the mean annual rate of the i-th earthquake source. Assuming a Poissonian model, the total exceedance probability over time T is then Prob T ( IMT IML) 1 e λ - T ( IMT IML) = (4-21) In the Equation (4-20), the probability of magnitude and the mean annual rate ν i can be obtained with the Gutenberg-Richter Relationship. As soon as the region to be evaluated is determined the probability of the source can be estimated. Therefore the joint hazard surface can be obtained based on the joint exceedance probability. When it is linked with the seismic loss function C, the expected loss can be easily calculated. The simplest examples are two cases, a series system and a parallel system shown in the Figure 4-5 with two members A 1 and A 2. The four domains of joint