5.1. Dashengguan Bridge
The Dashengguan Bridge is a long-span high-speed railway steel bridge crossing over Yangtze River in Nanjing, China.
Figure 2a show an actual image of this bridge. The construction of this structure began in 2006 and ended in 2010 to handle a speed of 300 km/h. The bridge consists of a large-span continuous steel arch truss with a total length of 1615 m. This research considers the six main parts of the bridge with the total length of 1272 m (i.e., 108, 192, 336, 336, 192, and 108 m), as depicted in
Figure 2b. These parts are separated by seven piers (Piers 4–10) mounted on deep piles. The two main spans over the major navigation channels of the Yangtze River are steel arch trusses with the lengths of 336 m and a maximum height of 74 m. The non-curved parts of the bridge have a constant height of 16 m [
45]. The arches are comprised of three truss planes above the deck. The main truss has a welded, monolithic joint. The members and gusset plates were welded together in the fabrication yard and then transported to the site and spliced outside the joint with high-strength bolts.
To analyze the longitudinal displacement of the bridge and investigate the effect of temperature variability, this paper utilizes the displacement samples extracted from 29 SAR images of Sentinel-1A acquired between 25 April 2015 and 5 August 2016. These displacement points were obtained by Huang et al. [
45] using the persistent scatterer interferometry (PSI) technique and determining the light of sight (LOS) deformation time series at the six piers (i.e., Piers 4–6 and 8–10). Moreover, some contact-based temperature sensors were installed in the bridge to record temperature data during the monitoring time.
Figure 3a shows the 29 displacement samples (i.e., in the unit of mm) at the six piers, and
Figure 3b indicates their corresponding air temperature (°C).
The process of input–output data normalization begins by training the GPR and SVR models, for which the recorded temperature and limited displacement sets are used as the input (predictor) and output (response) data, respectively. Bayesian hyperparameter optimization is utilized to optimize some hyperparameters of the GPR and SVR models, as listed in
Table 1 and
Table 2, respectively. Note that the ratios of 80% and 20% are considered for the training and testing sets. Once the GPR and SVR models have been trained, the residual function
E =
is employed to extract the normalized displacement data, where
denotes the predicted displacement data from the supervised regression models.
Figure 4 and
Figure 5 show the results of the thermal effect elimination by comparing the original and normalized displacement data at the six piers. As can be seen, the normalized displacement samples differ from the original ones, and most of them vary in the vicinity of the baseline with the displacement rate equal to zero [
20]. This means that the normalized displacements have been separated from the thermal effects.
Subsequently, the unsupervised reconstruction-based data normalization techniques based on the PCA and DAE are used to eliminate the temperature influences and other unmeasured environmental and/or operational conditions. The initial requirements for these techniques are their hyperparameters. By collecting the displacement samples of the six piers into a matrix with a size of 29 × 6, the optimal number of PCs with the threshold equal to 0.9 is identical to 1. On the other hand, to determine the hyperparameters of the DAE (i.e., the neuron numbers of the mapping/de-mapping layers
lm and encoding layer
lc), the grid search algorithm initially utilizes two different sets of sample neurons, which are equal to 1:20 for the mapping/de-mapping layers and 1:10 for the encoding layer. Note that as the proposed DAE has a symmetric configuration, the neuron numbers of the mapping and de-mapping layers are identical. In this regard, one needs to train a DAE by using each of the sample neurons, compute the
RMSE value as expressed in Equation (31), and store the computed value in a matrix.
Figure 6 illustrates the inverse of the stored
RMSE values of the sample neurons, which have been stored in a matrix with 20 rows and 10 columns. Notice that the inverted
RMSE is applied to better indicate the result. On this basis, the optimal numbers of
lm and
lc coincide with the sample neurons with the largest value of the inverted
RMSE quantities. With this description, it can be found that the optimal neuron sizes correspond to
lm = 6 and
lc = 7. Hence, a DAE with 6, 7, and 6 neurons for the mapping, encoding, and de-mapping layers is trained to reconstruct the original (multivariate) displacement data. Once the reconstructed (predicted) displacement data have been determined using the PCA and DAE models, the residual function
E =
–
is used to extract the normalized displacement data.
Figure 7 and
Figure 8 show the results of output-only data normalization based on the PCA and DAE methods, respectively. Similar to the supervised regression models, one can perceive that the normalized displacement data differ from the original data. Approximately, all normalized points coincide with the baseline, which demonstrates the appropriate separation of the displacement data from the thermal effects. This conclusion not only verifies the effectiveness of the unsupervised data normalization methods in removing the thermal effects but also indicates their great ability to implement the process of data normalization without measuring the temperature data. Finally, to show the quality of the normalized responses and also the performance of the supervised and unsupervised data normalization models,
Table 3 lists the R-squared values at the six piers of the Dashengguan Bridge. As can be observed, all quantities are close to one, which means that not only the data normalization models could predict the displacement responses properly but also the normalized responses are extracted correctly.
5.2. Rainbow Bridge
The Rainbow Bridge is a long-span structure in Tianjin, China that was designed and constructed as a concrete-filled steel tubular arch bridge. This structure was built in 1996 and then opened to traffic at the end of 1998.
Figure 9a shows an actual image of the Rainbow Bridge. The total length of this bridge corresponds to 1215.69 m, while the main bridge structure contains three spans with the lengths of 164, 168, and 164.7 m as shown in
Figure 9b. The bridge structure includes a rigid arch system with a simple supported down-bearing flexible tie rod. The upper and lower chords, along with the arch skewback, were filled with micro-expansive concrete. There are eight K-shaped transverse bracings for each span. Each arch contains 18 pairs of suspenders with a spacing of 8.3 m, and each suspender is composed of 91 galvanized prestressed steel wires. The deck system consists of a prestressed concrete middle cross girder, reinforced concrete T-shaped stiffened longitudinal girder, and T-shaped longitudinal girder [
62].
Due to the long-term passage of overweight vehicles exceeding the design load, serious damage patterns influenced the performance and serviceability of the Rainbow Bridge. In this regard, some cracks were detected at a longitudinal concrete beam of the bridge, which caused varying levels of damage in two adjacent longitudinal concrete beams. To avoid any catastrophic events, such as failure and collapse, all longitudinal concrete beams were replaced to increase the bridge structural performance. To evaluate the integrity of the bridge, long-term SAR-based SHM was implemented by Qin et al. [
46] using 53 descending SAR images from Sentinel-1A between 2015 and 2017. Using a multi-temporal differential synthetic aperture radar interferometry (DInSAR) technique,
Figure 10a,b illustrates the displacement samples in the main four piers (i.e., Piers 1–4) and middle spans (i.e., Span 1–3), respectively. Moreover,
Figure 10c displays the temperature records during the monitoring time.
Using the temperature and displacement data, both of them are divided into the training and testing sets based on the ratios of 80% and 20%, respectively, to prepare the predictor (input) and response (output) data for the GPR and SVR models. Bayesian hyperparameter optimization is employed to optimize the main hyperparameters, as listed in
Table 4 and
Table 5. Based on these hyperparameters, the supervised regression models are trained to predict the displacement data and then extract the normalized data.
Figure 11 compares the original and normalized displacement data points of the four piers and three spans regarding the process of data normalization via the GPR. Apart from the results associated with the second and third piers, as shown in
Figure 11b,c, the other plots demonstrate that this supervised regression technique could not present reliable normalized displacement data. As can be observed, the normalized samples of the first and fourth piers, as well as all three spans, conform with the original data. This means that the normalized displacement data of these locations are still influenced by the temperature variability and the GPR techniques could not be successful in data normalization.
The results of data normalization through the SVR technique are displayed in
Figure 12. As can be seen, this technique underperforms compared to the GPR. In all piers and spans, the normalized displacement data obtained from the SVR have the same forms as the original data. Accordingly, one can conclude that both supervised regression models failed in eliminating the temperature variability from the displacement data. Thus, the results verify that the use of temperature data as the input cannot be sufficiently effective for modeling influential supervised regression models and establishing the relationship between the displacement and temperature data. In other words, an unmeasured environmental and/or operational condition affected the displacement data of the Rainbow Bridge. Since this unmeasured condition was not included in training the GPR and SVR models, both of them could not yield reasonable outputs.
In order to investigate the environmental and/or operational effects on the limited displacement data via unsupervised data normalization techniques, the displacement samples are collected to make four multivariate datasets (matrices) regarding the piers and spans of the Rainbow Bridge. Using these matrices, the hyperparameter optimization of the PCA is carried out to choose the number of principal components with the threshold value equal to 0.9. According to the indicator
IPC in Equation (26), only one PC is required to model the PCA. Hence, once the PCA model for each multivariate dataset has been established, its residual function is considered to extract the normalized displacement data. In this regard,
Figure 13 shows the original and normalized displacement samples regarding the unsupervised data normalization via the PCA technique. From
Figure 13b, one can understand that the PCA model of the second pier could not properly extract the normalized data as it follows the same form of the original data. However, the other PCA models related to the first, third, and fourth piers, as shown in
Figure 13a,c,d, have been developed accurately, in which case their normalized points approximately coincide with the baselines. Regarding the three spans, as
Figure 13e,f reveals, all PCA models could properly remove the thermal and other environmental/operational effects from the displacement data.
To perform the unsupervised data normalization via the DAE, it is necessary to determine the neuron sizes of the mapping, encoding, and de-mapping layers. Utilizing the same sample neurons of the preceding structure, which are equal to 1:20 for the mapping/de-mapping layers and 1:10 for the encoding layer,
Figure 14 illustrates the inverted
RMSE values regarding the four piers and three spans of the Rainbow Bridge. Among the aforementioned sample neurons used in the grid search algorithm, the best DAE for modeling the displacement data of the piers requires 6 and 4 neurons for the mapping (de-mapping) and encoding layers, respectively, as these neurons provide the maximum inverted
RMSE values, which are equivalent to the minimum
RMSE values.
Furthermore, the optimal numbers of
lm and
lc regarding the DAE of the displacement samples of the three spans are equal to 9 and 8, respectively. It should be emphasized that we applied the inverted
RMSE to better indicate the result of hyperparameter optimization. The results of the DAE-based data normalization at the piers and spans of the Rainbow Bridge are shown in
Figure 15 and
Figure 16, respectively. To better show the results, the normalized displacement samples are separately plotted in
Figure 15e–h and
Figure 16d–f. In contrast to all the previous techniques, the DAE yielded the best performance in removing the thermal effects. As can be observed, all normalized displacement samples coincide with the baselines, which means that the DAE-based data normalization method could thoroughly eliminate the thermal or other environmental/operational effects.
To demonstrate the quality of the normalized responses and also assess the performance of the supervised and unsupervised data normalization models,
Table 6 presents their R-squared values at the piers and spans of the Rainbow Bridge. As can be discerned, the DAE-aided data normalization method provides the most reliable normalized responses because all R-squared values roughly correspond to one. In relation to the normalized responses of the four piers, the worst performance belongs to the SVR model since all R-squared quantities are far away from one. The GPR model also failed in extracting reliable normalized responses for the first and fourth piers. The PCA model is approximately successful; however, the normalized response of the second pier is not reasonable. Regarding the spans of the Rainbow Bridge, one can observe that the R-squared values of the supervised predictive models yielded moderate R-squared values varying within 0.45–0.55, which implies that the normalized responses could not be extracted appropriately. However, both the PCA and DAE methods could attain the best performances for extracting the normalized responses of the bridge spans.