The study on the ground motion is one of important parts in the earthquake engineering since the ground motion is the essential link between the earthquake source and the structural response. A seismic risk analysis, for example, can be partitioned into three elements: hazard analysis, vulnerability analysis and loss analysis, where the prediction of the ground motion is a key issue. These factors motive this study to improve the prediction of the ground motion with new perspectives. On the other hand, the accumulation of observations due to the deployment of the seismographic networks in the recent years makes this study possible.

The risk analysis is simplified into three basic cases. They are the cases of a single structure at a single site, multiple structures at a single site, and multiple structures at multiple sites. Although the empirical ground motion attenuation relations have been widely adopted to provide a median value and an uncertainty of prediction to the probabilistic seismic hazard analysis (PSHA), the required information of the probability distribution of the ground motion are different case by case, which cannot be provided only by the existing attenuation relation. This study aims to improve the prediction of the ground motion for each basic case of the risk analysis. The complex risk analysis can then be implemented straightforward through different combinations of three basic cases.

New perspectives were explored based on the empirical data observed from the recent earthquakes for the probabilistic prediction of the seismic ground motion, which is briefly described in the following.

In Chapter 1, the general background of this study was introduced and the objectives of this study were clearly stated. The past studies on the prediction of the ground motion associated with this study were intensively reviewed. Chapters 2 through 4 gave the preliminary required in the following chapters in this study.

In Chapter 2, some basic concepts that would be used in the analyses were defined and classified into four groups. One is the concepts of the risk and elements of the risk. One is the concepts associated with the earthquake, fault, ground motion, and so on. One involves the mathematical concepts such as the deterministic and probabilistic models, likelihood functions, and residual. The last group involves the uncertainty: aleatory and epistemic uncertainties. The former is associated with the randomness while the latter is associated with lack of the knowledge. The latter uncertainty is classified into model and statistical uncertainty.

Chapter 3 introduced the Bayesian methodology. The posterior and predictive analyses were described for noninformative and informative cases, respectively. The advantage of allowing for the prior information in the Bayesian methodology makes one incorporate the prior knowledge, such as knowledge from other disciplines and engineering judgment, etc., into the inference of the unknown parameters. Another advantage is that the statistical uncertainty of the parameters can be accounted for from the Bayesian posterior distribution of the parameters. Another distinctive advantage of the Bayesian methodology is the predictive analysis. Rather than a point estimate of the prediction, a predictive distribution is defined as the expectation of the ground motion on the posterior distribution of the unknown parameters. Therefore the total uncertainty associated with the prediction can be accounted for from the predictive distribution. The features of the Bayesian approach and differences from the traditional estimation were described in detail, which aims to make one familiar with this method.

Chapter 4 dealt with the ground motion. The characteristics of the ground motion are affected by the source, path and site effects, each of which shows large complexity to be fully characterized. After brief review of the two prediction methods of the ground motion: theoretical and empirical methods, the past development of the empirical method were described in detail for one to understand the limitations of the existing approach. These drawbacks include: the median value given by the existing attenuation relation cannot represent those of the specific site, the uncertainty represented in terms of a standard deviation is constant for any sites, and the predicted values are independent of each other. These drawbacks lie in the factors that the data observed from the multiple events and multiple sites are processed together in the development of the attenuation relations with assumption that the observations are independently identically distributed. At the end of this chapter, the uncertainty of the prediction of the ground motion was clarified according to the classification of the uncertainty in Chapter 2. The uncertainty expressed in terms of the standard deviation of the existing attenuation relation was discussed.

Chapters 5 through 7 proposed the new perspectives for the probabilistic prediction of the ground motion, which compensate the shortcomings of the existing attenuation relation mentioned above.

Chapter 5 involved the first case of risk analysis, that is, a single structure at a single site, in which the site-specific attenuation relation was developed for the prediction of the ground motion in lieu of the existing attenuation relation. The conventional site-specific hazard analysis is made with the existing attenuation relation on the reference baseline (a soil category, e.g., rock) and the site amplification factor of the specific site to the reference site. The predictions from this transformation procedure are inaccurate. First, the prediction could have a bias which is the difference between the medians of the observed and the calculated motions for the specific site, because the existing attenuation relation is fitted with observations from different sites and different events, and the site amplification factor cannot fully account for local site conditions. Second, the predictions could have an incorrect dispersion relative to observation. Because the existing attenuation relation is developed with different sites, the uncertainty of the relation only represents the average characteristic of uncertainty of multiple sites. Furthermore, the uncertainty of the site amplification factor is ignored although there is a broad range of soil category. For example, Site category B, rock, in NEHRP provisions is defined as AVS30 between 760 and 1500 m/s, where AVS30 is average shear wave velocity in the upper 30 m.

In this study, the ground motion at a specific site was predicted with the site-specific attenuation relation. Rather than developing new attenuation relations, we introduced a correction term to the existing past attenuation equation in common use. The correction term was constructed as the function of the magnitude and distance, and the unknown parameters in the correction term was estimated with Bayesian approach based on the observations at the specific site. The advantages of this analysis procedure are: (1) The use of Bayesian updating technique can incorporate our prior knowledge on the ground motion (e.g., seismological knowledge, engineering judgment) and the observations; (2) Bayesian method can account for the uncertainty of the unknown parameters, which contributes the statistical uncertainty to the total uncertainty of the prediction due to limited number of data observed at the specific site; (3) The predictive density of the ground motion, which is averaged over the posterior distribution of the parameters, is obtained with Bayesian method rather than point estimate of the ground motion, therefore, the prediction accounts for the total uncertainty; (4) The structure of the correction term makes it possible to examine the uncertainty of prediction for different magnitude and different distance, especially, for the area with larger magnitudes and closer distance which is of major interest in the engineering, whereas few observations are available.

In this study, the site-specific attenuation relations for PGA, PGV and Sa were developed, respectively, for the 1558 sites of K-NET and KiK-NET. Both the noninformative and informative priors were adopted in the framework of Bayesian updating. The effects of the noninformative and informative prior on the estimation of the parameters for large-and-moderate size and small size of samples were discussed, respectively. It shows the estimation tends to similar for the different priors when large-and-moderate sizes of observations are available. The estimate of the uncertainty shows they are different from site to site, which implies the assumption of the identical distribution adopted in the existing attenuation relation cannot be satisfied. Three applications were illustrated, including the prediction of the ground motion, the development site-specific attenuation relation for the Hongo campus of the University of Tokyo, the site-specific risk analysis for three buildings in the Hongo campus.

Chapter 6 was devoted to the second case of risk analysis, that is, multiple structures at a single site, in which the joint distribution of the ground motion intensity measures (Sa) for multiple structures is necessary. Under the mild assumption that the joint and conditional distribution of the ground motion is assumed as lognormal, the correlation coefficient between two spectral values at different periods is necessary in addition to the median and uncertainty given by the existing attenuation relation. It is easily understood that the response of the different structures at the same site are somewhat correlated, since they are produced by the same input ground motion, although the response of the structure is only represented by a 5% damping linear response spectral measure of SDOF system in this study.

The correlation model of the spectral accelerations at different periods was developed in this study based on empirical data observed from 31 earthquakes. The model was expressed as a linear function of the log of the ratio of two periods. The results showed the correlation coefficients predicted with the proposed model can meet well with those calculated from the empirical data. The simple form of the model makes the use in practice with great ease. The joint distribution of the ground motion can fully be characterized by the median vector and covariance matrix by using the correlation model proposed in this study as well as the existing attenuation relation. The risk analysis can then follow the conventional procedures. The effects of the different attenuation relations and different soil conditions were examined. It shows the correlation is insensitive to these effects. Three applications also illustrated, including the simulation of the ground motion, joint hazard analysis and estimation of the joint probability of failures.

Chapter 7 involved the third case of risk analysis, that is, multiple structures at multiple sites with one at each site, in which the joint distribution of the ground motion intensity measures, for multiple structures was necessary. Under the mild assumption mentioned above, in addition to the median and uncertainty given by the existing attenuation relation, only the spatial correlation coefficient of the ground motion between two separated sites is needed to fully define the joint distribution and proceed with PSHA. In this study, the macrospatial correlation model was developed for PGA, PGV and Sa, respectively. The model is expressed in a simple form of an exponentially decaying function of the separation distance between two different sites. The only one parameter in the model, called a correlation length, was estimated for 26 earthquakes. In spite of the different attenuation relations and different components of the ground motion, the correlation lengths are the same and most of them fall in the range of 10 to 30 km. Applications to the simulation of the ground motion, evaluation of the joint probability of exceedance and portfolio analysis are illustrated at the end of this chapter.

In Chapter 8, conclusions were drawn and the potential applications and future studies were addressed. The site-specific attenuation relation, the correlation model of spectral values at two periods, and the macrospatial correlation model were developed based on the empirical data, respectively. They are corresponding to the different requirement of the ground motion in three basic risk analyses. Some important reminders should be pointed out. First, three new perspectives are not isolated. They can be and must be combined into different prediction of the ground motion and different risk analysis. Second, the existing attenuation relation was not abandoned but was fully utilized in this study. New perspectives were proposed to compensate the shortcomings of the existing attenuation relation so that the ground motion can be appropriately predicted. Finally, one cannot be limited on the calculated results, but pay more attention to the methodology proposed in this study.

地震のリスク評価にはハザード評価、脆弱性評価、損失評価など3つの要素に分けられることができ、その中でも地震動予測は地震リスク評価に欠かせない重要な要素である。このような背景を踏まえて、王敏君は、従来から採用されている地震動の予測方法を大幅に改善し、地震動に対して適切な評価を行い、新たな観点から地震によるリスク評価を行う基盤を構築した。なお、この研究が実施可能となったのは、近年の地震観測網の拡大と、大量の観測データが蓄積かつ公表されるようになったからに他ならない。

まず、地震リスク評価は、次に示す3つのケースに分類できる。(1)特定サイトにおける単一建物の地震リスク評価、(2)特定サイトにおける複数建物の地震リスク評価、(3)複数サイトにおける複数建物の地震リスク評価(ただし、各サイトの単一建物を前提とする)。過去より地震動評価には、地震動距離減衰式が幅広く採用されている。その理由は、距離減衰式から得られた地震動の中央値とばらつきを確率論的ハザード評価に容易に取り込むことができる利点があるからである。しかし、上記に定義したように、地震リスク評価における基本ケースが異なると、必要情報量も異なり、既往手法では取り扱うことが不可能となる場合がある。本論では地震のリスク評価におけるこれらの3つの基本ケースに着目して、既往の地震動予測方法を改良し、これらの組合せを用いて更に複雑な地震リスク評価を実現させる基盤を築いている。

第1章では本論の研究背景・目的を明確にした上で、既往関連研究調査を行った。第2章では本論で使用する専門用語を定義する。中でも、不確定性に関わる重要な用語があることが紹介された。不確定性には、偶然的な不確定性と認識論的な不確定性があり、後者はさらにモデル不確定性と統計的不確定性に分類される。第3章ではベイズ予測方法論を紹介した。情報量が少ない場合に有効となるベイズ予測手法の基本についてレビューしている。第4章では地震動の特徴について述べている。地震動の特性には3つの影響因子、すなわち震源、伝播経路、サイト特性がある。各影響因子は非常に複雑なため、それに対するモデル化は現在においても研究途上である。ここでは、2種類の地震動の予測方法―すなわち理論的方法と経験的方法―を簡単に紹介した後に、経験的な距離減衰式の構築プロセスに基づいて経験的な予測方法の限界について考察している。

第5章では地震動のリスク評価における第1の基本ケース、つまり特定サイトにおける単一建物の地震リスク評価法を提案している。本論では、評価サイトに対する地震動予測はそのサイト固有の距離減衰式に基づいて行う。サイト固有の距離減衰式を構築する際には、既往の距離減衰式を基にして、新たな修正項を追加して既往の距離減衰式の予測偏差を無くし、その不確定性と評価サイトの観測データとの調和を実現している。この修正項は地震マグニチュードと距離の関数として表し、未知パラメータはベイズ更新方法を用いて観測データから推定する方法を提案した。この方法の利点として(1)地震動に対する主観的な事前情報(例えば、地震学、地質学的な知識や工学的判断など)と観測データを同時に考慮することができる。(2)パラメータの不確定性を説明することができる。つまり、全不確定性を評価する際に、データが限られたことによる統計的不確定性が考慮できる。(3)地震動の予測分布を算出する際に、未知パラメータを点推定するのではなく可能な範囲で積分している。(4)修正項は地震マグニチュードと距離の関数であるため、予測の不確定性は震度と距離によって変化する。また不確定性の評価にデータ数を反映することができ、データが多ければその予測精度も高まる効果が表現されている。つまり、大地震かつ近距離の観測データが非常に不足している領域では、予測精度は中地震かつ中等距離の地震による地震動の予測精度より低いことが表現できる利点がある

第6章では地震動のリスク評価における第2の基本ケース、つまり特定サイトにおける複数建物の地震リスク評価について検討している。この場合、複数建物が受ける異なる地震動の同時確率分布を評価する必要がある。ここで評価する地震動指標は異なる建物の加速度応答スペクトルである。地震動の同時条件確率分布が対数正規分布であると仮定すると、既往の距離減衰式が評価している地震動の中央値とばらつき以外に地震動指標間の相関係数を評価する必要がある。この点については、以下のように理解することができる。異なる建物の応答は違うが、同じサイトにある限り、受ける地震動は同じであるために、弾性応答加速度スペクトルを採用してもその応答値間に相関関係は存在すると考えられる。

第7章では地震動のリスク評価における第3の基本ケース、つまり複数サイトにおける複数建物の地震リスク評価について検討している。第2のケースと同じく、複数建物が受ける異なる地震動の同時確率分布を評価する必要があり、地動最大加速度、地動最大速度、応答スペクトル値の空間相関モデルを構築した。この相関モデルは指数関数形式を採用している。その変数は両地点の相対距離である。モデルの中で唯一のパラメータは相関距離であり、26の地震で観測したデータから得られている。異なる減衰距離式および地震動成分に対して分析しても相関距離は10~30km範囲に収まることがわかった。これらは、空間的に離れた複数建物を対象とするポートフォリオ解析に有効に応用可能である。

最後に、第8章では本論のまとめ及び耐震設計などの領域における応用分野と今後の研究について述べた。本論で提案する3種の地震動予測の改善方法は3種のケースの地震リスク評価と対応しており、これら3つの方法は独立な方法ではなく、これらの方法を組み合わせることで更に複雑な地震リスク評価ができる。最後に、既往の距離減衰式の利点を十分尊重しつつ、より新しい地震動予測手法を展開した所に、本研究の新しさと、将来への発展可能性があると言える。

よって本論文は博士(工学)の学位請求論文として合格と認められる。