International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Seismic hazard analysis with the Bayesian approach Wang, J. P. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1Seismic Hazard Analysis with the Bayesian Approach  J.P. Wang Assistant Professor, Dept. of Civil & Environmental Engineering, the Hong Kong University of Science and Technology, Kowloon, Hong Kong; jpwang@ust.hk (corresponding author) ABSTRACT: The Bayesian approach is of increasing popularity in engineering probability assessment. The key purpose of the method is to develop a new estimate by integrating observations or samples, along with other sources of prior data. By applying the approach to seismic hazard analysis, we developed and introduced the Bayesian seismic hazard analysis in this paper, including the algorithm, and a case study from Taipei City. The Bayesian analysis shows that based on earthquake strong-motion samples in the past 15 years (i.e., observation), and the return periods of 261 and 475 years reported in two studies (priors), the Bayesian estimate on the return period of PGA ≥ 0.25 g at the study site is equal to 339 years, a new estimate using the Bayesian approach to integrate limited earthquake strong-motion observations, with indirect evidence and estimates. 1.  INTRODUCTION In contrast to conventional statistical approaches, the Bayesian approach is to perform an estimating with multiple sources of information, usually referred to as prior data and observation (from samples). Implicitly, because a Bayesian estimate is an integration of prior data and observation, the method becomes very useful when observation is limited. For example, given no major earthquake is observed in the past 50 years, the rate of the earthquakes should be equal to zero, although such a best estimate might be neither representative nor realistic because a 50-year earthquake observation is a limited sample size considering the long return period of major earthquakes. Therefore, in order to compensate the situation, one possible solution is to add other indirect evidence in the analysis, such as the earthquake return period derived from some empirical model (e.g., Youngs and Coppersmith, 1985).         The Bayesian method has become more and more popular in engineering and science, for developing a new estimate with multiple sources/types of data. For instance, one of the earlier examples in material science was to evaluate the probability of defects in concrete, by integrating the data from nondestructive testing with a prior probability distribution obtained/designed in the first place (Tang, 1973).  Recently, Hapke and Plant (2010) employed the Bayesian approach to calculate the erosion rate of a cliff, with multiple sources/types of information from the cliff's height, slope, lithology, as well as the short-term erosion rate observed in the past decades. Similarly, Papaioannou and Straub (2012) used the Bayesian method to estimate the reliability of an engineered system, by combining the theoretical analysis (as prior) and structure monitoring data (as observation); moreover, Wang and Xu (2015) developed a new Bayesian algorithm for estimating the standard deviation of soil properties based on only one sample, along with a suggested variability range from the literature; similar example can go on and on (e.g., Ferni and Mannani, 2010; Zhang et al., 2012; Ramin et al., 2012; Vu et al., 2013).      As a result, the key scope of the study is to apply the Bayesian approach to seismic hazard analysis for the first time, developing a new Bayesian estimate on seismic hazard, based on limited earthquake observation and prior seismic hazard studies. In addition to the derivations of the algorithm, a case study of the Bayesian seismic hazard analysis was also reported in the paper.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  22. BAYESIAN ALGORITHM As mentioned previously, the objective of the Bayesian approach is to integrate prior data and (limited) observation to develop a new estimate. For a discrete case, the algorithm can be expressed as follows (Ang and Tang, 2007):    iiiiii )|Pr()(Pr')|Pr()(Pr')|('Pr'   (1) where )(Pr' i and )('Pr' i are the so-called prior and posterior probability for a prior estimate i ; )|Pr( i  is the likelihood function, or the probability for the observation   to occur given i .    3. REVIEWS OF SEISMIC HAZARD STUDY IN TAIWAN  3.1 The "2007" Study Cheng et al. (2007) conducted a comprehensive PSHA (probabilistic seismic hazard analysis) study for Taiwan, developing two PGA hazard maps in corresponding to return periods of 475 and 2,475 years. For example, by looking up the hazard map, the return period of PGA ≥ 0.25 g in Taipei (the geographical center of the city at 121.500 E and 25.050 N) is 475 years; by contrast, the return period of PGA ≥ 0.45 g is 2,475 years at the study site in Taipei based on the other hazard map developed. In addition to the hazard maps, the highlights of the PSHA study are summarized as follows: 1) the study employed a series of local ground motion models for better modeling hanging-wall and foot-wall effects in the analysis; 2) characteristic earthquake modeling was also used in the analysis to evaluate seismic hazard contributed from active faults; 3) a huge, complicated logic-tree analysis (more than 100 scenarios) was used in the study for consolidating the so-called epistemic uncertainties, from b-value, to maximum earthquake magnitude, to dip angle of faulting, to fault slip rate, etc.  3.2 The "2013" Study In contrast to the previous study focusing on the region around Taiwan, the study of Wang et al. (2013) was aimed at investigating seismic hazard in Taipei in much more detail, including the developments of hazard curves, two design response spectra, hazard deaggregation analysis, and time history selections. In short, the key difference between the two studies is as follows: the "2007" study provided two seismic hazard maps for the region around Taiwan, while the "2013" study was a much more detailed investigation focusing on Taipei, the most important city in Taiwan.  Also, the highlights of the "2013" study are summarized as follows: 1) it used another series of local ground motion models (Lin et al., 2010) for developing the two design response spectra; 2) based on hazard deaggregation and the design spectra, several time histories were selected from the NGA (Next Generation Attenuation) database; 3) the logic-tree analysis used in the study was focused on the uncertainty of ground motion model selections.  3.3 The Two Estimates for Taipei Although the scopes are different, two seismic hazards, the return periods of PGA ≥ 0.25 and PGA ≥ 0.45 g, were all given in the studies: 1)  as far as PGA ≥ 0.25 g is concerned, the return periods could be 261 and 475 years in Taipei from the two independent studies; 2) for PGA ≥ 0.45 g, the return periods were estimated at 1,305 and 2,475 years in Taipei. Understandably, the seismic hazard estimates are a result of indirect earthquake evidence (e.g., seismicity) and an analytical procedure (i.e., PSHA). In other words, the seismic hazards are our best estimate from indirect evidence, with some inevitable engineering judgments involved in the process (e.g., logic-tree analysis). As a result, it is expectable that seismic hazards estimated by independent studies must be different, as the case studies from Taipei that were just summarized.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 4. SEISMIC HAZARD FROM OBSERVATION In addition, the third option to estimate seismic hazard is based on direct observation. That is, if the site experienced one event of PGA ≥ 0.25 g induced by major earthquakes in the past 100 years, its return period should be 100 years. As a result, this section would like to present seismic hazard estimates from direct observation for this case study, thanks to the Taiwan Strong Motion Instrumentation Program established in the 1990s. More introduction to the network is given in the following.          4.1 Earthquake Monitoring Network in Taiwan The region around Taiwan is known for high seismicity.  In order to collect more local earthquake data, the Taiwan Strong Motion Instrumentation Program (TSMIP) was established in the 1990s (Shin, 1993; Shin et al., 2003). Basically, the program is to setup many well-instrumented earthquake stations in Taiwan, with around 700 stations (Fig. 1) in operation as we speak. That is, if an earthquake around Taiwan is occurring now, many earthquake strong motions or time histories can be collected from different locations during the event. The TSMIP earthquake database is valuable to local earthquake studies. For example, Solokov et al. (2001) utilized the strong-motion database to study the basin effect on site response (or site amplification) in Taipei. Similarly, Lee et al. (2001) used the TSMIP database to perform site characterizations for earthquake-resistant design in Taiwan.  In addition, Lin et al. (2010) used the data from TSMIP to develop a series of local ground motion models, from PGA, to short-period SA, to long-period SA (spectral acceleration). As a result, the two design response spectra for Taipei given in the "2013" seismic hazard study would not have been conducted without the ground motion models developed with the TSMIP database.   4.2 Strong-Motion Observation at Study Site Similarly, we also utilized the TSMIP database to estimate the frequency of PGA ≥ 0.25 g at the study site in Taipei. In this study, we specifically used the data from Station TAP012 of the TSMIP, given the station is only 1 km away from the site, thanks to the "high-density" earthquake monitoring network as shown in Fig. 1.   Figure 1: The earthquake stations of the Taiwan Strong Motion Instrumentation Program; the distance between the nearest station and the site is about 1 km.   Accordingly, from years 1999 to 2013, the station recorded 37 time histories induced by 37 major earthquakes around Taipei with ML greater than 5.0 (local magnitude). As shown in Fig. 2a and Fig. 2b, the majority of the earthquakes were between ML 5.0 and ML 6.5, and around 85 percent of them were occurring in a distance within 150 km from the study site in Taipei. More importantly, based on the strong-motion data induced by the major earthquakes, Fig. 2c shows the distribution of PGA at the study site in the past 15 years. Accordingly, the 120.0 120.5 121.0 121.5 122.021.522.022.523.023.524.024.525.025.5Orchid IslandGreen IslandIslandHsiao LiouciouLatitude (N)Longitude (E) Distance = 1.05 km  The nearest station121.508E, 25.056N  The study site121.50E, 25.05N12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4maximum PGA at the site was about 0.1 g. In other words, no event of PGA ≥ 0.25 g was observed at the site in the past 15 years, although several major earthquakes have occurred around the city during the period of time.   It must be noted that as far as the specific location (121.500 E and 25.050 N) is concerned, the 15-year observation from the local TSMIP database is the best earthquake strong-motion data available to us. That is, although a well-established earthquake catalog listing more than 50,000 earthquakes since 1900 is also available (Chen et al., 2013; Wang et al., 2011, 2014), the data of the catalog, such as earthquake magnitude, depth, and location, are different from strong-motion data (i.e., time history), and they are not useful to this study in need of strong-motion data to estimate seismic hazards (e.g., PGA ≥ 0.25 g) from direct observation.     Figure 2: The statistics of the 37 earthquake strong-motion data recorded at the site, and those major events (ML ≥ 5.0) around the region: a) the probability distribution of earthquake magnitude, b) the probability distribution of source-to-site distance, and c) the probability distribution of PGA induced by the 37 events.   5. BAYESIAN SEISMIC HAZARD ANALYSIS From the two seismic hazard studies and the 15-year observation data, it is expectable that the three estimates would be different as we have seen. In the meanwhile, it raises a fundamental question as follows: Which one among the three should be more reliable or representative?  Understandably, the estimate from the direct observation would have been the most reliable, if the observation was not that limited. By contrast, the indirect estimates from two independent studies should be equally reliable or unreliable, with each presenting some kind of best estimate with engineering judgments involved (e.g., the selection of ground motion models, logic-tree analysis).       As a result, a possible solution to resolve the differences is to combine them with a well-established algorithm, such as the Bayesian approach. For the first time, we would like to apply it to seismic hazard analysis, and developed and introduced the framework of Bayesian seismic hazard analysis in the following.    0.00.10.20.30.45.5-6.0 6.0-6.5 6.5-7.0  7.0ProbabilityMagnitude (ML)(a)5.0-5.50.00.10.20.30.40.50.60.750-100 100-150 150-200  2000-50(b)ProbabilityDistance (km)0-0.02 0.02-0.04 0.04-0.06 0.06-0.08 0.08-0.100.00.10.20.30.40.50.60.7(c)ProbabilityPGA (g)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 5.1 Prior and Observation As mentioned previously, a Bayesian estimate is a result of some prior estimates and observation data. Therefore, the first step of a Bayesian seismic hazard analysis is to search for prior studies, as the two summarized earlier in this paper. Moreover, if each prior is considered equally reliable (or unreliable), the prior probability for each study is then equal to 1 / n, where n is the number of priors collected. As a result, the prior probability mass function for this case study can be developed as Fig. 3. That is, there should be a 50-to-50 probability for the return periods of PGA ≥ 0.25 g equal to 261 or 475 years in Taipei, based on the two independent studies relying on a specific analytical procedure and indirect evidence. Understandably, the next step of the Bayesian analysis is to look for some observation data. As for this case study, the observation is no event of PGA ≥ 0.25 g induced by any major earthquakes around the site in the past 15 years.  In short, this is a "0-event-in-15-year" observation for the given problem.  5.2 The Bayesian Updating  This section would like to present the general algorithm of the Bayesian seismic hazard analysis, and the key to this Bayesian analysis is the calculation of the likelihood function )|Pr( i , since the prior probabilities )(Pr' i  equal to 1 / n are given. For a given level of seismic hazard (e.g., PGA ≥ 0.25 g), we first write the likelihood function of the Bayesian analysis as follows:  )|:Pr()|Pr( ii kyearstineventsm         (2) Therefore, the likelihood function is to calculate the probability for such an "m-event-in-t-year" observation to occur given a prior return period ki. Considering seismic hazard is a random variable following the Poisson distribution as commonly accepted and used (e.g., Kramer, 1996; Cheng et al., 2007; Wang et al., 2013), the probability is then calculated as Eq. 3, on the basis of the model's probability mass function:    !)/()/exp()|Pr(mktktkperiodreturnyearstineventsmmiii                    (3) With the prior probabilities (i.e., 1 / n) and likelihood functions (i.e., Eq. 3), the next step is to calculate the posterior probability for each estimate, by substituting them into Eq. 1:      nimiimiiimktktnmktktn1 !)/()/exp(1!)/()/exp(1)('Pr'         (4)  As a result, the Bayesian estimate (denoted as B) can be developed based on the updated, posterior probability mass function, following the fundamentals of probability and statistics (Ang and Tang, 2007):   niiiB1)('Pr'     (5)  6. THE CASE STUDY FROM TAIWAN With the prior, observation, and the Bayesian algorithms (Eqs. 1 ~ 5), the prior and posterior probabilities for the return periods of PGA ≥ 0.25 g in Taipei are shown in Fig. 3. For this case study, the "long" prior (i.e., 475-year return period) had a larger posterior probability after updating, in contrast to the "short" prior (i.e., 261-year return period). Most importantly, based on the posterior probability function, the Bayesian seismic hazard analysis suggests the return period of PGA ≥ 0.25 g in Taipei should be equal to 339 years, based on the strong-motion observation in the past 15 years, along 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6with the prior estimates (i.e., 261 and 475 years) from two independent studies.  To sum up, the algorithms of the Bayesian seismic hazard analysis were developed on the following conditions: 1) n prior estimates on a given seismic hazard are available; 2) each prior is considered equally reliable; 3) m events of the seismic hazard were observed in the past t years; 4) seismic hazard is a random variable following the Poisson distribution. It is worth noting that the Bayesian seismic hazard analysis is applicable to other case studies with different prior and observation data.  Understandably, with new input data, the Bayesian seismic hazard estimates will change.  But as for this case study for Taiwan, the prior and observation used in the analysis are the “best” data available to us.  Therefore, it is difficult to examine the influence of different prior or observation on the Bayesian calculation at this moment.  Figure 3: The prior and posterior probability mass functions for the two prior estimates on the return periods of PGA ≥ 0.25 g in Taipei, given the "0-event-in-15-year" observation from a local database.   7. CONCLUSION AND SUMMARY The Bayesian approach has been increasingly used in engineering, for developing a new estimate based on multiple sources/types of data. In this study, we applied the method to seismic hazard analysis for the first time, developing the framework of Bayesian seismic hazard analysis.  In addition to the derivations of the algorithm, a case study from Taipei was also reported in this paper. The Bayesian analysis suggests the return period of PGA ≥ 0.25 g in Taipei should be equal to 339 years, based on the earthquake strong-motion observation in the past 15 years, and the prior return periods of 261 and 475 years given in two independent studies. To sum up, the Bayesian seismic hazard analysis is a new, robust option for seismic hazard assessment, similar to many other Bayesian applications, in order to develop a new estimate with prior data given direct observation is limited.  8. REFERENCES Ang AHS, Tang WH (2007) Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering. John Wiley & Sons, Inc., NJ Chen CH, Wang JP, Wu YM (2007) A study of earthquake inter-occurrence times distribution models in Taiwan. Natural Hazards; 69, 1335-1350 Cheng CT, Chiou SJ, Lee CT, Tsai YB (2007) Study on probabilistic seismic hazard maps of Taiwan after Chi–Chi earthquake. Journal of GeoEng 2(1): 19-28 Ferni G, Mannani G (2010) Bayesian approach for uncertainty quantification in water quality modeling: the influence of prior distribution. J Hydrol 392(1): 31-39 Hapke C, Plant N (2010) Predicting coastal cliff erosion using a Bayesian probabilistic model. 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