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Abstract

The effects of operational and environmental variability have been posed as one of the biggest challenges to transit structural health monitoring (SHM) from research to practice. To deal with that, machine learning algorithms have been proposed to learn from experience based on a reference data set. These machine learning algorithms work well if the operational and environmental conditions under which the bridge operates do not change over time. Meanwhile, climate change has been posed as one of the biggest concerns for the health of bridges. Although the uncertainty associated with the magnitude of the change is large, the fact that our climate is changing is unequivocal. Therefore, it is expected that climate change can be another source of environmental variability, especially the temperature. So, what happens if the mean temperature changes over time? Will it significantly affect the dynamics of bridges? Will the reference data set used for the training of algorithms become outdated? Are machine learning algorithms robust enough to deal with those changes? This is a pioneering work on the impact of climate change on the long-term damage detection in the context of bridge SHM. A classifier rooted in machine learning is trained using one-year data from the Z-24 Bridge in Switzerland and tested with current and future data. Three climate change scenarios are assumed, centered in three future periods, namely 2035, 2060, and 2085. This study concludes that climate change may be seen as another source of operational and environmental variability to be considered when using machine learning algorithms for long-term damage detection.

Keywords

Structural health monitoring climate change bridges machine learning damage detection

Introduction

In the last three decades, structural health monitoring (SHM) has been a promising tool in management activities of bridges as potentially it enables one to perform condition assessment aiming to reduce the uncertainty in the planning and designing of maintenance activities as well as to increase the service performance and safety of operation.¹ The general idea has been the transformation of massive data obtained from monitoring systems into meaningful information. To deal with large amounts of data and perform the damage identification automatically, SHM has been cast in the context of a statistical pattern recognition paradigm.²

The effects of operational and environmental variability have been posed as one of the biggest challenges to transit SHM from research to practice.³ Varying operational conditions include live loads such as traffic loads, speed of operation, and changing excitation sources. Varying environmental conditions are manifestations of weather in the form of temperature, relative humidity, wind, rainfall, and snow. The effects of the operational and environmental variability are particularly relevant on the damage-sensitive features, such as modal properties. In particular, daily temperature variations can impose significant changes in the natural frequencies. For instance, a study addressing the influence of the temperature effect on the modal parameters of the Z-24 Bridge concluded that the absolute value of percentage changes in the second natural frequency caused by ambient temperature variations (11.21%) is higher than those changes caused by damage (9.49%). Based on experimental data obtained from the Alamosa Canyon Bridge, in the United States, Farrar et al.⁴ noted that natural frequencies were found to vary, approximately, 5% during a 24-h period. In the same bridge, another study found an asymmetrical variation in the first mode shape observed throughout the day. This asymmetry along the longitudinal axis was correlated with the time of the day and associated solar heating. If not properly accounted for, such changes in the dynamic response characteristics can potentially result in false indications of damage. This outlier behavior could inappropriately be labeled as damage if the environmental variability associated with this feature was not considered in the outlier detection process.³ A review on the effects of operational and environmental variability in SHM can be found in the study by Sohn⁵; a bridge-guided review on those effects can be found in the study by Figueiredo.⁶

To deal with those effects, machine learning algorithms have been proposed to learn from experience based on a reference data set, which normally represents the bridge behavior for at least one year. These machine learning algorithms work well based on the premise that the basis of the reference data does not change significantly over time.

Meanwhile, climate change has been posed as one of the biggest concerns for the health of bridges.⁷ Recently, Figueiredo et al.⁸ have reviewed the work already done on the impacts of climate change to bridges and set a roadmap for an integrated assessment approach to the adaptation of concrete bridges to climate change. In this approach, SHM is proposed as a mechanism for assessing and continuously evaluating the structural condition of bridges and for triggering adaptation measures imposed by climate change. In this regard, SHM systems are viewed as an indispensable element of efficient climate change adaptation strategies that enable the early detection and prognosis of climate change impacts on bridges (see also Argyroudis et al.⁹). This would subsequently enable the timely implementation of appropriate adaptation measures. Adaptation is considered essential to advance climate resilient development, as pointed out recently by the Intergovernmental Panel on Climate Change (IPCC).¹⁰

Although the uncertainty associated with the magnitude of the change is large, the fact that our climate is changing is unequivocal.¹¹ Therefore, it is expected that climate change can be another source of environmental variability, especially the temperature.¹² Frangopol et al.¹³ presented a brief overview of the integration of risk, sustainability, and resilience into the life-cycle management of deteriorating infrastructure with an emphasis on bridges while considering climate-change effects. Rising global temperatures was one of the three greatest concerns regarding climate change for highway bridges (increased frequency of heavy precipitation events and increased levels of atmospheric CO₂ were the other two).

In recent decades, several studies have focused on analyzing and quantifying the impacts of climate change on infrastructure.^14–19 Noting that infrastructure deterioration is associated with severe consequences.^20,21 Bastidas-Arteaga et al.¹⁴ and Stewart et al.¹⁹ were among the first studies that focused on investigating the effect of climate change on the deterioration of reinforced concrete infrastructure. For instance, based on the results of Stewart et al.,¹⁹ up to a 400% increase in the risk of carbonation-induced damage of concrete infrastructure is possible in some regions in Australia by the end of the current century. Following the two studies of Bastidas-Arteaga et al.¹⁴ and Stewart et al.,¹⁹ several other researchers analyzed and quantified this effect in different regions and contexts (e.g., AL-Ameeri et al.²²; Chirdeep et al.²³; Köliö et al.²⁴; Mohamed and Jayadipta²⁵; Hamidreza et al.²⁶). Tidblad²⁷ was among the first researchers to quantify the impact of climate change on the durability of metallic infrastructure. One of the notable findings of Tidblad is that a significant effect of climate change on the corrosion of metals is expected in certain climate scenarios, especially, for coastal areas of southern Europe. Other studies that analyzed this impact include Chaves et al.,²⁸ Nguyen et al.,²⁹ Peng et al.,³⁰ and Zhang et al.³¹ The deterioration of timber infrastructure in a changing climate was also analyzed in several studies (Bjarnadottir et al.³²; Ryan and Stewart³³; Salman et al.³⁴; Wang and Wang³⁵). In particular, Wang and Wang³⁵ concluded that the median decay rate of timber elements may increase by up to 10% in some climate scenarios by the year 2080. Other climate change impacts on infrastructure that were analyzed by researchers include bridge scour (e.g., Dikanski et al.¹⁵ and Amro et al.,¹⁷ Kallias and Imam,³⁶ Omid and Mohamed,³⁷ Khelifa et al.³⁸ David and Frangopol³⁹), creep of reinforced concrete,¹⁶ and storm- and wind-induced damages to infrastructure,¹⁸ among others. For more elaborate reviews on the potential impacts of climate change on infrastructure, the reader is referred to Nasr et al.,⁷ Figueiredo et al.,⁸ Mishra and Sadhu,⁴⁰ and André et al.⁴¹

However, an important gap in previous literature relates to the effect of the varying environmental conditions that are induced by climate change on the long-term damage detection performance of SHM systems. Addressing this gap is necessary for unleashing the full potential of SHM systems in adapting to the impacts of climate change. Nonetheless, to the best of the authors’ knowledge, this issue has never been addressed so far.

So, what happens if the average temperature changes over time? Will it significantly affect the dynamics of bridges? Will the reference data set used for the training of algorithms become outdated? Are machine learning algorithms robust enough to deal with those changes? Outlier detection in a changing environment was studied by Worden et al.⁴² If the reference data set is only characteristic of a limited range of the environmental parameters, the measurements in an undamaged condition but from a different environmental state may cause false inference of damage; the paper demonstrated a potential solution to the problem via the construction of a reference set parameterized by an environmental variable. However, climate change was not one of the environmental variables.

Therefore, this paper, for the first time, analyzes the impact of climate change on long-term damage detection performance in bridges, using a classifier built with a machine learning algorithm trained with roughly one-year data set from the Z-24 Bridge in Switzerland. The objective of the paper is to assess whether a competent (well trained, highly effective) machine learning algorithm for damage detection may become less competent because of the temperature increase caused by climate change. The data set is from a reference period between 1997 and 1998. The performance is tested for climate change projections based on three different Representative Concentration Pathway (RCP) scenarios, namely, RCP2.6, RCP4.5, and RCP8.5, in three future periods centered in the years 2035, 2060, and 2085 to represent, respectively, the near future, the mid-century, and the end-of-century. Seasonal variability of the climate change impact is taken into account. To fully capture the uncertainty related to climate change projections, three estimates are considered to project changes in average annual temperature within each scenario, namely lower estimate (5th percentile), medium estimate (50th percentile), and upper estimate (95th percentile).

Besides this first section, this article is organized as follows. In the second section, the long-term outlier detection strategy is laid down and the machine learning algorithm used is briefly introduced. The third section presents and describes the Z-24 Bridge, outlines the data sets for the reference period, and explains the estimation process of the data for future periods as a function of projected changes in average annual temperature. Three key climate change estimates are defined for detailed analysis. The fourth section shows the results and discusses the main findings. Finally, the fifth section draws the main conclusions on the influence of climate change on the long-term damage detection process in bridges.

Long-term damage detection process

Problem, challenges, and assumptions

The damage detection process can be cast in the context of outlier detection. In an outlier detection perspective for feature classification, one possible approach for data mining is a clustering procedure combined with a (dis)similarity metric and a threshold. The goals of data mining for damage detection are:

1. learning clusters of observations extracted from monitoring data that correspond to normal structural conditions (undamaged condition) in the training phase;

2. estimating a threshold from training data (set of observations from the undamaged condition) to classify new observations;

3. computing a damage indicator (DI) for each new observation considering the (dis)similarity between this observation and the centers of the clusters; and

4. classifying each DI according to a defined threshold.

Herein, an observation is a feature vector composed of one or more types of damage-sensitive features at a given time.

The learning clusters play a crucial role for mining patterns and extracting knowledge from the training data, as they can be correlated with the physical states of the structure under operational and environmental influences.¹⁹ The better the clusters can represent the undamaged condition, the better is the classification performance.

Some of the challenges in the data mining for feature classification are discussed as follows:

updating statistical models as new data become available;

for a specific data mining task, the choice of the machine learning algorithm must be done as a function of the damage-sensitive features used and their distribution in the feature space;

the feature classification is often posed in the context of a binary classification, with a trade-off between false-positive (type I error) and false-negative (type II error) indications of outliers or damage. It recognizes that false-positive classifications may have different practical consequences than false-negative ones. Analytical approaches to defining threshold levels must balance trade-offs between false-positive and false-negative indications. One should minimize false-positives when economic concerns drive the SHM applications and minimize false-negatives when life safety issues drive the SHM systems.

As a first study and preliminary analysis, the authors assume that the undamaged condition (i) can be modeled as a combination of several Gaussian distributions, corresponding to different structural and environmental conditions; and (ii) the training process is static in time, and so machine learning algorithms are not updated as new data (i.e., observations) become available.

Machine-learning-based classifier: Gaussian mixture model

One of the first publications to use Gaussian mixture models (GMMs) for bridge SHM was performed by Figueiredo and Cross.⁴³ Other works using GMM on bridge data sets can be found in the literatures.^44–46 The GMM is a probabilistic model that assumes all features are generated from a mixture of a finite number of Gaussian distributions. GMM captures the main cluster/components present in the damage-sensitive features, which correspond to the normal and stable state conditions even when the structure is affected by extreme operational and environmental conditions. Afterward, an outlier detection strategy is implemented in relation to the main components. The damage detection is carried out based on multiple Mahalanobis squared distances (MSDs), where the covariance matrices and mean vectors are function of the main components.

The GMM-based algorithms have shown superior performance over other machine learning algorithms due to their capability to model singularities of the data sets, as demonstrated by Figueiredo and Santos.⁴⁷

For general purposes, one may assume a training data matrix (X $\in R^{n \times m}$ ) with m-dimensional feature vectors (also known as observations) from n different environmental and operational conditions, when the structure is undamaged, and a test data matrix (Z $\in R^{l \times m}$ ) where l is the number of observations from undamaged and/or damaged conditions. First, the machine learning algorithm is trained, and its parameters are adjusted using observations from X. Second, in the test phase, the algorithm transforms each new input observation from Z into a DI.

A finite-mixture model [p(X | Θ)] is defined⁴⁸ as the weighted sum of K > 1 components, or clusters in the context of SHM, [p(x | θ_k)] in $R^{m}$ ,

$p (X | Θ) = \sum_{k = 1}^{K} α_{k} p (x | θ_{k}),$ (1)

where x is an m-dimensional observation and α_k corresponds to the weight of each component. These weights are constrained to α_k > 0 with $\sum_{k = 1}^{K} α_{k}$ = 1. For a GMM, each component [p(x | θ_k)] is represented as a Gaussian distribution,

$p (x | θ_{k}) = \frac{1}{{(2 π)}^{m / 2} \sqrt{\det (Σ_{k})}} \exp {- \frac{1}{2} {(x - μ_{k})}^{T} {(Σ_{k})}^{- 1} (x - μ_{k})},$ (2)

where θ_k = {μ_k,Σ_k} is a parameter composed of the mean vector (μ_k) and the covariance matrix (Σ_k). The GMM is completely specified by the set of parameters Θ = {α₁, α₂, …, α_k, θ₁, θ₂, …, θ_k}.

The expectation–maximization algorithm is the most common technique used to estimate those parameters. This approach consists of an expectation step and a maximization step, which are applied until the log-likelihood (LogL), [log p(X | Θ) = log $Π_{i = 1}^{m} p (x_{i} | θ)$ ], converges to a local optimum, although not necessarily the global optimum. The performance of the expectation–maximization algorithm depends directly on the choice of the initial parameters Θ. Therefore, the technique is applied with various choices of initial parameters to improve the chances of finding the optimal solution, that is, for each number of components, k, the parameters are chosen based on the lowest error given for several runs with initial random parameters.

The most efficient GMM can be selected based on the Bayesian information criterion (BIC), which determines the appropriate number of components K,

$BIC = - 2 \log p (X | Θ) + {Km [(\frac{m + 1}{2}) + 1] + K - 1} \log (n) .$ (3)

Similar to Akaike information criterion (AIC), BIC uses the optimal LogL function value and penalizes for more complex models, that is, models with additional parameters. The penalty term is a function of the training data size and so it is often more severe than AIC.

The outlier detection is based on the MSD, or DI in the context of SHM, because it is well known that its applicability is almost insensitive to the effects of operational and environmental conditions as well as to reduce each observation (z) into a score.⁴² Basically, for each main component (k) of the data:

$D I_{k} (z) = (z - μ_{k}) {(Σ_{k})}^{- 1} (z - μ_{k})^{T} .$ (4)

The mean vector, $μ_{k}$ , and covariance matrix, $Σ_{k}$ , represent all the observations from the k component of the data, when the structure is undamaged even though under varying operational and environmental conditions. If the null hypothesis (undamaged condition) is true, that is, a new observation z is extracted from the same multivariate normal distribution as the undamaged data, then the test statistic DI will be approximately Chi-square distributed with m degrees-of-freedom, $χ_{m}^{2}$ . Finally, for each observation, the DI is given by the smallest DI estimated on each component,

$DI (z) = \min [D I_{1} (z), . . ., D I_{k} (z)] .$ (5)

Bridge and data set description

Structural description

The Z-24 Bridge (Figure 1) was a post-tensioned concrete box girder bridge composed of a main span of 30 m and two side spans of 14 m each. Before complete demolition, the bridge was instrumented and tested with the purpose of providing a benchmark for vibration-based SHM. A long-term monitoring program was carried out from November 11, 1997, until September 10, 1998, to monitor the changes of the natural modes of vibration of the bridge caused by environmental variability and controlled damage.⁴⁹ Every hour, eight accelerometers captured the vibrations of the bridge as sequences of 65,536 samples (sampling frequency of 100 Hz), and other sensors measured environmental parameters, including temperature at several locations (Figure 2).

Figure 1.

Z-24 Bridge in Switzerland.

Figure 2.

Longitudinal section (a) and the location and orientation of accelerometers (b) on the Z-24 Bridge. Sensors marked in red failed during the monitoring campaign.

The monitoring program consisted of two phases. From November 11, 1997, until August 5, 1998, the bridge was monitored under environmental variability only (the bridge was closed to traffic), using a coarse network of sensors. From August 5, 1998, a controlled damage campaign was conducted on the bridge. It involved the installation of hydraulic jacks into one of the piers, through which pier settlements and foundation tilts were enforced, followed by concrete spalling, landslide at abutment, concrete hinge failure, failure of anchor heads, and rupture of tendons. The damage scenarios were carried out to prove that realistic damage has a measurable influence on the bridge dynamics. Those scenarios were performed in a sequential manner, which caused a cumulative degradation of the bridge. The various scenarios of the monitoring program are listed in Table 1, along with the periods when the measurements took place. A total of 4915 observations were obtained in this way (for technical reasons, the monitoring system was down at times, which caused periodic failures in the data acquisition). The observations correspond to hourly measurements.

Table 1.

Description of the condition scenarios and observations.

Observations	Start-end dates	Condition	Scenario description
1–4339	11 November 1997 to 05 August 1998	Undamaged	Environmental variability
4340–4475	11 August 1998 to 19 August 1998	Damaged	Pier foundation settlement
4476–4580	19 August 1998 to 25 August 1998	Damaged	Settlement recovered
4581–4626	25 August 1998 to 27 August 1998	Damaged	Spalling of concrete
4627–4718	27 August 1998 to 31 August 1998	Damaged	Landslide
4719–4765	31 August 1998 to 02 September 1998	Damaged	Failure of concrete hinges
4766–4808	02 September 1998 to 07 September 1998	Damaged	Failure of anchor heads
4809–4915	07 September 1998 to 10 September 1998	Damaged	Failure of tendon wires

Figure 7(a) shows the observations composed of the first two natural frequencies estimated using a reference-based stochastic subspace identification method on vibration measurements from the accelerometers,⁵⁰ along with colored markers reflecting the measured ambient temperature. As the controlled damage campaign started on August 5, 1998, a considerable number of hot days preceded the introduction of damage. This can be visually confirmed by checking the amount of red and yellow markers in Figure 7(a) that are located before the vertical line that marks the onset of damage. They are no less than those observation located in the damage time range, and enough to leave a strong influence on the machine learning process. However, to improve the understanding of the temperature variability, a time history of the ambient, asphalt, and concrete temperatures over the monitoring period is shown in Figure 3.

Figure 3.

Time histories of the ambient, asphalt, and concrete temperatures over the monitoring period.

Climate change data and scenarios

Several sets of climate change scenarios have been introduced in the literature. The current study uses climate change projections based on RCP2.6, RCP4.5, and RCP8.5. The number identifying each scenario represents the radiative forcing (i.e., the change in energy flux per surface area) in W/m² at the end of century, or at stabilization afterward, relative to the preindustrial era. Hence, RCP8.5 depicts a relatively pessimistic scenario with high greenhouse gas (GHG) emissions, while RCP2.6 reflects a scenario with stringent mitigation measures resulting in low GHG emissions. It is worth noting that both the fifth and sixth assessment reports of the IPCC adopt scenarios with forcings equivalent to the three scenarios considered herein.^10,11 RCP4.5 is a scenario lying between the previous two extremes.

The RCP can be used to obtain projections from several climate change variables, like temperature, relative humidity, precipitation, and so on. However, as shown in previous publications,⁴⁹ the temperature is in fact the main driving factor of global stiffness changes.

For the sake of clarity, Table 2 summarizes and Figure 4 shows the projected changes in average annual temperature (°C) for lower estimate (5th percentile), medium estimate (50th percentile), and upper estimate (95th percentile) for Bern (in Switzerland) and for the years 2035, 2060, and 2085 based on the three RCP scenarios. One can clearly observe that RCP8.5 yields a sharper increase of temperature than RCP4.5; for RCP2.6, a stagnation is observed after 2060.

Table 2.

Projected changes in average annual temperature (°C) for three climate change scenarios (Bern, Switzerland).

Scenario	2035			2060			2085
Scenario	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate
RCP2.6	0.6	1	1.6	0.7	1.3	2	0.7	1.2	1.9
RCP4.5	0.7	1.2	1.6	1.2	1.8	2.6	1.5	2.2	2.9
RCP8.5	0.9	1.4	1.8	2	2.5	3.2	3.3	4.3	5.3

Figure 4.

Projected changes in average annual temperature (°C) for three climate scenarios (RCP2.6, RCP4.5, and RCP8.5) for Bern, in Switzerland.

On the other hand, the average annual temperature changes are not uniformly distributed over the four seasons of the Swiss climate. Table 3 summarizes the projected changes in average seasonal temperature (spring, summer, autumn, and winter). These data were acquired from the CH2018 project database⁵¹ and are based on simulations involving seven different regional climate models with boundary conditions from 12 different global climate models. Quantile mapping was used for downscaling and the correction of bias in the projections. More details on the method used for producing these data are available in reference.⁵² These temperature data sets were used to project future observations of the natural frequencies based on the procedure described in the next section.

Table 3.

Projected changes ( $Δ T$ ) in average seasonal temperature (°C) for three climate change scenarios (Bern, Switzerland).

Scenario	Change in average spring (March–May) temperature for Bern (°C)
	2035			2060			2085
	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate
RCP2.6	0.3	1	1.3	0.2	1.1	1.5	0.3	0.9	1.6
RCP4.5	0.4	1	1.5	0.8	1.5	2.1	1.1	1.8	2.5
RCP8.5	0.5	1	1.6	1.4	2.2	3	2.7	3.7	4.8
Scenario	Change in average summer (June–August) temperature for Bern (°C)
	2035			2060			2085
	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate
RCP2.6	0.6	1.2	1.9	0.9	1.4	2.1	0.8	1.4	2
RCP4.5	0.9	1.3	1.8	1.6	2.2	2.9	1.8	2.5	3.3
RCP8.5	1	1.6	2.3	2.2	2.8	3.9	3.7	4.3	6.5
Scenario	Change in average autumn (September–November) temperature for Bern (°C)
	2035			2060			2085
	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate
RCP2.6	0.6	1	1.7	0.9	1.3	2.1	0.6	1.3	2.1
RCP4.5	0.8	1.2	1.8	1	1.8	2.8	1.7	2.1	3.1
RCP8.5	1	1.3	2.1	1.9	2.7	3.6	3.2	4.2	5.6
Scenario	Change in average winter (December–February) temperature for Bern (°C)
	2035			2060			2085
	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate	Lower estimate	Medium estimate	Upper estimate
RCP2.6	0.6	0.9	1.8	0.6	1.3	2	0.7	1.4	2.1
RCP4.5	0.7	1.2	1.7	1.2	1.8	2.6	1.4	2.1	2.9
RCP8.5	0.6	1.5	1.8	1.7	2.3	2.9	3	4	4.9

RCP: Representative Concentration Pathway.

It should be highlighted that the RCP scenarios are commonly used by the scientific community for projecting climate change variables as they describe plausible trajectories of the future climate under different forcing conditions. Using these scenarios for studying the impact of climate change demonstrates the effect of low, average, and high radiative forcings. This is useful for planning efficient adaptation strategies in both SHM systems and bridges themselves. However, it is worth mentioning that scenarios other than the ones used herein may also be plausible.¹⁰

It should also be noted that the different uncertainties involved in projecting climate change (i.e., scenario uncertainty, climate model uncertainty, and internal variability, see, e.g., the study by Figueiredo et al.⁵³) are reflected in the current study. Scenario uncertainty is accounted for by using three different RCP scenarios, 12 different climate models to represent climate model uncertainty, and averaging over 30-year periods to capture internal variability.

Projections of frequency measurements under climate change

To assess the impact of climate change on the damage detection performance of machine learning algorithms trained with climatically outdated data, a projection of the frequency readings corresponding to the scenarios listed in Table 3 is performed. The projection considers that the ambient (or air) temperature recorded during the monitoring campaign of 1997–1998 is incremented with the changes listed in Table 3, according to the season during which they were measured. Then, the natural frequencies corresponding to the new ambient temperatures are obtained from the Gaussian process regression (GPR) of the frequencies recorded during the monitoring campaign. The variability of the projected frequencies is also calibrated to that of the historical data. The methodology used for the projection of future natural frequencies ( ${\hat{f}}_{i}$ ) is summarized in the flowchart shown in Figure 5.

Figure 5.

Flowchart for the projection of the natural frequencies as a function of temperature.

The natural frequencies ( $f_{0}$ ) measured for the undamaged condition during the 1997–1998 campaign exhibit two distinct trends based on whether the ambient temperature ( $T_{0}$ ) is above or below 0°C.⁴⁹ As the data have high degree of variability and some outliers that could influence the quality of the regression model, an outlier detection and removal method was employed to cleanse it before performing GPR. In particular, the local outlier factor (LOF) method was used with a contamination fraction of 5%, that is, the method assumes that 5% of the data are outliers and detects and removes them based on the relative density of an observation with respect to its neighbors, reducing the total number of observations from 4915 to 4698. Figure 6 shows the outlier-free behavior for the first two frequencies after the LOF method was applied (blue dots). Even though these plots clearly show the two previously mentioned trends for negative and positive temperatures, a segmented linear regression would not model adequately the behavior of the data either (i) in areas with a higher degree of variability or (ii) in the tails, which follow a more curved shape. Hence, a GPR model is fitted to the undamaged data set; Figure 6 also shows the GPR models obtained for the first two frequencies.

Figure 6.

Outlier-free original undamaged natural frequencies along with the GPR models. (a) First natural frequency. (b) Second natural frequency.

The projected changes in average seasonal temperature for the three climate scenarios were obtained by adding the average season estimates (temperature increments, $Δ T_{i}$ ) presented in Table 3 to the original measured temperatures, that is, $T_{i} = T_{0} + Δ T_{i}$ . Preliminary projections of the frequencies, ${\tilde{f}}_{i}$ , are obtained using these increased temperatures on the fitted GPR models, resulting in a data set falling onto the GPR curves.

In order to simulate a more realistic set of future frequencies and to maintain the same degree of variability as in the original data, the residuals in GPR models are assumed to follow a Gaussian distribution with zero-valued mean. Following this hypothesis, the following steps are performed:

1. the standard deviations of the original undamaged frequencies are estimated ( $σ_{0}$ ) and used to generate random samples for the residuals, $Δ f_{i}$ , as shown in Figure 5;

2. the generated residuals are added to the preliminary estimations of the frequencies, allowing one to obtain the final projected frequencies: ${\hat{f}}_{i} = {\tilde{f}}_{i} + Δ f_{i}$ . The generated residuals were carefully chosen to ensure that the relative differences between the standard deviations of the finalprojected frequencies ( ${\hat{f}}_{i}$ ) and the original data ( $f_{0}$ ) are less than 2%.

The natural frequencies measured under damaged state conditions are also projected into the future considering the temperature increments, but no additional variability is added, meaning that the data falls directly onto the GPR curves.

Note that the temperature is assumed as the driving factor of frequency changes, but (i) the influence of heating and cooling of the bridge’s materials and surrounding soil was taken into account through the GPR interpolation of the frequencies that were measured when these heating and cooling effects actually occurred (Figure 6); (ii) the normal variability of the frequencies around the GPR interpolation was considered as explained in this section, so that the future frequency projections have the same variance as the original monitored data.

Figure 7(a) shows the observations (composed of the first two natural frequencies) for the reference period (1997–1998) as a function of ambient temperature. For completeness and illustration purposes, Figure 7(b), (c), and (d) show, respectively, the projected observations for three key climate change estimates: a lower-case (RCP2.6, lower estimate), an intermediate-case (RCP4.5, medium estimate), and the worst-case (RCP8.5, upper estimate) scenarios. These key scenarios are chosen to represent the most favorable, the median, and the most unfavorable of the 27 tested scenarios listed in Table 3. However, similar analyses are performed for all scenarios, as discussed in the next section.

Figure 7.

Monitoring observations for (a) the reference period (1997–1998) and projected observations for the three key climate change estimates (b, c, and d). (a) 1997–1998. (b) 2035 (RCP2.6, lower estimate). (c) 2060 (RCP4.5, medium estimate). (d) 2085 (RCP8.5, upper estimate).

Results and discussion

Outlier detection

The first two natural frequencies of the bridge are taken as damage-sensitive features. The reduced number of frequencies enhances the visibility of the clustering and classification plots with minimal loss of relevant information. Indeed, it is noted that the first and third frequencies are strongly correlated,^54,55 and that the first two frequencies can cluster the various conditions that the bridge experienced during monitoring quite consistently. To illustrate this feature, Figure 8 presents the clustering of the data in the frequency space obtained with a GMM and with the K-medoids method. Both techniques yield similar results, identifying two dense clusters on the lower-left side of the plots and another cluster which is spread out over the higher frequency range. The clusters are mostly related to the ambient temperature ranges and consist of readings made at temperatures roughly above 10°C (green cluster), below −2°C (blue cluster), and in-between these values (red cluster). The GMM clustering procedure is adopted for the ensuing analysis as the distribution of the frequencies in the denser clusters is relatively close to Gaussian. For instance, Figure 9 presents the warm temperatures cluster and the histograms of the first two frequencies distributions, along with their respective skewness and kurtosis measures. It is recalled that a purely Gaussian distribution has null skewness and kurtosis equal to three.

Figure 8.

Undamaged condition defined by three clusters using GMM and K-medoids.

Figure 9.

Gaussian-like distributions of the first two frequencies.

A GMM-based classifier is built assuming a training data set consisting of 80% of the monitoring observations (composed of the first two frequencies) from the undamaged bridge selected randomly. Three clusters are assumed as shown in Figure 8(a). For the outlier detection (or test) phase, the threshold is defined for a level of significance of 5% of a chi-squared distribution. The 5% level of significance is a common choice in the literature (e.g., the study by Bud et al.⁵⁶), as it strikes a good balance between the risks of the algorithm yielding false positives and false negatives.

For the reference period (1997–1998), the test phase was performed on the remaining 20% of the monitoring observations made on the undamaged bridge along with all observations from its damaged condition. For each of the three climate change scenarios (RCP2.6, RCP4.5, and RCP8.5) in each future period (2035, 2060, and 2085), the test phase was carried out with all projected observations. Figure 10 shows the outlier detection for the reference period and for the three key climate change scenarios, by assigning a DI to each observation.

Figure 10.

Outlier detection for the reference period (a) and for the three key climate change estimates (b, c, and d). (a) In 1997–1998. (b) In 2035 (RCP 2.6, lower estimate). (c) In 2060 (RCP 4.5, medium estimate). (d) In 2085 (RCP 8.5, upper estimate).

Tables 4 and 5 summarize the classification performance in terms of a binary classification, that is, type I (false-positive) and type II (false-negative) errors. For a better visualization of the trends, Figure 11 shows the evolution of those errors (type I and type II) as a function of each climate change scenario.

Table 4.

Summary of false positives (type I errors) in percentage (%).

(a) Lower estimate.
Scenario	1997–1998	2035	2060	2085
RCP2.6	2.18	2.84	2.28	2.6
RCP4.5		2.54	2.23	1.92
RCP8.5		2.69	1.65	1.21
(b) Medium estimate.
Scenario	1997–1998	2035	2060	2085
RCP2.6	2.18	2.43	1.94	1.92
RCP4.5		1.92	2.01	1.75
RCP8.5		2.01	1.7	0.99
(c) Upper estimate.
Scenario	1997–1998	2035	2060	2085
RCP2.6	2.18	1.82	1.65	1.77
RCP4.5		2.38	1.72	1.58
RCP8.5		1.67	1.77	1.02

RCP: Representative Concentration Pathway.

Table 5.

Summary of false negatives (type II errors) in percentage (%).

(a) Lower estimate.
Scenario	1997–1998	2035	2060	2085
RCP2.6	0.69	0.69	0.87	0.69
RCP4.5		0.87	0.87	0.87
RCP8.5		0.87	1.22	1.74
(b) Medium estimate.
Scenario	1997–1998	2035	2060	2085
RCP2.6	0.69	0.87	0.87	0.87
RCP4.5			1.22	1.22
RCP8.5			1.39	1.91
(c) Upper estimate.
Scenario	1997–1998	2035	2060	2085
RCP2.6	0.69	0.87	1.04	0.87
RCP4.5		0.87	1.39	1.39
RCP8.5		1.22	1.74	3.13

Figure 11.

False-positives (type I errors) and false-negatives (type II errors) indications for each climate change scenario.

Despite the relatively low amount of test data corresponding to the 1997–1998 monitoring campaign, the type I and II errors obtained for these years are in good agreement with other studies of the authors.^54,57 Regarding the future evolution of the errors, the results show a general trend to decrease the type I errors and increase the type II errors toward the worst-case scenario (RCP8.5) by the end of the century in relation to the reference period (1997–1998).

For instance, in the case of RCP8.5, the type II error may increase from 0.69% to 3.13% for the upper estimate.

For RCP4.5, the most plausive climate change scenario, a decrease in the number of type I errors is expected from 2.18% to 1.58% for the upper estimate; regarding type II errors, they tend overall to increase until 2060 and stabilize afterward.

For the RCP2.6, the low-emission and unlikely scenario, significant changes in the percentage of errors are not likely; for instance, for the medium estimate, by the end of the century, the type I errors may decrease from 2.18% to 1.92% and the type II errors may increase from 0.69% to 0.87%.

To understand the underlying causes and thus the generalization value of this behavior, a detailed discussion of the correct and erroneous classifications in the natural frequency space is given in the next section for the case of the monitoring data and for the three key climate change scenarios.

Analysis in the frequency space

To better understand the influence of climate change on both types of errors (type I and type II), the classification performance of the monitoring observations and projected observations are analyzed directly in the frequency space.

Figures 12, 13, 14, and 15 show the classification performance of all observations in the frequency space consisting of the first two frequencies, for the monitoring data (1997-1998) and the three key climate change estimates reflecting the lower (RCP2.6, 2035), medium (RCP4.5, 2060), and upper (RCP8.5, 2085) estimate impacts, respectively. It is recalled that the first two frequencies were used to train the GMM-based machine learning algorithms.

Figure 12.

Frequency space with classification performance of observations (negative = undamaged condition and positive = damaged condition) during the reference period (1997–1998).

Figure 13.

Frequency space with classification performance of observations (negative = undamaged condition and positive = damaged condition) in the near future (2035) and assuming the lower estimate for the RCP2.6.

Figure 14.

Frequency space with classification performance of observations (negative = undamaged condition and positive = damaged condition) in the mid-century period (2060) and assuming the medium estimate for the RCP4.5.

Figure 15.

Frequency space with classification performance of observations (negative = undamaged condition and positive = damaged condition) in the end-century period (2085) and assuming the upper estimate for the RCP8.5 (worst-case scenario).

The training observations (monitoring data from the undamaged condition) of the GMM-based machine learning algorithm are plotted in light blue. They form three Gaussian clusters centered in the points marked with red dots (cluster seeds), corresponding to observations obtained in very low (<−2°C), low (−2 to 2°C), and warm temperature ranges (>2°C). The test observations corresponding to the undamaged condition are plotted with squares and those corresponding to the damaged condition are plotted with stars. Both squares and stars are green, if the observation is correctly classified or red, if the observation is ill classified.

The false-positive classifications (type I errors) occur in three regions. They are associated with very low temperatures, leading to higher natural frequencies (red squares in the upper-right part of the plots), with very high temperatures (red squares in the center-left part of the plots), and to the region where the Gaussian clusters of the warm and low temperatures overlap. False-negative classifications (type II errors) occur exclusively in the region where data corresponding to the damaged and undamaged conditions overlap.

The main effect observed as temperature increases with climate change is the trend of the observations to group in the two clusters corresponding to warm and low temperatures, as the cluster of very low temperatures becomes less populated. This causes the false-positive readings (type I errors) that correspond to very low temperatures to gradually decrease between the plots in Figures 13, 14, and 15. Conversely, the false positives that correspond to very high temperatures (red squares in the center-left part of the plots) increase between the same plots, but to a much lesser extent, as the transition between warm and warmer temperatures cause a lesser structural effect than the transition between very low and low temperatures. This eventually leads to a gradual decrease in the number of false negatives from 2.84% for the lower climate change estimate, to 2.01% for the medium estimate, to 1.02% for the upper estimate impacts.

Regarding the false negatives (type II errors), it must be recalled that the controlled damage campaign was conducted during the summer, so the plots can only reflect the impact of the warm temperatures (recorded in 1998) becoming warmer, which cause, as seen above, a limited impact in the natural frequencies.

A particular feature of the GPR model used to obtain frequency estimates for given temperature data (see Figure 6) is that while the first natural frequency tends to decrease as one moves from warm to warmer temperatures, the second natural frequency actually increases for temperatures above 20°C. This feature causes increased overlapping between data corresponding to the undamaged structure under warm temperatures (under 20°C) and data corresponding to the slightly damaged structure under warmer temperatures, which leads to a gradual increase in the number of type II errors from 0.69% for the lower estimate, to 1.22% for the medium estimate, to 3.13% for the upper estimate impacts.

These conclusions obviously depend on the bridge, the climate it is inserted into, and the ambient temperature when damage occurs. However, it is clear from the results that climate change has the potential to alter the long-term damage detection performance of algorithms trained with historically past data.

Conclusions

The results based on an outlier detection strategy show that climate change may have an impact on SHM systems, by affecting the damage detection process in complex structures, like bridges, especially when there is an increment of global temperature, which causes changes in the dynamics of structures. Assuming a GMM-based machine learning algorithm trained with historical or past structural monitoring data (1997–1998), the overall results show that type I errors (false positives) may decrease, as the influence of climate change tends to group the observations from the undamaged condition into a unique central cluster, as a result of the increase of average temperature and the reduction of the number of observations in the freezing zone. On the other hand, the type II errors (false negatives) may increase with the severity of the climate change scenario, as the average temperature increases toward the end of the century and the observations associated with the lower level of damage tend to be misclassified due the proximity of the boundary region between the undamaged and damaged clusters.

These results suggest that machine learning algorithms may not be robust enough to deal with effects of climate change and that an SHM system must consider potential changes in the ambient temperature caused by climate change, especially for climate change scenarios RCP4.5 and RCP8.5, in order to minimize false-negative indications of damage, when life safety issues drive the SHM systems. Therefore, the results of this study suggest that SHM systems, and in particular the machine learning algorithms, may be adapted to climate change. Such understanding will assist bridge owners to plan efficient adaptation strategies in SHM systems in order to exploit the insight provided by the historical data while avoiding misclassifications. One such adaptation measure could be the periodic adaptation of the historical data to the new average temperature using, for instance, transfer learning.

This study has permitted one to conclude that climate change may be seen as another source of operational and environmental variability when using machine learning algorithms for long-term damage detection.

Finally, this paper stands as a novel study in terms of the impact of climate change on long-term damage detection in the context of bridge SHM. However, future publications should cover some limitations of this work, like:

1) the reference data set covers one-year measurements (1997–1998), which is the minimum period to characterize the seasonal variability caused by ambient temperature. Besides, it considers one-month data from the damaged condition, which is unique among the scientific community. However, it is not enough to cover the full range of temperature variations. It could be appropriate to run similar analysis on undamaged reference data sets from roughly three decades, which is the monitoring time needed to acknowledge that the mean value obtained from a data set represents the predominant value of the location, and a reference against which observations or climate projections are compared⁵⁸;

2) even though the GMM-based algorithm has demonstrated a superior behavior in terms of damage detection on the data sets from the Z-24 Bridge, other machine learning algorithms could be tested to perform sensitivity analysis in the context of climate change. Additionally, alternative approaches should be taken to develop algorithms capable to learn periodically or to take into account the average temperature in the damage detection process;

3) the natural frequencies were used as damage-sensitive features in this study. There is no guarantee that the results would be similar with other types of features with different relationships with the temperature;

4) even though the study was carried out on data sets from a concrete bridge, the results depend on the influence of temperature on the natural frequencies, rather than on the material used. Therefore, the results are valid for steel and composite steel-concrete bridges, for instance, as long as the temperature influences the natural frequencies.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The work was funded by the Portuguese Foundation for Science and Technology (FCT) through national funds (PIDDAC) under the R&D Unit “Civil Engineering Research and Innovation for Sustainability (CERIS),” reference UIDB/04625/2020.

ORCID iD

Eloi Figueiredo

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