Reliability-adaptive probabilistic imaging and severity assessment of axial cracks in steel pipelines

2.1. Steel pipeline specimen and damage scenarios

A straight Q235 steel pipeline specimen was used to evaluate the proposed reliability-adaptive probabilistic imaging and severity-assessment framework under controlled laboratory conditions. The specimen had an outer diameter of 219 mm, a wall thickness of 5 mm, an inner diameter of 209 mm, and a total length of 6.0 m. The material parameters used in the analysis were Young’s modulus $E=$ 206 GPa, Poisson’s ratio $\nu =$ 0.30, and mass density $\rho =$ 7850 kg/m³. The specimen geometry and crack arrangement are shown in Fig. 1, and the corresponding geometric and material parameters are summarized in Table 1.

One intact case, denoted as N0, and three damaged cases, denoted as C1-C3, were considered. The damaged cases were machined as external axial rectangular notches on the outer surface of the pipe. In all damaged cases, the axial location, circumferential location, crack length, and crack width were kept unchanged, while only the crack depth was varied. This controlled design was adopted to isolate the influence of crack depth on the guided-wave response. Therefore, the present experiment should be interpreted as a controlled-condition verification rather than a full generalization to arbitrary crack geometries, crack orientations, or pipe environments.

The crack length and width were fixed at $l=$ 20 mm and $w=$0.8 mm, respectively. The crack center was located at $x_c=$ 2.20 m and ${\theta }_c=$180°. The relative crack depth was defined as $d/t$, where $d$ is the notch depth and $t$ is the pipe wall thickness. The three damaged cases corresponded to depth ratios of 0.30, 0.50, and 0.70, representing mild, moderate, and severe axial surface cracks, respectively. The detailed definitions of the damage cases are listed in Table 2.

Fig. 1Geometry of the steel pipeline specimen and crack configuration: a) overall geometry of the Q235 steel pipeline specimen; b) pipe cross-section and crack circumferential position; c) definitions of the three crack-depth cases C1, C2, and C3

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2.2. Measurement setup, excitation configuration, and sensor arrangement

The guided-wave measurement system was configured to capture the incident wave packet, the crack-reflected response, and the subsequent boundary-related echoes. The actuator position, receiver layout, excitation waveform, and acquisition parameters were kept unchanged for all intact and damaged cases to ensure that the measured differences were mainly caused by crack-depth variation. Since guided-wave measurements in pipes are sensitive to transducer configuration, modal content, sensor spacing, and coupling conditions [6], [10], [13], these experimental parameters were controlled throughout the tests.

The axial coordinate $x$ was defined from the left end of the pipe, and the circumferential coordinate $\theta$ was measured from the reference generatrix of the pipe. The excitation transducer was mounted on the outer surface of the pipe at $x_a=$ 0.05 m. Sixteen receiving sensors were arranged along the same circumferential side as the crack to improve the sensitivity to the back-scattered crack response. The receiver positions were $x_i$ = 0.40, 0.80, 1.20, 1.60, 2.00, 2.40, 2.80, 3.20, 3.60, 4.00, 4.40, 4.80, 5.20, 5.40, 5.60, and 5.80 m, corresponding to channels S1-S16, respectively. Channels S5-S8 were located closest to the crack region and were selected as the principal observation channels for representative signal and feature analysis, while all available receiving channels were used in the imaging and fusion procedure.

A five-cycle Hanning-windowed sinusoidal burst was used as the excitation signal. Four center frequencies, namely 40, 50, 63, and 80 kHz, were adopted to investigate the frequency dependence of the crack-reflected response and to provide the basis for multi-frequency fusion. This low-frequency guided-wave range provides a practical compromise among propagation attenuation, spatial resolution, and modal complexity in pipeline inspection [14], [20-22]. The excitation voltage was fixed at 100 V for all test conditions. The signals received were sampled at 10 MHz with a record length of 12 ms, which was sufficient to cover the incident wave, the crack-reflected component, and later reflections from pipe boundaries.

The excitation configuration was designed to obtain a torsional-mode-dominated guided-wave response in the selected low-frequency range. In the revised manuscript, this response is referred to as the T(0,1)-dominated wave packet rather than an ideally pure single-mode wave, because weak residual modal components may still exist in practical measurements. The dominance of the target wave packet was verified by comparing the measured arrival time with the group velocity predicted from the dispersion analysis in Section 2.3. This cautious description avoids overclaiming pure single-mode excitation while still clarifying the dominant mode used for reflection-window prediction and probabilistic imaging.

Each measurement condition was repeated 20 times. These repeated measurements were used to evaluate signal repeatability, noise-interference sensitivity, and the stability of extracted features. They were not treated as independent pipe specimens or independent crack configurations. The sensor layout is illustrated in Fig. 2, and the detailed acquisition parameters are listed in Table 3.

Fig. 2Guided-wave measurement setup and layout of the actuator and receiving sensors

2.3. Guided-wave mode selection and dispersion analysis

Guided waves in cylindrical structures generally contain longitudinal, torsional, and flexural modes. These modes have different propagation velocities, dispersion characteristics, and sensitivity to defect orientation. Therefore, the target wave mode should be clarified before reflection-window extraction and probabilistic imaging. In this study, the T(0,1)-dominated torsional guided-wave response was selected as the target wave packet for subsequent analysis.

The selection of the T(0,1)-dominated response was based on two considerations. First, the torsional T(0,1) mode is nearly non-dispersive in the investigated low-frequency range, which is beneficial for predicting the arrival time of the crack-reflected wave packet [1], [14], [17], [20]. Second, the relatively stable group velocity of the torsional response reduces the uncertainty caused by frequency-dependent wave-packet spreading. This property is important for multi-frequency feature extraction and probabilistic imaging because the same physical reflection event must be isolated consistently at different excitation frequencies.

To verify the selected mode and the group velocity used in this study, a dispersion analysis was performed for the Q235 steel pipe using the geometric and material parameters listed in Table 1. The calculated phase-velocity and group-velocity curves are shown in Fig. 3. The four excitation frequencies used in the experiment, namely 40, 50, 63, and 80 kHz, are marked on the dispersion curves. In this frequency range, the selected T(0,1)-dominated response shows a group velocity close to 3260 m/s. Therefore, $c_g\approx$ 3260 m/s was used for theoretical prediction of the crack-reflection arrival time.

The addition of dispersion analysis also addresses the reproducibility of the experimental configuration. Recent guided-wave studies on complex pipe-like structures have similarly used phase- and group-velocity dispersion curves to guide mode selection and inspection-parameter design [32]. In the present study, the dispersion result provides the physical basis for both the reflection-window calculation and the later reliability-adaptive fusion strategy. Although the same target wave packet is considered at all excitation frequencies, the received signal quality, modal purity, and scattering sensitivity may still vary with frequency. Therefore, the later fusion procedure does not assign equal importance to all frequencies but determines their relative contributions according to measurement reliability.

Fig. 3Dispersion curves and mode selection for the Q235 steel pipe: a) phase-velocity dispersion curves; b) group-velocity dispersion curves. The four excitation frequencies of 40, 50, 63, and 80 kHz are marked, and the selected T(0,1)-dominated wave packet shows a group velocity close to in the investigated frequency range

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2.4. Signal preprocessing and reflection-window extraction

The raw guided-wave measurements contained the incident wave packet, crack-reflected components, pipe-end reflections, residual modal components, and broadband environmental noise. Since the crack-reflected response was much weaker than the directly transmitted wave packet, signal preprocessing was required before feature extraction and probabilistic imaging. The preprocessing procedure included direct-current removal, amplitude normalization, narrow-band filtering, and wavelet denoising. Signal-processing strategies such as multimode signal interpretation, dispersion-related analysis, filtering, and wavelet-based denoising have been widely used to improve the interpretability of guided-wave measurements [8-10], [19].

Let $s_{i,f}^{\left(m\right)}\left(t\right)$ denote the raw signal measured at receiver i, excitation frequency f, and repeated measurement m. After direct-current removal and amplitude normalization, the normalized signal was obtained as:

where $\stackrel{-}{s_{i,f}^{\left(m\right)}}$ denotes the mean value of the raw signal over the full record, and the denominator denotes the maximum absolute amplitude of the zero-mean signal. This normalization constrains the full-record amplitude to the range of [–1, 1], thereby reducing amplitude fluctuations caused by channel sensitivity and repeated-measurement variation. Accordingly, the revised signal-preprocessing figure should use a normalized-amplitude axis covering [–1, 1]. If a local reflection segment is enlarged for clarity, it should be explicitly labelled as a zoomed view.

After normalization, a zero-phase band-pass filter centered at the excitation frequency was used to suppress out-of-band noise and non-target components. Wavelet denoising was then applied to further improve the signal-to-noise ratio of the crack-related response while preserving the main waveform characteristics. The preprocessing parameters for different excitation frequencies are summarized in Table 4.

The crack-related information was mainly concentrated in the reflected wave packet. Therefore, a dedicated reflection window was extracted from each preprocessed signal. The expected arrival time of the crack-reflected wave was estimated from the actuator-crack-receiver propagation path as:

where $L_{a,c}$ is the propagation distance from the actuator to the crack, $L_{c,i}$ is the propagation distance from the crack to receiver $i$, and $c_{g,f}$ is the group velocity of the selected T(0,1)-dominated wave packet at frequency f. In the present frequency range, $c_{g,f}$ was approximately 3260 m/s, as indicated by the dispersion analysis in Section 2.3.

A time window centered at $t_{r,i,f}$ was then extracted for each channel and frequency. The crack-reflected signal segment was defined as:

where $\mathrm{\Delta }{T}_f$ is the window length used at frequency $f$. A frequency-dependent window length was adopted to account for the different wave-packet durations at different excitation frequencies. The window was selected to include the main crack-reflected packet while reducing the influence of pipe-end echoes and non-target wave components.

A representative example of the preprocessing and reflection-window extraction is shown in Fig. 4. The full-record signal is displayed using the normalized-amplitude range of [–1, 1], while the extracted crack-reflection segment is presented as a zoomed view to better illustrate the local waveform characteristics around the expected reflection time. This representation allows the overall signal evolution and the weak crack-reflected wave packet to be observed simultaneously.

The resulting reflection-window signals were used for subsequent time-domain, frequency-domain, and time-frequency-domain feature extraction. By combining mode-guided arrival-time prediction, amplitude-normalized preprocessing, and frequency-dependent reflection-window extraction, the weak crack-reflected response could be isolated more consistently from the complex guided-wave background.

Fig. 4Example of signal preprocessing and reflection-window extraction for C2 at 50 kHz in channel S6: a) raw full-record signal after normalization, with the normalized amplitude displayed in the range of [–1,1]; b) preprocessed signal after band-pass filtering and wavelet denoising, with the reflection window highlighted. (c) Zoomed view of the extracted crack-reflected wave packet

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3.1. Overall processing flow

The proposed framework transforms the measured guided-wave responses into probabilistic crack-localization images and semi-quantitative severity-grading results. As shown in Fig. 5, the procedure consists of six main steps: reflection-window signal extraction, multi-domain feature extraction, single-frequency probabilistic imaging, reliability-adaptive multi-frequency fusion, confidence-enhanced probabilistic imaging, and posterior-distribution-based severity assessment.

After preprocessing and reflection-window extraction, each receiver channel and excitation frequency provides a crack-related signal segment. Time-domain, frequency-domain, and time-frequency-domain descriptors are then extracted from this segment to characterize the crack-reflected response from complementary perspectives. For each excitation frequency, a single-frequency probability field is constructed on the imaging grid by combining reflection intensity and travel-time consistency. The probability fields obtained at different frequencies are then fused using reliability-adaptive weights. These weights are determined by echo quality, inter-sensor consistency, and modal purity, so that frequency components with higher measurement reliability contribute more strongly to the final image.

This design follows the general guided-wave inspection logic in which signal preprocessing, modal interpretation, feature extraction, and spatial imaging are combined to improve defect detectability in pipe-like structures [2-4], [15], [29], [31]. After multi-frequency fusion, a confidence-enhancement step is introduced to suppress uncertain secondary hotspots in the probability field. Finally, several posterior-distribution features are extracted from the confidence-enhanced image and combined into a severity index for semi-quantitative grading. Compared with direct equal-weight fusion, the proposed framework emphasizes the reliability and physical consistency of the measured information rather than treating all frequencies and channels as equally informative.

Fig. 5Overall processing flow of the proposed reliability-adaptive probabilistic imaging and severity-assessment framework

3.2. Multi-domain feature extraction

Guided-wave crack responses may exhibit changes in amplitude, energy, arrival time, spectral distribution, and local time-frequency concentration. Such multi-domain signal variations are closely related to multimodal propagation, dispersion, defect scattering, and time-frequency characteristics of guided-wave measurements [8-10], [18], [19]. A single descriptor is therefore insufficient to represent the weak and frequency-dependent crack-reflected response. For this reason, a multi-domain feature vector was constructed from the reflection-window signal.

Let $s_{r,i,f}^{\left(m\right)}\left(t_n\right)$ denote the extracted crack-reflected signal segment measured at receiver $i$, excitation frequency $f$, and repeated measurement $m$, where $t_n$ is the discrete sampling time and $n=\mathrm{1,2},\dots ,N_w$. The reflected peak amplitude is defined as:

The reflected energy is calculated as:

where $\mathrm{\Delta }t$ is the sampling interval. The peak-based arrival time of the reflected wave packet is defined as:

where $t_{start,i,f}$ is the start time of the extracted reflection window. The arival-time descriptor is mainly used to evaluate whether the measured crack-related response is consistent with the theoretical actuator-crack-receiver propagation path.

Frequency-domain and time-frequency-domain descriptors were introduced because defect scattering may redistribute signal energy in the spectrum and produce localized time-frequency variations that are not fully captured by time-domain peak features alone [18], [19], [29]. Let $S_{r,i,f}^{\left(m\right)}\left(f_k\right)$ be the amplitude spectrum of the reflection-window signal at frequency bin $f_k$. The dominant frequency is defined as:

and the spectral centroid is defined as:

To further describe the local time-frequency organization of the crack response, a time-frequency representation $W_{i,f}^{\left(m\right)}\left(t_n,f_k\right)$ was computed using a short-time or wavelet-based time-frequency transform. The total time-frequency energy is given by:

The time-frequency concentration factor is defined as:

where $\epsilon$ is a small positive constant used to avoid division by zero. A larger TFC value indicates that the crack-related energy is concentrated in a more compact time-frequency region.

The extracted descriptors were stacked into a multi-domain feature vector:

Before probabilistic imaging and fusion, each feature component was standardized to zero mean and unit variance across the corresponding dataset. This normalization prevents features with large numerical scales from dominating the subsequent analysis. The adopted multi-domain feature descriptors are summarized in Table 5, and a representative feature-extraction example is illustrated in Fig. 6.

Fig. 6Example of multi-domain feature extraction from the crack-reflected wave packet

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3.3. Single-frequency probabilistic imaging formulation

To clarify the generation of the frequency-specific probability field, an imaging grid $\mathrm{\Omega }$ was defined in the axial-circumferential plane of the pipe. Each candidate pixel is denoted by $r=\left(x,\theta \right)$. For a given excitation frequency $f$, the theoretical arrival time of a crack-reflected wave associated with receiver $i$ and candidate position $r$ is expressed as:

where $L_a\left(r\right)$ is the propagation distance from the actuator to the candidate pixel, $L_i\left(r\right)$ is the propagation distance from the candidate pixel to receiver $i,$ and $c_{g,f}$ is the group velocity of the selected T(0,1)-dominated wave packet at frequency $f$. This path-consistency concept is consistent with guided-wave imaging studies in which candidate damage locations are evaluated according to the agreement between measured wave arrivals and theoretically predicted propagation paths [13], [15], [18].

The travel-time consistency between the measured response and the candidate pixel is evaluated by:

where ${\tau }_{p,i,f}$ is the measured peak arrival time and ${\sigma }_t$ controls the tolerance of arrival-time deviation. A larger value of $K_{i,f}\left(r\right)$ indicates that the candidate pixel is more consistent with the measured reflection arrival time.

The response strength of each receiver channel is described by a normalized damage-sensitive feature score:

where the tilde denotes normalized feature values. The coefficients ${\lambda }_1$-${\lambda }_4$ are non-negative and satisfy:

In this study, the same feature-score definition was used for all frequencies to avoid frequency-specific manual tuning.

The single-frequency probabilistic indication is then calculated as:

where $N_s$ is the number of receiving channels used for imaging. The normalized single-frequency probability field is obtained by:

Thus, $P_f\left(r\right)\in \left[\mathrm{0,1}\right]$, and a larger value indicates a stronger probability that the candidate pixel corresponds to the crack-related reflection source. This formulation links the probability image to both reflected-response strength and travel-time consistency, rather than generating the image only from empirical visual smoothing.

3.4. Reliability-adaptive multi-frequency fusion

The single-frequency probability fields may differ significantly because guided-wave scattering, modal purity, signal-to-noise ratio, and sensor consistency vary with excitation frequency. Previous studies on torsional guided-wave excitation and pipeline defect interaction have shown that the received response may vary with excitation frequency, modal condition, propagation path, and defect geometry [14], [20-23]. Equal-weight fusion assumes that all frequencies are equally reliable, which may introduce unstable or weakly informative frequency components into the final image. To address this issue, a reliability-adaptive fusion strategy was used to combine the probability fields obtained at different excitation frequencies.

Let $P_f\left(r\right)$ be the single-frequency probability field at frequency $f$. The fused probability field is defined as:

where $F=$ 40, 50, 63, 80 kHz is the set of excitation frequencies, and $w_f$ is the weight assigned to frequency $f$. The weights satisfy:

To evaluate the contribution of the proposed weighting strategy, two baseline fusion strategies were also considered. The first baseline is equal-weight fusion, in which all excitation frequencies are assigned the same contribution, i.e., $w_f^{EW}=1/N_f$, where $N_f$ is the number of excitation frequencies. This strategy is simple and commonly used as a direct baseline, but it assumes that all frequency components have identical reliability.

The second baseline is SNR-weighted fusion. In this baseline, the contribution of each frequency is determined only by the echo-to-noise level, i.e., $w_f^{SNR}=Q_f/{\sum }_{f^{\text{'}}\in F}{Q}_{f^{\text{'}}}$, where $Q_f$ denotes the echo-quality indicator defined below. This baseline is used to examine whether considering noise level alone is sufficient for multi-frequency fusion. Compared with equal-weight fusion, SNR-weighted fusion can suppress frequencies with weak crack-related echoes or high pre-reflection noise. However, it does not explicitly consider whether the defect-sensitive response is spatially consistent among sensors or whether the reflected wave packet is dominated by the target guided-wave mode.

In the proposed reliability-adaptive fusion strategy, three measurement-oriented reliability indicators were used to determine $w_f$: echo quality, inter-sensor consistency, and modal purity. The echo-quality indicator is defined as:

where $\overline{A}r,f$ is the mean reflected peak amplitude over the selected receiver channels and repeated measurements, and $\sigma n,f$ is the standard deviation of the pre-reflection noise at frequency $f$. A larger $Q_f$ indicates a stronger crack-related echo relative to background noise.

The inter-sensor consistency indicator is defined as:

where $CV\left(\cdot \right)$ denotes the coefficient of variation. This term assigns a larger value to a frequency whose damage-sensitive responses are more consistent across receiver channels.

The modal-purity indicator is defined as:

where $E_{tar,f}$ is the energy contained in the target modal band around the selected T(0,1)-dominated wave packet, and $E_{tot,f}$ is the total energy in the reflection window. This term penalizes frequency components with stronger non-target modal contamination.

The adaptive weight is calculated by multiplicative normalization:

where $\alpha$, $\beta$, and $\gamma$ control the relative importance of echo quality, inter-sensor consistency, and modal purity, respectively. Unless otherwise stated, $\alpha =\beta =\gamma =1$ was used to avoid over-parameterization. In this way, a frequency receives a large contribution only when it simultaneously exhibits strong echo quality, stable inter-sensor response, and acceptable modal purity.

Compared with equal-weight fusion and SNR-weighted fusion, Eq. (23) incorporates three physically interpretable reliability factors. Therefore, the proposed fusion strategy is better described as a reliability-informed multi-frequency integration mechanism rather than a purely empirical weighting scheme. A representative example of the single-frequency probability images, adaptive weights, and fused probability image is shown in Fig. 7.

Fig. 7Reliability-adaptive multi-frequency fusion results for the representative case of C2: a)-d) show the single-frequency probabilistic images reconstructed at 40, 50, 63, and 80 kHz, respectively; e) presents the adaptive weights determined by echo quality, inter-sensor consistency, and modal purity; f) shows the fused probability image obtained by reliability-adaptive multi-frequency fusion

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3.5. Confidence-enhanced probabilistic imaging

Although reliability-adaptive fusion improves the utilization of frequency-dependent information, the fused image may still contain secondary hotspots when several non-damage regions exhibit moderate probability. To suppress uncertain indications without excessively smoothing the dominant crack response, a confidence-enhancement step was introduced after multi-frequency fusion.

For each candidate pixel $r$, the uncertainty of the fused probability field is quantified using binary entropy:

The corresponding confidence coefficient is defined as:

The confidence-enhanced probability image is then obtained as:

According to Eqs. (24-26), pixels with high fused probability and low uncertainty are retained, whereas pixels with moderate but ambiguous probability are attenuated. This operation is used as an uncertainty-control step rather than a purely visual image-sharpening procedure. The posterior uncertainty, confidence coefficient, and confidence-enhanced probability image are illustrated in Fig. 8.

Fig. 8Illustration of posterior uncertainty and confidence-enhanced imaging for the representative case of C2: a) shows the fused probability image; b) shows the posterior uncertainty distribution; c) shows the confidence coefficient distribution; d) shows the confidence-enhanced probability image

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3.6. Severity index for crack assessment

Before defining the proposed severity index, it is necessary to clarify its relationship with existing severity-assessment indicators. In guided-wave inspection, reflected peak amplitude, reflected energy, and reflection-coefficient-related quantities are commonly used to describe defect severity because they are directly associated with defect-induced scattering strength [18], [30]. Feature-based methods further incorporate waveform, spectral, or time-frequency descriptors to improve sensitivity to damage-related signal variations [19], [29]. However, these conventional indicators are usually extracted from individual signals or selected channels, and they may be affected by sensor coupling, propagation distance, noise level, and frequency-dependent modal behavior. In contrast, the severity index used in this study is extracted from the posterior probability image after reliability-adaptive multi-frequency fusion and confidence enhancement. Therefore, it does not replace physical amplitude or energy indicators, but integrates them into a spatial probabilistic representation for semi-quantitative crack grading.

The final objective of the proposed framework is not only to localize the crack but also to provide a semi-quantitative assessment of crack severity. Since reflected amplitude, reflected energy, and extracted signal features are commonly used to describe defect-induced guided-wave scattering, the severity index in this study is constructed from posterior-distribution features rather than from a single waveform descriptor [18], [29], [30]. This design allows the severity assessment to consider peak posterior intensity, regional response level, high-probability spatial extent, and cross-frequency stability.

The crack candidate location is first determined by:

A region of interest ${\mathrm{\Omega }}_{ROI}$ is then defined around $r^{*}$. The peak posterior response is:

The regional mean response is calculated as:

The high-response area is defined as:

where $I\left[\cdot \right]$ is the indicator function, $\eta$ is the relative high-response threshold, and $\mathrm{\Delta }A$ is the area represented by one imaging pixel.

The cross-frequency consistency is defined as:

This term evaluates whether different excitation frequencies provide consistent evidence at the estimated crack location.

Before constructing the severity index, $P_{max}$, $\stackrel{-}{P_{ROI}}$, $A_{high}$, $C_{freq}$ were normalized to the range of [0, 1]. The overall severity index is defined as:

where:

In the present controlled experiment, the weighting coefficients were selected to balance local posterior intensity, regional distribution, high-response extent, and cross-frequency stability. The influence of these coefficients is further discussed in the ablation and limitation analysis in Section 4.6. The damage grade is finally determined by two thresholds $T_1$ and $T_2$:

It should be emphasized that the proposed index is designed for semi-quantitative severity grading rather than absolute crack-depth inversion. The thresholds $T_1$ and $T_2$ are calibrated under the present controlled experimental configuration and should be recalibrated when additional crack geometries, locations, orientations, or sensor-coupling conditions are introduced. The complete severity-grading procedure based on posterior distribution features is summarized in Fig. 9.

Fig. 9Flowchart of crack severity grading based on posterior distribution features

4.1. Response of multi-domain features to crack severity

The response of representative multi-domain features to crack severity is shown in Fig. 10 and summarized in Table 6. Under the present controlled experimental configuration, the reflected peak amplitude $A_r$, reflected energy $E_r$, and time-frequency energy $E_{TF}$ increased progressively from the intact case N0 to the severe crack case C3. This trend indicates that deeper axial surface notches produced stronger back-scattered guided-wave responses and higher local time-frequency energy. Such amplitude- and energy-related variations are consistent with the general physical interpretation of guided-wave scattering from notch-like defects [18], [19], [29].

By contrast, the peak arrival time ${\tau }_p$ changed only slightly among C1-C3. This is expected because the crack location was fixed in all damaged cases, and the dominant actuator-crack-receiver propagation path remained unchanged. Therefore, in the present experiment, the arrival-time descriptor mainly supports localization consistency rather than severity discrimination. This also confirms the necessity of using different descriptors for different assessment purposes: arrival-time information is more relevant to spatial localization, whereas amplitude- and energy-related features are more sensitive to crack-depth variation.

The frequency-domain descriptors showed a relatively moderate trend. The dominant frequency $f_p$ and spectral centroid $f_c$ varied within a narrow range, suggesting that crack deepening did not fundamentally shift the central frequency content of the reflected wave packet. However, the increase in $E_{TF}$ and TFC indicates that the crack-related energy became more concentrated as the crack depth increased. These observations support the use of a multi-domain feature vector rather than a single waveform descriptor.

It should be noted that the observed monotonic trends were obtained with a fixed crack position, fixed crack length, fixed crack width, and fixed crack orientation. Therefore, the feature response should be interpreted as depth-sensitive behavior under controlled conditions, rather than as a complete representation of all possible axial crack configurations.

Fig. 10Comparison of representative multi-domain features under different crack conditions: a)-d) show the variations of the reflected peak amplitude, reflected energy, time-frequency energy, and time-frequency concentration factor TFC, respectively, for the intact case N0 and the three crack cases C1-C3

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4.2. Single-frequency probabilistic imaging and frequency dependence

Fig. 11 compares the single-frequency probabilistic images obtained at 40, 50, 63, and 80 kHz for the three damaged cases C1-C3. All four excitation frequencies produced a damage-sensitive indication in the vicinity of the actual crack position, indicating that each frequency retained a certain degree of axial-crack sensitivity. However, the compactness of the hotspot, the level of background response, and the stability of localization differed among the four frequencies.

The 40 kHz results generally produced broader probability distributions. This behavior can be explained by the relatively longer wavelength and lower spatial resolution at lower frequency. Although the low-frequency response is usually more stable for long-distance propagation, its ability to concentrate the probability field around a small crack region is limited. The 80 kHz results showed stronger local sensitivity in some cases, but they were also more susceptible to background fluctuation and possible modal interference. This is consistent with the frequency-dependent nature of guided-wave propagation and scattering in pipe-like structures [14], [20-23].

For the present specimen and crack configuration, 50 and 63 kHz provided a better balance between crack-response strength, localization compactness, and background suppression. Nevertheless, the results also show that no single frequency was uniformly optimal for all crack depths. A frequency that provides a clear response for a severe crack may not necessarily provide the most stable response for a mild crack. Therefore, relying on a fixed single-frequency inspection strategy may lead to inconsistent imaging quality when the crack severity is unknown.

These single-frequency results provide the experimental motivation for multi-frequency fusion. The purpose of fusion is not simply to accumulate all available frequency responses, but to integrate complementary information while suppressing frequency components with lower measurement reliability.

Fig. 11Single-frequency probabilistic imaging results at 40, 50, 63, and 80 kHz for axial crack cases C1-C3. The actual crack position is marked in each image

4.3. Comparison with equal-weight and SNR-weighted fusion

Fig. 12 and Table 7 compare three multi-frequency fusion strategies: equal-weight fusion, SNR-weighted fusion, and the proposed reliability-adaptive fusion. The equal-weight fusion method assigns identical contributions to all excitation frequencies, whereas the SNR-weighted baseline determines the frequency weights only from the echo-to-noise level. The proposed method further incorporates inter-sensor consistency and modal purity in addition to echo quality.

All three methods produced damage-sensitive indications near the actual crack region, indicating that multi-frequency combination is beneficial compared with relying on a single frequency alone. However, the equal-weight fusion images showed broader high-probability regions and stronger background artifacts, especially for the mild and moderate crack cases. This behavior is expected because all frequency components contribute identically even when their echo quality, modal purity, or inter-sensor consistency is relatively poor.

The SNR-weighted fusion baseline reduced part of the background response because frequency components with lower echo-to-noise levels were assigned smaller weights. For example, the localization error decreased from 42 mm to 30 mm for C1, from 35 mm to 25 mm for C2, and from 27 mm to 18 mm for C3. The background fluctuation also decreased compared with equal-weight fusion. Nevertheless, some secondary probability distributions remained because SNR weighting does not explicitly evaluate whether the defect-sensitive response is spatially consistent among sensors or whether the reflected wave packet is dominated by the target guided-wave mode.

Compared with both baselines, the proposed reliability-adaptive fusion produced more concentrated probability distributions around the actual crack position. For C1, the localization error decreased from 42 mm using equal-weight fusion and 30 mm using SNR-weighted fusion to 21 mm using the proposed method. For C2, the localization error decreased from 35 mm and 25 mm to 16 mm. For C3, it decreased from 27 mm and 18 mm to 11 mm. The high-response area and background fluctuation were also reduced, while the image contrast increased. These results indicate that the proposed weighting strategy improves localization concentration not simply by amplifying high-SNR frequency components, but by suppressing frequency components with poor echo quality, unstable inter-sensor response, or stronger non-target modal contamination.

Although the present comparison is still limited to fusion-based baselines and does not include RAPID-type probability imaging, delay-and-sum imaging, or tomographic reconstruction, the addition of the SNR-weighted baseline provides a more direct evaluation of the methodological contribution of the proposed reliability-adaptive weighting strategy.

Fig. 12Comparison of equal-weight fusion, SNR-weighted fusion, and reliability-adaptive fusion imaging results for C1-C3. The left column shows equal-weight fusion, the middle column shows SNR-weighted fusion, and the right column shows reliability-adaptive fusion. The actual crack position is marked in each image

4.4. Confidence-enhanced imaging under noise interference

The effect of the confidence-enhancement step was examined under different noise levels. As shown in Fig. 13, the rows correspond to SNR levels of 20, 15, 10, and 5 dB, while the left and right columns represent the conventional probabilistic images and the confidence-enhanced results, respectively. The quantitative comparison is summarized in Table 8.

As the SNR decreased, the conventional fused probability images exhibited more secondary hotspots and stronger background fluctuations. This behavior is expected because random noise and unstable wave-packet fluctuations can produce moderate probability values at non-damage locations. When the confidence coefficient defined in Eq. (25) was applied, these uncertain responses were attenuated, while the dominant crack-related indication remained visible.

The improvement became more obvious under lower SNR conditions. At 20 dB, the false-hotspot number decreased from 2 to 1, and the image contrast increased from 5.38 to 6.27. At 5 dB, the false-hotspot number decreased from 11 to 5, and the background fluctuation decreased from 0.241 to 0.139. These results indicate that the confidence-enhancement step is especially useful when the probability field contains competing uncertain regions.

It should be emphasized that the confidence-enhancement step is not intended to create new crack information. Instead, it acts as an uncertainty-control operation applied to the fused probability field. Regions with high probability and low uncertainty are retained, whereas regions with moderate and ambiguous probability are suppressed. Therefore, the observed improvement should be interpreted as improved noise-interference stability under the tested controlled condition, rather than as general robustness against arbitrary pipeline environments.

Fig. 13Comparison of conventional probabilistic imaging and confidence-enhanced imaging under different noise levels. Rows correspond to SNR levels of 20, 15, 10, and 5 dB. The left column shows conventional probabilistic imaging, whereas the right column shows the corresponding confidence-enhanced results

a) Conventional probabilistic imaging

b) Confidence-enhanced imaging

4.5. Severity grading performance

The severity index defined in Eqs. (27-34) was evaluated for the three axial crack depths. As shown in Fig. 14, the severity index increased progressively from C1 to C3. Fig. 15 further shows that the three crack cases could be separated into mild, moderate, and severe regions using the calibrated thresholds. The corresponding severity-related posterior features, including $P_{max}$, ${\bar{P}}_{ROI}$, $A_{high}$, $C_{freq}$, and the final severity index $S$, are listed in Table 9, and the confusion matrix is summarized in Table 10.

For comparison with conventional severity-assessment logic, the proposed severity index should be interpreted against three typical benchmark indicators: reflected peak amplitude, reflected energy, and peak posterior probability. Reflected peak amplitude and reflected energy provide direct descriptions of scattering strength, and their monotonic increase from C1 to C3 confirms that the measured guided-wave response is sensitive to crack-depth variation under the present controlled configuration. However, these two indicators are single-signal or feature-level quantities and do not directly describe whether the corresponding response is spatially concentrated around the actual crack position. The peak posterior probability provides a spatially interpretable indicator, but it may still be insufficient when two cases have similar peak values but different high-response areas or frequency consistency. The proposed severity index combines peak posterior response, regional mean probability, high-response area, and cross-frequency consistency; therefore, it provides a more comprehensive benchmark for semi-quantitative grading than a single amplitude-, energy-, or peak-probability-based criterion.

Fig. 14Distribution of the proposed severity index for different crack cases

Fig. 15Grading results of mild, moderate, and severe cracks based on the proposed severity index

The grading results indicate that the confidence-enhanced probability image contains severity-related distributional information under the present controlled configuration. Mild cracks produced lower peak response and smaller high-response area, whereas severe cracks produced stronger local probability and a larger consistent high-response region. This suggests that crack severity is reflected not only in the maximum probability value, but also in the spatial distribution of the posterior field.

The proposed severity index is therefore more informative than a simple peak-value criterion for the tested cases. By combining $P_{max}$, ${\bar{P}}_{ROI}$, $A_{high}$, $C_{freq}$, the index considers local intensity, regional distribution, spatial extent, and cross-frequency consistency. This posterior-distribution-based design provides a semi-quantitative grading indicator under the present controlled experimental condition, but it should not be interpreted as a general crack-depth inversion model.

The confusion matrix shows an overall grading accuracy of 93.3 % for the tested repeated measurements. Nevertheless, this value should be interpreted carefully. The repeated measurements were obtained from the same pipe specimen and the same crack geometry, and they should not be regarded as independent specimens. Therefore, the grading result demonstrates repeatability under the present controlled configuration rather than general classification capability across arbitrary crack locations, lengths, widths, orientations, or sensor-coupling conditions.

4.6. Ablation and limitation analysis

To further clarify the contribution of different components in the proposed framework, an ablation comparison was conducted based on the quantitative results of multi-frequency fusion, confidence-enhanced imaging, and severity grading. The analysis focused on three key components that can be evaluated from the present experimental results: SNR-weighted fusion, reliability-adaptive weighting, and confidence enhancement. The mean localization error, mean high-response area, mean background fluctuation, and mean image contrast were calculated by averaging the results over the three crack cases C1-C3.

The equal-weight fusion result was used as the basic baseline. The SNR-weighted fusion baseline was introduced to evaluate whether echo-to-noise information alone is sufficient for frequency weighting. The contribution of reliability-adaptive weighting was evaluated by comparing the SNR-weighted baseline and the proposed reliability-adaptive fusion. The contribution of confidence enhancement was estimated using the 20 dB noise condition as the representative low-noise case, because this condition is closest to the original experimental measurement state. The severity accuracy was obtained from the final grading confusion matrix. The corresponding ablation results are summarized in Table 11.

As shown in Table 11, the equal-weight baseline produced a mean localization error of 34.67 mm, a mean high-response area of 2161.67 pixels, a mean background fluctuation of 0.173, and a mean image contrast of 3.73. After SNR-weighted fusion was introduced, the mean localization error decreased to 24.33 mm, the mean high-response area decreased to 1733.00 pixels, the mean background fluctuation decreased to 0.135, and the mean image contrast increased to 4.65. This confirms that echo-to-noise information is useful for suppressing weak or noisy frequency components.

However, SNR-weighted fusion was still inferior to the proposed reliability-adaptive fusion. When inter-sensor consistency and modal purity were further incorporated into the weighting strategy, the mean localization error decreased to 16.00 mm, the mean high-response area decreased to 1381.67 pixels, the mean background fluctuation decreased to 0.103, and the mean image contrast increased to 5.40. These results indicate that the improvement of the proposed method is not caused only by higher echo amplitude or lower noise level. Instead, the additional use of response consistency and modal-purity information helps suppress frequency components that are not physically reliable for crack localization.

The confidence-enhancement step further reduced background fluctuation and improved image contrast. When confidence enhancement was applied to the equal-weight baseline, the mean background fluctuation decreased from 0.173 to 0.128, and the mean image contrast increased from 3.73 to 4.34. When it was applied after reliability-adaptive fusion, the mean background fluctuation decreased from 0.103 to 0.076, and the mean image contrast increased from 5.40 to 6.29. These results show that confidence enhancement mainly acts as an uncertainty-control step for reducing ambiguous secondary probability regions, rather than as an independent source of crack information.

The proposed framework achieved a severity grading accuracy of 93.3 % for the tested repeated measurements. This result suggests that the confidence-enhanced posterior probability distribution contains useful severity-related information under the present controlled configuration. Nevertheless, this value should not be interpreted as general classification performance for arbitrary pipeline defects. The repeated measurements were obtained from the same pipe specimen and the same crack geometry, and they were used mainly to evaluate repeatability and noise-interference stability under the tested controlled condition. Broader validation with different crack positions, crack lengths, crack widths, crack orientations, sensor-coupling states, and environmental conditions is still required.

The present comparison is also limited in terms of baseline methods. Although equal-weight fusion and SNR-weighted fusion provide two useful fusion-based baselines, additional comparisons with RAPID-type probability imaging, delay-and-sum imaging, tomographic reconstruction, and data-driven regression methods should be conducted in future work when comparable sensor-network configurations and datasets are available. Therefore, the present results should be regarded as controlled-condition verification of the proposed reliability-adaptive probabilistic imaging and semi-quantitative severity-assessment framework.