SITE-SPECIFIC PREDICTION OF SEISMIC GROUND MOTION WITH BAYESIAN UPDATING FRAMEWORK

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1 th International Conference on Earthquake Engineering Taipei, Taiwan October -3, 6 Paper No. SITE-SPECIFIC PREDICTION OF SEISMIC GROUND MOTION WITH BAYESIAN UPDATING FRAMEWORK Min Wang and Tsuyoshi Takada ABSTRACT Prediction of the ground motion with the past attenuation relation is widely adopted in the seismic hazard analysis. However, it provides biased estimates of the ground motion for specific sites, and its proposed uncertainty is a fixed value common for any sites. The site-specific attenuation relation is developed in this study with Bayesian framework based on the data observed at a specific site. Rather than developing a new attenuation relation, a correction term in a linear model is developed to the existing past attenuation relation in common use. The relative uncertainties associated with the parameters inferred by Bayesian approach are also accounted for. Applications of the Bayesian updating at two actual sites are illustrated as examples. Keywords: Uncertainty, Site-specific, Attenuation relation, Ground motion, Bayesian framework INTRODUCTION Prediction of ground motion is one of primary interest in earthquake engineering. Past empirical attenuation relations are widely used for their convenience and efficiency, providing a vital link to the seismic hazard analysis. Over the past decades, many attenuation relations have been proposed using different sets of data, independent variables, functional forms, and regression methods. Persistent efforts have been continuously expended on studying attenuation relations of ground motion, which may be reviewed in the recent works (Compell, 985; Abrahamson and Shedlock, 997; Boore et al., 997). Apart from general independent variables chosen for attenuation relations: magnitude and distance despite of various definitions on magnitude and distance, recent attenuation relations have focused on the specific seismo-tectomic regions (Abrahamson and Silva, 997; Youngs et al., 997), i.e., active tectonic regions, subduction zones and stable continental regions. Some attenuation relations have been developed based on the ground motion records in Japan (Fukushima and Tanaka, 99; Annaka et al., 997). Also derived are attenuation relations for crustal, interplate and intraplate earthquakes based on strong ground motion records (Si and Midorikawa, 999), and for different focal depth (Midorikawa and Ohtake, ). The attenuation relation provides two important components to PSHA (Probabilistic Seismic Hazard Analysis). One is a median component and the other is an uncertainty component. Although it is asserted to be unbiased and gives a value of uncertainty, how much truth are they for a specific site? As is known, the past attenuation relation is developed by processing different records observed at different sites for regression. Then, it is unbiased for multi-sites, in other words, it may be biased for a specific site (Wang and Takada, 6). The bias may at least arise from the inability to account for the detailed local site conditions in the past attenuation relation. The term local site conditions is often Ph.D. student, Graduate School of Engineering, The University Of Tokyo, Tokyo, Japan, oh@load.arch.t.u-tokyo.ac.jp Professor, Graduate School of Engineering, The University Of Tokyo, Tokyo, Japan, takada@load.arch.t.u-tokyo.ac.jp

2 taken to mean soil conditions, but it can also encompass effects such as near-surface weathering and fracturing of rock or topographic effects that influence the character of ground motion. In general, any local geologic conditions that differ from the assumptions inherent in the ground motion equations used in the PSHA must be handled as local site conditions. In this context, the site-specific attenuation relation developed in this study is only applied for the specific site not others. It should be distinguished from the site-specific procedures for estimate of ground motion (Fallgren et al., 97; Guzman and Jennings, 976). The latters are in fact the soil-specific attenuation relations for the specific site category (or soil condition) with similar site characteristics. The prediction of ground motion from such relations can vary considerably depending on which equation is used. What does not seem to so vary is the magnitude of the uncertainty associated with these relations. This uncertainty, expressed as a factor of standard deviation, associated with almost all attenuation relations, is about from. to.7 in natural logarithm (Abrahamson and Shedlock, 997), and there has been little or no decrease in this uncertainty even through the use of more data or more complex methods of analysis (Douglas and Smit, ). Aforementioned, the median of ground motion from the past attenuation represents the average characteristics of multi-sites, so does the uncertainty. It may underestimate the uncertainty of ground motion for some sites or overestimate for others. Furthermore, the model parameters related to independent variables may be differently regressed with different data set. Event for the same site, they may be likely different for the prediction of ground motion by future earthquakes. This kind of uncertainty associated with the parameters is not taken into account in the past attenuation relation. The site-specific attenuation relation developed by Bayesian approach in this study can account for the uncertainty associated with the specific site and can deal with relative uncertainty associated with these parameters. Application of Bayesian approach to the attenuation relation was attempted on Mexico City (Ordaz et al., 99). A new attenuation relation of the Fourier spectrum was developed by estimating all the regression coefficients with the Bayesian approach. As is known, it need a sufficient number of data evenly distributed in the space of the variables while only 3 data were observed between 6 and 7 km away from the station CU for which the attenuation relation was developed. Additional constrained conditions had to be imposed on the estimate to avoid the unacceptable regression coefficients. In our study, a correction term is suggested as in a linear function of earthquake magnitude and distance with more flexibility to avoid the aforementioned problem. If there are few data available, the variables of the correction term should be reduced, say only the constant term remains. If there are sufficient data in the space of the variables, the correction term in a linear model of magnitude and distance is suggested. Model Selection SITE-SPECIFIC ATTENUATION RELATION The strong motion observation networks, such as K-NET and KiK-NET etc., were initiated in 996 in light of the lessons from the 995 Kobe Earthquake. It becomes possible to develop the site-specific attenuation relation for the observatories. With the aim of facilitating the use in practice, rather than developing new attenuation relations, we develop correction term to existing deterministic attenuation model in common use. Bayesian approach is adopted to assess the model parameters with observations. The site-specific attenuation relation in the general univariate model form is adopted as in Eq.. Without a special description, the natural logarithmic scale is used in this paper. y( x, θ) = gˆ ( x, β) + γ ( x, θ ) + ε () where y denotes ground motion intensity in natural logarithm, gˆ( x, β) is a selected deterministic model, x denotes the independent variables representing those characteristics of source, path and local conditions, β is regression coefficients in gˆ( x, β ), γ(x, θ) is the correction term for the bias inherent in the deterministic model that is expressed as a function of the variables x related to parameters θ, θ denotes the set of unknown parameters, ε is a random variable with zero mean and variance σ ε. Note that for given x, β, θ and σ ε, we have Var[y(x, β, θ, σ ε )] = σ ε as the variance of the model.

3 The function γ(x, θ) in Eq. corrects the bias in the deterministic model gˆ( x, β ). Without loss of generality, the linear model of the basic explanatory variables of magnitude m and distance r are selected for the correction term γ(x, θ) i.e. γ mrθ θ θ m θ r () (,, ) = + m + r where θ = (θ, θ m, θ r ), m is an earthquake magnitude and r denotes distance. Unknown parameters θ and σ ε in Eq. can be inferred by the Bayesian updating approach with the data observed at the site. The prediction of the ground motion then can be estimated from Eq.. Adopting Eq. is based on the considerations: () because the selected deterministic model ĝ (x, β) gives the common regression coefficients of m and r for different sites, the bias due to m and r can exist for a specific site; () since the probability density functions of model parameters θ m and θ r are estimated by Bayesian approach, the statistical uncertainty with different m and r can be examined, especially on the range of large magnitude and short distance which is of major interest in risk analysis. To update the parameters θ in Eq., it is ideal to have sufficient data evenly distributing over the m- r space. If this is not the case, the reduced form of γ(x, θ) is suggested, such as excluding m or r from the function, or the simplest form Eq. 3 γ ( θ ) = θ (3) In the following analysis, Eqs. and 3 are referred to as model A and model B, respectively. Model B cannot account for the statistical uncertainty associated with variables m and r. Uncertainty Associated with the Prediction Prediction model Eq. is then used to predict ground motion. The uncertainty associated with the prediction includes three specific types, inherent variability, model uncertainty and statistical uncertainty (Benjamin and Cornell, 97), which are described in the following. Uncertainty due to inherent variability is of aleatory nature and is associated with randomness of nature produced because the exact realization of a phenomenon cannot be determined with certainty, manifesting as variability over time for phenomena that takes place at a single location (temporal variability) or as variability over space for phenomena that takes place at different locations but at a single time (spatial variability), or as variability over both time and space. Even though the correction term γ(x, θ) is introduced to account for the local site conditions, Eq. is just a simplification of the real model, that is, inexactness. The uncertainty due to imperfections is called model uncertainty. This kind of uncertainty arises from missing variables and from our simplifying the function form. Model uncertainty is of epistemic nature since the model can be refined with the increasingly understanding of physical law. One can introduce more predictive variables such as site category, fault type, building size (the size of building where the instrument was located), or one can use better form of equations such as mixed magnitude-distance term to achieve magnitude-dependent distance scaling and distance-dependent magnitude scaling (McGuire, ). σ ε, standard deviation of ε, represents the inherent viability and model uncertainty in Eq.. Prediction error due to missing variables includes a systematic component (i.e. bias due to missing variables) and a random component (statistical uncertainty associated with its bias) (Ang and Tang, 98). Therefore, introducing the variables to refine the model will reduce the model uncertainty in σ ε. The site-specific attenuation relation Eq. refines the past attenuation relation, taking into account the local site conditions with the correction terms γ(x, θ) which correct the bias due to local site conditions in the past attenuation relation. The uncertainty σ ε can therefore be reduced. Ground motion predicted with the past attenuation relation is commonly assumed following the lognormal distribution with mean ĝ (x, β) and variance σ ε given magnitude and distance. It should not be forgotten that the lognormal distribution of ground motion is also conditioned on β and σ ε. That is, the uncertainty associated with the mean value is usually neglected. This kind of uncertainty can be categorized as statistical uncertainty present in the regression coefficients β and be of epistemic

4 nature. Statistical uncertainty lies in the statistics of the parameters from limited data. Past attenuation relation uses the results of a regression analysis of historical ground motion data for forecasting future ground motion. In extrapolating from the past to future prediction, there is not only uncertainty from the imperfect fit to the past data, but also uncertainty about how much the future will be like the past (Morgan and Henrion, 3). The accuracy of estimation of the model parameters β depends on the observation sample size. The smaller the sample size is, the larger the uncertainty in the estimated values of the parameters is. The uncertainties associated with the parameters can be dealt with Bayesian approach in this study. Bayesian Updating Theorem BAYESIAN PARAMETERS ESTIMATION The development of the site-specific attenuation in this study is based on the Bayesian model assessment technique. The Bayesian approach is capable of incorporating all type of available information, including laboratory test data, field observations and subjective engineering judgment as well as accounting for all the relevant uncertainties. The inference of the parameters Θ is based on the well known Bayesian updating rule (Box and Tiao, 973) f( Θ) = κ L( Θ) p( Θ ) () where f(θ) is the posterior distribution representing our updated state of knowledge about Θ; L(Θ) is the likelihood function representing the objective information on Θ contained in a set of observations; p(θ) is the prior distribution reflecting our state of knowledge about Θ prior to making the observations; and κ = [ L(Θ)pΘ)dΘ] - is a normalizing factor. The likelihood function is proportional to the conditional probability of making the observations for a given value of Θ. The prior distribution may incorporate any information about Θ that is based on our past experience or engineering judgment. When no such information is available, one should use a prior distribution that has minimal influence on the posterior distribution, so that inferences are unaffected by information external to the observations. Parameter Estimation With Baye s theorem expressed in Eq., the model parameters Θ = (θ, σ ε ), θ = (θ, θ m, θ r ) in Eq. can be estimated with the observational data. For the set of parameters Θ = (θ, σ ε ), it is generally assumed that θ and σ ε are approximately independent so that p(θ) p(θ)p(σ ε ). Using Jeffrey s rule (Jeffreys, 96), it is shown that the noninformative prior for the parameters θ is locally uniform and p(σ ε ) /σ ε for parameter σ ε (Box and Tiao, 973). The prior distribution of (θ, σ ε ) is p( θ, σ ε ) (5) σ ε For large or even moderate-sized samples, fairly drastic modification of the prior distribution may only lead to minor modification of the posterior density. Thus the likelihood function will tend to increasingly overwhelm the prior as the sample size increases (Box and Tiao, 973). Given the observations, the likelihood function for Eq. can be written as in Eq. 6 n / ( γ xθ)'( γ xθ) L(, σε ) ( σε ) exp σ ε θ (6) where γ = y g ˆ obtained from Eq., x = (, m, r) is an n 3 matrix, n is the sample size, and denotes an n vector of ones.

5 By substituting Eqs. 5 and 6 into Eq., the joint posterior density distribution of θ and σ ε can be yielded. The marginal posterior densities of θ and σ ε, Eqs. 7 and 8, are obtained from their joint posterior density distribution, respectively. n / ( θ θˆ)' xxθ ' ( θˆ) f ( θ γ ) + (7) ν s ( v /+ ) ν s f ( σε ) ( σε ) exp σ ε γ (8) where ˆ θ = ( xx ' ) xy, ' ν is the degree of freedom, ν=n 3, s = (/ ν )( γ γˆ) '( γ γ ˆ) and γˆ = xθ ˆ. The posterior distribution of θ is thus the three dimensional student t distribution, denoted here as ˆ θ~[, t θ s ( xx '),] v. Then the marginal distribution of any subset of θ can be obtained from Eq. 7 and also follows the student t distribution. σ ε /(νs ) has the χ - ν inversed chi-squared distribution with ν degrees of freedom. The point estimators of θ and σ ε can be made from their posterior distributions. They may be unbiased estimators, maximum likelihood estimators or means of the posterior distributions, etc. In this study, we adopt the modes of the posterior distribution as the estimators of θ and unbiased estimator for σ ε. Predictive Ground Motion A point estimate of the correction term can be obtained by ignoring the uncertainty in the model parameters and using a point estimates ˆΘ in place of Θ. This option clearly does not account for the statistical uncertainties associated with the model parameters. To incorporate these uncertainties, we must consider Θ as random variables. The predictive distribution for a new output y *, based on a new observation x * and previous observations is defined as (Press, 98) * * * * f ( y ) L( y, ) f( ) d x = x Θ Θ Θ (9) Θ As can be seen from Eq. 9, since the predictive distribution is parameters free, it is useful for making inferences about the magnitude of future observations. The uncertainty associated with the prediction in Eq. 9 accounts for all uncertainties, while it only accounts for the model uncertainties and inherent variability with the past attenuation relations. Point estimator or any fractile of ground motion can be obtained from the predictive PDF in Eq. 9. Although the above result is obtained for model A, the estimation of model B can be made straightforward. Database ASSESSMENT OF SITE-SPECIFIC ATTENUATION RELATION Site-specific attenuation relation can be assessed for any observatories with observational data. As an example, two sites are selected to be illustrated. One is a site HKD of K-NET, the other is a site EKO.ERI of SK-NET. The deterministic past attenuation relation adopted in Eq. is the Si- Midorikawa attenuation relation for PGA (Si and Midorikawa, 999). gˆ( x, β) = ln [.5m+.3h+ d s +.6.5m log ( r.55 ).3 r] + i i () where m is a moment magnitude, h is a source depth in km and r is a minimum distance to the fault plane in km. Dummy variable d =.,.,. for fault type s such as crustal, interplate and

6 intraplate earthquakes respectively. The earthquake source parameters are adopted from the data published by Japan Meteorological Agency (JMA). Moment magnitude is adopted from CMT solutions of JMA and moment tensors of F-net. For simplicity, the hypocenter distances are adopted for r. The uncertainty associated with the prediction with Eq. is suggested to be.7. data and 7 data observed during 997~5 are selected for analysis at the site EKO.ERI and HKD as shown in Fig., respectively. The selected records meet with PGA larger than gal, m larger than 5. and r smaller than 5 km. The observed data of site HKD distribute more evenly on m-r space than those of site EKO.ERI from Fig.. Model A is applied to site the HKD while model B is suitable to the site EKO.ERI. Hypocenter Distance r (km) Site Code : EKO.ERI Number of Observations : Hypocenter Distance r (km) Site Code : HKD Number of Observations : Moment Magnitude m Moment Magnitude m (a) Site EKO.ERI (b) Site HKD Figure. Observations of earthquakes. The logarithmic deviations, γs, of the difference between the log of the observation and that of the estimation from the Si-Midorikawa attenuation relation are shown in Fig.. It shows that the logarithmic deviations of the site EKO.ERI distribute from -. to 3. and those of the site HKD distribute between -. and. with a relative small spread. The logarithmic deviations mostly bias positively at these two sites. 3 Number of data n = 7 Logarithmic Deviation γ - - EKO.ERI HKD Site Code Figure. Plot of logarithmic deviation. Parameter Estimation The site-specific attenuation relations of the site HKD and EKO.ERI are assessed with the observations respectively. The parameters in Model A for the site HKD and model B for the site EKO.ERI are estimated with Bayesian updating approach described above. Table lists the posterior

7 statistics of the parameters Θ = (θ, θ m, θ r, σ ε ) of model A for site HKD. Their posterior distributions of Θ are shown in Fig. 3. Table. Posterior statistics of parameters Θ (site HKD, model A, n=7) Parameters Estimator Standard deviation θ θ m.9.9 θ r 3.78e-3. σ ε f (θ ). f (θ m ). f (θ r ) θ θ r f (σ ε ) θ m σ ε Figure 3. Posterior distributions of θ, θ m, θ r, σ ε (n=7). The following observations are noteworthy: () The negative mean of θ indicates that independent of the variables x, the past attenuation relation gˆ( x, β ) tends to overestimate the ground motion. () The positive estimate of θ m indicates that gˆ( x, β ) tends to underestimate the contribution of the moment magnitude. (3) The positive estimate of θ r indicates that gˆ( x, β ) tends to underestimate the effect of the anelastic attenuation. () The unbiased estimator σ ˆε of uncertainty in Eq. for the site HKD is.5 smaller than that is suggested by the Si-Midorikawa attenuation relation,.7 (Si and Midorikawa, 999). The site-specific attenuation relation of the site EKO.ERI is assessed with model B by Bayesian approach. It is also done for the site HKD likewise. Table lists the posterior statistics of the parameters Θ = (θ, σ ε ) of model B for the two sites. Fig. shows their posterior distributions of Θ. Table. Posterior statistics of parameters Θ (model B) Site EKO.ERI (n=) Site HKD (n=7) Parameters Estimator Standard deviation Estimator Standard deviation θ σ ε The results of posterior estimators in Fig. and Table show that the uncertainty of the site EKO.ERI associated with parameters is larger than those of the site HKD due to sparse data. The

8 unbiased estimator σ ˆε with model B is.8 and.58 for the sites EKO.ERI and HKD, respectively. Although σ ˆε of the site EKO.ERI is larger than.7, it does not imply the worse prediction than the past attenuation relation, since the uncertainty of the latter represents the average characteristics of different sites. σ ˆε of the site HKD with model B is slightly larger than that with model A, since model A is more refined model. f (σ ε ) f (θ ) θ σ ε f (θ ) f (σ ε ) θ 6.5 σ ε Prediction of the Ground Motion (a) Site EKO.ERI (n=) (b) Site HKD (n=7) Figure. Posterior distributions of θ, σ ε. The site-specific attenuation relation developed by Bayesian updating approach can be applied to prediction of the ground motion. Fig. 5(a) shows the predictions of ground motion with model B at the site EKO.ERI for the earthquake with moment magnitude m = 6., occurred at central Chiba prefecture on July 3, 5, compared with the prediction with the Si-Midorikawa attenuation relation and as well as the observation. Fig. 5(b) shows the predictions of ground motion with model A at the site HKD for the earthquake with moment magnitude m = 7.3, occurred at SE off Erimomisaki on September 6, 3, compared with the prediction with the Si-Midorikawa attenuation relation and as well as the observation. The comparisons show the predictions by this study are closer to the observations than those predicted by the Si-Midorikawa attenuation relation. The uncertainty associated with this study is.7 and.58 for the sites EKO.ERI and HKD, respectively. They accounts for all uncertainty of the prediction, while the uncertainty.7 suggested by the Si- Midorikawa attenuation relation does not account for the statistical uncertainty associated the model parameters, as may lead to underestimation of the uncertainty of prediction of ground motion. Discussions The site-specific attenuation relation is developed by Bayesian updating approach with observation illustrated with two examples above. In the classical regression approach, the model parameters are assumed to be fixed but unknown, and the sample statistics are used as the estimators of these parameters. In the Bayesian approach, the parameters are considered as random variables, and inferences are made by considering their distributions conditional on the fixed data, whereas the statistical uncertainty associated with these parameters can be accounted for. The prediction of the ground motion with Bayesian approach accounts for all uncertainty, since the prediction is parameters free and conditional on the sample as in Eq. 9. The past attenuation relation is developed by pooling different records observed at different sites. The suggested uncertainty only represents an average characteristic of uncertainty of multi-sites so that it cannot represent the uncertainty of the specific site. The site-specific attenuation relation by

9 Bayesian updating approach with observations corrects the bias due to local site conditions, which can naturally reduce the model uncertainty. As shown in Fig. 5, the medians of the prediction by this study are closer to the observations than those by the past attenuation relation. Although Bayesian updating procedure is discussed with the linear model A in Eq., its reduced forms are also suggested for more flexibility in estimates according to the observation, such as model B in Eq. 3, since biased estimates of the regression coefficients will be obtained if the data are not distributed evenly among the parameters, for example, if magnitude and distance are statistically correlated (Campbell, 985; Boore and Joyner, 98). If data are evenly distributed, model A is preferred, while the reduced forms are desirable if data are sparse or with strong correlation on m-r. f PGA (pga) Si-Midorikawa Observed: 3. (gal) Si-Midorikawa: (gal), σ ε =.7 This study: 9.98 (gal), σ all =.7 Site Code : EKO.ERI (35.7N, 39.76E) CENTRAL CHIBA PREF 5/7/3 Observation Median This study f PGA (pga) 8 x Si-Midorikawa Median Observed: 5.9 (gal) Si-Midorikawa: 39.9 (gal), σ ε =.7 This study: 35. (gal), σ all =.58 Site Code : HKD (.8N, 3.3E) SE OFF ERIMOMISAKI 3/9/6 Observation This Study PGA (gal) PGA (gal) (a) Site EKO.ERI with model B (b) Site HKD with model A Figure 5. Prediction of ground motion. CONCLUSONS A comprehensive Bayesian framework for developing the site-specific attenuation relation based on the observations is formulated. The relation is unbiased and explicitly accounts for all the prevailing uncertainties. In Bayesian Framework, the uncertainties associated with unknown parameters can be expressed in terms of their posterior probability density distributions. For facilitating the use in practice, the site-specific attenuation relation is developed by adding the correction term to the existing attenuation relation in common use. Two models are exemplified for the correction term with more flexibility in Bayesian updating according to the observations. The correction term accounts for the bias of ground motion at the specific site predicted with the past attenuation relation. The results shows that the prediction with the site-specific attenuation relation is better than that with the past attenuation relation, and the accuracy of prediction is improved. The model uncertainty and statistical uncertainty are of epistemic nature, which tends to be reduced with the accumulation of the observations, while the inherent variability is of alearoty nature. However, the inherent variability and the model uncertainties due to the factors not incorporated into the model remain unchanged no matter how much data increase. In PSHA, it is expected to take into account all uncertainties including statistical uncertainty. ACKNOWLEDGMENTS The first author would like to acknowledge the support from the st Century Center of Excellence (COE) under Program B KS7. This study is a part of a research program on the establishment and application of probabilistic inter-related fragility assessment methodology for urban structures against earthquakes. The support provided by the MEXT (Ministry of Education, Culture, Sports, Science and Technology) under Grant Scientific Research A 73 is also gratefully acknowledged. The ground motion data used in this study are provided by K-NET and SK-NET.

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