1. Introduction
The rapid uptake of high-speed rail (HSR) has been largely due to its superior economic, social, and environmental benefits in comparison with other transport modes [
1], and China’s HSR has witnessed significant development over the past 15 years. For the HSR ballastless track system, one type of low maintenance track [
2,
3] which was extensively used for HSR lines in China (e.g., Beijing-Shanghai HSR, Lanzhou-Xinjiang HSR, Beijing-Shenyang HSR, etc.), track slab infrastructure assets are crucial elements, and their sustained efficient and safe operation is essential for ensuring the running safety of high-speed trains. Structural health monitoring methods that can detect performance changes and precursors to failure are a critical tool for guaranteeing the efficient and safe operation. Generally, the main interests in the monitoring of the track slab are the whole-life-cycle deterioration [
4,
5,
6] and the deformation under external excitation (e.g., natural disaster, dynamic train loading, etc.) or environmental effects during the operation of HSR lines [
7,
8,
9]. This study focuses on the latter issue—deformation monitoring.
Maintaining the alignment of a rail track is essential for the smooth and safe passage of the rail vehicles, and ideal rail track guides the vehicles along a smooth path, following the design curvature. The deformation of track slab will inevitably have an influence on the track alignment and result in track irregularity since there is a rigid connection between the rail track and track slab, which can undermine the ride comfort and running safety of the vehicles. HSR track irregularity can be measured several times every month by specially designed rail track inspection vehicles and the versine (or mid-chord offset), where the lateral offset of the rail from the center of a string that is stretched between two ends touching the rail side is generally used to describe the track alignment or irregularity [
10]. For a straight track, the versine should be zero, and the versine is a constant value for constant curves. However, these methods cannot provide real-time and long-term monitoring.
The monitored segment of the ballastless track slab is located inside a large-section deep-buried railway tunnel of a high-speed railway (
Figure 1). Cracks appear in the railway tunnel walls, and spalling of the concrete happens, especially in the arch vault and foot of the tunnel, due to the geological disaster of landslide induced by earthquakes. According to the measurement of CP III track control network, both lateral and vertical deformation occurs in the railway tunnel. The range of lateral displacement is +22 mm ~ +104 mm. The deformation of the whole tunnel would directly lead to the displacement of the track slab, which results in large lateral track irregularity. To restore the railway operation, the railway management departments immediately carried out the reconstruction of the ballastless track bed in the section with large deformation. It is necessary to deploy a real-time online deformation monitoring system for rail section in concern, so as to achieve real-time warning, when the track irregularity reaches a certain level in order to ensure the safe operation of high-speed railway.
Although there have been many studies on deformation monitoring for civil infrastructure while using various techniques and devices, e.g., tape extensometers [
11,
12], total stations [
13,
14], photogrammetry [
15,
16], and three-dimensional (3D) laser scanners [
17,
18], these methods have their own disadvantages for HSR track slab deformation monitoring: the tape extensometer can only be used to measure the distance changes relative to a fixed point and must be manipulated manually; total station is vulnerable to environmental interferences; and, the laser scanning technique has low accuracy with an error level of around ±5 mm. Therefore, this study will firstly develop a novel online SHM system that is enabled by fiber Bragg grating (FBG) technology (immunity to electromagnetic interference in HSR) for continuous and long-term monitoring of track slab deformation. The devised FBG bending gauge sensor does not require an external power supply, and its sensing principle is converting a wavelength shift of FBG into a physical quantity to be measured, and thus will not be affected by electromagnetic interference. The traditional piezoelectric sensor converts the electrical signal into a physical quantity of interest, so it is subject to electromagnetic interference, and the phenomenon of frequency multiplication occurs in the frequency domain of the original measurement data. In addition, the technical advantages of the developed FBG sensor include the following: (i) high measurement accuracy: the strain accuracy can reach 1 με; (ii) suitable for harsh environment in HSR; (iii) suitable for remote sensing; (iv) small size (diameter of FBD is about 0.15 mm); and, (v) cheap price.
In terms of SHM of railways specifically, the data noise and uncertainty are mainly from (i) the dynamic train loading and the ground-borne vibration arising from the wheel/rail interface, train weight, change in supporting stiffness (e.g., regularly spaced sleepers), and irregularities in the wheel/rail geometry [
19,
20]. Besides, the vibration will be elevated if the running speed of the train is comparable with the natural Rayleigh wave speed in the supporting soil [
21,
22]; (ii) the HSR contact network of electric locomotives, with a voltage of 27,000, provides power for the train, which will inevitably generate electromagnetic interference to the piezoelectric sensors. The devised SHM in this study, which is based on FBG, in contrast, is immune to the electromagnetic interference. It will greatly reduce the data uncertainty and ensure the reliability of measurement data; (iii) the fiber Bragg grating wavelength of the FBG-based sensor is sensitive to the temperature change. Large temperature variation can create large uncertainty in the measurement data. The monitoring system is designed for years of continuous monitoring. The maximum temperature difference in summer can be 20 degrees, and the maximum temperature difference throughout the whole year can reach up to 50 degrees. Although the developed FBG sensor considers the temperature compensation, such a large temperature difference can still cause daily and seasonal fluctuation in measured deformations; and, (iv) other factors, such as demodulation error of demodulator and the influence of periodic inspection activities on the high-speed railway (maintenance personnel may kick on the sensor, which will cause data fluctuation). The aforementioned four types of uncertainties will result in heteroscedastic noise in HSR SHM data at different time periods. Therefore, developing a rational data modelling method to deal with such non-stationary SHM data with input-dependent noise is highly desired.
Gaussian process (GP) modelling [
23,
24,
25] is a powerful statistical modelling framework that puts forward a structure for the covariance matrix of input variables to calculate a predictive distribution for output variable. The GP model is capable of capturing complex nonlinear relationships between input and output by employing Bayesian framework for covariance hyper-parameters training, and it can offer mean prediction for testing points with associated uncertainty levels (e.g., 95% confidence interval). A limitation of the standard (homoscedastic) GPs is the assumption of constant noise power throughout the input space, i.e., homoscedastic noise [
26]. To release this assumption, Heteroscedastic Gaussian process (HGP), which was first proposed by Goldberg [
27], on the other hand assumes that observation noise can vary and it aims to model the mean and variance distributions. In the framework of HGP, two-GP models have been proposed: one estimates the mean while the other captures the log-noise power. Making inferences for HGP is not as easy as standard GPs, because the predictive posterior distribution also accounts for the noise rates as the independent latent variables, which makes it intractable to analytically deal with the integral for the predictive posterior distribution. Therefore, a number of numerical methods have emerged, including Markov Chain Monte Carlo (MCMC), most likely HGP, sparse Gaussian process (referred to as GPz), and
maximum a posterior (MAP) HGP. Goldberg et al. [
27] proposed MCMC sampling to approximate the posterior distribution over the aforementioned two GPs. The solution converges to the exact posterior when the number of samples tends to infinity, but the computational cost is very high and it makes the model expensive to compute on large data sets. Kersting et al. [
28] employed a most likely noise approach to approximate the posterior noise variance. The likelihood is maximized while using an iterative procedure that is similar to Expectation Maximization updates. However, the algorithm is not guaranteed to converge and it may instead oscillate. In most recent related works, Almosallam et al. [
29] proposed a sparse Gaussian process (GPz) for heteroscedastc uncertainty estimation. In this framework, a Bayesian machine learning approach is employed to jointly optimize the model with respect to both the predictive mean and variance. The variance is an input-dependent function and it consists of two terms that capture different sources of uncertainty. The first term represents the intrinsic uncertainty about the mean function because of data density, while the second term is the uncertainty that is due to the intrinsic noise or the lack of precision/features in the training data. In order to achieving accurate predictions, GPz incorporates a sparsity-inducing prior to minimize the number of basis functions, to produce a sparse model representation. The MAP approach has been performed to maximize a penalized likelihood for the HGP approximation [
30]. Le et al. [
31] presented a non-parametric approach to estimate heteroscedastic noise by performing
maximum a posteriori for regression with exponential families. However, MAPHGP provides a point estimate for noise, while the Variational Heteroscedastic Gaussian Process (VHGP) that is explored in this study is a fully Bayesian approach that variationally integrates out the GP for noise term. The VHGP firstly proposed by Lazaro-Gredilla and Titsias [
32], is based on variational Bayes and the Gaussian approximation for accurate inference in HGP. The computational cost for HGP is greatly reduced and at the same time a high accuracy can be guaranteed. These features make VHGP particularly attractive for modelling and forecasting of SHM data corrupted with uncertainties, which can help to assess the amount and resolution of data required to reach desired uncertainty levels.
In this paper, a novel online SHM system is developed for monitoring the lateral deformation of track slabs by deploying an array of FBG bending gauges, which is able to continuously detect the deformation of track slabs and provide an automatic condition assessment of in-service railway. FBG bending gauges with temperature self-compensation are designed and fabricated, and the sensor array is verified in laboratory experiments before the implementation of the devised system in an in-service HSR line. This study explores a VHGP approach with variational Bayesian and Gaussian approximation for data modelling to handle the different sources of uncertainty in SHM data of track slabs, which quantifies the uncertainty level of the monitoring data and further makes forecasting on SHM data. Previous studies have proved that VHGP is superior to MAPHGP and MLHGP in various applications [
32,
33]. MCMC HGP typically needs several thousands of sample-generating runs before achieving stable results; thus, the computational cost is extremely high. HGP and MAP HGP both most likely provide a point estimate for noise in the iterative solution, while the Variational Heteroscedastic Gaussian Process (VHGP) gives rise to a full distribution result instead of a single point estimate. In this regard, the GPz method is quite similar to VHGP method, because it also generates full distribution results through a Bayesian machine learning approach. This paper will compare VHGP with the latest HGP model, i.e., GPz, to illustrate its advantages in dealing with SHM data for modelling and forecasting of the deformation of HSR track slabs. Laboratory and field tests will both be made to validate the proposed method.
The contributions of this study include:
- (i)
A novel online SHM system enabled by FBG bending gauges is developed for monitoring the lateral deformation of HSR track slabs, with the capacity of temperature self-compensation.
- (ii)
The study innovatively applies VHGP to deal with the SHM data for treatment of input-dependent noise, uncertainty qualification, and forecasting. The performance of two HGP models, namely VHGP and GPz, is compared for the first time.
- (iii)
A specific example of M&F for deformation of an in-service HSR track slab is provided. Uncertainty characteristics of SHM data for HSR specifically are discussed and the uncertainty levels of all in-situ sensors in the monitored segment are obtained, with a comprehensive analysis of uncertainty sources.
2. VHGP Framework
2.1. Standard GP Methodology
The GP is a supervised nonlinear regression algorithm lying within the class of Bayesian non-parametric models. Given a set of inputs
and a set of target outputs
, where
N represents the number of input or output. The underlying assumption is that
y is generated by a function of the input
x, plus additive noise
where the independent noise term
is assumed a zero-mean and
σ2-variance Gaussian prior.
y has a Gaussian prior distribution with a zero-mean function (without loss of generality) and a positive-definite covariance function
with the hyperparameters
.
= [
f1, …,
fN]
T is defined as the evaluation of
on the inputs
. Subsequently, a multivariate Gaussian prior has the form
, where
. Thus, the Gaussian likelihood is
, where
I is a unit matrix. While integrating out the latent function values, the marginal likelihood can be obtained as
where
represent hyperparameters. A maximum for the logarithm of marginal likelihood in Equation (2), also known as log-evidence, is sought to choose hyperparameters
and
. Finally, the predictive posterior distribution for a new sample
is distributed normal with the following mean and variance
(
represents the target output for a new sample
;
is the mean value of
, and
is the variance of
):
where
,
and
.
2.2. Heteroscedastic GP Model
To define the heteroscedastic GP (HGP) model, a Gaussian prior is placed on the noise term :, in which the variance r(x) of the noise can vary at each input x. is parametrized to ensure positivity and a GP prior is placed as .
Once the parametric forms of and are determined, the HGP model will be specified and it only depends on and the covariance hyperparameters: and . For VHGP, since is explicitly considered, the scale of the noise term can be controlled.
2.3. MV Bound for HGP and Optimization
VHGP is more flexible when compared with the standard GP. However, it is analytically intractable. Thus, variational approximation is proposed to deal with the computational difficulty. While it is impossible to calculate the marginal log-likelihood analytically for HGP, it can be lower bounded by variational approximation.
The definition of standard variational approximation follows:
Since Kullback–Leibler (KL) divergence [
34,
35,
36] is non-negative, it is obvious that the evidence
is lower bounded by
F, i.e.,
for any variational distributions
and
. The objective is to maximize the bound
F with respect to
and
, and it is equivalent to minimize the KL divergence with the fact that the evidence has no dependence on
and
, which is to find the best approximation to the posterior distribution, according to the definition of the KL divergence.
F depends on both
and
. For simplification, the Marginalized Variational (MV) bound will be obtained by marginalizing out
to remove its dependence. The optimal distribution
can be obtained through the variational Bayesian theory to maximize
where
is a constant, i.e.,
. The MV bound can be obtained by inserting
back to
:
which removes the dependence on
.
If
is restricted to be a multivariate normal distribution, i.e.,
, the MV bound for HGP model can be rewritten as
where
Kf and
Kg represent the covariance matrices of
f and
g. After the simplification, the MV bound for VHGP can be obtained, as follows:
where
R is a diagonal matrix with elements
.
The MV bound depends on
N+N(N+1)/2 free variational parameters (defining
and
). According to Gaussian approximation theory, the stationary equations
and
must be satisfied at any local or global maximum, and the two equations are obtained as follows after manipulation:
for some semi-positive definite matrix
. Thus, both
and
depend on
, and only
N diagonal elements are needed to define
. Therefore, after reparametrization, the number of free variational parameters for optimization is reduced to be
N. Eventually, the MV bound
needs to be maximized with respect to the
N variational parameters in
. This is advantageous, both from a computational point of view and an optimization point of view. At the same time,
F can be maximized with respect to the hyperparameters
through Type-II Maximum Likelihood. The whole optimization is non-linear and gradient-based procedures, and the derivatives with respect to
can be analytically computed.
2.4. Predictive Posterior Distribution for VHGP
Given training data, represents the predictive distribution for a new test output . is viewed as a good approximation to the posterior distribution , the mean and variance of p(y*|x*,D) can be analytically calculated.
can be calculated according to Equation (6):
where
. Through the variational approximation, the posterior distribution for
is
where
and
. Similarly, the posterior distribution of
is obtained by:
where
and
. Consequently, the distribution for
is estimated by:
Although the above expression is not analytically tractable, the mean and variance of the posterior distribution can be analytically calculated: and .
5. Conclusions
A novel online SHM system using FBG sensing technology has been developed, tested, and implemented for HSR track slab deformation monitoring. FBG bending gauges with temperature self-compensation capacity have been devised specifically for the deformation monitoring task and they are validated by laboratory test before the implementation of this system in an in-service HSR line. The SHM system can greatly eliminate the data uncertainty caused by electromagnetic interference in HSR applications since FBG is immune to electromagnetic interference.
To properly deal with the different sources of uncertainty, the study innovatively applies VHGP for SHM data modelling, which enjoys full Bayesian, non-parametric, probabilistic modelling, principled learning of free variational parameters with dimension reduction, and heteroscedasticity. The uncertainty level of monitoring data is quantified, and forecasting based on history data is carried out. The experimental results on laboratory and field test data sets in M&F process show significant improvement in terms of regression with respect to the state-of-the-art GPz algorithm. VHGP can better capture the heteroscedastic variances of the noise, and the estimated confidence interval varies accordingly. For in-situ sensors, the accuracy improvement level varies, with an average increase of 66.1% on MSE. In regard to forecasting, the improvement is not as high as that of regression. VHGP is slightly superior to GPz in general, with a forecasting accuracy that increased by 21.24% on average for all sensors. However, the VHGP framework is more accurate for maximum noise position prediction, the forecasting mean of VHGP is more consistent with the real measurement values, and the method can yield more robust forecasting at the peaks for the field monitoring data. Finally, for the specific example of M&F for deformation of in-service HSR track, the uncertainty levels of all sensors are estimated with associated deformation profiles for the monitoring segment on the basis of in-situ measurement data, and three typical types of uncertainty, i.e., high uncertainty induced low data density, periodic uncertainty, and relatively stable uncertainty, are summarized during the M&F process of HSR track slab deformation. The purpose of uncertainty analysis is to quantify the uncertainty level of each sensor, to figure out the sensors with the maximum data volatility and the sensors with the most stable performance. During the modelling and forecastingfor the deformation of in-service HSR track slabs, the data of sensors with good stability shall be fully considered to obtain more accurate measurement results.
Future work will expand the monitoring system to deploy more FBG bending gauges along the longitudinal direction to cover the full length of the track slabs in the tunnel. The accuracy of multi-step ahead forecasting by VHGP method is not as high as that of one-step ahead forecasting. Further research will be pursued regarding how the sparse theory can be introduced into VHGP to improve its generalization capacity with the intention of achieving more accurate multi-step forecasting. The possible influence of covariance matrix selection on the result will also be studied. In addition, the HGP state-space spatiotemporal model will be investigated to deal with the multiple output problem.