A Method to Detect Concealed Damage in Concrete Tunnels Using a Radar Feature Vector and Bayesian Analysis of Ground-Penetrating Radar Data

Abstract

Many machine learning (ML)-based detection methods for interpreting ground-penetrating radar (GPR) data of concrete tunnels require extensive labeled damage-state data for model training, limiting their practical use in concealed damage detection of in-service tunnels. This study presents a probabilistic, data-driven method for GPR-based damage detection, which exempts the requirement in the training process of supervised ML models. The approach involves extracting a radar feature vector (RFV), building a Bayesian baseline model with healthy data, and quantifying damage severity with the Bayes factor. The RFV is a complex vector obtained by radargram data fusion. Bayesian regression is applied to build a model for the relationship between real and imaginary parts of the RFV. The Bayes factor is employed for defect identification and severity assessment, by quantifying the difference between the RFV built with new observations and the baseline RFV predicted by the baseline model with new input. The probability of damage is calculated to reflect the influence of uncertainties on the detection result. The effectiveness of the proposed method is validated through simulated data with random noise and physical model tests. This method facilitates GPR-based hidden damage detection of in-service tunnels when lacking labeled damage-state data in the model training process.

1. Introduction

The long-term operation of tunnels is influenced by various environmental factors, operational loads, and material degradation, leading to the gradual emergence of hidden defects [1,2]. This may lead to the development of subsurface issues, such as void formation, internal cracking, reinforcing steel corrosion, and delamination, which are often not easily detectable through visual inspection. Non-destructive inspection techniques, including ultrasonics, GPR, and radiography, have demonstrated significant potential in detecting structural damage [3]. Among these, Rayleigh and Lamb waves—two types of ultrasonic waves—are commonly used in concrete structures to identify surface and near-surface defects [4]. Rayleigh waves are particularly effective for detecting cracks and other surface-level damages, as their energy is concentrated at the surface [5,6,7]. In contrast, Lamb waves are employed to inspect thin concrete structures, such as slabs or plates, where they can identify both surface and internal defects at shallow depths [8,9]. GPR, on the other hand, can penetrate thicker layers of concrete or rock, facilitating deeper structural inspections and making it especially suitable for identifying defects in underground structures and geological formations. GPR has proven invaluable for uncovering hidden structural defects in civil structures such as tunnels and roads [10]. The non-invasive and non-destructive nature of this method further underscores its applicability in scenarios where traditional inspection methods fall short and are thus widely applied in the detection of voids/cavities in tunnels [11].

The application of GPR in detecting hidden defects has primarily focused on generating and analyzing grayscale images. Lu et al. [12] utilized simulated images to characterize concealed cracks in asphalt pavements. Then, they accomplished the recognition and localization of these cracks by GPR-based field testing. The data processing and interpretation in this study were almost manual. Similarly, Xu et al. [13] completed the detection of voids and cracks under hydraulic projects using GPR, using data processes that were complex and artificial. Khamzin et al. [14] used an air-launched GPR system to assess highway structures and defects, while the classification of defects based on GPR images was completed manually. Furthermore, Zhang et al. [15] pointed out that variations in material properties and environmental conditions affect the accuracy of image-based GPR interpretations, necessitating advanced processing techniques to mitigate these effects. Finally, Ma et al. [16] stressed the importance of combining multiple data sources and processing algorithms to improve the reliability of GPR assessments.

Many ML-based methods for interpreting GPR tunnel data have been developed. For instance, Jin and Duan [17] demonstrated the utility of GPR in identifying voids within tunnel linings, using image processing techniques to enhance the visibility of subsurface anomalies. This approach was further supported by Li et al. [18], who employed deep learning algorithms to interpret GPR images, effectively distinguishing between different types of structural defects. Additionally, Liu et al. [19] applied adaptive particle swarm support vector machines (SVMs) for the classification of GPR images, yielding high accuracy in the identification of defects in highway layers. There are more examples of ML-based methods for interpreting GPR tunnel data, for example, concealed damage identification of tunnels through a “You Only Look Once” (YOLO)-based approach [20], tunnel defect segmentation based on a convolutional neural network (CNN) [21], and identification of tunnel cavities using a generative adversarial network (GAN) for reducing the influence of rebar clutter [22].

However, studies on interpreting GPR data for ML-based damage detection generally require collecting different types of data for a specific object, with corresponding labels (e.g., “healthy” and “damaged”) [23]. These data are subsequently converted into grayscale images via signal processing and utilized to train algorithms. However, in real-world scenarios, the amount of available damage data with labels is sometimes limited, thereby posing challenges to the practical applicability of these methods [24]. Furthermore, many previous approaches to interpreting GPR data have been inadequate in assessing the severity of the damage effectively. Therefore, it is necessary to develop an ML method applicable to the situation where labeled damage-state data are not available in the model training process for GPR-based damage detection and assessment.

The extraction of structural safety-related information from sensed signals is crucial for damage detection [25]. An extraction method was successfully applied for monitoring rail damage using acoustic emission signals [26], but its application for identifying hidden defects using GPR data remains unexplored. To implement this approach in damage detection, various machine learning techniques can be utilized, including support vector machines [27], artificial neural networks [28,29,30], Gaussian process regression [31,32], and Bayesian modeling [33,34]. Bayesian inference is a robust tool for quantifying anomalies under uncertainty, and interest in applying Bayesian methods to damage detection has been steadily increasing [35]. Bayesian inference for structural damage assessment using modal data information was introduced by Vanik et al. [36,37] and Sohn and Law [38] in the early days. Additionally, Wang [39] developed a Bayesian machine learning approach to extract patterns from subsurface data, which provided a probabilistic framework for modeling spatial and statistical patterns inherent in GPR data. Similarly, Chen et al. [40] proposed a method combining random forest and extreme gradient boosting with Bayesian optimization for pavement density prediction using GPR data, significantly improving the prediction accuracy. Li et al. [41] employed Bayesian optimization alongside XGBoost for recognizing moisture damage in asphalt pavement from GPR time-frequency features, yielding high classification accuracy. Furthermore, Lähivaara et al. [42] addressed the challenges of model uncertainties in GPR data through a Bayesian approximation error approach, effectively estimating pipeline locations in the presence of such uncertainties. Liu et al. [43] extended this approach by utilizing Gaussian processes for recognizing typical geological anomalies, such as karst cavities and fault zones. Qin et al. [44] proposed a sliding window probabilistic inversion method using Bayesian inference and Markov Chain Monte Carlo (MCMC). This method was used to estimate the grouting layer thickness in shield tunnels, providing a method for assessing hidden defects in tunnels. Given the Bayesian framework’s strengths in handling uncertainty, integrating prior knowledge, and analyzing incomplete data, we intend to apply this method to detect and assess hidden defects of tunnels using GPR data.

The main contributions of this study are as follows: (1) The method realizes concealed damage detection and severity assessment by quantifying the relative change between the real and imaginary parts of RFV. The baseline model of the relationship between them is trained with healthy data, thus exempting the need for labeled damage-state data in the training process of many radargram-based supervised ML models. (2) The correlation between the Bayes factor and cavity size is explored to reveal how the extent of hidden damage within a tunnel structure affects the Bayes factor. (3) The proposed method is potentially applicable to the radargram-based assessment of both shield and other types of tunnels since the developing process of this method is not based on the assumption of a specific tunnel structure.

The remainder of this paper is structured as follows: Section 2 outlines the methodology, starting with the construction of a radar feature vector (RFV) by transforming radargrams into a column vector. It then details the training process of a Bayesian regression model to capture the relationship between the components of normal RFV. The Bayes factor is subsequently applied to assess the extent of hidden damage in tunnels by examining the difference between the new and baseline RFVs. Section 3 validates the proposed method by acquiring GPRMAX-simulated data and by testing on a physical tunnel model. Finally, Section 4 concludes the paper by summarizing the key findings and suggesting potential directions for future research.

2. Methodology

The proposed approach in this paper consists of three computation stages. Firstly, a one-dimensional RFV is generated by fusing a radar data matrix in the normal state of a tunnel with a transformation vector. Following this, the Bayesian baseline model is trained using this RFV to capture the relationship between its real and imaginary components. When new radar data are received, a new RFV is constructed by fusing the new data matrix with the same transformation vector. The next step involves conducting a quantitative analysis between the new RFV and the baseline RFV predicted by the Bayesian model with one component of the new RFV as its input. This analysis calculates both the degree and probability of concealed damage of a tunnel. A flowchart depicting this radargram-based method for identifying damage can be found in Figure 1. The process for constructing an RFV is detailed in Section 2.1, Section 2.2 describes the formulation of how to create the baseline model, and Section 2.3 describes the quantitative analysis of damage.

The establishment of the proposed method does not involve assumptions about the types of voids, and thus, it can potentially be applied to the voids occurring behind the primary lining, the separation between the secondary lining and primary lining, and the cavities present within the geological formations. The applicability of this method to the concealed defects similar to the separation between the secondary lining and primary lining and the cavity in the surrounding formation has been examined in Section 3.1 and Section 3.2, respectively.

2.1. Establishment of the Radar Feature Vector

This study introduces a radar feature vector (RFV) that incorporates frequency-domain analysis and matrix transformation, building upon the linear transformation technique which was developed for constructing a generic metric in the time domain [25]. Radargrams undergo discrete Fourier transform to obtain complex Fourier series in the frequency domain. These complex Fourier series are then used to construct an N × M matrix $Q_n$. Here, N represents the number of frequency bins of interest and M denotes the different spatial positions along a survey line. To convert this complex-valued matrix $Q_{n0}$ into a one-dimensional complex array, we design a transformation vector $P$ with dimensions M × 1.

$$Q_{n0}P=R_0$$

(1)

Here, $Q_{n0}$ is derived from the data in a state of health; $R_0=(a+ib)/\sqrt{a^2+b^2}$ signifies an undamaged state of the structure and is represented as a complex vector with dimensions N × 1. Furthermore, the transformation vector $P$ can be acquired by applying the pseudo-inverse operation to the matrix.

$$\overline{P}={\left(Q^{H}Q+\mathit{\lambda }I\right)}^{-1}{Q}^H{R}_0$$

(2)

Here, λ serves as a small positive value for fine-tuning regularization, $I$ represents a unit matrix, $\overline{P}$ is an approximate solution of the transformation vector $P$, and the superscript “H” stands for a complex conjugate transpose. The solution of Equation (1) is obtained by using the L₂-regularized least-squares method and the solution is $\overline{P}$.

$$R=Q_n\overline{P}$$

(3)

In the cases where there is an anomaly in the radar feature vector (RFV), it can be solely attributed to $Q_n$; the transformation matrix $\overline{P}$ used here remains the same and it originates from normal data. Moreover, the extent to which the two parts of RFV change relatively serves as an indicator for assessing the seriousness of structural abnormalities.

In addition, damage localization is achieved by dividing column vectors into a data matrix into several groups and combining the column vectors at adjacent circumferential positions for constructing each RFV. An anomaly that is detected within a specific tunnel segment indicates that potential damage occurs at the corresponding circumferential location. For depth estimation, the data associated with the circumferential location can be divided into row-based segments for building RFVs. After comparing the deviation of the RFV at each segment from its baseline RFV, the segment with the largest deviation is identified, potentially indicating the damage depth. This two-level segmentation approach facilitates the identification of both the circumferential location and depth of damage.

2.2. Construction of Bayesian Regression Model

Bayesian learning is employed to develop a regression model. It is a baseline model for the relationship between real and imaginary parts of an RFV built with normal GPR data. With this baseline model, potential damage is identified during the detection phase by making a comparison between the RFV built by new observations and the RFV predicted by the baseline model with new input. When the deviation between the RFVs exceeds the prescribed threshold, a damage alert is triggered. This model not only investigates the potential characteristics of RFV but also enables damage assessment by measuring relative changes between the real and imaginary components. The Bayesian regression modeling procedure is outlined as follows: The real component of RFV is represented by $R_{rl}$, while the imaginary component is denoted as $R_{im}$. A predictive model is established to estimate the imaginary part based on the real part, expressed as $R_{rl}\left(R_{im}\right)$. The method employs the Bayesian generalized linear regression model, as described in the study of Robert et al. [45].

$$R_{rl}\left(R_{im}\right)=f(R_{im})+ϵ={\displaystyle \sum _{k=1}^L{\omega }_k}{\varphi }_k(R_{im},R_{i{m}_k})+ϵ={\Phi }^T\omega +ϵ$$

(4)

Here, L represents the total number of elements in RFV; the feature vector consists of fundamental functions denoted by $\Phi$, while $\omega$ represents the weight vector; and the error term $ϵ$ is characterized by independent Gaussian noise, following a distribution $ϵ~\mathcal{N}(0,{\sigma }^2)$. Then, we have the following:

$$R_{rl}(R_{im})~\mathcal{N}({\Phi }^T\omega ,{\sigma }^2)$$

(5)

where $\mathcal{N}({\Phi }^T\omega ,{\sigma }^2)$ represents a Gaussian distribution with mean ${\Phi }^T\omega$ and variance σ², indicating the statistical relationship between the real and imaginary components of the RFV. In Equation (4), ${\omega }_k$ represents the model parameters (weights), and ${\varphi }_k(R_{im},R_{i{m}_k})$ is the Gaussian kernel function, defined as

$${\varphi }_k(R_{im},R_{i{m}_k})=\exp \left(-{\displaystyle \frac{{(R_{im}-R_{i{m}_k})}^2}{2{\sigma }_k^2}}\right)$$

(6)

The prior distribution of the weight vector $\omega$ is assumed to be Gaussian with zero mean and variance ${\tau }^2$ which can be expressed as

$$\omega ~\mathcal{N}\left(0,{\tau }^{2}I\right)$$

(7)

Considering the advantages of using conjugate prior distribution in Bayesian inference, we assume that the prior distribution for the variance ${\tau }^2$ of the Gaussian distribution mentioned above, which has a zero mean, follows an inverse Gamma distribution. By applying Bayes’ theorem, we can easily update $\omega$ with the training data. After obtaining the posterior distribution $\mathcal{N}\left({\mu }_p, {\tau }_p^2\right)$ for $\omega$, we can express the function $f\left( \right)$ as follows:

$$f_{Prd}(R_{im})={\Phi }^T\omega$$

(8)

The prediction process for $R_{im}\left(R_{rl}\right)$ is identical to $R_{rl}\left(R_{im}\right)$ and is therefore not repeated herein.

2.3. Damage Severity Evaluation Using Bayes Factor

The evaluation of damage is conducted based on Bayesian hypothesis testing of residuals. The Bayes factor, initially proposed by Jeffreys [46] as part of Bayesian hypothesis testing, was introduced to further quantify the extent of damage [47]. Equations (9) and (10) below express the residuals between the model predictions (${\hat{\mathbf{R}}}_{rl}=f_{prd}(R_{im})$ or ${\hat{R}}_{im}=f_{prd}(R_{rl})$) and the corresponding observations ($R_{rl}$ or $R_{im}$). ${\hat{\mathbf{R}}}_{rl}$, ${\hat{\mathbf{R}}}_{im}$, $R_{rl}$, and $R_{im}$ are the vectors, respectively, combining the values of ${\hat{R}}_{rl}$, ${\hat{R}}_{im}$, $R_{rl}$, and $R_{im}$.

$${\epsilon }_{rl}=R_{rl}-{\hat{\mathbf{R}}}_{rl}=R_{rl}-f_{prd}(R_{im})$$

(9)

$${\epsilon }_{im}=R_{im}-{\hat{\mathbf{R}}}_{im}=R_{im}-f_{prd}(R_{rl})$$

(10)

To evaluate the mean of these residuals, Bayesian hypothesis testing is employed. The null hypothesis ($H_0$) posits that the mean of the residual vector is approximately zero, indicating consistency between the two real (imaginary) parts in this study. The structure is considered to be in a healthy state when this hypothesis is supported. Conversely, the alternative hypothesis ($H_1$) suggests that the mean of the residual vector is not zero, implying a difference between the two real (imaginary) parts. This embodies the fact that the structure is in a damaged state. The Bayes factor is utilized to quantify the ratio of likelihoods for the two hypotheses that state whether the discrimination between two signals exists or not [48]; meanwhile, they are considered the damaged and healthy hypotheses in this study.

$$B={\displaystyle \frac{p\left(\left.D\right|H_1\right)}{p\left(\left.D\right|H_0\right)}}={\displaystyle \frac{p\left(\left.D\right|H_1:{\epsilon }_{rl}\ne 0\right)}{p\left(\left.D\right|H_0:{\epsilon }_{rl}=0\right)}}$$

(11)

The term D refers to observations and p is the probability. To avoid the problem that the denominator probability is close to zero and the calculation encounters singularity, a logarithmic operation on both sides of Equation (11) is conducted. Thus, the relationship between the numerator and denominator becomes a subtraction between two terms. Another technique for avoiding this problem is to use the probability of exceedance of a certain low-damage state to define the severity of damage [49]. A Bayes factor below 1 indicates the absence of damage and supports the hypothesis of a healthy state. Conversely, it suggests the occurrence of damage when the Bayes factor exceeds 1. To assess the significance of discrimination or severity of damage, Bayes factors greater than 1 are categorized into different intervals [46]: the values between 1 and 3 are considered insignificant, those between 3 and 10 are substantial, those between 10 and 30 are strong, those between 30 and 100 are very strong, and those above 100 are decisive. This classification was initially developed to assess the degree of support that observed data provide for one hypothesis over its alternative when taking into account uncertainties [50,51]. It has since been applied in probabilistic structure damage identification to distinguish between damaged and healthy states [26,52]. Clear definitions of these thresholds make it easy to understand the implications of a Bayes factor value.

The calculation of two distinct Bayes factors based on the reference models defined in Equations (9) and (10) can present a challenge in decision-making. In such scenarios, one may opt to select the higher value between the two weighted Bayes factors. By utilizing this approach, we derive the synthetic Bayes factor $B_s$, as given in Equation (12) with $B_{rl}$ for ${\hat{\mathbf{R}}}_{rl}\left(R_{im}\right)$ and $B_{im}$ for ${\hat{\mathbf{R}}}_{im}\left(R_{rl}\right)$.

$$B_s=\max (B_{rl},B_{im})$$

(12)

The probability or confidence associated with the Bayes factor for Gaussian residuals [53] is given as

$$p(H_1\left|D\right.)=1-p(H_0\left|D\right.)={\displaystyle \frac{B{B}_r}{1+B{B}_r}}$$

(13)

where p stands for the probability or confidence, and B and B_r are the previous and the current Bayes factors obtained by Equation (11).

In this way, this method can not only effectively identify concealed damage but also quantify its extent using the synthetic Bayes factor. Therefore, the correlation between the Bayes factor and ground-penetrating radar data potentially affected by concealed damage is then built.

3. Case Study

The proposed method is initially evaluated using simulated GPR data, followed by its application to a physical model of a tunnel with steel mesh in the lining. This simulation provides a controlled environment where the presence and extent of damage can be precisely manipulated, thus establishing a benchmark for evaluating the accuracy and robustness of the proposed approach. Subsequently, real-world data from a specific metro tunnel model is utilized to test the proposed method, demonstrating its practical applicability and relevance in authentic scenarios.

3.1. GPRMAX Simulation for Validation

In the absence of ground truth data, we used the GPRMAX simulator, a Python-based tool that employs the Finite Difference Time Domain (FDTD) method to simulate electromagnetic wave propagation across a spatial grid. A tunnel cross-section with a thickness of 0.002 m is simulated. In the meshed domain, each cell can have distinct electromagnetic properties (e.g., permittivity and conductivity), allowing the model to capture the heterogeneous nature of the surrounding materials while enabling the simulation of GPR responses in various defect scenarios.

For examining the patterns of GPR response affected by a concealed defect, it is crucial to consider the structure and materials involved. The simulation involves a tunnel structure comprising a primary lining with a thickness of 0.2 m, a secondary concrete lining with a thickness of 0.3~0.4 m and two-layer steel reinforcement, and a surrounding rock layer. The simulated data used for building an RFV are the GPR data generated by DEM plus random noise. The noise follows a Gaussian distribution, of which the mean is zero and the standard deviation is 5% times that of the DEM-based data. Butterworth band-pass filtering was then performed.

It has been demonstrated that GPRMAX simulations can produce results consistent with actual GPR tests, particularly for detecting subsurface defects within concrete structures [54]. We adopted a grid size of 0.002 × 0.002 × 0.002 m. This grid size and the information in

Table 1. Electromagnetic properties of different materials used in the simulation.

Structure	Material	Permittivity (εr)	Conductivity (σ) (Ohms/m)
Surrounding rock	Soil	8	0.001
	Rock	7.5	0.001
	Water	80	0.01
Primary lining	Concrete I	6	0.005
Second lining	Steel bar	0	∞
	Concrete II	7.5	0.005
Void	Air	1	0

are chosen according to the verified model in a previous study [20]. Generally, it is chosen based on the comparison of accuracy and computational cost when using different grid sizes. The surrounding rock layer is modeled with dimensions of 3.2 m × 0.4 m × 0.002 m, incorporating an average particle size of 0.05 m. The primary lining is simulated by a homogeneous concrete structure with dimensions of 3.2 m × 0.2 m × 0.002 m. The secondary lining consists of steel bars and concrete, with an overall dimension of 3.2 m × 0.4 m × 0.002 m and a bar radius of 0.01 m.

Constructing an accurate model poses challenges due to the heterogeneous nature of the surrounding rock material that consists of stochastically distributed dispersive materials such as sand particles, water, and clay. To effectively address this challenge, we employ the Discrete Element Method (DEM). The DEM is widely used for simulating particulate materials like soil and rock by considering their size, shape, and distribution characteristics at particle-level motion analysis. By utilizing the DEM in our study, we are able to construct a realistic spatial model of tunnel lining that reflects real-world conditions through graded particle sizes. The voids with randomly generated irregular shapes and varying widths and lengths are used in simulation, to reflect the complexity of real-world conditions. Subsequently assigned were electromagnetic properties (EM), including relative permittivity (ε_r), conductivity (σ in Siemens/m), relative permeability (μ_r), and magnetic loss (σ_r in Ohms/m) values for each particle. A previous study [20] has validated the values of these properties in GPRMAX simulations for concealed defect detection of tunnels. It is shown that the spectra of the GPR signal from the simulated model with these parameters in Table 1 were consistent with that acquired from a real tunnel. The relative permeability (μ_r) values for all materials in the Table 1 are assumed to be 1, and the magnetic loss (σ_r in Ohms/m) values are assumed to be 0. The numerical model incorporating a cavity, depicted in Figure 2, is constructed to represent a damaged condition.

It has been demonstrated that DEM-FDTD simulations using GPRMAX can effectively generate simulated radargrams that closely resemble real-world measurements [20,55]. In contrast, this study solely utilizes radar signals in normal states during the training process of a machine learning model, eliminating the necessity for damage-state image datasets in this process. By employing the methodology described in Section 2, this study can detect and assess concealed anomalies without depending on defect-specific models or image interpretation. This streamlined approach enhances efficiency and adaptability across various conditions. The proposed procedure follows the subsequent steps.

Initially, GPRMAX is employed to simulate an intact region of a tunnel without concealed damage. These simulated data are then fused to generate a historical RFV in the undamaged state, where its real and imaginary components are utilized to train two baseline models. Subsequently, the baseline models are applied to evaluate other groups of datasets with unknown health conditions. When any defect is found, both the severity and probabilistic confidence levels of the defect are quantified.

In the beginning, radargrams obtained from GPRMAX simulation of the intact region of the tunnel (as depicted in Figure 3) are utilized to establish the healthy-state RFV and train two distinct baseline models: $R_{rl}\left(R_{im}\right)$ and $R_{im}\left(R_{rl}\right)$. The correlation between the real and imaginary components of the RFV is examined. Figure 4a illustrates the relationship between observed and predicted patterns based on the imaginary-to-real relationship, while Figure 4b showcases the relationship based on the real-to-imaginary relationship. Importantly, there exists a strong concurrence between predicted and original patterns, affirming the suitability of the baseline models. Furthermore, these relationships exhibit an arcuate pattern.

In the first scenario for condition assessment, another set of data is used to establish a new RFV with an unknown state of the tunnel. The real and imaginary components of the RFV are separately provided to two baseline models to generate the corresponding predicted patterns. Figure 5a illustrates the correlation between the newly observed and predicted patterns based on the imaginary-to-real relationship, while Figure 5b demonstrates the real-to-imaginary relationship. It can be observed that both the observed and predicted patterns consistently maintain their characteristic arcuate features. Despite the difference at the ends of this arcuate shape, there remains a high level of consistency between the new predicted patterns and the original ones. Quantitative assessment using Bayes factors reveals the small deviations between these observed and predicted patterns. The calculated Bayes factor values for discrepancies between blue and orange patterns are 0.9 and 0.8, respectively, with the damage probabilities (confidence levels) of 6.1% and 5.3%. By combining these results with the synthetic Bayes factor, it supports the hypothesis of “without void”, which corresponds to an intact condition behind the tunnel surface. This additional dataset can be incorporated into the database to update the baseline models.

In the second scenario for condition assessment, the data collected from a region of the tunnel that has a void in its primary lining is utilized to generate a new RFV. Figure 6 illustrates the specific location and size of the void. The new RFV is then fed into both baseline models. As depicted in Figure 7, although the observed and predicted patterns still exhibit an arcuate shape, there is a noticeable deviation between the current observed pattern and the predicted one. The Bayes factor values for these deviations are calculated as 3.6 and 3.1, respectively, accompanied by the damage probabilities (confidence levels) of 65.3% and 60.8%. By applying Equation (12), we obtain a synthetic Bayes factor of 3.6 along with a damage probability of 65.3%. The pair of Bayes factor and probability provide support for the hypothesis of damage, and the damage severity is “substantial”.

The third scenario involves utilizing radar data collected from a region of the tunnel with a bigger void in its primary lining to generate a new RFV. Figure 8 illustrates the specific location and size of the bigger void. The real and imaginary parts of this RFV are then provided to the two baseline models. As shown in Figure 9, although the pattern keeps an arcuate shape, there are noticeable deviations between the observed pattern and the predicted pattern, surpassing those depicted in Figure 7. The deviations are calculated, producing Bayes factor values of 4.7 and 4.5, respectively, corresponding to damage probabilities of 74.3% and 71.8%. By employing Equation (12), we obtain a synthetic Bayes factor of 4.6 along with a damage probability of 74.3%. The pair of Bayes factor and probability supports the hypothesis of a “damaged state” and identifies the severity as a “substantial” level.

To reveal the relationship between the Bayes factor and void size, more simulations with different sizes of voids at the same depth are conducted. As shown in Figure 10, the Bayes factor increases with the size of concealed damage, and both of the two curves corresponding to the baseline models exhibit similar upgoing trends. The severity level of the concealed damage is gradually higher, from “normal” and “substantial” to “strong”. To make the assessment result of the tunnel safer, a larger value between the two curves can be chosen for assessing damage severity.

The effectiveness of the proposed method in detecting and evaluating concealed damage has been successfully demonstrated through simulated experiments. The robustness of the reference models and the accuracy of the Baye factor-based evaluation have been confirmed by analyzing different scenarios, including those with varying degrees of damage. These findings underscore the potential practical applicability of this approach. The method is validated through simulations, before its implementation in real-world situations.

3.2. Validation Through Physical Model Testing

To further advance the progress achieved in simulated experiments of a tunnel, it is imperative to validate the proposed method through physical model testing. This phase examines the effectiveness of the method within practical limitations and complexities. The evaluation is conducted on a tunnel model located at Guangzhou University, as depicted in Figure 11. The IDS GPR system from Italy was employed for data collection and void detection of the tunnel model. The survey line was positioned circumferentially alongside the tunnel wall. Components comprising the GPR system encompass an antenna, computer, odometer wheel, and main unit.

In a practical scenario, the ground-penetrating radar system was used to collect data from the physical model. The data acquired in the intact tunnel model serve as the foundation for constructing the historical RFV (as depicted in Figure 12) and training the baseline models. Similarly, two baseline models are established to investigate the relationship between the real and imaginary components of the RFV in this practical context: $R_{rl}\left(R_{im}\right)$ and $R_{im}\left(R_{rl}\right)$. Figure 12a showcases the correlation between observed and predicted patterns based on the imaginary-to-real relationship, while Figure 12b illustrates the real-to-imaginary relationship. Similar to GPRMAX simulations, there is strong agreement between predicted and original patterns, confirming the reliability of these baseline models. Furthermore, these relationships exhibit arcuate patterns.

A new RFV is subsequently constructed using GPR data obtained from the physical model in an unknown state. Subsequently, the real and imaginary components of the new RFV are utilized as inputs to the two reference models for generating predicted patterns. Figure 13a showcases the correlation between newly observed and predicted patterns. Similarly, Figure 13b illustrates the relationship between real and imaginary parts. Despite the differences at the ends of the arcuate shape, there remains a significant level of consistency between the newly predicted and original patterns. To quantitatively evaluate the small deviations between observed and predicted patterns, Bayes factors are employed. The resulting Bayes factors for differences between blue and orange patterns were found to be 1.2 and 0.9, respectively, with corresponding damage probabilities of 6.4% and 1.1%. Given the low probability of damage existence, the hypothesis of a “normal state” is supported, indicating that the tunnel is almost healthy. This is consistent with the current condition of the physical model.

To examine the fitting degree of the baseline models, linear regression analysis is conducted between the actual imaginary part of RFV and the predicted imaginary part. The fitting performance is evaluated by using the Root Mean Square Error (RMSE) and coefficient of determination (R²). The fitting degree can be seen in Figure 14. The RMSE value is very small, indicating a good fitting performance. A larger value of R² in [0, 1] indicates a stronger correlation between the two variables.

Finally, additional data from the physical model are utilized to construct a new RFV. The real and imaginary components of the new RFV are once again provided to the two baseline models to generate their corresponding patterns. As illustrated in Figure 15, despite maintaining their characteristic arcuate shape, there are noticeable deviations between the current observed and predicted patterns. The Bayes factors for these deviations were determined to be 2.9 and 3.2, respectively, with damage probabilities of 55.7% and 64.6%. By employing Equation (12), we obtained a synthetic Bayes factor of 3.2 along with a damage probability of 64.6%. The pair of Bayes factor and probability provides support for the hypothesis that the state is damaged, which aligns with the finding from the physical model with a cavity.

The assessment results in the simulation and physical model test highly agree with actual situations, demonstrating the effectiveness of the proposed method. Sometimes, the degrees of damage may be different from those defined in an industrial standard [56]. Therefore, it is desirable to improve the approach for calculating the Bayes factor and one possible solution may be adding an automatically adjustable weight to the Bayes factor.

4. Conclusions

The novel application of a probabilistic, data-driven method for detecting and assessing concealed tunnel damage using GPR data has been investigated, to realize quantitative evaluation when the labeled damage-state data are unavailable in the ML model training process. The main findings are summarized as follows:

(1)

The proposed method incorporated a Bayesian baseline model for building the relationship between the real and imaginary parts of the RFV in a healthy state. The case study based on the simulated and physical model tests showed that the concealed damage was effectively identified and quantitatively assessed, by analyzing the deviation of the RFV built with new observations from the baseline RFV predicted by the Bayesian model with new input.

(2)

It was also demonstrated that effective damage identification was realized in the absence of pre-existing damage-state data during the ML model training process, highlighting its potential adaptability to more real-world scenarios.

(3)

It is implied that the relative change in the relationship between the real and imaginary parts of RFV is an effective representation of tunnel-healthy conditions.

(4)

The Bayes factor exhibited a positive correlation with cavity size, indicating its appropriateness in quantifying damage severity.

A limitation of this method is the incapability of identifying the geometry of concealed damage in a tunnel, as this method focuses on the identification of damage existence and severity by taking advantage of Bayesian updating which provides a small amount of new data to an offline Bayesian model to guarantee efficiency. A possible solution to this limitation is to build a supervised ML model using GPR data labeled with various geometries of defects in the model training process.

The method has the potential to be applied in other scenarios of radargram-based assessments, as the development of this method is not based on the assumption of a specific tunnel structure. Future work may focus on the automatic adjustment in the thresholds of the Bayes factor such that the identified severity level can be highly consistent with industrial standards.