A Bayesian approach for site-specific extreme load prediction of large scale bridges

Abstract To measure uncertainties of site-specific extreme loads for a signature bridge, a probabilistic method is developed based on structural health monitoring data in the Bayesian context. Compared with respective design loads, the probability of the predicted extreme load exceeding the design one is used to determine whether the design load is sufficient. The generalized Pareto distribution was first employed to model extreme value information. Then, Bayesian estimation was applied to predict probabilistic distributions of extreme loads, accounting for uncertainties within the prediction process. Finally, the effectiveness of the proposed method was validated through wind speed and temperature measurements from the Nanjing Dashengguan Yangtze River Bridge. The extreme wind speed and four types of thermal loads were predicted based on site-specific monitored loading data by using Bayesian estimation of the generalized Pareto distribution. As a result, the design wind speed and design change in uniform temperature are sufficient for the signature bridge. However, the probability that the predicted extreme vertical girder temperature difference exceeds the design one is 99.96%, and the probabilities subject to the transverse girder temperature difference and tower temperature difference are almost 100%.


Introduction
Loads (e.g. wind speeds, temperatures and vehicle loads) used to design a signature bridge are determined according to related design standards or codes, where the design load is derived from generalized investigations of loading conditions all over a country or region (AASHTO, 2017;MOT, 2015). However, since loading circumstance and structural resistance vary with the evolution of society, change of environment and degradation of material, loading and resistance factors within the design course may be insufficient to evaluate existing bridges. In addition, with the evolution of sensing and information technologies, structural health monitoring (SHM) systems have been devised and applied to many bridges worldwide, especially large scale bridges, to meet the demand on functionality and safety (Z. Riches et al., 2019;Y. L. Xu & Xia, 2011). SHM systems are used to monitor external loads, environmental factors and structural responses of a signature bridge in real time X. Xu, Huang, Ren, Zhao, Yang, & Zhang, 2019). Furthermore, SHM systems are able to quantify local uncertainties and time-dependent variations in loading demand and resistance capacity through site-specific measurements, which will facilitate evaluation, operation and maintenance of the signature bridge.
Abundant researches are available to evaluate functionality and safety of a signature bridge by using structural response measurements (e.g. girder deflections, strains and cable forces), where the measured structural responses are compared with the resistance capacity to determine whether the capacity is sufficient for the signature bridge . In addition to structural responses, the external load is the other critical indicator to impact structural functionality and safety. It is significative and instructive to predict the site-specific extreme load (SEL) based on SHM data, and the predicted extreme loads are compared with respective design loads to determine whether the design loads are sufficient for the signature bridge. Based on the obtained site-specific loading information, related maintenance and management plans of the signature bridge should be updated. Moreover, site-specific loading information of a signature bridge provides reference for the updating of design loads in the design code.
Extreme value analysis (EVA) is always employed to predict extreme loads in practical engineering. In the EVA, the data acquisition strategy needs to be determined in advance, mainly including block maxima series and peak-over-threshold (Coles et al., 2001). Based on the block maxima series (e.g. annual maxima series), the generalized extreme value distribution (GEVD) is used to fit the extracted series, including Type I extreme value distribution, Type II extreme value distribution, and Type III extreme value distribution (Beirlant et al., 2006;Fisher & Tippett, 1928). The Type I extreme value distribution (i.e. the Gumbel distribution) is always utilized to estimate the extreme wind speed based on the annual maxima from meteorological data (Gumbel, 2004). Moreover, the Gumbel distribution is also applied to estimate extreme thermal actions based on long-term meteorological information (Tong et al., 2002). However, the GEVD requires long-term loading data available to predict a reliable extreme load. For instance, to obtain a reliable extreme wind speed value, more than 20-year wind speed data are requested (Palutikof et al., 1999). In reality, bridge SHM systems are difficult to provide such long-term sitespecific loading data due to the relatively short history of applications. Based on the peak-over-threshold data acquisition strategy, the generalized Pareto distribution (GPD) is adopted to fit the exceedance (Leadbetter, 1991). The tails of another distribution are always modeled by using the GPD through the peak-over-threshold approach, which could make full advantages of extreme value information (Pickands, 1975). The GPD has been applied to estimate extreme wind speeds, temperature actions and trafficinduced structural responses based on site-specific SHM data (Holmes & Moriarty, 1999;X. Xu et al., 2021;Zhou et al., 2017). Considering that the block maxima strategy only focuses on the maximum data within a determined time window and dismisses the remaining data in the group, the GPD is adopted to predict extreme loads in this paper owing to the limited site-specific monitored loading data available from the SHM system.
Although SHM-based SEL estimation has been well investigated, all the previous studies are under the deterministic context. For instance, a deterministic algorithm, namely maximum likelihood estimation, is always used to predict SELs based on bridge SHM measurements. However, uncertainties inevitably exist in the course of extreme load estimation due to environmental variability, measurement noise and parameter estimation. The deterministic methods are incapable to interpret the uncertainties within the prediction process. In reality, the ground truth of SELs may be hidden, leading to inaccurate decisions of loading condition for a signature bridge. In this regard, probabilistic methodologies are more suitable in predicting SELs under the uncertain context. Bayesian inference, as a powerful analytic tool in quantifying uncertainties, has been increasingly adopted to update model, identify system, detect damages, diagnose sensor faults, reconstruct monitored data and assess structural condition (Gu et al., 2014;Huang et al., 2017;Ni et al., 2020;Wang et al., 2019;Wan & Ni, 2019;Zhang et al., 2021). In this regard, Bayesian inference is promising to account for uncertainties in the extreme load estimation process. To the best of the authors' knowledge, rare investigation has been carried out to resolve the SEL estimation of bridges under Bayesian inference.
In this study, a Bayesian approach for SEL estimation is developed based on bridge SHM measurements to consider uncertainties within the prediction process. Based on probabilistic distributions of estimated extreme loads and respective design values, probabilities that predicted extreme loads exceed design ones are calculated to determine whether the design loads are sufficient for the signature bridge. Based on site-specific monitored loading data, the GPD is first used to model the measured limited extreme loading data, where the analysis data are required to be independently and identically distributed. Then, Bayesian estimation is employed to predict probabilistic distributions of SELs, where Metropolis-within-Gibbs (MG) sampler, one of Markov Chain Monte Carlo (MCMC) algorithms, is used to obtain posterior distributions in this study. Finally, based on site-specific SHM loading data of the Nanjing Dashengguan Yangtze River Bridge (NDB), the probabilistic distributions of extreme wind speeds and thermal actions are predicted and compared with the design ones.

Probabilistic estimation of site-specific extreme loads
In this paper, site-specific monitored loading information from bridge SHM systems will be first re-sampled to make the studied extreme value information independent. Then, the GPD is used to depict the distribution of extreme values, where the extreme value is defined as peaks over a determined threshold. Bayesian estimation is employed to estimate probability density functions of parameters and extreme loads. If the probability that the estimated extreme loads exceed the design ones is high, it illustrates that the design loads are not adequate for the signature bridge and vice versa. The general procedure of the proposed probabilistic estimation of SELs is shown in Figure 1.

GPD analysis
In statistics, the GPD is a family of continuous probability distributions, which is often used to model the tails of another distribution. The GPD is on the foundation of the peak-over-threshold technique, which has the potential to make full use of extreme value information. Considering limited SHM-based site-specific loading information, the GPD is promising to predict extreme loads. The GPD is specified by two parameters: scale parameter r and shape parameter f, and its cumulative density function subject to a variate x is as follows: where the support is x ! 0 when f ! 0, and 0 x Àr=f when f < 0: The probability density function (pdf) of X $ GPDðr, fÞ is for x ! 0 when f > 0, and 0 x Àr=f when f < 0: When f ¼ 0, the pdf is for x ! 0: The basic procedure to determine SELs corresponding to a T-year return period based on bridge SHM measurements by using the GPD is divided into four steps, namely re-sampling, threshold determination, parameter estimation and extreme load prediction. The specific descriptions of the four steps are summarized as follows: (1) Re-sampling. To predict SELs subject to a determined return period through the GPD, the studied samples are required to be re-sampled to satisfy the request of independent and identical distribution.
(2) Threshold determination. Determination of a proper threshold is critical for the GPD analysis. When the threshold stands in a high level, the statistical analytic is short of samples, leading to uncertainties in statistics. In contrast, small value of the threshold causes a biased estimator, where the out-of-samples do not follow the extreme value distribution. In this regard, the threshold is always determined by the mean excess function of the GPD, where the mean excess function is expressed as in which N u is the number of samples larger than the threshold, x i is the ith studied sample, and u is the threshold. According to Equation (4), the linear relation between the threshold u and mean excess e(u) is observed (Gilli & K€ ellezi, 2006). Previously, the threshold is often determined by making the mean excess function linear. However, there will be a group of candidate thresholds to make the mean excess function linear. To determine the optimal one, the root mean square error (RMSE) is defined here where L e ðuÞ is the linear fit of the measured mean excess function over a given potential threshold. According to Equation (4), the optimal threshold makes a linear between u and e(u). In this regard, the ideal situation is the calculated points ½u, eðuÞ lie on the linear fitting function L e ðuÞ, resulting in RMSE ¼ 0. Thus, the minimum value of the RMSE indicates the best linear fit, which corresponds to the optimal threshold.
(3) Parameter estimation. According to the studied extreme value data, the posterior distributions of scale and shape parameters of the GPD are figured out by using Bayesian estimation method. One can refer to Section 2.2 for the detailed description of Bayesian estimation. (4) Extreme load prediction. The extreme point subject to the design quantile is estimated as the SEL in this study. Within the design reference time (e.g. T ¼ 100 years), the cumulative probability is calculated as where P r is the probability of exceedance. According to Chinese bridge design code (MOT, 2015), design loads are always subject to a 100-year return period. In order to in line with the design code, the same return period (i.e. 100 years) is used to predict SELs in this paper. The estimated extreme load is calculated by using where u 0 is the threshold;r is the estimated scale parameter;f is the estimated shape parameter; n is the number of samples; and N u is the number of exceedance.
Once obtaining the pdf of estimated SELs, the site-specific loading situation for a signature bridge is evaluated by comparing the estimated extreme load with the design value. The probability that the predicted SEL exceeds the design one is calculated by using the following equation where x d is the value of design loads, and f ðx p Þ is the pdf of the estimated SEL. The probability Pðx p > x d Þ serves as an indicator to measure the loading condition of a signature bridge under an uncertain context. In detail, if the probability is high, it indicates that the design load for the signature bridge is insufficient and vice versa.

Bayesian estimation of the GPD
Bayesian estimation is one of probabilistic ways to predict extreme values. Compared with deterministic methods, the estimated parameters are treated as random variables rather than constants. In this regard, the predicted extreme values derived from the parameters are in the form of probabilistic distribution to take uncertainties within the estimation process into account. Based on Bayes' theorem, the posterior distribution of the parameter is where h ! ¼ ðr, fÞ is the parameter vector of the GPD, Under Bayesian inferences, the prior distribution indicates prior knowledge regarding the parameters. In general, two types of prior distributions are available, including informative and non-informative priors. The informative is determined on the foundation of the prior knowledge usually obtained from existing literature or experts. While, the non-informative is employed when litter or no prior knowledge is known. Owing to the lack of prior information regarding parameters, a flat distribution is always used for the non-informative prior distribution to cover a wide range of possible values. In this paper, the non-informative prior distribution is employed since rare knowledge regarding parameters of the GPD for SEL estimation is known. The selected flat distribution is N(0, 100), which will cover from À400 to 400.
Considering the complexity of the Bayesian model and rare knowledge of prior distributions, MCMC sampling is adopted herein to figure out posterior parameter distributions of the GPD. MCMC generates samples from a continuous random variable with probability density proportional to a known function. These samples are used to evaluate an integral over that variable. Various algorithms exist for constructing chains in the MCMC methods. Herein, Metropolis-within-Gibbs (MG) sampler, one of MCMC algorithms, is employed in this study. The general idea for the MG sampler is first to generate a proposal by applying Gibbs sampling to the prior, then uses the Metropolis step to incorporate data information (Q. Liu & Tong, 2020). In detail, it generates iterations through the following steps: (1) Assigning initial values: a transformation of the shape parameter f of the GPD to make it positive, and r is the scale parameter. In this paper, initial values of parameters could refer to the estimated values from the point estimation method.
(2) For the iteration index i which starts from 1, a candidate of the parameter / Ã is first generated from the normal proposal distribution conditional on / iÀ1 , that is, / Ã $ Nð/ iÀ1 , s / Þ, where N represents normal distribution, and s / is the scale of the proposal distribution. Then, the acceptance ratio for / Ã is calculated by using the following equation where function f is the unnormalized full conditional distribution of the parameter that is expressed as the product between the likelihood function and the prior distribution, which is Subsequently, sample u from the uniform distribution U(0, 1) is generated. If u < minð1, sÞ, accept / Ã , otherwise, reject / Ã : (3) Generate a candidate for the scale parameter r Ã from the normal proposal distribution conditional on r iÀ1 , that is, r Ã $ Jðr Ã jr iÀ1 Þ ¼ Nðr iÀ1 , s r Þ, where s r is the scale of the proposal distribution. Then, the acceptance ratio for r Ã is calculated by using where the unnormalized full conditional distribution of the parameter is Then, sample u from the uniform distribution U(0, 1) is generated. If u < minð1, sÞ, accept r Ã , otherwise, reject r Ã : (4) Increase the iteration index from i to i ¼ i þ 1, and repeat steps (2) to (3) until i reaches N, where N is the total iteration number.
After a period of iterations, Markov chain converges to a stationary distribution. The period of iterations before the Markov chain convergence is called the burn-in period. The simulated values in the burn-in period cannot be treated as samples to form the posterior distribution. Excluding the burn-in period, the remaining chain, termed as stationary period, is used to discuss the stochastic characteristics of parameters of the GPD. Once obtaining the posterior distributions of the parameters, the posterior distribution of the predicted SEL can be figured out as where x p is the predicted extreme load, g is the pdf of the GPD, and p is the posterior distributions of the parameters. In practice, the posterior predictive distribution is generated by using the following procedure: (1) For loop index i initialed with 1, a pair of shape and scale parameters of the GPD are first randomly generated from their posterior distributions. Then a sample of extreme load is calculated according to Equation (7).
(2) Increase the loop index from i to i ¼ i þ 1, and repeat step (1) until i reaches M, where M is the total loop number. The obtained samples will be a realization of size M from the posterior distribution of the predictive extreme loads.
As stated above, a distribution is possible to fit the histograms derived from the samples of estimated extreme loads. Then, the predicted extreme load is in a form of probabilistic distribution, which could offer uncertain information regarding the estimated extreme load.

Case study
3.1. The Nanjing Dashengguan Yangtze River Bridge and its SHM system The NDB (formerly the third Nanjing Yangtze River Bridge), opened to public traffic in 2005, is a key transportation link crossing the Yangtze River and connecting Liuhe District with Nanjing City as shown in Figure 2. The steel cable-stayed bridge has a total length of 1288 m, where the main span is 648 m. The configuration of the NDB is shown in Figure 3. The bridge deck is supported by a total of 167 stay cables, and each cable consists of 109 to 241 7 mm-diameter wires. A sophisticated SHM system was applied to measure the structural related information in 2006. A total of 599 sensors were installed and their layout is shown in Figure 4, which are anemometers, temperature transducers and so forth. Wind characteristics (i.e. wind speeds and directions) are recorded by using two deck level ultrasonic anemometers: one set at the middle of main span, and the other located at the south side span. The frequency and resolution of the ultrasonic anemometer are 10 Hz and 0.1 m/s, respectively. Bridge temperature distribution is measured through thermometers in the girder and tower as shown in Figure 5. In detail, temperature distribution of a girder section is monitored using 16 thermal sensors, and a tower leg section is measured by eight sensors. The sampling frequency of the thermal sensor is relatively low, namely one datum for 30 minutes, and the resolution is ±0.5 C.
Wind speeds, temperatures, traffic loads and seismic actions are major loads for bridge structures. However, estimation of extreme seismic actions is difficult to investigate since insufficient site-specific monitored data are available. Moreover, it is still a challenge to predict site-specific extreme traffic loads owing to the obstacle to obtain the spatial and temporal distribution of vehicle loads (Chen et al., 2016). Current studies focus on estimation of extreme traffic load-induced structural responses (OBrien & Enright, 2011;. Thus, in this case study, site-specific extreme wind speeds and temperature loads are predicted by using the relatively long-term monitored data.

Data preparation
Wind speed is a major design load for large scale bridges, especially those built in a wind-prone region. For the NDB, the design wind speed (i.e. ten-min mean wind speed at 10 m high over the sea level) is 31.7 m/s based on the design reference. Since the studied deck level ultrasonic anemometer is 43.5 m high over the sea level, the deck level design wind speed for the signature bridge is 39.0 m/s based on the mean wind speed profile in the code (MOT, 2015;Zhu et al., 2007). In this regard, the site-specific extreme wind speed estimation in this study is carried out based on the ten-min mean wind speed at the deck level. Ten-min wind speed samples (from 00:00:00 to 00:10:00 July 1, 2007) are plotted in Figure 6(a). The ten-min mean wind speed is highlighted in Figure 6(a) and prepared for the further discussion. In view of the requirement of independent and identical distribution for the studied samples, fourday maxima are extracted as the extreme value database for the GPD analysis (Cook, 2013). An example is given by Figure 6(b) to show the four-day maximum value which is derived from the four-day (July 1 to July 4, 2007) ten-min mean wind speed data.
Four-day maximum wind speed data in ten years (from 2006 to 2015) build the data foundation for the prediction of site-specific extreme wind speed in this study, which are plotted in Figure 7. According to Figure 7, around 1700-day samples are obtained rather than 3650-day data since missing samples and abnormal samples are deleted in advance.

Probabilistic estimation of extreme wind speeds
As stated above, determination of threshold is prior to the GPD analysis, which could be determined through two indicators -mean excess function and RMSE. Based on the database shown in Figure 7 for estimation of extreme wind speed, mean excess function and RMSE are figured out and plotted in Figure 8. According to Figure 8, a candidate threshold of 20.6 m/s was first picked to make the value of RMSE smallest. Whereas, when the threshold was set as 20.6 m/s, only four samples were above the threshold causing statistical uncertainty. To lower the uncertainty, a better threshold of 16.3 m/s in this case was selected eventually. A strong linear fit is still observed at the threshold of 16.3 m/s, and abundant samples are available for the GPD analysis. Furthermore, RMSE achieves its local lowest value when the threshold is set as 16.3 m/s, which also supports the selection of this threshold. Thus, the optimal threshold of the four-day maximum ten-min mean wind speed for the GPD is set as 16.3 m/s in this study.
Wind speed exceedance is used to estimate parameter distributions of the GPD. Following the steps of Bayesian estimation, parameters of / and r are calculated over 40,000 iterations. For convenience, / is transformed to shape parameter f of GPD. As a result, the iteration processes for shape and scale parameters are plotted in Figures  9(a) and 10(a), where the burn-in period and stationary period are apparently observed. For conservative consideration, it is suggested to abandon some stationary samples which are next to the burn-in period, termed as fuzzy area as shown in Figures 9(a) and 10(a). Furthermore, according to a large number of trails and errors, it is found that this strategy will not disturb the parameter distribution significantly. In this case, the stationary period for the shape parameter is determined from iteration 7,750 to 40,000, and the stationary period subject to the scale parameter is from 9,900 to 40,000.
All measures of goodness-of-fit suffer from the same serious drawback -the large sample size produces statistically significant lack of fit. In view of the large number of samples in this case, the calculated p-values from the K-S test, Chi-square test and Shapiro-Wilk test are almost equal to zero -rejecting the null hypothesis. In this regard, goodness-of-fit test will not be applied to evaluate the fitting performance in this case. Moreover, to discuss the impact of the selection of stationary period, Gaussian function is used to fit different scale stationary period for shape and scale parameters. For the shape parameter, in addition to the stationary interval [7,750,40,000], other intervals are determined, i.e. [6,000, 40,000], [8,000,40,000], and [9,000, 40,000]. As a result, the fitting Gaussian functions are N(À0.3427,0.0212), N(À0.3420,0.0216), and N(À0.3419,0.0219), respectively. For the scale parameter, in addition to the stationary interval [9,900,40,000], other intervals are determined, i.e. [9,000,40,000], [11,000, 40,000], and [11,500, 40,000]. As a result, the fitting Gaussian functions are N(1.8193,0.0634), N(1.8280,0.0556), and N(1.8295,0.0549), respectively. The fitting Gaussian functions are extremely similar with each other even though the selected stationary periods are discrepant. Finally, based on the distributions of the two parameters and Equation (15), the pdf of extreme wind speeds is predicted and compared with the design wind speed as shown in Figure 11.

Discussions
According to Figure 11, the estimated site-specific extreme wind speed mainly varies between 20 m/s and 23 m/s, which is far smaller than the design one. Furthermore, Gaussian function is used to fit the extreme wind speed histogram. As a result, the extreme wind speed obeys the Gaussian distribution with a mean of 21.3377 and standard deviation of 0.3306. Since the standard deviation (i.e. 0.3306) is small, the uncertainty of the estimated wind speed is low in this case. Based on the pdf of extreme wind speeds, the probability that the estimated site-specific extreme wind speed exceeds the design one is calculated as Based on above calculations, it is almost impossible for the estimated site-specific extreme wind speed exceeding the design one, which indicates that the design wind speed is sufficient for the signature bridge. Compared with the estimated extreme wind speed of 21.6 m/s using the point estimation (Xu et al., 2021), the Bayesian estimation not only gives the extreme value, but the probabilistic information, which will help stakeholders make decisions. Since Nanjing is not located at a wind-prone region, the wind speed is always not the critical design factor to bridges located in this area. Thus, the conclusion is acceptable based on the engineering experience. However, the estimated extreme wind speed in this study is derived from the ten-year monitored wind speed data. With the accumulation of new monitored wind speeds, the pdf of   extreme wind speed should be updated for the concern of safety. In this regard, the distribution of extreme wind speed will change with the new-coming data.

Data preparation
Due to solar radiation and ambient temperature, large scale bridges suffer from thermal loads. Temperatures not only introduce structural deformations but internal forces or stresses owing to structure redundancy. Thus, the temperature field is carefully considered in the design stage of large scale bridges. In general, a total of four temperature loads are taken into account for long scale bridges, including uniform girder temperature change dT u , vertical girder temperature difference T vg , transverse girder temperature difference T tg , and tower temperature difference T t . The uniform girder temperature change is associated with the expansion and contraction of the bridge girder to guide the design of bridge expansion joints and bearings. Whereas, the temperature differences might generate secondary stresses due to structure redundancy. According to the bridge design code, the four design temperature loads, namely the uniform girder temperature change, vertical girder temperature difference, transverse girder temperature difference and tower temperature difference, are determined as 55.0 C, 20.0 C, 4.5 C and 5.0 C, respectively.
The uniform girder temperature (i.e. the effective girder temperature) is defined as (Deng et al., 2018) where E and a are the Young's modulus and coefficient of thermal expansion, respectively; T is the temperature distribution over the cross section; and the integration is carried out over the whole cross section of the bridge girder. Based on the bridge construction record, the uniform girder temperature at the instant of bridge completion is 20.1 C, which serves as the baseline to determine the change in uniform girder temperature. If the uniform girder temperature exceeds the baseline, the bridge girder will expand; while if the uniform girder temperature is lower than the baseline, the girder will contract. The measured temperatures in 2012 are used to predict the site-specific extreme uniform girder temperature change, where the daily maxima and minima are obtained and plotted in Figure 12. Owing to data missing phenomenon that is often observed in SHM systems, only 265-day samples are available in 2012.
Since the deck is straightforwardly exposed to solar radiation, the temperature of deck always exceeds that of the bottom plate of the girder. The temperature difference between the deck and bottom plate is termed as the vertical temperature difference of the bridge girder section. According to the thermal transducers installed along the girder section, five pairs are selected to calculate the vertical girder temperature difference. Specifically, they are T D2B1 , T D3B2 , T D4B3 , T D5B4 and T D6B5 , where T DiBj ¼ T Di -T Bj . In view of the fluctuation of the vertical girder temperature difference crossing the transverse direction, the effective temperature difference is taken into consideration, which is calculated by where T DiBj is the temperature difference distribution along the transverse direction of the girder cross section. The measured temperatures in 2012 are used to predict the sitespecific extreme vertical girder temperature difference, where the daily maxima are obtained and plotted in Figure  13. Owing to data missing, only 265-day samples are available in 2012.
The transverse girder temperature difference is defined as T D7D1 ¼ T D7 À T D1 in this case. One-year daily maxima are used to predict the site-specific extreme transverse girder temperature difference, which are plotted in Figure 14.
Owing to data missing, only 265-day samples are available in 2012. The temperature differences of T U8U1 , T U7U2 , T U6U3 and T U5U4 are used to predict site-specific extreme  temperature difference of the tower leg section. The effective temperature difference of the tower leg is regarded as the representative value. One-year daily maxima are used to predict the site-specific extreme tower temperature difference, which are plotted in Figure 15. Owing to data missing, only 240-day samples are available in 2012.

Probabilistic estimation of extreme temperatures
To introduce the prediction process of site-specific extreme temperatures by using the GPD, the vertical girder temperature difference is chosen as an instance, and the related calculation steps are presented. Similarly, the threshold should be determined first. The mean excess function and RMSE of the vertical girder temperature difference are figured out and shown in Figure 16. The RMSE achieves its lowest point when the threshold equals to 7.15 C, which supports that the best linear fit of the mean excess and threshold is achieved at the threshold of 7.15 C. Moreover, adequate samples are available over the determined threshold. Thus, the threshold for prediction of extreme vertical girder temperature difference is set as 7.15 C in this case.
By applying the Bayesian estimation to the studied data, the iteration processes of the two parameters (i.e. shape and scale parameters) are figured out as shown in Figures 17(a) and 18(a). As suggested, some samples in the early stationary period are abandoned, which are highlighted by using the fuzzy area. In this case, the stationary period for the shape parameter is determined as [13,000,40,000], and the stationary period of the scale parameter is [16,500,40,000]. Then, the samples within the stationary period are used for statistical analysis. Unimodal and bimodal Gaussian distributions are used to fit the histograms. As a result, f $ Nðmean ¼ À0:3375, std ¼ 0:0188Þ, f $ BNðmean1 ¼ À0:3691, std1 ¼ 0:0050, mean2 ¼ À0:3362, std2 ¼ 0:01728, alpha ¼ 0:06547Þ, r $ Nðmean ¼ 5:5087, std ¼ 0:1461Þ, and r $ BNðmean1 ¼ 5:5356, std1 ¼ 0:0:1441, mean2 ¼ 5:2577, std2 ¼ 0:03578, alpha ¼ 0:9588Þ: In the bimodal Gaussian fitting, although the two peaks of both the shape and scale parameters are not extremely close to each other, the parameter alpha is extremely high or low, which indicates that one of the two peaks within the bimodal distribution plays a dominant role. Fortunately, means of both the dominant peaks for shape and scale parameters are similar to those of the unimodal Gaussian distribution. In this regard, the parameters are prone to be distributed following the unimodal Gaussian function, and the fitting results are shown in Figures 17(b) and 18(b). Eventually, 40000 random samples of extreme vertical girder temperature difference are generated based on the parameters distributed in Figures  17(b) and 18(b). The histogram and pdf of the extreme vertical girder temperature difference are plotted in Figure 19, which is compared to the design one.
Similarly, the probability density functions of extreme uniform girder temperature change (dT u ), extreme transverse girder temperature difference (T tg ), and extreme tower temperature difference (T t ) were all predicted using the same procedure as stated above. The threshold (u), distributions of shape and scale parameters (f and r), distributions of predicted extreme loads (x p ) and respective design values (x d ) for the four temperature loads are summarized in Table  1. It is noted that the threshold, distributions of shape and scale parameters of uniform girder temperature change are inapplicable in Table 1 since this thermal load is calculated   from the extreme maximum and minimum uniform temperatures. In detail, extreme value distributions of the maximum and minimum uniform temperatures were first predicted. As a result, the estimated extreme maximum uniform temperature follows a Gaussian distribution of Nð46:04, 0:12Þ, while the extreme minimum uniform obeys a Gaussian distribution of NðÀ1:52, 0:18Þ: Then, the extreme value distribution of uniform girder temperature change Nðl, rÞ is derived, where the mean and standard deviation were calculated by using where l max and r max are mean and standard deviation of the distribution of extreme maximum uniform temperature, and l min and r min are mean and standard deviation of the distribution of extreme minimum uniform temperature. As   a result, the extreme uniform girder temperature change follows a normal distribution of Nð47:56, 0:30Þ as listed in Table 1.

Discussions
According to Figure 19, majority of the estimated site-specific extreme vertical girder temperature difference samples are larger than the design one, which concludes that the design vertical girder temperature difference is insufficient for this signature bridge. Moreover, Gaussian distribution is employed to interpret the uncertainty of extreme vertical girder temperature difference, where T tg $ Nð22:91, 0:87Þ: The estimated extreme values fluctuate within a relatively wide range, specifically between 20 C and 26 C. Based on the pdf of extreme vertical girder temperature difference, the probability that the estimated site-specific extreme value exceeds the design one is calculated as As a result, there is a 99.96% confidence to conclude that the design vertical girder temperature difference is insufficient for this signature bridge. If the point estimation is applied, the predicted extreme vertical girder temperature difference is 21.25 C. Although the point estimation method could conclude the designed vertical girder temperature difference is insufficient, stakeholders lack information of confidence to draw this conclusion. The vertical girder temperature difference may yield a vertical bending of the girder, where the higher temperature side will bear extra tension stress owing to the constraints of girder ends. Since the probability that the estimated site-specific extreme vertical girder temperature difference is larger than the design one is high, more attention is suggested to be paid to the tension stress of deck plate. However, mean of the extreme vertical girder temperature difference is 22.91 C, which slightly exceeds the design value 20.0 C. Since the estimated vertical girder temperature difference is not significantly higher than the design value, which will not generate tremendous stress of the girder. Thus, it is suggested to take this situation into account when planning bridge maintenance and management activities. Furthermore, the discussion in this study could offer valid information for the updating of the bridge design code.
According to Table 1, the probability that the estimated extreme uniform girder temperature change exceeds the design value is almost 0, which means the design load is sufficient for the signature bridge. However, the probabilities that the estimated extreme transverse girder temperature difference and tower temperature difference exceed the design ones are almost 100%, which remind that the design values of the two thermal loads are insufficient for the signature bridge. Similarly, the identical conclusions could be drawn by using the point estimation, where the extreme uniform girder temperature change, extreme transverse temperature difference, extreme tower temperature difference are 43.26 C, 11.26 C, and 9.21 C, respectively. However, compared with the Bayesian estimation, the probabilistic information is dismissed. For the transverse girder temperature difference, it produces a transverse girder bending. For the tower temperature difference, it yields a displacement of the tower top. Since the transverse girder width significantly exceeds its depth and the strong connection exists between the two tower sections, the two temperature differences might not introduce secondary stresses compared with the vertical girder temperature difference in this case. However, it is still suggested to pay attention to the design values of the transverse temperature difference of the girder and the temperature difference of the tower section in the design codes and documents.

Conclusions
In this paper, a Bayesian approach is proposed to estimate SELs of bridges based on SHM data, which takes uncertainties within the estimation process into consideration. The probability that the predicted SEL exceeds the design one is calculated to determine whether the design load is sufficient for the signature bridge. Wind speed and temperature measurements from the NDB are used for estimation of SELs to verify the effectiveness of the proposed method. The conclusions drawn from this study are summarized as follows: 1. The GPD is proper to predict SELs on the condition that the related loading information is limited, which requests that the extreme data used for the GPD analysis should be independently, identically distributed. 2. Bayesian estimation explicitly accounts for uncertainties arising from environmental variability, measurement noise and parameter estimation, which is used to figure out probabilistic distributions of shape and scale parameters of the GPD, and extreme loads. MG sampler, one of MCMC algorithms, is employed to figure out posterior distributions of the GPD parameters. 3. Based on ten-year wind speed data from the NDB, there is a nearly 100% confidence to conclude that the design wind speed is sufficient for the signature bridge. Nevertheless, more data are needed in the coming years ahead to update the results. 4. Based on one year temperature measurements, four types of site-specific extreme thermal loads (i.e. uniform girder temperature change, vertical girder temperature difference, transverse girder temperature difference, and tower temperature difference) are predicted. In total, the design value of uniform girder temperature difference is sufficient in this study. However, the other three design thermal loads are insufficient. In specific, the probability that the estimated extreme vertical girder temperature difference exceeds the design one is 99.96%, and the probabilities subject to the transverse girder temperature difference and tower temperature difference are almost 100%. It is suggested to pay attention to the design values of the vertical temperature difference of the bridge girder, the transverse temperature difference of the girder and the temperature difference of the tower section in the design codes and documents.